Time-Independent Gaussian channel

For channel coefficients 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, M is the FIR channel length and 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)

($\doteq$ denotes equivalent for optimization purposes).

The time-recursive form for (6) can be expressed as

$$\Lambda _{n} = \Lambda _{n-1} + \lambda _{n} ~ ,$$ (8)

where $\Lambda _{n}$ is the cumulative cost up to time n. From (6), $\lambda _{n}$ is the incremental cost from time n-1 to n:

$$\lambda _{n}= \frac { \vert r_n - {\bf a}_n^T {\bf d}_n \vert^2 } { \sigma_n^2 } + \log ( \sigma_n^2 ) ~ .$$ (9)

Since the incremental cost only depends on the state transition at time n, the Viterbi algorithm can be used directly to minimize (8). Therefore, (8) represents a computationally efficient, time-recursive, exact MAP (optimum) algorithm.