Communication System Model

In the discrete-time FIR model of a time-varying noisy communications channel with inter-symbol interference, for a block of N received data values, 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 { $a_{k-i+1}, i=1,\ldots,L$}, L is the FIR channel length, ${\bf h}_k$ is a complex column vector containing the channel impulse response coefficients (which are unknown) at time k, and n_k is the white Gaussian complex noise at time k with variance $\sigma^2$. For $j \le 0$, the transmitted data a_j may be either known (e.g. all 0's), unknown, or estimated with some associated probabilities from the end of a previous data block.

Let ${\bf H} = [{\bf h}_1, \ldots , {\bf h}_N]$ represent the matrix of channel coefficient vectors over time arranged by columns, ${\bf A} = [{\bf a}_1, \ldots , {\bf a}_N]^T$ the matrix of transmitted data arranged by rows, and ${\bf r} = [ r_1 , \ldots , r_N ]^T$ the column vector of received data. 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)