I think you are mixing up time- and spatial- domain sampling.
For what you describe, you have to do it in two steps:
1. For each antenna array waveform, Isolate each frequency tone, regardless of which source they come for. This is done by taking into account that each tone (disregarding from which source) is narrowband, so you can apply MUSIC/ESPRIT in *time* domain, for each antenna array waveform.
2. For each processed / isolated tone, take spatial snapshots in *spatial* domain, across your (uniform) antenna array. Then apply MUSIC/ESPRIT in *spatial* domain, to get the arrival angles.
If you want to then identify which sources sent which tones, you need a third step where you associate tones and DOAs.
It may be possible to do both steps at once, but I am not aware of any references.
In summary, in what you are describing you have two domains: temporal and spatial. MUSIC and ESPRIT works in one dimension at a time. For them to work in either dimension, the other dimension has to be "narrowband". Thus, you have to pick one domain to use to isolate its components, such that when you switch to the other dimension, the narrowband assumption applies.
Recall the original Prony model, and write down your problem simultaneously in time- and spatial-domains, keep track of your indices.
Hope this explains it better. If you need a more elementary explanation, feel free to let me know, but if you do then you have to forget either the spatial domain or the time domain. Given your way of writing S1 and S2 I think we can stick to time domain first, ignore the spatial part.
At any rate, for any signal that has components cos(2pi f0 n), you can write as 1/2* [ \exp(j 2pi f0 n)+\exp(-j2pi f0 n) ] which fits into the original Prony problem.
Julius
On Friday, November 1, 2013 5:55:01 PM UTC-4, rudykeram wrote:
> Julius,
>
> Thank you veyr much for taking the time and explaining it. You answer make
>
> it cleaner, but I am still little unsure about the whole idea that why and
>
> how the algorithm actually works.
>
>
>
> Because for example, let's say that I have two different sources.
>
> Let's say the first souce generates the following waveform:
>
> S1 = sin(pi/4*n) + cos(pi/3*n) + sin(pi/5*n)
>
> So, my first source is a waveform composed of three different frequencies.
>
>
>
>
>
> And, let's say my second waveform runs at completely different
>
> frequencies.
>
> S2 = sin(pi/6*n) + cos(pi/2*n) + sin(pi/7*n)
>
>
>
> At the input of the the ADC of each of my antennas, I am going to see the
>
> combination of these signals (S1+S2).
>
>
>
> So, each ADC will see a phase shifted version of (S1+S2), but in fact it
>
> will see a combination of waveform comprising 6 different freqauency terms.
>
> Then how would the MUSIC algorithm be able to even distinguish between the
>
> two sources ?!?
>
>
>
> We have a mixutre of six different frequencies. How would the algorithm
>
> works itself out such that it knows that which three frequencies belong to
>
> S1, and which three frequencies belong to S2 ?
>
>
>
>
>
> Could you please explain in simple terms, why the algorithm works the way
>
> it works. What is it about breaking down the covarience matrix that will
>
> determine what direction these six signals are coming from ?
>
>
>
> I would really appreciate some explanation.
>
>
>
> Thanks,
>
> --Rudy
>
>
>
>
>
> >MUSIC, ESPRIT, and all related methods basically "sample" a signal
>
> spatiall=
>
> >y, and use an appropriate Fourier transform to transform direction of
>
> arriv=
>
> >al onto a frequency term.=20
>
> >
>
> >The actual prototype problem they solve is the classical Prony
>
> problem.=20
>
> >y_n =3D \sum_k a_k exp(b_k n), n =3D 0, ..., N-1.=20
>
> >
>
> >In MUSIC/ESPRIT papers written in the 90s, the derivation follows: a
>
> narrow=
>
> >band signal arrives at a uniform array, such that when a snapshot of the
>
> re=
>
> >ceived signal is taken, one gets:
>
> >y_n =3D \sum_k a_k exp(j \theta_k n), n =3D 0, ..., N-1.=20
>
> >
>
> >In this case \theta_k are the directions of arrival, up to a scaling
>
> factor=
>
> >, and n indexes the antenna arrays.=20
>
> >
>
> >So to recap you need two things: 1. The signal is narrowband, 2. The array
>
> =
>
> >is uniformly spaced.=20
>
> >
>
> >What you are trying to solve is actually much easier. Since you say your
>
> tw=
>
> >o signals are at different frequencies, you can simply separate them
>
> first,=
>
> > and then for each of the two frequencies apply MUSIC/ESPRIT or what have
>
> y=
>
> >ou.=20
>
> >
>
> >The amplification part that papers of that era are fixated on can in fact
>
> h=
>
> >elp you. Both MUSIC/ESPRIT families of algorithm first solve for the b_k,
>
> a=
>
> >nd then do a linear fit to get a_k. The a_k values can be used to estimate
>
> =
>
> >signal strengths, or for optimizing how to best combine the incident
>
> signal=
>
> >s to get the best SNR.=20
>
> >
>
> >If your source signal is not narrowband in the first place, then the
>
> typica=
>
> >l approach is to break it down via filterbank or FFT bank, then for each
>
> bi=
>
> >n one can apply MUSIC/ESPRIT. Then one typically applies a higher-level
>
> alg=
>
> >orithm to pick and choose and combine the outputs of the different
>
> bins.=20
>
> >
>
> >However, the usual caution applies: in the general non-narrowband case,
>
> the=
>
> >re is no one method that can be applied universally and trivially at the
>
> sa=
>
> >me time. You have to look at the specific case, and make sure the whole
>
> sys=
>
> >tem works for whatever your goal is.=20
>
> >
>
> >Anyway, I hope the above helps. Your questions aren't elementary.=20
>
> >
>
> >Julius=20
>
> >
>
> >
>
> >
>
> >On Thursday, October 31, 2013 3:26:37 PM UTC-4, rudykeram wrote:
>
> >> Hi,=20
>
> >>=20
>
> >> I am trying to understand the MUISC (Multiple Signal Classification)
>
> >>=20
>
> >> algorithm. I am new in this topic. So, I am sorry if my questions seem
>
> so
>
> >>=20
>
> >> elementry.
