Performance Evaluation of M-ary Frequency Shift Keying Radio

Modems via Measurements and Simulations

Submitted by

ERIC NII OTORKUNOR SACKEY

Department of Electrical Engineering

Blekinge Institute of Technology

Karlskrona, Sweden

September 2006

This thesis is presented as part of the Degree of Master of Science in Electrical

Engineering with emphasis on Telecommunications/Signal Processing.

Abstract M-ary Frequency Shift Keying is a power efficient modulation scheme that is currently used by manufacturers of low power low data rate data transmission equipment. The power efficiency of this modulation increases as the signal alphabet increases at the expense of increased complexity and reduced bandwidth efficiency. There is, however, a gap between the performance of real world systems employing Frequency Shift Keying (FSK) and that of theoretical FSK systems. To investigate the nature of this gap, a comparison is needed between the performance of real world systems using FSK and that of theoretical FSK systems. This thesis investigates the nature of this gap by simulating 2, 4 and 8-level FSK systems in additive white Gaussian noise channel using MATLAB, measuring of the performance of commercially available data transmission equipment manufactured by RACOM s.r.o of the Czech Republic, and comparison of their performances. Some important results have been illustrated and also, it is observed that the gap between the performance of theoretical and real world systems using FSK is about 1 dB at a bit error rate (BER) of 10-3 and widens as BER decreases.

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Acknowledgements The completion of my studies at Blekinge Institute of Technology, Sweden would have been impossible without the generous help of others. Foremost, I give credit to the almighty God. I also extend my gratitude to my supervisors, Dr. Abbas Mohammed of the School of Engineering and Mr. Jiri Hruska of RACOM s.r.o of the Czech Republic, for their encouragement and support. I also thank the staff of RACOM s.r.o., especially Karel and Marek, who have helped me during the period of measurements by answering questions and providing a conducive environment for my work. Finally, I wish to express my sincere appreciation to Mr. Kwaku Boadu for his support, guidance and encouragement throughout my studies at Blekinge Institute of Technology, Sweden.

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Table of Contents Abstract 2 Acknowledgements 3 Table of Contents 4 List of Figures 6 List of Tables 8 1 Introduction 9 2 Modulation 10 2.1 Modulation Format 10 2.2 Digital Modulation – an Overview 10 2.3 Factors that Influence the Choice of Digital Modulation 11 2.4 Additive White Gaussian Noise Channel 12 3 FSK Background 15

3.1 Constant Envelope Modulation 15 3.2 Binary Frequency Shift Keying 15 3.2.1 Binary FSK signal Modulator 15 3.2.2 Coherent Demodulation and Error Performance 19 3.2.3 Noncoherent Demodulation and Error Performance 21 3.2.4 Power Spectral Density 24 3.3 M-ary FSK 27 3.3.1 Modulator, Demodulator and Error Performance 27 3.3.2 Coherent Versus Noncoherent 36 3.3.3 Power Spectral Density 38 3.3.4 Bandwidth Efficiency 42

4 Simulations and Results 44 4.1 Simulation of AWGN Channel 44 4.2 Simulation of Binary FSK System in AWGN Channel 45 4.2.1 Coherent System 45 4.2.2 Noncoherent System 46 4.3 Simulation of 4 and 8-level FSK Systems in AWGN Channel 48 5 Measuring of Performance of Commercially Available Radio Modems 52

5.1 RACOM s.r.o 52 5.2 Measurements 52 5.2.1 Extraction of BER from measured PER 54 5.2.2 Calculation of Noise Power 55 5.2.3 Processing of Measurement Data on MR400 Radio Modem 56 5.2.4 Processing of Measurement Data on MX160 Radio Modem 58

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6 Comparison of Performance of 2, 4 and 8-level FSK Systems 62 6.1 Bounds on Communication 62 6.2 Comparison of Performance of Simulated FSK Systems 64 6.3 Theory versus Reality 66 6.4 Comparison of Performance using Shannon’s Capacity Curve 67

7 Conclusions 69 8 References 70 A Acronyms 71

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List of Figures 3.1 Noncoherent BFSK modulator 16 3.2 Coherent BFSK modulator 17 3.3 Coherent BFSK demodulator: correlator implementation 19 3.4 Coherent BFSK demodulator: matched filter implementation 20 3.5 Probability of error of coherently demodulated BFSK signal 21 3.6 BFSK noncoherent demodulator: correlator-squarer implementation 22 3.7 BFSK noncoherent demodulator: matched filter implementation 23 3.8 Probability of error of noncoherently demodulated BFSK signal 24 3.9 Coherent M-ary FSK modulator 27 3.10 Coherent M-ary FSK demodulator: correlator implementation 28 3.11 Coherent M-ary FSK demodulator: matched filter implementation 28 3.12 Bit error probability of coherently demodulated M-ary FSK signals 30 3.13 Symbol error probability of coherently demodulated M-ary FSK signals 30 3.14 Noncoherent M-ary FSK modulator 31 3.15 Noncoherent M-ary FSK demodulator: correlator-squarer implementation 32 3.16 Noncoherent M-ary FSK demodulator: matched filter implementation 33 3.17 Noncoherent M-ary FSK demodulator: envelope detector implementation 34 3.18 Bit error probability of noncoherently demodulated M-ary FSK signals 35 3.19 Symbol error probability of noncoherently demodulated M-ary FSK signals 35 3.20 Comparison of BER for coherent and noncoherent BFSK 36 3.21 Comparison of BER for coherent and noncoherent 4-FSK 37 3.22 Comparison of BER for coherent and noncoherent 8-FSK 37 3.23 Power-density spectrum of BFSK signal (h = 0.5, 0.6, 0.7) 39 3.24 Power-density spectrum of BFSK signal (h = 0.8, 0.9, 0.95) 39 3.25 Power-density spectrum of 4-FSK signal (h = 0.2, 0.3, 0.4) 40 3.26 Power-density spectrum of 4-FSK signal (h= 0.5, 0.6, 0.7) 40 3.27 Power-density spectrum of 8-FSK signal (h = 0.125, 0.2, 0.3) 41 3.28 Power-density spectrum of 8-FSK signal (h = 0.4, 0.5, 0.6) 41 3.29 Power-density spectra of M-ary FSK signals for M = 2, 4 and 8 (h = 0.5) 42 4.1 Simulation model for coherent BFSK system 45 4.2 Performance of simulated coherent BFSK system 46 4.3 Simulation model for noncoherent BFSK system 47 4.4 Performance of simulated noncoherent BFSK system 48 4.5 Simulation model for coherent 4-level FSK system 49 4.6 Performance of simulated coherent 4-level FSK system 50 4.7 Performance of simulated noncoherent 4-level FSK system 50 4.8 Performance of simulated coherent 8-level FSK system 51 4.9 Performance of simulated noncoherent 8-level FSK system 51 5.1 MR400 radio modem 53 5.2 MX160 radio modem 53 5.3 PER versus packet length at -108 dBm for MR400 radio modem 56

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5.4 Performance curve of MR400 radio modem 58 5.5 PER versus packet length at -104 dBm for MX160 radio modem 59 5.6 Performance curve of MX160 radio modem 60 5.7 Comparison of performance of MR400 and MX160 radio modems 61 6.1 Energy versus spectral efficiency of an optimum system 63 6.2 Comparison of simulated symbol error probabilities for coherent M-ary FSK 64 6.3 Comparison of simulated bit error probabilities for coherent M-ary FSK 65 6.4 Comparison of simulated symbol error prob for noncoherent M-ary FSK 65 6.5 Comparison of simulated bit error probabilities for noncoherent M-ary FSK 66 6.6 Comparison of theoretical and practical BER for 2 and 4-level FSK systems 67 6.7 Comparison of theoretical and practical orthogonal modulation techniques at BER of 10-5 68

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List of Tables 3.1 Bandwidth efficiency of coherent M-ary FSK signals 43 5.1 Technical data for MR400 and MX160 radio modems 53 5.2 Raw data from measurements on MR400 radio modem 54 5.3 Raw data from measurements on MX160 radio modem 54 5.4 Processed data from measurements on MR400 radio modem 56 5.5 Processed data from measurements on MX160 radio modem 58 6.1 Spectral efficiency and S/N required to achieve BER of 10-5 67

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Chapter 1 Introduction M-ary frequency shift keying (FSK) is a power efficient modulation scheme whose efficiency improves as the number of frequencies employed (M) increases at the expense of additional complexity and smaller bandwidth efficiency. This scheme has been found advantageous in low rate low power applications. There is a difference, however, between the performance of theoretical M-ary FSK systems and that of real world systems employing M-ary FSK modulation schemes. The primary objective of this thesis is to simulate 2, 4 and 8-level FSK systems in additive white Gaussian noise (AWGN) channel using MATLAB, compare their performance to that of real world systems employing Gaussian minimum shift keying (GMSK) and 4-level FSK modulation techniques and explain the difference. Basic simulations of both coherent and noncoherent 2, 4 and 8-level FSK systems are considered and measurements of performance of real world systems are focused on commercially available data transmission equipment, manufactured by RACOM s.r.o of the Czech Republic, using GMSK and 4-level FSK modulation schemes. In order to establish the difference between the performance of theoretical and practical FSK systems, only basic simulations are considered since their performance is close to that of theory. In the following chapters, we discuss general modulation in brief, factors that influence the choice of a particular digital modulation scheme, how the comparison of performance of different digital modulation types are made and a model of AWGN channel. Next we follow with the theoretical background of binary and M-ary frequency shift keying (FSK). The next chapter would be simulations of 2, 4 and 8-level FSK systems, for both coherent and noncoherent demodulation, followed by measurements of performance of commercially available data transmission equipment (radio transceivers), by RACOM s.r.o. of Czech Republic, using GMSK and 4-level FSK. Finally we compare the performance of the commercially available data transmission equipment to theory and end with the conclusions.

