# More on "beamforming"

Started by September 28, 2003
```Hi!
I am beginner in signal processing areas. I have few questions about
beamforming. From paper by Krim and Viberg in IEEE Signal
Processing Magazine published in 1996, Capon's beamformer is
alternative to overcome the limitations of the conventional beamformer
but with increased complexity. Can I know in what criteria its
improves the performance as compared the conventional method? Also
what makes it more complex? According to the some sources, it stated
that the significantly lower resolution threshold reduced the spectral
leakage from closely spaced signal sources. I am not pretty sure what
does this means?

Thanks.

Linda
```
```lindah74uk@yahoo.co.uk (Linda) wrote in message news:<124fd8f7.0309280146.4a4e1b37@posting.google.com>...
> Hi!
> I am beginner in signal processing areas. I have few questions about
> beamforming. From paper by Krim and Viberg in IEEE Signal
> Processing Magazine published in 1996, Capon's beamformer is
> alternative to overcome the limitations of the conventional beamformer
> but with increased complexity. Can I know in what criteria its
> improves the performance as compared the conventional method? Also
> what makes it more complex? According to the some sources, it stated
> that the significantly lower resolution threshold reduced the spectral
> leakage from closely spaced signal sources. I am not pretty sure what
> does this means?

I'll just recap that the conventional beamformer tries to map the recieved
signal onto what some people call the "array manifold". You take the
measured data vector (call that d) and then construct a steering vector
(call it s) that takes frequency, array geometry and proposed Direction
of Arrival (DoA) into account. Then, you just compute the inner product
between the column vectors d and s as

x=d'*s

where x is the spatially filtered signal and "'" is the complex conjugate
transposed.

So what you do, then, is to find the coefficients of the steering vector
s, and use them as a completely trivial FIR filter. You make one such
filter for each direction you want to look in.

Now, this is a very simple procedure. If the array is uniform linear, and
the medium is "reasonable", the above procedure essentially amounts to a
Fourier transform. If you are in enclosed cavities (under water or inside
a room), things become a bit more cumbersome, but leave that for now.

As you probably know, the Fourier transform is a bit awkward in that it
suffers form resolution problems. The Heisenberg inequality states that
the closer two sources are in wavenumber domain, the longer array is needed
to separate the two (at a given frequency). So if you want to "see" two
sources that lie at almost the same bearing, you need a long array to
separate them. On the other hand, the Fourier transform works with
arbitrary signals.

So, the model-based beamformers Krim and Viberg review, all intend to
overcome the Fourier resolution limit. The way these model-based beamformers
(Capon among them) work, is to assume that the recieved signal follows a
specific parmetric form, usually what I call the "sum of sines",

Q
d(n)=  sum  A_q*exp(j*k_qx_n)
q=1

where d(n) is the signal measured at the n'th sensor of N in the array,
Q is the number of signals present, k_q is the wavenumber of the q'th
such sine, and x_n is the spatial coordinate of the n'th sensor. For
simplicity, I leave the frequency out.

Once you know what to look for, you can be way more specific in how you
process the data, than if you use a general method. However, being specific
requires much more cumbersome computations. If you compare Capon's method
with the conventional beamformer, you will see that where the conventional
beamformer is a mere inner product with fixed coefficients, Capon's method
requires that you first estimate a signal covariance matrix, and then also
inverts this matrix. That's way more complicated form a computational point
of view that the simple FIR filter method, and takes significantly more
time to do, for the computer.

Now, *if* your signals are on the "sum of sines" form, a model-based
naive Fourier-based methods can't. On the other hand, Fourier-based
methods are much more robust and will not fail as often and as catastrophic
as the model-based methods do.

HTH,

Rune
```
```lindah74uk@yahoo.co.uk (Linda) wrote in message news:<124fd8f7.0309280146.4a4e1b37@posting.google.com>...
> Hi!
> I am beginner in signal processing areas. I have few questions about
> beamforming. From paper by Krim and Viberg in IEEE Signal
> Processing Magazine published in 1996, Capon's beamformer is
> alternative to overcome the limitations of the conventional beamformer
> but with increased complexity. Can I know in what criteria its
> improves the performance as compared the conventional method? Also
> what makes it more complex?

Hi again.

I just wrote an answer where I discussed the basic Fourier-based
beamformer
and the parametric DoA estimators. What I wrote there is basically
true,
but does, unfortunately, not apply to Capon's beamformer. For some
reson

Anyway, Capon's beamformer is intended not as much as a high
resolution
beamformer (like e.g. Burg's method) as an interference-suppressing
beamformer. This means that the user chooses a steering direction
(like
the Fourier-based beamformers) and Capon's beamformer shows a main
lobe
that may or may not be narrower than the Fourier-based beamformers.

The crux of Capon's beamformer is that it minimizes the interference
from directions "far from" the steering direction. If you have a
moderatly
strong source in the steering direction and a strong source in the
direction
of a sidelobe, the output of the Fourier-based beamformer may be
dominated
by the strong spurious source. Capon's method tries to suppress that
unwanted signal.

> According to the some sources, it stated
> that the significantly lower resolution threshold reduced the spectral
> leakage from closely spaced signal sources. I am not pretty sure what
> does this means?

I don't understand what you mean by "low resolution treshold". I
would, at
least in the outset, not think of Capon's method as a high-resolution
method.
I am not at all sure that a spurious signal that is separated from the
"useful" signal by less than the width of a (conventional) main lobe
would
be suppressed, or the two signals resolved in the spectrum. This is
one of
those questions where a matlab simulation would come in handy to find
out
what's going on. And I don't have matlab available right now.

I'm sorry for the confusion I'm causing.

Rune
```
```Linda wrote:
> Hi!
> I am beginner in signal processing areas. I have few questions about
> beamforming. From paper by Krim and Viberg in IEEE Signal
> Processing Magazine published in 1996, Capon's beamformer is
> alternative to overcome the limitations of the conventional beamformer
> but with increased complexity. Can I know in what criteria its
> improves the performance as compared the conventional method? Also
> what makes it more complex?

You need to find a willing rooster and then you need to hold it down ;)

According to the some sources, it stated
> that the significantly lower resolution threshold reduced the spectral
> leakage from closely spaced signal sources. I am not pretty sure what
> does this means?
>
> Thanks.
>
> Linda

Capon's method is essentially MVDR (See Johnson & Dudgeon's book for
details).  A beamformer is a spatial filter.  If you plot a beampattern
as a function of bearing, you will note the nulls of the beampattern.  A
null is a direction where a plane wave from that direction will be
canceled.  A canceled plane wave has zero leakage. A MVDR beamformer
attempts to place nulls in the direction of each interferer. A MVDR
beamformer can place nulls fairly close to its steering direction, much
closer than the beam width of a conventional beamformer.  In fact if you
are a bit mismatched in steering direction to an actual source, there is
an undesirable effect known as "squint".

```