Data 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 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, and n_k is the white Gaussian complex noise with variance $\sigma^2$. Let ${\bf H} = [{\bf h}_1, \ldots , {\bf h}_n]$ represent the matrix of channel coefficient vectors over time, and let ${\bf A} = [{\bf a}_1, \ldots , {\bf a}_n]^T$ represent the matrix of transmitted data. Also let ${\bf r} = [ r_1 , \ldots , r_n ]^T$and ${\bf n} = [ n_1 , \ldots , n_n ]^T$. 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 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 $f({\bf r}\vert{\bf A})$ is used. 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)