Concatenated codes using multilevel structures for PSK signalling over the Rayleigh fading channel

H.G. Jardak J.N. Livingston

Indexing terms: Communication system. Fading enrironment, Decoders, Trellis coded modulation, Multilevel codes

Abstract: The authors investigate the application of concatenated codes to the Rayleigh fading channel using M-PSK modulation. To obtain a variety of diversity levels, the authors propose several concatenated schemes where the inner codes are multilevel codes. This approach yields high diversity with low complexity, essentially providing soft decision decoding performance on Reed-Solomon codes.

1 Introduction

To enhance the performance of a communication system in a fading environment, multiple replicas of the information-bearing signal may be provided for reception (diversity transmission ) [ 13. Perhaps a better alternative to repetition coding is to achieve diversity through trellis coded modulation (TCM). A subclass of TCM, known as multilevel codes, has attracted the attention of various researchers over the last few years [2-41. This approach is an attractive coding technique for the fading channel [5-71, where their flexibility of design and simple decod- ing strategies lead to simple but powerful systems.

In this work, our concern is to provide a high level of diversity for fading channels, while keeping the decoders simple. Our approach is to use concatenated codes, where the inner code is multilevel in design. In [SI, con- catenated codes are used to provide large gains on the Gaussian channel. These codes are complex, and require many levels of decoding. In Reference 9 concatenated codes are used on the fading channel. However, the inner codes were general 4- or 16-state trellis codes, and the outer code was usually the (255,223) Reed-Solomon code. Our outer codes are a variety of Reed-Solomon codes, and the inner codes are various multilevel codes. We are able to generate high diversity orders, while maintaining small decoding complexity. Indeed, we effec- tively provide soft decision performance for decoding the Reed-Solomon codes.

2

2.1 Channel model The channel model we use is the Rayleigh fading channel. The received symbol is given by

(1)

Channel model and code performance

Y , = a, x, + z, ~

0 IEE, 1994 Paper 13121 (E5, EX), first received 20th July 1993 and in revised form 12th April 1994 H.G. Jardak is at P.O. Box 16-5361, Beirut, Lebanon J.N. Livingston is at Texas A&M University, College Stanon, TX 77843-3128, USA

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where x, is the transmitted symbol, a, is a sequence of independent Rayleigh distributed random variables and z , is additive white Gaussian noise with zero mean and variance N 0 / 2 . We normalize the density function of the Rayleigh distributed random variables to have unit power. Throughout this work we assume ideal inter- leaving.

The performance of TCM on such channel has been derived by Wilson and Leung [lo] (among others) to be:

where P , is the probability of symbol error, C is a con- stant that depends on the products of Euclidean distance in the minimum weight error events, E, is the average symbol energy (over all fading statistics), N0/2 is the variance of the Gaussian noise, and d , is the minimum Hamming distance of the code in terms of PSK symbols (also referred to as the effective length of the shortest error event) [ l l , 121. The diversity of the system is then

However, in this work we are also interested in hard decisions, since we will use a block code for the outer code in the concatenated coding scheme. The per- formance of a block code employing hard decisions over a Rayleigh fading channel is derived in the Appendix. Using these results, we find the effective diversity at high signal-to-noise ratio (SNR) is

D = d , .

D = ( t + l)d, (3) where t is the guaranteed error correcting capability of the outer block code, and d , is the effective diversity of the inner code.

2.2 Multilevel codes The concept of multilevel coding consists in partitioning the original signal space into sets of symbols called subsets. In an L-level scheme, the original constellation is partitioned L times to form successively smaller parti- tions. The multilevel code employs an L-level code, C = [C,, ..., CL], where the Ci are component codes. It can easily be shown that d , of the multilevel code is given by

d , = min {dl, d , , . . . , dL} (4) where di, i = 1, 2, ..., L, is the minimum Hamming dis- tance of the respective component code [SI. This assumes the use of soft decision decoding.

Throughout this paper, we will assume the use of multistage decoding of the multilevel codes. This limits the complexity of decoding to that of the most complex code used, and simplifies the decoding process. For more discussion of multistage decoding, see, for example, Ref- erence 3.