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Chapter 2 Modulation This chapter describes what modulation is, and an overview of digital modulation. It also describes the factors that influence the choice of a particular digital modulation scheme, and a model for additive white Gaussian noise (AWGN) channel. 2.1 Modulation Format A modulation format is the means by which information is encoded unto a signal. Information, or data, can be carried in the amplitude, frequency, or phase of a signal. A modulation scheme can be analog (where data is contained in a set of continuous values) or digital (where the data is contained in a set of discrete values). The information signal is called modulating waveform. When the information is encoded on a signal, the signal is called a modulated waveform. The process of bringing the bandpass signal down to baseband is denoted as demodulation. We distinguish detection from demodulation by denoting detection as the process of extracting the information from the baseband demodulated signal. A receiver consists of a demodulator and a detector. Noncoherent techniques (where the reference phase is unknown; discussed in chapter 3.2.4) can often be implemented in demodulation or detection, such that the two terms can be used interchangeably. 2.2 Digital Modulation – an Overview Modern communication systems use digital modulation techniques. Advancement in very large-scale integration (VLSI) and digital signal processing (DSP) technology have made digital modulation more cost effective than analog transmission systems. Digital modulation offers many advantages over analog modulation. Some advantages include greater noise immunity and robustness to channel impairments, easier multiplexing of various forms of information (for example, voice, data, and video), and greater security. Further more, digital transmissions accommodate digital error-control codes which detect and/or correct transmission errors, and support complex signal conditioning and processing techniques such as source coding, encryption, and equalization to improve the performance of the overall communication link. New multipurpose programmable digital signal processors have made it possible to implement digital modulators and demodulators completely in software. Instead of having a particular modem design permanently frozen as hardware, embedded software implementations now allow alterations and improvements without having to redesign or replace the modem. In digital wireless communication systems, the modulating signal (e.g., the message) may be represented as a time sequence of symbols or pulses, where each symbol has m finite states. Each symbol represent n bits of information, where n = log2m bits/symbol. Many digital modulation techniques are used in modern wireless

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communication systems, and many more are sure to be introduced. Some of these techniques have subtle differences between one another, and each technique belongs to a family of related modulation methods. For example, Frequency Shift Keying (FSK) may be either coherently or noncoherently detected; and may have two, four, eight or more possible levels per symbol, depending on the manner in which information is transmitted within a single symbol [1]. 2.3 Factors That Influence the Choice of Digital Modulation Several factors influence the choice of a digital modulation scheme. A desirable modulation scheme provides low bit error rates at low received signal-to-noise ratios, performs well in multipath and fading conditions, occupies a minimum bandwidth, and is easy and cost effective to implement. Existing modulation schemes do not simultaneously satisfy all of these requirements. Some modulation schemes are better in terms of bit error rate performance, while others are better in terms of bandwidth efficiency. Depending on the demands of a particular application, tradeoffs are made when selecting a digital modulation. The performance of a modulation scheme is often measured in terms of its power efficiency and bandwidth efficiency. Power efficiency describes the ability of a modulation technique to preserve the fidelity of the digital message at low power levels. In a digital communication system, in order to increase noise immunity, it is necessary to increase the signal power. However, the amount by which the signal power should be increased to obtain a certain level of fidelity (i.e., an acceptable bit error probability) depends on the particular type of modulation employed. The power efficiency (sometimes called energy efficiency) of a digital modulation scheme is a measure of how favorable this tradeoff between fidelity and signal power is made, and is often expressed as the ratio of the signal energy per bit to noise power spectral density (Eb/No) required at the input of the receiver for a certain probability of error (say 10-5). Bandwidth efficiency describes the ability of a modulation scheme to accommodate data within a limited bandwidth. In general, increasing the data rate implies decreasing the pulse width of a digital symbol, which increases the bandwidth of the signal. Thus, there is an unavoidable relationship between data rate and bandwidth occupancy. However, some modulation schemes perform better than others in making this tradeoff. Bandwidth efficiency reflects how efficiently the allocated bandwidth is utilized and is defined as the ratio of the throughput data rate per Hertz in a given bandwidth. If R is the data rate in bits per second, and B is the bandwidth occupied by the modulated radio frequency signal, then bandwidth efficiency ηB is expressed as

ηB = BR

bps/Hz (2.1)

The system capacity of a digital communication system is directly related to the bandwidth efficiency of the modulation scheme, since a modulation with a greater value of ηB will transmit more data in a given spectrum allocation. There is a fundamental upper bound on achievable bandwidth efficiency. Shannon’s channel coding theorem states that for an arbitrary small probability or error,

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the maximum possible bandwidth efficiency is limited by the noise in the channel, and is given by channel capacity formula. The Shannon’s bound for additive white Gaussian noise (AWGN) non-fading channel is given by;

ηBmax = BC = log2(1 +

NS ) (2.2)

where C is the channel capacity in bits per second, B is the radio frequency (RF) bandwidth, and S/N is the signal-to-noise ratio. In design of a digital communication system, very often there is a tradeoff between bandwidth efficiency and power efficiency. For example, adding error control coding to a message increases the bandwidth occupancy (and this, in turn, reduces the bandwidth efficiency), but at the same time reduces the required power for a particular bit error rate, and hence trades bandwidth efficiency for power efficiency. On the other hand, higher level modulation schemes (M-ary keying), except M-ary FSK, decrease bandwidth occupancy but increase the required received power, and hence trades power efficiency for bandwidth efficiency. While power and bandwidth considerations are very important, other factors also affect the choice of a digital modulation scheme. For example, for all personal communication systems which serve a large user community, the cost and complexity of the subscriber receiver must be minimized, and a modulation which is simple to detect is most attractive. The performance of a modulation scheme under various types of channel impairments such as Rayleigh and Ricean fading and multipath time dispersion, given a particular demodulator implementation, is another key factor in selecting a modulation. In wireless systems where interference is a major issue, the performance of a modulation scheme in an interference environment is extremely important. Sensitivity to detection of time jitter, caused by time-varying channels, is also an important consideration in choosing a particular modulation scheme. In general, the modulation, interference, and implementation of the time-varying effects on a channel as well as the performance of the specific demodulator are analyzed as a complete system using simulation to determine relative performance and ultimate selection [1]. 2.4 Additive White Gaussian Noise Channel Additive white Gaussian noise (AWGN) channel is a universal channel model for analyzing modulation schemes. In this model, the channel does nothing but add a white Gaussian noise to the signal passing through it. This implies that the channel’s amplitude frequency response is flat (thus with unlimited or infinite bandwidth) and phase response is linear for all frequencies so that the modulated signal pass through it without any amplitude or phase loss or distortion of frequency components. Fading does not exist. The only distortion is introduced by the AWGN. The received signal is then equal to

r(t) = s(t) + n(t) (2.3) where n(t) is the additive white Gaussian noise, and s(t) is the modulated signal.

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The whiteness of n(t) implies that it is a stationary random process with a flat power spectral density (PSD) for all frequencies. It can be observed, however, that if Sn(f) = C for all frequencies, where C is a constant, then

( )dffSn∫∞

∞−

= = ∞ ∫∞

∞−

Cdf

so that the total power is infinite. Obviously, no real physical process can have infinite power and, therefore, a white process may not be a meaningful physical process. However, quantum mechanical analysis of thermal noise shows that it has a power-spectral density given by [2]

S(f) = ( )12 / −kThfehf (2.4)

in which h denotes Planck’s constant (equal to 6.6 x 10-34 Js), k is Boltzmann’s constant (equal to 1.38 x 10-23 J/K), and T denotes temperature in Kelvin. This spectrum achieves it maximum at f = 0, and the value of this maximum is kT/2. The spectrum goes to zero as f goes to infinity, but the range of convergence to zero is very slow. For instance, at room temperature (T = 300 K), S(f) drops to 90 % of its maximum at about f = 2.0 x 1012 Hz, which is beyond the frequencies employed in conventional communication systems. From this we conclude that thermal noise, although not precisely white, can be modeled for all practical purposes as a white process with the power spectrum equaling kT/2. The value kT/2 is usually denoted by N0; therefore, the power spectral density of additive white Gaussian noise is usually given as Sn(f) = N0/2 and is sometimes referred to as the two-sided power spectral density, emphasizing that this spectrum extends to both positive and negative frequencies. According to the Wiener-Khinchine theorem, the autocorrelation function of the AWGN is

R(τ) = E{n(t)n(t-τ)} = ( ) dfefS fjn

τπ2∫∞

∞−

= dfeN fj τπ20

2∫∞

∞−

= ( )τδ2

0N (2.5)

where δ(τ) is the Dirac delta function. This shows that noise samples are uncorrelated no matter how close they are in time. The samples are also independent since the process is Gaussian. At any time instance, the amplitude of n(t) obeys a Gaussian probability density function given by

p(η) = ⎟⎟⎠

⎞⎜⎜⎝

⎛ −2

2

2 2exp

21

ση

πσ (2.6)

13

where η is used to represent the values of the random process n(t) and σ2 is the variance of the random process. It is interesting to not that σ2 = ∞ for AWGN process since σ2 is the power of the noise, which is infinite due to its “whiteness”. However, when r(t) is correlated with an orthonormal function φ (t), the noise in the output has a finite variance. In fact,

r = = s + n dtttr )()( φ∫∞

∞−

where s = ∫ ∞

∞−

dttts )()( φ

and n = ∫ ∞

∞−

dtttn )()( φ

The variance of n is

E{n2} = E{[ ]∫∞

∞−

dtttn )()( φ 2}

= E{ ∫ ∫ } ∞

∞−

∞

∞−

ττφτφ dtdnttn )()()()(

= ∫ ∫ ∞

∞−

∞

∞−

ττφφτ dtdtntnE )()()}()({

= ττφφτδ dtdttN )()()(2

0 −∫ ∫∞

∞−

∞

∞−

= dttN )(2

20 ∫∞

∞−

φ

= 2

0N

Then the probability density function of AWGN can be written as

p(n) = ⎟⎟⎠

⎞⎜⎜⎝

⎛ −

0

2

0

exp1Nn

Nπ

Strictly speaking, the AWGN channel does not exist since no channel can have an infinite bandwidth. However, when the signal bandwidth is smaller than the channel bandwidth, many practical channels are approximately an AWGN channel. For example, the line-of-sight (LOS) radio channels, including fixed terrestrial microwave links and fixed satellite links, are approximately AWGN channels when the weather is good. Wideband coaxial cables are also approximately AWGN channels since there is no other interference except Gaussian noise [3].

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Chapter 3 FSK Background This chapter talks briefly about constant envelope modulation; its advantages and disadvantages, and then discusses binary and M-ary FSK in detail. 3.1 Constant Envelope Modulation Many practical radio communication systems use nonlinear modulation methods, where the amplitude of the carrier is constant, regardless of the variation in the modulating signal. The constant envelope family of modulations has the advantage of satisfying a number of conditions [1], some of which are:

• Power efficient class C amplifiers can be used without introducing degradation in the spectrum occupancy of the transmitted signal.

