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Suppose that we have the following model

$y(t) = A_1\sin(\omega_1 t+\phi_1) + A_2\sin(\omega_2 t+\phi_2) + ... + A_p\sin(\omega_p t+\phi_p) + z(t)$

My question is not related to how to determine the number of components, but I am interested in the maximum. For example, let us assume that we have sampled this model and get sample with size let's say $294$, the question arises what can be the maximum number of components? For instance, can it be more then $147$? or more then half the size of the signal sample? Or can it be in general something $20-50$, for instance? I am not expecting an exact number, but a possible maximum number which can be in this case.

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  • $\begingroup$ 294 sample points give you 294 equations. That means the maximum number of real parameters that can be uniquely determined is 294. $\endgroup$
    – Jazzmaniac
    May 6, 2014 at 10:44
  • $\begingroup$ no no let say we have parameter amplitudes,phases let say 4,i mean number of sine $\endgroup$
    – user350
    May 6, 2014 at 10:46
  • $\begingroup$ let say that we are going singular spectrum analysis,we need window length right?so in this case what should be number of component? $\endgroup$
    – user350
    May 6, 2014 at 10:47
  • $\begingroup$ It shouldn't be hard to see that you have three real parameters per sinusoidal component. Assuming z(t) vanishes, that would make 294/3 = 98 components tops. $\endgroup$
    – Jazzmaniac
    May 6, 2014 at 10:50
  • $\begingroup$ so to take window length 98 should be enough right?what about if white noise does not vanishes? $\endgroup$
    – user350
    May 6, 2014 at 10:51

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I guess that your problem can be modeled as a harmonic retrieval problem. You can check the Kung's method.

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For the extremely special case where you can use a Fourier transform(*),

When you collect p real sample points (in this case, p = 294), and then you feed that though a discrete Fourier transform, it gives you 2 components at n = p/2 (in this case, n = 147) different frequencies. (Internally we practically always calculate a in-phase I and a quadrature Q component at each frequency, but many times we take the I and Q at each frequency and transform them to the amplitude A and the phase phi at that frequency).

Those 2 components at n=p/2 different frequencies can be used to exactly reconstruct the original series of sample points -- there is no more information.

In practice, we practically always want to have anti-aliasing filters and we usually have DC-blocking filters, so we usually find that the first few and the last few of those n=p/2 frequency components have zero amplitude, and in some situations we find that only 1 or 2 or 16 (DTMF) frequencies ever have amplitudes significantly more than zero. In those cases you only need 1 or 2 sinusoids (or however many the Fourier transform indicates have nonzero amplitude) in order to exactly reconstruct the original series of sample points.

But in other situations we find that every one of the n=p/2 (in this case, 147) components has a large amplitude -- in those cases, you need all those amplitudes to exactly reconstruct the signal.

(*) the sample-time values are evenly spaced from t0=0 to tp, the lowest non-zero frequency w1 = 1/(tp), the frequencies are evenly spaced integer multiples of w1, etc.

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