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Figure 11.12 The turbo decoder for the encoder of Figure 11.11. The symbol sents a deinterleaver. (Courtesy of A Burr and the IEEE.) repre Error Control Coding
where p(r/1) is the probability of receiving value r, given that a 1 was transmitted and p(r/0) the probability of receiving value r, given that a 0 was transmitted. If the voltage levels are normalized so that 1V represents a probability of 1, a certainty and 0V, zero probability, then for r 0.9V for example, p(r/1) 0.9 and p(r/0) 1 .9 0.1, so that LLR 2.197. With r 0.3, LLR 0.847. In general, LLR yields a positive number for r closer to 1 and a negative number for r closer to 0. The magnitude of LLR is a measure of how close. These two pieces of information are included in the soft sequences that form a part of the output of the multiplexer and which is the input to the decoders. The outputs from the decoders are also soft and the system is referred to as softinput softoutput (SISO). As shown in Fig. 11.12, the switches are in position 1 for the first iteration of the decoding step. Following the first iteration the switches are switched to position 2, and each decoder makes use of the soft information obtained from the other decoder to obtain a better estimate of bit values. Recall that two independent parity sequences are available for a given data sequence. The decoded data is adjusted to take into account the new estimates, and the process is repeated a number of times, typically for 4 to 10, before a final hard decision is made. The information that is obtained from the received data bits is termed intrinsic information, the intrinsic information flow paths being shown by the solid line in Fig. 11.12. The information that is passed from one decoder to the other is termed extrinsic information, the paths for the extrinsic information flow being shown by the dotted line. After the final iteration the output of the second decoder is switched to the output line (not shown in Fig. 11.12). It will be a 1 for a positive LLR and a 0 for a negative LLR. A more detailed description of the encoder of Fig. 11.11 and the decoder of Fig. 11.12 will be found in Burr (2001). 11.11.1 Low density parity check (LDPC) codes
LDPC refers to the fact that the parity check matrix (Sec. 11.2) is sparse, that is, it has few binary 1s compared to binary 0s. The LDPC codes were first introduced by Gallagher (1962) who showed that a low density parity check matrix resulted in excellent minimum distance properties (as defined in Sec. 11.2), and they are comparatively easy to implement. As mentioned above a feature common to LDPC codes and turbo codes is that SISO decoding is employed, and a series of iterations performed to (hopefully) improve the probability estimate of a bit being a 1 or 0. Only after a predetermined number of iterations is a hard decision arrived at. Eleven
An example of a parity check matrix for a LDPC code (Summers, 2004) is 1 0 0 H 5 G1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0W 0 0 1 (11.25) As shown in connection with Eq. (11.5) the number of rows in H is equal to the number of parity bits n k, and the number of columns is equal to the length n of the codeword. In this case n k 7 and n 16, hence k 9, and the H matrix represents a (16, 9) code. From Eq. (11.7) the syndrome is obtained on multiplying the received codeword by HT, the transpose of H, and ideally, an errorfree codeword is indicated by an allzero syndrome. Standard practice is to index bit positions starting from zero, thus a 16bit codeword would have the bits labeled c0, c1, c2, . . . c15. Likewise, elements in the H matrix are labeled hpq where the first element (top lefthand corner) is h00. In general the row number (indexed from zero) gives the number of the syndrome element, and the 1s in the columns indicate which codeword bits are used. The seven parity check equations obtained from the H matrix, are, on setting the syndrome equal to 0. c0 { c1 { c2 { c9 c3 { c4 { c5 { c10 c6 { c7 { c8 { c11 c0 { c3 { c6 { c12 c1 { c4 { c7 { c13 c2 { c5 { c8 { c14 c12 { c13 { c14 { c15 0 0 0 0 0 0 0 (11.26) As noted in connection with Eq. (11.3) a systematic code has the dataword at the beginning of the codeword, thus it follows that the columns 0 to 8 of the H matrix operate on the datawords. The fact that each column has two 1s means that two of the dataword bits appear in each parity check equation determined by these columns. A standard way of showing the parity check equations and the codeword bits is by means of a Tanner graph (Tanner, 1981) in which circles represent the bit nodes and squares represent the parity check equations, Fig. 11.13. The

