JOURNAL OF TELECOMMUNICATIONS, VOLUME 31, ISSUE 2, AUGUST 2015 1

System Model of TH-UWB Using LDPC Code Implementation

M. Marjanović-Jakovljević

Abstract— A Low-Density Parity- Check (LDPC) code is an error-correcting code in a noisy-channel transmission that closely approaches the Shannon limit, also called channel capacity. Time Hopping-Ultra Wideband (TH-UWB) is a relatively new technology that might have a huge impact on improving wireless communications. Since that current TH-UWB systems apply convolutional codes as their channel coding scheme, it is very usefull to investigate LDPC codes performance for those systems. This paper presents a mathematical model in order to simulate TH-UWB systems with LDPC code implementation. Using this implementation, it is shown that for low Signal to Noise Ratio (SNR) in the real multipath channel environment, the difference between LDPC coded and non-coded system is negligible, but for BER=10-3, the gain of the coded system compared to the non-coded system is approximately 4 dB. Owing to this implementation, in this accurate and flexible system model, BER performance of a TH-UWB System in different scenarios is presented and good performance in terms of BER versus SNR is achieved. An additional result is the validation of the simulated results with performance formula for TH-UWB systems when LDPC is implemented.

Index Terms— TH-UWB, LDPC codes, Multiuser interference (MUI), BER, Additive White Gaussian Noise (AWGN).

—————————— u ——————————

1 INTRODUCTIONTH-UWB technology, introduced in [1], [2] and [3], pre-sents some very attractive features for future indoor wire-less systems in terms of achievable transmission rate and multiple access capabilities. Thus, it is very important and useful to continuously work on performance improve-ment of those systems. TH-UWB systems, as any other system, deal with the problems with signal transmission over a noisy commu-nication channel. The job of the encoder and decoder is to transmit information about the source across the noisy channel. Numerous studies have attempted to obtain the minimum channel capacity, needed to almost surely as-ymptotically observe and stabilize the system [4]. In [5] it is demonstrated that coding techniques for noisy channels have near optimal performance in wireless sys-tems. Therefore, a comprehensive description of the cod-ing techniques, including convolution, trellis, concatenat-ed, turbo and LDPC codes is given. The Noisy Channel Coding Theorem is discovered by C. E. Shannon [6] in order to reduce error rate on noisy channels to negligible levels without affecting the data rates. In order to reach BER performance close to the Shannon limit, in [7] the low density parity check codes were developed by Robert Gallegar. Current TH-UWB systems apply convolution codes as their channel coding scheme. Thus, it is very usefull to investigate LDPC codes performance for TH-UWB.

Based on this accurate and flexible model, a description on how to simulate system in a multipath environment

through employing RAKE receiver [5] is presented. Addi-tionally, the TH-UWB system model with LDPC code implementation is described and the influence of LDPC codes on TH-UWB system performance is presented. The impact of different factors on TH-UWB system perfor-mance over AWGN channel is shown.

Section II describes the system and signal model used for the purposes of this paper. Sections III and IV elabo-rate on the implementation of LDPC coding and decoding schemes into this system model. In Section V, the theoret-ical value of Error Probability of LDPC decoding in AWGN is calculated. Section VI depicts simulation re-sults, whereas Sections VII offers conclusions from pre-sent work.

2 SYSTEM AND MODEL 2.1 Signal Model

Bit structure of the TH-UWB System for the kth user is

shown in Figure 1. The total number of bits is Nb. Each bit is subdivided into Nf frames, and each frame is subdivid-ed into Nh chips.

Signal transmitted through the kth link can be presented as follows:

( )( ) ( ) ( )( )ckjf

kj

jtr

k TcjTdtwts −−−= ∑∞

−∞=

λ (1)

where trw represents the transmitted waveform. In [8], some possible waveforms have been proposed. In this work, we select the pulse shaper to be the second deriva-tive of the Gaussian function that has been normalized to have unit energy. In order to normalize its energy, we consider that

———————————————— • M. Marjanović-Jakovljević  is Associate Professor at Singidunum Univer-

sitz, Belgrade, Serbia.