• Low out-of-band radiation of the order of -60 dB to -70 dB can be achieved. • Limiter-discriminator detector can be used, which simplifies receiver design and

provides high immunity against random frequency modulation noise and signal fluctuations due to Rayleigh fading.

While constant envelope modulations have many advantages, they occupy a larger

bandwidth than linear modulation schemes. In situations where bandwidth efficiency is more important than power efficiency, constant envelope modulation is not well-suited. 3.2 Binary Frequency Shift Keying 3.2.1 Binary FSK Signal and Modulator In binary frequency shift keying (BFSK), the frequency of a constant amplitude carrier signal is switched between two values according to the two possible message states, corresponding to a binary 1 or 0. Depending on how the frequency variations are imparted into the transmitted waveform, the FSK signal will have either a discontinuous phase or a continuous phase at bit transmissions. In general, a BFSK signal may be represented as

s1(t) = )2cos(211 φπ +tf

TE

b

b , 0 ≤ t ≤ Tb , for binary 1

s2(t) = )2cos(222 φπ +tf

TE

b

b , 0 ≤ t ≤ Tb , for binary 0 (3.1)

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where 1φ and 2φ are initial phases at t = 0, Tb is the bit period of the binary data, and Eb is the transmitted signal energy per bit. One obvious way to generate a FSK signal is to switch between two independent oscillators according to whether the data bit is a 0 or a 1, as shown in Figure 3.1. Normally this form of FSK generation results in a waveform that is discontinuous at the switching times, and for this reason this type of FSK is called discontinuous of noncoherent FSK. Equation (3.1) represents a discontinuous FSK signal since 1φ and 2φ

are not the same in general.

Oscillator 1

s1(t) =

Figure 3.1 Noncoherent BFSK modulator.

ince the phase discontinuities pose several problems, such as spectral spreading and

The second type of FSK is the coherent one where the two signals have the same

Sspurious transmissions, discontinuous FSK is generally not used in highly regulated wireless systems. initial phase φ at t = 0;

s1(t) = )2cos(21 φπ +tf

TE

b

b , 0 ≤ t ≤ Tb , for binary 1

2(t) = s )2cos(22 φπ +tf

TE

b

b , 0 ≤ t ≤ Tb , for binary 0 (3.2)

Control line

Binary data input, ak

)2cos(211 φπ +tf

TE

b

b

Oscillator 2

s2(t) = )2cos(222 φπ +tf

TE

b

b

Multiplexer

f1, Ø1

f

f2, Ø2

i, Øi

16

This type of FSK can be generated by the modulator as shown in Figure 3.2. The

or coherent demodulation of the coherent FSK signal, the two frequencies are so chosen

= 0

That is

)2cos()2cos( 20

1 φπφπ ++∫

=

frequency synthesizer generates two frequencies, f and f1 2, which are synchronized. The binary input data controls the multiplexer. The bit timing must be synchronized with the carrier frequencies. If a 1 is present, s (t) will pass and if a 0 is present, s1 2(t) will pass. s1(t) and s2(t) are always there regardless of the data input.

Figure 3.2 Coherent BFSK modulator.

Fthat the two signals are orthogonal:

dttstsbT

)()( 20

1∫

dttftfbT

[ ] [{ }dttfftffT

∫ −+++0

2121 )(2cos2)(2cos21 πφπ ]

= [ ]{ } bTtffff 021

21

2)(2sin)(4

1 φππ

+++

+ bTtffff 021

21

)(2sin)(4

1−

−π

π

s1(t) =

Control line

Binary data input, ak

)2cos(21 φπ +tf

TE

b

b

Frequency Synthesizer

s2(t) = )2cos(22 φπ +tf

TE

b

b

Multiplexer

φ,1f

φ,if

φ,2f

17

[ ] bTtfftffff 02121

21

)(2cos2sin)(2sin2cos)(4

1+++

+πφπφ

π+ =

bTtffff 021

21

)(2sin)(4

1−

−π

π

{ }φφπφπ

2sin)cos(2sin)(2sin2cos)(4

12121

21

−++++ bb TffTff

ff+ =

bTffff

)(2sin)(4

121

21

−−

ππ

= 0

This implies that

bTff )(2 21 +π nπ2 = (3.3) and

bTff )(2 21 −π πm = (3.4) where and are integers. Solving equations (3.3) and (3.4) simultaneously leads to m n

bTmn

42 +

bTmn

42 −

1f and = 2f =

fΔ2 =bT

m221 ff − =

bT41Thus we conclude that for orthogonality and must be an integer multiple of 1f 2f

and their difference must be integer multiple ofbT2

1 fΔ. Using we can rewrite the two

frequencies as

bTn

2221 ff +

1f = and =ffc Δ+ ffc Δ−2f , which leads to = cf , =

bT21where is the nominal carrier frequency which must be an integer multiple of cf for

orthogonality.

bT1 When the separation is chosen as , then the phase continuity will be maintained

at bit transitions, and the FSK is called Sunde’s FSK [3]. As a matter of fact, if the

separation isbTk , where k is an integer, the phase of the coherent FSK of equation (3.2) is

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bT21always continuous. The minimum separation for orthogonality between and is1f 2f .

However, this separation cannot guarantee continuous phase. A particular form of FSK called minimum shift keying (MSK) not only has the minimum separation but also has continuous phase. However, MSK is much more than ordinary FSK, it has properties that ordinary FSK doesn’t have. It must be generated by methods other than the one described in Figure 3.2. MSK is an important modulation method scheme but not included in the scope of this thesis. 3.2.2 Coherent Demodulation and Error Performance in AWGN Channel Coherent demodulation requires knowledge of the reference phase or exact phase recovery, meaning local oscillators, phase-lock-loops, and carrier recovery circuits may be required, adding to the complexity of the receiver. The demodulator can be implemented with two correlators as shown in Figure 3.3, where the two reference signals are )2cos( 1tfπ )2cos( 2tfπ and . They must be synchronized with the received signal.

)2cos( 2tfπ

received signal, r(t)

dtbT

∫0

)2cos( 1tfπ

dtbT

∫0

∑

l1

l2

l

Threshold detector

0

1 0

Figure 3.3 Coherent BFSK demodulator: correlator implementation.

The receiver is optimum in the sense that it minimizes the error probability for equally likely binary signals. When signal s1(t) is transmitted, the upper correlator yields a signal l1 with a positive signal component and a noise component. However, the lower correlator output l2, due to the signal’s orthogonality, has only a noise component. Thus the output of the summer is most likely above zero, and the threshold detector will most likely produce a 1. When signal s2(t) is transmitted, opposite things happen to the two correlators and the threshold detector will most likely produce a 0. However, due to the noise nature that it values range from -∞ to +∞, occasionally the noise amplitude might overpower the signal amplitude, and then detection errors happen.

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An alternative to Figure 3.3 is to use matched filter implementation of the demodulator that matches )2cos( 1tfπ )2cos( 2tfπ- (Figure 3.4). Both correlator and matched filter implementation are equivalent in terms of error performance.

h(Tb - t) 1

Sample at t = Tb

l

Threshold detector

0 0

received signal, r(t)

h(t) = )2cos()2cos( 21 tftf ππ −

Figure 3.4 Coherent BFSK demodulator: matched filter implementation. In the presence of AWGN channel, the received signal is

(t) + n(t), i = 1, 2 r(t) = si

where n(t) is the additive white Gaussian noise with zero mean and a two-sided power spectral density N0/2. The bit error probability for an equally likely binary signal is given by

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −+

0

21221

22N

EEEEQ

ρ = (3.5) [3] Pb

where E1 and E are the energies of the binary signals, 12ρ2 is the correlation co-efficient of the binary signals, and Q(z) is the Q-function defined as

dzzz

)2/exp(21 2−∫

∞

πQ(z) =

For Sunde’s FSK signals, E1 = E2 = Eb, and 12ρ = 0 since the signals are orthogonal. Thus the error probability is

⎟⎟⎠

⎞⎜⎜⎝

⎛

0NEQ b = (3.6) Pb

where Eb is the average transmitted bit energy of the FSK signal. A plot of equation (3.6) is shown in Figure 3.5.

20

0 5 10 1510-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Bit

erro

r pro

babi

lity,

Pb

Figure 3.5 Probability of error of coherently demodulated BFSK signal.

3.2.3 Noncoherent Demodulation and Error Performance in AWGN Channel Noncoherent demodulation techniques do not require knowledge of the reference phase, eliminating the need for phase-lock-loops, local oscillators, and carrier recovery circuits. Non coherent demodulation techniques are generally less expensive and easier to build than coherent techniques (since coherent reference signals do not have to be generated), and are often preferred, though they can degrade performance under certain channel conditions. Coherently generated FSK signals can be noncoherently demodulated to avoid the carrier recovery. Noncoherently generated FSK can only be noncoherently demodulated. Both are referred to as noncoherent FSK and the demodulation problem becomes a problem of detecting signals with unknown phases. It can be shown that the optimum receiver for noncoherent demodulation is a quadrature receiver. It can be implemented using correlators or equivalently, matched filters. With the assumption that the binary noncoherent FSK signals are equally likely and of equal energies, the demodulator using correlators is shown in Figure 3.6.

21

r(t) Comparator

dtbT

∫0

∑

dtTb

∫0

Squarer

Squarer

dtbT

∫0

∑

dtTb

∫0

Squarer

Squarer

)2cos( 1tfπ

)2sin( 1tfπ

)2cos( 2tfπ

)2sin( 2tfπ

21l

22l

If 21l > 2

2l choose 1 If 2

1l < 22l

choose 0

Figure 3.6 BFSK noncoherent demodulator: correlator-squarer implementation.

The received signal (ignoring noise for the moment) with unknown phase can be written as

),2cos( φπ +tfA i(t) = i = 1, 2 si

)2sin(sin)2cos(cos tfAtfA ii πθπθ − = )2cos(cos tfA iπθThe signal consists of inphase and quadrature components and

)2sin(sin tfA iπθ )2cos( tfiπ respectively. Thus the signal is partially correlated with and partially correlated to )2sin( tfiπ . The outputs of the inphase and quadrature correlators

will be 2cosθbAT

2sinθbAT and , respectively. Depending on the value of the unknown

phase, these two outputs could be anything in ⎟⎠⎞

⎜⎝⎛ −

2,

2bb ATAT . Fortunately the squared

sum of these two signals is not dependent on the unknown phase. That is

22

2sin

2cos

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ θθ bb ATAT

2

22bTA =

22

This quantity is actually the mean value of the statistics when the signal s2il i(t) is

transmitted and noise is taken into consideration. When si(t) is not transmitted the mean value of is zero. The comparator decides which signal was transmitted by checking

these .

