MULTY BINARY TURBO CODED WOFDM PERFORMANCE IN FLAT RAYLEIGH FADING CHANNELS

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1 Volume 49, Number 3, 28 MULTY BINARY TURBO CODED WOFDM PERFORMANCE IN FLAT RAYLEIGH FADING CHANNELS Marius OLTEAN Maria KOVACI Horia BALTA Andrei CAMPEANU Faculty of, Timisoara, Romania Bd. V. Parvan, nr.2, Timisoara, Pone: , Abstract: In tis paper, te application of Multi Binary Turbo Codes (MBTCs) to two multicarrier modulation tecniques used for transmission in te flat radio fading cannel is studied. Tus, te performance of te coded versions of Ortogonal Frequency Division Multiplexing (OFDM) and of its wavelets based version (WOFDM) is evaluated under several simulation scenarios. Tese scenarios take into account te time variability of te radio cannel, by te mean of te Doppler sift parameter. An important conclusion is tat, under te assumptions made in tis paper, WOFDM outperforms OFDM. By te oter and, tis tecnique sows little influence on te coice of te wavelets moter used in te modulator. Key words: Duo binary Turbo Codes, WOFDM, OFDM, Flat Fading. I. INTRODUCTION Wavelets represent a successful story of te last decade in signal processing applications. Tus, tese signals, wit some igly desirable properties, are widely used in domains as compression, denoising, segmentation, inpainting or classification. By te oter and, in data communications, te same successful story can be assigned to multi-carrier modulation tecniques. Te principle of te OFDM is employed in a large number of transmission standards, over wired and wireless cannels: WiFi, WiMAX, DAVB or ADSL. Te wavelet based OFDM (WOFDM), sometimes referred to as wavelet modulation is te point were te above concepts meet wit eac-oter. Altoug tey are widely used in signal processing, few wavelets applications are known in data transmission. Te idea tat gaters te two concepts is to use wavelet signals as carriers in a multicarrier data transmission. Recent researc as sown tat, by associating te multicarrier concept and te wavelet signals, some of te OFDM s classical drawbacks can be counteracted. Tere are different forms of multi-carrier wavelet-based transmission [-3] and most of tem use te Inverse Discrete Wavelet Transform (IDWT) for te implementation of te multicarrier transmitter. Te BER performance of WOFDM was extensively studied in previous papers, in AWGN and flat fading cannels [4]. Briefly, our investigations sowed tat, wile tere is no difference between OFDM and its wavelet-based version for AWGN cannels, it may be wortwile to carry a deeper researc on te fading cannels case. Tus, WOFDM provided significantly better results tan OFDM, no meter wat was te wavelet use as carrier. Furtermore, te coice of te wavelets moter is meaningful for te BER performance of WOFDM: Haar wavelet provided, by far, te best results [4]. In general, our empirical researc led us to te conclusion tat, better te time localization of te carrier, iger te resilience to te time-variant caracter of te fading and lower te BER of te system. All te above investigations were conducted in a simple scenario, witout considering any additional issues as syncronization or cannel coding. Tis time, our goal is to associate te WOFDM transmission wit a very powerful cannel coding tecnique, namely te turbo-codes. Turbo codes are a class of ig-performance error correction codes developed in 993, [5] wic are finding use in deep space satellite communications and oter applications were designers seek to acieve maximal information transfer over a limited-bandwidt communication link in te presence of data-corrupting noise. Tey are particularly attractive for cellular communication systems and ave been included in te specifications for bot te WCDMA (Wideband Code Division Multiple Access)/UMTS (Universal Mobile Telecommunications System) and cdma2 tirdgeneration cellular standards. Te Multi Binary Turbo-Codes (MTBC) are a parallel concatenation of two binary recursive systematic convolutional (RSC) codes based on multiple-input (r-input) linear feedback sift registers (LFSRs) and tey provide several advantages versus te classical turbo-codes. MBTC as already been adopted in te digital video broadcasting (DVB) standards for return cannel via satellite (DVB-RCS) and te terrestrial distribution system (DVB-RCT), and also in te 82.6 standard for local and metropolitan area networks. Forward Error Correction (FEC) coding, for wireless applications on fading cannels, is an important Manuscript received September, 28; revised October 5, 28

