New Applications of the Simple Bound: Random Codes, Serial Turbo Codes and Punctured Codes A. Martinez 1 , S. Morosi2 , B. Ponticelli2 , 1
European Space – ESA/ESTEC – P.O. Box 229, 2200 AG Nordwijk, The Netherlands 2 El. & Tlc. Dept., University of Florence, Via di S. Marta 3, I-50139 Firenze, Italy Corresponding authors e-mail addresses:
[email protected],
[email protected]
Abstract- This paper deals with the application of the “Simple” Bound technique to a new family of serially concatenated turbo codes and to random codes with the goal of a fair comparison of the ML (Maximum Likelihood) performance in AWGN channel. Reported results confirm that outer code improvement causes great advantages both for the waterfall and the floor. In particular, it is shown that, for given rate and block size, better outer encoders make concatenated codes very close to random codes down to the floor. Moreover, the BER floor is shown to be related to the weight distribution. As this becomes more and more binomial (random-code-like), the BER and FER curves improve. Finally, punctured communications are taken into account and a new upper bound of the ML performance is introduced; in particular, a new strategy, called “uniform puncturing”, is considered with remarkable results for random codes and, also, for the considered turbo-like codes. Keywords: Simple Bound, Serially Concatenated Codes, Random Codes, Uniform Puncturing.
I. INTRODUCTION One of the earliest practical applications of channel codes can be identified in deep space digital communications, where a large free space propagation loss between spacecraft and earth based receiver has to be overcome. Historically, this task was achieved by means of channel codes with considerable bandwidth expansion and the required overhead was not a big complication in this kind of communication because of bandwidth availability. Recently, with the growth of the digital communications and the advent of multimedia wireless application, as video broadcasting, wireless networking and mobile telephony, the situation has dramatically changed due to remarkable increase of the data traffic. As a consequence, a new situation has to be faced with higher bit rates to be transmitted on limited bandwidth. This problem can be overcome by choosing higher order modulations schemes but this solution causes sensible performance loss due to the decreased distance between signals: this is a strong motivation to introduce more powerful codes The need of coupling large coding gains with high spectral efficiency naturally leads to resort to the potentiality of turbo codes. In fact, turbo codes have shown very good performance in terms of Bit Error Rate (BER) at Signal-to-
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Noise Ratios (SNR) very close to the Shannon limit, and this is particularly important in a satellite environment, characterized by limited transmitted power. In order to define fair criteria to select turbo codes that can be effectively coupled with higher order modulation schemes, it is important to identify efficient evaluation technique of codes behavior: this paper deals with the application of “Simple” Bound technique to different codes family: in particular, an evaluation of theoretical performance, i.e. an estimate of the optimum decoder behavior, is provided for a new family of turbo codes characterized by a relatively low complexity and by BER performance very close to random codes one.
II. “SIMPLE” BOUND FOR ML DECODING We use an upper bound to estimate the ML performance for a turbo decoder. For a (n,k) block code, the classical Union Bound becomes meaningless below the channel cut-off rate, so that other bounding techniques are needed. One of them is the “Simple” Bound derived by Divsalar [1]. It is based on an original technique by Gallager [2], who obtained an upper bound for the codeword error probability ML decoding. This bound is linear in the codewords, and can be applied to a code (average over a code’s set of codewords), or to an ensemble of codes (averaging over several set of codewords). The code (resp. ensemble of codes) is represented by means of a (k+1)x(n+1) matrix
Awh , whose element (w,h) equals
the number (resp. ensemble average number) of codewords with input Hamming weight w and output weight h. Other names for this matrix Awh are IOWC matrix (Input Output Weight Coefficient) or code spectrum. For any binary (n,k) block code, and binary antipodal modulation (BPSK, QPSK), the Simple Bound estimates the Bit and Frame Error Rate (BER and FER) as: Pe ≤
h max
∑
{
(
min e− nE(c, h ) , e nr(δ )Q 2ch
h = h min
)}
(1)
with 1 cf(c) E (c, h) = 2 ln[ 1−2c0( δ ) f (c)]+1+ f (c) −r( δ ) +δ c
c0( δ ) < c
