PERFORMANCE ANALYSIS OF SUBOPTIMAL SOFT DECISION DS/BPSK RECEIVERS IN PULSED NOISE AND CW JAMMING UTILIZING JAMMER STATE INFORMATION
JUHANIJUNTTI
Department of Electrical andInformation Engineering,
University of Oulu
OULU 2004
JUHANI JUNTTI
PERFORMANCE ANALYSIS OF SUBOPTIMAL SOFT DECISION DS/BPSK RECEIVERS IN PULSED NOISE AND CW JAMMING UTILIZING JAMMER STATE INFORMATION
Academic Dissertation to be presented with the assent ofthe Faculty of Technology, University of Oulu, for publicdiscussion in Kajaaninsali (Auditorium L6), Linnanmaa, onJune 17th, 2004, at 12 noon.
OULUN YLIOPISTO, OULU 2004
Copyright © 2004University of Oulu, 2004
Supervised byProfessor Pentti Leppänen
Reviewed byResearch Professor Aarne MämmeläProfessor Branka Vucetic
ISBN 951-42-7385-0 (nid.)ISBN 951-42-7386-9 (PDF) http://herkules.oulu.fi/isbn9514273869/
ISSN 0355-3213 http://herkules.oulu.fi/issn03553213/
OULU UNIVERSITY PRESSOULU 2004
Juntti, Juhani, Performance analysis of suboptimal soft decision DS/BPSK receiversin pulsed noise and CW jamming utilizing jammer state information Department of Electrical and Information Engineering, University of Oulu, P.O.Box 4500, FIN-90014 University of Oulu, Finland 2004Oulu, Finland
AbstractThe problem of receiving direct sequence (DS) spread spectrum, binary phase shift keyed (BPSK)information in pulsed noise and continuous wave (CW) jamming is studied in additive white noise.An automatic gain control is not modelled. The general system theory of receiver analysis is firstpresented and previous literature is reviewed. The study treats the problem of decision making aftermatched filter or integrate and dump demodulation. The decision methods have a great effect onsystem performance with pulsed jamming. The following receivers are compared: hard, soft,quantized soft, signal level based erasure, and chip combiner receivers. The analysis is done using achannel parameter D, and bit error upper bound. Simulations were done in original papers using aconvolutionally coded DS/BPSK system. The simulations confirm that analytical results are valid.Final conclusions are based on analytical results.
The analysis is done using a Chernoff upper bound and a union bound. The analysis is presentedwith pulsed noise and CW jamming. The same kinds of methods can also be used to analyse otherjamming signals. The receivers are compared under pulsed noise and CW jamming along with whitegaussian noise. The results show that noise jamming is more harmful than CW jamming and that ajammer should use a high pulse duty factor. If the jammer cannot optimise a pulse duty factor, a goodrobust choice is to use continuous time jamming.
The best performance was achieved by the use of the chip combiner receiver. Just slightly worsewas the quantized soft and signal level based erasure receivers. The hard decision receiver was clearlyworse. The soft decision receiver without jammer state information was shown to be the mostvulnerable to pulsed jamming. The chip combiner receiver is 3 dB worse than an optimum receiver(the soft decision receiver with perfect channel state information).
If a simple implementation is required, the hard decision receiver should be used. If moderatecomplex implementation is allowed, the quantized soft decision receiver should be used. The signallevel based erasure receiver does not give any remarkable improvement, so that it is not worth using,because it is more complex to implement. If receiver complexity is not limiting factor, the chipcombiner receiver should be used.
Uncoded DS/BPSK systems are vulnerable to jamming and a channel coding is an essential partof antijam communication system. Detecting the jamming and erasing jammed symbols in a channeldecoder can remove the effect of pulsed jamming. The realization of erasure receivers is rather easyusing current integrated circuit technology.
Keywords: channel coding, decision metrics, direct-sequence, interference cancellation,spread spectrum
Dedicated to my family
Preface
This thesis is based on the work that was carried out both in the spread-spectrum research group in the Telecommunication Laboratory, Department of Electrical Engineering, University of Oulu, Finland, during the years 1986-1991 and in the Aircraft and Weapon Systems Department of Finnish Air Force Headquarters, during the years 1992-2003. During the years 1986-1991 the basic ideas were discovered and the simulations were done. During the years 1992-2003 the analysis of the systems was done and the thesis was completed.
I want to express my gratitude to Professor Pentti Leppänen for providing me the opportunity to do research in the Telecommunication laboratory, and his guidance and support throughout the work, as well as to the chiefs of Aircraft and Weapon Systems Department General Markku Ihantola (1995-2000) and Colonel Niilo Kansanen (2001-2004) for providing me the opportunity to do research work while working in the Finnish Air Force Headquarters.
I would like to thank the reviewers Professor Branka Vucetic and Professor Aarne Mämmelä for a thorough examination of the thesis, and their useful recommendations and Dr. Harri Saarnisaari, from the Telecommunication Laboratory of University of Oulu, for thorough guidance, recommendations, and pre-review of the thesis.
I wish to thank M.Sc. Kari Jyrkkä for his contribution in preparing some original publications and doing simulations of the continuous wave jamming cases. I wish also to thank Lic. Tech. Ari Pouttu for many useful discussions and recommendations and M.Sc. Matti Raustia for help in Matlab programming, both from the Telecommunication Laboratory. There are numerous other members of the spread-spectrum research group in the Telecommunication Laboratory of University of Oulu, to whom I am very grateful for their support and comments. I would like to thank computer system manager M.Sc. Pekka Nissinaho for the support in running the simulations.
The financial support provided by Finnish Air Force, University of Oulu, Elektrobit Ltd., Nokia Mobile Phones Ltd., Nokia Telecommunications Ltd., the Technology Developement Center of Finland, Finnish foundations: Emil Aaltosen säätiö, Nokia Foundation, Oulun yliopiston tukisäätiö, Tauno Tönnigin säätö, Tekniikan Edistämissäätiö enabled this work and are thus gratefully acknowledged.
I wish to express my deepest thanks to my wife Päivi, and my children Hanna-Maria, Juho-Pekka, Jukka, Mikko, and Eero for all the patience and understanding they have shown during the course of this work.
List of original papers
The receiver structures studied in this thesis are presented in the original papers. Preliminary performance of the receivers was studied by simulations and the results are shown in the papers. These papers are referred to in the text by their Roman numerals. I J. Juntti, P. Leppänen, Performance of a Convolutionally Coded Hard-Decision DS
Receiver in Pulsed Noise Interference, IEEE International Symposium on Spread Spectrum Techniques and Applications (ISSSTA’90), King’s College, London, Sep. 24-26, 1990.
II J. Juntti, K. Jyrkkä, P. Leppänen, Performance of a Convolutionally Coded Hard-Decision DS Receiver in Pulsed CW Interference, Military Communications Conference, McLean, Virginia, Nov. 4-7, 1991.
III J. Juntti, P. Leppänen, K. Jyrkkä, Performance of a Convolutionally Coded Soft Decision Limitter DS Receiver in Pulsed Interference, The 14th Symposium on Information Theory and Its Applications (SITA‘91), Ibusuki, Japan, Dec. 11-14, 1991.
IV J. Juntti, Quantized Soft Limitter DS/BPSK Receiver in Pulsed Noise Interference, IEEE Second International Symposium on Spread Spectrum Techniques and Applications (ISSSTA’92), Yokohama, Japan, Nov. 29 - Dec. 2, 1992.
V J. Juntti, P. Leppänen, Performance of a Convolutionally Coded Signal Level Based Erasure DS Receiver in Pulsed Noise Interference, International Conference Communication Systems (ICCS´90), Singapore, Nov. 5-9, 1990.
VI J. Juntti, P. Leppänen, Performance of a Convolutionally Coded Chip Combiner and Signal Level Based Erasure DS Receivers in Pulsed Noise Interference, Military Communications Conference, McLean, Virginia, Nov. 4-7, 1991.
VII J. Juntti, K. Jyrkkä, P. Leppänen, Performance of a Convolutionally Coded Chip Combiner DS Receiver in Pulsed Noise and CW Interference, IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’91), King’s College London, Sep. 23-25, 1991.
Papers I and II present the hard decision receiver and papers III and IV the soft decision limiter receiver. In papers I - IV performance analysis and simulations of the receivers are presented in pulsed noise and continuous wave (CW) jamming.
Paper V presents bit error simulations of the signal level based erasure receiver with pulsed noise channel. Paper VI presents bit error simulations of the chip combiner receiver with pulsed noise channel. Paper VI also compares the signal level based erasure and the chip combiner receivers with pulsed noise jamming. Paper VII presents bit error rate simulations of the signal level based erasure receiver and the chip combiner receivers under pulsed noise and CW jamming and compare the performance of the receivers.
The author laid the idea of the research work and prepared research and publication plan of all the papers. The author carried out analysis and performed simulations in pulsed noise jamming cases. Kari Jyrkkä performed analysis and simulations in CW jamming cases under the guidance of the author. The author wrote Papers I, IV, V and VI. Papers II, III and VII were written under the guidance of the author by Kari Jyrkkä. Pentti Leppänen guided the research work and gave valuable comments in the preparation of all the papers.
List of symbols and abbreviations
a1 the first bit of a code word a a2 the second bit of a code word a a3 the third bit of a code word a an the n:th bit of a code word a A/D analog to digital AGC automatic gain control AJ amplitude of the jamming signal AWGN additive white gaussian noise Bi ith quantization interval of the received signal y b1 the first bit of a code word b b2 the second bit of a code word b b3 the third bit of a code word b bn the n:th bit of a code word b B(R0) an unique function of the cut-off rate R0 for a code b channel gain BER bit error rate BPSK binary phase shift keying C group of all possible codewords C1, C2 constant c(t) PN function c(z) weighting coefficient, where z is side-information of the detected
symbol y cb code sequence unbalance CCR chip combiner receiver cdf cumulative density function CW continuous wave D channel parameter DA channel parameter D, effect of the AWGN channel DJ channel parameter D, effect of the jamming channel d received data bit DPSK differential phase shift keying
DMC discrete memoryless channel DS direct sequence DSP digital signal processing DSSS direct sequence spread spectrum E{•} expectation Eb information bit energy Es channel symbol energy f0 center frequency of the information signal fJ center frequency of the jamming signal g1 the first bit of a code word g g2 the second bit of a code word g g3 the third bit of a code word g gn the n:th bit of a code word g G(D) bit error upper bound function determined by a code i transmitted information bit i index J1-J6 argument of Q-function in jamming J jammer power JN noise jammer power Jpeak jammer peak power J(t) deterministic jamming signal j effect of the deterministic jamming in the output of the demodulator jc the effect of the deterministic jamming in the output of chip combiner K constraint length of convolutional code K number of the information bits in block code word k constant k index L number of the quantization levels LA1- LA6 quantization threshold in AWGN channel LJ1- LJ6 quantization threshold in jamming channel M code symbol alphabet size m(x,y) decision metric of a symbol decision m(x,y) decision metric of a sequence MATLAB Matrix Laboratory mathematical simulation program MF matched filter ML maximum likelihood N length of the spreading code sequence (one channel symbol) N0 amount of zeros in spreading code sequence N0 one sided channel noise spectral density N1 amount of ones in spreading code sequence NJ one sided jamming noise spectral density Ntot total one sided noise spectral density n length of block code n effect of AWGN in the output of the demodulator nj effect of the noise jamming in the output of the demodulator
ntot combined effect of AWGN and noise jamming in the output of the demodulator
n(t) AWGN noise signal nJ(t) noise jamming signal NTT noise-threshold test P probability
)( xx )→P pair-wise error probability, where x is transmitted code sequence and x) is an other code sequence detected by a receiver
Pb information bit error probability PFA jamming detection false alarm probability PinA probability that received signal is between determined thresholds in
AWGN channel PinJ probability that received signal is between determined thresholds in
jamming channel PMD probability to miss detecting jamming Ps channel symbol error probability PsN channel symbol error probability in AWGN channel PsNJ channel symbol error probability in noise jamming PsJ channel symbol error probability in CW jamming PsJrandom channel symbol error probability in CW jamming when code is random p(cb) probability density function of code unbalance pI (y xm) probability of received signal y to be in region I when xm was
transmitted PG processing gain PDF probability density function PN pseudo noise Pr probability
),( nnn zxyp channel symbol transition probability, where yn is received symbol, xn is transmitted symbol and zn is the side information related to these symbols
),( zxyNp channel sequence transition probability, where y is received code sequence, x is transmitted code sequence and z is the side information sequence related to these symbols
p(t) transmitted DS/BPSK signal R code rate R0 cut-off rate, bits/channel use RS channel data rate, bits/second r(t) channel output function q(t) received signal function RAM random access memory RTT ratio-threshold test S signal power s(t) transmitted BPSK signal SFH slow frequency hop SNR signal to noise ratio SJR signal-to-jamming ratio (S/J)
SNRJ signal-to-noise ratio in jamming channel (ES /NJ) SNR0 signal-to-noise ratio in AWGN channel (ES /N0) SNRtot total signal to noise ratio (ES /Ntot) TOPSIM Torino Polytecnico Simulator, communication system simulation
program Tb information bit duration, second Tc chip duration, second Ts channel symbol duration, second t error correcting capability of a block code t time u output of the chip combiner uLA lower threshold of chip combiner output in AWGN channel uUA upper threshold of chip combiner output in AWGN channel uLJ lower threshold of chip combiner output in jamming uUJ upper threshold of chip combiner output in jamming Q(x) 1-erfc(x/2), erfc=complementary error function
)( nzq jamming state probability on the symbol decision time, where zn is the side information related to these symbols
)(ZqN jamming state sequence probability, z is the side information sequence related to these symbols
w(x, $x ) the number of places where Nnxn ,...,2,1,x̂n =≠ . WJ noise jamming signal bandwidth WSS spread spectrum signal bandwidth WGN white gaussian noise x(t) transmitted data function x transmitted bit x coded symbol sequence xn n:th element of the code sequence x xc the effect of the information in the output of chip combiner y(t) received signal y received signal y channel output code sequence yn n:th element of the code sequence y yj demodulator output when there is jamming in the channel z jammer state random variable Z the group of the integer numbers z side information sequence zi i:th element of the sequence z φJ phase difference between jamming and information signal λ parameter to be optimized in Chernoff bound σ2 noise variance ν auxiliary variable ρ probability that a code symbol is affected by jamming θ erasing threshold ω0 angular frequency of the information signal, 2πf0 ωJ angular frequency of the jamming signal, 2πfJ
Contents
Abstract Preface List of original papers List of symbols and abbreviations Contents 1 Introduction ...................................................................................................................17
1.1 Background, motivation and aim of the thesis........................................................17 1.2 System description..................................................................................................18
1.2.1 Interleaving techniques....................................................................................22 1.2.2 Automatic gain control ....................................................................................24 1.2.3 Interference cancellation..................................................................................24
1.3 Author’s contribution..............................................................................................25 1.4 Outline of the thesis................................................................................................26
2 Decision metrics ............................................................................................................29 2.1 Coding channel model ............................................................................................29 2.2 Literature review ....................................................................................................32
2.2.1 Decision metrics ..............................................................................................32 2.2.2 Side information ..............................................................................................37
2.3 Coded bit error rate bound......................................................................................39 3 Communication system model ......................................................................................42
3.1 Jamming signals .....................................................................................................44 3.2 Communication system in jamming .......................................................................45
3.2.1 CW jamming....................................................................................................46 3.2.2 Channel symbol and coded bit error probabilities ...........................................50
4 Hard decision receiver ...................................................................................................52 4.1 Decision metric and channel parameter..................................................................52 4.2 Performance in jamming ........................................................................................53
5 Soft decision receiver ....................................................................................................61 5.1 Optimum soft decision receiver, known jammer state............................................61
5.1.1 Decision metric and channel parameter...........................................................61 5.1.2 Performance in jamming..................................................................................62
5.2 Soft decision receiver, unknown jammer state .......................................................67 5.2.1 Decision metric and channel parameter...........................................................67 5.2.2 Performance in jamming..................................................................................69
6 Quantized Soft Decision Limiter Receiver....................................................................73 6.1 Decision metric and channel parameter..................................................................73 6.2 Performance in jamming ........................................................................................75
7 Signal Level Based Erasure Receiver............................................................................81 7.1 Decision metric and channel parameter..................................................................81 7.2 Performance in jamming ........................................................................................85
8 Chip Combiner Receiver ...............................................................................................93 8.1 Decision metric and channel parameter..................................................................93 8.2 Performance in jamming ........................................................................................96
9 Comparison of the Receivers.......................................................................................109 9.1 Pulsed noise jamming...........................................................................................109 9.2 Pulsed CW jamming............................................................................................. 116
10 Discussion and Conclusions ......................................................................................125 10.1 Discussion...........................................................................................................125 10.2 Conclusions ........................................................................................................126 10.3 Future research directions...................................................................................129
References Appendices
1 Introduction
1.1 Background, motivation and aim of the thesis
In some cases communication systems are required to operate over a channel that is affected by intentional jamming, especially against military systems. The jamming signal may also be another user in the same communication system or another communication system causing unintentional jamming. The jamming signals may have different waveforms: noise, continuous wave, modulated, pulsed, or continuous signal. The jammer cause severe performance degradation for all types of uncoded communication systems ([1] Vol. 1 p. 148-151). In some cases jamming converts an exponential dependency of the bit error rate on the signal-to-jamming ratio into a linear one on logarithmic scale.
The use of a spread-spectrum modulation with error correcting coding is an effective technique to combat the effects of jamming ([1] Vol. 1 p. 151-167). Usually hard or soft decision metrics in receivers are studied for coded anti-jam systems. Jammer state information may be measured from received signal and it can be used with both metrics. The soft decision metrics outperform hard decision metrics in an additive white Gaussian noise (AWGN) channel, but in jamming soft decision decoders either require jammer state information, or must have some method to obtain information concerning the reliability of received code symbols for satisfactory performance [1 Vol. 1, p. 157-160]. Often it is enough to know if the received symbol was jammed or not jammed. It is shown, that soft decision metric can give better performance if reliable jammer state information is available. The measurement of the jammer state information is studied in this thesis. An effective soft decision metric was studied in the original Papers III and VI for a direct sequence spread spectrum system with binary phase shift keying (DS/BPSK) operating in the presence of pulsed jamming. If jammer state information is not available, one should use the hard decision metric, because the use of the soft decision metric may lead poor performance in some jamming cases. The hard decision metric is robust and very simple to implement.
In the area of cellular communication a lot of research work has been done in recent years to optimize system performance in a very dense signal environment. Many
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multiuser interference cancellation and multiuser detection techniques have been proposed. These techniques are often very complex to implement and in many cases jamming signals are assumed to be known. The receiver structures studied in this thesis assume no knowledge of the jamming signal and are simple to implement. They are not so effective against known interference, but work well in most cases. They can be considered to be robust receiver structures. The presented methods can be used in many situations including cellular communications to reduce the effect of interference. One goal of the study was to find out robust, simple receiver structure in jamming environment.
The interference cancellation techniques are proposed to be used in spread spectrum systems to combat the effect of the jamming and interference (e.g. [2] pp. 291-333). These techniques are still expensive to implement if chip rate of the signal is high. Hybrid FH/DS systems are even more demanding while the adaptation time of the receiver is short. It is thus important to optimize receiver structure in jamming without interference cancellation techniques.
In this thesis binary phase shift keying (BPSK) modulation is considered, but a similar type of analysis could be done for other coherent phase and frequency modulation techniques. The metrics must be modified to fit the modulation method. For non-coherent modulations metrics are different and the analysis is different, because the effect of the noise and interference in receiver output differs from that of coherent demodulation [3], [4]. Also mentioned ad hoc metrics are specially designed to be used in BPSK demodulated systems. In reference [5] differential phase shift keying (DPSK) modulated systems are analyzed.
In this thesis quantization of the signals are modeled in the receivers. Automatic gain control (AGC) is assumed to be perfect. The DS/BPSK receivers are studied within a pulsed jamming channel with the assumption of no fading. The bits are assumed to be jammed independently, which can be implemented using interleaving techniques.
1.2 System description
In basic detection theory [6], ([7] p. 105-117), the optimal receiver in an AWGN channel is shown to be a parallel bank of matched filters (MF). In the case of BPSK modulation only one MF or a correlation receiver is needed. If noise is Gaussian but not white, the use of a whitening filter before the MF forms the optimum receiver. If the noise is non-Gaussian, the optimum receiver becomes difficult to implement. Reference [8] presents the optimum receiver in the case of a summed sine wave and noise. Thus, non-optimal receivers are often used for simplicity. This has lead to the use of interference cancellation techniques, which are shortly reviewed in Section 1.2.3. The approach in this thesis is to consider decision-making techniques to improve a receiver’s performance.
References [9], [10], [11], and [12] discuss the analysis of DS/BPSK systems in the presence of pulsed jamming. The analysis is well summarized in reference ([1] Vol. 2 p. 3-60). Most of the receivers analyzed in this thesis are presented in the previously mentioned references. Because in this thesis notations and receiver structures are a little
19
bit different than in the references, some of the equations are repeated to give to a reader a clear understanding of the systems and analysis.
The general anti-jam communication system is illustrated in Figure 1 ([1] Vol. 1 p. 189). In general, a channel can include noise, jamming, multiuser interference, and distortion such as fading and dispersion. In this thesis only AWGN and jamming are added to the signal in the channel. The modulator is assumed to be memoryless and the signal is spread by a DS-spreading code in bandwidth. The system model does not depend on a spreading code. At the receiver, despreading is done first, followed by detection, which can be hard decision, soft decision, threshold, limiter, possibility of erasures, etc. In some receivers channel measurements may provide side information to the decoder.
Fig. 1. General anti-jam system overview ([1] Vol. 1 p. 189).
Traditional communication receiver analysis supposes that received signals are corrupted by additive white Gaussian noise (AWGN). Analytically this is a convenient assumption, because it makes analysis easier. The optimum receiver for AWGN is the matched filter (MF) receiver (or Integrate and Dump receiver for BPSK). In some cases the Gaussian assumption is quite right, for example in satellite or airborne communication. But in many cases, for example in mobile and military communication, there are a lot of other factors corrupting the transmitted signal. The most common factors are for example fading and interference. Interference may be intentional or it may be generated by other communication systems. The intentional interference is called jamming. A receiver designed for an AWGN channel is still the most commonly used, which can then be further developed to perform better in a real signal environment. The same approach is usually used also in receiver design for an interference channel. This is often a good approximation because noise, interference, and fading are so strong non-idealities that the
DESPREADERDEMODULATORDEINTERLEAVERDECODER
CHANNEL- Noise- Fading- Jamming
SPREADER
SIDEINFORMATION
DATABITS
DATABITS
JAMMER STATE
MODULATORINTERLEAVERENCODER
PROBELONG-TERM
MEASUREMENTLONG-TERM
CODING CHANNEL
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degradation of other factors (e.g. scintillation, multipath and intersymbol interference etc.) is so small, that they can be ignored.
