antenna-theory.com

[Mrcelloman99]

In the MMSE optimal weighting derivation on the website, why is the actual output Y(t) equal to W conjugate transpose times X(t) and not simply W transpose times X(t)?

Also, how is the gradient of the MSE equation taken?

Thanks

[bigSteve]

That's just convention. It doesn't really matter. The output should be:

y = w1*x1 + w2*x2 +....

The hermitian does introduce the complex conjugate, but again, that doesn't really matter as is just convention. Kind of like e(-jt) or e(+jt).

The gradiant of the MSE equation is the vector derivative with respect to the weighting vector, detailed on this page:
http://www.antenna-theory.com/arrays/weights/mmse2.php

It seems complicated, but if you write it out in terms of scalar components (w1, w2, ....) and take the single-variable derivatives you will see how this enables the vector derivatives on that page. It's a nice shortcut. Anyway, the whole point is that at an optimal value for the weight vector, the gradiant or rate of change of the MSE should be zero (since the functions are continuous). If you go through that page and think about each step, it becomes pretty clear.