Communication System Model

In the discrete-time FIR model of a 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:

$$r_k = {\bf a}_k^T {\bf h} + n_k, \ \ k = 1,\ldots,N ,$$ (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, ${\bf h}$ is a complex column vector containing the channel impulse response coefficients (which are unknown), 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. Using matrix-vector notation, the above equations can then be expressed more simply as:

$${\bf r} = {\bf A h} + {\bf n} ,$$ (2)

and

$$f({\bf r}\vert{\bf h},{\bf A}) = \frac {1} {(\pi \sigma^{2})^N} ~ e^{ - \frac {\vert{\bf r}-{\bf Ah}\vert^2} {\sigma^2} } .$$ (3)