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 discrete Fourier transform (DFT) X(k/N), k = − N/2, …, N/2 − 1 (N even) has frequency bins uniformly spaced in the main frequency interval [− 1/2, 1/2] with frequency spacing 1/N.

Let ψ(ν) be a real-valued strictly-increasing differentiable possibly nonlinear function defined in [− 1/2, 1/2] and with values contained in [− 1/2, 1/2]. The process

(4.308)

is a frequency-warped version of x(n). Due to the nonlinear behavior of ψ(·), X(ψ(k/N)) is a DFT of x(n) with nonuniform (or unequal) frequency spacing (Oppenheim et al. 1971), (Oppenheim and Johnson 1972), (Braccini and Oppenheim 1974), (Makur and Mitra 2001), (Franz et al. 2003). That is, spectral analysis with nonuniform frequency spacing of x(n) is equivalent to spectral analysis with uniform frequency spacing of a frequency-warped version of x(n).

Relationship (4.308) describes a linear (time-variant) transformation of x(n) into its frequency-warped version y(n). In the following, the effects of this frequency warping transformation on the nonstationarity properties of x(n) is analyzed with reference to the class of the ACS processes. WSS processes are considered as special case.

Let x(n) be an ACS process with Loève bifrequency spectrum (4.202) with x₁ ≡ x₂ ≡ x and let us denote by the periodic replication with period 1 of ψ(ν), that is, . From (4.308) and (4.202) and reasoning as in the proof of Theorem 4.8.3 (Section 5.9), it follows that

(4.309)

where is the periodic replication with period 1 of ϕ(·), the inverse function of ψ(·), and the variable change property in the argument of the Dirac delta (Zemanian 1987, Section 1.7) is used. From (4.309) it follows that the Loève bifrequency spectrum of y(n) has spectral masses concentrated on a countable set of support curves in the bifrequency plane. That is, y(n) is a SC process. The Loève bifrequency cross-spectrum of y(n) and x(n) is

(4.310)

that is, y(n) and x(n) are jointly SC processes.

In the special case where x(n) is WSS, the set contains the only element . Thus,

(4.311)

that is, y(n) is in turn WSS accordingly with the results of (Franaszek and Liu 1967) and (Liu and Franaszek 1969). However, x(n) and y(n) are jointly SC with Loève bifrequency cross-spectrum

(4.312)

An illustrative example is presented to show the effects of spectral analysis with nonuniform frequency spacing on an ACS process. A sampled PAM signal x(n) with raised cosine pulse with excess bandwith η = 0.85 and symbol period T_p = 4T_s, where T_s is the sampling period is considered. It is a discrete-time ACS process with three cycle frequencies . Its frequency-warped version y(n) is also considered, with frequency warping function ψ(ν) = B_m tan ⁻¹(ν/B_s) which is typical in spectral analysis with non uniform frequency spacing (Oppenheim et al. 1971), (Oppenheim and Johnson 1972), (Braccini and Oppenheim 1974). In the example, B_m = B_s = 0.2 are assumed. In the figures, supports of impulsive functions are drawn as “checkerboard” plots with gray levels representing the magnitude of the density of the impulsive functions. In addition, for notation simplicity, the bifrequency SCD of signals a and b is denoted by . In Figure 4.33, (top) magnitude and (bottom) “checkerboard” plot of the byfrequency spectral correlation density function of x(n) are reported as functions of ν₁ and ν₂. In Figure 4.34 (top) magnitude and (bottom) “checkerboard” plot of the byfrequency spectral correlation density function of y(n) are reported as functions of ν₁ and ν₂. The frequency-warping operation transforms an ACS process into a SC process. The support of the power spectral density (PSD) of the process, which is contained in the main diagonal, remains contained in the main diagonal even if the shape of the PSD is modified. This result is in accordance with the fact that frequency warping transforms WSS processes into WSS processes (Franaszek 1967), (Franaszek and Liu 1967), (Liu and Franaszek 1969).

Figure 4.33 PAM signal x(n) with raised cosine pulse. Bifrequency spectral cross-correlation density as a function of ν₁ and ν₂.(Top) magnitude and (bottom) support of S_xx*(ν₁, ν₂)

Figure 4.34 Frequency-warped version y(n) of a PAM signal x(n) with raised cosine pulse. Bifrequency spectral correlation density as a function of ν₁ and ν₂. (Top) magnitude and (bottom) support of S_yy*(ν₁, ν₂)