I am multiplying my time domain sparse signal,x with orthonormal matrix,i.e.,Y=dftmtx(N)*x;but in frequency domain this signal,Y is not sparse.
This is not surprising. A signal that is sparse in the time domain does not have to be sparse in the frequency domain. And a sparse signal that is sparse in the frequency domain does not have to be sparse in the time domain. In fact, one could even say that a signal cannot be too sparse in both domains simultaneously. For example, if we have a discreet signal of length n containing only k frequencies, then the (inverse) Fourier transform satisfies a linear recurrence relation of degree k, which implies that the signal cannot have k consecutive zeroes in the time signal. So at least n/k entries in the time domain have to be nonzero. For the discrete cosine transform one can give similar arguments.
Respected Sir,
According to the definition of a sparse signal as given below:
A signal is called sparse when it is represented by a small number of nonzero coefficients in any convenient domain. More specifically, N×1 signal X is
called K-sparse if X is represented by the multiplication of any N×N transform matrix Y and N×1 coefficient vector s, and s has only K nonzero coefficients. The definition of sparsity is not limited to the orthogonal transform domain (i.e., Fourier transform or wavelet transform).
So,I considered NxN transform matrix to be DFT and multiplied with a Nx1 signal that has a only 5 nonzero coefficients (K=5) which should result in K-sparse signal in other domain. But, this is not representing as a sparse signal in my case.
I am not sure where that definition came from, but I think what was meant is: A signal is sparse if it has few nonzero entries in some convenient domain (not every convenient domain). So signal that has few nonzero entries in the time domain could be called sparse, and a signal that has only few frequencies could also be sparse. But to be more accurate one should say in which domain the signal is sparse. Also, the definition is a bit vague because what is a "convenient domain". It is probably in the eye of the beholder. For a given nonzero signal, one can always choose a basis such that the signal is sparse with respect to that basis. And one can also choose a basis such that that signal has no nonzero entries when expressed in that basis (domain).
Doctor Harm Derksen is correct. A sparse signal (in one domain) does not have to be sparse in any domain.
For instance, denote by "s" the time-domain signal (vector) known as the kronecker delta (it has only one nonzero coefficient in the time domain). If you take its Fast Fourier Transform (FFT), then you will obtain a vector z as
z = F s ,
where F is the FFT matrix and s is a vector in which only the i-th entry is nonzero and equal to 1. Therefore, z is equal to the i-th column of matrix F, whose elements are all different from zero. Therefore, a signal which is sparse in the time domain is not sparse in the frequency domain.
In addition, observe that the example I gave above is related to the Heisenberg's uncertainty principle or Gabor's limit, which states that a signal cannot be limited in both time and frequency domain.
You can check the following link for a brief explanation: https://en.wikipedia.org/wiki/Uncertainty_principle#Harmonic_analysis
Respected Sir,
In the following paper that I have attached, it was mentioned in Section 5.2. Simulation example 2: arbitrary sparsity basis that the channel impulse response (CIR) W is sparse in a sparsity domain denoted by Ψ, i.e.,WΨ=ΨW, where Ψ be the NxN orthonormal matrix.
For example, a CIR of length N=16 with the sparsity level of S=2 is being estimated which is sparse in the discrete cosine transform (DCT) domain (given below equation 53- page.no.77).
But in your paper it was shown that DCT doesn’t preserve the sparsity after transformation.
Sir, Can you please clarify then how CIR is made sparse in DCT domain in this attached paper?
Dear Rakesh Pogula,
Observe the subsequent phrase of the paper: "The positions of nonzero taps in the DCT domain are chosen randomly.", which means that the channel was generated directly in the DCT domain and then converted back to time domain. Observe also that the time-domain representation of the channel is not sparse!
Therefore, DCT does not preserve sparsity, meaning that a signal that is sparse in one domain (e.g., time or frequency) will not be sparse in the other domain (e.g., frequency or time).
Best.
Respected Sir,
From the explanation given so far, can i believe that W should be a non-sparse vector in time domain which is multiplied with N*N DCT matrix to obtain WΨ which is a sparse vector in DCT domain, where WΨ=ΨW.
This implies W to be non-sparse in time domain and WΨ to be sparse in the DCT domain.
Is my understanding right?
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