Convolutional Coding and Decoding in Hybrid Type-II ARQ Schemes on Wireless Channels Sorour Falahati, Tony Ottosson, Arne Svensson and Lin Zihuai Communication Systems Group, Department of Signals and Systems, Chalmers University of Technology, SE-412 96 Goteborg, Sweden phone: +46 31 772 1767, fax: +46 31 772 1748 email:fSorour.Falahati, Tony.Ottosson, Arne.Svensson,
[email protected]
Abstract | In this work we studied the performance of the hybrid type-II Automatic Repeat reQuest (ARQ) system on Rayleigh fading channels. This system is based on High Rate Optimized Rate Compatible Punctured Convolutional (HRO-RCPC) codes combined with simple repetition codes. The performance is investigated for di erent packet sizes and constraint lengths of the convolutional encoder. Additionally the e ect of packet size on the performance in di erent fading environments is studied in more detail. We have also studied the performance of the HRO-RCPC codes combined with optimized Rate Compatible Repetition Convolutional (RCRC) codes as another alternative for channel coding in hybrid type-II ARQ schemes. The simulation results show that HRO-RCPC codes combined with optimized RCRC codes with high parent code rate perform almost as good as the HRO-RCPC codes with simple repetition codes at lower parent code rate. Also a tailbiting decoder using the Circular Viterbi Algorithm (CVA) is examined for di erent hybrid type-II ARQ schemes and packet sizes. It is shown that the improvement of the performance due to tailbiting is noticeable only for short packets. For long packets, the sub-optimality of CVA decoding compared to the General Viterbi Algorithm (GVA) degrades the performance considerably. Keywords | Rayleigh fading channels, BPSK modulation, rate compatible convolutional codes, hybrid type-II ARQ, tailbiting decoder.
I. Introduction
In data communications where a return channel is available, ARQ schemes can be employed in order to provide almost error free reception. By applying channel coding in ARQ schemes, data bits can be protected against channel impairments. However, on time varying channel, it is reasonable to adapt the redundancy bits for error correction to the channel variations. Therefore, transmission of unnecessary redundancy bits can be avoided. This scheme is called hybrid type-II ARQ where RCPC codes are wellknown candidates for the channel coding [1{10]. Based on the results of our previous study presented in [10], a hybrid type-II ARQ scheme named Scheme 5 is proposed which provides the highest throughput in all the cases compared to other examined hybrid ARQ schemes. In this work we examine the performance of Scheme 5 based on the HRO-RCPC codes combined with simple repetition codes for di erent constraint lengths and information packet sizes. The system performance is measured as the throughput of the system de ned as the inverse of the average number of transmitted symbols per error free detected data bit [1, 2]. We have also studied the perfor-
mance of the HRO-RCPC codes combined with optimized RCRC codes as another alternative for channel coding in hybrid type-II ARQ schemes. Furthermore a tailbiting decoder based on the Circular Viterbi Algorithm (CVA) [11] is simulated for di erent hybrid type-II ARQ schemes and packet sizes. The following two Sections II and III, contain brief explanations of the system model and the di erent ARQ schemes. Section IV describes the hybrid type-II ARQ schemes based on HRO-RCPC codes in combination with simple repetition codes. It is followed by Section V which explains the hybrid type-II ARQ schemes based on HRORCPC codes in combination with optimized RCRC codes. A special case of tailbiting decoding, named CVA decoding, is described in Section VI. The corresponding numerical results of Sections IV-VI are presented in Section VII. Finally some conclusions are drawn in Section VIII. II. System Model
The hybrid ARQ schemes that are presented in this paper are based on the RCC codes [6]. The candidate code word for transmission is antipodal modulated where the energy per coded bit is denoted by Ec . Then channel symbols are interleaved and packed into blocks of xed length, which are referred to as channel blocks, and then transmitted over a Rayleigh fading channel. The channel block length is denoted by Lc. The statistical characteristics of the channel are assumed to be independent for each channel block transmission. The fading process is generated by the Jakes model [12]. The received symbols are deinterleaved and fed into the maximal ratio combining receiver [13] which is assumed to have perfect channel state information (CSI). Soft decoding is then performed by a Viterbi decoder. More details may be found in [10]. III. Description of Hybrid Type-II ARQ Schemes
