Temporally Independent Signal Amplitudes

If the signal amplitude vectors ${\bf s}_k$ are independent over time, then $f({\bf S})$ can be written as:

$$f({\bf S}) = ~ \prod_{k=1}^{N} f({\bf s}_k) ,$$ (15)

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

$$f({\bf S}\vert{\bf X}, \phi ) = ~ \prod_{k=1}^{N} f({\bf s}_k\vert{\bf X}, \phi ) \; \; .$$ (16)

From (10) and (3) we have that

$$f({\bf s}_k\vert{\bf X}, \phi) \doteq f({\bf s}_k) ~ e^{ - \f... ...k -{\bf A} ( \phi ) {\bf s}_k\vert\vert^{2}} {\sigma^{2}} } .$$ (17)

For temporally independent Gaussian signal amplitudes, let $f({\bf s}_k)$ be a joint Gaussian distribution with mean ${\bf c}_k$ and covariance ${\bf C}_k$:

$$f({\bf s}_k) = \frac {1}{\pi^N \vert{\bf C}_k\vert} e^{- ({\b... ... - {\bf c}_k )^{H} {\bf C}_k^{-1} ({\bf s}_k - {\bf c}_k ) } .$$ (18)

In this case (17) specifically becomes

 

$$f({\bf s}_k \vert{\bf X}, \theta ) \doteq e^{- \frac{\vert\vert {\bf x}_k - {\bf A} ( \phi ) {\bf s}_k \ver... ... ~ ~ e^{- ({\bf s}_k -{\bf c}_k )^{H} {\bf C}_k^{-1} ({\bf s}_k - {\bf c}_k ) }$$

$$\doteq e^{- ({\bf s}_k - {\bf g}_k )^{H} {\bf G}_k^{-1} ({\bf s}_k - {\bf g}_k)} ,$$ (19)

with

  

$${\bf G}_k \left[\frac {{\bf A}^H ( \phi ) {\bf A} ( \phi )} {\sigma^2} + {\bf C}_k^{-1} \right]^{-1}$$ = (20)

$${\bf g}_k {\bf G}_k \left( \frac {{\bf A}^H ( \phi ) {\bf x}_k} {\sigma^2} + {\bf C}_k^{-1} {\bf c}_k \right) \; \; .$$ = (21)

To initialize $\phi$ in step 1, we could, for example, initialize ${\bf G}_k = {\bf C}_k$ and ${\bf g}_k = {\bf c}_k$, $k = 1 , \ldots , N$.