The channel model used in this thesis is AWGN channel added by broadband Gaussian noise or continuous wave (CW) jamming signal. It is assumed, that the effects of fading can be compensated by automatic gain control (AGC) to set signal to the dynamic range of the receiver ([13] p. 265). The dynamic range of the receiver is the area where received signal can vary without it to be distorted by amplifier or some other signal processing circuit including analog-to-digital (A/D) conversion. The received signal level may vary on the order of 100 dB, so the received signal must be amplified by variable gain. This is especially important in digital receivers where the signal must be in the dynamic range of the A/D-converter ([14] p. 80). In the interference case it is usually a good approximation to assume that there are only noise and interference present in the channel. So in the thesis it is assumed that channel model includes only interference and white Gaussian noise.
Figure 2 is a block diagram of a studied coded DS/BPSK anti-jam system, where also the transmitter and the receiver structures are shown ([1] Vol. 1, p. 142-148). The receiver is a so-called integrate-and-dump receiver, which is equivalent to the matched filter receiver. The receiver is assumed to be perfectly time, frequency, and phase synchronized. These assumptions are done to ease analysis.
In the transmitter information bits i are first encoded to form channel bits xn before BPSK modulation. After that the signal s(t) is modulated by a spreading code before transmitting to the channel. In the channel jamming and noise is added to the signal.
Fig. 2. Coded DS/BPSK system.
In the receiver signal r(t) is first despread (demodulated by the spreading code). The spreading codes of the transmitter and the receiver are assumed to be time synchronized. After that signal q(t) is demodulated and integrated by integrate and dump receiver. After demodulation decision metric is used to make a decision which bit (0 or 1) was transmitted. The decision metric can be as simple as a hard decision or it may be a soft
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decision (analog signal) using side information from a channel state. The received bits are then decoded to recover information bits.
The part of the system shown inside the dotted lines in Figure 1 is called the coding channel, which is the effective channel seen by the encoder and the decoder. The coding aspect of the system is decoupled from the remaining part of the communication system in the analysis. In this thesis the coding channel is mainly considered. The channel coding is not an issue in the thesis. As a channel code a binary convolutional code with constraint length K = 7 and code rate R = ½ is used and as a decoding scheme is used Viterbi decoder [15]. Binary phase shift keying (BPSK) is used as a modulation method. A channel is usually characterized using signal-to-noise ratios (SNR). SNRs used in the thesis are equivalent channel symbol energy to jammer noise and equivalent channel symbol energy to noise ratio
00 NE
RNE
NE
RNE
bs
J
b
J
s
=
=
(1.1)
energybit n informatio therate code the
density spectral noise channel sided onedensity spectral noise jammig sided one
energy symbol channel thewhere
0
==
==
=
b
J
s
ERNNE
Processing gain PG is defined to be the improvement of SNR in the receiver output compared to SNR in the receiver input and for BPSK it can be shown to be ([1] Vol.1 p. 137-138)
s
ss
RW
PG = (1.2)
.duration symbol channel theT
BPSKfor secondper bitsin rate data channel=1bandwidth spread
where
s =
=
=
ss
ss
TR
W
The information bit energy to jammer noise ratio is
, JSPG
NE
J
b = (1.3)
22
,power jammer =2
power signal=
where
2===
==
Jsss
JJ
s
s
b
b
NWTEA
J
TE
TE
S
where Tb is the information bit duration and AJ is the amplitude of the jamming signal.
1.2.1 Interleaving techniques
Most error correcting codes can correct random errors. If errors tend to appear in bursts interleaving can randomize them. Burst error correcting codes are difficult to utilize in a jamming channel where the burst length and distribution is not known ([16] p. 442-443). Thus using a random error correcting code and interleaving is a more robust way to correct bit errors. If bits are interleaved randomly or far enough from each other, the errors can be assumed to be independent of each other. In this thesis errors are assumed to appear randomly and interleaving is not used. Interleaving techniques are well presented for example in references ([1] p. 182-186), ([14] p. 114-116), ([4] p. 440-442), and ([17] p. 679-680), ([18] p. 348-354). Usual techniques used to interleave are block and convolutional interleaving.
If for an (n,k) block code, which can correct t symbol errors, L code words are interleaved, the result is an (Ln,Lk) interleaved block code ([17], p. 679-680), ([4], p. 440-441). Instead of transmitting code words one by one, they are first grouped as in Figure 3. The words are written in row by row and read out column by column. This process is repeated for the next L code words. In decoding interleaving is removed first. It is easily noticed, that the interleaved block code can correct any combination of t bits error burst of length L.
Fig. 3. Block interleaving ([4] p. 442).
In convolutional interleaving each code symbol out of the encoder is inserted into one of the L tapped shift registers of the interleaver bank ([14] p. 114-116), see Figure 4. The zeroth element of this bank provides no storage, while each successive element provides i symbols more storage than the preceding one. The input commutator switches from one
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register to the next until the (L-1)th after which the commutator returns to the zeroth element. The output commutator feeds to the channel one code symbol at a time, switching from one register to the next after each symbol, synchronously with the input commutator. The minimum separation for any two code symbols is L. For convolutional code i is usually one and for block codes I=iL should be at least equal to the block length. The deinterleaver must invert the action of the interleaver. Usually demodulated symbols are stored digitally and this means that they must be quantized.
Fig. 4. Convolutional interleaving ([14] p. 111).
The convolutional interleaver requires just half of the memory and adds half the delay to that of the block interleaver. The block interleaver may, however, be easier to implement, because it can be implemented using random access memory (RAM) while the convolutional interleaver must be implemented using shift registers, which requires more integrated circuit area ([14] p. 116).
In this thesis interleaving method needs not to be defined. The simulations and analysis applies for all kinds of the interleaving methods. The pulsed jamming is produced randomly for each symbol in the channel, so that interleaving needs not to be done any more. So, instead of jamming consecutive symbols and then interleaving these symbols to randomize errors, errors are produced randomly to simplify analysis. Same principle was applied in simulations of the original Papers. In the simulations and bit error calculations a convolutional code with rate R=1/2 and length K=7 is used, which is random error correcting code. User can select an interleaving method, which best suites to his application. The interleaving must only be long enough to randomize errors. In real applications, where jamming may cover several symbols, interleaving must be used if a random error correcting code is used.
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1.2.2 Automatic gain control
To implement multilevel quantized receiver, a demodulator must have automatic gain control ([14] p. 80]. In general, the purpose of an automatic gain control (AGC) circuit is to maintain the output level of an amplifier at an almost constant value, even though its input signal level may change substantially [19]. A feedback loop to control the output level is usually used. The feedback loop parameters determine the behavior of the AGC circuit. It is possible to select the parameters so that short pulses or fading will not be followed if desired ([20] p. 167-204), ([21] p. 147-171). In that case higher dynamics in sampling and A/D conversion is required. This type of system design is out of the scope of this thesis. System design is considered in ([20] p. 167-204), ([21] p. 147-171) and circuit design in ([22] p. 369-382). One should be careful in the system design to avoid the problems arising from clipping of the input signal in A/D conversion [13]. Clipping may cause severe performance degradation.
In a DS spread spectrum receiver the feedback signal can be obtained from a wide or narrow band signal. In ([13] p. 265-277) two types of AGC are proposed to be used: a presync AGC, where the feedback is generated by a wide band signal, and a postsync AGC, where the feedback is taken from a narrow band despread signal. Usually some cascade combination of these two AGC is used [23]. This configuration is not optimal, and it can be optimized by making it adaptive as proposed in [23]. One should optimize the combination of AGC and decision metric.
In this thesis AGC circuit is not modeled. It is assumed that AGC keeps signal level constant before despreading and demodulation.
1.2.3 Interference cancellation
Reference ([2] p. 291-333) presents the most usual interference suppression techniques used in spread spectrum systems. In these techniques interference cancellation is done before despreading and demodulation in two steps. In the first step the interfering signals are estimated and coefficients of the suppressor are calculated. In the second step the filtering of the received signal is carried out. Interference cancellation techniques have been classified into three main categories: adaptive linear transversal filtering, frequency domain filtering [24], [6], [25], [26], [27] and adaptive CDMA linear receivers [28], [29]. Recently use of nonlinear transversal [30] and space [31] or space-time filtering has also been introduced [25], [32]. In CDMA systems multiuser detection techniques can also be shown to be effective against signals, which are digitally modulated [32].
The adaptive linear transversal filter tunes a notch filter to the frequency band of the interference. This technique works well when interference is a stationary narrow-band signal. The filter needs some time to adapt its coefficients.
The transform domain processing usually uses a real-time Fourier transformer. A notch is implemented in the frequency domain and the signal is inverse transformed back to the time domain. This does not require any adaptation time and thus it is suitable for use
25
when interference is rapidly changing. Wide-band interference cannot be cancelled using transform domain processing.
An adaptive linear spread spectrum receiver consists of a least mean square error (LMSE) transversal digital filter and a decision device. The receiver estimates the filter coefficients and adjusts them continuously. The receiver does not require a separate despreading device in addition to interference suppression filter. A nonlinear receiver uses a nonlinear front-end before linear receiver [30].
There still exist a challenge to cancel wide-band interference. One solution to this is space and space-time filtering [25]. Filtering in the space domain places an antenna null to the direction of interference. This area has been under large interest in recent years.
Interference cancellation techniques can reduce influence of jamming several tens of decibels. The use of interference cancellation makes receivers more complex and costly. The use of interference cancellation does not reduce the importance of optimizing the decision metric. Interference cancellation techniques are often compromises between performance and complexity and there may still be interference after cancellation. This residual interference can be minimized by choosing optimal metric. The decision metrics presented in this thesis can be used at the same time with interference cancellation. In this thesis interference cancellation is not used.
1.3 Author’s contribution
The equations of D for the hard and the soft decision receivers are shown in the literature, but the analysis of the quantized soft decision BPSK receiver have not been published earlier and no equations for the bit error upper bounds have been presented. The simulation results published in the original Papers I - VII during 1990-1992 give clear indication that these methods may be good in a pulsed jamming channel. The general theory to calculate the bit error upper bound as well as the optimization of receivers and jamming parameters will be presented. In this thesis bit error upper bound calculation and parameters for robust receivers are determined under pulsed noise and continuous wave jamming channels.
The thesis is based on research work done by the author. The work has been done under the supervision of Professor Pentti Leppänen. The basic ideas of the receiver structures are presented in the literature. The performance analysis of the quantized soft decision limiter (Chapter 6), the signal level based erasure (Chapter 6) and the chip combiner receivers (Chapter 8) in pulsed noise and CW jamming are the author’s own contribution. The detailed derivations of the equations are shown in the Appendixes 4 – 6.
The idea of separating random and deterministic interference in Chapter 3 is the author’s own contribution. Jamming is usually supposed to be a noise like signal after despreading. In this thesis jamming is divided into two parts: deterministic and random signals. The effect of the deterministic jamming in the output of the demodulator is calculated accurately instead of approximating it by noise. Superposition principle is applied when deterministic and random (noise) jamming are separated. By using this method it is possible to get more accurate results and the effect of the deterministic
26
jamming signal can be accurately calculated. A continuous wave (CW) signal is used as a deterministic and white Gaussian noise as a random signal.
A method to optimize channel measurement parameters (erasure levels) by minimizing the channel parameter D (i.e. bit error upper bound), instead of using optimization of false alarm rates as usually, in Chapters 7 and 8 is the author’s own contribution. It is shown that minimizing the channel parameter gives the smaller bit error upper bound than optimizing the false alarm rate.
The simulations of the receivers were studied in the Licentiate thesis of the author [33] and in the Master’s Thesis of Kari Jyrkkä [34]. M.Sc. Jyrkkä did his Master’s Thesis under supervision of the author and the author presented all the ideas of the work. Papers I – VII have been made as a team work where the author has the main contribution in all of the them. The author performed simulations of the Papers I, IV, V and VI using TOPSIM and MATLAB simulation software. In Paper II M.Sc. Jyrkkä made the simulations under the guidance of the author. In Papers III and VIII the simulations of the CW jamming were made by M.Sc. Jyrkkä under the guidance of the author, and the simulations of noise jamming were made by the author.
In the original Papers III, V and VII the practical quantized soft-decision receivers are simulated with pulsed noise and CW jamming. The proposed systems are presented in detail in the thesis. In simulations, shown in the original Papers I – VII, channel state measuring receivers turned out to be better than the hard decision or quantized soft decision receivers with pulsed noise and CW jamming. The performance is calculated analytically in the thesis, because simulations are cumbersome to calculate when there may be millions of the simulation cases. By analysis it can also be calculated the bit error upper bounds and optimize erasure levels of the signal level based erasure (Chapter 7) and chip combiner receivers (Chapter 8). As a result of the studies, different metrics can be analyzed in jamming and the receivers can easily be compared. The worst case pulse duty factor can be calculated as well as the worst case bit error upper bound. The worst case performance of the communication systems can be determined and for an intentional jammer the optimum pulse duty factor can be calculated. Erasure levels of the signal level based erasure (Chapter 7) and the chip combiner receivers (Chapter 8) can also be calculated. The values of the results of the thesis are in the analysis method and in the comparison the receivers in jamming.
1.4 Outline of the thesis
In Chapter 1 the system model will be introduced, in Chapter 2 previous work will be summarized, literature will be reviewed, and in Chapter 3 receiver models will be discussed. The goal of this thesis is to derive the general equations of a channel parameter D and the bit error upper bound for quantized soft decision receivers with different channel state estimators. The channel parameter D is defined in detail in Section 2.1. D is directly proportional to channel bit error rate and it is independent of the channel coding used in the communication system, but it is dependent on used modulation type, the decision metric, jammer state information and jamming. The bit error upper bound of the
27
communication system can be calculated using D when channel coding is known as is discussed later in Chapter 3. So it can easily be calculated how much it is possible to gain using different metrics by comparing the values of the channel parameter D. By simulations these kinds of comparisons are not easy to do and there is still effect of the channel code included in the bit error rates. The author has done Monte Carlo simulations using TOPSIM simulation program. The simulation results was published in the original Papers I – III and V - V II. The simulations are very cumbersome to calculate. For these reason the channel parameter D and the bit error upper bounds are calculated.
The effects of the AGC or synchronization circuits are not analyzed, it is assumed that they work ideally. Fading and other jamming signals (e.g. chirp, modulated etc.) are also outside the scope of this work.
In Chapter 4 the hard decision receiver is studied. In the hard decision receiver bit decisions are made by quantizing the signal into two regions. The quantization threshold is set to zero and the receiver is assumed not to have any side information about channel state. This receiver structure is very simple and widely used in many applications. The analysis of the hard decision receiver is presented for example in reference ([1] Vol. 1, p. 198). The performance is shown in Chapter 4 with pulsed noise and CW jamming, mainly to have a reference system.
In Chapter 5 the unquantized soft decision receivers are studied. Analysis is presented in two cases: with perfect channel state information and without any channel state or side information. Both systems are analyzed and performance results are shown. The results are compared to the hard decision case. The soft decision receiver with perfect channel state information is an optimum receiver in interference. So this gives a reference for comparison of the other studied receivers.
In Chapter 6 quantized soft decision receivers are studied. The quantized soft decision receiver is like the hard decision receiver with more than one threshold. It limits the matched filter output to some maximum value, i.e., it clips the detected signal. After that the signal is quantized and for each quantization interval a reliability value is given. The determination of the reliabilities is studied and the performance of the receiver is analyzed with pulsed noise and CW jamming.
In Chapter 7 signal level based erasure receivers are studied. The signal level based erasure receiver has one added quantization level compared to the quantized soft decision receiver presented in Chapter 6. When the added quantization level is exceeded, the receiver makes the decision that there is jamming in the channel and the receiver erases this symbol. The added quantization level is optimized and the performance of the receiver is analyzed with pulsed noise and CW jamming.
In Chapter 8 chip combiner receivers are studied. In this receiver channel state is measured using a so called chip combiner circuit. When the chip combiner circuit gives indication that there is jamming in the channel, that symbol is erased. The parameters of the chip combiner are optimized and the performance of the receiver is analyzed with pulsed noise and CW jamming.
The receivers in Chapters 4 and 6 - 8 are all rather easy to implement using digital electronics and are thus interesting to compare. The unquantized soft decision receivers in Chapter 5 are presented mainly for reference purposes. The performances of the receivers in Chapters 4 - 8 are analyzed with pulsed noise and CW jamming. The worst case pulse duty factor and the worst case channel parameter D and the bit error upper bounds for
28
different receivers are calculated. In Chapter 9 the different receivers in jamming are compared and in Chapter 10 the work is summarized.
2 Decision metrics
The used system and channel models were presented in Chapter 1 in Figure 2. In references [1], [5], [9], [10], [11], [12], [35], [36], [37], and [38] jamming is studied in DS receivers. Many useful ideas are presented and will be considered in this chapter. First, decision metrics are introduced and then generation and use of side information is presented.
2.1 Coding channel model
Figure 5 shows the general memoryless coding channel, a basic model for the analysis ([1] Vol.1 p. 191). The coding channel includes modulator, the spectrum spreading, the channel, despreading, demodulator, and the channel measuring circuit. The used communication system is shown in Figure 2. A channel coded symbol sequence x of length K is denoted as
),...,,( 21 Kxxx=x , (2.1)
where xn is a transmitted code symbol.
Fig. 5. General memoryless coding channel ([1] Vol. 1 p. 191).
This symbol xn is then modulated and spread in bandwidth before transmitting to the channel as shown in Figure 2. The additive white Gaussian noise (AWGN) channel is modeled with summing junction ([14] p. 51)
30
sTttntpty ≤≤+= 0)()()( , (2.2)
where p(t) is transmitted signal, y(t) is received signal and n(t) is a stationary random process whose power is spread uniformly over a bandwidth much wider than the signal bandwidth. In the thesis interference is added in the channel and thus the received signal y(t) is
sJ TttJtntntpty ≤≤+++= 0)()()()()( , (2.3)
where nJ(t) is noise jamming and J(t) deterministic jamming (e.g. CW jamming in this thesis). While transmitted code sequence was x, the channel output sequence y is respectively
),...,,( 21 Nyyy=y , (2.4)
where yn is a received code symbol.
If the decoder is implemented digitally, the received signal must be quantized before decoding ([14] p. 78 – 79). The coding channel input symbols xn are normally elements of a finite set, and in this thesis they are binary symbols with values 1 or –1. The outputs of the coding channel are continuous Gaussian variables. An example of an eight-level uniform quantizer is shown in Figure 6 (a) ([14] p. 78). This quantizer transforms the AWGN channel (where jamming may be summed) to a finite-input, finite-output alphabet channel. An example of the BPSK modulated AWGN channel with output quantized to eight levels is shown in Figure 6 (b).
a) b) c)
Fig. 6. a) Uniform eight-level quantizer, b) quantized channel model, c) binary symmetric channel (BSC) ([14] p. 78 – 80).
Denoting the binary input alphabet by {a1, a2}, where a1 = -a2 = 1 and denoting the output alphabet by {b1, b2, b3, . . . , b8}, the channel can be described completely by the conditional probabilities or likelihood functions ([14] p. 78 – 79)
31
( )
==
−−=== ∫
∈8 , . . . 1,2,
2 1,,exp1)(
0
2
0 ik
dyN
ayN
axbypiBy
kkknin
π (2.5)
where k is modulation alphabet size index, i is the channel output alphabet size index, y is received unquantized signal, N0 is one sided channel noise spectral density, yn is received code symbol, and Bi ith quantization interval. As defined, ak has the numerical value, but bi is an abstract symbol.
In AWGN channel all code symbols are identically distributed and this kind of channel is called memoryless. The coding channel is memoryless when error statistics of the bits are independent. When channel input and output alphabets are finite it is called discrete memoryless channel (DMC) ([14] p. 79). A multilevel quantizer is called a soft quantizer shown in Figure 6 (a) and (b). The simplest discrete memoryless channel is the one with binary input and output ([14] p. 80). It utilizes a two-level quantizer and it is called a hard decision quantizer or receiver and a channel is called binary symmetric channel (BSC), which is shown in Figure 6 (c). The conditional transition probabilities of the BSC are p = p{y=b2x= a1} = p{y=b1x= a2} and 1-p. The conditional probability p is called the crossover probability and it is the same as the symbol error probability ([14] p. 80). The advantage of the hard decision is that no knowledge is needed of the signal energy, while the soft decision requires that information and hence must employ an AGC. In this thesis the DMC is used for soft decision and BSC for hard decision as a channel model.
For a binary code sequence of length K the likelihood of a channel code transition probability can be described as ([14] p. 79)
2 1,)|(),|(1
== ∏=
kxyppK
nknnK zxy . (2.6)
The implementation of an effective multilevel quantizer requires that demodulator
incorporates automatic gain control (AGC) ([14] p. 80). Proakis ([4] p. 485) shows that variations in the AGC level of the order of 20 % will cause only little degradation to performance in the case of an eight-level uniform quantizer. Increasing the number of quantization levels can also increase tolerance for AGC errors.
If channel state is measured, the decoder may use this information in decoding. In pulsed jamming the received signal is jammed with probability ρ. Thus, the received signal is ([1] Vol. 1 p. 148-149)
sJ TttJtnztntpty ≤≤+++= 0))()(()()()( (2.7)
where z is jammer state random variable ([1] Vol. 1 p. 156)
=ion. transmisssymbol during offjammer ,0
ion transmisssymbol duringon jammer ,1z (2.8)
The variable z is independent of jamming and z has probability ([1] Vol. 1 p. 149),
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{ }{ } .10Pr
1Prρ
ρ−==
==zz
(2.9)
While transmitted code sequence was x and channel output sequence y, a corresponding side information sequence z is
),...,,( 21 Kzzz=z (2.10)
where zn is side information corresponding received signal yn.
In this thesis it is assumed that pulsed jamming jams each symbol independently with equal probabilities. If this is not the case in real signal environment, one must use interleaving described in the Section 1.2.1 to randomize the pulsed jamming. From these assumptions follows that the channel transition probabilities satisfy in pulsed jamming channel ([1] Vol. 1 p. 191)
∏=
=K
nnknnK zxypp
1
),|(),|( zxy (2.11)
and
∏=
=K
nnK zqq
1
)()(z (2.12)
where p(yn | xkn, zn) is the transition probability when xkn was transmitted, yn was received, and while channel state is zn. The code transition probability is pK(y|x,z) when a sequence x was transmitted, a sequence y was received, and while a channel state sequence was z. The jamming state probability on the symbol decision time is q(zn), and qK(z) is the jamming state sequence probability for a code length of K.
2.2 Literature review
2.2.1 Decision metrics
In this thesis the input to the channel model is binary and the output is quantized to 2L levels. The uniform output quantizers shown in Figure 6 (a) are usually used, although non-uniform quantization levels may slightly improve performance ([14] p. 78-82). It can be shown that the performance loss is negligible when uniform quantization is used ([14] p. 78-82), ([39] p. 29-33), [40]. A step size of the quantizer should also be optimized in an optimum receiver. In [41] is stated that no single step size is globally optimal. In [42] it is shown that in an AWGN channel optimum uniform quantization spacing of the
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demodulated signal is 0.275 times nominal received signal level when 8 level quantization is used and ES/N0 = 0.5 dB. The optimum quantization is dependent on signal-to-jamming ratio and on jamming. Clark and Cain ([39] p. 29-33) propose uniform quantization in AWGN channel with spacing 2/(2L-1) where 2L is the number of the channel output quantization levels. The absolute values of the metrics are of no concern so, for convenience, the metric will be set to the region from -1 to +1 in this thesis and in numerical examples L is equal to 8 for the soft decision and equal to 1 for the hard decision receivers.