2il

2il

The matched filter equivalence of Figure 3.6, which has the same error performance, is shown in Figure 3.7.

r(t)

Envelope Detector

Envelop Detector

Comparator

Sample at t = Tb

Sample at t = Tb

1l

2l

)(2cos 1 tTf b −π

)(2cos 2 tTf b −π

if 1l > 2l choose 1 if 1l < 2l choose 0

Figure 3.7 BFSK noncoherent demodulator: matched filter implementation. The probability of error of noncoherent, orthogonal binary FSK signals can be shown to be

P ⎟⎟⎠

⎞⎜⎜⎝

⎛ −

02exp

21

NEb = (3.7) b

The plot of equation (3.7) is shown in Figure 3.8 below.

23

0 5 10 1510-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Bit

erro

r pro

babi

lity,

Pb

Figure 3.8 Probability of error of noncoherently demodulated BFSK signal. It is worth noting that the demodulators in Figures 3.6 and 3.7 are good for equiprobable, equal-energy, noncoherent signals. They do not require the signals to be orthogonal. However, equation (3.7) is only applicable for orthogonal, equiprobable, equal-energy, noncoherent signals. It was shown in chapter 3.2.1 that the minimum frequency separation for coherent FSK signals is 1/2T (where T is the symbol period). It can similarly be shown that that the minimum separation for noncoherent FSK signals is 1/T instead of 1/2T. Thus the separations for noncoherent FSK is double that of coherent FSK. Hence more system bandwidth is required for noncoherent FSK for the same symbol rate. 3.2.4 Power Spectral Density of BFSK The Sunde’s FSK signal, assuming the initial phase to be zero, can be written as

tT

fTE

bc

b

b )21(2cos2

+π bTt ≤≤s1(t) = , 0 , for binary 1

tT

fTE

bc

b

b )21(2cos2

−π bTt ≤≤s2(t) = , 0 , for binary 0 (3.8)

24

It is obvious from equation (3.8) that Sunde’s FSK signal can be further simplified as

tT

fTE

bc

b

b )21(2cos2

±π s(t) =

)2cos(2

bc

b Tttf

TE ππ ± =

)2cos(b

kc TtatfA ππ + = (3.9)

where

b

b

TE2 A = , and = ka 1±

Expanding equation (3.9) leads to

)2sin()sin()2cos()cos( tfTtaAtf

TtaA c

bkc

bk ππππ

− s(t) =

)2sin()sin()2cos()cos( tfTtAatf

TtA c

bkc

b

ππππ− bTt ≤≤0 , =

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+ − tfjk

b

ceTb

tjAaTtA πππ 2)sin()cos(Re =

= (3.10) ⎥⎦

⎤⎢⎣

⎡ − tfj cets π2)(~Re

)(~ tswhere is the complex envelope of the bandpass signal s(t), and defined as

)sin()cos(b

kb T

tjAaT

tA ππ+)(~ ts =

It can be shown that the power spectral density (PSD) of a bandpass signal

[ ]tfj cets π2)(~Re −s(t) = is the shifted version of the equivalent baseband signal or the complex envelope )(~ ts ’s PSD . )( fSB

[ ]cBcB ffSffS )()(41

++−)( fSs , =

where is the power spectral density of the bandpass signal s(t). Therefore it suffices to determine the PSD of the equivalent baseband signal

)( fSs

)(~ ts . Since the inphase and quadrature component of the FSK signal of equation (3.10) are independent of each other, the PSD for the complex envelope is the sum of the PSDs of these two components.

25

)( fSB = )()( fSfS QI +

)( fS I can easily be found since the inphase component is independent of data. It is defined on the entire time axis. Thus

2

cos⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛

bTtAF π = )( fS I

= ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛−

bb Tf

TfA

21

21

4δδ

where F stands for Fourier transform. It is seen that the spectrum of the inphase part of the Sunde’s FSK signal are two delta functions. It can also be shown that the PSD of a binary, bipolar, equiprobable, stationary, and uncorrelated digital waveform is just equal to the energy spectral density of the symbol shaping pulse divided by the symbol period. The symbol shaping pulse of the

quadrature component is )sin(bTtA π , and therefore

bbb

TtT

tAFT

≤≤⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛0,sin1

2π)( fSQ =

dteT

tA fjT

b

bππ 2

0

sin −∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛

bTtAF πsin =

= ])2(1[)cos(2

2fTfTAT

b

bb

−ππ

Thus,

⎟⎟⎠

⎞⎜⎜⎝

⎛− ])2(1[

)cos(212fT

fTATT b

bb

b ππ

=)( fSQ

The complete baseband PSD of the binary FSK signal is the sum of and ; )( fSQ)( fS I

2

2

2

])2(1[)cos(2

21

21

4 ⎟⎟⎠

⎞⎜⎜⎝

⎛−

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛−

fTfTA

TT

fT

fA

b

bb

bb ππ

δδ)( fSB = (3.11)

26

3.3 M-ary FSK 3.3.1 Modulator, Demodulator, and Error Performance in AWGN Channel The coherent modulator of binary FSK in Figure 3.2 can easily be extended to coherent M-ary FSK (Figure 3.9). Here the frequency synthesizer generates M signals with the designed frequencies and coherent phase, and the multiplexer chooses one of the frequencies, according to the n = bits. M2log

1f

Multiplexer

Frequency Synthesizer

2f if

.

.

.

Mf

Figure 3.9 Coherent M-ary FSK modulator.

The coherent M-ary FSK demodulator falls in the general form of detector for M-

ary equiprobable, equal-energy signals with known phases. The demodulator consists of a bank of M correlators or matched filters (Figure 3.10 and Figure 3.11). At sample times t = kT, the receiver makes decisions based on the largest output of the correlators or matched filters. It is worth noting that the coherent M-ary FSK receivers in Figures 3.10 and 3.11 only require that the M-ary FSK signals be equiprobable, equal energy, and do not require them to be orthogonal.

S/P

Converter

1b 2

b . . .nb

Control lines

Binary input data

27

Figure 3.10 Coherent M-ary FSK demodulator: correlator implementation.

Figure 3.11 Coherent M-ary FSK demodulator: matched filter implementation.

)(2cos 1 tTf −π

)(2cos 2 tTf −π

)(2cos tTfM −π

Sample t = bT

Sample at t = bT

Sample at t = bT

.

.

.

If il > jl ij ≠∀

choose im

im

1l

2l

Ml

.

.

.

Received

signal, r(t)

)2cos( tfMπ

Received signal, r(t)

dtbT

∫0

dtbT

∫0

dtbT

∫0

)2cos( 1tfπ

)2cos( 2tfπ . . .

.

.

.

1l

If il > jl

ij

2l

im ∀ ≠ choose im

.

.

.

Ml

28

The exact expression for the symbol error probability for symmetrical signal set, equal energy and equiprobable, is given as

P( ) ( )[ ] dxxQ

NEx Ms 12

0 12

/2exp

211 −

∞

∞−

−⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −−− ∫ π

= (3.12) [3] s

This expression does not require the signal set to be orthogonal, and cannot be analytically evaluated. If the signal set is equal-energy and orthogonal (not necessarily equiprobable), all distances between any two signals are equal. The distance d = sE2 , and the upper bound obtained from equation (3.12) is

⎟⎟⎠

⎞⎜⎜⎝

⎛−≤

0

)1(NE

QM ssP (3.13)

where = , and Q(z) is the Q-function defined in chapter 3.2.2. sE ME b 2log

/NFor fixed M this bound becomes increasingly tight as Es 0 is increased. Infact, it becomes a good approximation for P ≤ 10-3. sFor equally likely orthogonal M-ary signals, all symbol errors are equiprobable. That is, the demodulator may choose any one of the )1( −M erroneous orthogonal signals with equal probability. Hence it can be shown that the average bit error probability is given by

sn

n

P12

2 1

−

−

s

n

PM 12 1

−

−

=bP = , where n = log M. 2

) and symbol error probability (PBit error probability (Pb s) for coherently demodulated,

equal-energy, equiprobable, and orthogonal M-ary FSK signals are shown in Figures 3.12 and 3.13.

29

0 5 10 1510-6

10-5

10-4

10-3

10-2

10-1

Eb/No (dB)

Bit-

erro

r pro

babi

lity,

Pb

Bit-error Probability of Coherent M-ary FSK

BFSK4-FSK8-FSK16-FSK32-FSK64-FSK

Figure 3.12 Bit error probability of coherently demodulated M-ary FSK.

0 5 10 1510-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Eb/No (dB)

Sym

bol-e

rror p

roba

bilit

y, P

s

Symbol-error Probability of Coherent M-ary FSK

BFSK4-FSK8-FSK16-FSK32-FSK64-FSK

Figure 3.13 Symbol error probability of coherently demodulated M-ary FSK.

30

/NIt can be seen from Figures 3.12 and 3.13 that for the same Eb 0, error probability reduces when M increases, or for the same error probability, the required E /Nb 0 decrease as M increases. However, the speed of decrease in E /Nb 0 slows down when M gets larger. The noncoherent modulator for binary FSK in Figure 3.2 can also be easily extended to noncoherent M-ary FSK by simply increasing the number of independent oscillators to M (Figure 3.14).

Multiplexer

Oscillator 1 )2cos( 11

11,φf

22 ,

π φ+tfA

Figure 3.14 Noncoherent M-ary FSK modulator

The noncoherent demodulator for M-ary FSK falls in the general form of detector for M-ary equiprobable, equal-energy signals with unknown phases as described in many communication books. The demodulator can be implemented in correlator-squarer form, or matched filter-squarer or matched filter-envelope detector form (Figures 3.15, 3.16 and 3.17).

Oscillator 2 )2cos( 12

φf φπ +tfA

Oscillator M )2cos( MM tfA φπ +

.

.

.

.

.

.

MMf φ,

S/P

Converter

1b 2b . . .nb

Control lines

Binary input data

iif φ,

31

dtbT

∫0

dtbT

∫0

Squarer

Squarer

∑)2cos( tfMπ

)2sin( tfMπ

2Ml

)2sin( 1tfπ

dtbT

∫0

dtbT

∫0

Squarer

Squarer

∑)2cos( 1tfπ

21l

.