2 Volume 49, Number 3, 28 tool for improving communications reliability. MBTCs ave been sown to perform near te capacity limit on te additive wite Gaussian noise (AWGN) cannel, [6]. An association between WOFDM and turbo-codes is not extensively treated in te literature. Wen presented [7], te results are owever very promising, and tey sow tat it is wortwile to carry a researc on tis topic. Te autors of tis paper will associate te two concepts, encoding te data bits wit a MBTC, before passing tem to te WOFDM modulator. Te transmission cain obtained will be called MBTC-WOFDM, and its performance is studied in flat Rayleig fading cannels. In section 2, te multi-binary turbo-codes and te wavelet modulation principles are briefly presented. Next, te transmission cain used for simulations is described. Simulation results are presented in section IV, wereas te final conclusions are drawn in te last section. II. BASIC CONCEPTS OF MBTC-WOFDM A. Multi-binary turbo codes Te general sceme of a r/(r+)-rate recursive systematic convolutional (RSC) multi binary codes is presented in [8], were r represents te number of inputs for RSC encoders. In tis work we consider te value of r set to 2. Terefore, a sceme of te 8-state duo binary encoder is represented in Figure, [9]. u 2 u 2 u u 2,,,3, S Figure : Duo binary recursive systematic convolutional code wit m=3 memory units (8-state), wic is te building block for duo binary turbo encoding, wit polynomials 5 (feedback) and 3 (redundancy) in octal form (DVB-RCS constituent encoder). t t t Denoted by = [ s s s ] T t t and [ u u ] T St 3 2 Ut 2 = are related to te encoder state. Te full generator matrix of te 3 memory multi input encoder as te following form: H = = 3,3 3,2 3, 2,2 2,3 2,2 2, S 2,3,2, = 3,2 2,3,3,2, = [ 6 7 5] S 3 3,3 c =u () By skipping te weigts for te redundant bit and te weigts for te recursive part, we obtain te matrix H as follows: 2,3,3 H = = 2,2,2 (2) 2,, Te feedback vector and te output vector ave te form: F [ ] T [ ] T,3,2, = [ ] T [ ] T H = H = out 3,3 3,2 3, = (3) were te operator (.) T denotes te transpose of a vector. Te main equation of te circuit in Figure is: ( t + ) 3 = ( H ) 3 2 ( Ut ) 2 + ( T) 3 3 ( St ) 3 S (4) were te matrix T is defined as: T = = 2 I2 H F Te redundant output is equal to: t = Hout St + W St + (5) c (6) were te vector W is to [ ]. A parallel concatenation of two identical r-ary RSC encoders wit an interleaver ( ) is presented in Figure 2. k/r data binary couple Permutation r RSC r-ary RSC r-ary Figure 2. Te r-ary turbo-encoder. Blocks of k bits (k being a multiple of r) are encoded twice by te bi-dimensional code, wose rate is r/(r+2) (for r=2, it results /2 rate), to obtain te multi binary turbo encoder sceme from figure 2. In tis paper we used te S-interleaver,, []. Te S- interleaver is a random type interleaver. Terefore, unlike te pure random interleaver, a minimum interleaving distance equal wit S is forced by construction. Te interleaving mapping construction algoritm will be next presented. We select a possible future position for te current bit. Tis position is compared to te last S positions already selected. If te condition: u c c 2 2

