MUSIC and DOA Estimation

Started by June 13, 2009
```Hi,

In the literature, there is a common claim made that the performance
of MUSIC for direction of arrival estimation (DOA) is affected when
the wrong number of directional sources is assumed. There is no
argument presented to support this claim, nor any references given,
just the claim.

I am curious what the basis for this claim is. Although there are a
variety of MUSIC-like methods and a this topic can get complicated,
overall, it seems pretty fair to say that when you underestimate the
number of sources, it is due to the presence of sources whose signals
are weak (relative to the noise floor). In this case, the error would
seem to be conservative and not affect the performance of MUSIC much
(since the remaining signal subspace will still span the array
manifold of the sources which are counted as being present). If you
were using Spectral MUSIC, you would have more noise evrs to take the
projection with (which is not a bad thing) and if you were using ROOT-
MUSIC, the roots of the signals counted as being present would not be
affected either. If you overestimate the number of sources, this would
not be a problem for Spectral MUSIC, since the signal subspace will
span the subspace of the steering vectors for the sources which are
actually present. The only problem I can see is with ROOT-MUSIC, where
if you factor a root from a spurious source, it will affect the
subsequent roots of the actual sources present.

Does this reasoning sound like the justification for the claim about
MUSIC I presented, or is there something else I am missing?

TIA,

Matt

If one is using MUSIC, it would seem reasonable to assume that one is
probably using some information theoretic method (such as the MDL or
AIC) to estimate the number of sources. Now suppose that the number of
directional sources is mis-estimated. The three major sources I can
see that would contribute to the mis-estimation are:
1) You have correlated signals
2) Directional sources are missed because their signals are weak
(relative to the noise floor)
3) Spurious directional sources are counted (due to the fact that
MUSIC sometimes
```
```On 13 Jun, 21:33, junoexpress <MTBrenne...@gmail.com> wrote:
> Hi,
>
> In the literature, there is a common claim made that the performance
> of MUSIC for direction of arrival estimation (DOA) is affected when
> the wrong number of directional sources is assumed. There is no
> argument presented to support this claim, nor any references given,
> just the claim.

The claims may or may not be true, depending on exactly
how  they are phrased. Do you have any pointers to where

> I am curious what the basis for this claim is. Although there are a
> variety of MUSIC-like methods and a this topic can get complicated,
> overall, it seems pretty fair to say that when you underestimate the
> number of sources, it is due to the presence of sources whose signals
> are weak (relative to the noise floor). In this case, the error would
> seem to be conservative and not affect the performance of MUSIC much
> (since the remaining signal subspace will still span the array
> manifold of the sources which are counted as being present). If you
> were using Spectral MUSIC, you would have more noise evrs to take the
> projection with (which is not a bad thing) and if you were using ROOT-
> MUSIC, the roots of the signals counted as being present would not be
> affected either. If you overestimate the number of sources, this would
> not be a problem for Spectral MUSIC, since the signal subspace will
> span the subspace of the steering vectors for the sources which are
> actually present. The only problem I can see is with ROOT-MUSIC, where
> if you factor a root from a spurious source, it will affect the
> subsequent roots of the actual sources present.

There are several causes for inaccuracies in the MUSIC-type
estimators. One is the SNR, others are related to wrong order
estimates, like if there in reality are 4 signals present,
but the order estimator only finds 3 of them. In such cases,
it is likely that at least some of the DoAs are off, either
because the SNR is low or the 'missing' signal lies very close
to one of the others. If the 'missing' signal has a significantly
smaller amplitude than the others, the remaining DoA estimates
might be accurate anyway.

> Does this reasoning sound like the justification for the claim about
> MUSIC I presented, or is there something else I am missing?

Yes, there is.

MUSIC-type estimators (meaning all estimators which
implicitly or explicitly use a covariance matrix of
order P to estimate the parameters of D signals) *fail*
*unconditionally* when D >=P.

Just try it and see: Use your favourite MUSIC implementation
and specify it to run with a covariance matrix of order P.
Simulate a noise-free signal with N = 3P samples (to make
sure the covariance matrix is well-behaved) as follows:

First use *one* sinusoidal with DoA cos(phi_1) = 2*pi*1/P.
Use your MUSIC estimator and verify that you find the
DoA to within numerical precision (remember, noise-free
signal).

Next, *add* a signal component with the same amplitude
and with DoA cos(phi_2) = 2*pi*2/P. Repeat the MUSIC
analysis to verify that you find the two DoAs.

*Add* successive signals with DoAs cos(phi_n) = 2*pi*n/P
for n=3,4,...,2P. Make sure the signals have the same

Just to be clear: The first time around the noise-free
signal contains one sinusoidal. The scond time it contains
two sinusodials. The n'th time it contains n sinusoidals.

At some point, either near n = P/2 or near n=P (the
details depend a bit on your MUSIC implementation and
if you use real-valued or complex-valued signals),
the DoA estimates break down. This is because the
number of signal components becomes as large as the
dimension of the column space of the signal covariance
matrix. Once that happens, the rank of the null space
is 0 (it needs to be at least 1 for MUSIC to work),
so MUSIC breaks down.

If you want to, repeat the excercise with various SNRs,
and with different MUSIC implementations.

Rune
```
```On 13 Jun, 22:00, Rune Allnor <all...@tele.ntnu.no> wrote:

> Just try it and see: Use your favourite MUSIC implementation
> and specify it to run with a covariance matrix of order P.
