# Is it common to impose the sparsity on the Fourier coefficient itself?

In compressive sensing, I see many works to impose the sparsity on the wavelet coefficients (e.g., by minimizing the L1 norm of such coefficients.) Another example in MRI is to impose sparsity on the gradient of the unknown image (total variation).

However, I'm interested in imposing the sparsity on the Fourier coefficients itself. I couldn't find for the moment. If there is any reference, could anyone let me know? Or, I'm not sure whether it doesn't make sense.

• yes, that's very common in many signal models. Assuming narrowband signals kind of "suggests" such assumptions as "only few occupied DFT bins". – Marcus Müller Jan 10 at 21:38

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal $$f\in \mathbb{C}^N$$ and a randomly chosen set of frequencies $$\Omega$$. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set $$\Omega$$?