Consider a signal that is sparse in frequency, but it measured in the time domain, for example (in matlab):
let's say that some fraction of the signal is missing, for example the 120-130 measurement:
using standard interpolation methods (nearest neighbors, spline, etc) will fail to reproduce the missing signal, because these methods do not make use of frequency information of the signal.
I thought that by using compressed sensing (L1-magic for example) and random sampling of the signal outside the missing part I'll be able to recover the signal and the missing part, but looking into it I saw that I still need the full signal to create a measurement vector via
A is the sensing matrix.
NaN values then
y will be all
NaNs, and the entire process is compromised.
So how can I "compress" my sampling by not having the full signal even though it has the information needed? (assuming the sparsity above)...
I'm aware I may use other techniques but I'm curious how to use CS.