> Simulate a noise-free signal with N = 3P samples (to make
> sure the covariance matrix is well-behaved) as follows:
>
> First use *one* sinusoidal with DoA cos(phi_1) = 2*pi*1/P.
> Use your MUSIC estimator and verify that you find the
> DoA to within numerical precision (remember, noise-free
> signal).
>
> Next, *add* a signal component with the same amplitude
> and with DoA cos(phi_2) = 2*pi*2/P. Repeat the MUSIC
> analysis to verify that you find the two DoAs.
>
> *Add* successive signals with DoAs cos(phi_n) = 2*pi*n/P
> for n=3,4,...,2P. Make sure the signals have the same
> amplitudes, and add *no* noise.
>
> Just to be clear: The first time around the noise-free
> signal contains one sinusoidal. The scond time it contains
> two sinusodials. The n'th time it contains n sinusoidals.

Sorry, I got the DoAs wrong: Use a DoA for the n'th
signal as

cos(phi_n) = pi*n/P     (deleted factor 2).

Rune
```
```On Jun 13, 4:00&#2013266080;pm, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 13 Jun, 21:33, junoexpress <MTBrenne...@gmail.com> wrote:
>
> > Hi,
>
> > In the literature, there is a common claim made that the performance
> > of MUSIC for direction of arrival estimation (DOA) is affected when
> > the wrong number of directional sources is assumed. There is no
> > argument presented to support this claim, nor any references given,
> > just the claim.
>
> The claims may or may not be true, depending on exactly
> how &#2013266080;they are phrased. Do you have any pointers to where
>
This claim is found in a good many papers on the topic of signal
enumeration. Two that make the statement clearly are:
1) IEEE Trans Acc, Speech, and Signal Processing, vol. 38, no. 11,
1990
"On Information Theoretic Criteria...." by Wong, Zhang, Reilly, and
Yip
(see text after eqn 5 in article)
2) IEEE Trans Signal Processing, vol. 39, no. 8, 1991
"A Parametric Method for Determining the Number of Signals in Narrow-
Band Direction Finding"
by Wu and Fuhrman (see first paragraph)

>
>
>
> There are several causes for inaccuracies in the MUSIC-type
> estimators. One is the SNR, others are related to wrong order
> estimates, like if there in reality are 4 signals present,
> but the order estimator only finds 3 of them. In such cases,
> it is likely that at least some of the DoAs are off, either
> because the SNR is low or the 'missing' signal lies very close
> to one of the others. If the 'missing' signal has a significantly
> smaller amplitude than the others, the remaining DoA estimates
> might be accurate anyway.
>
Your line of reasoning is pretty much what makes me doubt this claim.
A source that is missed, is most likely missed because its SNR is very
small. The components of the stronger sources on the evr that gets
dropped from the signal subspace and gets put into the noise sub-space
cannot be that large, so when you run say Spectral MUSIC, the location
where the reciprocal of the projection of the steering vectors onto
the noise subspace has its max value should not change appreciably.

>
> > Does this reasoning sound like the justification for the claim about
> > MUSIC I presented, or is there something else I am missing?
>
> Yes, there is.
>
> MUSIC-type estimators (meaning all estimators which
> implicitly or explicitly use a covariance matrix of
> order P to estimate the parameters of D signals) *fail*
> *unconditionally* when D >=P.
>
The noise in the system (i.e. AWGN in the channels) will guarantee
that the covariance matrix has full rank. This condition is certainly
satisfied in all of the professional treatments of MUSIC you see in
the lit.
I am assuming the are are less sources than antennas here.
And there are other problems of course that could arise (such as the
presence of correlated sources, or an array whose antenna spacings are
greater than half a wavelength, or signals which are wide-band, etc
which will also act to screw up MUSIC), but neither I, nor the
articles to which I am referring, assume any of these conditions hold.
>
> If you want to, repeat the excercise with various SNRs,
> and with different MUSIC implementations.
>
Been there, done that.

> Rune

Thanks for your input Rune. It seems that you're the only person who
responds to array signal processing questions. I kinda thought there
would be some people on the group who could understand this stuff.
Maybe few professionals post here or are in another group.

TA,

Matt
```
```On 14 Jun, 05:30, Junoexpress <MTBrenne...@gmail.com> wrote:
> On Jun 13, 4:00&#2013266080;pm, Rune Allnor <all...@tele.ntnu.no> wrote:> On 13
Jun, 21:33, junoexpress <MTBrenne...@gmail.com> wrote:
>
> > > Hi,
>
> > > In the literature, there is a common claim made that the performance
> > > of MUSIC for direction of arrival estimation (DOA) is affected when
> > > the wrong number of directional sources is assumed. There is no
> > > argument presented to support this claim, nor any references given,
> > > just the claim.
>
> > The claims may or may not be true, depending on exactly
> > how &#2013266080;they are phrased. Do you have any pointers to where
> > such claims are made?