3 Volume 49, Number 3, 28 π s (i) π s (j) > S for i and j wit i j < S (7) is satisfied we go furter. If te condition is not true, we select anoter position for te current bit, wic will also be tested. Te process will be repeated up to te moment wen all te positions for te N s bits will ave been found. Te simulations demonstrated tat, if S< N s 2, ten te process will converge. B. Wavelet modulation and te multicarrier concept Te wavelet modulation, referred to as WOFDM relies on a multicarrier modulation concept, exactly as OFDM. Despite its excellent beavior in unfriendly cannel environments, OFDM raises some practical problems wic are difficult to overcome. Tus, te bandwidt is increased by te use of a cyclic prefix and te transmission is igly sensitive to te Doppler spread introduced by te time variant cannels. Next, te OFDM systems are very sensitive to te narrow band interferences. Furtermore, te syncronization in time and frequency is critical for te system performance and te peak-to-average ratio of te signal is very large due to te non-constant nature of te envelope. Finally, te out-of-band rejection of suc a signal is not satisfactory, since te OFDM spectrum is made of sinc functions, wom sidelobes contain an important amount of energy. Some of tese sortcomings can be eliminated by using wavelet carriers instead of te OFDM s complex exponentials. In any multi-carrier approac, data is transmitted using several parallel substreams. In OFDM, every suc a stream modulates a different complex exponential subcarrier, te used subcarriers being ortogonal to eac oter. Te ortogonality is te key point tat allows subcarrier separation at receiver. Tis modulation is implemented computing an Inverse Discrete Fourier Transform, (IDFT). Te crucial idea wic links te classical OFDM and te wavelet-based tecnique is tat, in te same manner tat te complex exponentials define an ortonormal basis for any periodic signal, te wavelet family as defined in (8) forms a complete ortonormal basis for L 2 ( R ) :, if j = m and k = n < ψ j, k ( t), Ψm, n ( t) >= (8), oterwise Equation (8) indicates tat all te members of te wavelet j / 2 j { Ψ j, k ( t) = 2 Ψ(2 t k)} k Z family (we considered s=2 and τ=) are ortogonal to eac oter. Consequently, if instead of complex exponential waveforms we use wavelet carriers, we will still be able to separate tese subcarriers at receiver, due to teir ortogonality. As for te classical OFDM, te WOFDM symbol can be generated by digital signal processing tecniques, suc as te Inverse Discrete Wavelet Transform (IDWT). In tis case, te transmitted signal is syntesized from te wavelet coefficients w j, k =< s( t), ψ j, k ( t) > located at te k-t position from scale j (j=,, J) and approximation coefficients aj, k =< s( t), ϕj, k ( t) >, located at te k-t position from te coarsest scale J: J s (t) = w j,kψ j.k (t) + a J,kϕJ,k (t) (9) j= k = k = Note tat, in practice, a sampled version of te signal s(t) is generated by means of Malat s Fast Wavelet Transform (FWT) algoritm []. In tis case, te input data vector can be seen as follows: data = [{ a J, k },{ wj, k },{ wj, k },...,{ w, k}] () Tis input data sequence is modulated onto a contiguous finite set of dyadic frequency bands and onto a finite number of time positions k witin eac scale. Te implementation using IDWT is sown in figure 3: Figure 3: Syntesizing te WOFDM symbol using IDWT: practical implementation. Note tat te output in figure 3 is represented by a sequence of samples, and not by an analog signal. Tis is because te Mallat s algoritm is based on digital filter banks and operates wit discrete-time inputs and outputs. III. SIMULATION SCENARIO Te transmission cain used for simulation purposes is sown in figure 4. a n Cannel Coding & Modulation Input data {a J } {w J } {w J- } {w } IDWT/ IDFT s[n] ray[n] IDWT p[n] Figure 4: Te baseband simulation cain for MBTC- WOFDM. Most of te parameters used in simulations are displayed in Table. We will now detail te most important features of te simulations described in tis paper. A. Te transmitter Te data source a n is a sequence of binary data symbols. Tis sequence is encoded using a MTBC and next transformed into a bipolar sequence, by te BPSK baseband s[n], n=,,n- DWT/ DFT Cannel Decoding a n Decision 3

