Hybrid Type-II ARQ Schemes for Rayleigh Fading Channels

Transcription

1 Hybrid Type-II ARQ Schemes for Rayleigh Fading Channels Sorour Falahati and Arne Svensson Dept. of Signals and Systems, Communication Systems Group Chalmers University of Technology, E Göteborg, Sweden Phone: , Fax: Sorour.Falahati@s2.chalmers.se, Arne.Svensson@s2.chalmers.se ABSTRACT In this paper the throughput performance of Simple Automatic Repeat request (ARQ) and Hybrid type-ii ARQ schemes over Rayleigh fading channels are studied. Some new hybrid type-ii schemes based on Rate Compatible Punctured Convolutional (RCPC) codes are proposed which can be regarded as a combination of simple ARQ and traditional hybrid type-ii ARQ schemes. It is shown that the suggested scheme provides a significantly better performance compared to other schemes. The transmitted bits are packed into channel blocks of fixed length. The effect of the channel block length on throughput is studied. Numerical results for normalized Doppler frequencies. and. are presented. I. INTRODUCTION In data communications, error free reception is strongly required. In ARQ strategies, when an erroneous packet is detected, a retransmission request is sent to the transmitter until an error free detection is done. A simple ARQ strategy does not apply channel coding while in hybrid ARQ strategies, channel codes for error correction are utilized. Since the channel is time varying the redundancy bits for error correction can be adapted to the channel condition and transmission of unnecessary redundancy bits can be avoided. This property provides a subset of hybrid schemes known as hybrid type-ii ARQ schemes. Analysis of such schemes for binary symmetric channels (BSC) and AWGN channels have been previously presented []-[5]. On fading channels, which is the case for wireless communications, fading of the received signal causes errors to appear more frequently when the channel SNR is poor. Some researchers have studied the performance of different ARQ strategies on the fading channel [6]-[9]. We propose some new hybrid schemes based on RCPC codes and compare their performance on fading channels with previously studied schemes. The system performance is measured by throughput of the system defined as the inverse of the average number of transmitted bits per each error free detected data bit [],[2]. A true fading channel modelled by the Jakes model [] has been used during simulations while the analysis has been based on the perfectly interleaved fading channel. Furthermore, transmitted bits have been packed into channel blocks which are sent over the channel. The effect of channel block length on the system performance is shown by studying different channel block lengths. The main application considered here is UMTS or IMT2, i.e. third generation wireless personal communication systems. These systems will carry both circuit switched and packet data traffic. In this paper we are interested in finding good channel coding methods for the packet data schemes, especially for Rayleigh fading channels. In general the methods and results are valid for all applications where data blocks are transmitted over fading channels. The results show that the proposed scheme using five different coding rates significantly improves the throughput. The largest improvements are obtained for short channel blocks on slow fading. In the following, section 2 explains the system model. Section 3 describes the different ARQ schemes studied in this work. Section 4 gives the analytical description of the throughput performance. Section 5 presents the simulation model and the numerical results. Conclusions and future work are described in section 6. II. SYSTEM MODEL The ARQ schemes that are studied in this paper employ RCPC codes based on a rate /3 mother code similar to the ones presented in [6]. The transmitted signal samples are packed into blocks, referred to as channel blocks of fixed length. The channel block length is denoted by L c. These blocks are sequentially transmitted over the channel while the statistic characteristics of the channel are independent for each transmission. A discrete model for the channel is shown in Fig.. The input sample to the channel is x mn, where m is the index for the channel block and n for the sample within the block, with antipodal modulation equal to ± E c, where E c is the energy per coded bit. α mn is the sample of the fading envelope which is a random variable having a Rayleigh distribution with the probability density function α mn 2 f α ( α mn ) = () σ 2 exp( α mn 2σ 2 ); α mn 2 where E( α mn ) = 2σ 2 is the average power of the fading envelope. The fading process is generated by the Jakes model []. Its autocorrelation function is given by [] E α mn e jθ mn α mn ( + p) e jθ mn ( + p) = J ( 2π f d Tp) E α mn e jθ mn α ( m + q) ( n + p) e jθ ( m+ q) ( n + p) = ; q where θ mn is the phase of the fading process, J ( x) is the zero:th order Bessel function, f d is the maximum Doppler frequency and T is the channel symbol interval. Since perfect coherent detection is assumed in this paper, θ mn is considered perfectly known and therefore it is not included in Fig.. The term n mn, is a sample of an AWGN with variance N o 2. The INT and DINT blocks in Fig. represent the interleaving and deinterleaving (2)

