Iterative Decoding vs. Viterbi Decoding: A Comparison Santosh Shah and V. Sinha The LNM Institute of Information Technology, Jaipur 303012, India Email: [email protected], [email protected] Abstract- Decoding algorithms employed in wireless communication use either hard decision decoding or soft decision decoding. Quantization is use in soft decision e.g. in Viterbi decoding. Similarly in iterative decoders (Turbo Codes) either serially concatenated or parallely concatenated decoders with interleavers use soft-input soft-output (SISO) and additive loglikelihood version (L-SISO) algorithms, to compute a posteriori probability (APP). Even though soft decision is more powerful than hard decision decoders, many systems can not use soft decision algorithms, e.g. in GSM. In GSM channel decoding is used after de-interleaving. So, if one were to use iterative decoding, one is constrained to hard decision decoders. We propose the uses of soft decision iterative decoders (Turbo codes) in place of hard decision data (i.e. data in binary format). We compare the performance of hard decision Viterbi decoding with iterative decoders.

I. INTRODUCTION We assume that the channel receives data in the binary format. There are various techniques to implement channel coding and decoding, with either hard decision decoding or soft decision decoding algorithms. There may be two situations for GSM application, one in which soft decision is used [1], the other in which conventional hard decision is used. The authors [1] use soft decision Viterbi decoding algorithm. They have used matched filter and performed the soft decoding Viterbi decoding algorithm. Thus they have proposed one kind of receiver structure for GSM system, which is not conventional. Since soft decision provides improved performance we would like to investigate the soft decision algorithm but using a turbo code. We study Serially Concatenated Convolutional Codes (SCCC) [5], [6], an algorithm based on soft-input soft-output (SISO) and additive log-likelihood version (L-SISO), to compute a posteriori probability (APP) with a random interleaver [3] between convolutional coders [2]. APP decoder is a two-input two-output device that accepts the quantity Lu(I), Lc(I) as inputs, and outputs the quantity Lu(O), Lc(O). The input Lu(I) represents the sequence of log-likelihoods of

encoder input bits, while the input Lc(I) represents the sequence of log-likelihoods of code bits. The outputs Lu(O) and Lc(O) are updated versions of these sequences, based on information about the encoder. It can be shown that this gives better result than hard and soft decision Viterbi decoders (we use hard decision type decoders only, for the channel, where data is in the binary format only), and convert the hard decision patameters suitably, as given in the system model in section II. We demonstrate our result by two convolution encoders in cascade, the first has K (constraint length) =7 and rate r = 1/2, and second has K = 7 and rate r = 2/3. This iterative decoder gives a rate r = 1/3, whereas Viterbi decoder has K = 7 and rate = 1/2. We show that bit-error rate (BER) in our scheme is comparatively better than the Viterbi decoder when number of iteration, and block size of turbo codes are increased. We show these by presenting simulation results (bit-error rates vs. Eb/No) with the AWGN channel using MATLAB and SIMULINK. The comparison has been carried out between serially concatenated convolution codes (SCCC) i.e. Turbo Codes with the Viterbi decoder, for various existing phase modulation techniques particularly BPSK, QPSK, and 8-PSK. Parameters for comparison are constraint length K = 7, rate r = 1/3, ½, number of iteration (2 and 8), block size (512 and 2048), and bit rate (9.6 kbps). II. SYSTEM MODEL A system for SCCC turbo encoding [6], [7] is given in the fig.1, where outer convolution encoder is chosen with constraint length K=7, and rate r = 1/2, inner convolutional encoder followed by random interleaver has constraint length K=7, rate r=2/3. Similarly for Viterbi decoder K = 7, rate r = ½ is taken. In fig.1 input data sequence dk is in the binary format which is encoded at rate r = 1/3 and finally sequence ck data is then fed to the modulator (which uses various phase modulation techniques) Similarly, on the other hand, sequence dk is convolved by the rate r = ½ for Viterbi decoder and converted to sequence ck.

Output of the modulator is y(t) (noiseless complex valued channel sequence).