.

.

.

.

.

.

.

.

If 2

il > 2jl , ij ≠∀

choose im

im

dtbT

∫0

dtbT

∫0

Squarer

Squarer

∑)2cos( 2tfπ

22l

)2sin( 2tfπ

r(t)

Figure 3.15 Noncoherent M-ary FSK demodulator: Correlator-squarer implementation.

32

Sample at t = bT

)(2sin tTf bM −π

.

.

.

.

.

.

.

.

.

If 2

il > 2jl ,

ij ≠∀ choose im

im r(t)

)(2cos 1 tTf b −π

)(2sin 1 tTf b −π

Squarer

Squarer

∑2

1l Sample at t = bT

)(2cos tTf bM −π Squarer

Squarer

∑2

Ml

Sample at t = bT

Sample at t = bT

)(2cos 2 tTf b −π

)(2sin 2 tTf b −π

Squarer

Squarer

∑2

2l

Sample at t = bT

Sample at t = bT

Figure 3.16 Noncoherent M-ary FSK demodulator: matched filter-squarer implementation.

33

If il > jl ij ≠∀

choose im im

)(2cos 1 tTf −π

Envelope Detector

Sample at t = bT

1l

)(2cos 2 tTf −π

Envelope Detector

Sample at t = bT

2l

)(2cos tTfM −π

Envelope Detector

Sample at t = bT

.

.

.

.

.

.

.

.

.

r(t)

Ml

Figure 3.17 Noncoherent M-ary FSK demodulator: matched filter-envelope detector implementation. The expression for symbol error probability of noncoherently demodulated, equiprobable, equal-energy, and orthogonal M-ary FSK is given as

⎥⎦

⎤⎢⎣

⎡+

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−∑−

=

+

0

1

1

1

)1(exp

11

)1(Nk

kEk

Mk

sM

k

k

sP = (3.14) [4]

where =⎟⎟⎠

⎞⎜⎜⎝

⎛ −k

M 1!)!1(

)!1(kkM

M−−− , is the binomial co-efficient.

The first term of the summation in equation (3.14) provides an upper bound as

⎥⎦

⎤⎢⎣

⎡−

−

02exp

21

NEM s ≤sP (3.15)

as E /NFor fixed M this bound becomes increasingly close to the actual value of Ps s 0 is increased. Figures 3.18 and 3.19 show bit and symbol error probabilities for noncoherently demodulated, equiprobable, equal-energy, and orthogonal M-ary FSK. The behavior of the curves with values of M is similar to that of the coherent case.

34

0 2 4 6 8 10 12 14 1610-6

10-5

10-4

10-3

10-2

10-1

Eb/No (dB)

Bit-

erro

r pro

babi

lity,

Pb

Bit-error Probability of Noncoherent M-ary FSK

BFSK4-FSK8-FSK16-FSK32-FSK64-FSK

Figure 3.18 Bit error probability of noncoherently demodulated M-ary FSK signals.

0 2 4 6 8 10 12 14 1610-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Eb/No (dB)

Sym

bol-e

rror p

roba

bilit

y, P

s

Symbol-error Probability of Noncoherent M-ary FSK

BFSK4-FSK8-FSK16-FSK32-FSK64-FSK

Figure 3.19 Symbol error probability of noncoherently demodulated M-ary FSK signals.

35

3.3.2 Coherent Versus Noncoherent Advantages and disadvantages exist for both coherent and noncoherent techniques. The most important advantage of noncoherent demodulation is that it is simple and thus requires a simple receiver, as opposed to coherent demodulation where the receiver is relatively complex. Coherent demodulation leads to better symbol error performance in the presence of AWGN.

Tfff 2/121 =−=ΔFor coherent demodulation, the minimum frequency separation is necessary to ensure orthogonality of signals over a signaling interval of length T. If noncoherent demodulation is used (such as envelope or square-law detection of FSK signals), the minimum frequency separation required for orthogonality of the signals is

in the presence of AWGN. This separation is twice as large as that required for coherent detection. This accounts for the performance degradation when noncoherent demodulation is used instead of coherent demodulation. For example, at BER = 10

Tf /1=

-4, coherent M-ary FSK is about 1 dB better than noncoherent M-ary FSK in terms of E /Nb 0 (Figures 3.20 – 3.24).

0 5 10 1510-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Bit-

erro

r pro

babi

lity,

Pb

Cohernt vrs Noncoherent Bit-error Probability for BFSK

Coherent BFSKNoncoherent BFSK

Figure 3.20 Comparison of BER for coherent and noncoherent BFSK.

36

0 5 10 1510

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Bit-

erro

r pro

babi

lity,

Pb

Cohernt vrs Noncoherent Bit-error Probability for 4FSK

Coherent 4-FSKNoncoherent 4-FSK

Figure 3.21 Comparison of BER for coherent and noncoherent 4-FSK.

0 5 10 1510-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Bit-

erro

r pro

babi

lity,

Pb

Cohernt vrs Noncoherent Bit-error Probability for 8FSK

Coherent 8-FSKNoncoherent 8-FSK

Figure 3.22 Comparison of BER for coherent and noncoherent 8-FSK.

37

3.3.3 Power Spectral Density The most important parameter of FSK modulation is not the frequency shift itself, but rather the relationship of the frequency shift with symbol rate Rs (or equivalently the

symbol period, T). The modulation index is defined as fTR

fhs

Δ=Δ

= 22 [5], where

is the separation between two adjacent frequencies. The modulation index is so important because it determines the ease of demodulation of the scheme, and its spectral characteristics.

fΔ2

In an M-ary FSK modulation, the binary data stream is divided into n-tuples of bits. All the M possible n-tuples are denoted as M messages; = 1, 2, …,

M. There are M signals with different frequencies to represent these M messages, and the expression for the signal is

imi ,Mn 2log=

),2cos()( iii tfAts φπ += ,0 Tt ≤≤ith for , where T is the symbol period which is n times the bit period. If the initial phases are the same for all i, the signal set is coherent. The initial phase is most of the time assumed to be zero for coherent M-ary FSK.

im

The derivation of the power spectral density of M-ary FSK scheme is very complicated, and not the purpose of this thesis work, thus I’ll just quote the expression. The power spectral density (PSD) expression of the complex envelope of M-ary FSK signal, for equiprobable messages, is given as

∑ ∑= = ⎥

⎥⎦

⎤

⎢⎢⎣

⎡+=

M

i

M

j j

j

i

iij

i

is A

MMTA

1 12

22

~sinsin1

2sin

γγ

γγ

γγψ (3.16) [3]

where; A is the signal amplitude and all signals have equal energies,

,)2cos(21

)2cos()cos(2 fTCC

fTCA

aa

jiajiij π

πγγγγ

−+

−+−+=

( ) MihmfT ii ,...,2,1,2/ =−= πγ

([ ],12cos2 2/

1∑=

−=M

ia ih

MC π ) and

MiMimi ,...,2,1),1(2 =+−=

Plots of equation (3.16) for various values of h for M = 2, 4, and 8 are shown in Figures 3.23 – 3.28.

38

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized frequency, (f-fc)T

Nor

mal

ized

pow

er s

pect

ral d

ensi

ty

PSD for BFSK

h = 0..5h = 0.6h = 0.7

Figure 3.23 Power-density spectrum of BFSK signal (for h = 0.5, 0.6, 0.7).

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized frequency, (f-fc)T

Nor

mal

ized

pow

er s

pect

ral d

ensi

ty

PSD for BFSK

h = 0.80h =0.90h = 0.95

Figure 3.24 Power-density spectrum of BFSK signal (for h = 0.8, 0.9, 0.95).

39

-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized frequency, (f-fc)T

Nor

mal

ized

pow

er s

pect

ral d

ensi

ty

PSD for 4-FSK

h = 0.20h = 0.35h = 0.40

Figure 3.25 Power-density spectrum of 4-FSK signal (for h = 0.2, 0.35, and 0.4).

-1.5 -1 -0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Normalized frequency, (f-fc)T

Nor

mal

ized

pow

er s

pect

ral d

ensi

ty

PSD for 4-FSK

h = 0.5h = 0.6h = 0.7

Figure 3.26 Power-density spectrum of 4-FSK signal (for h = 0.5, 0.6, and 0.7).

40

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized frequency, (f-fc)T

Nor

mal

ized

pow

er s

pect

ral d

ensi

ty

PSD for 8-FSK

h = 0.125h = 0.2h = 0.3

Figure 3.27 Power-density spectrum of 8-FSK signal (for h = 0.125, 0.2, and 0.3).

-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

Normalized frequency, (f-fc)T

Nor

mal

ized

pow

er s

pect

ral d

ensi

ty

PSD for 8-FSK

h = 0.4h = 0.5h = 0.6

Figure 3.28 Power-density spectrum of 8-FSK signal (for h = 0.4, 0.5, and 0.6).

41

Curves are presented for various values of h to show how the spectral shape changes with h. For small values of h, the spectra are narrow and decrease smoothly towards zero. As h increases towards unity, the spectrum widens and spectral power is increasingly concentrated around -0.5 ≤ (f – fc) ≤ 0.5 and its odd multiples. These are the frequencies of the M signals in the scheme. For coherent orthogonal case, h = 0.5 (Figure 3.29), most spectral components are in a bandwidth of M/2T. Thus the transmission bandwidth is set as BT = M/2T. Similarly, for noncoherent orthogonal case, h = 1, and then BT = M/T.

