Performance of Turbo Codes with Matrix Interleavers Maan A. Kousa

PERFORMANCE OF TURBO CODES WITH MATRIX 

INTERLEAVERS 

Maan A. Kousa* 

Department of Electrical Engineering 

King Fahd University of Petroleum and Minerals 

Dhahran, Saudi Arabia 

: ﺔﺻﻼﺨﻟا 

ﺪﻨﻋ ﺮـهﺎﺒﻟا 

ﺎﻬﺋادأ 

ﺐﺒﺴﺑ 

ﻦﻴﺜﺣﺎﺒﻟا ﻦﻣ ﺮﻴﺜﻜﻟا مﺎﻤﺘها ﻰﻠﻋ ًاﺮﺧﺆﻣ ﻮﺑﺮﺘﻟا 

تاﺮﻔﻴﺷ تذﻮﺤﺘﺳا ﺪﻘﻟ 

ﻢﻈﻌﻣ ضﺮﺘﻔﺗو . تاﺮﻔﻴﺸﻟا 

ﻩﺬه ﻢﻴﻤﺼﺗ ﻦﻣ ًﺎﻣﺎه ًاءﺰﺟ ﺮﺜﻌﺒﻤﻟا ﺮﺒﺘﻌﻳو .( SNR) 

ـﻟ ﺔﻀﻔﺨﻨﻤﻟا ﻢﻴﻘﻟا 

ماﺪﺨﺘﺳﺎﺑ ﻮﺑﺮﺗ تاﺮﻔﻴﺷ ءادأ 

سرﺪﻨﻓ ﺚﺤﺒﻟا اﺬه ﻲﻓ ﺎﻣأ 

،ﺔﻴﺋاﻮﺸﻋ تاﺮﺜﻌﺒﻣ لﺎﺠﻤﻟا اﺬه ﻲﻓ ثﺎﺤﺑﻷا 

لﻮﻃ ﺮﻴﺛﺄﺗو ،ﺔﻴﻔﺼﻟا ةﺮﺜﻌﺒﻤﻟا دﺎﻌﺑأ 

ﻢﻴﻤﺼﺗ ﻞﺜﻣ ،ﺎﻬﺋادﺄﺑ 

ﺔﻘﻠﻌﺘﻤﻟا ﻞﻣاﻮﻌﻟا ﻦﻣ ﺪﻳﺪﻌﻟاو 

،ﺔّﻴﻔﺻ 

تاﺮﺜﻌﺒﻣ 

ﻞﺧﺪﻤﻟا ﻦﻴﺑ ةﺮﺜﻌﺒﻤﻟا لدﺎﺒﺗ ﺮﻴﺛﺄﺗو ،ﺮﻴﻔﺸﺘﻟا ﻚﻓ تارود دﺪﻋو ةﺮﺜﻌﺒﻤﻟا ﻦﻴﺑ ﻞﻋﺎﻔﺘﻟاو ،ةﺮﺜﻌﺒﻤﻟا 

. جﺮﺨﻤﻟاو 

لاﻮﻃﻸﻟ 

ﺔﻴﺋاﻮﺸﻌﻟا تاﺮﺜﻌﺒﻤﻠﻟ 

ﺔﺴﻓﺎﻨﻣ نﻮﻜﺗ نأ 

ﻦﻜﻤﻳ 

ﺔﻴﻔﺼﻟا تاﺮﺜﻌﺒﻤﻟا نأ 

ةﺎآﺎﺤﻤﻟا ﺞﺋﺎﺘﻧ 

∗ 

Address for correspondence: 

KFUPM Box 470 

King Fahd University of Petroleum and Minerals 

Dhahran, 31261 Saudi Arabia 

E-mail: makousa@kfupm.edu.sa 

ﺮﻬﻈﺗو 

. ةﺮﻴﺼﻘﻟا 

Maan A. Kousa 

October 2003 The Arabian Journal for Science and Engineering, Volume 28, Number 2B 211

212 


ABSTRACT 

Recently, turbo codes have attracted many researchers because of the codes 

astonishing performance at low BER. The interleaver is an important part in the 

design of a turbo code. Most of the work done on turbo codes assumes random 

interleaving. In this paper, we study the performance of turbo codes with matrix 

interleaving. Various issues related to the code performance are investigated. 