>
> This claim is found in a good many papers on the topic of signal
> enumeration. Two that make the statement clearly are:
> 1) IEEE Trans Acc, Speech, and Signal Processing, vol. 38, no. 11,
> 1990
> "On Information Theoretic Criteria...." by Wong, Zhang, Reilly, and
> Yip
> (see text after eqn 5 in article)
> 2) IEEE Trans Signal Processing, vol. 39, no. 8, 1991
> "A Parametric Method for Determining the Number of Signals in Narrow-
> Band Direction Finding"
> by Wu and Fuhrman (see first paragraph)
>
>
>
> > There are several causes for inaccuracies in the MUSIC-type
> > estimators. One is the SNR, others are related to wrong order
> > estimates, like if there in reality are 4 signals present,
> > but the order estimator only finds 3 of them. In such cases,
> > it is likely that at least some of the DoAs are off, either
> > because the SNR is low or the 'missing' signal lies very close
> > to one of the others. If the 'missing' signal has a significantly
> > smaller amplitude than the others, the remaining DoA estimates
> > might be accurate anyway.
>
> Your line of reasoning is pretty much what makes me doubt this claim.
> A source that is missed, is most likely missed because its SNR is very
> small.

Not necesarily. There can be any number of reasons why a
source is missed. Small SNR is just one of them.

There is one aspect of order estimators you need to be
aware of: For AR models, order estimators are formally
are derived from statistical analyses of the reflection
coefficients that represent signal residuals during the
Levinson recursion.

MUSIC can not be represented as a Levinson recursion,
it's based on eigenvector decompositions. So the *formal*
statistical analysis, which works on the reflection
coefficients, don't work with the eigenvalues.

Now, this does *not* mean that oprder estimators do not
work with eigenvector decompositions - they do. It means
that one has bodged a tool to work in a context it was
not designed for. Which also goes a long way to explain
why order estimates might be off from time to time, with
no apparent reason.

> The components of the stronger sources on the evr that gets
> dropped from the signal subspace and gets put into the noise sub-space
> cannot be that large, so when you run say Spectral MUSIC,

Spectral MUSIC is...? If you are talking about the MUSIC
pseudo spectrum, keep in mind that there is no information
about the signal spectrum encoded in the pseudo spectrum.

> the location
> where the reciprocal of the projection of the steering vectors onto
> the noise subspace has its max value should not change appreciably.

Remember, you are working D-manifolds in an P-dimensional
complex-valued vector space. Don't expect your intuition
to be of the best help...

>
> > > Does this reasoning sound like the justification for the claim about
> > > MUSIC I presented, or is there something else I am missing?
>
> > Yes, there is.
>
> > MUSIC-type estimators (meaning all estimators which
> > implicitly or explicitly use a covariance matrix of
> > order P to estimate the parameters of D signals) *fail*
> > *unconditionally* when D >=P.
>
> The noise in the system (i.e. AWGN in the channels) will guarantee
> that the covariance matrix has full rank. This condition is certainly
> satisfied in all of the professional treatments of MUSIC you see in
> the lit.

You need to think one step further: The basis for the
noise subspace needs to be of rank at least 1 for MUSIC
and friends to work. Once you have a signal that contains
P (or P/2) or more sinusoidals, this no longer holds and
MUSIC fails unconditionally.

> I am assuming the are are less sources than antennas here.

You can certainly do that for academic purposes. That's
a very stupid thing to do in the real world.

> And there are other problems of course that could arise (such as the
> presence of correlated sources, or an array whose antenna spacings are
> greater than half a wavelength, or signals which are wide-band, etc
> which will also act to screw up MUSIC),

Most of those can be handled. Tedious, but not very
difficult.

> but neither I, nor the
> articles to which I am referring, assume any of these conditions hold.
>
> > If you want to, repeat the excercise with various SNRs,
> > and with different MUSIC implementations.
>
> Been there, done that.
>
> > Rune
>
> Thanks for your input Rune. It seems that you're the only person who
> responds to array signal processing questions. I kinda thought there
> would be some people on the group who could understand this stuff.

I am sure there are.

> Maybe few professionals post here or are in another group.

yourself: they just don't want to learn about the
pathological shortfalls of these types of methods.

The 'professionals' - people who work for DSP for a living
in the real world - just don't use MUSIC. For the very
reasons I've done my best to point out for you. If you don't
believe me, just look around for yourself and try and find
out how many real-world applications actually use these
sorts of methods.

There aren't too many - I'm not aware of a single one.

Rune
```
```On 13 Jun, 22:03, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 13 Jun, 22:00, Rune Allnor <all...@tele.ntnu.no> wrote:
>
>
>
>
>
> > Just try it and see: Use your favourite MUSIC implementation
> > and specify it to run with a covariance matrix of order P.
> > Simulate a noise-free signal with N = 3P samples (to make
> > sure the covariance matrix is well-behaved) as follows:
>
> > First use *one* sinusoidal with DoA cos(phi_1) = 2*pi*1/P.
> > Use your MUSIC estimator and verify that you find the
> > DoA to within numerical precision (remember, noise-free
> > signal).
>
> > Next, *add* a signal component with the same amplitude
> > and with DoA cos(phi_2) = 2*pi*2/P. Repeat the MUSIC
> > analysis to verify that you find the two DoAs.
>
> > *Add* successive signals with DoAs cos(phi_n) = 2*pi*n/P
> > for n=3,4,...,2P. Make sure the signals have the same
> > amplitudes, and add *no* noise.
>
> > Just to be clear: The first time around the noise-free
> > signal contains one sinusoidal. The scond time it contains
> > two sinusodials. The n'th time it contains n sinusoidals.
>
> Sorry, I got the DoAs wrong: Use a DoA for the n'th
> signal as
>
> cos(phi_n) = pi*n/P &#2013266080; &#2013266080; (deleted factor 2).
