System model

In the discrete-time FIR model of a time-varying noisy communications channel with inter-symbol interference, for the received data sequence up to time n, the complex received data r_k at time k is given by

$$r_k = {\bf a}_k^T {\bf h}_k + n_k, \ \;\;\;\;\;\;\;\;\;\;\; \ k = 1,\ldots,n ~ ,$$ (1)

where ${\bf a}_k^T$ is a complex row vector containing transmitted data { $a_{k-i+1}, i=1,\ldots,M$}, M is the FIR channel length, ${\bf h}_k$ is a complex column vector containing the channel impulse response coefficients at time k, and n_k is the white Gaussian complex noise at time k with variance $\sigma^2$. Let ${\bf H} = [{\bf h}_1, \ldots , {\bf h}_n]$ represent the matrix of channel coefficient vectors over time arranged by columns, and let ${\bf A} = [{\bf a}_1, \ldots , {\bf a}_n]^T$ represent the matrix of transmitted data arranged by rows. Also let ${\bf r} = [ r_1 , \ldots , r_n ]^T$ represent the column vector of received data, and ${\bf n} = [ n_1 , \ldots , n_n ]^T$ represent the column vector of noise over time. With this notation, the probability density function of the received data, given $\bf H$ and $\bf A$, is

$$f({\bf r}\vert{\bf H},{\bf A}) = \frac {1} {(\pi \sigma^2)^n... ... \frac {\vert r_k-{\bf a}_k^T{\bf h}_k\vert^2} {\sigma^2} } .$$ (2)

To minimize BER, we must find $\bf A$ to maximize $f({\bf A}\vert{\bf r})$ (i.e. the MAP estimator). This is equivalent to maximizing $f({\bf r}\vert{\bf A})f({\bf A })$. If $f({\bf A })$ is unknown or assumed to be uniform, then the ML estimator which maximizes the likelihood function $f({\bf r}\vert{\bf A})$ can be used instead. Referring to (2), if the channel is known, the Viterbi algorithm can be used directly to estimate the data sequence.

If the channel is unknown, the dependence of $f({\bf r}\vert{\bf A})$ on the random channel coefficients is

$$f({\bf r}\vert{\bf A}) = E[f({\bf r}\vert{\bf H},{\bf A})] = ... ...} f({\bf r}\vert{\bf H},{\bf A}) f({\bf H }) d{\bf H } \; \; .$$ (3)

Evaluating this multidimensional integral and then maximizing the result with respect to $\bf A$ seems to be an intractable problem in general. However, if the statistics of the channel coefficients can be assumed, it is shown below that in some cases this problem is solvable and that the Viterbi algorithm or GVA can be applied directly to obtain the optimum or suboptimum results. In the following two sections, we will show how the problem is solved over time-independent Gaussian channels and Gauss-Markov channels.