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More on "beamforming"

Started by Linda 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?
Before attemting to answer your question about the Capon beamformer, 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 beamformer like Capon's method can help you resolve signals that the 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 I thought about Burg's method when I wrote my first reply. 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".