Optimum metrics can be calculated for an AWGN channel ([39] p. 28)
( )[ ]xypCCyxm log),( 21 −−= (2.13)
where C1 and C2 are constants and p(yx) is the transition probability of the received signal to be in region Bi when x was a transmitted symbol and y is the received signal after demodulator in (2.5). The optimum metric is dependent on the signal-to-noise ratio and quantization. Figure 7 shows a uniformly spaced (as in Figure 6 (a)) decision metric’s optimum values as a function of signal-to-noise ratio in an AWGN channel when L=8 (i.e. 16 level quantization), C1 and C2 are arbitrary constants (C1=-0.9, C2=-1/7.5) and logarithm log10 is used. It can be seen, that the optimum values are close to linear ones. In this study uniform spacing is 1/L, which is for large L almost equal to that Clark and Cain proposed. It is not practical to calculate the optimum metric for every case. Instead, integer values for the metric are usually used.
Fig. 7. The optimal uniform decision metrics in an AWGN channel.
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The hard decision receiver makes a decision on a symbol (+1 or -1 in the binary case). A maximum likelihood (ML) decoder minimizes the “distance” between a code word and a received sequence. In the hard decision receiver the distance function is a measure of symbol errors between a code word and the received sequence. For a hard decision decoder the distance can be calculated by summing errors. As can be noted from BSC description in Figure 6 (c) the metric can be 1 when no error is made and 0 when error is made. A hard decision metric can thus be defined to be ([1] Vol. 1 p.198)
≠=
=xyxy
xym,0,1
),( (2.14)
where y is the received and x is the transmitted symbol, +1 or –1. The quantizer in the receiver is implemented as shown in Figure 8 (b). After each symbol decision a decoder stores decisions in memory. When all symbols of the code word are received the decoder uses the metric and compares the received symbols y to the symbols of possible transmitted code words. If y is for example +1 clearly the decoder comparison to x=+1 gives the metric value 1 and to x=-1 gives the metric value 0. This is repeated for all code word symbols. A code word metric is achieved by calculating the sum of all the symbol metrics. The decoder chooses the code word sequence, which has the largest code word metric (Hamming distance) ([14] p. 81). This clearly minimizes code error probability. The metrics are used by same way in the soft decision decoder. The only difference is that a metric has different definitions, but the basic idea of code word metric calculation stays the same.
If a soft decision receiver is used, a suitable distance function must first be defined. An unquantized soft decision metric is defined in reference ([1] Vol. 1 p. 202), see Figure 8 (a)
yxxym =),( (2.15)
where the transmitted signal x is +1 or –1 and y is the unquantized received (voltage) signal output of the matched filter receiver. This metric has clearly the property that a large received symbol value y results in a large positive or negative metric. The code word metric is achieved, as in the hard decision case, by calculating the sum of the symbol metrics. The optimum decoder chooses the sequence that has the largest code word metric. It has been shown that the use of the soft decision metric may result in a coding gain of approximately 2 dB in an AWGN channel when the bit error probability is 10-5 ([39] p. 35-36). The symbol metrics are used by decoding algorithms (for example the Viterbi algorithm) and that is why they are sometimes called decoding metrics.
The decoding process uses a metric of the form m(y,x;z) if side information is available and m(y,x) if it is not available ([11] p.78-81), where y is channel output code sequence, x is coded symbol sequence and z is side information sequence defined in (2.1), (2.4) and (2.10). This means that side information has effect on a metric, which will be discussed more in detain in Chapters 7 and 8.
The metric has an additivity property, which means that the metric for a sequence is the sum of the symbol metrics ([1] Vol. 1 p. 191). The total metric for a sequence is thus
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∑=
=N
nnnn zxymm
1);,()( zx;y, (2.16)
when side information is available or
∑=
=N
nnn xymm
1),()( xy, (2.17)
when side information is not available and where yn is n:th element of the sequence y, xn is n:th element of the sequence x and zn n:th element of the sequence z.
Fig. 8. Metrics for (a) the soft decision, (b) the hard decision, (c) the clamped or limiter and (d) the Dempster-Shafer receiver [37].
If fading or jamming is present, the soft decision receiver either requires channel state information or information concerning the reliability of the received code symbols, for satisfactory performance ([11] p. 46), [43]. Soft decision can be used under jamming by introducing a limiter, shown in Figure 8 (c) [5], to scale the detector output. In some papers this is called a clamped metric or receiver [37]. This limiter places an upper limit on the value of a decoding metric, so that jamming strategies with high power levels are thwarted but a soft decision decoding gain can still be obtained. These soft-limiter
36
receivers will always outperform the hard decision without jammer state information, but they will always perform worse than a soft decision maximum likelihood (ML) receiver with perfect jammer state information.
The optimum soft decision metric in a fading channel is [36]
2/),,,( σyxbnbxym = (2.18)
where y is the received signal, x is a transmitted bit (+1 or –1) to which received signal y is compared in decoder , b is channel gain (or the effect of fading) and σ 2 is the noise variance. With pulsed noise jamming, the noise variance should be measured separately for each symbol. In a fading channel b should be estimated. If there is no fading and noise is stationary, then the noise variance and the channel gain can be set equal to one and equation (2.15) results. In jamming (and particularly with pulsed jamming) the performance of the limiter receiver (or clamped metric, see Figure 8 (c)) is much better than that of the unquantized soft decision one without jammer state information [5], [37]. The limiter receivers are more robust against different kinds of jamming signals ([1] Vol. 2 p. 52-55).
Usually quantized soft decision metrics are used in practical receivers ([14] p. 78-82). These receivers can be considered as a special case of the hard decision receiver with more than one quantization level or as a modification of the limiter soft decision receiver. Detailed analysis and a decision metrics are shown in Chapter 6. The quantized soft decision receivers place an upper limit on the value of a decoding metric, as did soft decision limiter receiver in Figure 8 (c). In [44] it has been shown that a 3-bit quantization of metric results in a 0.25 dB loss in an AWGN channel. Because of signal amplitude variations 4-bit quantization is proposed in [36] for a quantized limiter receiver in a fading channel to cover the larger dynamic range. With pulsed jamming at least 4-bit quantization is also reasonable for the same reason. If implementations do not become too complex, even more bits may be beneficial. The more bits used in decision making, the less accurate automatic gain control (AGC) can be tolerated with less performance degradation.
In [37] Dempster-Shafer theory is applied to find an optimum decision metric in broad band noise jamming for a direct sequence spread spectrum (DSSS) system. In Dempster-Shafer theory unknown jammer state information is treated as uncertainty. This uncertainty is used to solve a hypothesis testing problem like that in the derivation of Bayesian receivers shown in many textbooks (e.g. ([3] p. 365-376), ([17] p. 594-593)). Dempster-Shafer theory is a generalized Bayesian theory. It provides a decision making method that discounts the degree of confidence on observations when the factor of uncertainty exists when jammer state information is unknown [37]. The decision metric for the BPSK receiver becomes [37]
( )[ ] )sgn(/2exp1),( 2 yyxyxm σ−−= . (2.19)
The decision metrics of soft, hard, clamped and Dempster-Shafer receivers are shown in Figure 8. Clearly the Dempster-Shafer receiver has the advantages of the hard decision
37
receiver when jamming is included in the signal as well as the advantages of the soft decision receiver for small signal levels. Dempster-Shafer receivers can smooth the clamped receiver and thus it does not require knowledge of jammer state (i.e., jammer power) to determine a clamping threshold. The Dempster-Shafer receiver is robust with respect to the signal-to-jamming ratio.
In [37] the Dempster-Shafer receiver is analyzed with pulsed noise jamming. The performance of the Dempster-Shafer receiver is between the soft and hard decision receiver in an AWGN channel. In worst case jamming without jammer state information the Dempster-Shafer receiver has about the same performance as the hard and clamped receivers when signal-to-jamming ratio (S/J) is large (10 dB), but when S/J is small (3 dB) the Dempster-Shafer receiver outperforms the others. In the case jammer state information is not available the Dempster-Shafer receiver is the best choice of the receivers shown in Figure 8 in pulsed noise jamming channel [37].
2.2.2 Side information
There are several methods to acquire side information about the channel state. Some methods use test symbols while others do not require any additional information to be transmitted. Usually the methods to acquire the side information are ad hoc. In this section the methods to acquire side information are reviewed.
Rahnema and Antia [36] propose for fading channels that the fading coefficient b in (2.18) can be estimated by sending reference symbols and estimating their amplitude in the receiver. This estimation can then be utilized in calculation of the decision metric. In a fading channel and convolutional coding (K=5, R=3/4) with Viterbi decoding this side information gives an advantage of 0.24-0.52 dB when soft decision is used. The gain achieved over hard decision decoding is 0.52-2.02 dB. This same method cannot be used to estimate interference if it changes from bit to bit, because estimation is based on the measurement of the sequence of test symbols.
In [35] four methods to acquire side information from channel state are presented for a slow frequency hop (SFH) system: test-symbol, parity-check, ratio-threshold test (RTT), and noise-threshold test (NTT) methods. In the test-symbol method a sequence of known symbols are transmitted in each frequency hop interval. The channel state is determined by measuring the number of errors made in demodulating the test symbols. In the parity-check method parity bits are associated with certain subset of code symbols in order to provide error detection capability. The channel state is based on the number of failed parity checks. These two methods require additional information to be transmitted to obtain side information. In DSSS systems the same principles can be applied when information is added frequently enough. However, it may be difficult to predetermine the period of information addition in intentional jamming, especially when short jamming pulses can appear.
In [35] two other methods are proposed to obtain side information without adding any information. In the ratio-threshold test (RTT) method side information is derived from the soft decision demodulator outputs. In [35] the use of the symbol block codes in the SFH
38
system is studied and all bits of a code symbol are transmitted on a different frequency. In the RTT method the ratio of the smallest and the largest envelope detector output for each of the binary symbols in a code symbol is determined. A code symbol is erased if, for any bit in the code symbol, the ratio exceeds a predetermined threshold. This threshold is a parameter to be optimized. The optimum value depends on the channel and interference model. The same approach can be directly applied to DS systems as well, if interference of the bits in a code symbol is independent. This can be guaranteed by interleaving bits. Another method is the noise-threshold test (NTT) method. In the NTT method, interference is estimated from the soft decision outputs of the demodulator before or after the signal reception in a frequency slot of a SFH system. If the interference level is larger than a predetermined threshold, the signal on that frequency is erased. This method is logically almost equivalent to the chip combiner receiver for DS system presented in Chapter 7.
In a Rayleigh fading and interference channel the parity-check method gives lower error probability than the test-symbol, RTT, and NTT methods [35]. The RTT give larger throughput than those that employ the parity-check method. Throughput means here higher data rate using same power and bandwidth. Each method provides a significant performance improvement over systems that do not erase and use errors-only decoding. If the SNR is good then the RTT and NTT give the best throughput [35].
In [45] Matis and Modestino propose two practical and robust methods to obtain side information without adding any information in DSSS systems. The first is based on the measurement of the demodulated soft decision signal level. When the signal level is larger than a predetermined threshold level, it is assumed that there is interference in the channel and that symbol is erased. The second is a chip combiner receiver where a separate receiver called a chip combiner measures the channel state. The chip combiner integrates the chips on the bit time from the wide-band signal before despreading. If the integration result is larger than a predetermined threshold level, it is assumed that there is interference in the channel and that symbol is erased [45], [33]. Both of these methods require optimization of the threshold level.
Torrieri [38] shows that soft decision decoding with maximum-likelihood (ML) metric works well with pulsed jamming. He shows that the ML metric is 4.5 dB better than the hard decision metric at bit error probability 10-4 when R=1/2 convolutional code is used. Torrieri also proposes to use an AGC metric, which measures the average power within the frequency band of the desired signal during each channel-symbol interval. This measurement result is then used to weight the demodulator output instead of noise variance σ2 shown in (2.18). The performance of this AGC metric is assumed to be worse than the optimal metric in (2.18) because energy measurement inaccuracies during jamming pulses. Torrieri proposes to use the same kind of channel measuring receivers as Matis and Modestino in [45].
The methods, presented in the literature, to get side information are ad hoc methods, and are not optimal ones. Optimization of channel state measuring receivers for many varieties of the jamming signals is neither possible nor practical. Therefore some ad hoc, nonlinear method may be a good way to measure the channel state. Some of the receivers introduced here use nonlinear decision metric (e.g. limiter receivers). It seems to be so that, a good receiver in nonlinear jamming situation is often nonlinear.
39
In this thesis methods presented by Matis and Modestino [45] are used. They are designed to be used with BPSK modulation and are shown to be effective against pulsed jamming, but the performance of the non-jammed situation is not either degraded. These methods do not require any additional information to be transmitted and these receivers are rather easy to implement.
2.3 Coded bit error rate bound
It is assumed that the sequence x in (2.1) is transmitted. There are many other sequences, which could also have been transmitted. Another coded sequence is denoted
)ˆ,...,ˆ,ˆ(ˆ 21 Kxxx=x (2.20)
where nx̂ is a transmitted code symbol and K is the length of the code sequence. When the channel output sequence is y in (2.4) and the side information is z in (2.10), the decoder decides that x̂ is the transmitted sequence if );ˆ,( zxym is the largest of all possible code word metrics and so it satisfies
xxzxyzxy ˆ);,();ˆ,( ≠∀≥ mm . (2.21)
If x was the actually transmitted code sequence, the probability that the decoder decides incorrectly that a sequence x̂ was transmitted is given by the pair-wise error probability
{ }xzxyzxyxx |);,();ˆ,(Pr)ˆ( mmP ≥=→ . (2.22)
Applying the Chernoff bound ([1] Vol. 1 p. 192, 248-249), this can be bounded as
[ ]( ){ }∏=
−≤→K
nnnnnnn zxymzxymEP
1
|);,();ˆ,(exp)ˆ( xxx λ (2.23)
for any 0≥λ . For nn xx =ˆ it is clear that
[ ]( ){ } 1|);,();ˆ,(exp =− xnnnnnn zxymzxymE λ . (2.24)
Antipodal and orthogonal waveforms and all metrics of interest have the important property ([1] Vol. 1 p. 193)
[ ]( ){ } )(|);,();ˆ,(exp ˆ λλ DzxymzxymE
nn xxnnnnnn =− ≠x (2.25)
40
where function D(λ) is independent of xn and nx̂ as long as nn xx ≠ˆ . Thus, the pair-wise error probability is bounded by
[ ] ),()()ˆ( xxxx
)wDP λ≤→ (2.26)
where w(x, x̂ ) is the number of places where Nnnn ...,,2,1 , ˆ =≠ xx . This is called the Hamming distance. D is defined as
[ ]( ){ }
nn xxnnnnnn zxymzxymEDD ≠≥≥−== ˆ00
|);,();ˆ,(expmin)(min xλλλλ
. (2.27)
The Chernoff bound of the pair-wise error probability is
),(
)ˆ(xx
xx)w
DP =→ . (2.28)
By using the union bound ([1] Vol. 1 p. 194) the coded bit error bound has the form
( )∑ ∑∈
≤c
xxwb DpaP
xxxxx
ˆ,
),()(ˆ,)
(2.29)
where a(x, x̂ ) is the number of bit errors occurring when x is transmitted and x̂ is chosen by the decoder, p(x) is the probability of transmitting sequence x and C is a group of all possible codewords. The bit error bound has the general form
)(DGPb ≤ (2.30)
where G is a function determined solely by the specific code, whereas the parameter D depends only on the coding channel and decoding metric. D is a quality measure of the receiver and it can be used to compare different receivers and decision metrics.
A bound on the coded bit error probability has the form ([1] Vol.1 p. 195)
)( 0RBPb ≤ (2.31)
where B is a unique function of the cut-off rate R0 for each code. The cut-off rate R0 (bits/channel use) represents the practically achievable reliable data rate per coded symbol ([1] Vol. 1 p. 194). Because R0 is independent of the code used, it is possible to decouple coding from the rest of the system. Thus, to evaluate various anti-jam communication systems, they are first compared using the cut-off rate or the channel parameter D. Codes are then evaluated separately. In general, there is a one-to-one relationship between R0 and D ([1] Vol.1 p 195)
41
symbol code / bits)1(log11
12
20
0
DRM
MDR
+−=−
−=
−
(2.32)
where M is code symbol alphabet size. In this thesis BPSK is used and thus M = 2. For convolutional codes the bit error upper bound can be obtained by finding the transfer function of the state diagram, and then upper bounding the error event probabilities by using a union-Chernoff bound [43]. An example of this will be given in Chapter 3.
In some cases the bit error probability can be calculated exactly, but in most cases accurate calculation is not possible or it is very cumbersome. In these cases upper bounding techniques have been used. The upper bounding techniques give in some cases meaningless results for soft decision decoding without jammer state information. In [46] the exact pair-wise bit error probability is calculated instead of the Chernoff bound. There the error probability is calculated over all possible code word errors. This analysis shows that hard decision decoding without side information outperforms the case with channel state information for high pulse duty factor (ρ) jamming, where ρ is on time of the jamming and 1- ρ is off time of the jamming. There is little benefit from jammer state information when the pulse duty factor is small. Generally, hard decision decoding without side information exhibits more robust behavior than with side information. The Chernoff bound gives very pessimistic results when the pulse duty factor is small. So it can give misleading results in some cases (usually with the small pulse duty factor). The Chernoff and union bound techniques are used in this study and it is good to remember this anomaly. However, as it will be seen in later chapters, this is not usually a problem because the studied systems turn out to have the most interesting performance evaluations when the pulse duty factor is high or even one. Because Chernoff and union bound techniques are inaccurate in some cases it is necessary to conform the bit error probability of the system by simulation in selected cases. The simulation results of the analyzed receivers are shown in original Papers. However, the analytical calculations help a designer to limit the simulation cases to the worst (from communication point of view) or the optimal cases (from jammer point of view). Analysis can also be used to evaluate the optimal receiver parameters or the worst case jamming parameters.
3 Communication system model
The DS/BPSK receivers are studied within a pulsed noise or CW jamming channel with the assumption of no fading. The bits are assumed to be jammed independently, which can be implemented using interleaving techniques if interference bursts occur in the channel. In bit error upper bound calculations and in simulations convolutional code with rate R = ½ and constraint length K = 7 is used. The code is decoded using maximum likelihood decoder implemented by Viterbi algorithm.
In Figure 2 the block diagram of a coded DS/BPSK anti-jam system with the transmitter and the receiver structures are shown ([1] Vol. 1, p. 142-148). A BPSK modulated signal has the form
[ ]
integer,,)1(,cos22/sin2)( 00
=+<≤=+=
nTntnTtSxxtSts
ss
nn ωπω (3.1)
where s(t) is transmitted BPSK signal, S is signal power, ω0 = 2πf0, angular frequency of the information signal, f0 is the center frequency of the information signal, xn is n:th bit of the code sequence x, Ts is channel symbol duration, and
−=
.21y probabilitwith ,121y probabilitwith ,1
nx (3.2)
The power spectrum of the BPSK signal is sin2x/x2 shaped with a one-sided first null bandwidth equal to 1/Ts ([1] Vol. 1, p. 143). DS spreading is done with a pseudo noise (PN) binary sequence {ck} whose elements have the values ±1 and are generated by the PN sequence generator N times faster than the data rate. In this thesis maximum length
43
sequence (m-sequence) of the length N = 63 bits long is used. The duration of a PN binary symbol refereed to as a “chip” is
NT
T sc = . (3.3)
It is easily seen that processing gain defined in (1.2) is ([1] Vol. 1, p. 143)
NR
WPG
s
ss == . (3.4)
The DS/BPSK signal has the form
[ ] tScxcxtStp knNnknNn 00 cos22/sin2)( ωπω ++ =+= (3.5)
.integer 1, . . .,2,1,0
)1(where
=−=
++<≤+
nNk
TknTtkTnT cscs
The data function is continuous time function, which is modulated by BPSK
modulator
integer,)1(;)( =+<≤= nTntnTxtx ssn , (3.6)
and PN function is continuous time function, which modulates BPSK modulated signal s(t)
integer , )1(;)( =+<≤= kTktkTctc cckn , (3.7)
where ckn is +1 or –1.Thus, the DS/BPSK signal can be expressed as
[ ]
.)()(cos2)()(
2/)()(sin2)(
0
0
tstctStxtc
txtctStp
==
+=
ω
πω (3.8)
In the receiver the signal p(t) is multiplied synchronously by spreading sequence c(t). Because
ttc allfor 1)(2 = , (3.9)
it follows that
44
)()()( tstptc = , (3.10)
and it can be seen that the original BPSK modulated signal is recovered. Information is then recovered by coherent BPSK demodulator (integrate and dump receiver), as shown in Figure 2.
3.1 Jamming signals
Jamming signals may be deterministic or random and they are denoted by J(t) and nJ(t), respectively. An AWGN noise signal n(t) is also assumed to be present, as usual since it models receiver noise.
It is assumed that a receiver is subject to pulse jamming. With probability ρ a code symbol is affected by jamming, background, and thermal noise. With probability 1-ρ a code symbol is affected by background and thermal noise only. The pulse jammer is assumed to affect either all of PN chips comprising a BPSK symbol or none of them. This is worst case jamming because the jammer effect is in that case maximized for the bit under jamming. This assumption also largely simplifies the analysis. The variable z is the jammer state random variable with probabilities, defined in (2.9)
{ }{ } ρ
ρ−==
==10Pr
1Przz
(3.11)
and the power of the pulsed jammer when pulse is on is
ρJJ peak = (3.12)
where J is the average power of the jammer. In the case of pulsed jamming, D will be calculated as shown in (2.25) ([1] Vol. 1, p. 202)
( ) ( )( )[ ]{ }
( )( )[ ]{ }( ) ( )( )[ ]{ } xxzxymzxymcE
xxzxymzxymcE
xxzxymzxymzcED
≠−−+
≠−=
≠−=
))
))
))
),,(),,()0(exp1
),,(),,()1(exp
),,(),,()(exp
λρ
λρ
λλ
(3.13)
where c(z) is a weighting coefficient. If the receiver would have jammer state information (knowledge of z) then c(z) could be optimized for the best performance. If the receiver has no side information, then the metric is independent of z and weighting is c(0) = c(1), which can be normalized to unity.
In the jamming channel the parameter D(λ) is a function of the signal energy, noise power, jamming signal waveform and power, parameter λ, pulse duty factor ρ and a decision metric. The channel parameter D(λ) must first be minimized by choosing λ and
45
then maximized by choosing ρ (a jammer has objective of maximizing D) and, if feasible, optimizing then decision metric parameters [43]. After that, the value of D can be calculated for specific values of signal energy, noise power, jamming signal waveform and jamming power. As a random jamming signal nj(t) AWGN jamming is used in this thesis.
Pulse jamming may introduce channel memory as discussed earlier. Many codes are designed for use over memoryless channels. In this thesis, it is assumed that errors appear in the channel randomly and hence the coding channel can be assumed to be memoryless.
