Can I test any frequency with DFT?

Question

Ladies, Gentlemen,

Please let me the subject question, for I see in DFT equations testing frequencies sub-multiples of sampling frequency. Regards.

Answer 1

Yes, what you've observed is correct:

The $N$-point DFT only "compares" a signal to the set of complex oscillations $e^{j2\pi f n}$ for frequencies $f$ that fit an integer number of times into the DFT length $N$.

Any other frequency will not be mapped "sharply" to a single DFT bin, but will have some energy in the surrounding bins and spectral repetitions.

We use windows on the input signal of a DFT to shape that "bleeding" a bit.

It is very common to increase the "resolution" by having a greater $N$. To do that, you simply take $K<N$ samples of your input, and attach $N-K$ zeros, so that you end up with a vector of $N$ samples. That way, you can have a finer "grid" of frequencies.

Answer 2

The relation of a DFT to an actual Fourier transform is tenuous.

To arrive at a DFT starting with a continuous signal, the following steps significantly changing the signal are taken:

a) low-pass filtering is applied in the analog domain to remove any frequency content outside of the frequency range to be examined. If the signal is essentially a lowpass signal already, this should not cause significant changes. This step limits the information content to something representable by sampling. In the frequency domain, it windows a finite section of signals.

b) the signal is sampled at equidistant sampling points, discarding everything else. This fundamentally changes the character of the signal even if not the information content. In the frequency domain, this replicates the windowed section of step a) across the whole spectrum. The replicas are placed at multiples of the sampling frequency.

c) a finite span of samples is taken and everything else thrown away. In the frequency domain, this corresponds with smoothing/lowpass filtering with the limiting "frequency" being the time length of the finite span taken (frequencies and times are inverses and duals).

d) the finite span of samples is replicated across all the time domain. In the frequency domain, this corresponds with sampling the Fourier transform at multiples of the inverse of the finite span length (which is also the replication length) and discarding all intermediate values.

After that process, we have the DFT. Most of these steps can cause serious artifacts and bleedovers in time/frequency domain if the analyzed material does not just consist of sinoids with a period that wholly divides the analysis length. If the assumptions are met, step d) will only discard values that are zero anyway. If not, the combination of the last two steps significantly changes the character of the analysis.

Now FFT is an efficient implementation of a DFT and that is a great building block for a lot of analysis tools as long as you don't confuse it with a finished tool. To create actual analysis tools, one works with windowing, warping, smoothing, binning, and statistics in order to get a reasonable tradeoff between the various artifacts introduced by the operations performed in time and frequency domain by DFT itself as well as the additional tools one groups around it. A naked unadorned DFT tends to have rather bad tradeoffs regarding either changing signal characteristics or bad matches of signal frequency to analysis period.