antenna-theory.com

[lsmeng]

Hi,

I think this website is really amazing in the sense that it makes the theory really easy to understand.

I have been following the pages about the MMSE and adaptive LMS weighting method for the arrays.It was really helpful. But I noticed that the introductions are based on narrow band cases. Can anyone show me how to deal with a broadband case of the MMSE and LMS ? It would be even better if there is an example.

[admin]

The broadband case is more tricky.

Recall that the antenna array is a spatial filter - it operates on the relative phase variation received at each element. The phase variation that each element receives (from a particular signal direction) is a function of the frequency and antenna spacing/geometry of the array.

As we change frequency, the steering vector for a particular direction (with represents the relative phase delays) is altered as well. Hence, our math becomes a little tricky immediately.

A couple notes:
(1) You can still use the narrowband assumption, and then test how well it works by analyzing your results across the frequency range. Simply to the narrowband analysis at the center frequency, and see what happens away from this frequency. A couple hundred MHz bandwidth isn't a big deal if the center frequency is a few GHz.

(2) If your array is looking for a desired signal that is at theta=90 degrees for a linear array (or at theta=0 degrees for a planar array in the x-y plane), then the steering vector is constant for all frequencies (this is because the signal will come in broadside to the array, and hence will hit each element at the same point and be in phase). Hence, if you have this condition, a minimum sidelobe level method would work nicely.

--------

If you want to do some mathematical analysis, here is a suggestion:

Assume you have weights that are constant - i.e. not frequency dependent. Then assume you have some frequency range you are interested in, and sample that range at two points, say f1 and f2.

Then let your new MSE be:
MSE = MSE(f=f1) + MSE(f=f2)
i.e., the new MSE is the sum of the MSEs at the two individual frequencies. You should be able to factor everything out and perform the exact same analysis as on:
http://www.antenna-theory.com/arrays/weights/mmse2.php

From here, you can derive your new optimal weights, will probably be a function of the sum of the autocorrelation matrices and such. You can then see how well this works, and extend the number of frequency sample points as you like.

I don't know if this has been done before, but you could probably screw around with these ideas for a few days, come up with something intelligent, print out some results and present it at the IEEE Antennas Conference sometime.

---
Hope this helps