We assume a selective repeat (SR) ARQ protocol with in nite bu ers in the transmitter and the receiver. The feedback channel is assumed to be error free. In all the schemes the erroneously received channel blocks are not discarded but combined in an optimum way with the newly received block(s) for a given packet. The considered schemes are brie y described in the following. The relation between the data packets and the channel blocks are shown in Fig. 1. Scheme 1 is a simple ARQ scheme in the sense that no
divided into three channel blocks of length Lc. All the incremental code words in Scheme 4 and Scheme 5 have a length of Lc except for the rst code word C1 in Scheme 5 which has a length of 2Lc that is divided into two channel blocks after interleaving. In all the hybrid schemes, the transmission starts with the code of the highest rate. If the code word is received in error, a retransmission is requested and the transmitter sends the incremental code word of the next lower rate. This code word is combined at the receiver with previously erroneously received incremental code words. Then it is decoded and checked for existence of a detectable error. This procedure repeats until error free reception is achieved. If the received code word of rate 1=3 still fails in correcting all the detectable errors, the code of lower rates are used. The details about the code words at lower rates at this stage are more clari ed in Sections IV and V. IV. HRO-RCPC Codes in Hybrid Type-II ARQ Schemes
As a result of extensive computer search, good convoFig. 1. The diagrams of the data blocks and channel blocks in the lutional codes for parent rates 1=2; 1=3; 1=4 and constraint ARQ schemes. channel coding for error correction is applied. Each information block is concatenated by np parity bits for error detection to form the code word C0 with length L which is transmitted over the channel. The received packet is combined with the previously erroneously received packets, if they exist. In case of error detection a retransmission is requested until an error free reception is achieved. Schemes 2 to 5 are type-II hybrid ARQ schemes. In all the hybrid schemes C0 with length L is the input to the convolutional encoder. It contains the information bits and np parity bits for error detection. A zero tail corresponding to the memory of the convolutional encoder is included in C0 when the (general) Viterbi algorithm is used for decoding (see Section VI). By puncturing or repeating the encoded bits in a proper manner [5, 6] rate compatible code words at higher or lower code rates can be obtained, respectively. As it is shown in Fig. 1, each hybrid scheme contains a set of rate compatible codes with corresponding code rates Rk where k 1. The code rates are given in decreasing order such that Rk > Rk+1 . Ck contains the incremental redundancy bits in the code word at the rate Rk which are not included in the code words of the higher rate codes and sometimes is referred as incremental code word. The interleaving is done over the incremental code word. Schemes 2 and 3 apply codes of rates 2=3 and 1=3. Scheme 4 uses codes of rates 1, 1=2 and 1=3. The code rates 1, 2=3, 1=2, 2=5 and 1=3 are used in Scheme 5 where the step between two consecutive code rates is smaller compared to the other schemes. The encoder input C0 is twice as large in Schemes 3 and 5 compared to Schemes 2 and 4. In Scheme 2 each incremental code word has a length of 1:5Lc which is transmitted in one channel block. In Scheme 3 each incremental code word after interleaving is however
length 3 ? 