For a soft decision maximum likelihood (ML) receiver perfect jammer state information is needed ([1] Vol. 1, p. 202). It is assumed that the receiver has some circuit to produce this information for the ML receiver.
In this thesis the deterministic jamming signal used in the numerical analysis is the CW signal
( )JJJ tAtJ φω += cos)( (3.14)
where AJ is the amplitude and ωJ angular frequency of the jamming signal and φJ the phase difference between the jamming and information signal. The phase is assumed to be random and uniformly distributed over 0 and 2π. The power of the deterministic jamming signal is
2/2
JAJ = . (3.15)
The noise jamming signal used in numerical analysis is zero mean white Gaussian noise with power spectral density NJ and bandwidth WJ. So, the received noise jamming power is
JJN WNJ = . (3.16)
3.2 Communication system in jamming
In the communication channel the transmitted signal is corrupted by jamming J(t) or nJ(t) and noise n(t) shown in Figure 2. The channel output is, in general
)()()()()( tntntJtptr J +++= . (3.17)
The channel output r(t) is multiplied by the PN sequence c(t) to obtain
)()()()()()()()( tntctntctJtctstq J +++= . (3.18)
46
The BPSK detector output is obtained after demodulation where q(t) is multiplied by local oscillator signal and integrated, see Figure 2 ([1] Vol. 1, p. 142-148)
nnjEdy js +++= (3.19)
where d is the data bit for this Ts second interval, Es = STs is the channel bit energy,
dttTtJtcjsT
s∫=0
0 )cos(/2)()( ω , (J(t) is defined in (3.14)) the effect of the
deterministic jamming and
∫=sT
sJj dttTtntcn0
0 )cos(/2)()( ω is the effect of the noise jamming and
∫=sT
s dttTtntcn0
0 )cos(/2)()( ω is the effect of the noise signal.
The effect of the noise signals nj and n can be taken into account as in ([1] Vol1. p. 146-148) and bit error probability can be determined adding power spectral densities of both signals. The calculation of the effect of the CW jamming is studied in the Section 3.2.1 and BER calculation is done in the Section 3.2.2.
3.2.1 CW jamming
There are several ways to approximate the effect of CW jamming. In reference ([1], Vol. 2, p. 12) it is proposed that if the spreading factor N (or PG) is large enough the performance of DS/BPSK is essentially the same regardless of the type of the jammer. This is based on the assumption that the PN sequence is approximated as an independent binary sequence and the application of the Central Limit Theorem ([1] Vol. 2, p. 9). So the signal to jamming noise ratio for CW jamming would be the same as for noise jamming
SJN
NE
J
s
/= . (3.20)
Ziemer and Peterson ([7] p. 592) show that CW jamming with the same center
frequency than the information signal can be taken into account by substituting
Js
JcJ JNSE
JTN φφ 22 cos/
2cos2 == (3.21)
where the spreading code is assumed to be an infinite sequence of independent equally likely random binary digits. So the signal to jamming noise ratio is
47
JJ
s
SJN
NE
φ2cos21
/⋅= (3.22)
where φJ is the phase difference between the jamming and information signal. By comparing to equation (3.20) it is noticed that there is a constant k, which depends on the phase difference of the information and jamming signals, where k is
Jk
φ2cos21
= . (3.23)
In Figure 9, k is shown as a function of phase difference. The average value of k over all phases is one, so that when (3.22) is averaged over all phases equation (3.20) results.
Torrieri ([47] p. 117-118) deals with the effect of frequency offset. He shows that the effect of CW jamming can be calculated as
( )( )( )( ) ( )( ) ( )( )πωωωωφ
πωωπωω 2/sinc2cos
2/2sinc2/2sinc1 0
20
0
0cJsJJ
cJ
sJcJ TT
TTJTN −
−+
−−
+= (3.24)
and k is
( )( )( )( ) ( )( ) ( )( )πωωωωφ
πωωπωω
2/sinc2cos2/2sinc2/2sinc
11 02
00
0cJsJJ
cJ
sJ TTTT
k −
−+
−−
+= . (3.25)
Fig. 9. k as a function of φ J .
48
Equations (3.22) and (3.23) are just special cases of those presented by Torrieri. The constant k as a function of frequency and phase difference is shown in Figure 10. It is seen that Figure 9 is the special case when frequency difference is zero.
It is usually assumed that the phase difference is randomly and equally distributed over all phases. When k is averaged over all phases, Figure 11 results. It is noted from Figure 11 that the jamming is most effective on the same carrier frequency than the information signal. A small variation (less than the inverse of the symbol time) in the jammer center frequency does not have any impact. If frequency difference is (fJ- f0)/Ts=1 the effect of the phase difference ΦJ variation is removed and jammer has the same effect regardless ΦJ. If the worst case is analyzed, it must be assumed that the jammer has the same carrier frequency than the information signal.
Fig. 10. k as a function of φJ and fJ - f0 when N = 63.
49
Fig. 11. Average k as a function of fJ - f0 when N = 63.
The effect of deterministic jamming (marked by j) can be modeled also as another signal summed to the information signal. The analysis can be applied for any jamming signal, the effect of which can be calculated at the output of the receiver and is marked by amplitude value j. CW jamming with the same center frequency as the information signal is considered because it gives worst case performance.
CW jamming is shown in (3.14) with the same center frequency as the communication signal in (3.1). The effect of jamming j can be expressed after the integrate-and-dump filtering when the carrier phase of the information signal is synchronized in receiver (see Appendix 1), as
( )
JbJ
JbsJbJs
T
JJs
cE
cJTcAT
dtttAtcT
js
φ
φφ
φωω
cos2
cos2cos2
coscos)(2
000
=
==
+= ∫ (3.26)
where cb is the spreading code unbalance
NNNcb
01
bitin chips ofnumber zeros ofnumber - ones ofnumber −
== (3.27)
50
where N1 is the number of ones, N0 is the number of zeros and N is the total number of chips in spreading code on symbol time. The result (3.26) is presented also in reference [48] for a conventional receiver. If the code unbalance cb is not known, the average unbalance of the code family can be used.
The result in Figure 11 proposes that CW jamming should have the same effect as noise jamming. But as will be seen in Chapters 4 - 8, this is not always the case. When more accurate analysis is done using equation (3.26) to count the jammer effect, it will be seen that CW jamming is not so effective as noise jamming when m-sequence is used as a spreading code. The code unbalance is the most important factor, which changes the effect of CW jamming.
3.2.2 Channel symbol and coded bit error probabilities
The noise component nj is a zero-mean Gaussian random variable with variance NJ/2 ([1] Vol.1 p.146-148) and n a zero-mean Gaussian random variable with variance N0/2. The channel bit error probability is ([1] Vol.1 p. 148-149)
( ) ( )[ ]
( ) ( )[ ].21
101021
jEnnPjEnnP
xyPxyPP
sjsj
s
−>++−−<+=
−=>+=<= (3.28)
For convenience, in the first part of (3.28) we can consider also the inequality
jEnn sj +>+ because noise is zero mean WGN process. By using the Q-function
the bit error probability is ([17] p.375, 515; [1] Vol. 1, p.148)
( ) ( )[ ]
.2221
21
00
+
−+
+
+=
−>+++>+=
NN
jEQ
NN
jEQ
jEnnPjEnnPP
J
s
J
s
sjsjs
(3.29)
Equation (3.29) is general in the sense that in the case of no jamming signals, the terms j and NJ are just zero.
In the thesis the worst case channel parameter D, defined in (3.13), is calculated as a function of the pulse duty factor and signal to jamming ratio for different receivers. This is done to see the performance of the different receivers and the receivers are then easy to compared. The bit error probabilities are also calculated for an optimal convolutional code of length 7 and rate ½, which is widely used in literature and also in applications. The equation of the bit error upper bound for that code is ([15] p. 64)
51
( )...1163314042113621 16141210 ++++= DDDDPb (3.30)
where D is the channel parameter defined in (2.27). The generation, decoding and selection of the error correcting codes is not considered in this thesis, but can be found from many textbooks and papers, e.g, [14], [39], [16], [15], [49] - [52].
The bit error upper bound for the used code is shown as a function of the channel parameter D in Figure 12. It is seen that bit error probability is a monotonous function of D. The channel paramerer D values from 0.1 to 0.6 are in the most interesting bit error region.
In the following Chapters 4-8 the worst case D and the optimum pulse duty factor, ρ as a function of signal to jamming ratio are plotted separately to see clearly those optimum values. The power of the equations shown in the thesis is largely in the fact that the worst case can be determined. Accurate bit error rate can then be obtained by simulation. The simulation work is remarkably reduced when worst values of the pulse duty factors can be directly used in simulations.
Fig. 12. Bit error upper bound as a function of D for convolutional code K=7 and R=1/2.
4 Hard decision receiver
4.1 Decision metric and channel parameter
The hard decision receiver is considered as a basic reference system. The used communication system is described in Figure 2. The decision metric block in Figure 2 performs the operation described in Figure 8 (b). The hard decision receiver is widely used, easy to implement and was shown to be rather robust in a jamming channel in the literature reviewed in Chapter 2. The receiver is assumed not to have any information on channel state. The metric of the hard decision receiver is ([1] Vol.1, p.198)
≠=
=xyxy
xym,0,1
),( (4.1)
where y is the received and x is the transmitted symbol, +1 or –1, and the channel parameter D is ([1] Vol.1, p.198)
( )ss PPD −= 14 (4.2)
where Ps is channel symbol error probability. In an AWGN and jamming channel, D can be calculated using (4.2). Ps can be calculated using (3.29), where the effect of the CW jamming is calculated using (3.26). If the phase of the jamming signal φJ is assumed to be random, then D must be averaged over all values of φJ. In the case of pulsed jamming, D can be calculated as shown in (3.13).
53
4.2 Performance in jamming
In an AWGN (or barrage jamming) channel, D can be calculated from equation (4.2) by substituting Ps = PsN which can be calculated by using (3.29) (j=0 and NJ =0) ([1] Vol.1, p. 146), [44]
=
0
2NE
QP ssN . (4.3)
In noise jamming, the channel bit error probability PsNJ is calculated using (3.29) when j=0
+=
0
2NN
EQP
J
ssNJ . (4.4)
With pulsed noise jamming average Ps must first be calculated as
sNJsNs PPP ρρ +−= )1( (4.5)
and D is calculated using equations (4.2).
In CW jamming the channel bit error probability PsJ is calculated using (3.26) and (3.29) when NJ =0
( )
−+
+=
00
/2
/2
21
N
jEQ
N
jEQP ss
JsJρρ
φ (4.6)
and averaging over all phases
( )∫=π
φφπ
2
021
JJsJsJ dPP . (4.7)
With pulsed CW jamming Ps is calculated first
sJsNs PPP ρρ +−= )1( (4.8)
and D is calculated using equations (4.2).
D (and PsJ) can also be calculated for a random code by calculating the mean over code unbalance, which results in another integration over PsJ ([13] p.17)
( )∫∞
∞−= bbbsJsJrandom dccpcPP )( (4.9)
54
where p(cb) is the probability density function of the code unbalance and PsJ(cb) is the bit error probability of CW jamming with code unbalance cb.
The simulation results of the hard decision receiver with pulsed noise and CW jamming are presented in the original Papers I, and II respectively. In Figure 13 are shown simulated and upper bound bit error rate in convolutionally coded (code R=1/2, K=7) DS/BPSK when ρ = 0.1 and Eb/NJ = 0 dB. As a reference it is shown also uncoded bit error rate (the curve A) and AWGN channel bit error rates (the curve B). It can be seen that upper bound is rather loose in this case. In Figure 14 are shown simulated and upper bound bit error rates in CW and noise jammed channel. Also here it can be seen that the bit error bound is loose when S/J is small (-30 - -40 dB). It is thus necessary to confirm the analytical results (BER) by simulation.
Fig. 13. The bit error rate of the BPSK and DS/BPSK systems in pulsed noise jamming, when ρ = 0.1 and Eb/NJ = 0 dB. The curve A is uncoded jammed channel, the curve B is uncoded AWGN channel, the curve C is hard decision coded case upper bound bit error rate, the curve D is narrow band BPSK simulated system, and the curve E is DS/BPSK simulated system. (Reprinted from Original Paper I).
55
Fig. 14. The bit error rate of the DS/BPSK systems in pulsed noise and CW jamming, when ρ = 0.1 and S/J = -10 - -40 dB. The curve A is BPSK in AWGN channel, the curve B is upper bound bit error rate of the CW jammed channel, the curve C is simulated bit error rate of the coded system in CW jamming, the curve D is bit error rate of the uncoded DS/BPSK system in CW jamming, and the curve E is the simulated bit error rate of the coded DS/BPSK system in noise jamming. (Reprinted from Original Paper II).
D is calculated using (4.2) and substituting Ps as defined in (4.3) – (4.9) depending on
the jamming. D is first calculated as a function of Eb/NJ and ρ while Eb/N0 is as a parameter. In Figure 15 the case where Eb/N0 = 10 dB is shown as an example. The pulse duty factor has large impact on the performance of the receiver when signal-to-jamming ratio is small. So it is important to jammer to select right ρ. The jammer should know signal-to-jamming and signal-to-noise ratios in the input of the communication receiver to be able to optimize ρ. Usually this is not realistic assumption and jammer must select ρ without that knowledge ([53], p. 12).
56
a)
b)
Fig. 15. The hard decision receiver, D as a function of the pulse duty factor and signal-to-jamming ratio when Eb/N0 = 10 dB a) noise jamming b) CW jamming cb=1/63.
57
From Figure 15 it can be seen that there is an optimum value of the pulse duty factor ρ, which gives the worst D. The pulse duty factor ρ is optimized by computer search for each Eb/N0 and Eb/NJ or S/J, and the results are shown in Figures 16 and 17. In optimization it is assumed that the jammer has knowledge of Eb/N0 and Eb/NJ or S/J in the input of the receiver. In practice this is not a realistic assumption, but it gives us the worst case performance. A designer of the communication system can use this as a reference; the communication system will not work worse than the worst case shows. From Figure 17 it is seen that jammer should use rather high pulse duty factor if jamming power is high compared to signal power or if signal-to-noise ratio is low. So coding and the hard decision metric can force the jammer to use ρ = 1 to optimize the use of jamming power when Eb/NJ or S/J is small. The jammer is usually power limited and thus the only parameter to choose is the pulse duty factor ρ. When ρ is less than 0.01 errors caused by jamming are so rare that they can be corrected by many of the commonly used error correcting codes, e.g. convolutional (R=1/2 K=7) code [15]. From Figure 12 it can be seen that if jammer goal is to degrade bit error rate worse than 10-2 it should have D > 0.4. From Figure 16 it is seen that this requires signal-to-jamming ratio to be low (or signal-to-noise ratio low). Within this signal-to-jamming area ρ is also high, roughly in the region 0.4-1.
By using the worst case ρ of Figure 17 the bit error upper bound can be calculated for a specific error correcting code. Figure 18 shows, as an example, the worst case bit error upper bound for an optimal convolutional code R=1/2, K=7 and 63 bit m-sequence spreading code, whose PG=63 (18 dB). D is substituted to (3.30) and S/J is calculated for nose jamming using (1.3). The bit error rate of non-jammed case is for example 8•10-6 when Eb/N0 = 7dB. To degrade performance significantly from BER 10-4 by noise jamming S/J must be lower than 0 dB and in the case of CW jamming lower than –10 dB when Eb/N0 = 7 dB. Noise jamming seems to be more than 10 dB worse than CW jamming. The main reason is that the effect of CW jamming is decreased by a well selected spreading code balance of a m-sequence. As a result of this code selection, PG in CW jamming is doubled (36 dB in this case). The same effect applies to all studied receivers.
58
a)
b)
Fig. 16. The worst case D as a function of signal to jamming and signal to noise ratio a) noise jamming b) CW jamming cb=1/63.
59
a)
b)
Fig. 17. The worst case ρ as a function of signal to jamming and signal to noise ratio a) noise jamming b) CW jamming cb=1/63.
60
Fig. 18. The worst case bit error upper bounds in CW and noise jamming as a function of S/J, cb = 1/63.
5 Soft decision receiver
The communication system for soft decision receivers is described in Figure 2. The block decision metric in Figure 2 performs the operation described in Figure 8 (a). Both known and unknown jammer state receivers are studied in the case of the unquantized soft decision metric. In Chapter 2 the decision metric of the optimum receiver was introduced in (2.18). The optimum receiver acquires the perfect information on channel state including jamming signal. The optimum receiver gives thus a reference for the comparison of the other receivers. Usually perfect channel state is not possible to acquire, and as an opposite of optimum receiver an unknown unquantized soft decision receiver’s performance is also introduced. In the following chapters quantized soft decision metrics are based on the unquantized soft decision receiver simply by quantizing the receiver’s decision metric.
5.1 Optimum soft decision receiver, known jammer state
5.1.1 Decision metric and channel parameter
For a soft decision channel without fading the metric is ([1] Vol. 1, p. 202)
yxzczxym )();,( = (5.1) where c(z) is a weighting coefficient defined in (3.13) and z is jammer state random variable defined in (2.8). Compared to equation (2.18) c(z) replaces fading coefficient b. Now it is assumed that there is no fading in the channel and noise is also assumed to be stationary, so that σ2 may be omitted from (2.18). This metric is optimum in an AWGN channel. The optimum receiver is assumed to have perfect knowledge of the jammer state, which means that receiver knows weather jammer is on or off. The influence of noise jamming is considered in references ([1] Vol. 1 p. 158-160) and ([11] p. 29-30). The
62
soft decision metric assumes that in the input of the demodulator, the signal is unquantized and perfect AGC is used so that the signal level is constant. In the optimum receiver the jammer state is known, c(0) (jammer off) can be as large as possible, and c(1)=1 (jammer on). This means that jammed symbols are not accepted to give any information for decisions, instead only non-jammed symbols are utilized. From (3.13) it follows that in a pulsed jamming channel the parameter D is
[ ]{ } ( ) [ ]{ }
( )( )[ ]{ }( ) ( )[ ] ( ) ( )[ ]{ }.ˆexpˆexp
ˆexp(
ˆ)0(exp(1ˆ)1(exp()(
nnxxEjEdxx
xxnnjEdE
yxxycExyxycED
jS
jS
jj
+−+−=
−+++=
−−+−=
λλρ
λρ
λρλρλ (5.2)
The expectation is calculated in Appendix 2. As a result, D(λ) is
( ) ( ) ( )[ ].ˆexp)( 022
JS NNdjEdxxD +++−= λλρλ (5.3)
The result is that the worst case D is (Appendix 3)
( )
−=+>+=−<
−=+≤+=−≥
++−
=
.1,,1,,
1,,1,,
exp0
2
dEjdEj
dEjdEj
NNjEd
D
S
S
S
S
J
S
ρ
ρ (5.4)
In CW jamming D is calculated setting NJ =0 in (5.4) and in the pulsed noise case j=0. So D is in pulsed noise jamming
( )( )ρρ //exp 0 JS NNED +−= . (5.5)
In an AWGN channel D can be calculated using equation (2.27) by minimizing D(λ) with a result ([1] Vol.1, p. 196)
)/exp( 0NED s−= . (5.6)
5.1.2 Performance in jamming
The parameter D is calculated as a function of ρ, Eb/N0 and Eb/NJ or S/J (S/J is signal-to-noise power ratio defined in (1.3)). Figure 19 shows the cases where Eb/N0 = 10 dB. The pulsed noise jamming case is calculated using (5.5) and CW jamming case using (5.4),
63
where j is calculated using (3.26). The pulse duty factor has large impact on the performance of the receiver when signal-to-jamming ratio is small and the jammer should use ρ =1 to optimize jamming effect. The noise jamming is more harmless than CW jamming against optimum soft decision receiver.
The pulse duty factor ρ is optimized for each Eb/N0 and Eb/NJ or S/J by computer search. The results are shown in Figure 20, the worst case channel parameter D, and in Figure 21, the optimum pulse duty factor ρ. It is assumed, that the jammer has knowledge of Eb/N0 and Eb/NJ in the input of the receiver. From Figure 21 it is seen that the jammer should use the pulse duty factor equal to one in most jamming cases. This shows that it is possible to force the jammer to use continuous time jamming by using optimum soft decision receiver. By comparing Figures 17 and 21 it is seen that the optimum soft decision receiver has wider area of signal-to-noise and signal-to-jamming ratio where optimum pulse duty factor is one than the hard decision receiver.
The worst case bit error upper bound for an optimal convolutional code R=1/2, K=7 and 63 bit m-sequence spreading code, whose PG=63 (18 dB) is calculated by substituting D to (3.30) and calculating S/J for nose jamming using (1.3). In Figure 22 bit error upper bounds are shown when Eb/N0 is 7 and 15 dB. The bit error rate chances on very short signal-to-jamming area from 10-8 to 1 in noise jamming. A comparison of noise and CW jamming shows that noise jamming is more harmless than CW jamming as was the case with the hard decision receiver. The difference between noise and CW jamming is about 15 dB as can be seen from Figure 22. The main reason is that the effect of CW jamming is decreased by a well selected code balance of a m-sequence. By comparing Figures 18 and 22 it is seen that the hard decision receiver is more than 5 dB, in many cases more than 10 dB, worse than the optimum soft decision receiver. This gives a motivation to try to design better receiver than the hard decision receiver.
64
a)
b)
Fig. 19. The optimum soft decision receiver, D as a function of the pulse duty factor and signal to jamming ratio when Eb/N0 = 10 dB a) noise jamming b) CW jamming cb=1/63.
65
a)
b)
Fig. 20. The worst case D as a function of signal to jamming and signal to noise ratio a) noise jamming b) CW jamming cb=1/63.
66
a)
b)
Fig. 21. The worst case ρ as a function of signal to jamming and signal to noise ratio a) noise jamming b) CW jamming cb=1/63.
67
Fig. 22. The worst case bit error upper bounds in CW and noise jamming as a function of S/J, cb = 1/63.
5.2 Soft decision receiver, unknown jammer state
5.2.1 Decision metric and channel parameter
The soft decision receiver with the channel state information gives good performance, but perfect channel state is not normally available. For that reason, the unknown channel state receiver is considered.