>
> Rune

You might want to use

cos(phi_n) = pi*n/2P.

This ensures that all 2P signal components are in
the [0,pi] range.
```
```On Jun 14, 6:23&#2013266080;am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 14 Jun, 05:30, Junoexpress <MTBrenne...@gmail.com> wrote:
>
>
>
> > On Jun 13, 4:00&#2013266080;pm, Rune Allnor <all...@tele.ntnu.no> wrote:> On 13
Jun, 21:33, junoexpress <MTBrenne...@gmail.com> wrote:
>
> > > > Hi,
>
> > > > In the literature, there is a common claim made that the performance
> > > > of MUSIC for direction of arrival estimation (DOA) is affected when
> > > > the wrong number of directional sources is assumed. There is no
> > > > argument presented to support this claim, nor any references given,
> > > > just the claim.
>
> > > The claims may or may not be true, depending on exactly
> > > how &#2013266080;they are phrased. Do you have any pointers to where
> > > such claims are made?
>
> > This claim is found in a good many papers on the topic of signal
> > enumeration. Two that make the statement clearly are:
> > 1) IEEE Trans Acc, Speech, and Signal Processing, vol. 38, no. 11,
> > 1990
> > "On Information Theoretic Criteria...." by Wong, Zhang, Reilly, and
> > Yip
> > (see text after eqn 5 in article)
> > 2) IEEE Trans Signal Processing, vol. 39, no. 8, 1991
> > "A Parametric Method for Determining the Number of Signals in Narrow-
> > Band Direction Finding"
> > by Wu and Fuhrman (see first paragraph)
>
> > > There are several causes for inaccuracies in the MUSIC-type
> > > estimators. One is the SNR, others are related to wrong order
> > > estimates, like if there in reality are 4 signals present,
> > > but the order estimator only finds 3 of them. In such cases,
> > > it is likely that at least some of the DoAs are off, either
> > > because the SNR is low or the 'missing' signal lies very close
> > > to one of the others. If the 'missing' signal has a significantly
> > > smaller amplitude than the others, the remaining DoA estimates
> > > might be accurate anyway.
>
> > Your line of reasoning is pretty much what makes me doubt this claim.
> > A source that is missed, is most likely missed because its SNR is very
> > small.
>
> Not necesarily. There can be any number of reasons why a
> source is missed. Small SNR is just one of them.
>
> There is one aspect of order estimators you need to be
> aware of: For AR models, order estimators are formally
> are derived from statistical analyses of the reflection
> coefficients that represent signal residuals during the
> Levinson recursion.
>
> MUSIC can not be represented as a Levinson recursion,
> it's based on eigenvector decompositions. So the *formal*
> statistical analysis, which works on the reflection
> coefficients, don't work with the eigenvalues.
>
> Now, this does *not* mean that oprder estimators do not
> work with eigenvector decompositions - they do. It means
> that one has bodged a tool to work in a context it was
> not designed for. Which also goes a long way to explain
> why order estimates might be off from time to time, with
> no apparent reason.
>
If you consider this from a math point of view, the basic fact is that
the notion of a "signal subspace" and a "noise subspace" are fiction.
point of view is not well understood, and probably should be
investigated more.

> > The components of the stronger sources on the evr that gets
> > dropped from the signal subspace and gets put into the noise sub-space
> > cannot be that large, so when you run say Spectral MUSIC,
>
> Spectral MUSIC is...? If you are talking about the MUSIC
> pseudo spectrum, keep in mind that there is no information
> about the signal spectrum encoded in the pseudo spectrum.
>
There are two basic types of MUSIC: "spectral" MUSIC (which is what
was originally proposed by Schmidt) and "ROOT" MUSIC (which is a
simple extension of spectral MUSIC that works a bit better).
Estimates about the source AOAs can be made (if again we make a boat-
load full of assumptions ;>)) from both.

> > the location
> > where the reciprocal of the projection of the steering vectors onto
> > the noise subspace has its max value should not change appreciably.
>
> Remember, you are working D-manifolds in an P-dimensional
> complex-valued vector space. Don't expect your intuition
> to be of the best help...
>
>
>
> > > No easy answers.
>
> > > > Does this reasoning sound like the justification for the claim about
> > > > MUSIC I presented, or is there something else I am missing?
>
> > > Yes, there is.
>
> > > MUSIC-type estimators (meaning all estimators which
> > > implicitly or explicitly use a covariance matrix of
> > > order P to estimate the parameters of D signals) *fail*
> > > *unconditionally* when D >=P.
>
> > The noise in the system (i.e. AWGN in the channels) will guarantee
> > that the covariance matrix has full rank. This condition is certainly
> > satisfied in all of the professional treatments of MUSIC you see in
> > the lit.
>
> You need to think one step further: The basis for the
> noise subspace needs to be of rank at least 1 for MUSIC
> and friends to work. Once you have a signal that contains
> P (or P/2) or more sinusoidals, this no longer holds and
> MUSIC fails unconditionally.
>
> > I am assuming the are are less sources than antennas here.
>
> You can certainly do that for academic purposes. That's
> a very stupid thing to do in the real world.
>
Not really. In many applications the cost of the array alone prohibits
you from using a large array size and you have to make that
assumption.

> > And there are other problems of course that could arise (such as the
> > presence of correlated sources, or an array whose antenna spacings are
> > greater than half a wavelength, or signals which are wide-band, etc
> > which will also act to screw up MUSIC),
>
> Most of those can be handled. Tedious, but not very
> difficult.
