The 2D MFFT , SFFT , and 2D SDFT  are chosen for the existing fast DFT/FFT algorithms.
We investigated the stability of the proposed VR-2 x 2 SFFT algorithm using a complex test signal, which was zero-mean Gaussian noise with a standard deviation of one.
n] values of the 1D mSDFT + 1D FFT, 1D gSDFT + 1D FFT, and VR-2 x 2 SFFT are presented in Figure 9.
In this letter, a new stable SFFT based on the VR-2 x 2 FFT algorithm was presented for 2D input data.
Caption: FIGURE 9: Numerical errors of the 1D mSDFT + 1D FFT and the VR-2 x 2 SFFT algorithm for 106 iterations.
The sFFT algorithm chooses at random the spectral permutation parameters a and a from a uniform distribution, then the spectral permutation with these pseudo-random parameters is related to a pseudo-random sampling scheme ,.
2 Flat filtering window The flat window is a mathematical tool that allows to reduce the FFT size from N to B points, this is accomplished by extending a flat passband region of width N/B around each sparse component; this approach replaces the filter bank of previous sFFT algorithms ,; additionally, the flat window avoids the use of non equispaced data FFTs .
5, the hash can be zero which could reduce the capability of the sFFT algorithm to locate a sparse component.
The sFFT algorithm, presented in  and described in Alg.
In this section we describe the Modified Nearly Optimal sFFT algorithm and its optimized software implementation named sFFT-4.
1 Description of the modified nearly optimal sFFT algorithm
The modified sFFT algorithm is accomplished by designing a different flat window and by performing several modifications to the procedures described in section 2.