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Hi,
I want some help with finding the derivative of Re(aW*b) and aWW*a',
wrt W* where a is a 1xN, W is NxN and b is Nx1 and W* is the conjugate
of W and a' is the transpose of a and Re() is real part.

Thanks,
Prateek

(ps. i would appreciate if somebody can suggest a reference for such
complex matrix calculus).

[email protected] (Prateek) wrote in message news:<[email protected] com>...
> Hi,
> I want some help with finding the derivative of Re(aW*b) and aWW*a',
> wrt W* where a is a 1xN, W is NxN and b is Nx1 and W* is the conjugate
> of W and a' is the transpose of a and Re() is real part.
>
> Thanks,
> Prateek
>
> (ps. i would appreciate if somebody can suggest a reference for such
> complex matrix calculus).

Hi Prateek.

The only place where I have seen such questions discussed in print
(although concerning gradients wrt a) is Appendix A in

Therrien: "Discrete Random Signals and Statistical Signal Processing"
Prentice-Hall, 1992.

Therrien imposes the distinct impression that gradients of complex
matrixes/vectors may not stand on very firm ground, from a mathematichal
point of view. Still, he points to a paper,

Brandwood: "A Complex Gradient Opperator and its Application in
adaptive Array Theory", IEE Proceedings, Vol 130(1), February
1983, p. 11

that apparently gives what mathematical basis there is, for these
kinds of things. I haven't seen this paper, so I can't comment
more specifically on it.

Rune

Rune Allnor wrote:
> [email protected] (Prateek) wrote in message news:<[email protected] com>...
>
>>Hi,
>>I want some help with finding the derivative of Re(aW*b) and aWW*a',
>>wrt W* where a is a 1xN, W is NxN and b is Nx1 and W* is the conjugate
>>of W and a' is the transpose of a and Re() is real part.
>>
>>Thanks,
>>Prateek
>>
>>(ps. i would appreciate if somebody can suggest a reference for such
>>complex matrix calculus).
>
>
> Hi Prateek.
>
> The only place where I have seen such questions discussed in print
> (although concerning gradients wrt a) is Appendix A in
>
> Therrien: "Discrete Random Signals and Statistical Signal Processing"
> Prentice-Hall, 1992.
>
> Therrien imposes the distinct impression that gradients of complex
> matrixes/vectors may not stand on very firm ground, from a mathematichal
> point of view. Still, he points to a paper,
>
> Brandwood: "A Complex Gradient Opperator and its Application in
> adaptive Array Theory", IEE Proceedings, Vol 130(1), February
> 1983, p. 11
>
> that apparently gives what mathematical basis there is, for these
> kinds of things. I haven't seen this paper, so I can't comment
> more specifically on it.
>
> Rune

Van Trees uses Brandwood's formulation as well. If I recall correctly
the difficulty is that the function conjugate() isn't analytic. There
is also a discussion about this stuff in Dudgeon and Johnson's book on
array processing.

[email protected] (Prateek) wrote in message news:<[email protected] com>...
> Hi,
> I want some help with finding the derivative of Re(aW*b) and aWW*a',
> wrt W* where a is a 1xN, W is NxN and b is Nx1 and W* is the conjugate
> of W and a' is the transpose of a and Re() is real part.
>
> Thanks,
> Prateek
>
> (ps. i would appreciate if somebody can suggest a reference for such
> complex matrix calculus).

Hi Prateek.

Some simple matrix identities are available at the URL below.
http://www.geocities.com/mentalrubbe...Identities.pdf

Rewriting Re(aW*b) as
Re(aW*b) = (aW*b + (aW*b)*)/2 = (aW*b + a*Wb*)/2
it should not be a problem to find the derivative w.r.t W*.

Regards, Alan T.