These include the design of the relative dimensions of the matrix interleaver, the 

effect of the interleaver length, the interaction between the interleaver and the 

number of decoding iterations, and the effect of interchanging the interleaver. 

Simulation results show that matrix interleavers can be very competitive to 

random interleavers for short frame lengths. 

The Arabian Journal for Science and Engineering, Volume 28, Number 2B October 2003

PERFORMANCE OF TURBO CODES WITH MATRIX INTERLEAVERS 


1. INTRODUCTION 

Turbo code (TC) is the name given to parallel concatenation with interleaving [1]. Simplified schematics of the turbo 

encoder and decoder are shown in Figure 1. There are two convolutional encoders in parallel, which are usually taken to 

be identical. Information bits are interleaved before being fed to the second encoder. The codeword in a turbo code 

consists of a frame of input bits (X0) followed by the parity check bits from the first encoder (X1) then the parity bits from 

the second encoder (X2). Padding is used to append the proper sequence of bits in order to force all the encoders at the 

end of the frame to the all-zero state. 

Information 

Received 

Sequence 

source 

DEMUX/ 

INSERT 

Padding 

Interleaver 

Enc 1 

Enc 2 

The decoder works in an iterative way. The first decoder will decode the sequence and pass the hard decision together 

with a reliability estimate of this decision to the next decoder after proper interleaving. The second decoder will utilize 

the reliability estimates produced by the first encoder, and thus will have extra information for decoding. After a certain 

number of iterations is executed, a hard decision is made and the estimated sequence is delivered to the user. Several 

decoding algorithms have been proposed. In this work, we opt for the Normalized SOVA (Soft Output Viterbi 

October 2003 The Arabian Journal for Science and Engineering, Volume 28, Number 2B 213 

X0 

X1 

X2 

Figure 1a. Basic turbo encoder structure 

(Interleaver) -1 

Decoder 1 Interleaver 

Decoder 2 

Figure 1b. Basic turbo decoder structure 

Puncturing 

& 

Parallel/Serial 

MUX 

(Interleaver) -1 

to the 

channel 

X2 X1 X0 

Estimated 

Sequence

214 


Algorithm) decoder of [10]. It was shown that SOVA approaches maximum likelihood (ML) decoding for large number 

of decoding iterations [3]. 

The interleaver is the most critical part in the design of a turbo code, and has thus attracted many researchers [2–8]. 

The original paper on turbo codes [7] assumed a random interleaver. The authors in [3] introduced the notation of the 

uniform interleaver, which was then adopted by other researchers. 

Conventionally, interleaving is used to spread out the errors occurring in bursts like those exhibited in fading 

channels [9]. In this regard, matrix interleaving, where bits are fed in a matrix row by row and read out column by 

column, is usually implemented. For turbo codes, the interleaver does other functions. At the encoder, the interleaver is 

used to feed the encoders with different permutations of the information sequence so that the generated parity sequences 

can be assumed independent. At the decoder, both the interleaver and the deinterleaver are used to randomize the 

symbols after every decoding step, thus making the iterative decoding more efficient. 

In this paper, we attempt to provide an in-depth study on the performance of turbo codes with matrix interleavers. 

The channel is assumed to be AWGN. In Section 2 the design of the dimensions of the matrix interleaver is considered. 

The effects of some interleaver-related issues are presented and explained in Section 3. These include interleaver length, 

number of decoding iterations, and interleaver/deinterleaver. Conclusions are drawn in Section 4. 

2. MATRIX INTERLEAVERS 

Most of the work on TCs is based on random interleavers. In random interleavers, the mapping is performed pseudorandomly. 

In [3] it was shown that large-frame random interleavers perform equally well on the average, independent to 

a large extent of the particular mapping used. 

BER 

1.E+00 

1.E-01 

1.E-02 

1.E-03 

1.E-04 

1.E-05 

0 0.5 1 

Eb/N0 1.5 2 2.5 3 

(16,12) matrix (32,6) matrix (2,96) matrix (1,196) no-inter uniform 

Figure 2. Performance of matrix interleavers for different values of m and n (i=6, k=192) 

Benedetto et al. [3] introduced the notion of uniform interleaver to simplify the analysis of the code weight 

distribution. A uniform interleaver of length k is a probabilistic device that maps a given input word of weight w into all 

k 

k 

distinct C w permutations of it with equal probability of 1 / Cw 

. The uniform interleaver is an abstract interleaver, but 

can be approximately realized by changing the interleaver map randomly at every frame. The performance of the 



uniform interleaver was found to be very close to that of the random interleaver. Therefore, only the uniform interleaver 

is used here as a reference for comparing the matrix interleaver. 

