turbo equalizer is introduced, the interpolation method for optimal filters and interpolated filter-ing coefficients are derived, and the computational complexity is also analyzed. In Section 5.4, the simulation results are reported and comparison to existing solutions are also made. Finally, we draw conclusions in Section 5.5.
§ 5.1 System Model and Notations
The system model of turbo equalization signal processing for coded data transmitted over ISI channel is shown in Fig. 5.1. The transmitter shown in the upper part consists of a convo-lutional encoder, an interleaver, and a signal modulator. The information bit stream am is encoded with a binary convolutional encoder to generate the coded bits stream bm,l, where the subscript stands for the l-th bit of the m-th codeword and l ∈ {1, 2, ..., r} for a code with rate 1/r (r ∈ Z, r ≥ 2). In other words, an information bit am will generate r coded bits {bm,1, bm,2, ..., bm,r}. Then, bm,l is fed into the interleaver bit-by-bit with the order of (m, l) = (1, 1), (1, 2) . . . , (1, r), (2, 1), (2, 2), . . .. The interleaver permutes and randomizes bm,l
and yields an independent interleaved bits stream ck,j, where the subscript stands for the j-th bit of the k-th modulated symbol and j is in {1, 2, ..., ρ} for a ρ-bit modulator. That is, ρ interleaved bits are grouped and mapped to a transmit symbol. Note that the interleaver considered here is a bit interleaver [57] and it also performs the grouping operation for the modulator. Generally, the last error correction codedword may not form a complete symbol and zero padding may be necessary in that case.
The function of a modulator includes symbol mapping and analog waveform conversion.
The symbol mapping function is denoted by mi ↔ χi, where mi = [mi,1, mi,2, · · · , mi,ρ]T is a ρ-bit vector and χi is a symbol belonging to one of the M-ary signal constellation set X. Here, M = 2ρ and X = {χ0, χ1, · · · , χM −1}. Let the interleaved codeword be ck = [ck,1, ck,2, . . . , ck,ρ]T. Then, ck is mapped to a symbol xk = χi and converted to an analog waveform transmitted over the ISI channel. Except for the ISI effect, the channel also introduces
channelISI
Figure 5.1: The system model of turbo equalization for coded data transmitted over ISI channel.
AWGN, nk. In the subject of turbo equalization, most of the previous works focus on wireless or magnetic channels. In this dissertation, we will focus on the wireline communication channel.
Here, we assume that the baseband modulation scheme is the M-ary pulse amplitude modulation (M-PAM). The symbol mapping for M-PAM can be defined as
χi = (M − 1) − 2i assume that the channel impulse response and the noise are also real valued. For complex-valued passband communication systems such as QAM and PSK, please see [45,58] for details.
The receiver shown in the lower part of Fig. 5.1 consists of a SISO equalizer, a deinter-leaver, a SISO decoder, and an interleaver. At first iteration, the SISO equalizer reduces the ISI channel effect without the assistance of any a priori information, and it outputs the soft decision Le(ck, j) as the demodulated data for the SISO decoder. A deinterleaver is inserted on the forward path to permute the demodulated data stream back into the original coded bit stream. The interleaver design is important and closely related with the performance of the
5.1. SYSTEM MODEL AND NOTATIONS 69 turbo equalizer. We use a random interleaver called S-random interleaver [59] in this disserta-tion. The main characteristic of the interleaver is that it can guarantee a minimum distance of two consecutive bits after interleaving. Then, the SISO decoder performs the error correcting function on the deinterleaved bits L(bm,l)and generates a series of more reliable soft decisions outputs Le(bm,l). These outputs are interleaved to yield Le(ck,j)fedback to the SISO equalizer as a priori information. Note here that the so-called intrinsic information L(bm,l)and L(ck,j) have been subtracted before yielding the extrinsic information Le(bm,l) and Le(ck,j), respec-tively. The process can be repeated until a stopping criterion is met. It has been found that there exists a certain SNR threshold [60] which the turbo equalizer can work properly. When the SNR is higher the threshold, the performance can be improved iteratively. However, if the SNR is below the threshold, no performance gain can be obtained.
It is well known that the optimal turbo equalizer uses the BCJR algorithm as the SISO pro-cessing unit for both the equalizer and the decoder. For convenient comparison with previous works [45], the SISO decoder is also implemented with the BCJR algorithm in this dissertation.
The computational complexity of the BCJR SISO equalizer is on the order of O(Mµ), which grows exponentially with the channel memory length µ [58]. Similarly, the complexity of a BCJR SISO decoder grows exponentially with the encoder memory length. Usually the mem-ory length of a convolutional code is constrained on a length compromising the computational complexity with performance gain. However, the channel memory length depends on applica-tions having a large variation; it ranges from few taps in wireless systems to hundreds of taps in wireline systems. It is apparent that with existing results, turbo equalization is difficult to apply for systems with long channel responses. The algorithms developed in this chapter is aimed to solve the problem.
Since turbo equalization is an iterative processing scheme, it must operate with a block-by-block manner. Here, we let the block-by-block length be equal to Kc bits, where Kc = rKi + Ko, Ki
is the information bits, Ko is the overhead needed to terminate the trellis states, and 1/r (r ∈ Z, r ≥ 2) is the rate of convolutional code. The trellis state of the convolutional code is both
algorithm is applied.
For convenience of later use, following notations are pre-defined. Pr (·) is the probability function, p (·) is the probability density function (PDF), L (·) is the log-likelihood ratio (LLR) function, ln (·) is the natural logarithm function, max (·) is the maximum function, sign (·) is the sign function, E (·) is the expectation function, cov(x, y) = E(xy∆ T) − E(x)E(yT)is the covariance matrix function, diag [v1, v2, ...]is a diagonal matrix with the vector elements in the diagonal, and kxk =q
P |xi|2is the Euclidean vector norm (or 2-norm).