Signal Processing Tools for Radio Astronomy - Signal Processing Systems - page 439

Plugging this solution back into the LS problem, we can first rewrite

C − 1 / 2

( r − M s s − M n ˆ n )

w

C − 1 / 2 C − 1 / 2 M n C − w M n ) − 1 M n C − 1

=[ − M n ( ] r −

w w

w

C − 1 / 2 C − 1 / 2 M n C − w M n ) − 1 M n C − 1

− [ − M n ( ] M s s

w w

w

P ⊥ C − 1 / 2

= ( r − M s s ) w

where

C − 1 / 2 M n C − w M n ) − 1 M n C − 1 / 2

P ⊥ = I − P , P = M n ( . w w

, we can show that P is an orthogonal projection. Hence, also P ⊥ =

Similar to ˘

P is an orthogonal projection, onto the complement of the range of C − 1 / 2

−

I M n , w

˙

i.e., the weighted range of the noise matrix. If n is diagonal, this is equivalent to

“projecting out the diagonal”, thus omitting these entries in the WLS fitting. It is
interesting to note that, in current telescopes, the autocorrelations (main diagonal of
R ) are usually not estimated. That fits in very well with this scheme, as the projection
would project them out anyway!
The resulting “compressed” WLS problem is

C − 1 / 2

2

ˆ =

s

( r − )

arg min M s s s s

where

C − 1 / 2 P ⊥ C − 1 / 2

C − 1

= .

: w w

s

(This is with some abuse of notation: C s is singular due to the projection, hence not
invertible. However, we will only need C − 1 , and will use the above definition for

s

it.) The solution ˆ s will be exactly the same as in the original WLS problem ( 27 ) ,

but now it is obtained in two steps: first ˆ s and then, if required, ˆ

n via ( 31 ) .
As the expression for the compressed problem is very similar to the original WLS
problem, we obtain similar results: the solution is

M s C − 1

) − 1 M s C − 1

ˆ =( r

M s

s

s s

which can also be written as

M − 1
ds

M s C − 1 M s C − 1 s = ˆ ds , ˆ M ds = M s , ds = ˆ r .

s s

M ds is the deconvolution matrix, and ˆ ds is the WLS dirty image. This time, the

dirty image is really a (vectorized) image, whereas in the previous discussion, the
vector ˆ d had an image component and a noise component.

The covariance of the image estimate ˆ s is, using C − 1 C w C − 1 C − 1

s

= ,

s s

M s C − 1 M s ) − 1 M s C − 1 C − 1

s

M s C − 1 M s ) − 1 M − 1 C s =( ( C w ) M s ( =

ds .

s s s