2

( )

( ) ( ) ( )( ) 1)1(2

12 =−−−= ∫∫

++

dtTcjTdtwdttw ckjf

kj

Tj

jTtr

Tj

jTtr

f

f

f

f

λ (2)

where ( )kjd represents a sequence of time-shifts in a PPM

modulation [9]. ( ){ } { }1,...,1,0 −∈ hkj Nc is the orthogonal sequence, where Nh

is the integer number that denotes the position within the frame where the monocycle should be transmitted in or-der to mitigate the Multi User Interference (MUI) as de-scribed in [10]. For the purposes of this paper, we use pseudorandom TH codes.

Figure 1. Bit structure of TH-UWB Symbol for the kth user

2.2 Channel Model The transmitted signal of the kth user through the multi-

path channel has the following structure:

)()(*)()( )(

1

)( tnthtstr kN

k

ku

+=∑=

(3)

where * denotes the convolution between transmitted

signal s(k)(t) and normalized channel response h(k)(t). n(t) represents the AWGN with mean zero and a dou-

ble-sided power spectral density 2/2nσ . Considering that multipath channel is parameterized as

a combination of L paths, each characterized by delay }{ )(k

lτ and amplitude }{ )(klβ , signal from (3) can be writ-

ten as

−⎢⎢

⎣

⎡−−= ∑∑∑

= =

∞

−∞=

uN

k

L

l jf

kjrec

kl jTdtwtr

1 1

)()( ()( λβ

] )()()( tnTc k

lckj +−− τ (4)

where wrec (t) represents the received pulse of the kth user after the multipath propagation. Received pulse can be presented as a convolution between the transmitted mon-ocycle and the distorted channel response hdist (t) as

)(*)()( thtwtw disttrrec = (5)

For the sake of simplicity, we consider the perfect chan-nel and signal estimation and the perfect synchronization. For the same reason, this analyze will be limited on ob-serving the only one symbol transmission. In order to simulate a UWB system assuming that channel is not per-fectly synchronized, synchronization might be achieved, i.e. as in [11] or [12].

2.3 Receiver In order to collect multipath energy and to recover the information, as a general case of receiver, in this simula-tor, we describe this simulator model using selective RAKE receiver as it is proposed in [13]. This receiver gives the correlation between the received signal r(t) and template signal that should be previously synchronized. The statistics for the ith frame on the qth receiver is given as follows:

( ) dtTciTtvtrt cqi

TcTi

TciT

fq

i

cqif

cqif

)()( )()1(

)(

)(

)(

−−⋅= ∫++

+

α (6)

where )()( tv q represents the template signal described as follows:

( ))(0

)()(max

)( qm

L

m

qm

q ttv τφβ −=∑=

(7) The signal φ (t) depends on the type of the modulation employed. Since we apply the binary PPM, this signal might be defined as:

)()()( λφ −−= twtwt recrec (8) Lmax represents the number of RAKE fingers with the am-plitudes )(q

mβ and the corresponding finger duration .)(qmτ Once the frame statistics has been calculated, a bit

decision should be taken. Supposing that )(twtr and )( λ−twtr are orthogonal, soft decision [14] is obtained as

⎩⎨⎧

<∀≥∀= 0 1,

0, 0,ααdecision (9)

where the bit statistic for soft decision is presented as ∑=

=fN

ii

1

αα (10)

3 PROPOSED MODEL FOR LDPC CODE IMPLEMENTATION

This chapter addresses the issue of robust and progres-sive transmission of TH-UWB signals encoded with LDPC codes over noisy channels. It demonstrates that this