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3

In this Section we propose systems that achieve a variety of diversity levels for 4-PSK and 8-PSK constellations. Our aim is to achieve high diversity with small decoding effort. In Fig. 1, a general system level block diagram for

Examples of high diversity systems procedure is a variant of Wagner decoding for parity check codes. Fig. 2 gives the performance of this code, with an overall diversity of four.

The complexity of this system is small, since the inner code can be decoded using Wagner decoding, and the

PSK symbol mapper

1 Raylogh fading

channel

output data

Fig. 1 System block diagram for examples 1-5

the examples to be considered is given. The outer code will be either a BCH code or a Reed-Solomon code. The interleaver is used to break up errors from the inner code. The multiplexer is used to match the data at the output of the outer code to the rate requirements of the inner code. For instance, in example 1, the outer code is a (15, 11) Reed-Solomon code, and the inner code a two-level rate 2/3 multilevel code. The outer code generates length 15 codewords, where each element can be represented by 4 bits. The multiplexer divides the 4 bits between the two codes of the inner multilevel encoder. The same general operation is performed in each example. The output of the inner code is used to assemble blocks of PSK symbols. The data is then transmitted over a Rayleigh fading channel, and then decoded.

3.1 Example 1 : rate 0.489 4-PSK, D = 9 In this example we propose a ninth-order diversity system for the 4-PSK set. The outer code is a Reed- Solomon code ( n = 15, k = l l) , and the inner code is a two-level multilevel code, with rate 213, and each code being a rate 213 four-state convolutional code. The overall rate of the concatenated code is equal to ( l l / 15)(2/3) = 0.489, suitable for 4-PSK signalling. The diver- sity can be found from the fact that t = 2 and d , = 3, yielding D = 9. The two-level inner code is decoded using multistage decoding.

The simulation results for the system proposed in Example 1 are shown in Fig. 2. For comparison, the per- formance of uncoded BPSK on the fading channel is included. The overall performance of the concatenated structure is measured by the outer code probability of a block error, P,.

3.2 Example2:rate 1124-PSK. D = 4 In this example we propose a simple system that trans- mits symbols from the 4-PSK constellation that achieves a diversity level of four. The outer code is a Hamming code, with (n = 7, k = 4, t = 1). The inner code is a two- level rate 718 even-parity block code, with d, = 2. Each of the codes for the inner multilevel code are rate 7/8 parity check codes. They operate together to select eight 4-PSK symbols for each 14 bits input to the inner code. The overall rate of the concatenated code is equal to (4/73(7/8) = 1/2, and D = 4. Note that the inner code can be decoded using a maximum likelihood decoder. The

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outer code can be decoded using table look-up. Thus decoding can be accomplished rapidly. In comparison, TCM schemes generally require a 16-state code to achieve fourth-order diversity for 4-PSK signalling.

One feature of the above system is the effective doub- ling of the outer code’s diversity. This is achieved through the simple inner code. The effective doubling of diversity is equivalent to using a soft decision decoder on a rate 1/2 outer block code, the penalty being rate loss, which for this example is 14%. Thus the overall system is equiva- lent to an (8,4,4) code over GF(4), using soft decision decoding. For comparison, an (8,4,4) code over GF(4) would achieve a diversity of D = 2 using hard decisions.

3.3 Example 3: rate 0.625 8-PSK. D = 4 In this example we design a system for 8-PSK transmis- sion. The outer code is a Reed-Solomon code with (n = 7, k = 5, t = I), and the inner code is a three-level rate 7/8 even-parity block code with d , = 2. The overall rate of the concatenated code is equal to (5/7)(7/8) = 0.625. The diversity order achieved by this system is

If we consider this concatenated code to be equivalent to a rate 0.625 block code using soft decision decoding, then we lose 14% in rate over the original outer code, while gaining an extra order of diversity. This is equiva- lent to an (8,5,4) code over GF(8), decoded with soft deci- sions. For comparison, an extended (8,5) Reed-Solomon code would have a diversity of D = 2 using hard deci- sions.

