Beamforming question

Hi,

(Accept my apologies if I am posting to the wrong newsgroup: I am not
totally sure what group would be best to address regarding my
question.)


I am working with general 2-D adaptive arrays right now and my
understanding (from EE's in this area) is that null-steering is
typically better than optimization of the SINR (constrained or
unconstrained).  Based on what I've read about these methods, it is not

totally clear to me what the basis for this statement is, or that it is

even true under all circumstances. The closest I've come is a cryptic
statement in "Adaptive Array Processing" by Allen and Ghavami in which
they state that main beam steering (which is their name for
unconstrained beamforming) "will not produce a maximum SNR in the
presence of directional interference". However, if you look at the
formula they use, you see that the covariance matrix they are using is
due only to the noise and is not the conventional "noise" covariance
matrix that one uses to perform unconstrained beamforming with a
directional source of interference. It is easy to prove mathematically
that using the total "noise covariance matrix as R"noise" = R"jammers"
+ R"randomnoise", last statement is patently false.


Does anyone know of a text or journal article that gives a good
rigorous mathematical justification for this claim? The only thing I
can see which would possibly support this statement is that in practice
some of the assumptions made (like the fact that signals source,
directional intereference sources and the noise are uncorrelated) are
not typically satisfied or that perhaps there are other issues (like
beamwidth and the presence of side lobes) that vitiate the efficacy of
SINR optimization relative to null-steering.

You can reply to this message directly or e-mail me.


MTIA, 


Matt

Hi,

This book might help: "Adaptive Filter Theory", from Simon Haykin.
They go over null steering briefly on pg. 119-125. They assume the
angle of the interfering source is known and poses
the problem as a constrained problem. More detail is given in later
chapters.

hth

Brenneman wrote:
> Hi,
>
> (Accept my apologies if I am posting to the wrong newsgroup: I am not
> totally sure what group would be best to address regarding my
> question.)
>
>
> I am working with general 2-D adaptive arrays right now and my
> understanding (from EE's in this area) is that null-steering is
> typically better than optimization of the SINR (constrained or
> unconstrained).  Based on what I've read about these methods, it is not
>
> totally clear to me what the basis for this statement is, or that it is
>
> even true under all circumstances.

A wild guess would be that given an arry, the gain is fixed. There is
only
so much that can be achieved to increase the S part in the S/N ratio.
To increase further, you need to reduce the N part, which is exactly
what is done in interference cancellers.

If you know the DoA of a noise (point) source, the best you can do is
to 
place a null in that direction. 

Rune

Hi Rune,

This is a nice argument, but I'm not sure that it is consistent with
what we know about these methods mathematically.
In unconstrained optimization, you _are_ optimizing the SINR, so I
don't see how null-steering can outperform the unconstrained
beamforming method. OTOH, I think you are right, in the sense that
null-steering does seem to be the method of choice and there is a
consensus that it is better than unconstrained beamforming. So it seems
that there must be some other function that null-steering must be
optimizing. Since it is not SINR, what is it?


Matt

Brenneman wrote:
> Hi Rune,
>
> This is a nice argument, but I'm not sure that it is consistent with
> what we know about these methods mathematically.
> In unconstrained optimization, you _are_ optimizing the SINR, so I
> don't see how null-steering can outperform the unconstrained
> beamforming method.

I reckon we are talking about different flavours of the same thing.
I sketched the "trivial" example when the ineterfering source
is a point source and the bearing is known. One then can
make the problem more realistic (and harder) by relaxing the
constraints, either by assuming a distributes source instead of
a point source, or by assuming the bearing to be unknown.
Or both.

Depending on the exact formulation of the problem, one arrives at
different optimum solutions.

> OTOH, I think you are right, in the sense that
> null-steering does seem to be the method of choice and there is a
> consensus that it is better than unconstrained beamforming. So it seems
> that there must be some other function that null-steering must be
> optimizing. Since it is not SINR, what is it?

Now you are asking very detailed questions I am not able to answer.
It might help if you defined exactly what you mean by "constrained
beamformers" etc, so that we are sure we discuss the same things.

Generally speaking, one has a choise between something that
is simple to implement, is robust, and works reasonably well; and
something that is complicated, requires prior information that could
be hard to come by, and workse very well but is sensitive to flaws
in prior information.

For instance, if you try to track one particular source in an
environment where several interfering sources exist, and all move
at similar bearing rates, it is not obvious how to design an adaptive
null-stearing beamformer that nulls the interfering sources while
keeping the desired source. In that sort of scenario, an unconstrained
beamfomer might be preferable, since the effect of the beamformer
is consistent from time to time, and an operator can be trained to
account for such effects. A constrained beamfomer that dynamically
reacts to the various interfering sources might require an operator
with a  PhD in adaptive filters to interpret the situation.

Again, it is a choise between simplicity and robustness on one hand,
and accurate results from "itchy" methods, on the other. 

Rune