y1

Interleaver

y3

Figure 9.13 Iterative turbo decoder

and check symbols of this code. As initial information it uses a uniform a priori distribution q11(bi) ¼ 1/2, bi ¼ 0, 1, since setting all data bit patterns a priori equiprobable is natural. After this, the decoder of the second code computing a posteriori probabilities p(bijy1, y3) may rest not only on appropriate observations y1, y3 but also on the information delivered by the first decoder, using its a posteriori distribution p1(bijy1, y2) as the a priori one: q21(bi) ¼ p1(bijy1, y2). The result is the first approximation p1(bijy) of a posteriori distribution p(bijy). Since at this step the first decoder was not supported by the information of the second one, this is done at the second iteration, where the first decoder again decodes the first component code but using a priori distribution q12(bi) ¼ p1(bijy). Going on in this way, after the nth step the next approximation pn(bijy) of p(bijy) is formed by the second decoder and used by the first one as the next a priori distribution q1nþ1(bi) to output pnþ1(bijy1, y2). This latter in its turn is used by the second decoder as the next a priori distribution q2nþ1(bi) to produce the next approximation of the desired a posteriori probabilities pnþ1(bijy), etc. Since the interleaver permutes data bits before inputting the second encoder the interleavers permute in the same way observations y1 and a priori distribution q2n(bi) ¼ pn(bijy1, y2) entering the second decoder. Similarly, the deinterleaver restores the original order of bits in a feedback passing on pn(bijy) from the second decoder output to the first decoder input. With these rearrangements, all data processed by both decoders are aligned properly.

A vast simulation has confirmed the convergence of these iterations experimentally, although theoretical justification remains a matter of question.

9.4.3 Performance

As mentioned before, turbo codes were the first regular codes to provide reliable data transmission over the band-limited channel at near-capacity rates and low energies per bit. To illustrate it by examples, let us first examine some fundamental limits on BPSK data transmission. On the strength of the sampling theorem any bandpass signal of bandwidth W is a vector of dimension 2WT (see Section 2.3). In the case of BPSK each component of such a vector may take on only two values, implying that within the time–frequency resource WT the number M of BPSK signals obeys the bound M 22WT , or equivalently, no more than 2WT data bits may be transmitted. This

R/W 2 bps/Hz. Now consider a binary code with rate Rc ¼ 1/2, meaning that only every second component of a signal vector carries data, the rest being assigned to check symbols (two signal samples are spent per data bit), so that the ratio between data rate and bandwidth R/W ¼ 1 bps/Hz. Turning to Shannon’s bound (1.2), we may see that the minimum bit energy normalized to noise Eb/N0 (half of power bit SNR) necessary to provide errorless transmission over an AWGN channel at such a rate equals 0 dB. We are, however, now dealing not with an arbitrary Gaussian channel but with one whose input symbols are limited to the BPSK alphabet (Gaussian channel with binary input). This restriction increases the minimal Eb/N0 corresponding to R/W ¼ 1 bps/Hz up to 0.19 dB [97]. The turbo code of constraint length 5 and block length 65 536 proposed in [95,96] secures bit error probability Pb < 10 5 (this figure is often referred to as the practical criterion of wireless errorless operation) at Eb/N0 0:7 dB, i.e. yielding only about 0.5 dB to Shannon’s limit. At the moment of their publication these results seemed fantastic, since a long unsuccessful history had made many experts believe that finding deterministic coding rules allowing operation near Shannon’s bound was a hopeless task. Following the original works [95,96], other efficient turbo as well as serial concatenation codes have been found (see bibliography in [97]).

It should be noted that the asymptotic (Eb/N0 ! 1) behaviour of turbo codes is not better than that of convolutional codes of the same rate and memory, since they do not possess any advantage in minimum distance. Turning to (2.23), we may see that asymptotically (with growth of SNR) the effect of multiplicity nmin, i.e. the number of signals with minimum Euclidean distance dmin from a transmitted one, plays a secondary role against dmin itself due to the exponential drop of the Q-function with SNR (take the logarithm of Pe to make sure of it). For this reason dependence of Pe on Eb/N0 sooner or later achieves a ‘floor’ character determined by dmin and analogous to the one typical of other codes with the same minimum distance. This, however, occurs at SNR values securing very small bit error probabilities, falling far beyond the range of practical needs. The explanation of why turbo codes guarantee such excellent operational quality at low bit SNR is not in their large minimum distance, but rather in the relatively small number of words lying from each other at small distances, in particular the small multiplicity nmin in (2.23). This redistribution of distances towards a greater number of bigger ones versus convolutional codes happens due to the pseudorandom interleaving of data bits encoded by the second component encoder. If the data bit pattern is unlucky to generate a small-weight word of the first component code, its permutation may appear different enough to produce the second component word of a remarkably higher weight.

9.4.4 Applications

Despite their short history, turbo codes are now in widespread use and enter the specifications of many systems. The most interesting in our context is their involvement in 3G mobile radio standards. The UMTS specification includes turbo codes of rate 1/3 based on two component convolutional codes of constraint length 4 and interleaver of variable length in the range from 40 to 5114 [92,97]. The cdma2000 standard also contains two-component turbo codes of constraint length 4 with interleaver size ranging from 250 to 4090. Appropriate puncturing allows rates of 1/2, 1/3, 1/4 or 1/5 to be obtained [69,97].