III. BLOCK DIAGRAMS AND FIGURES

The signal y(t) is transmitted through an AWGN (Additive White Gaussian Noise) channel. The corrupted signal through AWGN channel y`(t) (noisy complex valued channel sequence with noise n(t)), is demodulated using appropriate demodulation techniques. The received data sequence c`k is a binary data. To use turbo decoder on c`k , we have converted c`k into bipolar and multiplied by the AWGN variance gain = (2 / [1.5 × 10(-Eb/No/10)]), obtaining the data sequence x(t) in integer format, which is required in SISO or L-SISO of the turbo decoding algorithm. x(t) =(2 / [1.5 × 10(-Eb/No/10)]) × c`k We have implemented the algorithm given in [6], i.e. we have implemented APP algorithm for iterative decoders. While converting unipolar data sequence to bipolar quantization noise will occur due to the rounding of the data, but even then the result is better than the Viterbi decoder. Turbo decoder consists of two APP decoders: inner APP decoder and an outer APP decoder. x(t) is the coded input data sequence for inner APP decoder. Another input to this decoder is a random de-interleaved of the coded data sequence coming from outer APP decoder. Outer APP decoder uses all zero data sequence as un-coded input, and coded input to this decoder is a random interleaved of the un-coded data sequence coming from inner APP decoder. Finally d`k is a un-coded data sequence coming from outer APP decoder. Lui(O) = APP{I-1[Lco(O)m], x(t)m}n Inner APP decoder m = block size, n = no. of iteration. Luo(O) = APP{I[Lui(O)m], Z(t)m}n Outer APP decoder m = block size, n = no. of iteration. d`k = if [Luo(O)m > 0 ] = 1 else= 0 APP{x,y} is L-SISO or SISO algorithm of Turbo decoder between x and y. Lui(O) is the un-coded output of the inner APP decoder, Lco(O) is the coded output of the outer APP decoder, Luo(O) is the un-coded output of the outer APP decoder, Z(t) denotes set of zero values of the size of block size, I and I-1 denotes interleaver and deinterleaver respectively.

Fig.3: BER vs. Eb/No plots of BPSK Comparison of VIterbi (K=7, r=1/2) with Turbo decoder (no. of iteration = 2, 8, and block size = 512, 2048)

IV. RESULTS

Fig.4: BER vs. Eb/No plots of QPSK Comparison of VIterbi (K=7, r=1/2) with Turbo decoder (no. of iteration = 2, 8, and block size = 512, 2048)

We have used Viterbi decoder K= 7, rate r = ½; and serially concatenated convolutional code (Turbo code) of K = 7, overall rate r = 1/3, number of iteration = 2, 8, and block size = 512, 2048 at the data rate 9.6 kbps. Refer figures 1 and 2 for SCCC turbo code. Figure 3, 4, and 5 shows the bit error rate plots vs. Eb/No for the BPSK, QPSK, and 8-PSK for various configurations, and show the comparison with Viterbi decoding. The scales of all the figures vary according to the modulation techniques used. It is seen that for BPSK modem at 5 dB Eb/No, BER for Viterbi is 1.36 x 10-5; whereas for turbo decoder the values are 3.09 x 10-7, 8.00 x 10-8, and 4.35 x 10-8 for number of iteration = 2, block size = 512; number of iteration = 8, block size = 512; and number if iteration = 8, block size = 2048 respectively. Similarly comparisons for other modulation techniques (i.e. QPSK and 8-PSK) are evident from figures 4 and 5. All these results prove that the turbo decoder, which we have used, is more powerful than Viterbi decoder with ~103 times improvements in BER. Plots also demonstrate improvement in BER if the number of iteration and the block size are increased, as random interleaver and de-interleaver are more effective for the large block size. V. CONCLUSION In this paper we have shown that iteration decoders perform better than Viterbi decoders. The tradeoff between BER and Eb/No will always exist in the wireless communication world. This may help in reducing the transceiver power. However complexity of implementation and delay are the parameters of concern. REFERENCES [1]

[2]

[3]

[4]

Fig.5: BER vs. Eb/No plots of 8-PSK Comparison of VIterbi (K=7, r=1/2) with Turbo decoder (no. of iteration = 2, 8, and block size = 512, 2048)

[5]

Bjørn A. Bjerke, John G. Proakis, K. Y. Martin Lee, and Zoran Zvonar ,” A Comparison of GSM Receivers for Fading Multipath Channels with Adjacent- and Co-Channel Interference” IEEE Journal on Selected Area in Communication, pp-2211 - 2219, vol. 18 no. 11 November 2000. Jan Bajcsy, "On Iterative Decoding in Some Existence System" IEEE Journal on Selected Area in Communication, pp-883 - 890, vol. 19 no. 5 may 2001. Benedetto, Sergio and Guido Montorsi, "Performance of Continuous and Blockwise Decoded Turbo Codes" IEEE Communication Letter, Vol. 1, May 1997, 77 -79. Viterbi Andrew J., "An Intuitive Justification and a Simplified Implementation of the MAP Decoder for Convolutional Codes" IEEE Journal on Selected Area in Communications, Vol. 16, February 1998, pp-260-264. Beedetto, S., D. Divsalar, G. Montorsi, and F. Pollara, "Serial Concatenation of Interleaved Codes: Performance Analysis, Desigh,

[6]

[7]

and Iterative Decoding" JPL TDA Progress Report, Vol. 42-126, August 1996. G. Montorsi, "Design of Fixed-Point Iterative Decoders for Concatenated Codes with Interleavers" IEEE Journal on Selected Area in Communication, pp-871 - 882, vol. 19 no. 5 may 2001. Dariush Divsalar, “Iterative Turbo Decoder Analysis Based on Density Evolution” " IEEE Journal on Selected Area in Communication, pp-891 - 907, vol. 19 no. 5 may 2001.

Iterative Decoding vs. Viterbi Decoding: A Comparison

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