15 are obtained and presented in [14]. These codes are based on the Optimum Distance Spectrum (ODS) criterion where the codes have the maximum free distance and provide a low information error weight on each error path. A convolutional code of rate 1=n and constraint length K can be punctured periodically with period p to provide codes of higher rates. These codes will be rate compatible if the symbols of the lower rate codes are used in the higher rate codes as well. By searching for an optimum puncturing pattern which satis es both the ODS and rate compatibility criteria, optimum RCPC codes with higher rates are found. This search can be done in two directions, i.e. from low rate to high rate and vice versa which are referred as Low Rate Optimized RCPC (LRO-RCPC) and High Rate Optimized RCPC (HRO-RCPC) codes, respectively. Comparing the LRO-, and HRO-RCPC codes to OPTimum Punctured Convolutional (OPT-PC) codes shows that both on Rayleigh fading and Gaussian channels, the performance of RCPC codes are very close to that of the OPT-PC codes [14]. The HRO-RCPC codes of parent code rate 1=3 and puncturing period equal to 2 is given in Table I where K denotes the constraint length of the encoder, (g0 ; g1 ; g2 ) represents the generator polynomials in octal form and PRk is the puncturing matrix for the corresponding code rate Rk . The PRk elements are zeros or ones corresponding to deleting or keeping the corresponding bits. We would like to mention that since the RCPC codes with rate 1 are catastrophic, the ODS criterion can not be used for optimizing P1 . Therefore in order to select one of them, we have used both of them in our simulations for hybrid ARQ schemes and selected the one which provided higher throughput in the most cases. The results are given in Section VII. The transmission of the incremental code words starts with the code of the highest rate and continues with the incremental code words corresponding to lower rates un-
TABLE I HRO-RCPC codes of parent code rate
K 3
(g0 ; g1; g2 ) (5; 7; 7)
4
(13; 15; 17)
5
(25; 33; 37)
6
(47; 53; 75)
7
(133; 165; 171)
8
(225; 331; 367)
9
(575; 623; 727)
P=
2 3
0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0
1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 1
P=
1 2
1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1
1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 1
TABLE II
13
The RCC codes of parent code rate
=
P=
2 5
1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1
1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1
til error free reception is acknowledged. If the received code word of parent rate 1=3 is still not able to correct all the detectable errors, the already transmitted incremental code words are repeated in the same order until no error is detected. Obviously at this stage the received code words are simple repetition code words. Therefore in this method both the punctured codes and simple repetition codes are employed. V. Optimum RCRC Codes in Hybrid Type-II ARQ Schemes
As it is described in Section IV, HRO-RCPC codes in combination with simple repetition codes is a good candidate for channel coding in hybrid type-II ARQ schemes. Another approach is applying optimized RCRC codes instead of simple repetition codes at rates below the parent code rate. Since the RCRC codes are rate compatible with the code of parent rate, they are rate compatible with the corresponding RCPC codes of higher rates. Thus the two families of RCPC codes and RCRC codes form a family of RCC codes. The repetition pattern denoted by Q, is similar to the puncturing matrix P. The only di erence is that the elements of P are zeros and ones, corresponding to deleting or keeping the corresponding symbol, whereas the elements of Q are either equal to one or greater than one, indicating the number of duplications of the corresponding symbol [5]. Optimization of the RCRC codes are based on the ODS and rate compatibility criteria. The optimum RCRC codes are obtained from a computer search for di erent parent code rates [5, 15]. We would like to mention that in the search method for RCRC codes proposed by Kallel in [5], the maximum di erence among the elements in repetition matrix is zero or one. However this search limitation is removed in [15]. In Table II the search results for the puncturing/repetition patterns for the RCC codes of parent code
P=
2 3
1 0
1 1
P=
1 2
1 1
1 1
Q=
2 5
2 1
1 1
1 2, =
Q=
1 3
2 1
1 2
K
= 7 , (133 171) ;
Q=
2 7
2 1
2 2
Q=
1 4
2 2
2 2
rate 1=2 with generator polynomials (133; 171) in octal form, constraint length 7 and puncturing/repetition period 2 are given. VI. CVA Decoding in Hybrid Type-II ARQ Schemes