4 Volume 49, Number 3, 28 modulator. Te decoding algoritm used in tis paper is te MaxLogMAP algoritm, [2]. We consider te extrinsic in- Table : Te most important simulation parameters. Parameter Used variant Te turbo code Parallel configuration Te component RSC code wit: code 3 memory (H=[6 7 5]) Turbo coding rate /2 Punctured no Modulation BPSK Systems OFDM WOFDM, wit wavelets moter: Haar and Daubecies2 N=24 carriers Cannel Flat fading + AWGN fm=.5 and. Interleaver S-interleaver, S=9 Data block lengt 2 x 256 = 52 bits Decoding algoritms MaxLogMAP Iteration number 5 iterations wit a stop criterion iteration based on APP (A Posteriori Probability) distribution. Simulated block Inversely proportional wit te number logaritm of te BER. formation wit a scaling factor set to.75 [3]. Te influence of te number of iterations and of te lengt of te interleaver on te performance of TCs over Rayleig fading cannels is already studied in [7]. Te multi-carrier symbol syntesizer is eiter an IDWT or an IDFT processor, for WOFDM and OFDM respectively. For te wavelet based approac, we tested two different wavelets: Haar and Daubecies-2. Previous simulations made for te notcoded system sowed tat Haar wavelets moter provide te best results. We tried to ceck if tis conclusion remains valid for te coded case, and to wat extent. Te number of carriers for bot systems is N=24. B. Te cannel We simulate te beavior of a radio cannel, an environment were te multicarrier approac suites te best. Suc a cannel exibit small scale fading, leading to two independent caracteristics: time variance and frequency selectivity of te cannel [4]. Te variance in time of te radio cannel's beavior can be expressed by te mean of te Doppler sift parameter, wic depends on te relative motion between transmitter and receiver (v) and on te transmission wavelengt (λ). Te maximum value of tis parameter is: f d = v λ () Te autors use in tis paper a normalized version of te Doppler sift: fm = fd TS (2) were T S is te time dedicated to te transmission of one data symbol, or te symbol time. We considered two values for te f m parameter, as detailed in Table. In slow fading scenarios, T S must be muc smaller tan te coerence time of te cannel expressed as: T C.423 = (3) f d Taking into account (2,3), our best case scenario (f m =.5) leads to a coerence time T C wic is approximately 8 times iger tan T S. In te oter case te coerence time is only 4 times longer tan te symbol duration. Tus, te cannel stays uncanged for te duration of a few serial symbols. Toug, wen evaluating te cannel beavior, one sould take into account tat in te multicarrier communications te transmitted symbol is longer. Usually, since te wole N samples data vector is required at demodulator to identify te transmitted symbols, we can consider tat te multicarrier symbol duration (an OFDM or a WOFDM block) is N times longer tan te serial symbols brougt at modulator's input. Note tat under tese conditions, te cannel response canges during te transmission of one symbol, or block, and can be considered as a fast fading cannel. From te frequency selectivity point of view, te scenario taken into account refers to flat fading model, were te frequency response of te cannel is considered approximately constant in te transmission band. Tis means flat frequency response of te cannel, wic can be implemented as a one-tap filter. Te small scale fading envelope can be modeled wit a Rayleig distribution, if tere is no line of sigt in transmission. Te impact of te Rayleig flat fading is given by te multiplicative ray[n] (see figure 4). A wite noise p[n] is ten added to te signal above, obtaining te sequence r[n] to be processed by te receiver: r [ n] = s[ n] ray[ n] + p[ n] (4) C. Te receiver Te main block of te receiver is composed of a DWT/DFT demodulator. Te data is next decoded by te turbodecoder. Note tat te turbo-decoder operates wit a soft input. Tis means tat te data samples at decoder input are neiter binary nor bipolar symbols on wic a ard decision as been taken a-priori. Instead, te decoder operates wit signal samples aving continuous values, resulted from te cannel impact (multiplicative Rayleig and additive wite noise) on te bipolar transmitted samples. Practically, te decoding algoritm leads to a ard decision on te 4

