Gauss-Markov Signal Amplitudes

For a Gauss-Markov time-varying signal amplitude vector, let $\alpha$ represent a complex first-order Markov factor. Let the signal amplitude vector follow a Gauss-Markov distribution such that, at time k:

$${\bf s}_k = \alpha {\bf s}_{k-1} + {\bf v}_k ,$$ (22)

where ${\bf v}_k$ is complex, white, and Gaussian with mean $\bf c$ and covariance $\bf C$. In this case, $f({\bf s}_k\vert{\bf s}_{k-1})$ is Gaussian with mean ${\bf c} + \alpha {\bf s}_{k-1}$ and covariance $\bf C$. The $Q({\bf\theta}\vert{\bf\phi})$ from (12) with the ${\bf g}_k$ and ${\bf G}_k$ from (13,14) still apply. $f({\bf S})$ can be written as:

$$f({\bf S}) = f({\bf s}_1) ~ \prod_{k=2}^N f({\bf s}_k\vert{\bf s}_{k-1}) ,$$ (23)

so from (10) and (3) $f({\bf S}\vert{\bf X},{\bf\phi})$ becomes:

$$f({\bf S}\vert{\bf X},{\bf\phi}) \doteq f({\bf s}_1) ~ \prod_... ...bf A}_k ({\bf\phi}) {\bf s}_k\vert\vert^{2}} {\sigma^{2}} } .$$ (24)

To determine $f({\bf s}_k\vert{\bf X},{\bf\phi})$, and thereby ${\bf g}_k$ and ${\bf G}_k$, we must integrate (24) over ${\bf s}_i, i = 1,\ldots,N, i \ne k$. Due to space limitations, the derivation of this integration result is not shown here. A parallel derivation for a similar model and problem (a Gauss-Markov FIR intersymbol interference channel model for a digital communication problem) appears in [10]. The result is that $f({\bf s}_k\vert{\bf X},{\bf\phi})$ is Gaussian, with mean ${\bf g}_k$ and covariance ${\bf G}_k$ determined in the E-step by the following recursive algorithm:
1) Initialize: for $k=N, N-1, \ldots , 2$

 

$${\bf\hat{G}}_k \left[ \frac { {\bf A}_k^H ({\bf\phi}) {\bf A}_k ({\bf\phi}) } {\... ...\bf C}^{-1} - {\bf C}^{-1} {\bf\hat{G}}_{k+1} {\bf C}^{-1} \right) \right]^{-1}$$ =

$${\bf\hat{q}}_k \frac { {\bf A}_k^H ({\bf\phi}) {\bf x}_k } {\sigma^2} + (1 - \al... ...f C}^{-1} {\bf c} + \alpha^* {\bf C}^{-1} {\bf\hat{G}}_{k+1} {\bf\hat{q}}_{k+1}$$ = (25)

2) Iterate: for $k=1, 2, \ldots, N$

 

$${\bf G}_k \left[ \frac { {\bf A}_k^H ({\bf\phi}) {\bf A}_k ({\bf\phi}) } {\sigma^2} + {\bf C}^{-1} \right.$$ =

$$\vert\alpha\vert^2 \left( \left. {\bf C}^{-1} - {\bf C}^{-1} {\bf... ...\bf C}^{-1} - {\bf C}^{-1} {\bf\hat{G}}_{k+1} {\bf C}^{-1} \right) \right]^{-1}$$ +

$${\bf q}_k \frac { {\bf A}_k^H ({\bf\phi}) {\bf x}_k} {\sigma^2} + (1 - \alpha^*) {\bf C}^{-1} {\bf c}$$ =

$$\alpha {\bf C}^{-1} {\bf\hat{G}}_{k-1} {\bf\hat{q}}_{k-1} + \alpha^* {\bf C}^{-1} {\bf\hat{G}}_{k+1} {\bf\hat{q}}_{k+1}$$ +

$${\bf g}_k {\bf G}_k {\bf q}_k \; \; .$$ = (26)

• ${\bf G}_k$ and ${\bf g}_k$ from above will be used in $Q({\bf\theta}\vert{\bf\phi})$ for the M-step cost function for time k.

3) Update for the next EM iteration:

 

$${\bf\hat{G}}_k \left[ \frac { {\bf A}_k^H ({\bf\phi}) {\bf A}_k ({\bf\phi}) } {\... ...\bf C}^{-1} - {\bf C}^{-1} {\bf\hat{G}}_{k-1} {\bf C}^{-1} \right) \right]^{-1}$$ =

$${\bf\hat{q}}_k \frac { {\bf A}_k^H ({\bf\phi}) {\bf x}_k } {\sigma^2} + (1 - \al... ...f C}^{-1} {\bf c} + \alpha {\bf C}^{-1} {\bf\hat{G}}_{k-1} {\bf\hat{q}}_{k-1} ,$$ = (27)

with ${\bf\hat{G}}_0 \equiv {\bf\hat{G}}_{N+1} \equiv {\bf C}$, ${\bf\hat{q}}_0 \equiv \left( \frac {1 - \alpha^*} {1 - \alpha} \right) {\bf C}^{-1} {\bf c}$, and ${\bf\hat{q}}_{N+1} \equiv {\bf C}^{-1} {\bf c}$. It is apparent from the above algorithm that $f({\bf s}_k\vert{\bf X},{\bf\phi})$ at any time k depends on all of the received data and estimated symbols from time 1 to N.
The detailed EM algorithm for Gauss-Markov ${\bf S}$ is:

1.

Initialize ${\bf\phi}$ to an estimate of ${\bf\theta}$.

2.

E-step: use the algorithm from (25-27) to estimate ${\bf G}_k$ and ${\bf g}_k$, $k = 1 , \ldots , N$.

3.

M-step: use the cost function (12) to estimate ${\bf\theta}$.

4.

Set ${\bf\phi} = {\bf\theta}$ and repeat steps 2 and 3 until converged.

To initialize ${\bf\phi}$ in step 1, we could, for example, initialize ${\bf G}_k$ and ${\bf g}_k$using ${\bf\phi} = 0$ in (25-27).