In general there are three classes of Viterbi algorithm decoders for convolutional codes. The rst class is continuous decoding with a nite path memory. The second class is block-wise decoding with a terminating tail which is known to the decoder and the third class is the block-wise decoding with no tail. The latter class is known as the decoding of tailbiting convolutional codes. In this case the starting and ending states are the same in the encoder but unknown to the decoder. Di erent algorithms for tailbiting decoding are introduced in the literature (see e.g. [11]). These algorithms are used in the block-wise transmission to save the overhead of a known tail. One of the algorithms is the Circular Viterbi Algorithm (CVA) which is proposed by R. Cox et. al [11] and is investigated in this study. The basic idea of CVA for decoding tailbiting convolutional codes is as follows: The received block is recorded at the receiver. Then the Viterbi algorithm for continuous decoding is applied to a sequence of repeated recorded blocks, thus simulating the situation where the same information is transmitted repeatedly in a continuous form. By connecting repeated versions of the same block together, the correct decoded path will automatically satisfy the condition that the ending and starting states are the same. Fixed or adaptive stopping rules to stop the decoding have been proposed. We have studied the second adaptive stopping rule proposed by R. Cox et. al [11]. The second stopping rule is brie y described in the following. Let N denote the length of the received block, m the memory size of the convolutional encoder and s(tk) the Viterbi metric for state k after the update for the received symbol at time t where k = 0; 1; : : : ; 2m ? 1. Also one complete metric update for all states in the trellis at one time unit is referred to a Viterbi Update (VU). After each VU, there is one survival path out of the two paths for every state in the trellis. Therefore the result of the decision at each state can be stored by one bit where zero or one corresponds to whether the lower or higher previous state is chosen. Since there are 2m states at each time unit, the decision results of each VU is stored in a word containing 2m bits where each bit corresponds to one state. This word is called as decision word. The decoding begins with the initial conditions, i.e.
1
0.9
0.9
0.8
0.8
0.7
0.7
Normalized Throughput
Normalized throughput
1
0.6 0.5 K=3 K=4 K=5 K=6 K=7 K=8 K=9
0.4 0.3 0.2 0.1 0
100
200 300 400 500 The length of the convolutional encoder input − L
0.6 0.5 0.4 0.3
SNR = 20 dB SNR = 15 dB SNR = 10 dB SNR = 5 dB SNR = 0 dB
0.2 0.1 600
0
100
200 300 400 500 The length of the convolutional encoder input − L
600
Fig. 2. Simulated normalized throughput for Scheme 5 with nor- Fig. 3. Simulated normalized throughput for Scheme 5 with conmalized Doppler frequency 0.001. Solid and dash-dotted lines straint length 7. Solid and dash-dotted lines corresponds to norcorresponds to 15 dB and 5 dB SNR respectively. malized Doppler frequencies 0.001 and 0.01 respectively.
s(0k) = 0 for all k, k = 0; 1; : : : ; 2m ? 1. After the rst N the following Subsections VII-A, VII-B and VII-C, respec-
VUs are completed, a comparison of decision words which are separated by N symbol times, begins. When m consecutive decision words have been observed to be same as their predecessors, the updating is stopped. Then the trace back procedure starts. We start from the state with the best metric at the stopping time (the ending state) and trace N trellis section back to nd the corresponding starting state by using the decision words. If the ending and starting states are the same, the decoding has converged and the information bits can be released. Otherwise we repeat same procedure starting from the state with the second highest metric and so on. If the above check fails for all the states at the stopping time, we try tracing back one more symbol and compare the two new states which are N symbols apart. Eventually convergence is assured. It is obvious that when the CVA decoder is applied, the encoder input does not contain any tail. Therefore in our hybrid schemes C0 just includes the information bits and parity bits for error detection. VII. Numerical Results
The constraints on simulations are as the following. We assume that an equal number of parity bits for error detection for a given size of the data blocks should be used for all the channel conditions in order not to have any undetected errors. Therefore the shortest parity bits which ful ll this requirement are selected. BPSK modulation and block interleaving over the incremental code words are applied. The Rayleigh fading process with unit average power for the fading envelope is generated by using the Jakes model [12]. Furthermore the maximum ratio combining receiver with perfect CSI and soft decisions uses the Viterbi algorithm decoder at the parent code rate. Each simulation was continued until 1000 data blocks were received correctly. The details about the simulations and the numerical results of the Sections IV, V and VI are given in
