Adaptive Channel Estimation in Wireless Communications - Adaptive Signal Processing in Wireless Communications - page 48

algorithm [42]. Seshadri [47] presented blind trellis search techniques. Reduced-state
sequence estimation was proposed in [12]. Raheli et al. proposed a per-survivor process-
ing technique in [44].
The convergence of such approaches is not guaranteed in general. Interesting exam-
ples have been provided in [5] where two different combinations of H and s lead to the
same cost:

2

Y −TH () s .

2.3.2.2 The Methods of Moments
Although the ML channel estimator discussed in section 2.3.2.1 usually provides better
performance, the computation complexity and the existence of local optima are the two
major difficulties. Therefore, simpler approaches have also been investigated.

2.3.2.2.1 SISO Channel Estimation
For baud-rate data, second-order statistics of the data do not carry enough informa-
tion to allow estimation of the channel impulse response as a typical channel is non-
minimum phase. On the other hand, higher-order statistics (in particular, fourth-order
cumulants) of the baud-rate (or fractional-rate) data can be exploited to yield the chan-
nel estimates to within a scale factor.
Given the mathematical model (2.21), there are two broad classes of direct approaches
to channel estimation, the distinguishing feature among them being the choice of the
optimization criterion. All of the approaches involve (more or less) a least squares error
measure. The error definition differs, however, as follows:

• Fitting error : Match the model-based higher-order (typically fourth-order)

statistics to the estimated (data-based) statistics in a least squares sense to
estimate the channel impulse response, as in [54] and [55], for example. This
approach allows consideration of noisy observations. In general, it results in a
nonlinear optimization problem. It requires availability of a good initial guess
to prevent convergence to a local minimum. It yields estimates of the channel
impulse response.
Equation error • : This is based on minimizing an “equation error” in some equa-

tion that is satisfied ideally. The approaches of [17] and [60] (among others) fall
in this category. In general, this class of approaches results in a closed-form
solution for the channel impulse response so that a global extremum is always
guaranteed provided that the channel length (order) is known. These approaches
may also provide good initial guesses for the nonlinear fitting error approaches.
Quite a few of these approaches fail if the channel length is unknown.

Further details may be found in [14, 52, 56] and references therein.

2.3.2.2.2 SIMO Channel Estimation
Here we will concentrate upon second-order statistical methods, but first a few com-
ments regarding indirect SIMO channel estimation. As noted in section 2.1, linear