So I have signals for the form:
$$x_{k}(t) = \sum_{n=0}^{N} a_{k,n} \cdot \delta(t - nT_k)$$
that I receive as a superposition:
$$x(t) = \sum_{k=0}^{K} \sum_{n=0}^{N} a_{k,n} \cdot \delta(t - nT_k)$$
We know that (if we let $N \rightarrow \infty$) the Fourier transforms of the $x_k(t)$ are sparse, and therefore so would that of $x(t)$. In my application, my $N$ is fairly large so I say this holds rather well.
Question:
I want to compute a discrete Fourier transform that encourages the result to be as sparse as possible.
Can I use the sparse Fourier transform algorithm to compute the FFT of these signals? Will it produce a spectrum that is sparse in this case?
The sparse Fourier transform seems to rely on the OFDM trick, and, since the majority of my samples are zero, this isn't going to work. Should I just use a L1 regularization technique like the LASSO instead? Or should I just add a bit of noise to the signals to prevent the catastrophic numerical errors?
Thanks!
Clarification:
Note that I really just want an 'clean' FFT because I'm doing all of this on a computer. Even though I am constrained to set a fixed length for my FFT, the FFT of these signals, after the proper resampling, gives me an approximate pulse train contaminated with noise. An NUFFT gives me something similar. I want to know if I can reduce this noise using regularization or the sparse Fourier transform algorithm.
