MAP Sequence Estimation

Here we summarize results from [2]. For channel coefficients which are independent over time, $f({\bf H})$ can be expressed as

$$f({\bf H}) = \ \prod_{k=1}^{n} f({\bf h}_k).$$ (4)

If $f({\bf h}_k)$ is Gaussian with mean ${\bf d}_k$ and covariance ${\bf C}_k$, then

$$f({\bf h}_k) = \frac {1} {\pi^M \vert{\bf C}_k\vert} e^{- ({\bf h}_k-{\bf d}_k)^{*T} {\bf C}_k^{-1} ({\bf h}_k-{\bf d}_k) } ,$$ (5)

where * denotes conjugate. We assume that ${\bf d}_k$ and ${\bf C}_k$ are known. Substituting (2) and (4) into (3), taking the negative natural logarithm of the integration result and ignoring the terms that are irrelevant to the minimization, we obtain:

$$- \log ( f({\bf r}\vert{\bf A})) \doteq \sum_{k=1}^n \frac { ... ... {\bf d}_k \vert^2 } { \sigma_k^2 } + \log ( \sigma_k^2 ) ~ ,$$ (6)

where

$$\sigma_k^2 = \sigma^2 + {\bf a}_k^T {\bf C}_k {\bf a}_k^* ~$$ (7)

and $\doteq$ denotes equivalent for optimization purposes. The time-recursive form for (6) is obvious, and since the incremental cost only depends on the state transition at the current time, the Viterbi algorithm can be used directly as an efficient, exact MAP (optimum) algorithm.

For a fast time-varying channel with Gauss-Markov fading parameters, the optimum solution is derived in [2] where a computationally effective suboptimum algorithm based on PSP and GVA is also proposed.

Concerning PSP, for each trellis survivor path the quantities involved in the transition cost are computed by channel model aided estimation as dictated by the MAP formulation. As per GVA, at each state of each stage of the trellis, we keep $L \ge 1$ survivors. We show that for reasonable L the algorithm approaches optimum results.