3

coding technique has near optimal performance in wire-less systems close to the Shannon limit [6]. In LDPC en-coding, a form of parity-checking is used, where extra bits are added to the transmission. In this way, the decoder in receiver enforces the constraint check for each received bit. LDPC are codes that are specified by a matrix that con-tains mostly 0’s with few 1’s. LDPC codes use parity-check sparse matrix H with dimensions MXN. Matrix H can be either regular, meaning that there is a specific number of 1’s per row or per column, or irregular, where there is no constraint on the number of 1’s. In this paper, we are going to use irregular matrix H, which means there is no specific number of 1’s per row and col-umn. The sparseness of matrix H means that there is a very low number of 1’s in H compared to its total size. Figure 2 presents a block diagram of the system for the kth user. The matrix H is used to encode a message of codeword )(kd . The columns in H correspond to the bits of the transmitted mes-sage, and the rows of matrix H correspond to the parity checks of the codeword. When LDPC are used, the coder would first take binary sequence and map our transmitted sequence into a redundant sequence, i.e. codeword.

In this work, we use binary PPM and the vector

{ }121)( ,,, −=

bNk ddd …d contains information on infor-

mation bits. If di=0, transmitted bit is si=0, while when di=1, transmitted bit is si=1, where { }bNi ,...,2,1∈ .

As an example, in Figure 3, Tanner graph is presented with its corresponding matrix, where the number of bits is Nb =7.

In general, this paper defines the rate R of the LDPC as a ratio of the length of the information sequence to the length of the codeword. Since, we are transmitting a sequence with

bN bits, there are bN2 possible sequences from the source that are mapped into n length codeword. Therefore, a frac-tion )1(2 Rn −− of the n2 different n length sequences can be used as codewords.

Figure 2. Block diagram of the proposed System model for the kth user

Figure 3. LDPC Matrix with its corresponding Tanner graph

The parity-check matrix is used to encode an imput message d. A valid codeword will satisfy Hd=0, where H is MXNb dimension matrix. M = n(1-R), represents the number of check bits.

For instance, when a codeword }1011{=d is received, each check bit performs the binary XOR operation with the corresponding bit in ( )kd . If all check nodes generate 0, the

4

codeword is correct. In case of the example shown in the Figure 3., we obtain the following equations:

1111)(4

)(2

)(1

)(1 =⊕⊕=++= kkkk dddC (11)

1001)(5

)(3

)(2

)(2 =⊕⊕=++= kkkk dddC (12)

)(6

)(4

)(3

)(2

)(1

)(3

kkkkkk dddddC ++++= (13)

011011)(3 =⊕⊕⊕⊕=kC (14)

1111)(7

)(6

)(5

)(4 =⊕⊕=++= kkkk dddC (15)

4 LDPC PARITY CHECK DECODING In the receiver, the decoder contains knowledge on which sequences are considered as codewords. Therefore, the re-ceiver is capable of separating the transmitted n length codeword from the channel noise. As a result, the codeword is mapped back into the Nb information bits. Numerous de-coding schemes were used to decode the codeword. In this paper, we have used LDPC decoding based on the Bit-Flipping decoding algorithm [7]. This decoding scheme takes an intermediate decision and operates with the a poste-riori probabilities of the input symbols.

In case shown in Figure 4, we assume that the codeword is ( )kα ={1111}T. Compared to ( )kd ={1011}T, which is as-

sumed as a correct codeword, we can see that error is located in the second bit of the received codeword ( )kα .

Figure 4. Bit flipping algorithm

The steps are the following:

a) Received codeword is ( )kα ={1011}T.

b) For each variable node, the nonzero parity check sums are counted.

c) The bit of variable node having the largest number of nonzero parity check sums is flipped.

d) Steps from a), b) and c) until all parity checks are zero should be repeated.