2 x mn INT. α mn n mn Fig.. The discrete model of the channel. DINT. y mn operations respectively. The unquantized received signal sample with coherent detector y mn, is equal to y mn = x mn α mπ ( n) + n mn (3) where π denotes the deinterleaving operation. Note that since the interleaved AWGN sample is also AWGN, the same indices can be used for the noise sample in (3). Our main interest in this paper is to design and analyse channel coding schemes that maximize the throughput over the channel. Therefore we assume optimum decoding of the received samples. The decoder is a Viterbi decoder for the rate /3 mother code used throughout the paper. When the channel code is punctured, a metric increment of zero is used for the punctured symbols. Soft decoding is applied and we assume that perfect channel state information (CSI) is available at the decoder. We will also present results for two branch diversity with ideal maximal ratio combining [2]. III. DESCRIPTION OF ARQ SCHEMES We assume a selective repeat (SR) ARQ protocol with infinite buffers in transmitter and receiver. The feedback channel is error free. In all the schemes the erroneously received channel blocks are not discarded but combined with the new received block(s) for the given packet. The combining is done in an optimum way by using known CSI, time diversity and maximum ratio combining. This is equivalent to hybrid ARQ schemes employing repetition coding. The considered schemes are described in the following. Scheme is a simple ARQ scheme. By adding n p parity bits to each n -bit information packet, a code word with length L = n p + n and denoted by C is obtained and transmitted. The received packet is combined with the previously erroneous received packets if they exist. In case of error detection a retransmission is requested until an error free detection is done. Schemes 2 to 5 are type-ii hybrid ARQ schemes. To every n-bit information packet, n p parity bits for error detection and m zero tail bits corresponding to the memory of the convolutional encoder are appended to form the code word C with length L = n + n p + m, which is the input to the RCPC encoder with code rates R k for k =,, K where K is the number of code rates. The code rates are given in decreasing order such that R > R 2 > > R K. The mother code for all codes is a (3,,6) convolutional code. The period of the puncturing matrix is two. The code word symbols of the rate R k code is given by C C 2 Ck, where C k denotes the code word symbols that are punctured in the rate R k code and C C 2 denotes the combination of code symbols in C and C 2. C k is sometimes referred to as the incremental code symbols or information. The data packets, and the channel blocks for the studied schemes are presented in Fig. 2. In scheme the packet of L c information plus parity bits form a channel block C of the same length. This block is retransmitted L = Lc Lc C C = C R = Scheme L = Lc.5Lc C R = 2/3, /3 C L = 2Lc 3Lc C C R = 2/3, /3 L = Lc Lc C C R =, /2, /3 C3 L = 2Lc 2Lc C C R =, 2/3, /2, 2/5, /3 C3 C4 C5 Fig. 2. The data and channel blocks relation diagram. until the packet can be detected free of errors. without packet combining is proposed in [7]. Here a packet of L c bits is encoded in a rate 2/3 encoder to obtain.5l c encoded bits which form the first channel block C. If the packet is detected in error after channel decoding of this block, the second block C 2 of.5l c encoded bits is sent. This block contains the coded bits of the rate /3 code that are punctured in the rate 2/3 code. The received samples corresponding to both blocks are combined and decoded in the Viterbi decoder of the rate /3 code. If the packet is still in error, C followed by C 2 are repeated until the packet is detected without errors. is a small modification of scheme 2. The data packet length is now 2L c. The coded symbols in C and C 2, are however divided into three channel blocks. These longer code words transmitted in several blocks, make the interleaving more effective which may improve the error correcting capability of the code. It also reduces the overhead in the information packet. Another modification of scheme 2 is scheme 4 with three code rates, /2 and /3 where a punctured code is used also for rate. Therefore the channel bits are not the same as the bits in the data packet. Its main difference with schemes 2 and 3 is that C includes very little redundance for error correction. In this way it has the possibility to improve throughput when the channel is good, since it resembles simple ARQ in the first step. In the second and third steps, redundant bits are available for error correction, which is helpful in noise and fading. is a combination of schemes 3 and 4. It starts with code rate and includes four additional code rates to obtain incremental information. The data packet and C are both of length 2L c, while the incremental code words C k for k > are of length L c. Thus C is divided into two channel blocks, while C k for k > is transmitted as it is. For all schemes interleaving is performed by a block interleaver operating over one single incremental code word only. This means that there is no interleaving between code words and interleaving does not add any extra delay.

