A fast frequency domain notch periodogram algorithm

Abstract

The notch periodogram is an algorithm which may be used iteratively for detection and super-resolution frequency estimation of multiple sinusoids in noise (harmonic analysis). A general notch periodogram algorithm has been described for an arbitrary number of notch frequencies; there is also an approximate algorithm for well separated notch frequencies or well separated clusters of notches, which has reduced computation load. However, the computation load of these algorithms is high, especially because almost all the computation is repeated for each iteration. This paper describes frequency domain notch periodogram algorithms which greatly reduce the computation load. After a single FFT of the data, iterated notch periodograms are computed by operations in the frequency domain. When refining notch frequencies, the notch periodogram only has to be computed over a narrow frequency range, and the new algorithms do this efficiently. Further speedups are achieved using new DFT and periodogram interpolation techniques which may be used to reduce the required zero-padding factor, and by an algorithm for fast approximation of the denominator to any specified accuracy. Representative speedup factors from 10 to over 100 are achieved.