>
> > but neither I, nor the
> > articles to which I am referring, assume any of these conditions hold.
>
> > > If you want to, repeat the excercise with various SNRs,
> > > and with different MUSIC implementations.
>
> > Been there, done that.
>
> > > Rune
>
> > Thanks for your input Rune. It seems that you're the only person who
> > responds to array signal processing questions. I kinda thought there
> > would be some people on the group who could understand this stuff.
>
> I am sure there are.
>
> > Maybe few professionals post here or are in another group.
>
> yourself: they just don't want to learn about the
> pathological shortfalls of these types of methods.
>
Unfortunately, there are people on both sides of the fence who are
biased and insecure, which really does make it difficult for the two
sides to communicate in a meaningful and respectful manner. But I'm
sure neither of us wants to contribute to such counter-productive
exchanges. I know that I don't. Personally I find the real-life
problems you' re talking about exciting and I do think they have to be
addressed to get a real understanding of how things work.

> The 'professionals' - people who work for DSP for a living
> in the real world - just don't use MUSIC. For the very
> reasons I've done my best to point out for you. If you don't
> believe me, just look around for yourself and try and find
> out how many real-world applications actually use these
> sorts of methods.
>
> There aren't too many - I'm not aware of a single one.
>
> Rune

Good to hear: that's what keeps me in business. ;>)

M
```
```On 16 Jun, 01:35, Junoexpress <MTBrenne...@gmail.com> wrote:
> On Jun 14, 6:23&#2013266080;am, Rune Allnor <all...@tele.ntnu.no> wrote:
>
>
>
> > On 14 Jun, 05:30, Junoexpress <MTBrenne...@gmail.com> wrote:
>
> > > On Jun 13, 4:00&#2013266080;pm, Rune Allnor <all...@tele.ntnu.no> wrote:> On
13 Jun, 21:33, junoexpress <MTBrenne...@gmail.com> wrote:
>
> > > > > Hi,
>
> > > > > In the literature, there is a common claim made that the performance
> > > > > of MUSIC for direction of arrival estimation (DOA) is affected when
> > > > > the wrong number of directional sources is assumed. There is no
> > > > > argument presented to support this claim, nor any references given,
> > > > > just the claim.
>
> > > > The claims may or may not be true, depending on exactly
> > > > how &#2013266080;they are phrased. Do you have any pointers to where
> > > > such claims are made?
>
> > > This claim is found in a good many papers on the topic of signal
> > > enumeration. Two that make the statement clearly are:
> > > 1) IEEE Trans Acc, Speech, and Signal Processing, vol. 38, no. 11,
> > > 1990
> > > "On Information Theoretic Criteria...." by Wong, Zhang, Reilly, and
> > > Yip
> > > (see text after eqn 5 in article)
> > > 2) IEEE Trans Signal Processing, vol. 39, no. 8, 1991
> > > "A Parametric Method for Determining the Number of Signals in Narrow-
> > > Band Direction Finding"
> > > by Wu and Fuhrman (see first paragraph)
>
> > > > There are several causes for inaccuracies in the MUSIC-type
> > > > estimators. One is the SNR, others are related to wrong order
> > > > estimates, like if there in reality are 4 signals present,
> > > > but the order estimator only finds 3 of them. In such cases,
> > > > it is likely that at least some of the DoAs are off, either
> > > > because the SNR is low or the 'missing' signal lies very close
> > > > to one of the others. If the 'missing' signal has a significantly
> > > > smaller amplitude than the others, the remaining DoA estimates
> > > > might be accurate anyway.
>
> > > Your line of reasoning is pretty much what makes me doubt this claim.
> > > A source that is missed, is most likely missed because its SNR is very
> > > small.
>
> > Not necesarily. There can be any number of reasons why a
> > source is missed. Small SNR is just one of them.
>
> > There is one aspect of order estimators you need to be
> > aware of: For AR models, order estimators are formally
> > are derived from statistical analyses of the reflection
> > coefficients that represent signal residuals during the
> > Levinson recursion.
>
> > MUSIC can not be represented as a Levinson recursion,
> > it's based on eigenvector decompositions. So the *formal*
> > statistical analysis, which works on the reflection
> > coefficients, don't work with the eigenvalues.
>
> > Now, this does *not* mean that oprder estimators do not
> > work with eigenvector decompositions - they do. It means
> > that one has bodged a tool to work in a context it was
> > not designed for. Which also goes a long way to explain
> > why order estimates might be off from time to time, with
> > no apparent reason.
>
> If you consider this from a math point of view, the basic fact is that
> the notion of a "signal subspace" and a "noise subspace" are fiction.
> point of view is not well understood, and probably should be
> investigated more.

No, there isn't. The terms are well-defined, the maths simple.
As long as one is comfortable with N-dimensional complex-valued
vector spaces. But those are just a matter of familiarization,
like i = sqrt(-1). A big hurdle for the newbie, second nature
to the somewhat more experienced.

> > > The components of the stronger sources on the evr that gets
> > > dropped from the signal subspace and gets put into the noise sub-space
> > > cannot be that large, so when you run say Spectral MUSIC,
>
> > Spectral MUSIC is...? If you are talking about the MUSIC
> > pseudo spectrum, keep in mind that there is no information
> > about the signal spectrum encoded in the pseudo spectrum.