3 .2 Scheme.2 Scheme Fig. 3. Analytical results for throughput without packet combining for L c = 64. Now the decoding is shortly described. We denote the received samples corresponding to the i :th transmission i of the k:th code word by D k. When i is larger than one, D k denotes the samples that are obtained after maximum i ratio combining of D k to D k, otherwise D is equal to k D k. Perfect CSI while combining samples is used. Obviously, if C k is not yet transmitted for the i:th time, D k i does not exist and is set to zero. In the Viterbi decoder of the rate /3 code, the received word is formed by D D 2 D K. Note that D k is zero for some of the largest k values if all the code words have not been transmitted for the first time. The received word is decoded by soft decision decoding and perfect CSI. After decoding error detection is performed. If the transmitted packet is detected in error, retransmission of the next code word is requested. Otherwise the packet is accepted. IV. THROUGHPUT ANALYSIS In this section, we analyse the throughput for different ARQ schemes according to the description in section 3. Ideal selective ARQ protocol with error free feedback channel is assumed. The receiver applies the VA decoder with perfect interleaving, soft decision and ideal CSI. For simplicity the analysis is done for the case without packet combining. This means that the received word used by l l l the decoder is D D 2 D k where l is the number of transmissions of C for the given packet. The throughput for scheme x, is given by η x = n l av, x (4) where l av, x is the average number of transmitted bits per each correctly detected data packet. For scheme, l av, is given by 2 l av, = L c ( + PR ( d ) + PR ( d, Rd ) + ) (5) i where PR ( d ) is the probability that i :th received sequence has been detected in error and L c = n + n p. Assuming that the probability is the same for each transmission and that transmissions are independent from each other, (5)can be approximated by l av, L c PR ( d ) P 2 L = ( + + ( R d ) + ) = c.(6) PR ( d ) For the uncorrelated Rayleigh fading channel PR ( d ) is given by [2] Fig. 4. Simulated results of throughput for L c = 64 and normalized Doppler freq..5 without packet combining. PR ( d ) ( q) n + n p = (7) with q = -- ( δ (8) 2 c ( + δ c )) where δ c is the average signal to noise ratio E c N o per coded symbol. The throughput for the hybrid schemes is described in detail in [6]. In general, l av is given by K k n l av = ( l k + nl K )( P E ( l k )) P E ( l i ) P e n = k = i = K k = (9) P e ( P E ( l k ))l k P E ( l i ) k = i = P K k e ( P e ) 2 l K ( P E ( l k )) P E ( l i ) k = i = where K P e = P E ( l j ) () j = and P E ( l ) = are assumed in (9). l k for k =,, K is the number of available received bits at the k :th decoding attempt and P E ( l k ) is the probability that the decoded code word is detected in error which is upper bounded by [6] L K n + n p + m P E ( l k ) < -- a () P d ( l k ) P d d= d free where a d ( l k ) is the distance spectra and P d is the probability that a wrong path at distance d is selected. On an uncorrelated Rayleigh fading channel with BPSK modulation, perfect channel estimation and soft decision decoding, P d is given by [2] P d q d d d + k k = ( q). (2) k k = Note that for schemes 4 and 5, P E ( l ) is given by P E ( l ) ( q) n + n p + m =. (3) By using the upper bound in () for P E ( l k ), the

4 Scheme Fig. 5. Simulated results of throughput for L c = 64 with normalized Doppler freq Scheme Fig. 7. Simulated results of throughput for L c = 256 with normalized Doppler frequency...2 Scheme Fig. 6. Simulated results of throughput for L c = 64 with normalized Doppler frequency...2 Scheme Fig. 8. Simulated results of throughput for L c = 256 with normalized Doppler frequency.. throughput according to (4) can be evaluated approximately. The analytical results for 64-bit channel blocks are given in Fig. 3. Since the analytical results are for perfect interleaving, we compare them with simulated results for fast fading which is given in Fig. 4 for normalized Doppler frequency.5 without packet combining. From these two figures, we conclude that the agreement between simulation and analysis is quite good. V. NUMERICAL RESULTS The numerical results of throughput for different schemes are presented in this section. In general the simulations are done for two values of L c, 64 bits and 256 bits and for various Rayleigh fading channels. Note that L c is the channel block length except for scheme 2. To every n data bits, n p = 6 parity bits for error detection are added. We have to remark that for both choices of L c all ARQ schemes were supposed to use an equal number of parity bits. In order not to have any undetected errors, 6 CRC bits were found as the shortest one which fulfil this requirement. A 6-bit zero tail is appended to the error detection code for hybrid ARQ schemes to form the resultant sequence C. The convolutional encoder uses a (3,, 6) mother code with generator polynomials (33, 65, 7) given in octal [3]. The puncturing matrix is a 3 2 matrix with period 2.The puncturing position is specified relative to the previous lower code rate and the