3.4 Example 4 : rate 0.65 8-PSK, D = 10 The outer code is a Reed-Solomon code with (n = 63, k = 55, t = 4), and the inner code is a three-level rate 3/4 even-parity block code with d, = 2. The overall rate of the concatenated code is equal to (55/63)(3/4) x 2/3. The diversity order achieved by this system is (t + l)(dmim) = (4 + 1x2) = 10. As can be inferred from Fig. 3, the diver- sity level is achieved asymptotically. Between 17 and 18 dB, the diversity level is close to IO.

Again, if we consider this concatenated code to be equivalent to a rate 2/3 block code using soft decision decoding, then we lose 37% in rate over the original outer code, while again gaining an extra order of diver- sity. This is approximately equivalent to a (63,41,10) code over GF(64), decoded using soft decisions, where we can

D = 4.

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form two 8-PSK symbols from each code symbol at the output of the encoder. Rate loss can be decreased by using a higher rate inner code, say a rate 6/7 parity code,

16‘ ’ O 0 1

, 6 7 1 , , , , , - , , , , , , , . > - ,

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 EbINo

Fig. 2 uncoded BPSK - uncoded BPSK

Comparison of simulation results for examples I and 2 with

block error rate, example 1 block error rate, example 2

. . . . . . . .

l D Z l

1 6 ~ m

15 16 17 18 19 20 21 22 23 2h 25

EblNO

Fig. 3 Comparison of simulation results for examples 3-5 with uncoded 4-PSK _ ~ ~ _ block error rate, example 3 . . . , . . block error rate, example 4

block error rate, example 5 - uncoded 4-PSK

yielding an approximately equivalent (63,47,10) code over GF(64). For comparison, a (63,41) Reed-Solomon code yields a diversiiy of 12, and a (63,47) Reed-Solomon code yields a diversity of 9, using hard decisions.

3.5 Example 5: rate 213 8- PSK, D = 16 In this final example, the outer code is a Reed-Solomon code with (n = 63, k = 49, t = 7), and the inner code is a three-level rate 6/7 even-parity block code. The overall rate of the concatenated code is equal to (49/63)(6/7) = 2/ 3. Fig. 3 shows a diversity order of approximately 14 between an average signal-to-noise per information bit of 16 dB and 17 dB.

Again, if we consider this concatenated code to be equivalent to a rate 2/3 block code using soft decision decoding, then we lose 17% in rate over the original outer code, while again gaining an extra order of diver- sity. This is equivalent to a (63,42,16) code over CF(64), using a soft decision decoder. For comparison, a (63,41) Reed-Solomon code would achieve a diversity of 12 using hard decision decoding.

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4 Conclusions

We have considered the use of multilevel codes in con- catenated structures as an attractive and efficient means to provide a wide variety of diversity orders. One of the key points is the ability to use simple high rate parity check codes for the inner code in the concatenated scheme. This has the effect of doubling the effective diver- sity of the outer code. This is equivalent to using a single outer block code at the overall rate of the concatenated code, but using soft decision decoding. This appears to be a reasonable way to achieve the performance of soft deci- sion decoding for BCH and Reed-Solomon codes without having to actually use a full soft decision decoder. The only penalty is rate loss.

In comparison, standard trellis coded modulation schemes can achieve the same diversity, but at, usually, a much higher complexity. For example, to achieve diver- sity orders of 4, 5, and 6 using rate 2/3 TCM on 8-PSK, we require a trellis of size 64,256, and 2048 states, respec- tively [12]. In examples 3-5, we achieve diversity orders of 4, 10, and 16. The inner codes have little complexity, being even parity check codes. The outer codes can be decoded using standard Reed-Solomon decoder tech- niques, and can be decoded quite fast.

Comparing the performance of the proposed codes for each signalling class (4 or 8 PSK), we note that the main difference is the inner code. Several of the examples rely upon an even parity check inner code that is especially simple to decode using soft decisions. These codes appear to be the best choice to obtain simple, efficient decoding combined with good performance. In particular, exam- ples 2, 3, and 5 provide better performance than an equivalent rate BCH or Reed-Solomon code using hard decision decoding, while retaining a simple decoder struc- ture. The proposed codes have application in situations where the signal-to-noise ratio is high enough to achieve their designed diversity. Hence, low SNR applications are inappropriate for these codes.

