Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

4.13 Spectral Analysis with Nonuniform Frequency Spacing

Spectral analysis with nonuniform (or unequal) frequency spacing finds applications in several fields such as frequency estimation. This problem has been investigated with reference to deterministic signals in (Oppenheim et al. 1971), (Oppenheim and Johnson 1972), (Braccini and Oppenheim 1974), (Makur and Mitra 2001), (Franz et al. 2003). The nonuniform frequency spacing is obtained by frequency-warping techniques. For this purpose, in (Makur and Mitra 2001) and (Franz et al. 2003) the warped discrete Fourier transform is introduced. In this section, the problem of spectral analysis with nonuniform frequency spacing is addressed for some classes of discrete-time stochastic processes. Spectral analysis with nonuniform frequency spacing of a given process is equivalent to spectral analysis with uniform frequency spacing of a frequency-warped version of the original process. Since frequency-warping is a linear time-variant transformation, this operation modifies the nonstationarity properties of the original stochastic process under analysis. In the notable case of a WSS process, the frequency-warped process is still WSS, but jointly SC with the original process. A frequency warped ACS process is SC and jointly SC with the original ACS process.

(Cross-)spectral analysis techniques of a discrete-time process x(n) based on the Fourier transform X(ν) (4.185) and its inverse

(4.307)

have uniform frequency spacing. That is, the ...