additional puncturing position by the row and column number in the puncturing matrix. Then the positions (2,2), (2,), (3,2) and (3,) correspond to the code rates 2/ 5, /2, 2/3 and respectively. BPSK modulation and interleaving are applied. The Rayleigh fading process with unit average power for the fading envelope is generated by using the Jakes model []. The maximum ratio receiver with perfect CSI and soft decision uses the VA decoder of the rate /3 code. Each simulation was continued until data packets were received correctly. The comparisons of the throughput performance of all ARQ schemes with packet combining, L c = 64 and normalized Doppler frequencies. and. are given in Figs. 5 and 6. From Fig. 5 we can see that at slow fading, simple ARQ with code combining has high throughput at high SNR and only the amount of CRC bits limits the throughput to.75. At lower SNRs the throughput decreases since it becomes quite likely that blocks are in error at the receiver. improves the performance at very low SNRs compared to scheme because more powerful channel coding (compared to the repetition coding) is used and the rate /3 code is used in most cases. The code rate 2/3 provides the maximum throughput equal to 4 which is almost obtained at the right end of the figure. This scheme performs worse than scheme for the most interesting values of SNRs. Scheme

5 Correct received packets Freq..2 scheme scheme 2 scheme 3 scheme 4 scheme Number of transmitted code words Fig. 9. The histograms for normalized Doppler freq.., 5 db SNR and L c = 64. Correct received packets Freq..2 scheme scheme 2 scheme 3 scheme 4 scheme Number of transmitted code words Fig.. The histograms for normalized Doppler freq.., 5 db SNR and L c = 64. improves significantly with code combining while scheme 2 is almost unaffected since it uses error correction coding anyway. This can be seen in Fig. 4 although this is for faster fading. performs somewhat better than scheme 2 mainly because of a relatively smaller number of CRC bits. Therefore the maximum throughput now becomes.55 instead of 4. This difference helps at all SNRs. shows that starting transmission at higher rate and using a more powerful code at the first coding level (/2 instead of 2/3) provides large improvement in performance compared to scheme 2. performs almost as good as scheme 3 which has double size of data packet. It is also interesting to compare schemes 4 and since they use the same code rates when scheme is considered as repetition coding. is better at low SNRs due to the more powerful coding while scheme is better at high SNRs due to the absence of tail bits. uses a longer data packet and a larger number of coding rates but channel blocks of the same length as scheme 4 which improves the maximum throughput to almost. Also the average coding rate becomes larger due to the larger number of rates which improves the throughput over the whole range of SNRs. It is observed that scheme 5 outperforms all the other schemes at all SNRs. Please note that the maximum throughput of scheme 3, is obtained at an SNR of around 4 db with scheme 5. This is a significant improvement of about 6 db. may be further improved by maximal ratio combining of two antenna diversity branches as shown by the dotted curve. With faster fading errors appears more randomly. Therefore channel coding becomes more effective and the coding gain increases. This leads to a small increase in throughput of schemes 2 and 3. The throughput of scheme is reduced since error free transmission of uncoded blocks is less likely at faster fading. The degradation is even more significant without packet combining as can be seen in Fig. 4. The throughput of scheme 4 and 5 in the middle range of SNRs are reduced. The comparison of performance of schemes 4 and 3 in Figs. 5 and 6 shows the advantageous of dividing the long block into shorter blocks for transmission in fast fading environment. All the hybrid schemes have remained almost unaffected by increasing the fading rate at low and high SNRs. Longer packets generally improve the throughput since less CRC bits are used per information bit. The result of simulations for L c = 256 and for normalized Doppler frequencies. and. are given in Figs. 7 and 8, respectively. From these two figures we can see that with long packets (and slow fading) the difference between scheme 2 and 3, and 4 and 5 becomes smaller, since interleaving becomes more powerful and more powerful coding is generally needed (the probability of error free code words is reduced). The improvement of scheme 5 (and 4) compared to scheme 2 (and 3) also becomes smaller but the gain is still significant. With long channel blocks and faster fading, the improvement of scheme 5 (and 4) becomes even smaller compared to scheme 2 (and 3). At intermediate SNRs scheme 4 performs worse than scheme 2 and 3, but scheme 5 is still the best. At low SNRs, scheme 5 (and 4) still have much better throughput than all other schemes. A fast fading channel becomes less time varying in the sense that the average channel BER dominates over instantaneous channel BER. Therefore we expect the relative performance between schemes in a fast fading