>
> There are two basic types of MUSIC: "spectral" MUSIC (which is what
> was originally proposed by Schmidt) and "ROOT" MUSIC (which is a
> simple extension of spectral MUSIC that works a bit better).
> Estimates about the source AOAs can be made (if again we make a boat-
> load full of assumptions ;>)) from both.

MUSIC is Scmhidt's original method that in principle works
with arrays of any geometry. The Root MUSIC is an ad-hoc
adaption for the case of Uniform Linear Arrays. It might have
been an interesting alternative, were it not for other methods
that addressed the ULA directly, which turned out to be far
more computationally efficient.

> > > the location
> > > where the reciprocal of the projection of the steering vectors onto
> > > the noise subspace has its max value should not change appreciably.
>
> > Remember, you are working D-manifolds in an P-dimensional
> > complex-valued vector space. Don't expect your intuition
> > to be of the best help...
>
> > > > No easy answers.
>
> > > > > Does this reasoning sound like the justification for the claim about
> > > > > MUSIC I presented, or is there something else I am missing?
>
> > > > Yes, there is.
>
> > > > MUSIC-type estimators (meaning all estimators which
> > > > implicitly or explicitly use a covariance matrix of
> > > > order P to estimate the parameters of D signals) *fail*
> > > > *unconditionally* when D >=P.
>
> > > The noise in the system (i.e. AWGN in the channels) will guarantee
> > > that the covariance matrix has full rank. This condition is certainly
> > > satisfied in all of the professional treatments of MUSIC you see in
> > > the lit.
>
> > You need to think one step further: The basis for the
> > noise subspace needs to be of rank at least 1 for MUSIC
> > and friends to work. Once you have a signal that contains
> > P (or P/2) or more sinusoidals, this no longer holds and
> > MUSIC fails unconditionally.
>
> > > I am assuming the are are less sources than antennas here.
>
> > You can certainly do that for academic purposes. That's
> > a very stupid thing to do in the real world.
>
> Not really. In many applications the cost of the array alone prohibits
> you from using a large array size and you have to make that
> assumption.

That's the blunder you and everybody else who try to use these
methods make: The fact that you assume that the sensor will
never see more than a small number of sources, does not
prevent that from happening.

So when (not if) the case of D >=P happens, the system breaks
down. Your clients are left in a void where nothing works, and
you don't understand why.

Before you try and make a living out of this (your lack
of historical knowledge and the statement above are
certain give-aways), take some time to think through the
legal obligations that are activated once you charge
be liable to damage compensations for any glitch or problem

failure, I will waste no time in suggest you might be
guilty of professional misconduct or fraud, if I find
you used MUSIC and only assumed (as opposed to ensured)
that the D < P.

> > > And there are other problems of course that could arise (such as the
> > > presence of correlated sources, or an array whose antenna spacings are
> > > greater than half a wavelength, or signals which are wide-band, etc
> > > which will also act to screw up MUSIC),
>
> > Most of those can be handled. Tedious, but not very
> > difficult.
>
> > > but neither I, nor the
> > > articles to which I am referring, assume any of these conditions hold.
>
> > > > If you want to, repeat the excercise with various SNRs,
> > > > and with different MUSIC implementations.
>
> > > Been there, done that.
>
> > > > Rune
>
> > > Thanks for your input Rune. It seems that you're the only person who
> > > responds to array signal processing questions. I kinda thought there
> > > would be some people on the group who could understand this stuff.
>
> > I am sure there are.
>
> > > Maybe few professionals post here or are in another group.
>
> > yourself: they just don't want to learn about the
> > pathological shortfalls of these types of methods.
>
> Unfortunately, there are people on both sides of the fence who are
> biased and insecure, which really does make it difficult for the two
> sides to communicate in a meaningful and respectful manner.

There isn't. The only problem is that most people don't
know the excercise I pointed out for you earlier on.
If everybody did, everybody would agree with me.

> But I'm
> sure neither of us wants to contribute to such counter-productive
> exchanges. I know that I don't. Personally I find the real-life
> problems you' re talking about exciting and I do think they have to be
> addressed to get a real understanding of how things work.

Not what MUSIC is concerned. Just run the excercise I showed
you, and contemplate what would happen if a similar situation
occured after you installed and accepted payment for one of

> > The 'professionals' - people who work for DSP for a living
> > in the real world - just don't use MUSIC. For the very
> > reasons I've done my best to point out for you. If you don't
> > believe me, just look around for yourself and try and find
> > out how many real-world applications actually use these
> > sorts of methods.
>
> > There aren't too many - I'm not aware of a single one.
>
> > Rune
>
> Good to hear: that's what keeps me in business. ;>)

If you say so. Just make sure you are well insured against
claims of professional misconduct, if you decide to sell
these kinds of things.

Rune
```
```On Jun 16, 4:32&#2013266080;am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 16 Jun, 01:35, Junoexpress <MTBrenne...@gmail.com> wrote:
>
>
>
> > On Jun 14, 6:23&#2013266080;am, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > > On 14 Jun, 05:30, Junoexpress <MTBrenne...@gmail.com> wrote:
>
> > > > On Jun 13, 4:00&#2013266080;pm, Rune Allnor <all...@tele.ntnu.no> wrote:> On
13 Jun, 21:33, junoexpress <MTBrenne...@gmail.com> wrote:
>
> > > > > > Hi,
>
> > > > > > In the literature, there is a common claim made that the performance
> > > > > > of MUSIC for direction of arrival estimation (DOA) is affected when
> > > > > > the wrong number of directional sources is assumed. There is no
> > > > > > argument presented to support this claim, nor any references given,
> > > > > > just the claim.