tively. A. Numerical Results based on HRO-RCPC Codes In our previous study, presented in [10], hybrid typeII ARQ schemes based on the LRO-RCPC codes of the parent code rate 1=3 are investigated. The results show that Scheme 5 provides the highest throughput in all the cases. Here, we study the performance of Scheme 5 based on HRO-RCPC codes of the same parent code rate. We mainly focus on the e ects of the data packet size and the constraint length of the encoder on the performance. The code table for HRO-RCPC codes is given in Table I. Also we choose np = 8 for L = f80; 160g, np = 12 for L = f240; 320g and np = 16 for L = f400; 480; 560; 640g. Please note that a general Viterbi algorithm for block-wise decoding is applied here. Thus C0 is appended by a tail of K ? 1 zeros. The simulations have been done for slow and fast fading environments with the normalized Doppler frequencies 0:001 and 0:01 respectively. Some of the results are presented in Figs. 2-4. The comparisons of throughput performance of the proposed Scheme 5 for di erent packet sizes and constraint lengths for slow fading environment is given in Fig. 2. The solid lines show that the di erence in the throughput due to the di erent constraint lengths at fairly high SNR, e.g. 15 dB, is small. We can see that by increasing the packet length, the tail overhead becomes less e ective on the throughput than the strength of the code given by the constraint length. Also the dashed-dotted lines in Fig. 2 show that at lower SNR, where more of the received symbols are in error, codes with larger constraint length perform better than the codes with smaller constraint length, even for large packets. However the di erence becomes smaller for short packets due to the advantage of having short packets in fading environments. Furthermore, the codes with constraint length 6 or 7 perform equally well as the codes with the constraint length
1
L = 80 L = 160 L = 240 L = 320 L = 400 L = 480 L = 560 L = 640
0.9 0.8 0.7 0.6 0.5
0.9 0.8 Normalized throughput
Freq. of correct reception per information packet
1
0.4 0.3
0.7 0.6 0.5 0.4 0.3
0.2
0.2
0.1
0.1
0 0
1
2 3 4 5 Number of transmitted code words
6
7
0 0
Scheme 2 Scheme 3 Scheme 4 Scheme 5 5
10 Average Ec/No (dB)
15
20
Fig. 4. Simulated histograms of the number of transmitted code Fig. 5. Simulated normalized throughput for 128 bits information words for Scheme 5 with constraint length 7, 5 dB SNR and blocks, 12 CRC bits and normalized Doppler frequency 0.01. normalized Doppler frequency 0.001. Solid and dash-dotted lines correspond to RCC codes at parent rate 1/2 and 1/3 respectively.
9. Hence the tail overhead can be reduced without loosing throughput. The normalized throughput versus the packet length for di erent SNRs and fading rates are plotted in Fig. 3. These results are given for K = 7. It can be seen that in general the changes due to the packet length are quite small. In addition, at low SNRs, noise is more defective than the fading. Therefore the di erence in throughput due to the fast and slow fading, becomes quite negligible. However the di erence grows at higher SNRs. We have also plotted the histograms of the number of transmitted code words at 5 dB SNR and slow fading in Fig. 4 for K = 7. It can be seen that most of the transmissions at rate 1 are not successful except for short packets. The largest probability of error free receptions happens at rate 2=3 and the probability increases with increasing packet length. Most of the packets are received correctly at the parent code rate 1=3. These and similar simulation results are given in more detail in [16]. B. Numerical Results based on Optimum RCRC Codes Instead of using the RCPC codes combined with the simple repetition codes, the RCPC codes combined with the optimized RCRC codes can be used in the hybrid type-II ARQ schemes. Furthermore, the analytical and numerical results given in [10,16] and Section VII-A show that most of the data packets are received correctly at the parent rate 1=3 and retransmission at this rate is seldom requested. Therefore we decided to apply the HRO-RCPC codes combined with the optimized RCRC codes at parent code rate 1=2, and compare them with the HRO-RCPC codes combined with the simple repetition codes at the parent code rate 1=3, in our hybrid schemes. The constraint length of both codes is 7. The code table for the RCC codes at parent rates 1=3 and 1=2 are given in Tables I and II, respectively. The simulation results for L = 128, np = 12