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized frequency, (f-fc)T

Nor

mal

ized

pow

er s

pect

ral d

ensi

ty

Comparison of PSD of M-ary FSK at h = 0.5

BFSK4-FSK8-FSK

Figure 3.29 Power spectra of M-ary FSK signals for M = 2, 4, and 8 (h = 0.5). 3.3.4 Bandwidth Efficiency Channel bandwidth and transmit power constitute two very important communication resources, efficient utilization of which provides the motivation for the search of efficient schemes. The goal of any efficient scheme is achieve efficiency in bandwidth at a minimum practical expenditure of average transmit power or, equivalently a channel perturbed by AWGN, expenditure of average signal-to-noise ratio. The bandwidth efficiency is defined as the number of bits per second that can be transmitted in one Hertz of system bandwidth. With the data rate denoted by R and the channel bandwidth by Bb T, we may express bandwidth efficiency, ρ , as

T

b

BR

=ρ bits/sec/Hz

42

The data rate is well defined. Unfortunately, however, there is no universal satisfying definition for the bandwidth BT. For modulation schemes that have power density spectral nulls, defining the bandwidth as the width of the main spectral lobe is a convenient way of bandwidth definition. If the spectrum of the modulated signal does not have spectral nulls, as in general continuous phase modulation, null-to-null bandwidth no longer exists. In this case energy percentage bandwidth may be used. Usually 99 % is used, even though other percentages (e.g., 90 %, 95 %) are also used. The bandwidth efficiency of coherently demodulated M-ary FSK signal that consists of an orthogonal set of M frequency-shifted signals is given by;

)2/( TMRb=ρ (3.17)

where Rb is the data rate in bits per second, and T is the symbol period which is n times the bit period. Equation (3.17) can be re-written as

MM2log2

=ρ (3.18)

ρ calculated from equation (3.18) for M = 2, 4, 8, and 16. Table 3.1 gives the values of

Table 3.1 Bandwidth efficiency of coherent M-ary FSK signals

M 2 4 8 16 ρ 1.0 1.0 0.75 0.50

(bits/s/Hz)

43

Chapter 4 Simulations and Results This chapter presents the simulation of 2, 4, and 8-level FSK systems in AGWN channel and the results obtained. 4.1 Simulation of Additive White Gaussian Noise (AWGN) Channel From probability theory it is known that a Rayleigh distributed random variable R, with probability distribution function

⎩⎨⎧

≥−<

= − 0,10,0

)( 22 2/ ReR

RF R σ

is related to a pair of Gaussian random variables X and X through the transformation 1 2

θcos1 RX = (4.1) θsin2 RX = (4.2)

where θ is a uniformly distributed variable in the interval )2,0( π , and the parameter 2σ and X . is the variance of X1 2

Now, generating a Rayleigh distributed random variable with the computer, we have (4.3) MeRF R =−= − 22 2/1)( σ

where M is a uniformly distributed random variable in the interval )2,0( π . Solving equation (4.3) results in

⎟⎠⎞

⎜⎝⎛−

=M

R1

1ln2σ (4.4)

If we generate a second uniformly distributed random variable N, and define Nπθ 2= , then from equations (4.1), (4.2) and (4.4) we obtain two independent Gaussian distributed random variables X and X as 1 2

)2cos(1

1ln21 NM

X πσ ×⎟⎠⎞

⎜⎝⎛−

= and

)2sin(1

1ln22 NM

X πσ ×⎟⎠⎞

⎜⎝⎛−

= .

44

4.2 Simulation of Binary FSK System in AWGN Channel In this section, simulations in additive white Gaussian channel and results for both coherent and noncoherent (square-law detection) binary FSK systems are presented. 4.2.1 Coherent System A model for the simulation of coherent binary FSK system in AWGN channel is shown in Figure 4.1.

Gaussian RNG

Uniform RNG

Figure 4.1 Simulation model for coherent binary FSK system. Since the signals are orthogonal, when a 0 (signal s1(t)) is transmitted, the correlator outputs are 00 nEr b += , and 11 nr = . When a 1 (signal s2(t)) is transmitted, the

correlator outputs are , and 00 nr = 11 nEr b += . Figure 4.2 shows the results of the simulation for the transmission of 20,000 bits at several different values of Eb/N0 and how it compares with theory.

Binary data source

Gaussian RNG

0r

1r

Detector Output data

Compare

0n

1n

Error Counter

45

0 5 10 1510-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No in dB

Bit-

erro

r pro

babi

lity,

Pb

Simulated bit-error rateTheoretical bit-error rate

Figure 4.2 Performance of simulated coherent binary FSK system. 4.2.2 Noncoherent (square-law detection) System A model for the simulation of noncoherent (square-law detection) binary FSK system in AWGN channel is shown in Figure 4.3. Since the signals are orthogonal, when s1(t) is transmitted, the first demodulator output is

IbI nEr 11 cos += φ

QbQ nEr 11 sin += φ and the second demodulator output is II nr 22 = QQ nr 22 =

where are mutually statistically independent zero-mean Guassian random

variables with variance and QIQI nnnn 2211 ,,,

2σ φ represents the channel-phase shift. The square-law detector computes , and selects the information bit corresponding to the larger of these two decision variables.

21

211 qI rrr += 2

22

22 QI rrr +=

46

Uniform RNG

Gaussian RNG

Gaussian RNG

Figure 4.3 Simulation model for noncoherent binary FSK system. Figure 4.4 shows the results of the simulation for the transmission of 20,000 bits at several different values of Eb/N0 and how it compares with theory.

Error counter

Gaussian RNG

Gaussian RNG

FSK signal

selector

Detector

( )2

( )2 1r

( )2

( )2 2r

Compare

Output data

47

0 5 10 1510-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No in dB

Bit-

erro

r pro

babi

lity,

Pb

Simulated bit-error rateTheoretical bit-error rate

Figure 4.4 Performance of simulated noncoherent binary FSK system. 4.3 Simulation of 4 and 8-level FSK Systems in AWGN Channel A model for the simulation of coherent 4-level FSK system in AWGN channel is shown in Figure 4.5. The block diagram of the simulation of noncoherent 4-level FSK system in AWGN channel is similar to that in Figure 4.3, the only difference being the number of correlator (demodulator) outputs. The block diagram of the simulation of coherent 8-level FSK is similar to that depicted in Figure 4.5, whiles the model for simulation of noncoherent 8-level FSK systems is similar to that shown in Figure 4.3.

48

Gaussian RNG

Figure 4.5 Simulation model for coherent 4-level FSK system. Figures 4.6 - 4.9 illustrates the results of the simulations for the transmission of 20,000 symbols at several different values of Eb/N0 and how it compares with theory.

Uniform RNG

Mapping to signal points

Gaussian RNG

Gaussian RNG

Gaussian RNG

Detector Output data

sE 0r

1r

2r

3r

Compare

Error counter

49

0 2 4 6 8 10 1210-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Pro

babi

lity

of e

rror

Simulated bit-error rateSimulated symbol-error rateTheoretical bit-error rate

Figure 4.6 Performance of simulated coherent 4-level FSK system.

0 2 4 6 8 10 1210-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Pro

babi

lity

od e

rror

Simulated bit-error rateSimulated symbol-error rateTheoretical bit-error rate

Figure 4.7 Performance of simulated noncoherent 4-level FSK system.

50

0 1 2 3 4 5 6 7 8 9 1010-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Pro

babi

lity

of e

rror

Simulated bit-error rateSimulated symbol-error rateTheoretical bit-error rate

Figure 4.8 Performance of simulated coherent 8-level FSK system

0 1 2 3 4 5 6 7 8 9 1010-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

Pro

babi

lity

or e

rror

Simulated bit-error rateSimulated symbol-error rateTheoretical bit-error rate

Figure 4.9 Performance of simulated noncoherent 8-level FSK system.

51

Chapter 5 Measuring of Performance of Commercially Available Radios Modems by RACOM This chapter describes the measurements performed on commercially available data transmission equipment (manufactured by RACOM s.r.o. of the Czech Republic) that uses GMSK and 4-level FSK modulation schemes, the results obtained and an analysis of the results. A brief description of RACOM s.r.o is also given. 5.1 RACOM s.r.o. RACOM s.r.o., is a company situated in a small town, Nove Mesto na-Morave, in the Czech Republic. It developes and manufactures devices suitable for data networks, and is aimed at narrowband radio transmission of data in frequency bands from 140 – 900 MHz. MORSE (MOdems for Radio-based SystEms) is the name given by RACOM to the telecommunication systems it manufactures. MORSE is a packet communication system designed for data transfer on narrowband radio channels, and is possible to integrate other transfer medium (IP, GPRS, etc) into a MORSE network. RACOM is one of several world manufacturers of narrowband radios. MORSE is deployed in about 35 countries including both advanced and developing countries. Germany, Norway, Sweden, and the United States of America are some of the advanced countries where MORSE networks are deployed. Ghana, Nigeria, Sudan, and Latvia are examples of developing nations where MORSE networks are in use. Typical applications for data transmission using equipment manufactured by RACOM are data transmission in technological process control, transaction networks, security systems, mobile tracking and fleet management, and et cetera. The MORSE system is also suitable for building extensive networks in which the data transmission services are provided to end users. More information on RACOM s.r.o. can be found at www.racom.cz. 5.2 Measurements Measurements were made on two of RACOM’s products namely MR400 and MX160. The MR400 radio modem (picture shown in Figure 5.1) uses 4-level FSK modulation which is always filtered using root-raised cosine filters (α = 0.2). While the MX160 radio modem (picture shown in Figure 5.2) uses GMSK modulation ( Tβ = 0.4). The detailed technical parameters of MR400 and MX160, relevant to this thesis work, are shown in Table 5.1. More technical information can be found on RACOM’s website. MR400 radio modem is in deployment in many countries already. For example, it is used in Porto, Portugal for fleet management. However, MX160 is a new addition to RACOM’s products, which is to be deployed in Norway for a Telenor project.

52

Figure 5.1 MR400 radio modem.

Figure 5.2 MX160 radio modem.

