Cemali,

This is not a direct answer, but you can easily calculate the effects of fewer waves but looking at the problem this way: The underlying signal is multiplied by a window function in the time domain. In the frequency domain, this is a convolution operation. Just take a simple model of your signal and apply a time domain window, then see how much it spreads out in frequency. A wider window (more waves) will lead to less spreading and more resolution in the frequency domain. Decide how tight you need the spectrum to be and then pick a window just wide enough to provide that level of resolution.

That's just a place to start. Then add some noise to your simple model and see how that affects the results.

Just my two cents.....

I suppose that the interferometric signal is available as a train of equidistant samples. In such case, the most flexible and also precise method for the frequency analysis is the discrete Fourier transform (DFT), the most common algorithm is the fast Fourier transform (FFT). Its output is a discrete frequency spectrum ranging from 0 to the Nyquist frequency, i.e. the half of the sampling rate f_s (f_Nyquist = f_s/2). The frequency spectrum is discrete, the frequency steps have the magnitude of delta_f = 1/T, where T is the length of the acquired data set in seconds.

As Jim Prater wrote in his post, the acquired data set can be observed as a multiplication of the original data multiplied by a square window with a width equal to T. In the frequency domain, you will see a convolution of the original frequency spectrum and the Fourier transform of the square window. The latter one is the sinc function, i.e. a sharp frequency peak (a delta function) in the original spectrum will be converted to a sinc function. The width of such peak is about 2-3 x delta_f, this implies that just by looking at the spectral data, you can determine the signal frequency with a precision of delta_f.

If you have acquired just a few signal periods, the precision is not exciting. If the signal contains, for instance, 3 periods, the frequency spectrum will contain a peak with the maximum at the index of 3, i.e. the frequency is (3±0.5) x delta_f. People usually fit the sinc function to the spectral data and obtain the real frequency with a better precision than ±0.5 x delta_f. There is, however, a much more elegant method that can be used here, it is called zero filling. You simply append zeros to the original data and increase the length of the data set in this way before you run the FFT algorithm. It increases the frequency resolution and improves the precision significantly. The peak does not contain of a few points more so that one can easily find its maximum. One can even fit a parabolic function to the peak around its maximum to further enhance the resolution. What improvement can be reached in practice is dependent on the noise in the acquired signal. If the noise is weak, one can improve the frequency resolution by many orders of magnitude. To get the answer you want, one has to simulate it or use the real data and just look for the results.

There is one point more that has to be mentioned: The sinc function is not the best peak shape that one can imagine. It has a lot of side bands, i.e. a single peak in the original spectrum results in many peaks in the calculated data. The side bands can obscure real peaks in the original spectrum, but if just a single peak is expected, they should not matter much. To improve the peak shape if complex spectra have to be analyzed, dedicated apodization or window functions were developed. One multiplies the data with such function before the FFT is calculated, i.e. one replaces the default square window by another function. One of the best ones is the Blackman-Harris polynomial with 4 terms, its side bands are about 5 orders of magnitude weaker than the main peak, i.e. under normal conditions, they are below the noise floor and therefore, not more visible in the resulting spectrum. Since the apodization generally increases the peak width, it must be carefully chosen since it can decrease the frequency resolution or obscure double peaks.

There are also much simpler methods that can be used for frequency determination but they cannot reach the precision of the Fourier transform. One of the methods is often used by frequency counters, where the time points are analyzed where the signal crosses the zero line or another other level. One can fit a linear function to them and obtain the frequency from the line slope. Since such method uses just a few data points from the whole data set, it is much more sensitive to noise and cannot reach the precision of the FFT which uses all available data. The main drawback of the FFT analysis is its complexity, it requires a lot of computational power and data memory - especially if data sets containing millions of samples have to be converted. For shorter data sets, the required resources are usually not critical and the results are excellent.

Thank you, Ivo Cermak and Jim Prater for your answers. As an additional point of view, I would like to emphasize that only the number of discrete samples of signals generated during the period of the signal plays a role in frequency estimation. For frequency estimation of one harmonical signals you need 4 or more than samples on the period. If you have more signals then you need more samples inside signals period with maximal frequency.

As an example you can see formulas in this patent description.

Patent Method measuring frequency of harmonic oscillations

depends on what accuracy your application requires.

If you know that the time-domain signal is a periodic signal, then its spectrum is discrete, in the sense that the spectrum contains harmonic components only. In such case, the time-domain samples should exactly cover at least one or more complete periods with sufficiently high sampling rate (i.e., at least twice of the maximum significant harmonic frequency). Covering more complete periods is helpful to hit (or, catch) the harmonic frequency bins.

Otherwise, if the time-domain signal is not periodic, then the spectrum is not discrete but continuous (i.e., the spectrum is a continuous function of frequency), though the obtained spectrum maybe digitized (such as obtained by using FFT). In such case, regarding how fine the frequency can be identified, there is a accuracy problem of the (obtained) digitized spectrum represented by the frequency resolution (FR), which equals the reciprocal of the time duration of the time-domain samples. For instance, if FR of 0.1 Hz is required, time duration of the time samples should be 10 seconds.

The issues affecting the magnitude accuracy of the obtained spectrum includes the truncation of the time-domain signal, noise and randomness of the signal samples (if the time-domain signal is a stationary random process). Truncation results in the Gibbs effect which generates rings and ripples in the spectrum. This can be reduced by applying some proper window on the sampled data, if you really care about it. To deal with the effects of noise and randomness of the time samples, you may take multiple independent segments of signal samples, get the periodogram for each segment individually, and then average them. Furthermore, if you like, you may calculate a confidence interval (CI) of the spectrum associated with a required confidence level (CL), to describe the accuracy of the obtained spectrum (that means, the real spectrum will fall within the CI with a probability of 95% if CL is set to 0.95).

Depending on Fourier properties the segment it self have a sinc function spectrum. Sinsoidal function within this segment lead to displace the spectrum to the fundamental frequency of sine, thus for sine and for other periodic it need for a one cycle to see the spectum by Z-Transform that can be an outcome of sinc (conv.) function of frequency, then by (deconv.) it can see signal synthesis but it not easy it may fall in residual range avoid that by putting a header for signal.

Typically, three to five full waves are required to accurately estimate the frequency of a signal segment.