and normalized Doppler frequency 0:01 are given in Fig. 5. We would like to mention that in this case the general Viterbi algorithm is used for block-wise decoding. Thus a tail containing 6 zero bits is included in C0 (see Fig. 1). In Fig. 5 we compare the throughput of the proposed hybrid type-II schemes based on the RCRC codes (solid lines) and RCPC codes (dash-dotted lines). We can see that the di erence in throughput is small but with some advantage to the RCPC/simple repetition codes for fast fading environments. Similar results are found for slow fading channels [16]. C. Numerical Results using CVA decoding In order to study the e ect of the CVA decoding on the performance of the hybrid type-II ARQ schemes, we have simulated our hybrid schemes with the tailbiting RCC codes and CVA decoding. In Fig. 6, the simulated throughput performance for the tailbiting RCC codes of parent code rate 1=2 with K = 7, using CVA decoding are compared with the corresponding results of the same codes but with tail and General Viterbi Algorithm (GVA) decoding. More details about the codes are given in Table II. The results are given for L = 128, np = 12 and normalized Doppler frequency 0:01. In hybrid ARQ schemes with CVA decoding, the input of the encoder contains more information bits due to the lack of tail. This property improves the throughput. On the other hand, CVA decoders are sub optimum compared to GVA decoders where the starting and ending states are known to the decoder. In Fig. 6 we can see that for Schemes 3 and 5 where the encoder input blocks are larger, the GVA decoder gives the best performance. For Schemes 2 and 4 where the encoder input blocks are shorter, the CVA decoder gives a small gain at high SNRs. Simulation results have also shown that different puncturing matrices at rate 1 in schemes 4 and 5, may change the throughput signi cantly. In general the
for short packets, otherwise the suboptimality of CVA decoding compared to the General Viterbi Algorithm (GVA) decoding degrades the performance considerably.
1 0.9
Acknowledgment
Normalized throughput
0.8
This work has been performed in the framework of the project ACTS AC090 FRAMES, which is partly funded by the European Community.
0.7 0.6 0.5 0.4 0.3 Scheme 2 Scheme 3 Scheme 4 Scheme 5
0.2 0.1 0 0
5
10 Average Ec/No (dB)
15
20
Fig. 6. Simulated normalized throughput for 128 bits information blocks, 12 CRC bits and normalized Doppler frequency 0.01. Solid and dash-dotted lines correspond to CVA decoding and GVA decoding for RCC codes at parent rate 1/2 respectively.
CVA decoders do not perform well at code rate 1 but a good choice of the puncturing matrix at rate 1 might provide a considerable gain. The results show that the GVA decoder usually results in better performance compared to the CVA decoder in all the cases because the suboptimality of CVA decoding degrades the performance considerably. VIII. Conclusion
This work is a continuation of the previous studies for hybrid type-II ARQ schemes on Rayleigh fading channels [10] where a hybrid scheme named Scheme 5, was proposed. It was shown that Scheme 5 performs very well compared to the other hybrid schemes and provides the highest throughput in all the examined situations. In this work the performance of Scheme 5 based on HRO-RCPC codes in combination with simple repetition codes for di erent constraint lengths and packet sizes are investigated. The simulation results show that the HRO-RCPC codes with intermediate constraint lengths perform equally well as the ones with longer constraint lengths. Furthermore, we show that in most cases, short packets provide higher throughput in a fading environments. Our results also show that by applying HRO-RCPC codes in hybrid type-II ARQ schemes for long packets, good performance in fading environments can be still obtained. We have also studied the performance of the HRORCPC codes combined with optimized RCRC codes as another alternative for channel coding in hybrid type-II ARQ schemes. The simulation results show that these codes with high parent code rate almost perform equally well as the HRO-RCPC codes with simple repetition codes at lower parent code rate. Finally tailbiting decoding with a Circular Viterbi Algorithm (CVA) is examined for di erent hybrid type-II ARQ schemes and packet sizes. It is shown that the improvement in the performance due to tailbiting is noticeable only
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