>
> >>=20
>
> >> First of all, I would like to make sure that Source Localization is
>
> >>=20
>
> >> different than beamforming. Even though in Source Localization, we are
>
> >>=20
>
> >> dealing with phase arrays, but the goal is not to form (amplify) the
>
> beam
>
> >>=20
>
> >> in one direction and suppress it in the other directions, correct? And,
>
> >>=20
>
> >> instead we are trying to find the DoA of the sound sources, correct?
>
> >>=20
>
> >>=20
>
> >>=20
>
> >> After reading some articles, I tried to look at matlab pmusic
>
> algorithm:
>
> >>=20
>
> >> http://www.mathworks.com/help/signal/ref/pmusic.html
>
> >>=20
>
> >> =20
>
> >>=20
>
> >> And I looked at the Example 1 of the above link, here is where I am
>
> getti=
>
> >ng
>
> >>=20
>
> >> confused.
>
> >>=20
>
> >> Because my understanding is that with MUISC we can do source
>
> localization
>
> >>=20
>
> >> by finding the DoA.
>
> >>=20
>
> >> But all the examples that I looked on the web, including the one I
>
> listed
>
> >>=20
>
> >> above, they are all treating the input signal being a sinusoidal
>
> waveform
>
> >>=20
>
> >> having two different frequencies, with small seperation.=20
>
> >>=20
>
> >> But are they all assuming that each sound source producing only one tone
>
> =
>
> >(a
>
> >>=20
>
> >> single frequency singal) ?!?
>
> >>=20
>
> >>=20
>
> >>=20
>
> >> Because, my understanding of the Example 1 is that one source produces
>
> a
>
> >>=20
>
> >> waveform at a certain frequency, and the second source produces another
>
> >>=20
>
> >> waveform at a different frequency?
>
> >>=20
>
> >> Is this what MUSIC algorithm can handle (Having sources running at one
>
> >>=20
>
> >> frequency)?=20
>
> >>=20
>
> >>=20
>
> >>=20
>
> >> How about the real system? What if in real system, one speaker generates
>
> =
>
> >a
>
> >>=20
>
> >> sinusoidal waveform that has several different frequencies, which is a
>
> ve=
>
> >ry
>
> >>=20
>
> >> general and realiztic case. Just to simplify my question, let's just
>
> say
>
> >>=20
>
> >> that Sound Source "A" generates a sinusoidal wave with two different
>
> >>=20
>
> >> frequencies, and Sound Sounce "B" generates another sinusoidal wave
>
> with
>
> >>=20
>
> >> different two frequenceis.=20
>
> >>=20
>
> >> In this case, the input signal will be composed of four different
>
> >>=20
>
> >> frequencies when it arrives at the system.
>
> >>=20
>
> >>=20
>
> >>=20
>
> >> Then can we still apply MUSIC algorithm to determine the DoA of the two
>
> >>=20
>
> >> sources?
>
> >>=20
>
> >> I would appreciate some help on this, to remove my confusion.
>
> >>=20
>
> >>=20
>
> >>=20
>
> >> Thanks,=20
>
> >>=20
>
> >> --Rudy
>
> >>=20
>
> >> =20
>
> >>=20
>
> >> =20
>
> >>=20
>
> >>=20
>
> >>=20
>
> >> _____________________________ =09
>
> >>=20
>
> >> Posted through www.DSPRelated.com
>
> >
>
> >
>
>
>
> _____________________________
>
> Posted through www.DSPRelated.com