Table 5.1 Technical parameters for MR400 and MX160 radio modems Radio Modem

Technical Parameters MX160 MR400 Frequency range 136 – 180 MHz 380 – 470 MHz Modulation scheme used GMSK 4-leve FSK Transmit power 0.1 – 25 W 0.1 – 5 W

200 kHz 16 kHz Channel bandwidth, 0BChannel spacing 25 kHz 25 kHz Data rate 140 kbps 21.68 kbps Frequency deviation 40 kHz 4.7 kHz

TβFilters used Root-raised cosine (Gaussian ( = 0.4) α = 0.2) Receiver sensitivity at BER = 10

> -107 dBm > -107 dBm -3

In transmit mode, bits/symbols are passed through the filters, named in table 5.1, to eliminate the high frequency components which would otherwise cause interference into adjacent radio channels. In the receive mode, the filters are used to reject high frequency noise and to equalize the received signal to a form suitable for extracting the bits/symbols. The measurements were made by setting up a radio connection between a pair of radio modems through a variable signal attenuator. The received signal strength (RSS) at

53

the receiver was then varied to several values. Random data of different lengths were then sent at different RSS levels and the packet error rate (PER) recorded. The packet error rate is the probability that a packet is discarded either due to synchronization errors or due to a bit or bits in error. The received signal strength at the receiver were measured with an oscilloscope, while the other measurements (PER, random data length) were done using a proprietary software by RACOM called ‘Setr’. The random data were also generated using ‘Setr’. The unprocessed results of measurements on MR400 and MX160 radio modems are shown in Tables 5.2 and 5.3 respectively. Table 5.2 Raw data from measurements on MR400 radio modem

User data length in bytes ( Random data) 0 50 200 1000

RSS (dBm) PER PER PER PER -110 1/3.1 1/2.4 1/1.4 - -109 1/7.3 1/3.5 1/1.7 - -108 1/19.3 1/9.0 1/3.2 - -107 1/160.4 1/81.7 1/32.3 1/6.6 -105 1/409.5 1/322.0 1/93.1 1/21.0 -103 1/829.0 1/459.3 1/396.0 1/231.5

Table 5.3 Raw data from measurements on MX160 radio modem

User data length in bytes (Random data) 0 50 200 1000

RSS (dBm) PER PER PER PER -107 1/3.5 1/1.7 1/1.1 - -106 1/12.4 1/5.5 1/2.4 1/1.1 -104 1/69.8 1/27.6 1/7.8 1/2.1 -103 1/160.0 1/112.2 1/54.4 1/13.8 -102 1/532.2 1/343.0 1/235.6 1/68.1 -101 1/551.3 1/382.1 1/277.3 1/272.3

5.2.1 Extraction of Bit Error Rate (BER) form measured Packet Error Rate (PER) By definition, the packet error rate (PER) from the measurements can be stated mathematically as

)(RSSPER = )()( RSSPERRSSPER BERSYNC + (5.1) where PER(RSS) is the total packet error rate obtained from measurements for a particular RSS, is the packet error rate due to synchronization error, and is )(RSSPERSYNC

54

assumed to be constant at a given RSS, and is the packet error rate due to a bit or bits in error at a given RSS.

)(RSSPERBER

Now, a packet without synchronization errors will not be discarded if all bits in the packet are correctly received. Thus, we can write mathematically that

( )lBER−1 ,)(1 RSSPERBER− = and hence

( )lBER−− 11)(RSSPERBER = (5.2) where l is the number of bits per packet, and BER is the bit error rate at a given RSS. Substituting equation (5.2) into equation (5.1) yields

( )lSYNC BERRSSPER −−+ 11)()(RSSPER = (5.3) Equation (5.3) shows that BER and can be obtained from the measured values of and by graphical methods.

)(RSSPERSYNC

PER l 5.2.2 Calculation of Noise Power The total noise at the receiver input is equal to white noise (thermal noise) at the receiver input plus the noise introduced by the receiver itself (noise figure of the receiver) modeled as input to the receiver. Mathematically, the total noise at the receiver input is given by

N = Thermal noise + Noise figure of receiver (5.4) The receivers used by RACOM has noise figure of 8 dB, and the thermal noise is given by kT , where = 1.38 x 10-23k J/K is the Boltzmann’s constant and T is the temperature in Kelvin. Thus, the total noise power

N = kT + 8 dB. (5.5) At room temperature,

kT = Watts/Hz 231038.1300 −×× = ) dBm/Hz 323

10 101038.1300(log10 ××× −

= dBm/Hz 174− Therefore, N = dBm/Hz + 8 dB 174− N = dBm/Hz (5.6) 166−

55

5.2.3 Processing of Measurement Data on MR400 Radio Modem The processed data of measurements on MR400 radio modem is shown in Table 5.4. Table 5.4 Processed data from measurements on MR400 radio modem

RSS (dBm) S/N (dB) BER -110 12 5.053 × 10-4

-4-109 13 3.53 × 10-108 14 1.873 × 10-4

-107 15 5.955 × 10-5

-6-105 17 5.86 × 10-103 19 3.714 × 10-7

Sample calculation on how to arrive at Table 5.4 from Table 5.2; At RSS = dBm, a graph of versus (packet length in bytes) obtained using data from Table 5.2 is shown in Figure 5.3.

PER l108−

PER vrs Packet length at RSS of -108 dBm

y = 0.0013x + 0.0176

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 50 100 150 200 250

Packet length (bytes)

PER

Figure 5.3 PER versus packet length at -108 dBm for MR400 radio modem.

56

Each packet in the MORSE communication system is made up of;

i. Variable length user data, ii. 4 bytes cyclic redundancy check (CRC),

iii. 12 bytes header, iv. 2 bytes header CRC, v. 6 bytes frame synchronizer.

Thus, the total length of each packet is equal to 24 bytes plus the length of user data. For example, a user data of length 0 bytes results in a packet length of 24 bytes. From equation (5.3);

( )lSYNC BERRSSPER −−+ 11)()(RSSPER = , which means that is the intercept on the PER axis (assuming a linear relation) of Figure 5.3. Hence we can write

)108( dBmPERSYNC −

)108( dBmPER − = , from which an expression for BER is obtained as

( lBER−−+ 110176.0 )

BER = (5.7) {( ldBmPER /)108(0176.1log10101 −−− } )

Now, BER can be calculated using equation (5.7) and data from Table 5.2;

-4At = 24 bytes, = 1/19.3, and BER = 1.813 × 10)108( dBmPER −l . At = 74 bytes, = 1/9.0, and BER = 1.658 × 10-4)108( dBmPER −l .

-4At = 224 bytes, = 1/3.2, and BER = 1.95 × 10)108( dBmPER −l .

( )( )2247424

1095.122410658.17410813.124 444

++××+××+×× −−−

Average (weighted) BER =

= 410873.1 −× Signal-to-noise ratio (S/N in dB) calculation; From chapter 5.2.2, it was deduced that the total noise power at the input of the receiver of RACOM modems is given by;

N = dBm/Hz. 166− The receiver used for reception of MR400 transmissions has a bandwidth of 22 kHz. Hence the total noise power at the receiver input of the MR400 radio modem is 122− dBm, and signal-to-noise ratio is given as

NS

dBmdBm

122108

−−= = 14 dB.

57

The BER curve for MR400 radio modem is shown in Figure 5.4.

0 2 4 6 8 10 12 14 16 18 2010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

S/N in dB

Bit-

erro

r pro

babi

lity

(BE

R)

BER curve for MR400 radio modem

Figure 5.4 Performance curve of MR400 radio modem.

5.2.4 Processing of Measurement data on MX160 Radio Modem The processed data of measurements on MX160 radio modem is shown in Table 5.5. Table 5.5 Processed data from measurements on MX160 radio modem

RSS (dBm) S/N (dB) BER -4-107 5 5.518 × 10-4-106 6 2.756 × 10-5-104 8 4.313 × 10

-6-103 9 8.56 × 10-102 10 1.579 × 10-6

-101 11 2.266 × 10-7

Sample calculation o how to arrive at Table 5.5 from Table 5.3; At RSS = dBm, a graph of versus (packet length in bytes) obtained using data from Table 5.3 is shown in Figure 5.5.

PER l103−

58

PER versus Packet length at RSS of -103 dBm

y = 7E-05x + 0.0041

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 200 400 600 800 1000 1200

Packet length (bytes)

PER

Figure 5.5 PER versus packet length at -104 dBm for MX16000 radio modem.

)103( dBmPERSYNC −From Figure 5.5, it is seen that = 0.0041 and the expression for BER can be written as

BER = (5.8) {( ldBmPER /)104(0041.1log10101 −−− } )

Now, BER can be calculated using equation (5.8) and data from Table 5.3; At = 24 bytes, = 1/160, and BER = 1.121 × 10-5)103( dBmPER −l . At = 74 bytes, = 1/112.2, and BER = 8.1491 × 10-6)103( dBmPER −l . At = 224 bytes, = 1/54.4, and BER = 8.0275 × 10-6)103( dBmPER −l . At = 1024 bytes, = 1/13.8, and BER = 8.644 × 10-6)103( dBmPER −l . Average (weighted) BER ( )

( )1024224742410644.81024100275.8224101491.87410121.124 6665

+++××+××+××+×× −−−−

=

= 61056.8 −×

59

Signal-to-noise ratio calculation; From chapter 5.2.2, it was deduced that the total noise power at the input of the receiver of RACOM modems is given by;

N = dBm/Hz. 166− The receiver used for reception of MX160 transmissions has a bandwidth of 230 kHz. Hence the total noise power at the receiver input of the MX160 radio modem is 112− dBm, and signal-to-noise ratio is given as

NS

dBmdBm

112103

−−= = 9 dB.

The BER curve for MX160 radio modem is shown in Figure 5.6.

0 2 4 6 8 10 12 14 16 18 2010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

S/N in dB

Bit-

erro

r pro

babi

lity

(BE

R)

BER curve for MX160 radio modem

Figure 5.6 Performance curve of MX160 radio modem.

A comparison of the energy performance of MR400 and MX160 radio modems is shown in Figure 5.7. It seen from the graph that at the same BER, MX160 radio modem is typically 7 dB better than the MR400 radio modem in terms of signal-to-noise ratio required. The difference can be explained using Shannon’s capacity theorem, which

60

states that there is always a compromise between energy and bandwidth efficiency. MX160 radio modem uses GMSK (βT = 0.4) modulation scheme and has a bandwidth efficiency of 0.70, whiles MR400 radio modem uses 4-level FSK modulation scheme and has a bandwidth efficiency of 1.355. Thus MR400 radio modem trades energy efficiency for bandwidth efficiency while the opposite is true for MX160 radio modem.

0 2 4 6 8 10 12 14 16 18 2010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

S/N in dB

Bit-

erro

r pro

babi

lity

(BE

R)

Comparison of BER curves for MR400 and MX160 radio modems

MR400 (4-FSK,alpha = 0.2)MX160 (GMSK, BT = 0.4)

Figure 5.7 Comparison of performance of MR400 and MX160 radio modems. It can also be seen from Figure 5.7 that the curves start at BER of 10-3. This is because it was very difficult getting any packet through at BER of 10-2. Hence it can be said that for any reliable communication, the BER must be 10-3 or less.