>
> > > > > The claims may or may not be true, depending on exactly
> > > > > how &#2013266080;they are phrased. Do you have any pointers to where
> > > > > such claims are made?
>
> > > > This claim is found in a good many papers on the topic of signal
> > > > enumeration. Two that make the statement clearly are:
> > > > 1) IEEE Trans Acc, Speech, and Signal Processing, vol. 38, no. 11,
> > > > 1990
> > > > "On Information Theoretic Criteria...." by Wong, Zhang, Reilly, and
> > > > Yip
> > > > (see text after eqn 5 in article)
> > > > 2) IEEE Trans Signal Processing, vol. 39, no. 8, 1991
> > > > "A Parametric Method for Determining the Number of Signals in Narrow-
> > > > Band Direction Finding"
> > > > by Wu and Fuhrman (see first paragraph)
>
> > > > > There are several causes for inaccuracies in the MUSIC-type
> > > > > estimators. One is the SNR, others are related to wrong order
> > > > > estimates, like if there in reality are 4 signals present,
> > > > > but the order estimator only finds 3 of them. In such cases,
> > > > > it is likely that at least some of the DoAs are off, either
> > > > > because the SNR is low or the 'missing' signal lies very close
> > > > > to one of the others. If the 'missing' signal has a significantly
> > > > > smaller amplitude than the others, the remaining DoA estimates
> > > > > might be accurate anyway.
>
> > > > Your line of reasoning is pretty much what makes me doubt this claim.
> > > > A source that is missed, is most likely missed because its SNR is very
> > > > small.
>
> > > Not necesarily. There can be any number of reasons why a
> > > source is missed. Small SNR is just one of them.
>
> > > There is one aspect of order estimators you need to be
> > > aware of: For AR models, order estimators are formally
> > > are derived from statistical analyses of the reflection
> > > coefficients that represent signal residuals during the
> > > Levinson recursion.
>
> > > MUSIC can not be represented as a Levinson recursion,
> > > it's based on eigenvector decompositions. So the *formal*
> > > statistical analysis, which works on the reflection
> > > coefficients, don't work with the eigenvalues.
>
> > > Now, this does *not* mean that oprder estimators do not
> > > work with eigenvector decompositions - they do. It means
> > > that one has bodged a tool to work in a context it was
> > > not designed for. Which also goes a long way to explain
> > > why order estimates might be off from time to time, with
> > > no apparent reason.
>
> > If you consider this from a math point of view, the basic fact is that
> > the notion of a "signal subspace" and a "noise subspace" are fiction.
> > point of view is not well understood, and probably should be
> > investigated more.
>
> No, there isn't. The terms are well-defined, the maths simple.
> As long as one is comfortable with N-dimensional complex-valued
> vector spaces. But those are just a matter of familiarization,
> like i = sqrt(-1). A big hurdle for the newbie, second nature
> to the somewhat more experienced.
>
I doubt it. I'm a PhD in math and I know from my experience that there
is a fair amount of algebraic geometry involved in understanding array
manifolds. I've corrected the errors of many engineers who don't
understand what a complex Steifel manifolds (or even a manifold for
that matter), let alone its topology, so I know this from experience
to be the case.

> > > > The components of the stronger sources on the evr that gets
> > > > dropped from the signal subspace and gets put into the noise sub-space
> > > > cannot be that large, so when you run say Spectral MUSIC,
>
> > > Spectral MUSIC is...? If you are talking about the MUSIC
> > > pseudo spectrum, keep in mind that there is no information
> > > about the signal spectrum encoded in the pseudo spectrum.
>
> > There are two basic types of MUSIC: "spectral" MUSIC (which is what
> > was originally proposed by Schmidt) and "ROOT" MUSIC (which is a
> > simple extension of spectral MUSIC that works a bit better).
> > Estimates about the source AOAs can be made (if again we make a boat-
> > load full of assumptions ;>)) from both.
>
> MUSIC is Scmhidt's original method that in principle works
> with arrays of any geometry. The Root MUSIC is an ad-hoc
> adaption for the case of Uniform Linear Arrays. It might have
> been an interesting alternative, were it not for other methods
> that addressed the ULA directly, which turned out to be far
> more computationally efficient.
>
>
>
> > > > the location
> > > > where the reciprocal of the projection of the steering vectors onto
> > > > the noise subspace has its max value should not change appreciably.
>
> > > Remember, you are working D-manifolds in an P-dimensional
> > > complex-valued vector space. Don't expect your intuition
> > > to be of the best help...
>
> > > > > No easy answers.
>
> > > > > > Does this reasoning sound like the justification for the claim about
> > > > > > MUSIC I presented, or is there something else I am missing?
>
> > > > > Yes, there is.
>
> > > > > MUSIC-type estimators (meaning all estimators which
> > > > > implicitly or explicitly use a covariance matrix of
> > > > > order P to estimate the parameters of D signals) *fail*
> > > > > *unconditionally* when D >=P.
>
> > > > The noise in the system (i.e. AWGN in the channels) will guarantee
> > > > that the covariance matrix has full rank. This condition is certainly
> > > > satisfied in all of the professional treatments of MUSIC you see in
> > > > the lit.
