MLSE with Gauss-Markov Channels

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 symbol period and $\frac {\omega}{\pi}$ is the Doppler spread. Let the time-varying channel coefficients follow a Gauss-Markov distribution such that, at time i, ${\bf h}_i = \alpha {\bf h}_{i-1} + {\bf v}_i$, where ${\bf v}_i$ is complex, white, and Gaussian with zero mean and covariance ${\bf C}$. In this case, $f({\bf h}_i\vert{\bf h}_{i-1})$ is Gaussian with mean $\alpha {\bf h}_{i-1}$ and covariance ${\bf C}$. $f({\bf H}^n)$ can be written as

$$f ({\bf H}^n) = ~ \prod_{i=1}^n f({\bf h}_i\vert{\bf h}_{i-1}) ~ .$$ (11)

We assume that the initial channel coefficients are known, and they are represented as ${\bf h}_0$. Then

$$f({\bf r}^n\vert{\bf A}^n) \doteq ~ \prod_{i=1}^n \int_{{\bf h}_i} e^{ - Q_i } d {\bf h}_i ~ ,$$ (12)

$Q_i = \vert\vert {\bf r}_i - {\bf A}_i {\bf h}_i \vert\vert^2 / \sigma^2 - ({\bf h}_i - \alpha {\bf h}_{i-1})^H {\bf C}^{-1} ({\bf h}_i - \alpha {\bf h}_{i-1})$. We can marginalize over the channel coefficient distribution step by step. The integral over ${\bf h}_1$ is

$$I_1 = \int_{{\bf h}_1} e^{- E_1 } d{\bf h}_1 ~ ,$$ (13)

with

 

$$({\bf h}_1 - \alpha {\bf h}_0)^H {\bf C}^{-1} ({\bf h}_1 - \alpha {\bf h}_0)$$ E₁ =

$$+ ~ ({\bf h}_2 - \alpha {\bf h}_1)^H {\bf C}^{-1} ({\bf h}_2 - \alpha {\bf h}_1)$$ (14)

$$+ ~ \frac {\vert\vert {\bf r}_1 - {\bf A}_1 {\bf h}_1 \vert\vert^2} {\sigma^2} ~ .$$

E₁ can be reduced, dropping constant terms, to

 

$$E_1 \doteq({\bf h}_1 - {\bf g}_1)^H {\bf G}_1^{-1} ({\bf h}_1 - {\bf g}_1) - {\bf q}_1^H {\bf G}_1 {\bf q}_1 + {\phi}_1({\bf h}_2)$$ (15)

with

 

$${\bf G}_1 \left[ \frac {{\bf A}_1^H {\bf A}_1} {\sigma^2} + (1 + {\vert\alpha\vert}^2) {\bf C}^{-1} \right]^{-1}$$ =

$${\bf q}_1 \frac {{\bf A}_1^H {\bf r}_1} {\sigma^2}+ \alpha {\bf C}^{-1} {\bf h}_0$$ = (16)

$${\bf g}_1 {\bf G}_1 ( {\bf q}_1 + \alpha^* {\bf C}^{-1} {\bf h}_2 ) ~ ~ .$$ =

After integration over ${\bf h}_1$,

$$I_1 \doteq \vert{\bf G}_1\vert e^{ {\bf q}_1^H {\bf G}_1 {\bf q}_1} e^{ - {\phi}_1({\bf h}_2)} ~ ~ .$$ (17)

The term $e^{-{\phi}_1({\bf h}_2)}$ in (17) is a function of ${\bf h}_2$ and will be involved in the integral over ${\bf h}_2$. The rest of the terms in (17) are the results after the integration over ${\bf h}_1$ and are only decided by the input sequence and will be used to compute the maximum likelihood probability. The integral over ${\bf h}_2$ is

$$I_2 = \int_{{\bf h}_2} e^{- E_2 } d{\bf h}_2$$ (18)

with

 

$$({\bf h}_3 - \alpha {\bf h}_2)^{H} {\bf C}^{-1} ({\bf h}_3 - \alpha {\bf h}_2) + \frac {\vert r_2 - {\bf A}_2 {\bf h}_2\vert^2 } {\sigma^2}$$ E₂ =

$$~ - \vert\alpha\vert^2 {\bf h}_2^{H} {\bf C}^{-1} {\bf {G}}_1 { \bf C}^{-1} {\bf h}_2 - \alpha {\bf {q}}_1^{H} {\bf {G}}_1 {\bf C}^{-1} {\bf h}_2$$

$$~ - \alpha {\bf h}_2^{H} {\bf C}^{-1} {\bf {G}}_1 {\bf {q}}_1 + {\bf h}_2^{H} {\bf C}^{-1} {\bf h}_2$$ (19)