5 BER PERFORMANCE OF LDPC CODES IN AWGN CHANNEL

In (19), a theoretical analysis of the performance of a TH-UWB systems under certain restrictions is presented, based on the one presented in [17]. In order to validate our approach, we made comparison between simulated results and the theoretical response. If we consider Nu independent users, the MUI can be mod-eled as Gaussian and the BER for soft decision can be ex-pressed as (18). We have assumed likelihood receiver which output is given by (since one of two waveforms is transmitted):

[ ][ ])(1Pr

)(0Prln )(

)(

trs

trsk

k

j=

==α (16)

In the block diagram in Figure 2, decision block is located after LDPC decoder. This is because if the output is converted into binary digit prior to attempting to the block the data, some information about transmitted sequence might be destroyed.

If one of the two waveforms is transmitted every T seconds, these signals appear in the receiver with equal energy.

∫∫ ==TT

c dttxdttxE0

21

0

20 )()( (17)

Let n(t) be a sample of white Gaussian noise of power densi-ty 0

2 Nn =σ . Then, the probability of error on the input is

[ ])( uNSNRQBER = (18)

where

∑ ∫

∫

=

∞+

∞−

+∞

∞−

−+

=uN

krecffn

recf

u

dsdttstwTN

dtttwN

NSNR

2

22

2

])()([)/(

))()((

)(

φσ

φ (19)

00

2 )1(4)1(4N

REN

Ecn

ρρσ

−=

−= ; (20)

and

dttxtxRE

dttxtxE

TT

c)()(1)()(1

10

010

0 ∫∫ ==ρ (21)

5

E is available energy per information digit. In [7] it is

shown that the rate 21

=R has the best performance com-

pared to other rates due to time duration of block length and we have used this value for the purpose of this paper.

7 SIMULATION RESULTS

Since an accurate and flexible simulation model is ob-tained, this chapter analyzes the influence of different factors (number of users, number of chips, waveform de-signs, frame duration, LDPC coding influence). Simula-tion results are obtained using MATLAB Monte Carlo simulations [15]. In Figure 5, it can be seen how theoretical value describes exactly the behavior of the simulated response. It is shown that for low Signal to Noise Ratio (SNR) in the presence of reach multipath environment, the difference between LDPC coded and non-coded system is negligible, but for BER=10-3, the gain of the coded system compared to the non-coded system is approximately 4 dB. Unfortunately, the development of a software simulator for UWB has several difficulties derived from the extremely large sam-pling rate necessary to process these UWB signals. Since the length of the array that contains the samples of a sin-gle bit can be very large. Therefore, in a real channel mul-tipath environment, in order to achieve low BER, a long time simulation process is required. In Figure 6, it is shown how BER performance decreases as duration of frame increases. Since, the real TH-UWB channel has large number of mul-tipath components and considering several users, the necessary computational requirements to evaluate pro-duce high simulation time, especially for low BERs. Therefore, for the rest of result, we are going to present simulation curves taking into account AWGN channel. In Figure 7, monocycle shape influence on BER perfor-mance employing Single User Receiver is shown. It is demonstrated that under the same scenario, the type of the monocycle does not have a considerable impact on the system performance. Anyway, for the rest of the results, we have decided to use the Second Derivative of the Gaussian Monocycle. In Figure 8, number of chips influence on BER perfor-mance employing Single User Receiver where Nu=64, Nf=64, fs=200/Tc is presented. The results are expected, since when the number of chips gets bigger, the perfor-mance becomes better. As we have already mentioned in I section, Nh is the integer number denoting the position within the frame where the monocycle should be trans-mitted in order to mitigate the Multi User Interference (MUI). Since this number is bigger, the MUI is lower and BER performance is better. In Figure 9, we have shown the LDPC influence on BER performance for different number of users. We have con-sidered two systems, one coded at rate

21 and other one

that is not coded, both transmitting the same number of

information symbols per second. BER curve for the sys-tem that is not coded is presented with the solid line, and the corresponding BER curve of the coded system (for the same number of users) is presented with the same color dashed line. It is shown that for BER=10-3, for the same number of users, the gain of the coded system compared to the non-coded system is approximately 4 dB.

