EM Algorithms for Specific Random Channels

Using (3) in (10), (8) becomes:

$$Q({\bf A}\vert{\bf B}) \doteq E[-\vert{\bf r}-{\bf Ah}\vert^2... ...f r}-{\bf Ah}\vert^2 f({\bf h}\vert{\bf r},{\bf B}) d{\bf h} ,$$ (11)

with

$$% latex2html id marker 804 f({\bf h}\vert{\bf r},{\bf B}) = K... ...frac {\vert{\bf r}-{\bf Bh}\vert^2} {\sigma^2} } f({\bf h}) ,$$ (12)

where $K_{\ref{eq:f(h\vert r,B)}}$ can be determined if needed by setting the integral of $f({\bf h }\vert{\bf r},{\bf B})$ to 1.

To evaluate (11) we only need the mean and covariance of ${\bf h}$ given ${\bf r}$ and ${\bf B}$using the distribution of (12). We will derive these for specific cases of $f({\bf h})$ subsequently; for now, we define them symbolically as ${\bf g}$ and ${\bf G}$:

  

$${\bf g} \stackrel {\triangle} {=} E[{\bf h}\vert{\bf r},{\bf B}]$$ (13)

$${\bf G} \stackrel {\triangle} {=} E[({\bf h} - {\bf g}) ({\bf h} - {\bf g})^{*T} \vert{\bf r},{\bf B}] .$$ (14)

The noise variance $\sigma^2$ is needed in general to compute ${\bf g}$ and ${\bf G}$. If $\sigma^2$is not known it could be estimated from the variances of the transmitted and received data.

The norm-squared term in (11), for each candidate ${\bf A}$, can be represented as a path through a trellis, whose states at a given time are the states of the of the ISI channel. A path represents a summation of incremental state transition costs. The minimum cost path can be determined by the Viterbi algorithm.

Rewriting (11) in terms of the incremental costs:

$$Q({\bf A}\vert{\bf B}) \doteq - \sum_{k=1}^N V_k ,$$ (15)

where V_k is:

$$V_k = E[\vert r_k - {\bf a}_k^T {\bf h}\vert^2 \ \vert{\bf r},{\bf B}] .$$ (16)

The expected value is taken over ${\bf h}$ using (12) for $f({\bf h }\vert{\bf r},{\bf B})$.

Expanding the norm squared:

 

$$\vert r_k - {\bf a}_k^T {\bf h} \vert^2 \vert r_k\vert^2 - r_k {\bf a}_k^{*T} {\bf h}^* - r_k^* {\bf a}_k^T {\bf h} + {\bf a}_k^T {\bf h}{\bf h}^{*T}{\bf a}_k^*$$ =

$$\vert r_k\vert^2 - r_k {\bf a}_k^{*T} {\bf h}^* - r_k^* {\bf a}_k^T {\bf h}$$ =

$$+ {\bf a}_k^{T} ({\bf h}-{\bf g})({\bf h}-{\bf g})^{*T} {\bf a}_k^*$$ (17)

$$+ \ {\bf a}_k^{T} {\bf h}{\bf g}^{*T} {\bf a}_k^* + {\bf a}_k^{T} {\bf g}{\bf h}^{*T} {\bf a}_k^* - {\bf a}_k^{T} {\bf g}{\bf g}^{*T} {\bf a}_k^* .$$

Taking the expected value over ${\bf h}$:

 

$$\vert r_k\vert^2 - r_k {\bf a}_k^{*T} {\bf g}^* - r_k^* {\bf a}_k... ...{\bf a}_k^{T} {\bf G}{\bf a}_k^* + {\bf a}_k^{T} {\bf g}{\bf g}^{*T}{\bf a}_k^*$$ V_k =

$$\vert r_k - {\bf a}_k^T {\bf g}\vert^2 + {\bf a}_k^{T} {\bf G}{\bf a}_k^*.$$ = (18)

Equation (18) represents the general form of the Viterbi algorithm incremental cost function for random ISI channels. To use (18) in an EM algorithm, ${\bf g}$ and ${\bf G}$ from (13) and (14) are computed in the M-step; these computations depend on the specific form of the apriori channel distribution $f({\bf h})$. The following subsections examine the specific cases of deterministic, uniform, and Gaussian $f({\bf h})$.