I am looking for a method to choose the minimum number of sampling points N of a function f(t) over a time interval [0, T] to be sure to correctly estimate with a discrete Fourier transform the magnitude and phase of a certain number of harmonics whose frequencies are already known, for instance frequencies f_{i} for i=1 to P (P discrete different frequencies, where f_{1}
Dear Jean
The Sampling theorem is the key answer for your question, if you broke it for computational purposes and acceptable results you have the so called the Aliasing on that case you have to control it. A lot of papers on the research gate of mine you can down load it easily covers this topic.
Good Luck.
Prof. S. EL-Rabaie
Definitely Nyquist criteria (double bandwidth), unless other factors exist that may influence the final result, such as analog filtering prior to sampling (sampling rate may have to be extended if filtering is poor).
As others said, your sampling frequency fs should be higher than double of the maximum frequency fmax. A reasonable choice is fs = 2.1 x fmax.
However, if you expect that your system is non-linear, there might be harmonics of fmax in the response. Therefore it is worthwhile to choose fs more than 4 times higher than fmax. Than you can check whether there is the second harmonic of flax in the response.
As for choosing T, you probably want to capture at least 2 full cycles of your lowest frequency of interest. If this is f_0, then T (final time) needs to be > 2*(1/f_0). Choosing a T and a sampling frequency (which other answers have addressed) will determine the number of sampling points N.
Well, the sampling frequency must be at least the double of the maximum frequency. It's recommended to use a sampling frequency between 4 to 10 times the highest frequency.
The number N of total grid points must be large enough in order to capture your signal in both time and frequency domain with minial losses, i.e., with minimal aliasing in both domains.
For that, it is required that your signal is both time and frequency limited in the sense that most of its energy in concentrated in the intervals [-T/2,+T/2] in time domain and [-B/2,+B/2] in frequency domain. If this is not the case, then your signal might be considered being periodic with period T in time domain or/and with period B in frequency domain which is equivalent of catching your signal in [-T/2,+T/2] and [-B/2,+B/2], respectively.
Now, the sampling theorem must be obeyed twice, in time and frequency domain, and this is done by collecting samples at distances of less than 1/B in time domain and of less than 1/T in frequency domain. It actually corresponds of sampling your signal at distances of less than 1/B in time domain and setting your signal beyond [-T/2,+T/2] or, alternatively, beyond [0,T] to zero. The number N of minimally required samples [f_0, ..., f_{N-1}] is now obtained as a result of choosing 1/B and choosing a suitable truncation [0,T].
The result of sampling a function in both time and frequency domain is a discrete periodic function satisfying N = TB because in time domain we have T = N (1/B) where (1/B) is the sampling step and in frequency domain we have B = N (1/T) where (1/T) is the sampling step. Now, in the discrete periodic case, T becomes N and B becomes 1. The final discrete periodic function is hence N- and 1-periodic simultaneously. (These considerations require using the so-called "unitary ordinary frequency" Fourier transform definition, see Wikipedia).
The entity N = TB is called the time-bandwidth product. So, for large time-bandwidth products, you will need a large N.
There are of course other strategies, non-uniform sampling for example, but they finally all obey twice the sampling theorem. The n-dimensional case is of course fully analog to the one-dimensional case. The n-dimensional sampling theorem is in fact a cartesian product of n times the one-dimensional sampling theorem. One follows twice the sampling theorem in each dimension, i.e., in x, y, z - direction for example.
More: https://www.researchgate.net/publication/325401098_There_Is_Only_One_Fourier_Transform
Desr Jean
based on Nyquist law for sampling, the sampling frequency must be at least twice the maximum available frequency.
Dear Jean,
do you know the following paper?Article Weak Coupling Between Electromagnetic and Structural Models ...
There, a method is presented, where the amplitudes and phases are determined for high frequencies, although the time signal is undersampled. However, this only works if the frequencies are known (e.g. analytically). But, when I understood your question correctly, that is the case in your case.
In the following work an undersampled signal is "reconstructed" with an analytically computed wave-number using a least square method:
Article Computation of Rotating Force Waves in Skewed Induction Mach...
Best regards,
Bernhard
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