I think, a useful method to observe the dominate frequency may the FFT (Fast Fourier Transform).

If the system can be modelled using some discrete methods, e.g., FEM, you can analyze the system using some softwares to obtain the eigenvalues or frequencies of the disrete system.

I think, you are asking for a so-called "instantaneouos" frequency Fi , correct?

In case, there is no periodic even with equal periods you have to consult the PHASE function PHI(t). In this case, you can use the relation Fi=(1/2Pi)*d[PHI(t)]/dt 

I believe that Short Time Fourier Transform (STFT) is what you need. STFT is Fourier transform of a windowed signal, where windows moves in time.

STFT(f,t) = Integrate_(-infinity)^infinity x(tau) w(tau-t) Exp(-2 pi f tau) d tau

You can use discrete form of STFT if you have discrete data.

You can try wavelet transform in MATLAB which gives you a unique toolbox for this purpose!!

It depends on what you want to use the analysis for - standard techniques like FFT and its moving window sub-types, or wavelets MIGHT give you what you need if your needs are very modest and the signal is non-stationary in a 'simple" way; but note your signal is breaking the assumptions underlying these techniques and artefacts of varying degrees are inevitable. If the signal is known to be a simple combination from different  sources, techniques like principle component analysis and independent component analysis might be of some use.

I know of no general technique that can analyse a non-stationary non-simple signal about which nothing, other than the signal itself is known. My long experience with attempting such analysis has been that you will find many different  models that can completely explain the data, and yet have no predictability of future data whatsoever.

I would suggest if it is a specific type of signal you are interested in (one about which you know something about its origin as compared to having just the signal itself), then you are best to come up with some differential equation that describes the physics of how the signal is produced, and solve that instead. This constrains the models available to just a subset of those capable of fitting the data well - and hopefully the predictive capability of such physical models is better, if you have done the physics right.

Finally for complex signals, you must be extremely careful about how you test generic models for predictive power. You ask for trouble to fit several models and then choose the one with the best predictive power - all you will end up with is a model that predicts the fitted data, and the predictive dataset - but which may well have no predictive power on any further "new" data. Meta analysis of groups of Climate Change models routinely fall into this trap - not out of ignorance but out of necessity (there not being multiple sets of actual data available). In their case though, it is the use of physics behind the models which gives them some validity according to the scientific method.

Thanks a lot for your thoughts. I have attached the snapshot of signal. Here the purpose is to calculate the number of particle and its size from the signal. The signal is generated by by passing the laser light on the sample . if i apply STFT  then what should be the window length? how can i decide that??

Ok the question did not describe the problem that well at all, and we are all way off the track. It does not look like FFT or SFFT is what you are after for this. Because the peaks come randomly wavelets won't work that well either.

Are you after all the little wiggles in the data as well, or just the big spikes?.

 I would take the first difference of the signal (maybe with a small weighted regression smoothing function depending on inherent equipment noise, and look along this series until you find it exceeds a certain threshold value. This becomes the position of a potentially detected particle.  I would then model the peak shape either analytically (probably some Gaussian function or variant thereof will do), or as an empirical probability density function based on known peak data , and then fit the peak model around the range where the first difference test triggered a detection using a non-linear solver such as Rosenbrock.

If the peak fit to the data meets some statistical test of goodness of fit, it can be counted as a real particle, and the best fit size  will be some analytical function of the fitted parameters of your peak function (depending on what peak model you choose) This should allow coincident particles to be rejected, and also reject small wiggles that almost look like a particle but are just wandering of the signal.

You should explain, what are you looking for in the signal. What characterises a particle? Are the spikes evidence of a particle, or something else? What specifies the size of a particle? Is it frequency? Tel us little more.

It is really hard to find if you do not know what you are looking for.

hello,

from my experience you may try Wigner Ville algorithm which produces an assessment of the frequency depending on time. So you get a spectrogram with time evolution.

Didier Ferrand

@Saso Tomazic .  The number of particle is calculate from the signal which is nothing but the number of peaks(down ward direction) and the width of  the peak is size of particle. Here peaks are not smooth so how can we take the fft as in fft we are converting a signal to sinusoidal?? i am thinking to smooth the data first than take fourier. will it help??

Dear:

Uing power spectrum  such as:

a) continue wavelet  tranform

b) empirical mode decomposition combined with hilbert spectrum

best regard

I do not understand. If the only thing you need to do is counting the peaks and measure their width, why do you need frequency? Why not just detect peaks, i.e., local maxima, and then measure their widths (maybe at 90% of peak value)?

Taking into account the Prof. Tomazic's proposels and comments I would like to propose to redefine the task in terms of distribution of signal overshoots over some threshold. I mean distribution of the duration of exesses over the slow changing threshold, distribution of the intervals of fixed duration on the number of overshoots.

Use power spectrum,

Take the portion of signal that you want 

Convert it back to time domain 

Many of you should realise these peaks are RANDOM in time