Intersymbol interference (ISI) and channel noise are the major impairments in a communication system. Typically, the transmitter and receiver in the system must be designed to combat these effects. In the transmitter, there is an error correction encoder adding redundant information for bit protection, and a signal modulator converting the bit stream into a valid analog waveform for transmission. In the receiver, there is an equalizer for ISI compensation and demodulation, and a decoder for error bits correction. For a long period of time, the receiver performs equalization and decoding separately. Although this receiver architecture is popular and works well for many systems, the achievable capacity is far from Shannon’s bound. To have better performance or higher capacity, a turbo equalization [38] architecture, conducting equalization and decoding jointly and iteratively, was proposed [39]. The turbo equalizer enhances the performance itera-tively with the soft extrinsic information exchange between a soft-in/soft-out (SISO) equalizer and a SISO decoder. The extrinsic information is extracted from the equalizer and the decoder at an iteration and used as a priori information in the next iteration. It has been shown that the turbo equalization can greatly enhance the receiver performance and the achievable capacity can help to approach the Shannon’s bound.
In 1995, the first turbo equalizer [39] employing a soft output Viterbi algorithm (SOVA) [40]
was introduced. In this work, the relationship between original and the encoded bits in the error 63
of the channel. Two serially concatenated SOVAs were then used to detect and decode the received signal iteratively. The performance was improved iteratively toward the bound corre-sponding a coded transmission over an additive white Gaussian noise (AWGN) channel. From a standalone soft equalizer’s (or decoder’s) point of view, the SOVA is a suboptimal algorithm though it is simpler, and further performance improvement is possible. Later, a turbo equalizer with the maximum a posteriori probability (MAP) criterion [41] was reported. The equalizer was realized with the BCJR algorithm which is an optimal SISO processing algorithm for the turbo equalizer. Theoretically, the optimal receiver should consider a single but larger trellis that combines the encoder trellis and channel trellis. However, the size of combined trellis is usually so large that the computational complexity of the BCJR algorithm becomes prohibitive and unrealizable. Thus, theoretical optimal receiver is rarely considered. The optimality for a serially concatenated turbo equalizer is evaluated on a single SISO detector or decoder only.
Since both the SOVA and BCJR turbo equalizers are constructed based on the trellis structure, such category of turbo equalizers was classified as the trellis-based turbo equalizer.
The performance of trellis-based turbo equalizer was shown to be excellent but the com-putational complexity is a big penalty. The complexity depends on the number of processing iterations, the signal constellation size, the memory length of the encoder, and the memory length of the channel. The main problem is that the complexity increases exponentially with the memory length of the encoder and that of the channel. While we can control the memory length of the encoder, we cannot control that of the channel. For a channel with medium or large delay spread, it is easy for the complexity to become impractically high.
The high computational complexity problem of the optimal trellis-based turbo equalizers motivated the study of low complexity suboptimal turbo equalizers. In 1999, Wang et al. [42]
proposed a lower complexity filter-based turbo interference canceller, which consists of a soft multiuser detector and a soft channel decoder, for the multiuser coded CDMA system. In-stead of using complex SOVA or BCJR algorithm as the multiuser detector, a low-complexity
65 filter-based soft multiuser detector was developed. Based on this concept, a low-complexity filter-based turbo equalizer was first proposed in [43]. Although this approach does have lower computational complexity, the optimal filter weights are varying (even the channel is time-invariant) and they are updated per symbol. The complexity for the optimal weights calculation is still high.
Motivated by [42], Tüchler et al. [44–46] also proposed a category of filter-based turbo equalizers with much lower complexity than ever. The soft equalizer in their works modifies the conventional linear equalizer or the decision feedback equalizer (DFE) with soft inputs. It is well known that the optimal filter weights [35, 47] of a linear equalizer or DFE equalizer is a function of channel response and signal-to-noise ratio (SNR), and can be solved with corre-sponding Wiener equations. However, the Wiener solution involves matrix inversion operations.
For a direct matrix inversion, the computational complexity is on the order of O(N3), where N is the filter length. The complexity is still high especially when the filter tap weights are updated per symbol. In [44], a time-recursive updated algorithm for calculating the optimal time-varying filter was proposed; the complexity was reduced to the order of O(N2). Exploring special matrix properties in a time-varying soft equalizer, a less complexity block-invariant sub-optimal soft equalizer was proposed in [45] and the complexity is further reduced to the order of O(N ). Later, in [46], three different kinds of low-complexity time-invariant soft equalizer were proposed. The complexities are on the same order with [45], but the convergence behaviors are quite different.
The adaptive turbo equalizer [48–52] is another low-complexity alternative. The efficiency of an adaptive turbo equalizer depends on the convergence behavior of the adaptive algorithm.
The convergence of the adaptive algorithm can be very slow for a long-response channel such as the wireline channel. The training period, which is usually thousands time of the received block length, becomes much longer than the equalization period. Thus, the adaptive turbo equalizer cannot be directly used in the application. The frequency-domain turbo equalizers were also studied extensively [53–56]. Its complexity is usually an order of magnitude less than
equalization (SC-FDE) system, which is a block-by-block transmission communication system with periodically inserted guard intervals. It cannot be applied to the conventional single carrier modulation system such as wireline SHDSL and wireless GSM, etc.
In this dissertation, we focus on the problem of time-domain turbo equalization over static frequency selective channels. We propose a fast turbo equalizer with complexity an order of magnitude less than the conventional. The proposed algorithm is based on the structure of [45]
in which the most computationally intensive operations are the optimal filter coefficients cal-culation and filtering (equalization). We explore the characteristics of the wireline channel responses and the optimal equalizers and propose interpolation schemes to reduce the complex-ity. We found that the relationship between an optimal filter coefficient and reliability infor-mation, which indicates the correctness of soft bits from the decoder, is nearly one-to-one and monotonic. Once the reliability is estimated, the corresponding optimal filter response can be calculated easily by interpolating two pre-calculated known optimal filter responses. In other words, we can bypass the computationally intensive matrix operations to obtain optimal filter coefficients. Since the reliability function ranges from 0 to 1, we only have to pre-calculate a small number of filter coefficient sets corresponding to specific reliability values and store those sets during initialization. Then, we can interpolate all possible optimal filters in run-time. This will dramatically reduce the computational requirement of the filter-based turbo equalizer. If the channel response is long and changing smoothly, we can apply the interpolated filtering [15]
to reduce the complexity further. Combining above schemes, we are able to obtain a fast inter-polated turbo equalizer required very low-complexity. Simulation results show that while the computational complexity is reduced dramatically, the performance of the proposed algorithm is almost not affected; it is about the same as that in [45].
This chapter is organized as follows. In Section 5.1, we briefly describe the system model and define some notations. In Section 5.2, the optimal BCJR turbo equalizer and suboptimal filter-based turbo equalizers are summarized. In Section 5.3, the proposed fast interpolated