In noise and CW jamming D must be calculated using (3.13). The received signal y is substituted from (3.19) to (3.13). When there are jamming in the channel, y is marked as yj and when no jamming exists in the channel just y is used. If side information is not available (i.e., c(0) = c (1) = 1) and nn xx ≠ˆ , so the channel parameter D is
68
[ ]{ } ( ) [ ]{ }( )( )[ ]{ }
( ) ( )( )[ ]{ }( ) ( )[ ] ( ) ( )[ ]{ }
( ) ( ) ( )[ ] ( )[ ]{ } .ˆexpˆexp1
ˆexpˆexp
ˆexp(1
ˆexp(
ˆexp(1ˆexp()(
nxxEEdxx
nnxxEjEdxx
xxnEdE
xxnnjEdE
yxxyExyxyED
S
jS
S
jS
jj
λλρ
λλρ
λρ
λρ
λρλρλ
−−−+
+−+−=
−+−+
−+++=
−−+−=
(5.7)
The expectations are derived in Appendix 2 and as a result D(λ) is
( ) ( ) ( )[ ]( ) ( ) ( )[ ] .ˆexp1
ˆexp)(
022
022
NxEdxx
NNxjEdxxD
S
JS
λλρ
λλρλ
+−−+
+++−= (5.8)
The effect of jamming and AWGN are separated in (5.8)
( ) ( ) ( )[ ]( ) ( )[ ] . channel AWGN ,ˆexp)(
channel jammig,ˆexp)(
022
022
NxEdxxD
NNxjEdxxD
SA
JSJ
λλλ
λλλ
+−=
+++−= (5.9)
The derivative D’(λ) can now be calculated as
( )( ) ( )[ ]( ) ( ) ( )[ ] .2ˆ)(1
2ˆ)()('
02
02
NxEdxxD
NNxjEdxxDD
SA
JSJ
λλλρ
λλρλ
+−−+
+++−= (5.10)
In the equations above, j is calculated using (3.26). Zero points of D’(λ) cannot be solved analytically. Therefore they have been calculated numerically using the “Matlab” software subroutine “fzero”. It uses a search algorithm originally generated by T. Dekker with some improvements made later [54], [55], [56]. When the zero points are found (λ >0), the minimum D can be calculated.
In a special case ρ =1 (i.e., jamming on all the time), D can be calculated (see Appendix 3) to be
( )
−=+>+=−<
−=+≤+=−≥
+
+−
=
.1,,1,,
1
1,,1,,
exp0
2
xEjxEj
xEjxEj
NNjEx
D
S
S
S
S
J
S
(5.11)
(5.11) can be used always when j and NJ are known to be independent of the wave-form of the jamming.
69
5.2.2 Performance in jamming
With pulsed jamming D is calculated using (5.8) when optimum λ is first calculated numerically using (5.10) separately for every ρ, j, Eb/N0 and Eb/NJ. The parameter D is calculated as a function of Eb/NJ, ρ and Eb/N0. Figure 23 shows the case where Eb/N0 = 10 dB. It can be seen that an optimum value of the pulse duty factor ρ, which gives the worst D, is close to zero. The calculation of D was finished to pulse duty factors 10-3 and 10-5 in the case of noise and CW jamming, respectively. If the calculation had been continued to smaller values, a larger D would probably have resulted. The results show that the unlimited soft decision receiver without jammer state information can be jammed effectively by very short pulse duty factors and this can cause a high bit error probability.
The pulse duty factor ρ is optimized numerically for each Eb/N0 and Eb/NJ or S/J. In Figure 24 the worst case channel parameter D is shown as a function of Eb/N0 and Eb/NJ or S/J. The optimum pulse duty factor is very small (close to zero). Against the soft decision receiver the jammer should use a very small pulse duty factor. The CW jamming is shown here without a spreading code (i.e., code balance cb = 1). The results are only scaled down by the factor of the spreading rate N, but otherwise there is no difference. For example, when a 63 bits m-sequence is used, one should decrease S/J by 36 dB in the case of CW jamming and Eb/NJ by 18 dB in the case of noise jamming.
From Figure 25 it can be seen that in optimized noise jamming the hard decision receiver has superior performance compared to the unquantized soft decision receiver. Poor performance of the unquantized soft decision receiver with pulsed noise jamming results from the fact that it is optimized for an AWGN channel. In CW jamming some bits are received with a good error rate even at high jamming power when φJ is in a 90 degrees phase shift. CW jamming seems to give a little bit better performance than pulsed noise jamming.
It is well known that the performance of both soft and hard decision can be better if the receiver has side information about the channel state. In Section 5.1 it was shown that side information in the soft decision receiver gave a lot performance improvement. The same can be shown to be the case with the hard decision receiver ([1] Vol. 1, p. 157-167). If side information is not used, one should use the hard decision receiver in a pulsed jamming channel. In Chapter 2 it was shown that the limiter soft decision receiver gives good performance in a pulsed jamming channel. The following chapters introduce some new receiver concepts with and without channel measurement capability. A common factor in all of them is that the ideas are raised from practical implementation considerations and they all can be easily implemented using current technology.
70
a)
b)
Fig. 23. The unquantized soft decision receiver, D as a function of the pulse duty factor and signal-to-jamming ratio when Eb/N0 = 10 dB a) noise jamming b) CW jamming cb=1.
71
a)
b)
Fig. 24. The worst case D as a function of signal to jamming and signal-to-noise ratio a) noise jamming b) CW jamming cb=1.
72
a)
b)
Fig. 25. The worst case bit error upper bounds of the soft and the hard decision receivers as a function of Eb/Nj or S/J a) noise jamming b) CW jamming cb=1.
6 Quantized Soft Decision Limiter Receiver
6.1 Decision metric and channel parameter
In Chapter 5 it was shown that if the channel state is unknown, the noise jammer could always make D approach 1 as closely as desired for the unquantized soft decision receiver, regardless of λ, by decreasing ρ sufficiently. The extreme swings of the jamming in the input of the receiver cause the poor performance in the unquantized soft decision receiver. In the literature review in Chapter 2 the receivers, which limit the input to some maximum value, were studied with pulsed noise jamming. The simplest of them was the limiter receiver with a clamped input shown in Figure 8 (c). The limiter receiver was shown to recover the most of the degradation lost to the unknown jammer and is better than the hard decision receiver. Performance is maximized by optimizing the clipping level. The digitally implemented limiter receiver is called as a quantized soft decision limiter receiver and it will be analyzed in this Chapter. The performance is expected to be close to that of the unquantized clipping receiver. The communication system for the quantized soft decision limiter receiver is shown in Figure 2.
As discussed in Chapter 2, the receiver with a 3-bit quantizer degrades the performance by about 0.2-0.25 dB compared to the unquantized soft-decision receiver in AWGN channel [44]. However, 4-bit quantization was proposed when operating in a fading or interference channel or when perfect AGC is not assumed to cover the larger dynamic range [36]. Usually it is not profitable to have more than 4 bits/sample ([39] p. 30-33). The theoretical analysis of the soft decision receivers supposes that the signal level is always adjusted to the same nominal level. In practice this would require the AGC circuit to operate perfectly. In this thesis AGC is assumed to follow the long time (tens of symbols) average of the information signal level.
In [5] the DPSK-modulated quantized soft decision limiter receiver is analyzed in the presence of broad band pulsed jamming. The optimum quantization levels depend on ES/N0 and the used code rate. In practice quantization can be uniform and the highest quantization level can be close to the nominal received signal level, as was discussed in Chapter 2. The quantized soft decision limiter receiver was shown to outperform the hard decision receiver in Chapter 2, but on the other hand, it was not as good as the optimum
74
soft decision receiver with perfect jammer state information. Because of the similarity of maximum likelihood and the soft limiter metric for DS/BPSK signaling, the soft limiter receiver is also expected to be very effective in coded DS/BPSK systems.
In [57] a DS/DPSK system is analyzed with pulsed noise jamming. There it is proposed to use a quantized soft decision limiter metric. Metric quantization steps can be optimized if signal and jamming power, noise variance, and the pulse duty factor are known. Usually this is not a realistic assumption and in this thesis near optimum quantization is used.
In this thesis uniform output quantizer shown in Figure 26 is used. This function is used in the receiver shown in Figure 2 in the block decision metric. The transition probabilities for this quantized channel can be calculated. As discussed in Chapter 2, in this study uniform 1/L spacing is used. The absolute values of the metrics are of no concern so, for convenience, the metric will be set to the region from -1 to 1. In numerical examples L is equal to 8, which means that, 16 levels quantization is used.
Fig. 26. The uniform 2L-level quantizer and the decision metric. The general 2L-level soft decision limiter quantized metric is [5]
75
( )
−−≤−
<≤−
−−
−>
=
S
SS
S
EL
Lyd
ELiydE
Li
Li
EL
Lyd
xym
1,1
1,1212
1,1
, (6.1)
( ) ( )[ ] ZiLLi ∈−+−∈ , 1,2 where
and Z is the group of integer numbers. The channel parameter D(λ) can be calculated by using (3.13). Because the metric is non-continuous, expectation is evaluated in several discrete intervals (Appendix 4). As a result, D(λ) can be expressed as
( )
( )
( )
.
2/2/
1
1
2//
/
2//
/1
12122exp
2/
1
12//
/1
)2exp(
2/
12
12//
/12
1)2exp()(
1
2
00
00
00
00
∑−
+−=
−
−
−−
−+
+
−−
−
+
−−−
−−
−+
−−+
+
−−−+
+−
−−
+
−+−
−=
L
LiSS
J
S
J
S
S
J
S
S
J
S
N
EL
Li
QN
EL
Li
Q
NN
djEL
Li
QNN
djEL
Li
Q
Li
N
ELQ
NN
djELQ
N
ELL
QNN
djELL
QD
ρ
ρ
ρ
ρ
ρρ
λ
ρρ
ρρλ
ρρ
ρρλλ
(6.2)
The channel parameter D is calculated by minimizing D(λ) when λ > 0. Minimization must be done numerically for each d, L, ES, J(t), NJ and N0 separately.
6.2 Performance in jamming
In an AWGN channel D can be calculated using equation (6.2) by minimizing D(λ) numerically. The variables ρ, j and NJ are equal to zero. With pulsed noise jamming the variable j = 0. By using (6.1) and (6.2) D(λ) can be calculated. When CW jamming is used, j is shown in equation (3.26). D is calculated as a function of ρ, Eb/N0, and Eb/NJ or S/J. There is an optimum value of the pulse duty factor, which gives the worst D. Figure 32 shows the case where Eb/N0 = 10 dB.
76
a)
b)
77 Fig. 27. The quantized soft decision receiver, D as a function of the pulse duty factor and signal-to-jamming ratio, Eb/N0 = 10 dB and L=8 a) noise jamming b) CW jamming cb=1/63.
The pulse duty factor ρ is optimized for each Eb/N0 and Eb/NJ or S/J. The results are shown in Figures 28 and 29. It is assumed that the jammer knows Eb/N0 and Eb/NJ or S/J in the input of the receiver. It can also be seen that the jammer can gain using pulsed jamming. The optimum pulse duty factor of the quantized soft decision limiter receiver is smaller than that of the hard decision receiver. The jammer should choose the pulse duty factor from region 0.01 to 1 to optimize the jammer effect.
The quantized soft decision receiver seems to have smaller optimum ρ than the unquantized soft decision receivers. In the useful BER region ρ should be about 0.01 – 1. The behavior is almost equal to the case of the hard decision receiver. The performance of the quantized soft decision receiver changes more rapidly when ρ changes from its optimum value than that of the hard decision receiver. The quantized soft decision receiver has thus better performance than the hard decision receiver.
Figure 30 shows, as an example, the worst case BER upper bound for an optimal convolutional code R=1/2, K=7 when Eb/N0 = 7 dB and 15 dB. BER is calculated substituting optimal D to (3.30). To degrade performance significantly from BER 10-4 by noise jamming S/J must be lower than -7 dB and in the case of CW jamming lower than –20 dB when Eb/N0 = 7 dB. Noise jamming is about 10 dB worse than CW jamming in the case of the quantized soft decision receiver, as was in the hard decision receiver case, too. The quantized soft decision receiver behaves as it was supposed: it is better than hard decision receiver, but not yet as good as optimum receiver with perfect channel state information.
78
a)
b)
Fig. 28. The worst case D as a function of signal-to-jamming and signal-to-noise ratio a) noise jamming b) CW jamming cb=1/63.
79
a)
b)
Fig. 29. The worst case ρ as a function of signal-to-jamming and signal-to-noise ratio a) noise jamming b) CW jamming cb=1/63.
80
Fig. 30. The worst case bit error upper bounds in CW and noise jamming as a function of S/J, cb = 1/63.
7 Signal Level Based Erasure Receiver
7.1 Decision metric and channel parameter
The communication system for signal level based erasure receiver is shown in Figure 2. In a signal level based erasure receiver the same principles as in the quantized soft decision limiter receiver are used. It is used the same type of quantization and limiting as in Chapter 6. The difference is that the received symbols exceeding a predetermined threshold value θ, are erased. This is called a signal level based erasure receiver and it’s decision metric is shown in Figure 31 [45], ([33] p. 41). In the case of strong interference, demodulated high signal values are not necessarily correct symbols; instead the bit error probability is close to 0.5. However, the quantized soft decision limiter receiver will give to that jammed symbol the highest possible reliability value. The right way would be to give it the lowest possible reliability or totally erase it in the decoding process. In this thesis convolution codes with a Viterbi decoder are used as a coding/decoding technique. For the Viterbi algorithm, the erased symbol can be given the lowest possible reliability value. Therefore, the metric for the signal level based erasure receiver is
( )
−<−
−−≤≤−−
<≤−
−−
<≤−>−
=
,,1
1,1
1,1212
1,1
,1
,
S
SS
SS
SS
S
Eyd
EL
LydE
ELiydE
Li
Li
EydEL
LEyd
xym
θ
θ
θ
θ
(7.1)
( ) ( )[ ] ZiLLi ∈−+−∈ , 1,2 where
82
and Z is the group of integer numbers. The metric has the same value –1 if received signal is larger than θ or smaller than -θ. These decisions are assumed to be jammed and they are thus given the lowest reliability value.
Fig. 31. The signal level based erasure metric. The channel parameter D(λ) can be calculated by using (3.13). Because the metric is
non-continuous, the expectation is evaluated in several discrete intervals, as was the case in Chapter 6. Calculation is shown in Appendix 5 and as a result, D(λ) can be expressed as
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]
( ) ( )( )( ) ( ) ( )( )∑
−
+−=
−−+
−
−−
−+
−−+−−+−−+−+
−++−−−=
1
2 )(3)(21)(3)(2
12122exp
511512exp461462exp
5156161)(
L
Li iAQiAQiJQiJQ
Li
AQAQJQJQAQAQJQJQ
AQJQAQJQD
ρρ
λ
ρρλρρλ
ρρρρλ
(7.2)
where
83
LNE
A S 0/21
−= ,
( )L
NELA S 0/2
124 +−= ,
( )L
NELiiA S 0/2
1)(2 −−= , (7.3)
( )L
NELiiA S 0/2
)(3 −= ,
( ) 0/215 NEA S−= θ ,
( ) 0/216 NEA S−−= θ ,
and for noise jamming
( ) ( )JSS NENELJ
//1//1211
0 ρ+−
= ,
( ) ( )JSS NENELLJ
//1//12124
0 ρ++−
= ,
( ) ( )JSS NENELLiiJ
//1//121)(2
0 ρ+−−
= , (7.4)
( ) ( )JSS NENELLiiJ
//1//12)(3
0 ρ+−
= ,
( ) ( ) ( )JSS NENEJ
//1//1215
0 ρθ
+−= ,
( ) ( ) ( )JSS NENEJ
//1//1216
0 ρθ
+−−= ,
84
and for CW-jamming
sS
SJb
S
JTENE
cdL
NEJ
//2
cos2/2
1 00
ρφ−
−= ,
( )sS
SJb
S
JTENE
cdL
NELJ
//2
cos2/2
124 00
ρφ−+−= ,
( )sS
SJb
S
JTENE
cdL
NELiiJ
//2
cos2/2
1)(2 00
ρφ−−−= , (7.5)
( )sS
SJb
S
JTENE
cdL
NELiiJ
//2
cos2/2
)(3 00
ρφ−−= ,
( )sS
SJbS JTE
NEcdNEJ
//2
cos2/215 00 ρ
φθ −−= ,
( )sS
SJbS JTE
NEcdNEJ
//2
cos2/216 00 ρ
φθ −−−= .
The channel parameter D is calculated, as in Chapter 6, by minimizing D by selecting λ. After that the jammer tries to maximize the bit error rate, which is equal to maximizing D by optimizing ρ. As the last step the communicator can optimize the erasing level θ so that D is minimized. This min-max-min-process gives the worst case performance of the optimized signal level based erasure receiver. A more difficult task is to select an optimum θ if there is a possibility of different jamming signals to exist in the channel. One approach could be to take the worst case jamming signal and optimize θ for that signal and confirm that it works satisfactorily in other cases.
The erasure receiver tries to detect a high jamming power in the channel. The detected high jamming power symbols are then erased. There are two possible errors that may happen: miss of jamming detection PMD and false alarm PFA (no jamming although detected jamming) ([1] vol. 2 p. 56), ([11] p. 71), [12]. These probabilities can be determined to be
( ) ( )( )
( ) ( ) ( )( )5611156
AQAQPPJQJQPP
inAFA
inJMD
−−−=−=−==ρ
ρ (7.6)
where PinJ is the probability of not erasing the received signal in jamming and PinA is the probability of not erasing the received signal in AWGN channel . In references ([1] Vol. 2 p. 57-58) and [12] the authors state that miss of jamming detection should be very small (e.g. 10-8), while false alarm probability may be higher (e.g. 10-2). Both probabilities are dependent on the erasure level, jammer pulse duty factor, signal-to-noise, and signal-to-jamming ratios. The use of these probabilities in the determination of the erasure level does not result in optimum performance. This will be seen in this chapter. A better way to
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optimize the erasure level is to select it such that D is minimized. The optimization is done as follows using a computer program: 1. The channel parameter D is calculated for wide range values of θ and ρ. 2. The worst case D is chosen by selecting optimal ρ (from jammer point of view). 3. The minimum D is chosen by selecting optimal θ.
7.2 Performance in jamming
The channel parameter D is calculated numerically by minimizing D(λ) using (7.2) and (3.26). The channel parameter D is calculated as a function of Eb/NJ or S/J and ρ when Eb/N0 and θ are parameters. Figure 32 shows the case where Eb/N0 = 10 dB and θ =5.
The pulse duty factor ρ is optimized for each θ, Eb/N0, and Eb/NJ or S/J. The results are shown in Figures 33 and 34 when θ=5. It is assumed, that the jammer has the knowledge of Eb/N0 and Eb/NJ or S/J at the receiver. The jammer gains using pulsed jamming. When the quantized soft decision limiter receiver in Chapter 6 and signal level based erasure receivers are compared, it is seen that in noise jamming the optimum pulse duty factor is almost equal, but in CW jamming a smaller pulse duty factor should be used for the erasure receiver than for the limiter receiver.
The optimum pulse duty factor of the signal level based erasure receiver is about the same as that of the hard decision receiver in noise jamming, but much lower in CW jamming. The effect of the CW jamming is strongly dependent on the choice of the pulse duty factor. The jammer should choose a pulse duty factor from region 0.01 to 1 to optimize the jammer effect in CW jamming and from 0.1 to 1 in noise jamming. If the jammer cannot optimize the pulse duty factor, a choice of the value 0.1 could be a good compromise.
After the optimization of ρ, the channel parameter D is minimized by selecting an optimal erasure level θ. The channel parameter D and the optimum erasure level θ are shown as a function of Eb/N0 and Eb/NJ or S/J in Figures 35 and 36, respectively. For noise jamming θ values from 2 to 7 and for CW jamming from 4 to 20 were calculated. Figure 36 proposes that θ should be high, even as high as 20 to be optimal. The high optimum value for the erasure level θ means that jamming is so high that it can be detected very reliably. But by examining Figure 37 it can be seen that when θ is higher than 4, it does not have a large influence. On the other hand, when Eb/N0 and Eb/NJ are high, smaller θ values should be used. However, in this case the bit error rate is good enough for satisfactory performance independent of the θ values used. The erasure level θ should be selected by optimizing it in the area of low Eb/N0 and Eb/NJ values. In the original papers VI and VII simulations with pulsed noise and CW jamming proposes to select θ equal to 2 for pulsed CW jamming and 2 or 3 for pulsed noise jamming. This example shows how misleading simulations based on few points can be. The analysis presented produces a more reliable result by searching the worst case pulse duty factors and the optimum θ values.
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a)
b)
Fig. 32. The signal level based erasure receiver, D as a function of the pulse duty factor ρ and signal-to-jamming ratio Eb/NJ or S/J when Eb/N0 = 10 dB, θ =5 and L=8 a) noise jamming b) CW jamming cb=1/63.
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a)
b)
Fig. 33. The worst case D as a function of signal-to-jamming and signal-to-noise ratio, θ =2 a) noise jamming b) CW jamming cb=1/63.
88
a)
b)
Fig. 34. The worst case ρ as a function of signal-to-jamming and signal-to-noise ratio, θ =5 a) noise jamming b) CW jamming cb=1/63.
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a)
b)
Fig. 35. Optimum D as a function of Eb/N0 and Eb/NJ or S/J a) noise jamming b) CW jamming cb=1/63.
90
a)
b)
Fig. 36. Optimum θ as a function of Eb/N0 and Eb/NJ or S/J a) noise jamming b) CW jamming cb=1/63.
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a)
b)
Fig. 37. D as a function of erasure level θ and Eb/NJ or S/J a) noise jamming b) CW jamming cb=1/63.
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Figure 38 shows the worst case bit error upper bound for an optimal convolutional code R=1/2, K=7 using (3.30). Noise jamming is seen to be about 15 dB worse than CW jamming. The bit error rate is almost equal to that of the quantized limiter receiver in Figure 30.
Fig. 38. The worst case bit error upper bounds in CW and noise jamming as a function of S/J, optimal erasure level θ, cb = 1/63.
8 Chip Combiner Receiver
8.1 Decision metric and channel parameter
In the chip combiner receiver the jammed symbols are erased ([33] p. 42), [45] based on the measurement done from the wide-band, not yet despread received signal, see Figure 39. The chip combiner circuit integrates the wide-band signal from the channel. The result of this integrate-and-dump is sampled at the same time as the information signal at the end of the bit-interval. Because the information in the channel is in the form of a spread spectrum signal (noise like), the effect of the information signal in the output of the chip combiner is very small. This makes the deception of jamming more reliable than in the case of the signal level based erasure receiver. Especially, the measurement of the narrow-band noise or jamming signal can be made more reliable. The output of the chip combiner integrator, at the same time as bit decision in the receiver is made, is thus
nnjxu jcc +++= (8.1)
where jc and xc are the effects of deterministic jamming and the information signal, nj and n are the effects of jamming and background noise in the output of the chip combiner.
If the output of the chip combiner exceeds the threshold θ, it is given the lowest possible reliability or it will be totally erased in the decoding process. So the metric is
94
( )
≥≤−
<<−
−≤−
<<<≤−
−−
<<−
>
=
SS
SSS
SSSS
SSS
EuEu
EuEEL
Lyd
EuEELiydE
Li
Li
EuEEL
Lyd
xym
θθ
θθ
θθ
θθ
or -,1
-,1,1
-,1,1212
-,1,1
, (8.2)
( ) ( )[ ] . , 1,2 where ZiLLi ∈−+−∈
Fig. 39. Communication system using the chip combiner receiver.
