On the use of parametric methods of spectral estimation in radar - the MUSIC method. Addition to the article
I came across a good article about the method of spectral estimation, which is great for a short signal from the sum of low-noise harmonics. (-copy) Perhaps my comments will help the reader understand the essence of the method. What upset me a little was the incompletely implemented capabilities of the method. The method is used for radar - to quickly determine the direction of incoming signals (angle θ) with the subsequent goal of automatic, it must be understood, adaptation of the system. But - the author does not produce a numerical definition of this angle (and this is strange in context), although this definition is quite possible. We have only beautiful graphs, according to which, it turns out, the system still needs to “crawl” and “crawl”, determining the number and location of the maxima, which is not entirely good.
Illustration of the author of the mentioned article
“Problem Statement”
Briefly: we need to somehow determine where (at what angle) the signal comes from the trellis antenna. Then to adjust it in the direction - but this is no longer in this “song”.
“Modeling the received signal”
(it doesn’t matter - apparently, the “symbol” must be read everywhere as a “signal”)
Here - be careful. The author seems to be working with a certain complex signal (spatial). Although X , yes, could be, as written, a matrix of “complex amplitudes” (depending not on the coordinate, but on the spatial frequency), but, for example, XX H are “covariances” and not “spectral densities”.
The matrix of "complex amplitudes" is more like S, with the help of which harmonic components are modeled (useful signal). Neither additive noises, nor, it seems, even harmonic components are an analytical signal here. Although harmonics, with reservations, are very close to this.
“# General formula:
# sqrt (N0 / 2) * (G1 + jG2),
# where G1 and G2 are independent Gaussian processes.”
The main thing is how the imaginary component comes from from real measurements, somehow it is not clear. It is possible, in principle, to calculate the analytical signal.
It is possible that there is a “source” where they worked with real X (received signal). For example, the author, it seems, was very keen to “make” the resulting spectra symmetrical (even) - in all the cases considered, the test signals were given coming symmetrically to the left and right.
“Conditions”
We determined the range of angles θ of the arrival of the signal, in which it makes sense to look. True, then we build the graphics anyway for some reason from +90 to -90 degrees.
“A little theory about the method itself”
Supplement. MUSIC is obtained from autoregressive estimation (from the Yule-Walker equations) almost by itself, in the case when the variance of the conditional white noise is negligible. The results are almost the same. The SLAE solution is even somewhat more economical than the search for eigenvectors, but, by the way, for a number of reasons, the spectral decomposition of the covariance matrix with its poor conditionality would be very desirable in any way. EVD, in fact, is simply = "finding eigenvalues and vectors", and nothing more. Not an algorithm.
Why do we write a “pseudo-spectrum” - because the spectrum can only be determined from the eigenvectors of the covariance (correlation) matrix only up to a scale factor, i.e. the resulting absolute values do not make sense. But we need precisely and only the position of the maxima.
- This is the most interesting. Well, firstly, U 0 are already eigenvectors, only for the covariance matrix - and “save” on their search will fail. Further. The search for solutions will lead to the need to determine the roots of the power equation, which is absolutely equivalent to another spectral decomposition. The author, apparently, confuses the eigenvalues of completely different matrices.
But ... the main thing ... now (!), Finally (!), We could logarithm the roots and numerically determine the complex “impedances” (model poles) (in the equation, this is again θ, which is not very good), which their imaginary part will show this very angle at which the signal came. It is very unfortunate that the author did not.
"Modeling"
That's a bit worrying - first calculated the covariance matrix of the R = the XX-th H , about which for some reason, then at some time forgot and started all over again - is decomposed into singular values and the vector of the X . They promised by text to search for the eigenvalues and the vector R, which, of course, is the same, but as if more logical, when R has already been found. It is not clear what problem the author encountered.
AboutR we recall when we evaluate the spectrum using the minimum dispersion method MVDR. And here it’s also interesting - R , judging by the script, seems to have been reversed, in full accordance with this method, in a classical way, without any SVD (pseudo-inversion), although it seems to be low-ranking (highly degenerate). I mean, are our noises so small? Well, maybe.
Really confuses here that. The size of the "noise subspace" in the script seems to be assigned by the volitional order (equal to d). But in the real case, we don’t know how many harmonics there are in the signal, and how many noises. It was necessary to analyze these eigenvalues — which of them are negligibly small, which are not.