For the matrix interleaver, the information stream is written row by row in a matrix of n rows and m columns and 

read out column by column. The column size n is called the depth and the row size m is the span. Such an interleaver is 

completely defined by n and m and is thus refereed to as (n,m) matrix interleaver. At the deinterleaver, information is 

written column-wise and read out row-wise. 

The capability of burst error scattering for the matrix interleaver depends on the values of n and m. Since both the 

interleaver and the deinterleaver are utilized iteratively at the decoder, it is anticipated that the optimum design is where 

m and n are both large and comparable in value. 

To test this conjecture, consider an interleaver of size 192 symbols. This size can be arranged in (192,1), (96,2), 

(32,6), or (16,12) matrices. Figure 2 shows the performance of the code under these matrices. It is depicted from the 

figure that the performance improves as the matrix dimensions approaches that of a square. The figure also shows that 

the (16,12) matrix interleaver slightly outperforms that of the uniform interleaver of the same frame length. 

3. STUDY ISSUES 

In this Section, some issues related to the matrix interleaver are studied. These issues include the interleaver length, 

the effect of the number of iterations, and the effect of interchanging the interleaver and the deinterleaver. 

3.1. Interleaver Length 

Figure 3 shows the performance of TC as a function of interleaver length for the uniform and matrix interleavers. It is 

seen that the BER decreases sharply with increasing k for a certain range of k and saturates at large values of k. The trend 

of the curves is close to that of 1/k. This trend was observed in [3] for the uniform interleaver. 

BER 

1.E-01 

1.E-02 

1.E-03 

1.E-04 

1.E-05 

Frame Length k 

0 200 400 600 800 1000 1200 

Uniform Random Matrix 

Figure 3. Effects increasing the frame size k on the code performance (i=6, E b/N 0=2 dB) 


216 


The figure shows that the performance of the matrix interleaver is comparable to, and slightly better than, that of 

uniform interleaver for short and medium sizes (k

BER 

1.E+00 

1.E-01 

1.E-02 

1.E-03 

1.E-04 

1.E-05 

3.3. Interleaving (I) / Deinterleaving (I -1 ) 


0 0.5 1 

Eb/N0 1.5 2 2.5 3 

(96,2) matrix (1) (96,2) matrix (6) (32,6) matrix (1) 

(32,6) matrix (6) (16,12) matrix (1) (16,12) matrix (6) 

Figure 5. Effect of the number of iterations for different degrees of scattering 

In the papers surveyed, the researchers attempt to design the interleaver by considering its effect on the encoding 

process. However, it should be recalled that both the interleaver and the deinterleaver play a significant role in the 

decoding process. 

Burst errors may be available at the output of decoder 1 or decoder 2 due to the memory correlation inherent in 

convolutional codes. Therefore, it is desirable to have large spreading effect for both the interleaver and the deinterleaver 

in order to randomize the errors and hence improve the chance of being corrected in the next iteration. Unfortunately, the 

spread of interleaver and the deinterleaver are inversely related. 

The similarity of the roles played by interleaver (I) and the deinterleaver (I -1 ) at the decoder suggests that designing a 

code with I -1 as the interleaver should not affect the overall performance. This is strongly supported by Figure 6 where 

the performance of TC under two configurations of matrix interleavers with I and I -1 interchanged are shown. 

Our observation that I and I -1 can be interchanged without any significant effect on the code performance can be proved 

by the following argument. Consider the decoder first. The iterative decoding is depicted in Figure 7a. The similar role 

played by I and I -1 is quite evident. It is apparent that, for i iterations, the final output sequence would have been 

processed by I and I -1 for i+1 and i times respectively. For i reasonably large (i=6) the two boxes can be interchanged. 


