Deterministic Channel Coefficients

The case of deterministic channel coefficients is examined first because it is the simplest case, and because it corresponds to the assumptions made by previously published algorithms [3] which do not use any known apriori statistics for the channel. Unlike the other cases considered subsequently, for the deterministic channel we do not know the apriori mean of the coefficients, and as shown below, the E-step of the EM algorithm can not use $E[{\bf h}\vert{\bf r,B}]$.

If ${\bf h}$ is not random, then let $f({\bf h}) = \delta({\bf h}-{\bf g})$ where ${\bf g}$ is the deterministic but unknown value of ${\bf h}$, and $\delta ()$ represents an impulse function. In this case, $E[{\bf h}] = {\bf g}$, and using (12) to estimate ${\bf g}$ does not provide anything useful since it reduces to:

 

$$E[{\bf h}\vert{\bf r,B}] \int_{\bf h} {\bf h} ~ K_{\ref{eq:f(h... ...\vert{\bf r}-{\bf Bh}\vert^2} {\sigma^2} } \delta ({\bf h} - {\bf g}) d {\bf h}$$ =

$${\bf g} ~ K_{\ref{eq:f(h\vert r,B)}} ~ e^{- \frac {\vert{\bf r}-{\bf Bg}\vert^2} {\sigma^2} } = {\bf g} ,$$ = (19)

and ${\bf g}$ is unknown.

So to estimate ${\bf g}$ in this case, we must use an ad-hoc method. Note that given ${\bf r}$ and ${\bf B}$, ${\bf h}$ satisfies (2):

$${\bf r} = {\bf B h} + {\bf n} .$$ (20)

Taking the expected value of the right-hand-side of (20) with respect to ${\bf h}$ and $\bf n$ yields:

$${\bf r} = {\bf B g} ,$$ (21)

which can be solved for a linear least-square-error estimate of ${\bf g}$using the pseudo-inverse of ${\bf B}$:

$${\bf g} = {\bf B}^+ {\bf r} .$$ (22)

Note that this estimate of ${\bf g}$ is also a maximum-likelihood estimate of ${\bf h}$ given ${\bf r}$ and ${\bf B}$ since it maximizes $f({\bf r}\vert{\bf h},{\bf B})$(i.e. minimizes $\vert{\bf r} - {\bf B h}\vert^2$). Although producing an optimal estimate of the channel coefficients was not a goal in this EM algorithm, it does so anyway as a byproduct of the E-step.

In this case, ${\bf G} = 0$, and the Viterbi algorithm incremental cost function from (18) becomes:

$$V_k = \vert r_k - {\bf a}_k^T {\bf g}\vert^2 = \vert r_k - {\bf a}_k^T {\bf B}^+ {\bf r}\vert^2 .$$ (23)

Referring to the general EM algorithm described above, in the E-step use (22) to estimate ${\bf g}$ and in the M-step use the Viterbi algorithm with incremental cost function (23) to estimate ${\bf A}$.

In step 1 of the EM algorithm, we could initialize ${\bf B}$ randomly, or to all 1's, etc. But to reduce the number of iterations required, and to help the algorithm converge to a proper solution, it is better to initialize ${\bf B}$ with some reasonable estimate of ${\bf A}$. This can be done, for example, by initially using a pure delay channel, i.e. using ${\bf g} = [0 ~ \ldots ~ 0 ~ 1 ~ 0 ~ \ldots ~ 0]^T$ in (23), then using the Viterbi algorithm to estimate ${\bf A}$, then setting ${\bf B}$ to this estimate of ${\bf A}$.

This EM algorithm is equivalent to previously published algorithms for optimal joint sequence and channel estimation [3], but the derivation here provides a mathematical basis for understanding this algorithm in the context of MLSE by marginalizing over the channel parameters.