In general, the work is very interesting, and not only for radar. The method, I believe, has great potential, just for these types of signals. The author worked very well, and some annoying inconsistencies are not so difficult to fix. And the main thing is to supplement the article with the RootMUSIC method.
Illustration of the author of the mentioned article
“Problem Statement”
Briefly: we need to somehow determine where (at what angle) the signal comes from the trellis antenna. Then to adjust it in the direction - but this is no longer in this “song”.
“Modeling the received signal”
(it doesn’t matter - apparently, the “symbol” must be read everywhere as a “signal”)
Here - be careful. The author seems to be working with a certain complex signal (spatial). Although X , yes, could be, as written, a matrix of “complex amplitudes” (depending not on the coordinate, but on the spatial frequency), but, for example, XX H are “covariances” and not “spectral densities”.
The matrix of "complex amplitudes" is more like S, with the help of which harmonic components are modeled (useful signal). Neither additive noises, nor, it seems, even harmonic components are an analytical signal here. Although harmonics, with reservations, are very close to this.
“# General formula:
# sqrt (N0 / 2) * (G1 + jG2),
# where G1 and G2 are independent Gaussian processes.”
The main thing is how the imaginary component comes from from real measurements, somehow it is not clear. It is possible, in principle, to calculate the analytical signal.
It is possible that there is a “source” where they worked with real X (received signal). For example, the author, it seems, was very keen to “make” the resulting spectra symmetrical (even) - in all the cases considered, the test signals were given coming symmetrically to the left and right.
“Conditions”
We determined the range of angles θ of the arrival of the signal, in which it makes sense to look. True, then we build the graphics anyway for some reason from +90 to -90 degrees.
“A little theory about the method itself”
Supplement. MUSIC is obtained from autoregressive estimation (from the Yule-Walker equations) almost by itself, in the case when the variance of the conditional white noise is negligible. The results are almost the same. The SLAE solution is even somewhat more economical than the search for eigenvectors, but, by the way, for a number of reasons, the spectral decomposition of the covariance matrix with its poor conditionality would be very desirable in any way. EVD, in fact, is simply = "finding eigenvalues and vectors", and nothing more. Not an algorithm.
Why do we write a “pseudo-spectrum” - because the spectrum can only be determined from the eigenvectors of the covariance (correlation) matrix only up to a scale factor, i.e. the resulting absolute values do not make sense. But we need precisely and only the position of the maxima.
- This is the most interesting. Well, firstly, U 0 are already eigenvectors, only for the covariance matrix - and “save” on their search will fail. Further. The search for solutions will lead to the need to determine the roots of the power equation, which is absolutely equivalent to another spectral decomposition. The author, apparently, confuses the eigenvalues of completely different matrices.
But ... the main thing ... now (!), Finally (!), We could logarithm the roots and numerically determine the complex “impedances” (model poles) (in the equation, this is again θ, which is not very good), which their imaginary part will show this very angle at which the signal came. It is very unfortunate that the author did not.
"Modeling"
That's a bit worrying - first calculated the covariance matrix of the R = the XX-th H , about which for some reason, then at some time forgot and started all over again - is decomposed into singular values and the vector of the X . They promised by text to search for the eigenvalues and the vector R, which, of course, is the same, but as if more logical, when R has already been found. It is not clear what problem the author encountered.
AboutR we recall when we evaluate the spectrum using the minimum dispersion method MVDR. And here it’s also interesting - R , judging by the script, seems to have been reversed, in full accordance with this method, in a classical way, without any SVD (pseudo-inversion), although it seems to be low-ranking (highly degenerate). I mean, are our noises so small? Well, maybe.
Really confuses here that. The size of the "noise subspace" in the script seems to be assigned by the volitional order (equal to d). But in the real case, we don’t know how many harmonics there are in the signal, and how many noises. It was necessary to analyze these eigenvalues — which of them are negligibly small, which are not.
In general, the work is very interesting, and not only for radar. The method, I believe, has great potential, just for these types of signals. The author worked very well, and some annoying inconsistencies are not so difficult to fix. And the main thing is to supplement the article with the RootMUSIC method.