The metric is the same as that in Figure 26 for the quantized soft decision metric with the exception that there is an added condition based on the channel measurement u. The metric is conditional and it depends on the threshold θ and the effect of the jamming signal j. The same technique on the calculation of the D(λ) can be used as in the
95
preceding Chapters 6 and 7 when the integrating regions are determined. Analysis is presented in Appendix 6. As a result, D(λ) can be expressed as
( ) ( )( ) ( ) ( )( )[ ]
( ) ( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( )
( )( ) ( )ρρ
ρρλ
ρρλρρλλ
inJinA
L
LiinAinJ
inAinJ
inAinJ
PP
iAQiAQ
PiJQiJQPLi
AQPJQPAQPJQPD
−+−−+
−
−+−
−−
−+
−+−+−−−−=
∑−
+−=
111
)(3)(2
1)(3)(212122exp
1112exp411412exp)(
1
2
(8.3)
where A1 – A4 and J1 – J4 are defined in (7.3) – (7.5), PinA and PinJ are the probabilities of the chip combiner output u to be between thresholds SS EuE θθ <<- in AWGN channel and in jamming, respectively (see Appendix 6)
( ) ( ) ( )σσθθ //- UALASSinA uQuQEuEPP −=<<=
(8.4) ( ) ( ) ( )σσθθ //- UJLJSSinJ uQuQEuEPP −=<<=
where σ is variance of the random variable ntot = nj + n and thus 2σ2 = N0+NJ. When CW-jamming exist in the channel,
,cos2
,cos2
SbJJSUJ
SbJJSLJ
EdcEEu
EdcEEu
−−=
−−−=
φθ
φθ (8.5)
and when no jamming exist,
.
,
SbSUA
SbSLA
EdcEu
EdcEu
−=
−−=
θ
θ (8.6)
Two possible errors may happen: a miss of jamming detection PMD and false alarm PFA (no jamming although detected jamming) ([1] vol. 2 p. 56), ([11] p. 71), [12]. These probabilities can be determined to be (see Appendix 6)
.1 inAFA
inJMD
PPPP−=
= (8.7)
Again, in references ([1] Vol. 2 p. 57-58) and [12] the authors state that miss detection should be very small (e.g. 10-8), while false alarm probability may be higher (e.g. 10-2). But as will be seen in the next section, this does not result necessarily optimum performance. The better way to optimize is to use optimum channel parameter method
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used in this thesis. This is done by computer program on the same way as was described in Section 7.1.
The channel parameter D is calculated, as in the previous chapters, minimizing D by selecting λ. The worst case D is obtained maximizing D by selecting ρ. The erasing level θ can then be optimized by selecting it so that D is minimized. This min-max-min-process gives the worst case performance of the optimized chip combiner receiver. This optimization cannot be done analytically, but it is done numerically in the examples of the next section. When there is the possibility of different jamming signals existing in a channel, the worst case jamming signal should be taken and θ optimized for it.
8.2 Performance in jamming
With pulsed noise jamming the variable j=0 and in CW-jamming nj =0, and in AWGN channel j=0 and nj =0. The channel parameter D(λ) can be calculated by using equations (8.3) and when CW jamming is used, j is shown in equation (3.26). The numerical examples for D are calculated as a function of Eb/NJ or S/J and ρ, while Eb/N0 is as a parameter. In Figure 40 an example is shown as on the case where Eb/N0 = 10 dB and θ =100.
The pulse duty factor ρ is optimized for each θ, Eb/N0 and Eb/NJ or S/J. The results are shown in Figures 41 and 42 when θ=100. It is assumed, that the jammer has knowledge of Eb/N0 and Eb/NJ in the input of the receiver. In noise jamming the optimum pulse duty factor is about the same as that was for the quantized limiter and the erasure receivers. The performance in CW jamming is strongly dependent on the choice of the pulse duty factor. In Figure 12 in Chapter 3 it was shown relation between bit error rate and D while the convolutional code K=7 and R=1/2 is used. In this case it results that the interesting D values are in the range 0.1 to 0.6 responding bit error rates from 10-9 to 0,5. From Figures 40 – 43 it can be concluded that the jammer should choose the pulse duty factor in the region 0.01 to 1 to optimize the jammer effect in noise jamming and in the region 0.1 to 1 in CW jamming. If the jammer cannot optimize the pulse duty factor, a choice of the value 0.1 could be a good compromise in noise jamming and 1 in CW jamming. The chip combiner receiver can detect and remove the effect of pulsed CW jamming very well and thus the jammer should use continuous time jamming.
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a)
b)
Fig. 40. The chip combiner receiver, D as a function of the pulse duty factor ρ and signal-to-jamming ratio Eb/NJ (or S/J in case of CW jamming) when, Eb/N0 = 10 dB θ =100 and L=8 a) noise jamming b) CW jamming cb=1/63.
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a)
b)
Fig. 41. The worst case D as a function of signal to jamming and signal-to-noise ratio, θ =100 a) noise jamming b) CW jamming cb=1/63.
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a)
b)
Fig. 42. The worst case ρ as a function of signal-to-jamming and signalto-noise ratio a) noise jamming b) CW jamming θ = 100, cb=1/63.
100
Fig. 43. The worst case ρ as a function of signal-to-jamming and signalto-noise ratio, CW jamming, optimum θ and cb=1/63.
After optimization of the pulse duty factor ρ, the channel parameter D is minimized numerically using computer program by selecting the optimal erasure level θ. The optimal erasure level is selected by computer program calculating the channel parameter D for a wide range of θ values. The channel parameter D and the optimal erasure level θ are shown as a function of Eb/N0 and Eb/NJ or S/J in Figures 44 and 45. For noise jamming θ values from 1 to 20 and for CW jamming from 1 to 200 were calculated. In noise jamming θ should be selected higher than 3. It should, however, be noticed that the chip combiner receiver makes use of pulsed jamming almost useless as can be seen by comparing Figures 29, 42 and 43. A jammer should use continuous time jamming against a chip combiner receiver. On the other hand, when Eb/N0 and Eb/NJ are high, the bit error rate is anyway good enough for satisfactory performance if θ > 3. In CW jamming θ should be selected higher than 100. Although optimum θ varies a lot, θ should be optimized for high power jamming cases. When jamming power is low, processing gain can take care of jamming. One must take into account that the code unbalance in the calculations was 1/63, and the optimum value of θ depends on the code unbalance. High optimum value means that jamming is so high that it can be detected very clearly. Simulations made in Paper VII show that the chip combiner receiver can eliminate the effect of pulsed CW and noise jamming. The simulations with pulsed noise and CW jamming proposes to select θ equal to 3 for pulsed CW jamming and 1.6 for pulsed noise jamming. From the analysis it is seen that this is not an optimal selection. This shows the power of analysis and the weakness of simulating only a limited set of parameters. Simulations may lead to a wrong conclusion. If a robust value for θ should be selected, θ equal to 100 would be a good compromise.
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a)
b)
Fig. 44. Optimum D as a function of Eb/N0 and Eb/NJ, pulsed jamming a) noise jamming b) CW jamming cb=1/63.
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a)
b)
Fig. 45. Optimum θ as a function of Eb/N0 and Eb/NJ a) noise jamming b) CW jamming cb=1/63.
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a)
b)
Fig. 46. D as a function of erasure level θ and Eb/NJ or S/J a) noise jamming b) CW jamming cb=1/63.
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In Figures 48-49 PMD is shown for the noise jamming case and in Figures 50-52 for CW jamming. PMD is higher than reference [12] proposes for optimal θ. The reason is, at least partly, that the detection of jamming is not very reliable. When the phase difference is close to 90 degrees, detection of jamming is not possible to do by this detector. On the other hand, the demodulator will attenuate the effect of the jammer. In fact, it would cause more harm to erase these symbols. So PMD can be between 0.1 and 1 and the receiver can still work well. The receiver also attenuates the effect of jamming in despreading and rare random errors can be corrected by a channel code. When θ is decreased PMD is also decreased, as can bee seen from Figures 48 and 52. But decreasing θ to have smaller PMD will not result in the optimum selection of θ. Optimization of the channel parameter D gives more reliably correct value for parameter θ than optimization using miss and false alarm probabilities. PFA is between 0 and 10-3 in the case of noise jamming (Figure 49) and zero in the case of CW jamming, so there are just a few false alarms. Optimization of D by the use of PMD and PFA would allow PFA to be in the order of 10-2. However, when PMD is as high as seen in the Figures 47-48 and 50-51, PFA is of course low.
Fig. 47. PMD for noise jamming as a function of Eb/NJ and Eb/N0.
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Fig. 48. PMD for noise jamming as a function of Eb/NJ and θ.
Fig. 49. PFA for noise jamming as a function of Eb/NJ and Eb/N0.
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Fig. 50. PMD for CW jamming as a function of S/J and Eb/N0.
Fig. 51. PMD for CW jamming as a function of S/J and θ.
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Fig. 52. PMD for CW jamming as a function of S/J and θ.
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Fig. 53. The worst case bit error upper bounds in CW and noise jamming as a function of S/J, optimal erasure level θ , cb = 1/63.
BER are compared for CW and noise jamming in Figure 53. Noise jamming is about 15 dB worse than CW jamming for the BER of 10-6. The code unbalance of the m-sequence is 1/63, which is equivalent to a processing gain of 36 dB in CW jamming while a processing gain of noise jamming is only 18 dB.
9 Comparison of the Receivers
The receivers studied in Chapters 4-8 are compared in this Chapter. The bit error rate (BER) upper bound is calculated for different cases while the convolutional code of length 7 and rate ½ is used (see Chapter 3). The channel parameter D calculated using results obtained in Chapters 4-8 and the bit error upper bound is calculated using (3.30). The receivers are compared in pulsed noise and CW jamming.
9.1 Pulsed noise jamming
In Figure 54 BER of the receivers are compared in the worst case noise jamming. All the quantized soft decision receivers (chip combiner, signal level based erasure and quantized soft decision) perform almost equally. Coding gain of the quantized receivers compared to the hard decision receiver is 3 dB when Eb/N0 = 15 dB and the bit error rate is 10-5. At the same point, the optimum soft decision receiver with perfect side information is still 3 dB better than the quantized receivers. When Eb/N0 = 7 dB and the bit error rate is 10-4 the coding gain is 5 dB between the hard decision and the quantized soft decision receivers. One cannot gain lot by erasing the jammed channel symbols. The unquantized soft decision receiver gives poor BER if jammer can optimize its pulse duty factor. The optimum pulse duty factors, corresponding the BER curves shown in Figure 54, are shown in Figure 55. The pulse duty factor is one for the optimum soft decision receiver when Eb/N0 = 7 dB. When Eb/N0 = 15 dB there is not big difference in the optimum pulse duty factor between the receivers. The unquantized soft decision receiver has the lower optimum pulse duty factor than the other receivers. The optimum pulse duty factor is higher for the hard decision receiver than for the quantized soft decision, the chip combiner and the signal level based receivers.
The coding gain diminishes and pulse duty factor increases while Eb/NJ decreases. When Eb/NJ is less than 6 dB, coding gain is less than 1 dB between the quantized receivers and the optimum soft decision receiver. On that area ρ is also close to 1, so that jamming is almost continuous time jamming.
110
a)
b)
Fig. 54. The worst case bit-error rate upper bounds in noise jamming with optimal erasure level θ, when a) Eb/N0 = 7 dB and b) Eb/N0 = 15 dB (CCR = chip combiner receiver, erasure = signal level based erasure receiver).
111
a)
b)
Fig. 55. The worst pulse duty factor in noise jamming with optimal erasure level θ, when a) Eb/N0 = 7 dB and b) Eb/N0 = 15 dB (CCR = chip combiner receiver, erasure = signal level based erasure receiver).
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If jammer cannot optimize the pulse duty factor, the effect of jamming may be significantly reduced. The BER of the studied receivers with pulse duty factors 1 and 0.1 are shown in figures 56 – 62. When jammer power is high, it is critical to have right duty factor. One way to choose ρ is to determine first the BER jammer wants to cause to receiver. For example, if a jammer wants to have the BER higher than 10-4 for the hard decision receiver the jammer must use ρ = 0.1 and have Eb/NJ larger than 9 dB.
Fig. 56. The bit-error rate upper bounds of the hard decision receiver in noise jamming, when Eb/N0 = 7 dB, Eb/N0 = 15 dB, and ρ = 0.1, 1 or optimal.
In Figure 57 the BER of the optimal soft decision receiver is same when ρ=1 than with optimal ρ when Eb/N0 = 7 dB. When Eb/N0 = 15 dB the BER is more sensitive if ρ changes from the optimal value. When ρ=0.1 the optimal soft decision receiver has much better performance than the other studied receivers. The optimal soft decision receiver has 3-6 dB smaller BER than the quantized or hard decision receiver. This shows clearly, how much it is possible to gain with erasuring the jammed symbols.
The BER of the unquantized soft decision receiver without jamming state information is very high when the jammer is capable of optimizing the pulse duty factor as can be seen from Figure 59. But if the jammer uses continuous time jamming (ρ=1, which is in fact the AWGN channel), unquantized receiver has the lower BER than hard decision or the other quantized receivers. This result is obvious because unquantized soft decision receiver is the optimum receiver in the AWGN channel. But when the pulse duty factor becomes smaller, the BER of the unquantized soft decision receiver comes soon poorer than the quantized receivers.
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Fig. 57. The bit-error rate upper bounds of the optimum soft decision receiver (known jammer state) in noise jamming, when Eb/N0 = 7 dB, Eb/N0 = 15 dB, and ρ = 0.1, 1 or optimal.
Fig. 58. The bit-error rate upper bounds of the unquantized soft decision receiver (unknown jammer state) in noise jamming, when Eb/N0 = 7 dB, Eb/N0 = 15 dB, and ρ = 0.1, 1 or optimal.
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The quantized soft decision receiver has almost three decade better BER than hard decision receiver when Eb/N0 = 7 dB, ρ=0.1, Eb/NJ = 7 dB and the coding gain of 3 dB when Eb/N0 = 15 dB and the BER is 10-8. When the AWGN channel is used (ρ=1), the quantized soft decision receiver has 2 dB coding gain compared to hard decision receiver when Eb/N0 = 15 dB and the BER is 10-4 - 10-10. Thus, the quantization of the signal to more than two levels (16 levels in these examples) gives more gain in the pulsed noise jamming channel than in the AWGN channel.
Fig. 59. The bit-error rate upper bounds of the quantized soft decision receiver (L=8) in noise jamming, when Eb/N0 = 7 dB, Eb/N0 = 15 dB, and ρ = 0.1, 1 or optimal.
The signal level based erasure receiver has the same BER performance as the quantized soft decision receiver when worst case jamming is used, as was shown in Figure 55. But when ρ = 0.1 the soft decision receiver has much smaller BER (compare Figures 59 and 60). The situation is the same when the chip combiner (Figure 61) and the quantized soft decision receiver (Figure 59) are compared. These erasure receivers do not give significant BER improvement in pulsed noise jamming. If jamming is continuous time (ρ = 1, AWGN channel), the quantized soft decision receiver has the lower BER than these erasure receivers. The chip combiner and the signal level based erasure receivers have approximately same BER at all the pulse duty factors in Figures 60 and 61.
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Fig. 60. The bit-error rate upper bounds of the signal level based erasure receiver (L=8) in noise jamming with optimal erasure level θ, Eb/N0 = 7 dB or 15 dB, and ρ = 0.1, 1 or optimal.
Fig. 61. The bit-error rate upper bounds of the chip combiner receiver in noise jamming (L=8) with optimal erasure level θ, Eb/N0 = 7 dB or 15 dB, and ρ = 0.1, 1 or optimal.
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The erasure levels of the signal level based erasure and chip combiner receivers in noise jamming were also optimized as was discussed in the Chapters 7 and 8. The optimum erasure level parameters θ corresponding the BER curves of the Figure 55 are shown in Figure 62. The optimum changes when Eb/NJ, S/J or Eb/N0 are changed, so that the receiver should optimize θ for optimum performance. Usually this is not practical and so one fixed value should be selected. In Chapters 7 and 8 it was shown, that signal level based erasure receiver is not so sensitive for the selection of the erasure threshold θ than was the chip combiner receiver. In the BER calculations of the Figures 60 - 61 the signal level based erasure receiver was selected to have θ=5 and the chip combiner receiver θ=2 or optimal θ and ρ was optimal.
Fig. 62. The optimal erasure level θ in noise jamming, when Eb/N0 = 7 dB and Eb/N0 = 15 dB (CCR = chip combiner receiver, erasure = signal level based erasure receiver).
9.2 Pulsed CW jamming
In Figure 63 the receivers are compared in the worst case CW jamming. The jammer pulse duty factor and erasure levels of the signal level based erasure and chip combiner
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receivers are optimized and they are shown in Figures 64 and 65. The unquantized soft decision receiver has poor BER performance. The chip combiner receiver gives the best performance in CW jamming if an optimum θ is used and the measurement of the jammer state is more reliable than that of the signal level based erasure receiver. The quantized soft decision receiver has the same BER than the signal level based erasure receiver. Thus it is not worth using the more complex signal level based erasure receiver in stead of the simple quantized receiver. The same result was achieved by simulations in original Papers VI and VII. In noise jamming channel state measurement made by the chip combiner gave no advantage, the quantized limiter receiver performs equally well. The optimum pulse duty factors of the studied receivers are different, as can be seen from Figure 64. The pulse duty factors are on same order of magnitude than in the case of pulsed noise jamming. The jammer has the difficulty that it does not know the receiver structure and which decision metric is in use. If the jammer chooses a wrong pulse duty factor, it can loose a lot of its effect as can be seen from Figures 66-71. All the studied quantized soft decision receivers perform better than the hard decision receiver and neither of them is close to the optimal receiver (the soft decision receiver with perfect side information). The difference between the quantized soft decision receivers and the optimum soft decision receiver is much larger in CW jamming than in noise jamming. The coding gain of the quantized soft decision receivers compared to the hard decision receiver is 3 dB when Eb/N0 = 15 dB when the bit error rate is 10-5 and 4.5 dB between quantized and chip combiner receivers. The optimum receiver has still 3 dB better BER than the chip combiner receiver when BER is 10-5. There still exist a possibility to discover other receivers, which could give better performance than these. The coding gain between optimum and the other receivers increases while S/J is less than –25 dB (Eb/N0 =7 dB) or –30 dB (Eb/N0 =15 dB).
In pulsed CW jamming the optimal erasure levels of the signal level based erasure receiver are close to same values as in the case of noise jamming as can be seen comparing Figures 62 and 65. The chip combiner receiver should have much higher erasure level θ than was those in pulsed noise case. In the case of the chip combiner θ is selected to be 100 in the BER calculations of the Figures 66-71. The hard decision receiver is rather sensitive if ρ changes as can be seen from Figure 66. The situation is very much like that it was in the case of noise jamming in Figure 57.
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a)
b)
Fig. 63. The worst case bit-error rate upper bounds in CW jamming with optimal erasure level θ and cb=1/63, when a) Eb/N0 = 7 dB and b) Eb/N0 = 15 dB (CCR = chip combiner receiver, erasure = signal level based erasure receiver).
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a)
b)
Fig. 64. The worst pulse duty factor in CW jamming with optimal erasure level θ and cb=1/63, when a) Eb/N0 = 7 dB and b) Eb/N0 = 15 dB (CCR = chip combiner receiver, erasure = signal level based erasure receiver).
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a)
b)
Fig. 65. The optimal erasure level θ in CW jamming, Eb/N0 = 7 dB and Eb/N0 = 15 dB a) chip combiner (CCR) and signal level based erasure receiver b) signal level based erasure receiver.
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Fig. 66. The bit-error rate upper bounds of the hard decision receiver in CW jamming, when cb=1/63, Eb/N0 = 7 dB, Eb/N0 = 15 dB, and ρ = 0.1, 1 or optimal.
The BER of the optimal soft decision receiver is equal when ρ=1 and ρ is optimal while Eb/N0=7dB as was the case in noise jamming. When ρ=0.1 the optimal soft decision receiver has much better performance than the other studied receivers. The BER is 13-18 decades less at the optimal soft decision than the quantized soft or hard decision receivers in CW jamming while it was 3-6 decade in noise jamming. This shows that optimal soft decision receiver gives even more gain in CW jamming than was the case in noise jamming. The chip combiner receiver in Figure 71 is seen to have the best performance after optimal receiver while ρ=0.1. This shows that it can detect jamming cases rather reliably and thus erase the jammed symbols.
The BER of the unquantized soft decision receiver without jamming state information is very high when the jammer is capable of optimizing the pulse duty factor as can be seen from Figure 68. In most cases the hard decision receiver gives better BER than unquantized soft decision receiver with ρ=0.1 and ρ=1.
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Fig. 67. The bit-error rate upper bounds of the optimum soft decision receiver (known jammer state) in CW jamming, cb=1/63, Eb/N0 = 7 dB or 15 dB, and ρ = 0.1, 1 or optimal.
Fig. 68. The bit-error rate upper bounds of the unquantized soft decision receiver (unknown jammer state) in CW jamming, when Eb/N0 = 7 dB, Eb/N0 = 15 dB, and ρ = 0.1, 1 or optimal.
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The quantized soft decision receiver has about three decade better BER than hard decision receiver when ρ=0.1, and S/J = -35 dB as can be seen comparing Figures 66 and 69. Also when ρ=1 the quantized soft decision receiver is better than the hard decision receiver. It is thus possible to achieve large gain from the quantization of the signal more than 2 levels (16 levels in this example).
Fig. 69. The bit-error rate upper bounds of the quantized soft decision receiver (L=8) in CW jamming, when cb=1/63, Eb/N0 = 7 dB, Eb/N0 = 15 dB, and ρ = 0.1, 1 or optimal.
The signal level based erasure receiver in Figure 70 has about 2 decade higher BER at S/J=-35 dB than the quantized soft decision receiver in Figure 69 while ρ = 0.1. This shows that the erasure receiver cannot detect jammed bits reliable to give coding gain; instead it causes more harm to decoding. The chip combiner receiver in Figure 71 has on the same point 2 - 3 decades smaller BER than the quantized soft decision receiver. This shows that the chip combiner receiver can detect and erase jammed symbols.
124
Fig. 70. The bit-error rate upper bounds of the signal level based erasure receiver (L=8) in CW jamming, optimal erasure level θ, cb=1/63, Eb/N0=7dB or 15dB, and ρ = 0.1, 1 or optimal.
Fig. 71. The bit-error rate upper bounds of the chip combiner receiver in CW jamming (L=8) with optimal erasure level θ, cb=1/63, Eb/N0 = 7dB or 15dB, and ρ = 0.1, 1 or optimal.
10 Discussion and Conclusions
10.1 Discussion
The equations were presented and used to analyze different metrics in pulsed noise and CW jamming. The receivers were easily compared using the channel parameter D and the bit error rate upper bounds. The worst case pulse duty factor can also be calculated as well as the worst case bit error upper bound. The worst performance of the communication system was determined and for the intentional jammer the optimum pulse duty factor was calculated. The theory presented can be used more generally to analyze spread spectrum systems’ performance. One can use the presented theory for any arbitrary jamming waveform. The same type of analysis can be used for modulation methods other than DS/BPSK as well.
The result of the thesis shows that the channel state measuring receivers performed better than those without channel state measurement. These receivers are more complex, but in a jamming channel it may be worth using them anyway. For moderate bit rates these channel state measuring receivers can be implemented digitally and for a low bit rate using software.
