[SciPy-User] frequency components of a signal buried in a noisy time domain signal

[Anne Archibald]

On 27 February 2010 08:37, Ivo Maljevic <ivo.maljevic at gmail.com> wrote:
>
>> > What makes a signal weak/strong periodic ?
>>
>> By an exactly periodic signal I mean something like sin(f*t). Such a
>> signal produces a very narrow peak of a characteristic shape in an
>> FFT, and so can be recognized even in the presence of quite strong
>> noise. If your signal is only quasi-periodic - perhaps the frequency
>> is a slowly-varying function of time - you'll have a much broader
>> peak, which will be much lower for the same input signal amplitude,
>> and hence more difficult to distinguish from noise. If your signal is
>> only quasi-periodic, or comes and goes in the data, you may want to do
>> a series of FFTs on short, overlapping pieces of the data, so you can
>> look at time evolution of the signal's spectral properties.
>>
>
> In my opinion this is not quite so. Periodic signal, as you rightly pointed
> out,
> is a sinusoidal signal. Quasi-periodic signal behaves like a periodic one,
> even though
> it does not satisfy the periodic condition x(t) = x (t+To), where To is the
> period. Best known
> examples are when you add two sinusoidal signals with frequencies that are
> not a fractional integer of each other.
> For example: sin(2pi f t)+sin(2pi^2 f t). You would still see a "spike" in
> the frequency domain, but quasi-periodicity
> definitely does not relate to low frequency.

I guess we have a minor conflict of definitions. Of course the signal
you describe is not periodic, but, at least in terms of Fourier
transforms, you can treat the terms of a sum as completely independent
signals. So I would view this as simply two exactly periodic signals,
which you can indeed pull out as two separate peaks in the Fourier
transform.

What I had in mind in referring to quasi-periodic signals was
something you might get out of an oscillator with poor frequency
stability, or a not-very-good resonant cavity, or a sinusoid with some
modulation: a signal localized in frequency but not exactly periodic.
This will have a broader peak in the Fourier transform, but will still
be interesting in many contexts.

Anne