61

Chapter 6 Comparison of Performance of 2, 4, and 8-level FSK Systems. 6.1 Bounds on Communication. The celebrated Shannon’s formula for the capacity of an additive white Gaussian noise channel is

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

BNPBC0

2 1log (6.1)[4]

where, C is the channel capacity in bits per second, B is the channel bandwidth in Hz, P is the signal power in watts, and N is the noise power spectrum. 0From equation (6.1) it can be seen that the basic factors that determine the channel capacity are the channel bandwidth B, the noise power spectrum N0 and the signal power P. There exists a trade-off between P and B in the sense that one can compensate for the other. Increasing the input signal power obviously increases the channel capacity. However, the increase in capacity as a function of power is logarithmic and slow. This fact not withstanding, the capacity of the channel can be increased to any value by increasing the input power. The effect of channel bandwidth, however, is quite different. Increasing bandwidth B has two effects. On one hand, a higher bandwidth channel can transmit more samples per second and, therefore, increase the transmission rate. On the other hand, a higher bandwidth means higher input noise to the receiver and this reduces its performance. The effect of increasing bandwidth is seen when we let bandwidth B in equation (6.1) tend to infinity. Using L’Hospital’s rule we obtain

02

0

44.1loglimNPe

NPC == .

This means that, contrary to the power case, by increasing the bandwidth alone cannot increase the capacity to any desired value. In any practical communication system, the bit rate, Rb is always less than C. In AWGN channel we have

⎟⎟⎠

⎞⎜⎜⎝

⎛+<

BNPBRb0

2 1log (6.2)

bbb

b RETEP ==Now, power is defined as energy per unit time, so we can write , and

equation (6.2) becomes

⎟⎟⎠

⎞⎜⎜⎝

⎛+<

BNREBR bb

b0

2 1log

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

BR

NE

BR bbb

02 1log

62

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

02 1log

NEbηη (6.3)

where η is the bandwidth efficiency defined as R /B bits per second per Hertz. b

Equivalently, equation (6.3) can be written as

η

η 12

0

−>

NEb (6.4)

The plot of equation (6.4) is shown in Figure 6.1.

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

Bandwidth Efficiency (Bits/sec/Hz)

Ene

rgy

Effi

cien

cy, E

b/N

o (d

B)

Shannon Capacity Limit Curve

Figure 6.1 Energy efficiency versus spectral efficiency of an optimum system.

η

η 12

0

−=

NEbThe curve of Figure 6.1, defined by divides the plane into two regions. In

the region above the curve, reliable communication is possible and in the region below the curve, reliable communication is not possible. The performance of any communication system can be denoted by a point in this plane and the closer the point is to this curve, the closer is the performance of the system to an optimal system. From

equation (6.4), it is seen that (as 6.1693.02ln0

−≈=>NEbη tends to zero), dB is the

absolute minimum for reliable communication. In Figure 6.1, when 1<<η , we are dealing with a case where bandwidth is large (relative to Rb), and the main concern is limitation on power. This case is usually referred to as power-limited case. Signaling schemes, with high dimensionality, such as orthogonal, biorthogonal, and simplex are frequently used in these cases. The case where 1>>η , happens when the bandwidth of

63

the channel is small (relative to Rb), and therefore is referred to as the bandwidth-limited case. Low dimensional signaling schemes with crowed constellations, for example, 256-QAM (Quadrature amplitude modulation), are implemented in these case. 6.2 Comparison of Performance of the Simulated Systems. Comparison of simulated bit and symbol error probabilities for coherent and noncoherent 2, 4, and 8-level FSK systems is shown in Figures 6.2 – 6.5.

0 5 10 15

10-4

10-3

10-2

10-1

100

Eb/No in dB

Sim

ulat

ed s

ymbo

l-erro

r pro

babi

litie

s, P

s

Comparison of simulated Ps for coherent FSK

BFSK4-FSK8-FSK

Figure 6.2 Comparison of simulated symbol error probabilities for coherent M-ary FSK.

64

0 5 10 15

10-4

10-3

10-2

10-1

100

Eb/No in dB

Sim

ulat

ed b

it-er

ror p

roba

bilit

ies,

Pb

Comparison of simulated Pb for coherent M-ary FSK

BFSK4-FSK8-FSK

Figure 6.3 Comparison of simulated bit error probabilities for coherent M-ary FSK.

0 5 10 15

10-4

10-3

10-2

10-1

100

Eb/No in dB

Sim

ulat

ed s

ymbo

l-erro

r pro

babi

litie

s, P

s

Comparison of simulated Ps for noncoherent M-ary FSK

BFSK4-FSK8-FSK

Fig 6.4 Comparison of simulated symbol error probabilities for noncoherent M-ary FSK.

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0 5 10 15

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10-3

10-2

10-1

100

Eb/No in dB

Sim

ulat

ed b

it-er

ror p

roba

bilit

y, P

b

Comparison of simulated Pb for noncohrent M-ary FSK

BFSK4-FSK8-FSK

Figure 6.5 Comparison of simulated bit error probabilities for noncoherent M-ary FSK. It is seen from the graphs that at a particular error probability, the required energy efficiency (E /Nb 0) is lowest for 8-level FSK and largest for binary FSK. Or equivalently, at a constant E /Nb 0, 8-level FSK has the lowest error probability, and binary FSK the largest. Hence, the curves in Figures 6.2 – 6.5 confirms that M-ary FSK is a power efficient modulation scheme whose power efficiency increases as the number of frequencies employed increases. 6.3 Theory Versus Reality There is a gap between the performance of binary and M-ary FSK in theory and in reality. An example of this gap is depicted in Figure 6.6 by comparing the performance of theoretical binary and 4-level FSK to that of commercially available data transmission equipment which uses GMSK and 4-level FSK modulation schemes. GMSK is a special modulation technique (uses two orthogonal frequencies like in BFSK) which performs better than binary FSK because it has the minimum separation required for orthogonality and at the same time continuous phase at bit transitions. Figure 6.6 confirms this fact. Since GMSK is quite different from binary FSK, the comparison between theoretical performance and that of reality reduces to comparison of 4-level FSK. It is observed, from Figure 6.6, that the gap between performance of theoretical 4-level FSK and that of MR400 radio modem (which uses 4-level FSK modulation) is about 1 dB at a BER of 10-3 -4, 3 dB at a BER of 10 , and 5 dB at a BER of 10-7.

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0 2 4 6 8 10 12 14 16 18 2010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Signal-to-noise ratio (S/N) in dB

Bit-

erro

r pro

babi

lity,

BE

RBFSK (theoretical)4-FSK (theoretical)MR400 (4-FSK,alpha = 0.2)MX160 (GMSK, BT = 0.4)

Figure 6.6 Comparison of theoretical and practical BER for 2, and 4-level FSK systems. Typically, the main source of difference between theoretical performance and reality is the noise bandwidth used in the calculation of signal-to-noise ratio. Other sources of difference are filtering effects in transmitters and receivers of practical systems, and over simplification of assumption in theoretical derivations. 6.4 Comparison of Performance using Shannon’s Capacity Curve. A compact comparison of modulation methods is one that is based on the normalized data rate Rb/B (bits per second per Hertz of bandwidth) versus the signal-to-noise ratio per bit (Eb/N0) required to achieve a given error probability. Table 6.1 shows the spectral efficiency and signal-to-noise ratio per bit required to achieve a BER of 10-5 for different modulation types. Table 6.1 Spectral efficiency and S/N required to achieve BER of 10-5 for different modulation types.

Modulation type R /B (bits/s/Hz) S/N (dB) bTheoretical BFSK 1.0 10.0 Theoretical 4-FSK 1.0 12.6 GMSK (BT = 0.4) 0.7 9.0 4-FSK (alpha = 0.2) 1.335 17.0

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Figure 6.7 illustrates how the performance of MX160 radio modem, MR400 radio modem, theoretical binary and 4-level FSK modulations compare to that of an optimum system.

0.5 1 1.5 2 2.5 3 3.5 4

0

5

10

15

20

25

Bandwidth Efficiency (Bits/sec/Hz)

Ene

rgy

Effi

cien

cy, E

b/N

o (d

B)

Shannon capacity curveTheoretical BFSKTheoretical 4-FSKMX160 (GMSK, BT=0.4)MR400 (4-FSK, alpha=0.2)

Figure 6.7 Comparison of theoretical and practical orthogonal modulation techniques at BER of 10-5. It can be seen from Figure 6.7 that theoretical 4-level FSK performs better than practical 4-level FSK in terms of energy efficiency, while practical 4-level FSK performs better in terms of bandwidth efficiency. This trend can be attributed to the noise bandwidth used in the calculation of signal-to-noise ratio and filtering effects of the modulated signal respectively.

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Chapter 7 Conclusions In this thesis work, binary and M-ary FSK modulation techniques are extensively studied. The performance of 2, 4 and 8-level FSK systems in additive white Gaussian noise channel is evaluated and compared on the basis of the simulations in MATLAB. The primary objective of the thesis, the advantage of FSK modulation technique, factors influencing the choice of a particular digital modulation scheme, a model for AWGN channel, generation and detection of binary and M-ary FSK modulated signals, and error performance of binary and M-ary FSK modulation systems in AWGN channel, have been discussed in detail in the first three chapters of the thesis. Simulations in MATLAB and results are presented in chapter four. The results of measurements on commercially available data transmission equipment, using GMSK and 4-level FSK, and how performance curves are extracted from the measurement results are presented in chapter 5. The results show that the noise bandwidth used in the calculation of signal-to-noise ratio and the type of filters employed in modulators (transmitters) and demodulators (receivers) play a very important role in the performance evaluation of a modulation scheme. The practical measurements results and simulation results compared to theory in chapter 6 confirms that M-ary FSK is a power efficient modulation scheme whose efficiency improves as the number of frequencies employed increases at the expense of bandwidth efficiency. And also shows that the gap between the performance of theoretical and practical M-ary FSK systems widens as the bit error rate (BER) decreases, with theoretical FSK systems always performing better.

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Chapter 8 References [1] Theodore S. Rappaport, “Wireless Communications: Principles and Practice”, Prentice Hall, Second Edition. [2] J. G. Proakis, M. Saheli, “Communication Systems Engineering”, Prentice Hall, Second Edition. [3] Fuqin Xiong, “Digital Modulation Techniques”, Artech House Publishers, 2002. [4] J. G. Proakis, M. Saheli, “Digital Communication”, Prentice Hall, Third Edition. [5] Alister Burr, “Modulation and Coding for Wireless Communications”, Prentice Hall, 2001.

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Appendix A Acronyms AWGN Additive White Gaussian Noise. BER Bit Error Rate. BFSK Binary Frequency Shift Keying. CRC Cyclic Redundancy Check. DSP Digital Signal Processing. FSK Frequency Shift Keying. GMSK Gaussian Minimum Shift Keying. PER Packet Error Rate. PSD Power Spectral Density. VLSI Very Large-Scale Integration.

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