All the studied receivers performed better than the hard decision receiver and neither of them was close to the optimal receiver (soft decision receiver with perfect side information). The difference between channel state measuring and the quantized limiter or the hard decision receivers in the worst case is not large. The jammer cannot usually optimize its pulse duty factor and it must occasionally listen to the radio channel. This causes jamming to be pulsed. In pulsed jamming the chip combiner and quantized soft decision limiter erasure receivers performed better than quantized limiter or hard decision receiver.
In system design, receiver simulations should be done based on the analysis shown. The presented analysis helps to select optimal parameters for receivers (threshold θ ) and the worst case jamming. When coding is used, one should cover the pulse duty factor area from 0.1 to 1. In the case of intentional jamming, the channel coding scheme, the interleaving technique, and the interleaving depth (= separation of the consecutive bits
126
after interleaving) should be kept as classified information because the jammer can utilize these in the design of its jamming strategy.
10.2 Conclusions
Suboptimal soft decision DSSS receivers with error correcting coding in pulsed noise and CW jamming were considered. The literature on decision metrics in pulsed jamming was reviewed in Chapter 2. The system model was presented in Chapter 3.
The hard decision receiver was analyzed in Chapter 4. It was shown that the hard decision receiver works in pulsed noise channel and the jammer should have a pulse duty factor from 0.01 to 1. If the jammer does not want or cannot determine the received signal and the jamming powers in the receiver input, it can use a pulse duty factor values between 0.1 - 1 with good results. If the pulse duty factor is less than 0.1, most error correcting codes can correct the errors caused by jamming. This analysis showed that the hard decision receiver is robust in pulsed jamming. Noise jamming was more than 10 dB worse than CW jamming. The main reason is that the effect of CW jamming is decreased by a well selected spreading code balance of a m-sequence. As a result of this code selection, PG in CW jamming is doubled (36 dB in this case).
Unquantized soft decision receivers were considered in Chapter 5. It was shown that the soft decision receiver with perfect side information works very well in a pulsed noise channel and it is not useful for the jammer to have a smaller pulse duty factor than 0.01. If the jammer does not want or cannot determine the received signal and the jamming power in the receiver input, it can use a pulse duty factor equal to one with good results. The reason is that optimum receiver can detect jammed symbols and effectively erase those. The soft decision receiver with perfect side information is the optimum receiver in pulsed jamming and it formed a baseline reference to the other systems. Noise jamming was about 15 dB worse than CW jamming and the reason is that the effect of CW jamming is decreased by a well selected spreading code balance of a m-sequence. The soft decision receiver without side information performs poorly. The jammer can use short, powerful pulses, which cut communication links easily.
The quantized soft decision limiter receiver was considered in Chapter 6. The decision metric was introduced and the equation for channel parameter D was presented. The optimum pulse duty factor of the quantized soft decision limiter receiver was shown to be smaller than that of the hard decision receiver. The jammer should choose a pulse duty factor from the region 0.01 to 1 to optimize the jammer effect.
The signal level based erasure receiver was considered in Chapter 7. The decision metric for channel parameter D was presented. The optimum pulse duty factor of the signal level based erasure receiver was shown to be about the same as that of the hard decision receiver in noise jamming, but much higher in CW jamming. The effect of CW jamming was shown to be strongly dependent on the choice of the pulse duty factor. The jammer should choose the pulse duty factor from the region 0.01 to 1 to optimize the jamming effect in CW jamming and from 0.1 to 1 in noise jamming. The best choice of the erasure level θ was shown to be 4. If θ is less than 4, the channel parameter D is
127
increased on all the cases. If θ is larger than 4, performance gets better only slightly in some cases, but on the other hand, it gets worse in other cases.
The chip combiner receiver was considered in Chapter 8. The decision metric was introduced and the equation for channel parameter D was presented. The performance in CW jamming was shown to be strongly dependent on the choice of the pulse duty factor. The jammer should choose a pulse duty factor from the region 0.01 to 1 to optimize the jammer effect in noise jamming and 1 in CW jamming. If the jammer cannot optimize the pulse duty factor, a choice of 0.1 could be a good compromise in noise jamming and 1 in CW jamming. This is due to the fact that the chip combiner receiver cannot detect noise jamming reliably, but it can detect CW jamming more reliably and thus erase these symbols. The best choice of the erasure level θ was shown to be 100. This selection works well to both jamming cases. If θ is lower or higher, the performance decreases in CW jamming.
The bit error rates of the receivers were compared in Chapter 9. All the quantized soft decision receivers performed almost equally in noise jamming. Coding gain compared to the hard decision receiver was 3 dB when Eb/N0 = 15 dB and the bit error rate is 10-5 in noise jamming. At the same BER the soft decision receiver with perfect side information is still 3 dB better than these quantized receivers. When Eb/N0 = 7 dB and the bit error rate is 10-4 the coding gain is 5 dB between the hard decision and the quantized soft decision receivers.
In the worst case CW jamming, the chip combiner receiver performed best. The coding gain of the quantized soft decision receiver compared to the hard decision receiver is 3 dB when Eb/N0 = 15 dB and the bit error rate is 10-5 and 4.5 dB between quantized and chip combiner receivers. The optimum receiver has still 3 dB better BER than the chip combiner receiver. When Eb/N0 = 7 dB and the bit error rate is 10-4 the coding gain of the quantized soft decision receivers compared to the hard decision receiver is 5 dB, the chip combiner receiver is 2 dB better than the quantized soft decision receiver. The optimum receiver is 3 dB better than chip combiner receiver.
It can be concluded that the chip combiner circuit has benefit in CW jamming. In noise jamming the chip combiner do not have significant improvement compared to the other quantized soft decision receivers. The hard decision receiver is much more vulnerable to thermal noise effects than the quantized soft decision receivers, which can be seen comparing the coding gains of the cases Eb/N0 = 7 dB and Eb/N0 = 15 dB. Under jamming signal influence the coding gain is 3 dB in the case of Eb/N0 = 15 dB and 5 dB in the case of Eb/N0 = 7 dB. All the quantized receivers work better than the hard decision receiver.
The analysis showed that noise jamming is more effective against the DS/BPSK system than CW jamming on the same center frequency. There is variation in difference (10 to 15 dB), but one must remember that these results are upper bounds and in some cases they may be inaccurate. In Chapter 2 and in [46] it was argued that the Chernoff upper bound is inaccurate when the pulse duty factor is small. These results show that in coded systems only high pulse duty factors (0.01 – 1) are important. This is due to the fact that coding can correct the errors if they appear very seldom (e.g. ρ< 0.01). A small (order of (f0-fJ/Ts=1) change in the center frequency of CW jamming may increase the BER because the effect of phase difference is removed as was shown in Section 3.2.1. If the frequency error becomes large, the BER will decrease, because the power of the jamming signal moves out of the band of the received signal and is thus filtered out in the
128
receiver. The noise assumption used in many textbooks is good if spreading sequence assumptions mentioned in Section 3.2.1 applies, but otherwise it gives misleading results.
The jammer should use a very high pulse duty factor. If the jammer cannot approximate the S/J ratio in the receiver input and it do not know what kind of receiver is used, the jammer should use a pulse duty factor on the range 0.1 – 1 to achieve on average reasonable jamming effect. Usually the jammer must anyway occasionally receive signals to be sure jamming is on the right target. Under a receiving session the jamming must stop, causing interruption to the jamming transmission. The jammer should jam long enough to cause the signal reception to be cut off before switching to the receiving mode. In practice the jammer cannot use the same effective power in short pulses because of implementation problems arising from the transmitter design. This fact also favors the use of continuous time jamming instead of pulsed jamming.
The hard decision receiver is the easiest to implement and it has better performance than unquantized soft decision receiver in pulsed jamming. By using more quantization steps (quantized soft decision receiver) one can gain several decibels, but the implementation is more complex: the decisions are quantized by A/D-converter to several regions and receiver must be capable to handle bytes instead of single bits decisions in channel decoder and in deinterleaver. The signal level based erasure receiver is only slightly more complex than the quantized soft decision receiver: there are two more decision thresholds and a little logic to erasure jammed bits. On the other hand, it has practically seen same BER performance in worst case jamming. If jamming was not worst case, for example ρ=0.1, the signal level based erasure receiver had much higher BER than quantized soft decision receiver. Thus, in many cases it may be useless to implement signal level based erasure receiver instead of simple quantized soft decision receiver. The chip combiner receiver requires roughly twice the signal processing power (analog or digital) of the quantized soft decision receiver, because there is another receiver (chip combiner) operating on chip rate. The chip combiner receiver has much smaller BER in CW jamming compared to the quantized soft decision receiver, so that it may be worth using in the pulsed jamming channel.
As a final conclusions to a designer of the communication system could be as follows: 1. If a simple implementation is required, the hard decision receiver should be used. It
is better than unquantized soft decision receiver without jammer state information. But it is still several decibels (approximately 3-5 dB in many cases) worse than the quantized soft decision receiver.
2. If moderate complex implementation is allowed, the quantized soft decision receiver should be used. The signal level based erasure receiver does not give any remarkable improvement, so that it is not worth using, because it is more complex to implement. In CW jamming the quantized soft decision receiver is still couple of decibels (approximately 2-5 dB in many cases) worse than the chip combiner receiver.
3. If receiver complexity is not limiting factor, the chip combiner receiver should be used, because it gives the best performance in jamming channel. The chip combiner receiver is 2-5 dB better than quantized soft decision receiver and 3-8 dB better than the hard decision receiver.
129
10.3 Future research directions
The performance of channel state measuring receivers was not close that of the optimum receiver. Therefore there may exist other receivers, which give better performance in pulsed noise and CW jamming than those considered here.
The thesis has concentrated on BPSK modulated systems. In reference [5] DPSK is considered. Same type of the analysis can also be done for the other modulations. The other jamming signals could also be analyzed by calculating their effect in the receiver output.
In the thesis a convolutional code of length 7 and rate ½ was used. One topic for further research would be to optimize channel coding in the pulsed jamming case. It would be interesting to study, for example, the performance of turbo codes. They have not been considered in jamming. Fading and jamming often cause the same type of SNR variation and both require the use of interleaving if random error correcting codes are used. So it could be feasible to use turbo coding in a jamming channel.
The results presented in the thesis are mainly bit error upper bounds. One should simulate the studied receivers with the most interesting parameter values before proceeding to receiver design. For example, optimization of the erasure levels of the signal based erasure and chip combiner receiver should be simulated on the areas of the worst case pulse duty factors and the optimum θ values. There are also many other non-idealities (e.g. time, frequency, and phase synchronization errors, fading etc.), which should be studied. Simulation of all the non-idealities at the same time may be too demanding, but using proper parameter selection most of the phenomenon can be simulated or analyzed. The final test must be done using hardware.
The effects of the AGC or synchronization circuits were not analyzed. Fading and other jamming signals (e.g. chirp, modulated etc.) were neither analyzed. The analysis of these phenomena could be issues for further studies.
Under this research work some further research items were uncovered. It would be interesting to analyze a quantized Dempster-Shafer receiver presented in reference [37] and DS variations of those methods used in frequency hopping receivers in reference [35].
Performance can be even better when interference cancellation is used. However, this also makes a receiver more complex. These are anyway worth considering, because digital signal processors speed is increasing and prices are coming down.
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Appendix 1 Effect of CW-jamming
The effect of CW-jamming from (3.19) can be calculated as follows:
( )
( )( )∫
∫
∫
+−=
+=
=
s
s
s
T
JJJs
T
JJs
T
s
dttttcAT
dttAttcT
dttT
tJtcj
000
000
00
sin)sin(cos)(2
cos)cos()(2
)cos(2)()(
φωωφ
φωω
ω
(A.1.1)
The integrator in the receiver in Figure 2 behaves as a low pass filter and thus, the last term inside integral is filtered out, so the j is
( )
( )
JbsJbJs
sbJJs
N
kckJJ
s
N
k
Tk
kTkJJ
s
N
k
Tk
kTJJ
s
T
JJs
cJTcAT
TcAT
TcAT
dtcAT
dttcAT
dttcAT
j
c
c
c
c
s
φφ
φφ
φ
φ
φ
cos2cos2
cos2cos2
cos2
)(cos2
)(cos2
1
0
1
0
1
1
0
10
==
==
=
=
=
∑
∑ ∫
∑ ∫
∫
−
=
−
=
+
−
=
+
(A.1.2)
where cb is code unbalance defined in (3.27). The effect of the CW-jamming term
0/ NNj J + in (3.29) is
( )( )
( ) ( ).
//1//1cos//12
/cos/2
/
/cos2
00
00
Jss
Jb
sJ
Jb
sssJ
sJbs
J
NENEcJS
ENNcSJ
STEENN
EcJT
NNj
+=
+=
=+
=+
φφ
φ
(A.1.3)
Appendix 2 Expectation in derivation of D for unquantized soft decision receiver
Noise variables of (5.2) can be combined to be ntot = nj+n which is also an AWGN random variable. The variance of the random variable ntot is σ2 and so
JNN += 022σ . (A.2.1)
In BPSK signaling it must be
22)ˆ(1 m=−=−⇒±= dxxd (A.2.2)
The expectation in (5.2) is ([4] p. 17 and 23)
[ ]{ } [ ]{ }
[ ] [ ]
[ ]
( )[ ]
( )
[ ] ( )tot
tot
tottot
tottottot
tottottot
tottottot
tottot
dndn
d
dnddn
dnddndn
dnnnd
dnnnd
ndEnxxE
∫
∫
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
±−=
+±−=
−+−=
−=
−=
=−
2
22222
2222
22222222
22
22
22
exp212exp
222/exp21
222/222/exp21
2/2exp21
2/exp212exp
2exp)ˆ(exp
σσλ
σπσλ
σλσλσσπ
σλσλσσλσσπ
σλσπ
σσπ
λ
λλ
m
m
m
m
(A.2.3)
The last integral is equal to one because it is the cdf of a gaussian distributed random variable over all values of the variable. So expectation is
[ ]{ } [ ]222 2exp)ˆ(exp σλλ dnxxE tot =− . (A.2.4)
Appendix 3 Calculation of D for the unquantized soft decision receiver
To minimize D(λ) in (5.3) the derivative will be evaluated and zero point must be searched. Since
( ) ( ) ( )[ ]
( )( ) ( )[ ]
( )( )( )
.2
ˆand
02ˆ)()('then
ˆexp)(
02
02
022
J
S
JS
JS
NNd
jEdxx
NNdjEdxxDD
NNdjEdxxD
+
+−−=
=+++−=
+++−=
λ
λλλ
λλρλ
(A.3.1)
It is easily seen that
∞→∞=
===
λλ
λ)(limand
1)0exp()0(D
D (A.3.2)
Thus λ in (A.3.1) is minimum when it is positive; otherwise minimum is λ equal to zero. Inequality λ ≥ 0 must be solved. For that purpose d must be given a numerical value from equation (3.2). The first case will be d = +1
( )( )
.
followsit which from
always0;0 00
S
JJ
S
Ej
NNNN
jE
−≥
>+≥+
+=λ
(A.3.3)
The second case is d = -1
( )( )
.
followsit which from
allways statisfies0;0 00
S
JJ
S
Ej
NNNN
jE
≤
>+≥+
−=λ
(A.3.4)
136
Finally D(λ) in the zero of the derivative is
( ) ( )( )( )
( )
( )( )( )
( )
( )
+
+−=
+
+
+−−+
++
+−−−
=
J
S
JJ
S
SJ
S
NNjEd
NNdNNd
jEdxx
jEdNNd
jEdxxxx
D
0
2
02
2
02
02
exp
2ˆ
2ˆ
ˆ
exp
ρ
ρ
(A.3.5)
This can be simplified to
( )
−=+>+=−<
−=+≤+=−≥
+
+−
=
1,,1,,
1,,1,,
exp0
2
dEjdEj
dEjdEj
NNjEd
D
S
S
S
S
J
S
ρ
ρ (A.3.6)
To calculate D, the exponent in (A.3.5) can further be presented as
( ).
2
000
2
++
+−=
+
+−
JJ
S
J
S
NNj
NNE
dNN
jEd (A.3.7)
The first term is
( ) ( )JSSJ
S
NENEd
NNE
d//1//1
1
00 +=
+ (A.3.8)
and the second term (see (1.3) and (3.26))
( ) ( ) ( )[ ]JSSJb
J NENEJSc
NNj
//1//1/1cos2
00 +⋅=
+φ . (A.3.9)
Using (3.26) the inequalities are
.21cos
21cos
cos2cos21,1,
JSc
JSc
EEcEEcdEjdEj
JbJb
SJJbSJJb
SS
≤−≥
+≤−≥−=+≤+=−≥
φφ
φφ (A.3.10)
Appendix 4 Calculation of D for quantized soft decision limiter receiver
The metric used for the quantized soft decision limiter receiver is as shown in (6.1)
( )
( ) ( )[ ] . , 1,2 where
1,1
1,1212
1,1
,
ZiLLi
EL
Lyd
ELiydE
Li
Li
EL
Lyd
xym
S
SS
S
∈−+−∈
−−≤−
<≤−
−−
−>
=
(A.4.1)
The channel parameter D must be calculated using (3.13). Let us suppose that side information is not available (i.e. c(0) = c (1) = 1) and nn xx̂ ≠ . In this case
[ ]( ){ }( ) [ ]( ){ }
( ) ( )( )[ ] [ ]
( ) ( ) ( )( )[ ] [ ] ,2/exp21,ˆ,exp1
2/exp21,ˆ,exp
|);,();ˆ,(exp1
|);,();ˆ,(exp)(
22
22
ˆ
ˆ
dnnxymxym
dnnxymxym
zxymzxymE
zxymzxymED
tottotjj
xx
xxjj
σσπ
λρ
σσπ
λρ
λρ
λρλ
−−−+
−−=
−−+
−=
∫
∫∞+
∞−
∞+
∞−
≠
≠
x
x
(A.4.2)
where ntot = nj+n. The integral must be calculated in several regions, because the metric is non-continuous. The integration regions are calculated first and yj is obtained from (3.19).
The case 1:
.1
1
2 djEdL
Ldn
EL
Ldy
Stot
S
−
−
−>⇔
−>
(A.4.3)
Now d is +1 or -1, so that d2 is 1. So the first integration region is:
138
The case 1.1 d=1:
djLE
jLE
n SStot −
−=−
−>
The case 1.2 d=-1: (A.4.4)
djLE
jLE
njLE
n SStot
Stot −
−=+
−>⇔+
−>− .
Because ntot is a WGN variable with zero mean. So the result of 1.1 and 1.2 can be combined
1JS
tot LdjLE
n =−−
> .
When no jamming exist (A.4.5)
1AS L
LE
n =−
> .
The case 2:
jdEL
LidnjdEL
Li
ELidnjdEdE
Li
ELiydE
Li
StotS
StotSS
SS
−−
<≤−−−
⇔
<++≤−
⇔
<≤−
1
1
1
2
The case 2.1 d=1: (A.4.6)
.
1 jEL
LinjEL
LiStotS −
−<≤−
−−
The case 2.2 d=-1:
jEL
LinjEL
LiStotS +
−<−≤+
−−1 .
139
Because ntot is WGN variable with zero mean and the PDF is symmetric, the result of 2.1 and 2.2 can be combined and the region is
)(1)( 32 iLjdEL
LinjdEL
LiiL JStotSJ =−−
<≤−−−
=
where i is defined in (A.4.1). When no jamming exist
)(1)( 32 iLEL
LinEL
LiiL ASSA =−
<≤−−
= . (A.4.7)
The case 3:
( )
.12
1
1
djELLdn
EL
LnjEdd
EL
Ldy
Stot
StotS
S
−
+−
≤⇔
−−≤++⇔
−−≤
The case 3.1 d=1: (A.4.8)
.
12 jELLn Stot −
+−
≤ ,
The case 3.2 d=-1:
jELLn Stot +
+−
≤12 .
Because ntot is WGN variable with zero mean and the PDF is symmetric, the result of 3.1 and 3.2 can be combined and the region is
412
JStot LdjELLn =−
+−
≤ .
When no jamming exist (A.4.9)
412
AStot LELLn =
+−
≤ .
140
Using results shown in cases 1-3 in AWGN channel (A.4.1) becomes
( )
=−
+−
≤−
=−−
<≤−−−
=−−
=−−
>
=
4
32
1
12,1
)(1)(,1212
,1
,
AStot
AStotSA
AS
tot
LdjELLn
iLjdEL
LinjdEL
LiiLLi
LdjLE
n
xym
(A.4.10)
( ) ( )[ ] . , 1,2 where ZiLLi ∈−+−∈
When jamming channel is used
( )
=−
+−
≤−
=−−
<≤−−−
=−−
=−−
>
=
4
32
1
12,1
)(1)(,1212
,1
,
JStot
JStotSJ
JS
tot
LdjELLn
iLjdEL
LinjdEL
LiiLLi
LdjLE
n
xym
(A.4.11)
( ) ( )[ ] . , 1,2 where ZiLLi ∈−+−∈ Equation (A.4.2) becomes
141
( )( ) [ ]
( ) ( )( ) [ ]
( )( ) [ ]
( ) ( )( ) [ ]
[ ]
( ) [ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )( )
( ) ( )( )( ) ( ) ( )( ) .
/)(/)(1/)(/)(
12122exp
/1/2exp/1/12exp
2/exp21
1212
1212exp1
2/exp21
1212
1212exp
2/exp2111exp1
2/exp2111exp
2/exp21)1(1exp1
2/exp21)1(1exp)(
1
2 32
32
11
44
1
2
22)(
)(
1
2
22)(
)(
22
1
22
22
22
3
2
3
2
1
4
4
∑
∑ ∫
∑ ∫
∫
∫
∫
∫
−
+−=
−
+−=
−
+−=
∞
∞∞−
∞−
−−
+−
−−
−+
−+−+−−−=
−
−−
−−−
−−+
−
−−
−−−
−+
−−−−+
−−−+
−−−−+
−−−=
L
Li AA
JJ
AJ
AJ
L
Li
iL
iL
L
Litottot
iL
iL
L
tottotL
L
tottot
L
iLQiLQiLQiLQ
Li
LQLQLQLQ
dnnLi
Li
dnnLi
Li
dnn
dnn
dnn
dnnD
A
A
J
J
A
J
A
J
σσρσσρ
λ
σρσρλσρσρλ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρλ
(A.4.12)
Derivative of the D(λ) can be calculated and it is
( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )[ ]( ) ( )( )
( ) ( ) ( )( ) ./)(/)(1
/)(/)(12122exp
12122
/1/2exp2/1/12exp2)('
1
2 32
32
11
44
∑−
+−=
−−
+−
−−
−−−
−
−+−−−−−=
L
Li AA
JJ
AJ
JJ
iLQiLQiLQiLQ
Li
Li
LQLQLQLQD
σσρσσρ
λ
σρσρλσρσρλλ
(A.4.13)
142
The arguments of the Q-function for CW-jamming are (when NJ = 0) (3.26) and (A.2.1)
( )
sS
SJb
S
S
sSJb
S
sJb
SS
J
JTENE
cdL
NE
ENTEJcd
LNE
NJT
cdLNE
N
djLE
LJ
//2
cos2/2
///2cos2
/2
/2cos2
2
2/
1
00
0
0
0
0
0
1
ρφ
ρφ
ρφ
ρ
σ
−−
=
−−
=
−
−
=−
−
==
( )sS
SJb
SJ
JTENE
cdL
NEL
LJ
//2
cos2/2
124 004
ρφ
σ−+−== (A.4.14)
( )sS
SJb
SJ
JTENE
cdL
NELi
iLiJ
//2
cos2/2
1)(
)(2 002
ρφ
σ−−−==
( )sS
SJb
SJ
JTENE
cdL
NELi
iLiJ
//2
cos2/2)(
)(3 003
ρφ
σ−−== .