Figure 5. Comparison between the simulated response and the theo-retical expression (based on formula (18)) for a PPM-TH-UWB sys-tem for LDPC coded and non-coded system in the multipath channel environment. fs=25e9;tc=1ns;Tf=20ns;Nf=4; Nh=4; Ec=1;R=0.5;

Figure 6. Duration of frame influence on BER performance employ-ing Single User Receiver; multipath channel; fs=25e9;tc=1ns; Nf=4; Nh=4;R=0.5;

5 6 7 8 9 10 11 12 13 1410-3

10-2

10-1

SNR

BE

R

Second Derivative of the Gaussian MonocycleRayleigh MonocycleCubic Monocycle

Figure 7. Monocycle Shape Influence on BER performance employ-ing Single User Receiver; AWGN channel; Nu=64, Nh=64, Nf=8, fs=200/Tc; Ec=1;

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

SNR[dB]

BER

Tf=20n

s Tf=40n

s Tf=60n

s Tf=80n

s

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

SNR[dB]

BER

Theoretical curve for non-coded system Theoretical curve when LDPC is implemented Simulated curve for non-coded system Simulated curve when LDPC is implemented

6

Figure 8. Number of Chips Influence on BER performance employing Single User Receiver; Second Derivative of the Gaussian Monopulse; AWGN channel; Nu=64, Nf=64, fs=200/Tc;

Figure 9. Low Density Parity Codes influence on BER performance for different Number of users, employing Single User Receiver; Se-cond Derivative of the Gaussian Monopulse; AWGN channel with

12 =σ ; Nf=4, Nh=64, Tc=1ns, fs=250/Tc, R=1/2;

7 CONCLUSION presents an encoder-decoder design solution, for practical LDPC coding TH-UWB system implementation. Computer simulations have shown that the LDPC codes have significant error-correcting performance in those systems. We believe that the simulation model of TH-UWB systems with LDPC design approach will give communication system designers a unique opportunity to explore attractive features of TH-UWB Systems in many real-life applications. Since the simulation of TH-UWB systems in the multipath environment requires large sampling rates, our future work shall mostly be directed towards reducing the simu-lation time by considering LDPC codes implementation in Low Complexity Simulation Algorithm described in [16] and presenting BER performance of TH-UWB systems using the real channel model from [17] or [18].

With this low complexity simulation model; we might analyze the performance of the TH-UWB system and the impact of different factors of TH-UWB systems

(the number of users, waveform design, time-hopping codes, channel models, multiuser receivers [19]-[22]) and achieve a low BER in a real time application even in the presence of reach multipath environment.

