EM for Gauss-Markov Channels

Using (2) in (6) and dropping constants:

$$Q({\bf A}\vert{\bf B}) \doteq \int_{\bf H} \ - \sum_{k=1}^{N}... ...{\bf h}_k\vert^{2} \ f({\bf H}\vert{\bf r},{\bf B}) d{\bf H} ,$$ (7)

with:

$$f({\bf H}\vert{\bf r},{\bf B}) \doteq \ f({\bf H}) \ \prod_{... ... {\vert r_k-{\bf b}_k^{T}{\bf h}_k\vert^{2}} {\sigma^{2}} } .$$ (8)

As an intermediate development, first assume that the elements of $\bf H$ are independent over time. Then $f({\bf H})$ can be written as $f({\bf H}) = \prod_{k=1}^{N} f({\bf h}_k)$, and $f({\bf H}\vert{\bf r},{\bf B})$ becomes:

$$f({\bf H}\vert{\bf r},{\bf B}) = \ \prod_{k=1}^{N} f({\bf h}_k\vert{\bf r},{\bf B}) ,$$ (9)

with:

$$f({\bf h}_k\vert{\bf r},{\bf B}) \doteq f({\bf h}_k) ~ e^{ - ... ...{\vert r_k -{\bf b}_k^{T}{\bf h}_k\vert^{2}} {\sigma^{2}} } .$$ (10)

Using (9) and (10) in (7) we obtain:

$$Q({\bf A}\vert{\bf B}) \doteq - \sum_{k=1}^N V_k ,$$ (11)

with:

   

$${\bf g}_k \stackrel{\triangle}{=} E[{\bf h}_k \vert {\bf r},{\bf B}]$$ (12)

$${\bf G}_k \stackrel{\triangle}{=} E[({\bf h}_k - {\bf g}_k) ({\bf h}_k - {\bf g}_k)^{*T} \vert {\bf r},{\bf B}]$$ (13)

$$\stackrel{\triangle}{=} \vert r_k - {\bf a}_k^T {\bf g}_k\vert^2 + {\bf a}_k^{T} {\bf G}_k {\bf a}_k^* .$$ V_k (14)

V_k represents the Viterbi algorithm incremental cost function at time k.

Now, for a fast time-varying channel with Gauss-Markov fading parameters, let $\alpha$ represent a complex first-order Markov factor such that $\vert\alpha\vert = e^{-\omega T}$ where T is the sampling period and $\frac {\omega}{\pi}$ is the Doppler spread [17]. Let the time-varying channel coefficients follow a Gauss-Markov distribution such that, at time k:

$${\bf h}_k = \alpha {\bf h}_{k-1} + {\bf v}_k ,$$ (15)

where ${\bf v}_k$ is complex, white, and Gaussian with mean $\bf d$ and covariance $\bf C$. In this case, $f({\bf h}_k\vert{\bf h}_{k-1})$ is Gaussian with mean ${\bf d} + \alpha {\bf h}_{k-1}$ and covariance $\bf C$. Additional properties of this distribution are derived in the appendix.

(11) and the definitions from (12-14) still apply, with:

$$f({\bf H}\vert{\bf r},{\bf B}) \doteq f({\bf h}_1) \ \prod_{k... ... {\vert r_k-{\bf b}_k^{T}{\bf h}_k\vert^{2}} {\sigma^{2}} } .$$ (16)

To determine $f({\bf h}_k\vert{\bf r},{\bf B})$, and thereby ${\bf g}_k$ and ${\bf G}_k$, we must integrate (16) over ${\bf h}_i, i = 1,\ldots,N, i \ne k$. The derivation of this integration result is shown in the appendix. The result is that $f({\bf h}_k\vert{\bf r},{\bf B})$ is Gaussian, with mean ${\bf g}_k$ and covariance ${\bf G}_k$ determined by the following recursive algorithm:

Initialize: for $k=N, N-1, \ldots , 2$

 

$${\bf\hat{G}}_k \left[ \frac {{\bf b}_k^*{\bf b}_k^T} {\sigma^2} + {\bf C}^{-1} \right.$$ =

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

$${\bf\hat{q}}_k \frac {r_k{\bf b}_k^*} {\sigma^2} + (1 - \alpha^*) {\bf C}^{-1} {\bf d}$$ = (17)

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

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

 

$${\bf G}_k \left[ \frac {{\bf b}_k^*{\bf b}_k^T} {\sigma^2} + {\bf C}^{-1} + \vert\alpha\vert^2 \left( {\bf C}^{-1} \right. \right.$$ =

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

$${\bf q_k} \frac {r_k{\bf b}_k^*} {\sigma^2} + (1 - \alpha^*) {\bf C}^{-1} {\bf d}$$ = (18)

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

$${\bf g}_k {\bf G_k} {\bf q_k}$$ =

• ${\bf G}_k$ and ${\bf g}_k$ from (18) will be used in the Viterbi algorithm incremental cost function for time k.
• Update for next iteration:

 

$${\bf\hat{G}}_k \left[ \frac {{\bf b}_k^*{\bf b}_k^T} {\sigma^2} + {\bf C}^{-1} \right.$$ =

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

$${\bf\hat{q}}_k \frac {r_k{\bf b}_k^*} {\sigma^2} + (1 - \alpha^*) {\bf C}^{-1} {\bf d}$$ = (19)

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

with ${\bf\hat{G}}_0 \equiv {\bf\hat{G}}_{N+1} \equiv {\bf C}$, ${\bf\hat{q}}_0 \equiv \left( \frac {1 - \alpha^*} {1 - \alpha} \right) {\bf C}^{-1} {\bf d}$, and ${\bf\hat{q}}_{N+1} \equiv {\bf C}^{-1} {\bf d}$.

It is apparent from the above algorithm that $f({\bf h}_k\vert{\bf r},{\bf B})$ at any time kdepends on all of the received data and estimated symbols from time 1 to N. In the EM algorithm E-step, (17-19) are used to estimate ${\bf G}_k$ and ${\bf g}_k$, $k = 1 , \ldots , N$. In the M-step, the Viterbi algorithm with incremental cost function (14) is used to estimate $\bf A$.