E₂ can be reduced, dropping constant terms, to:

$$E_2 \doteq ({\bf h}_2 - {\bf {g}}_2)^{H} {\bf {G}}_2^{-1} ({\... ...- {\bf {q}}_2^{H} {\bf {G}}_2 {\bf {q}}_2 +{\phi}_2({\bf h}_3)$$ (20)

with

 

$${\bf ~ {G}}_2 \left[ \frac {{\bf A}_2^H {\bf A}_2} {\sigma^2} + {\bf C}^{-1} + ... ...left( {\bf C}^{-1} - {\bf C}^{-1} {\bf {G}}_1 {\bf C}^{-1} \right) \right]^{-1}$$ =

$${\bf {q}}_2 \frac { {\bf A}_2^H\bf {r_2}} {\sigma^2}+ \alpha {\bf C}^{-1} {\bf {G}}_1 {\bf {q}}_1$$ = (21)

$${\bf {g}}_2 {\bf {G}}_2 ( {\bf {q}}_2 + \alpha^* {\bf C}^{-1} {\bf h}_3 )$$ =

After integration over ${\bf h}_2$ we have:

$$I_2 \doteq \vert{\bf {G}}_2\vert e^{{{\bf {q}}_2}^{H} {\bf {G}}_2 {\bf {q}}_2} e^{-{\phi}_2({\bf h}_3)}$$ (22)

The last term in (22) will be involved in the integral over ${\bf h}_3$.

The general form of the results after integral over ${\bf h}_i$( i=2,...,n-1) will be

$$I_i \doteq \vert{\bf {G}}_i\vert e^{{{\bf {q}}_i}^{H} {\bf {G}}_i {\bf {q}}_i} e^{-{\phi}_i({\bf h}_{i+1})} ,$$ (23)

with

 

$${\bf ~ {G}}_i \left[ \frac {{\bf A}_i^H {\bf A}_i} {\sigma^2} + {\bf C}^{-1} + ... ...( {\bf C}^{-1} - {\bf C}^{-1} {\bf {G}}_{i-1} {\bf C}^{-1} \right) \right]^{-1}$$ =

$${\bf {q}}_i \frac {{\bf A}_i^H {\bf r}_i } {\sigma^2} + \alpha {\bf C}^{-1} {\bf {G}}_{i-1} {\bf {q}}_{i-1} .$$ = (24)

For the integral over ${\bf h}_n$,

$$I_n \doteq \vert{\bf {G}}_n\vert e^{{{\bf {q}}_n}^{H} {\bf {G}}_n {\bf {q}}_n}$$ (25)

with

 

$${\bf G}_n \left[ \frac {{\bf A}_n^H{\bf A}_n} {\sigma^2} + {\bf C}^{-1} -\vert\alpha\vert^2 {\bf C}^{-1} {\bf {G}}_{n-1} {\bf C}^{-1} \right]^{-1}$$ =

$${\bf q}_n \frac {{\bf A}_n^H{\bf r}_n} {\sigma^2} + \alpha {\bf C}^{-1} {\bf {G}}_{n-1} {\bf {q}}_{n-1}$$ = (26)

After the integration over all the channel coefficients, we have the likelihood function

$$f({\bf r^n}\vert{\bf A^n}) \doteq \prod_{i=1}^n \vert{\bf G}_i\vert e^{{{\bf q}_i}^{H} {\bf G}_i {\bf q}_i} .$$ (27)

If we take the negative natural logarithm of the above equation, we will obtain the time-recursive form of the cost function

$$- \log f({\bf r^n}\vert{\bf A^n}) \doteq ~ - ~ \sum_{i=1}^n \... ...i}^{H} {\bf G}_i {\bf q}_i}+ \log \vert{\bf G}_i\vert \right].$$ (28)

From the expressions of ${\bf G}_i$ and ${\bf q}_i$ we know the transition costs depend on all the previous states, so the standard Viterbi algorithm cannot be applied here. The optimum results can only be obtained by exhaustive search.