On Jun 16, 4:32�am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 16 Jun, 01:35, Junoexpress <MTBrenne...@gmail.com> wrote:
>
>
>
> > On Jun 14, 6:23�am, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > > On 14 Jun, 05:30, Junoexpress <MTBrenne...@gmail.com> wrote:
>
> > > > On Jun 13, 4:00�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 �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.
> > There is a lot about this simple picture that from a mathematical
> > 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
> somebody for your advice or systems: You might very well
> be liable to damage compensations for any glitch or problem
> that are caused by your advice or system.
>
> 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.
>
> > > Nope. The academics who ask about MUSIC react exactly like
> > > 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
> your systems.
>
> > > 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