218 


BER 

1.E+00 

1.E-01 

1.E-02 

1.E-03 

1.E-04 

1.E-05 

0 0.5 1 

E b /N 0 

1.5 2 2.5 3 

(16,12) matrix (12,16) matrix (2,96) matrix (96,2) matrix 

Figure 6. Code performance under interchanging I & I -1 

Next, let us consider the encoder. Figure 7b is obtained by replacing I in Figure 1 with I -1 . We will show that this change 

has almost no effect on the encoding process. Figure 7c is identical to Figure 7b and is obtained by adding I -1 and I at the 

front of the encoder. Distributing I over the three branches leads to Figure 7d. Note that I -1 at the front of the encoder and 

I in the systematic branch do not modify the performance since they are external to the encoding process. As a result, the 

TC in Figure 7d yields the same performance as that in Figure 1. 

4. CONCLUSION 

Matrix interleavers are commonly used in digital communication systems to randomize burst errors. This paper shows 

that the application of matrix interleavers to turbo codes provides a performance comparable to, and slightly better than, 

that of random interleavers for short frame lengths. This result is very significant as many communication systems 

necessitate the use of short frames to keep the delay time at an acceptable level. 

We also showed that the matrix interleaver yields the best performance when its dimensions approach those of a square. 

The effect of the number of decoding iterations and interchanging the interleaver and deinterleaver were simulated and 

discussed. 


(a) 

(b) 

(c) 

(d) 

Info. 

Info. 

Info. 

I -1 I 

Enc 1 

Enc 2 


Figure 7. (a) Decoder I, I -1 loop; (b) Turbo encoder (with interleaver I -1 ); (c) I and I -1 are added to (b); (d) Similarity of (b) to the 

turbo encoder with (I) as the interleaver 

I -1 

Decoder 1 I 

October 2003 The Arabian Journal for Science and Engineering, Volume 28, Number 2B 219 

I -1 

I -1 

I 

Decoder 2 

I -1 I 

Enc 1 

Enc 2 

Enc 1 

Enc 2

220 


ACKNOWLEDGMENT 

The author would like to thank KFUPM for supporting this research. 

REFERENCES 

[1] C. Berrou and A. Glavieux,: “Near Optimum Error Correcting Coding and Decoding: Turbo-Codes”, IEEE Transactions on 

Communications, 44 (1996), pp. 1261–1271. 

[2] Gerard Battail, “A Conceptual Framework for Understanding Turbo Codes,” IEEE Journal on Selected Areas in 

Communications, 16 (2), (1998), pp. 245–254. 

[3] Sergio Benedetto and Guido Montorsi, “Unveiling Turbo-Codes: Some Results on Parallel Concatenated Coding Schemes,” 

IEEE Transactions on Information Theory, 42, (2), (1996), pp. 409–428. 

[4] A. K. Khandani, “Design of Turbo-Code Interleaver using Hungarian Method,” Electronics Letters, 34 (1) (1998), pp. 

63–65. 

[5] W. J. Blackert, E. K. Hall, and S. G. Wilson, “Turbo Code Termination and Interleaver Conditions,” Electronic Letters, 31 

(24), (1995), pp. 2082–2084. 

[6] P. Jung and M. Nabhan, “Dependence of the Error Performance of Turbo-Codes on the Interleaver Structure in Short Frame 

Transmission Systems,” Electronics Letters, 30 (4) (1994), pp. 287–288. 

[7] C. Berrou, A. Glavieux, and P. Thitimajhima, “Near Shannon Limit Error Correcting Coding and Decoding : Turbo-Codes,” 

Proc. of ICC 93, Geneva, May (1993), pp. 1064–1070. 

[8] S. Dolinar and D. Divsalar, “Weight Distribution for Turbo Codes using Random and Nonrandom Permutations,” TDA 

Progress Report 42-122 , Augest (1995), pp. 56–65. 

[9] K. Andrew, C. Heegrad, and D. Kozen, “A Theory of Interleavers,” Technical report 97-1634, Computer Science 

Department, Cornell University, June, (1997). 

[10] L. Papke and P. Robertson, “Improved Decoding with the SOVA in a Parallel Concatenated (Turbo Code) Scheme,” IEEE 

Int. Conference on Communications ICC ’96, 1996, pp. 102–106. 

Paper Received 4 September 2000; Revised 15 December 2002; Accepted 21 May 2003.

Performance of Turbo Codes with Matrix Interleavers Maan A. Kousa

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