>
> > > You need to think one step further: The basis for the
> > > noise subspace needs to be of rank at least 1 for MUSIC
> > > and friends to work. Once you have a signal that contains
> > > P (or P/2) or more sinusoidals, this no longer holds and
> > > MUSIC fails unconditionally.
>
> > > > I am assuming the are are less sources than antennas here.
>
> > > You can certainly do that for academic purposes. That's
> > > a very stupid thing to do in the real world.
>
> > Not really. In many applications the cost of the array alone prohibits
> > you from using a large array size and you have to make that
> > assumption.
>
> That's the blunder you and everybody else who try to use these
> methods make: The fact that you assume that the sensor will
> never see more than a small number of sources, does not
> prevent that from happening.
>
> So when (not if) the case of D >=P happens, the system breaks
> down. Your clients are left in a void where nothing works, and
> you don't understand why.
>
> Before you try and make a living out of this (your lack
> of historical knowledge and the statement above are
> certain give-aways), take some time to think through the
> legal obligations that are activated once you charge
> be liable to damage compensations for any glitch or problem
>
> If your clients hire me to review your system after a
> failure, I will waste no time in suggest you might be
> guilty of professional misconduct or fraud, if I find
> you used MUSIC and only assumed (as opposed to ensured)
> that the D < P.
>
You have to be very clear what the limitations of your method are. I
don't know your background, but I don't know any professionals who
don't spell these type of things out up front.
>
>
> > > > And there are other problems of course that could arise (such as the
> > > > presence of correlated sources, or an array whose antenna spacings are
> > > > greater than half a wavelength, or signals which are wide-band, etc
> > > > which will also act to screw up MUSIC),
>
> > > Most of those can be handled. Tedious, but not very
> > > difficult.
>
> > > > but neither I, nor the
> > > > articles to which I am referring, assume any of these conditions hold.
>
> > > > > If you want to, repeat the excercise with various SNRs,
> > > > > and with different MUSIC implementations.
>
> > > > Been there, done that.
>
> > > > > Rune
>
> > > > Thanks for your input Rune. It seems that you're the only person who
> > > > responds to array signal processing questions. I kinda thought there
> > > > would be some people on the group who could understand this stuff.
>
> > > I am sure there are.
>
> > > > Maybe few professionals post here or are in another group.
>
> > > yourself: they just don't want to learn about the
> > > pathological shortfalls of these types of methods.
>
> > Unfortunately, there are people on both sides of the fence who are
> > biased and insecure, which really does make it difficult for the two
> > sides to communicate in a meaningful and respectful manner.
>
> There isn't. The only problem is that most people don't
> know the excercise I pointed out for you earlier on.
> If everybody did, everybody would agree with me.
>
> > But I'm
> > sure neither of us wants to contribute to such counter-productive
> > exchanges. I know that I don't. Personally I find the real-life
> > problems you' re talking about exciting and I do think they have to be
> > addressed to get a real understanding of how things work.
>
> Not what MUSIC is concerned. Just run the excercise I showed
> you, and contemplate what would happen if a similar situation
> occured after you installed and accepted payment for one of
>
> > > The 'professionals' - people who work for DSP for a living
> > > in the real world - just don't use MUSIC. For the very
> > > reasons I've done my best to point out for you. If you don't
> > > believe me, just look around for yourself and try and find
> > > out how many real-world applications actually use these
> > > sorts of methods.
>
> > > There aren't too many - I'm not aware of a single one.
>
> > > Rune
>
> > Good to hear: that's what keeps me in business. ;>)
>
> If you say so. Just make sure you are well insured against
> claims of professional misconduct, if you decide to sell
> these kinds of things.
>
> Rune

I do math, not engineering. All of my equations have worked so far,
and consulting has never been better. ;>)

M

```
```On 16 Jun, 22:22, Junoexpress <MTBrenne...@gmail.com> wrote:
> On Jun 16, 4:32&#2013266080;am, Rune Allnor <all...@tele.ntnu.no> wrote:

> > > If you consider this from a math point of view, the basic fact is that
> > > the notion of a "signal subspace" and a "noise subspace" are fiction.
> > > point of view is not well understood, and probably should be
> > > investigated more.
>
> > No, there isn't. The terms are well-defined, the maths simple.
> > As long as one is comfortable with N-dimensional complex-valued
> > vector spaces. But those are just a matter of familiarization,
> > like i = sqrt(-1). A big hurdle for the newbie, second nature
> > to the somewhat more experienced.
>
> I doubt it. I'm a PhD in math and I know from my experience that there
> is a fair amount of algebraic geometry involved in understanding array
> manifolds. I've corrected the errors of many engineers who don't
> understand what a complex Steifel manifolds (or even a manifold for
> that matter), let alone its topology, so I know this from experience
> to be the case.

Still, you don't undrestand the trivial basics of vector spaces.

> > If your clients hire me to review your system after a
> > failure, I will waste no time in suggest you might be
> > guilty of professional misconduct or fraud, if I find
> > you used MUSIC and only assumed (as opposed to ensured)
> > that the D < P.
>
> You have to be very clear what the limitations of your method are. I
> don't know your background, but I don't know any professionals who
> don't spell these type of things out up front.

You claim to be a professional. You actively disregard
precise and clear warnings about trivial pathologies
of the method you claim to build a business on.

That's gross professional misconduct at the best of times,
and pretty close to fraud.

Rune
```