5 Volume 49, Number 3, 28 transmitted symbol, incorporating tis way te final detector. By te oter and, neiter syncronization nor equalization issues were considered. IV. SIMULATION RESULTS AND DISCUSSIONS For te performance evaluation, BER and FER statistics are computed. Te results are syntesized in figures 5 and 6. Several conclusions can be drawn. Te most important remark is tat previous results obtained for te un-coded case [4] are confirmed. Tus, all te WOFDM transmissions ave smaller BER and FER tan in te OFDM for bot Doppler sifts taken into account. R E B R E F - fm=,5 WOFDM HAAR fm=,5 WOFDM DAUBECHIES2 fm=,5 OFDM fm=, WOFDM HAAR fm=, OFDM fm=, WOFDM DAUBECHIES SNR(dB) Figure 5: BER performance of MBTC-WOFDM and OFDM in te Rayleig flat fading cannel. spectacular as in te un-coded case, due to te errors correction power given by te turbo-code. Regarding te WOFDM metods, te only difference in te performance of te system issued by te wavelet moter used is a small gain of Haar-based WOFDM compared to Daubecies-2, only noticeable at f m =.. Surprisingly, wile tere is no Doppler-issued difference for te OFDM transmission, in te case of WOFDM, te performance is better at iger Doppler sifts. Our guess is tat tis beavior is generated by te way tat te erroneous bits are distributed over te data blocks, before te decoding algoritm is applied. Toug, a conclusion in tis direction sall be drawn only after carrying out a study of te error distribution. V. CONCLUSIONS AND FURTHER WORK In tis paper we evaluate te BER performance of MBTC associated wit two multicarrier modulation tecniques: OFDM and its wavelet based version, in a flat fading cannel. Te WOFDM provides better results tan OFDM, especially for ig values of te Doppler sift parameter, wic quantifies te time variability of te cannel. Furtermore, te coded WOFDM performance sows no significant dependency on te wavelets moter used for DWT computation. A surprising result of our simulations is tat, unlike for te un-coded system, te BER and FER performance for te MBTC-WOFDM is better for iger values of te Doppler sift. In order to rigorously explain te later result, a detailed researc will be conducted in te future, aiming to study ow te erroneous bits are distributed over te transmitted blocks. By te oter and, as noticeable from figures 5 and 6, te gap between OFDM and WOFDM enlarges wit increasing SNR values. An extended range for tis parameter will allow to better igligt tis effect in future works. We will also consider for our furter researc te second very important caracteristic of te radio cannels, besides teir time variability: te frequency selectivity. ACKNOWLEDGEMENT -2-3 fm=,5 WOFDM HAAR fm=,5 WOFDM DAUBECHIES2 fm=,5 OFDM fm=, WOFDM HAAR fm=, OFDM fm=, WOFDM DAUBECHIES SNR(dB) Figure 6: FER performance of MBTC-WOFDM and OFDM in te Rayleig flat fading cannel. Te OFDM curves represent te worst case plots from te figures above. Te WOFDM performance improvement is iger for f m =., were for a BER of., tis tecnique brings a gain of approximately.4db. Te remark is valid for te FER measure too, wit approximately te same gain for a FER of.5. Te gain brougt by WOFDM is not as Tis study was conducted in te framework of te researc contract for young P.D. students, no.4/27 sponsored by CNCSIS. REFERENCES [] M. J. Manglani and A. E. Bell, "Wavelet Modulation Performance in Gaussian and Rayleig Fading Cannels" Proceedings of MILCOM 2, McLean, VA, October 2. [2] F. Zao, H. Zang, D. Yuan, "Performance of COFDM wit Different ortogonal Basis on AWGN and frequency Selective Cannel ", Proeedings. of IEEE International Symposium on Emerging Tecnologies: Mobile and Wireless Communications, Sangai, Cina, May 3 June 2, 24, pp

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