channel to become more similar to a static channel. In Figs. 9 and we show histograms of the number of transmitted code words (not channel blocks) for each successful data packet. Please note that data packets are of different length. The number of transmitted code words per data packet is related to the delay of the scheme and is therefore interesting to study. The results are given for 5 db SNR, L c = 64 and normalized Doppler frequencies. and., respectively. It is clearly seen that the histogram is more concentrated to the left for schemes 2 and 3. The average number of code words per data packet is here.2 and., respectively, at the given Doppler frequencies. At low Doppler frequencies schemes, 4 and 5 are quite similar. The average number of code words per data packet are.9,.5 and.9, respectively. Therefore scheme 4 has some advantages here. At higher Doppler frequencies schemes 4 and 5 are better than scheme. The average number of code words per data packet are now 2.2,.6 and 2. for schemes, 4 and 5, respectively. From these results it is clear that starting the transmission at rate and using more rates leads to a somewhat higher probability for several code words per data packet but these schemes are not worse than simple ARQ with code combining in this context. Simple ARQ without code combining is much worse. Finally it is interesting to note that almost in all cases, even at low SNRs like 5 db, successful decoding is achieved before C is retransmit-

6 ted for the first time for schemes 2 to 5. Therefore code combining does almost not affect the performance. For scheme on the other hand, C is retransmitted several times. VI. CONCLUSIONS AND FUTURE WORK In this paper we propose some new coding schemes for packet transmission on Rayleigh fading channels. The throughput is analysed by an approximate theoretical analysis and also simulated for various channel block lengths and fading rates. A concatenation of a CRC code and an RCPC code is used for error detection and error correction, respectively. Decoding is done by using a Viterbi decoder and we assume soft decision with perfect channel state information. Fading is assumed independent between blocks but correlated within blocks. The Jakes model has been used to model the fading within a block. Interleaving is only done within each code word. The proposed scheme 5 significantly improves the throughput and/or SNR both with short and long channel blocks and both with slow and fast fading. The gains are however largest for short blocks and slow fading. The general conclusion is that short blocks in combination with rate compatible codes (or varying coding rates) seems to have some advantage in slow fading. Short blocks increase the probability for error free transmission in fading (without interleaving between blocks) and that leads to a higher average coding rate in the variable rate code, which improves the throughput. With faster fading the probability for errors in a channel block increase. Therefore more coding is needed, but still it is advantageous to try to maximize the average coding rate. This is done by using many code rates in the proposed schemes. The only disadvantage with short packets may be the CRC overhead. With longer packets, the packet error rate varies less between packets and the channel becomes more like a static channel. Then the improvements becomes smaller but are still there. There is never any need to code more than necessary and this may be obtained by using variable rate coding as done here. The improved throughput is at the expense of a larger number of retransmissions in the sense that more code words are needed on average to be transmitted per data packet. This causes extra delay in the system. The proposed schemes are still not worse (and sometimes better) than simple ARQ with code combining and much better than simple ARQ without code combining in this context. More studies are however needed on delay aspects. An open question is optimizing the number of CRC bits needed to obtain a low enough undetected error rate. Another issue that is to the disadvantage of short packets is the encoder tail bits. This can be solved by using tailbiting decoders and this is also an issue for future work. It is also not clear how to interleave the channel. Interleaving over many bits gives improved coding performance but it is difficult to combine with transmission of incremental information. It also causes longer delays which is a problem of a coding scheme using many code rates. ACKNOWLEDGEMENT We would like to acknowledge the fruitful discussions and help from Dr. Tony Ottosson. The work presented in this paper is partly financed by the projects ACTS AC9 FRAMES which is founded by the European community. REFERENCES [] S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamental and Applications, Englewood Cliffs, N. J., Prentice-Hall, 983. [2] S. B. Wicker, Error Control Systems for Digital Communications and Storage, Englewood Cliffs, N. J., Prentice-Hall, 995. [3] Y. M. Wang and S. Lin, A Modified Selective Repeat Type-II Hybrid ARQ System and Its Performance Analysis, IEEE Trans. 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