The arguments of the Q-function for an AWGN channel are (when NJ = 0 and J(t)=0)
LNELA SA 01 /2
1−
==σ
( )L
NELLA SA 04 /2
124 +−==σ
(A.4.15)
( )L
NELiiLiA SA 02 /2
1)()(2 −−==σ
.
( )L
NELi
iLiA SA 03 /2)()(3 −==
σ
143
The arguments of the Q-function for noise jamming are (when J(t)=0)
( ) ( ) ,//1//1
21
/212
/1
0
00
1
JSS
J
S
J
S
J
NENEL
NNE
LNNLE
LJ
ρ
ρρσ
+−
=
+−
=+
−
==
( ) ( )JSS
J
NENELLL
J//1//1
21240
4
ρσ ++−
== (A.4.16)
( ) ( )JSS
J
NENELLiiL
iJ//1//1
21)()(2
0
2
ρσ +−−
==
( ) ( )JSS
J
NENELLiiL
iJ//1//1
2)()(3
0
3
ρσ +−
== .
Derivative of the D(λ) can be presented as
( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )( )( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )( )( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( )[ ] ( ) ( ) ( )( )
( ) ( )( )( ) ( ) ( )( ) .
)(3)(21)(3)(2
124exp
12122
1112414112
4exp2
)(3)(21)(3)(2
12)12(2
12122exp
12122
111241414exp22exp)('
)(3)(21)(3)(2
12122exp
12122
1112exp241412exp2)('
1
2
12
1
2
1
2
further and
∑
∑
∑
−
+−=
−
−
−
+−=
−
+−=
−−
+−
−−−
−
−+−−−−
−=
−−
+−
−−
+−−
−−−
−
−+−−−−=⋅
−−
+−
−−
−−−
−
−+−−−−−=
L
Li
iL
L
L
Li
L
Li
iAQiAQiJQiJQ
LLi
AQJQAQJQL
iAQiAQiJQiJQ
LL
Li
Li
AQJQAQJQD
iAQiAQiJQiJQ
Li
Li
AQJQAQJQD
ρρλ
ρρρρλ
ρρλλ
ρρρρλλλ
ρρ
λ
ρρλρρλλ
(A.4.17)
When D’(λ) is set equal to zero, the zero points can be calculated for a polynomial, and the following variable change is made
−=
124expL
v λ (A.4.18)
Because λ >= 0 it follows that v >= 1. D(λ) can be presented by v as
144
( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )( )( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( )[ ] ( ) ( ) ( )( )( ) ( )( )
( ) ( ) ( )( ) .)(3)(21
)(3)(2
1114141
)(3)(21)(3)(2
12122exp
1112exp41412exp)(
1
2
212
212
212
1
2
∑
∑
−
+−=
−−
−−
−
−
+−=
−−
+−+
−++−−−=
−−
+−
−−
−+
−+−+−−−=
L
Li
i
LL
L
Li
iAQiAQiJQiJQ
v
AQJQvAQJQv
iAQiAQiJQiJQ
Li
AQJQAQJQD
ρρ
ρρρρ
ρρ
λ
ρρλρρλλ
(A.4.19)
Appendix 5 Calculation of D for a signal level based erasure receiver
The metric used for a signal level based erasure receiver is shown in (7.1)
( )
−<−
−−≤≤−−
<≤−
−−
<≤−>−
=
S
SS
SS
SS
S
Eyd
EL
LydE
ELiydE
Li
Li
EydEL
LEyd
xym
θ
θ
θ
θ
,1
1,1
1,1212
1,1
,1
, (A.5.1)
( ) ( )[ ] . , 1,2 where ZiLLi ∈−+−∈
The channel parameter D must be calculated using (3.13) as in Appendix 4. Let us suppose that side information is not available (i.e. c(0) = c (1) = 1) and nn xx̂ ≠ . In this case
[ ]( ){ }( ) [ ]( ){ }
( ) ( )( )[ ] [ ]
( ) ( ) ( )( )[ ] [ ]dnnxymxym
dnnxymxym
zxymzxymE
zxymzxymED
tottotjj
xx
xxjj
22
22
ˆ
ˆ
2/exp21,ˆ,exp1
2/exp21,ˆ,exp
|);,();ˆ,(exp1
|);,();ˆ,(exp)(
σσπ
λρ
σσπ
λρ
λρ
λρλ
−−−+
−−=
−−+
−=
∫
∫∞+
∞−
∞+
∞−
≠
≠
x
x
(A.5.2)
where ntot = nj+n. The integral must be calculated in several regions because the metric is non-continuous. The integration regions are calculated first. The received signal in jamming, yj, is obtained from equation (3.19).
146
The case 1:
( ) .1 5JStot
S
LdjEn
Eyd
=−−>⇔
>
θ
θ
When no jamming exist (A.5.3)
( )
51
AS L
LE
n =−
>θ
.
The case 2:
( ) .11
1
51 JStotSJ
SS
LdjEndjEL
L
EydEL
L
=−−<≤−−
=⇔
<≤−
θ
θ
When no jamming exist (A.5.4)
( ) 51 11ASSA LEnE
LL =−<≤
−= θ .
The case 3:
.)(1)(
1
32 iLdjEL
LindjEL
LiiL
ELiydE
Li
JStotSJ
SS
=−−
<≤−−−
=⇔
<≤−
When no jamming exist (A.5.5)
)(1)( 32 iLEL
LinEL
LiiL ASSA =−
<≤−−
= .
147
The case 4:
( ) .121
1
46 JStotSJ
SS
LdjELLndjEL
EL
LydE
=−+−
<≤−−−=⇔
−−≤≤−
θ
θ
When no jamming exist (A.5.6)
( ) 46121 AStotSA LE
LLnEL =
+−<≤−−= θ .
The case 5:
( ) .1 6JStot
S
LdjEn
Eyd
=−−−<
−<
θ
θ
When no jamming exist (A.5.7)
( ) 61 AS LEn =−−< θ .
Using the results of cases 1-5 the metric (A.5.1) in jamming channel becomes
( )
( )
( )
( )
( )
=−−−<−
=−
+−
≤≤−−−=−
=−−
<≤−−−
=−−
=−−<≤−−
=
=−−>−
=
,1,1
121,1
)(1)(,1212
1,1
1,1
,
4
46
32
51
5
JStot
JStotSJ
JStotSJ
JStotS
J
JStot
LdjEn
LdjELLndjEL
iLdjEL
LindjEL
LiiLLi
LdjEndjLE
L
LdjEn
xym
θ
θ
θ
θ
(A.5.8)
( ) ( )[ ] ZiLLi ∈−+−∈ , 1,2 where
and in AWGN channel
148
( )
( )
( )
( )
( )
=−−−<−
=−
+−
≤≤−−−=−
=−−
<≤−−−
=−−
=−−<≤−−
=
=−−>−
=
4
46
32
51
5
1,1
121,1
)(1)(,1212
1,1
1,1
,
AStot
AStotSA
AStotSA
AStotS
A
AStot
LdjEn
LdjELLndjEL
iLdjEL
LindjEL
LiiLLi
LdjEndjLE
L
LdjEn
xym
θ
θ
θ
θ
(A.5.9)
( ) ( )[ ] . , 1,2 where ZiLLi ∈−+−∈
So D(λ) is (as in Appendix 4)
( )( ) [ ]
( ) ( )( ) [ ]
( )( ) [ ]
( ) ( )( ) [ ]
[ ]
( ) [ ]
( )( ) [ ]
( ) ( )( ) [ ]
( )( ) [ ]
( ) ( )( ) [ ]dnn
dnn
dnn
dnn
dnnLi
dnnLi
dnn
dnn
dnn
dnnD
A
J
A
A
J
J
A
A
J
J
A
J
J
J
A
J
L
tottotL
L
L
tottot
L
L
L
Li
iL
iL
L
Litottot
iL
iL
L
L
tottot
L
L
L
tottot
L
22
22
22
22
1
2
22)(
)(
1
2
22)(
)(
22
22
22
22
2/exp21)1(1exp1
2/exp21)1(1exp
2/exp2111exp1
2/exp2111exp
2/exp21
12122exp1
2/exp21
12122exp
2/exp21)1(1exp1
2/exp21)1(1exp
2/exp21)1(1exp1
2/exp21)1(1exp)(
5
5
5
1
5
1
3
2
3
2
4
6
4
6
6
6
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρλ
−−−−−+
−−−−+
−−−−+
−−−+
−
−−
−−+
−
−−
−+
−−−−+
−−−+
−−−−−+
−−−−=
∫
∫
∫
∫
∑ ∫
∑ ∫
∫
∫
∫
∫
∞
∞
−
+−=
−
+−=
∞−
∞−
(A.5.10)
149
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]
( ) ( )( )( ) ( ) ( )( ) .
/)(/)(1/)(/)(
12122exp
//1//12exp//1//2exp
/1//1/1)(
1
2 32
32
515
4646
5566
∑−
+−=
−−+
−
−−
−+
−−+−−+−−+−+
−++−−−=⇔
L
Li AA
JJ
AAJJ
AAJJ
AJAJ
iLQiLQiLQiLQ
Li
LQLQLQLQLQLQLQLQ
LQLQLQLQD
σσρσσρ
λ
σσρσσρλσσρσσρλ
σρσρσρσρλ
Derivative of the D(λ) can be calculated and it is
( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]
( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]( ) ( )( )
( ) ( ) ( )( ) ./)(/)(1
/)(/)(12122exp
12122
//1//2exp2//1//2exp2)('
1
2 32
32
5151
4646
∑−
+−=
−−+
−
−−
−−−
−
−−+−−−−−+−=
L
Li AA
JJ
AAJJ
AAJJ
iLQiLQiLQiLQ
Li
Li
LQLQLQLQLQLQLQLQD
σσρσσρ
λ
σσρσσρλσσρσσρλλ
(A.5.11)
The arguments of the Q-function for CW-jamming are the same as in Appendix 4 added with two new arguments (when NJ = 0) (3.26) and (A.2.1)
sS
SJb
SJ
JTENE
cdL
NELJ
//2
cos2/2
1 001
ρφ
σ−
−==
( )sS
SJb
SJ
JTENE
cdL
NEL
LJ
//2
cos2/2
124 004
ρφ
σ−+−==
( )sS
SJb
SJ
JTENE
cdL
NELi
iLiJ
//2
cos2/2
1)(
)(2 002
ρφ
σ−−−== (A.5.12)
( )sS
SJb
SJ
JTENE
cdL
NELi
iLiJ
//2
cos2/2)(
)(3 003
ρφ
σ−−==
( )sS
SJbS
J
JTENE
cdNEL
J//2
cos2/215 00
5
ρφθ
σ−−==
( )sS
SJbS
J
JTENE
cdNEL
J//2
cos2/216 00
6
ρφθ
σ−−−== .
150
The arguments of the Q-function for noise jamming the same as in Appendix 4 added with two new arguments (when J(t)=0)
( ) ( )JSS
J
NENELL
J//1//1
2110
1
ρσ +−
==
( ) ( )JSS
J
NENELLL
J//1//1
21240
4
ρσ ++−
==
( ) ( )JSS
J
NENELLiiL
iJ//1//1
21)()(2
0
2
ρσ +−−
== (A.5.13)
( ) ( )JSS
J
NENELLiiL
iJ//1//1
2)()(3
0
3
ρσ +−
==
( ) ( ) ( )JSS
J
NENEL
J//1//1
2150
5
ρθ
σ +−==
( ) ( ) ( )JSS
J
NENEL
J//1//1
2160
6
ρθ
σ +−−== .
The arguments of the Q-function for AWGN channel are the same as in appendix 4 added with two new arguments
LNEL
A SA 01 /21
−==
σ
( )L
NEL
LA SA 04 /2
124 +−==σ
( )L
NELiiLiA SA 02 /2
1)()(2 −−==σ
(A.5.14)
( )L
NELi
iLiA SA 03 /2)()(3 −==
σ
( ) 05 /215 NE
LA S
A −== θσ
( ) 06 /216 NE
LA S
A −−== θσ
By multiplying D’(λ) by exp(2 λ) a change of variable is made as in Appendix 4 (A.4.18)
151
( ) ( )( ) ( ) ( ) ( )( )[ ]
( ) ( )( ) ( ) ( ) ( )( )( )( ) ( )( )
( ) ( ) ( )( ) .)(3)(21
)(3)(2
51151
461461)('
1
2
212
212
212
∑−
+−=
−−
−−
−
−−
+−+
−−+−+
−−+−−=
L
Li
i
L
L
iAQiAQiJQiJQ
v
AQAQJQJQv
AQAQJQJQvD
ρρ
ρρ
ρρλ
(A.5.15)
When D’(λ) is set equal to zero, the zero points can be calculated for a polynomial. Because 0≥λ it follows that 1≥ν . The channel parameter D(λ) can be presented by v
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]
( ) ( )( )( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( )( )[ ]
( ) ( )( ) ( ) ( ) ( )( )[ ]( ) ( )( )
( ) ( ) ( )( ) .)(3)(21
)(3)(2
51151
46146
5156161
/)(3/)(21/)(3/)(2
12122exp
511512exp461462exp
5156161)(
1
2
212
212
212
1
2
∑
∑
−
+−=
−−
−−
−
−
+−=
−−
+−+
−−+−+
−−+−+
−++−−−=
−−+
−
−−
−+
−−+−−+−−+−+
−++−−−=
L
Li
i
L
L
L
Li
iAQiAQiJQiJQ
v
AQAQJQJQv
AQAQJQJQv
AQJQAQJQ
iLAQiLAQiLJQiLJQ
Li
AQAQJQJQAQAQJQJQ
AQJQAQJQD
ρρ
ρρ
ρρ
ρρρρ
σσρσσρ
λ
ρρλρρλ
ρρρρλ
(A.5.16)
Appendix 6. Calculation of D for the chip combiner receiver
The metric used for the chip combiner receiver is shown in (8.2) as
( )
( ) ( )[ ] . , 1,2 where
or -,1
-,1,1
-,1,1212
-,1,1
,
ZiLLi
EuEu
EuEEL
Lyd
EuEELiydE
Li
Li
EuEEL
Lyd
xym
SS
SSS
SSSS
SSS
∈−+−∈
≥≤−
<<−
−≤−
<<<≤−
−−
<<−
>
=
θθ
θθ
θθ
θθ
(A.6.1)
In the receiver shown in Figure 39, the chip combiner measures the jamming signal. The information and noise signals disturb jamming measurement. The noise level in the chip combiner can be approximated to be the sum of the noise components N0 and NJ. Thus the equivalent noise power spectral density on the information signal bandwidth is
Ju NNN += 0 . (A.6.2)
The power of the jamming signal is J and so the energy of the jamming signal in the symbol interval is
SJ JTE = . (A.6.3)
The chip combiner is the integrator receiver, and the output of the integrator is
( )
( )
.)()(cos22
4
)()()cos(cos)(cos2)()(2
)cos()()()cos()()()cos(2
)()cos(2
0000
2
0000
00
tntnTAT
TT
Sdc
tntndtttAtStxtcT
dtttntnttJtptT
dttrtT
u
JJSJS
S
Sb
T
JJJS
T
JS
T
S
S
S
S
+++=
++++=
+++=
=
∫
∫
∫
φ
ωφωω
ωωω
ω
(A.6.4)
The energy of the jamming signal, total noise and the energy of the signal are, respectively,
153
.,
,2
2
SS
Jtot
SSJ
J
STEnnn
JTTA
E
=+=
==
(A.6.5)
The output of the chip combiner is
uSbJJ nEdcEu ++= φcos2 (A.6.6)
whereφ J is phase difference of the jamming signal and the local oscillator of the chip combiner receiver. The decision regions for u in equation (A.6.1) are as a function of nu as follows:
.cos2cos2
cos2
SbJJSuSbJJS
SuSbJJS
SS
EdcEEnEdcEE
EnEdcEE
EuE
−−<<−−−⇔
<++<−⇔
<<−
φθφθ
θφθ
θθ
(A.6.7)
When jamming exists, the limits are
SbJJSUJ
SbJJSLJ
EdcEEu
EdcEEu
−−=
−−−=
φθ
φθ
cos2
cos2 (A.6.8)
and when jamming does not exist,
.SbSUA
SbSLA
EdcEu
EdcEu
−=
−−=
θ
θ (A.6.9)
The probability of not erasing the received signal in jamming is
( ) ( ) ( )σσθθ //- UJLJSSinJ uQuQEuEPP −=<<= (A.6.10)
where σ is defined in (A.2.1). The probability of not erasing the received signal in AWGN channel is
( ) ( ) ( )σσθθ //- UALASSinA uQuQEuEPP −=<<= . (A.6.11) Two possible errors may happen: miss of jamming detection PMD, and false alarm of jamming PFA ([1] vol. 2 p. 56), ([11] p. 71), [12]. These probabilities are
154
.1 inAFA
inJMD
PPPP−=
= (A.6.12)
In noise jamming the limits are
( ) ( ) ( )
( ) ( ) ( ) ,//1//1
1/
//1//11/
0
0
JSSbUJ
JSSbLJ
NENEdcu
NENEdcu
ρθσ
ρθσ
+−=
+−−=
(A.6.13)
in CW jamming the limits are
( )
( ) , /
/cos2//
//
cos2//
00
00
JSNE
NEdcu
JSNE
NEdcu
SJSbUJ
SJSbLJ
ρφθσ
ρφθσ
−−=
−−−=
(A.6.14)
and in an AWGN channel
( )( ) .//
//
0
0
NEdcu
NEdcu
SbUA
SbLA
−=
−−=
θσ
θσ (A.6.15)
The probabilities in (A.6.12) are dependent on the jamming situation. Results must be calculated separately for jamming and AWGN cases. To calculate the channel parameter D(λ), the expectation shown in equation (2.27) must be taken. It results
155
[ ]( ){ }( ) [ ]( ){ }
( ) [ ]( ){ }( )( ) [ ]( ){ }
( ) ( )( )[ ] [ ]
( ) ( ) ( )( )[ ] [ ]
( ) ( ) ( )( )[ ] [ ]
( )( ) ( ) ( )( )[ ] [ ]dnnxymxymP
dnnxymxymP
dnnxymxymP
dnnxymxymP
zxymzxymEP
zxymzxymEPzxymzxymEP
zxymzxymEPD
inA
tottotjjinJ
inA
tottotjjinJ
xxinA
xxjjinJ
xxinA
xxjjinJ
22
22
22
22
ˆ
ˆ
ˆ
ˆ
2/exp21,ˆ,exp11
2/exp21,ˆ,exp1
2/exp21,ˆ,exp1
2/exp21,ˆ,exp
|);,();ˆ,(exp11
|);,();ˆ,(exp1|);,();ˆ,(exp1
|);,();ˆ,(exp)(
σσπ
λρ
σσπ
λρ
σσπ
λρ
σσπ
λρ
λρ
λρλρ
λρλ
−−−−+
−−−+
−−−+
−−=
−−−+
−−+
−−+
−=
∫
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
≠
≠
≠
≠
x
xx
x
(A.6.16)
where PinA and PinJ are the probability of not erasing the received signal in AWGN and in jamming channels, respectively. The two first terms of (A.6.16) are otherwise same as those in Appendix 4, except the added multiplier of Pin. The two last terms can be calculated using metrics shown in (A.6.1) and probabilities PinA and PinJ in (A.6.10) and (A.6.11). Thus
( ) ( )( ) ( ) ( )( )[ ]
( ) ( ) ( ) ( )( )( ) ( )( )
( ) ( ) ( )( )( )( ) ( )ρρ
σσρσσρ
λ
σρσρλσρσρλλ
inJinA
L
Li inA
inJ
inAinJ
inAinJ
PP
iLAQiLAQPiLJQiLJQP
Li
LAQPLJQPLAQPLJQPD
−+−−+
−−
+−
−−
−+
−+−+−−−−=
∑−
+−=
111
/)(3/)(21/)(3/)(2
12122exp
/11/12exp/411/412exp)(
1
2
(A.6.17)
where terms are defined in (A.4.13) –(A.4.15).
The derivative of D(λ) can be calculated and it is
( ) ( )( ) ( ) ( )( )[ ]( ) ( ) ( ) ( )[ ]
( ) ( )( )( ) ( ) ( )( ) .
/)(3/)(21/)(3/)(2
12122exp
12122
/11/12exp2/411/412exp2)('
1
2∑
−
+−=
−−
+−
−−
−−−
−
−+−−−−−−=
L
Li inA
inJ
inAinJ
inAinJ
iLAQiLAQPiLJQiLJQP
Li
Li
LAQPLJQPLAQPLJQPD
σσρσσρ
λ
σρσρλσρσρλλ
(A.6.18)
The arguments of the Q-function for CW-jamming, noise jamming and the AWGN channel are the same as in Appendix 4 (A.4.13) - (A-2-15). The derivative multiplied by exp(2λ) is
156
( ) ( )( ) ( ) ( )( )[ ]
( ) ( ) ( )( )( ) ( )( )
( ) ( ) ( )( ) .)(3)(21
)(3)(212
4exp12122
1112
4114112
4exp2)('2exp
1
2
12
∑−
+−=
−
−
−−
+−
−−−
−
−+−
−−−−
−=
L
Li inA
inJiL
inAinJ
inAinJ
L
iAQiAQPiJQiJQP
LLi
AQPJQP
AQPJQPL
D
ρρλ
ρρ
ρρλλλ
(A.6.19)
When D’(λ) is set equal to zero, the zeros can be calculated for a polynomial where the variable ν is now
−=
124expL
v λ . (A.6.20)
Because 0≥λ it follows that 1≥ν . The channel parameter D(λ) can be presented by v as
( ) ( )( ) ( ) ( )( )[ ]
( ) ( ) ( ) ( )( )( ) ( )( )
( ) ( ) ( )( )( )( ) ( )
( )( ) ( ) ( )( )[ ]
( ) ( ) ( )( )( ) ( )( )
( ) ( ) ( )( )( )( ) ( ) .111
)(3)(21)(3)(2
111
41141
111)(3)(21
)(3)(212122exp
1112exp411412exp)(
1
2
212
212
212
1
2
ρρ
ρρ
ρρ
ρρ
ρρ
ρρ
λ
ρρλρρλλ
inJinA
L
Li inA
inJi
inAinJ
LinAinJ
L
inJinA
L
Li inA
inJ
inAinJ
inAinJ
PPiAQiAQP
iJQiJQPv
AQPJQPv
AQPJQPv
PPiAQiAQP
iJQiJQPLi
AQPJQPAQPJQPD
−+−−+
−−
+−+
−++
−−−−=
−+−−+
−−
+−
−−
−+
−+−+−−−−=
∑
∑
−
+−=
−−
−−
−
−
+−=
(A.6.21)
