THE PROPOSED FAST INTERPOLATED TURBO EQUALIZER 99 parameters are used for all tap weights

In this dissertation, we apply the piecewisely linear Lagrange interpolation [70] scheme.

Assume that (¯v1, f (¯v1))and (¯v2, f (¯v2))are known as a priori with (5.90) and ¯v1 < ¯v < ¯v2. For a single tap weight, we can has the linear interpolation as

fj(¯v) = wf (¯v1) + (1 − w)f(¯v²) (5.94) where w is a weighting factor. The weight factor is given by

w = v − ¯v¯ ²

¯ v₁− ¯v2

. (5.95)

we can the partition the full range of reliability [0, 1] into a number of adjoined intervals and approximate fj(¯v)in each interval with a piecewisely linear function.

As mentioned, we use a suboptimal interpolation scheme. For an optimal filter, assume that (¯v1, f1 = ζ(¯v1))and (¯v2, f2 = ζ(¯v2))are known as a priori with (5.90) and 0 ≤ ¯v¹ < ¯v2 ≤ 1.

For a reliability ¯vi located between the interval ¯v1 < ¯vi < ¯v2, the corresponding optimal filter f_i = ζ(¯vi)can be interpolated by two given optimal filters.

f_i ≈ ˜fⁱ = ξ(f^∆ 1, f2) (5.96)

where ˜fiis the interpolated optimal filter and ξ(·) is the linear interpolated function as

ξ(f1, f2) = wf1+ (1 − w)f². (5.97) where w is a weighting factor. The weight factor is given by

w = ¯vi− ¯v²

¯

v1− ¯v². (5.98)

Thus, all coefficients of optimal filters are interpolated with the same set of parameters. From our experience, a better approximation may be obtained in terms of the square root of reliability (i.e., standard deviation of soft decisions). However, the square root operation is required and this will complicate the computation. To compensate the possible performance loss, we make the interpolation interval smaller, which will only increase implementation cost slightly.

NIE=^∆

˜f_i− fⁱ

2

kfik² . (5.99)

If ¯v1, ¯v2are close enough, the interpolated optimal filter ˜fi will close to the optimal filter fi, too. The reliability is within the range [0, 1] and we can select a set of reference reliabilities {¯v^j} = {0, ¯v¹, ¯v2, · · · , 1}. If the number of the reference reliabilities is large enough, the NIE will be small and the performance loss will be ignorable. Let the full range of reliability be partitioned by Z sub-intervals, i.e., the number of reference reliabilities be (Z + 1). We can then calculate (Z + 1) corresponding reference optimal filters and store the filter coefficients in a table. For any reliability, we can then look up the table and obtain the corresponding optimal filter through interpolation. This will dramatically reduce the computational requirement for the LSL equalizer. The next problem is how to determine the values of reference reliabilities.

This can be seen as a sampling problem and the simplest one is an uniform sampling scheme as

¯

vj = j∆, j = {0, 1, · · · , Z} . (5.100) where ∆ = 1/Z is the sampling spacing. However, simulations show that the NIE performance is satisfactory for large Z only. The reason can be explained below. For a specific tap weight, let its value corresponding to ¯v = 1 be fN and it value corresponding to ¯v = 0 be fP. A uniform sampling for the reliablity between ¯v = 1 does not give a uniform sampling for the tap weight between fN and fP.

Thus, the reference reliabilities should be non-uniformly sampled. The goal is to make the corresponding optimal filter weights be uniformly sampled (because of piecewise linear interpolation). Although we can formulate an MMSE cost function to the nonlinear sampling problem and find the optimal solution by the generalized Pontryagin’s maximum principle [71], it will be time-consuming. Here, we only select a common nonlinear function to the job. We observe that when the reliability is large, the variation of optimal weights are small. However, when the reliability is small, the variation will be large. We then need denser sampling when

5.3. THE PROPOSED FAST INTERPOLATED TURBO EQUALIZER 101 reliability is small. This motivate us to use an exponential function for sampling. The sampling scheme is given by

¯ v_j =







e^{−(Z−1λ )·j} j = {0, 1, · · · , Z − 1}

0 j = Z

, (5.101)

where λ is a factor controlling the decaying rate of the exponential function. The reliability with zero value is considered as the perfect a priori case. As we can see, the larger the entries are, the smaller the interpolated error will be. But, the the table size will become larger. There is a tradeoff between interpolation performance and table size.

From (5.101), we also see that the decaying factor also determines the minimum (except zero) reliability value, namely ¯v_Z−1 = e^−λ. The minimum reliability value is crucial to both NIE and BER performance. This is because when the number of iteration or the SNR is high, the corresponding reliability will be very small. It will fall into the region between ¯v_Z−1 = e^−λ and ¯vZ = 0. The interpolation in this region becomes critical. The decaying factor is determined through some trial-and-errors. This will be discussed in simulations section later.

The sampled reference reliability ¯vj begins at ¯v0 = 1corresponding no a priori case. Then, its value is exponentially reduced, and finally ends with ¯vZ = 0corresponding to the perfect a priori case. If the channel is time-invariant, it can be identified during the initialization stage.

Also in this stage, we can then calculate a set of reference optimal filters {f^j} = {f⁰, f1, · · · , f^Z} (5.90) according to the sampled reliability. Later, for any reliability ¯vⁿ_i, we can approximate the optimal filter f_iⁿ with the interpolation shown in (5.97). For any iteration and any block, we do not have to re-calculate the optimal filter using (5.90). Thus, the computational complexity can be very low. Before interpolation, we have to perform a binary search to locate the interpolation interval (¯v1 < ¯v_iⁿ < ¯v₂). The required operations are log₂(Z + 1) comparisons only. Our simulations shown that 16 sampling points are large enough for a channel with length up to hundreds of taps. That means only four comparison operators are required and the complexity is ignorable. The complexity for computing reference optimal filters, {fiⁿ}, is (Z + 1) · O(N³).

This is also ignorable since we only have to carry out the operations once. Unless the channel

§ 5.3.2 FDISL Equalizer

For the channel with hundreds of taps, the application of a turbo equalizer is difficult, if not impossible. Even for the LSL equalizer, we may need one million operations to compute an optimal filter. The complexity of LSLH equalizer is very low; however, the performance is usually not satisfactory. For a long response, the convergence behavior of a turbo equalizer become more sensitive and difficult to control. Despite the problem, the filter-based turbo equalizer is still the only possible candidate to apply. In the previous subsection, we have developed an interpolation scheme dramatically reducing the computational complexity of LSL equalizer. Note that the scheme is to interpolate the whole response of an optimal filter. We call this a whole response (WR) interpolation scheme.

Inspecting the structure of LSL equalizer in Fig. 5.13 or (5.72), we find that the equalizer consists of two separate linear filters. One is the optimal filter f we have been working with, and the other is a filter with soft-decisions as its inputs and channel responses as its coefficients. The length of optimal filter is usually on the same order of the channel length. Thus, the equalization operation will require high computational complexity when the channel length is long. If the response of the channel and optimal filter can be interpolated, the complexity can be reduced further. For wireline channels, this is indeed possible. Note that the scheme here is to interpolate an individual sample of a channel or a filter response and we call this an individual response (IR) interpolation scheme. Note that what discussed in the previous chapters belongs to the IR interpolation schemes. Combining the WR and IR interpolation schemes, we obtain the FDISL equalize with very low computational complexity.

Similar to the interpolated echo canceller in Chapter 3, the channel response used in (5.72) can be approximated by a low-complexity interpolated FIR (IFIR) filter. For convenience, we rewrite the soft observation again.

¯

y_k = H¯x_k. (5.102)