# Different ways of decomposing signals into sinusoids

For purposes of generating a spectrogram, the signal can be divided into overlapping chunks, each chunk multiplied by a window function and transformed to frequency domain using Discrete Fourier Transform (DFT).

But isn't DFT just one of infinitely many ways to decompose a finite-length discrete-time signal into a sum of sinusoids? What other methods are there to do such a decomposition? How would this choice of method affect the appearance of the spectrogram?

• The thing is that you are not asking a question, but rather making statements with a question mark at the end. Your statement is false and I have no idea where it is coming from. Apart from "No", there is nothing else to be added unless you improve your question. Being able to formulate a clear question is first step in getting a response. – Pier-Yves Lessard Sep 27 '18 at 3:58
• It is not obvious how to formulate this fundamental question in a way that we gain insight rather than get a trivial answer. At least in case of a finite sequence of uniform samples of a sinusoid that doesn't have the same frequency as any of the bins of the discrete Fourier transform (DFT) of that sequence, the DFT gives an alternate decomposition using the harmonic frequencies of the bins. That's already two possible decompositions. – Olli Niemitalo Sep 27 '18 at 8:42

Setting aside the issue of a Spectrogram or a sequence of windowed Fourier transforms.

A single windowed Fourier Transform can be shown to correspond to a Holder p norm minimization where p=2.

One can also use any p norm. so yes there are many possible Fourier types of compositions but the p=2 Holder norm can be bound in terms of the p=1 and p=infinity norms, and vice verse. This implies that these alternate spectra will not be radically different from the conventional DFT ( for discrete time).

A lot of sparse techniques depend on the zero norm.

The conventional Fourier Transform is a direct calculation while the others require iterative optimization.

In some cases like a sparse expansion, it is worth it but in general, the conventional Fourier transform is satisfactory. Also through Parsevals theorem, the conventional transform has a straightforward physical interpretation in terms of power.

Given the Wiener Khinchin theorem for random processes, the other norms would have seem to have less justification.

In a nutshell, yes you can have these other possible expansions but not without a lot of justification that the conventional Fourier series has.

There are cases where other expansions have been shown to be useful but not as a general approach

There are an infinite number of possible extensions of a discrete sequence of length $$N$$ to length $$N + M.$$ Calculating the DFT of an extended sequence gives one of an infinite number of possible decompositions of the length $$N$$ sequence. However, the number of sinusoids in the decomposition will be greater than what it would be by taking the DFT of the original sequence.

An equivalent arbitrary extension can be done to a continuous-time signal.

A generalized DFT allows to shift the frequencies of the bins by an arbitrary constant, and this gives an infinite number of decompositions into complex exponentials without increasing the decomposition length as compared to DFT. If the input signal is real, a shift of half a bin width gives an alternate separation into sinusoids without a 0 Hz and a Nyquist frequency bin. The generalized DFT can be implemented for example by multiplying the input signal by a complex exponential, thus shifting the frequencies, before regular DFT.