5 References

1 PROAKIS, J.G.. Digital Communications (Mdraw-Hill, New York, 1983)

2 IMAI, H., and HIRAKAWA, S.: ‘A new multilevel coding method using error-correcting codes’, IEEE Trans., 1977. IT-23, (3). pp. 371-377

3 CALDERBANK, A.R.: ‘Multilevel codes and multistage decoding’, IEEE Transactions on Communications, 1989,37, (5), pp. 222-229

4 POTTIE, G.J., and TAYLOR, D.P.: ‘Multilevel codes based on par- titioning’, IEEE Transactions on Information Theory, 1989, 35, (l), pp. 87-98

5 LIVINGSTON, J.N., and MULLIGAN, M.G.: ‘Multilevel codes and suboptimal decoding for PSK signalling on the fading channel’. Proceedings of international symposium on Information theory and its applications, Honolulu, HI., November, 1990, pp. 163-166

6 SESHADRI, N., SUNDBERG, C.E.W.: ‘Multilevel trellis coded modulations with large time diversity for the Rayleigh fading channel’. Proceedings CISS, Princeton, NJ., March, 1990, pp. 853- 857

7 SUNDBERG, C-E.W., and SESHADRI, N . : ‘Multilevel block coded modulations for the Rayleigh fading channel’. Proceedings Globecom ‘91, Phoenix, AZ., December, 1991, pp. 2.2.1-2.2.5

8 HERZBERG, H., BEERY, Y., and SNYDERS, J.: ‘Concatenated multilevel block coded modulation’, IEEE Trans. on Communica- tions, 1993,41, pp. 41-49

9 VUCETIC, B.: ‘Bandwidth efficient concatenated coding schemes for fading channels’, IEEE Trans. on Communications, 1993, 41, pp. 50-61

10 WILSON, S.G., and LEUNG, Y.S.: ‘Trellis-coded phase modula- tion on Rayleigh channels’. International conference on Communica- tions, Seattle, WA, June, 1987, pp. 21.3.1-21.3.5

I I DIVSALAR, D., and SIMON, M.K.: ‘Trellis coded modulation for 4800-9600 bits/s transmission over a fading mobile satellite

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channel', IEEE Journal on Selected Areas in Communications, 1981, 5, (Z), pp. 162-175

12 SCHLEGEL, C., and COSTELLO, D.J.. Jr.: 'Bandwidth efficient coding for fading channels: code construction and performance analysis', I E E E Journal on Selected Areas in Communications, 1, (9). pp. 1356-1368

6 Appendix

To understand the performance of a concatenated code, we will assume first that we are using a single block code with hard decisions on the Rayleigh fading channel, with BPSK as the means of signalling. Then the probability of a block error is given by

Pg= i (;)p,1 - P Y ' i = 1 + 1

where t is the error correcting capability of the code, n is the block length, and p is the probability of a symbol error. At high SNR, we may approximate this expression by its first term, yielding

On the Rayleigh fading channel with BPSK, the prob- ability of symbol error is approximated by p sz 1/(4EJN,) [ 11, yielding

(7)

Thus we see that we obtain a diversity order of

For nonbinary block codes, the main result is the D = t + l .

same, namely

P E % ( f + 1 )p:+'

where p, is the probability of a symbol error in the alpha- bet of the nonbinary code.

For concatenated codes, the probability of a symbol error at the input to the outer decoder depends on how the nonbinary symbols are assembled from the inner decoder. In our case, we will assume errors from the inner decoder are independent, and given by

P , % C ( E J i N J - (9)

Independence can be assured through proper interleaving between the outer coder and the inner coder, breaking up bursts of errors across block boundaries. Given the above for the probability of symbol error, this leads to the prob- ability of a block error:

Note that this gives us an effective diversity of

D = ( t + l)d, (1 1)

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