REFERENCES [1] R. A. Scholtz and M.Z Win, “Impulse Radio, How it works,” IEEE Communication Letters, vol. 2, pp. 36–38, Feb., 1998. [2] M. Z.Win, J. Ju, X. Quiu, V. O. K. Li and R. A. Scholtz, , “ATM based Ultra-Wide Bandwidth Multiple-Access Radio Networks for Multimedia PCS,” IEEE 4th Annual Networld+Interop Conference, pp. 101–108, May, 1997. [3] M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth timehop-ping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. Commun., vol. 48, no. 4, pp. 679–689, 2000. [4] S. Tatikonda and S. Mitter, “Control over Noisy Channels,” IEEE Transactions on Automatic Control, vol 49, issue 7. Feb., 2004. [5] A. Goldsmith, “Wireless Communications,” Cam-bridge. University Press, 2005. [6] D. J. C. Mackay and R. M. Neal, “Near Shannon limit perfor-mance of low-density parity-check codes,” Elect. Letter, vol. 42, no. 11, 1645–1646, Aug. 1996. [7] Gallager, R. G., “Low Density Parity Check Codes,” Number 21 in Research monograph series. MIT Press, Cambridge, Mass, 1996. [8] Chen X., Kiaei S., “Monocycle Shapes for Ultra Wideband Sys-tems,” IEEE International Symposium on Circuits and Systems 2002, ISCAS 2002, vol. 1, pp. I-597 – I-600, May 2002. [9] F. Ramirez-Mireles, “Performance of ultrawideband SSMA using time hopping and M-ary PPM,” IEEE J. Select. Areas Commun. vol. 19, no. 6, pp. 1186–1196, 2001. [10] M. S. Iacobucci and M.-G. Di Benedetto, “Time hopping codes in impulse radio multiple access communication systems,” in Proc. International Workshop on 3G Infrastructure and Services, pp. 171–175, Athens, Greece, July 2001. [11] M. Marjanovic, J. M. Páez Borrallo, “Analysis of Timing Offset Estimation Schemes in UWB“ EUSIPCO 2005, The 13th Signal Pro-cessing European Conference. [12] C. Falsi, D. Dardari, L. Mucchi, and M. Z. Win, “Time of Arrival Estimation for UWB Localizers in Realistic Environments,” EURA-SIP Journal on Applied Signal Processing, vol. 2006, Article ID 32082, 13 pages, 2006. doi:10.1155/ASP/2006/32082. [13] D. Cassioli, M. Z. Win, F. Vatalaro, and A. F. Molisch, “Perfor-mance of low-complexity RAKE reception in a realistic UWB chan-nel,” in Proc. IEEE International Conference on Communications (ICC ’02), vol. 2, pp. 763–767, April–May 2002. [14] M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time hop-ping spread-spectrum impulse radio for wireless multiple-access communications,” Communications, IEEE Transactions (Volume: 48, Issue: 4 ) [15] R. Casarin, “Monte Carlo Methods using Matlab,” University Ca’ Foscari, Venice Summer School of Bayesian Econometrics Peru-gia 2013 September 9, 2013. [16] G. N. Barrau and J. M. Páez-Borrallo, “A New Time-Hopping Multiple Access Communication System Simulator: Application to Ultra-Wideband,” EURASIP Journal on Applied Signal Processing - Vol. 2005, Issue 3, Pages 346-358 [17] A. Saleh and R. Valenzuela, “A Statistical Model for Indoor Multipath Propagation,” IEEE JSAC, vol. SAC-5, no. 2, February 1987, pp. 128–137. [18] J. Foerster, Q. Li, UWB channel modelling contribution from Intel, IEEE P802.15-02/279r0-SG3a, June 2002. [19] M. Marjanovic, J. M. Páez Borrallo, “A New Approach of MUD in UWB Systems,” ICUWB 2006, IEEE International Conference on Ultra-Wideband, Waltham, Massachusetts, USA, September, 2006. [20] E. Fishler and H. V. Poor, “Low-Complexity Multi-User Detec-tors for Time Hopping Impulse Radio Systems,” submitted for pub-lication at the IEEE Trans. On Signal Processing [21] S. Verdu, “Multiuser Detection,” Cambridge University Press, 1998. [22] M. Marjanovic, J. M. Páez Borrallo, “A Low Complexity Simula-tion Algorithm for TH-UWB MMSE RAKE Receiver,” ISSPIT 2006, The 6th IEEE International Symposium on Signal Processing and Infor-mation Technology, Vancouver, Canada, August, 2006.

5 6 7 8 9 10 11 12 13 1410-5

10-4

10-3

10-2

10-1

100

SNR

BER

Nh=2Nh=16Nh=32Nh=64

7

M. Marjanović Jakovljević received the Electrical Engineer degree from Belgrade University, Serbia, in 2002. In 2007. She received Ph.D. degree in Telecommunications at the Signal Processing Group of the Polytechnic University de Madrid (UPM). She has been awarded a Telefonica Moviles Fellowship to the best academic trajectory. She is currently working as an Associate Professor in the department of the Computer Engineering at the Singidunum University in Belgrade. Her research interests include Information Retreival Systems, UWB systems, ad hoc networks, and wireless communications.