4.5 Measurement of Spectral Correlation –Unknown Support Curves

In this section, for jointly SC processes, the time-smoothed bifrequency cross-periodogram is considered as an estimator of the bifrequency spectral cross-correlation density function when the location of the support curves is unknown. For such an estimator, bias (Lemma 4.5.5) and covariance (Lemma 4.5.6) are determined. Moreover, its asymptotic biasedness and consistency are discussed (Theorems 4.5.7 and 4.5.9 and Corollary 4.5.8) (Napolitano 2001, 2003).

Definition 4.5.1 Given two stochastic processes and , their time-smoothed bifrequency cross-periodogram is defined as

(4.123)

where Y_1/Δf(t, f₁) and X_1/Δf(t, f₂) are the STFTs of y(t) and x(t), respectively, defined according to (4.94a), and a_T(t) is a T-duration time-smoothing window.

Accordingly with the definition of STFT (4.94a), Y_1/Δf(t, f₁) represents the output of a low-pass filter with impulse-response function b_1/Δf(t) (assumed to be an even function) with bandwidth Δf when the input is the frequency-shifted version y(t)e^−j2πf₁t of y(t). An analogous interpretation holds for X_1/Δf(t, f₂). Therefore, the time-smoothed bifrequency cross-periodogram represents the finite-time-averaged cross-correlation (with zero lag) between the spectral components of y(t) and x(t) in the bands (f₁ − Δf/2, f₁ + Δf/2) and (f₂ − Δf/2, f₂ + Δf/2), respectively, normalized to 1/Δf. The spectral cross-correlation analyzer can be realized by frequency shifting y(t) by f₁ and x(t) by f₂, passing such frequency-shifted versions through two low-pass filters h_Δf(t) with bandwidth Δf and unity band-pass height, and then correlating the output signals (Figure 4.5) (h_Δf(t) = b_1/Δf(t) in (4.94a)).

Figure 4.5 Spectral cross-correlation analyzer

From Theorems 4.4.8 and 4.4.9 it follows that, for SC processes, the bifrequency cross-periodogram is an inconsistent estimator (with known bias only if the functions Ψ⁽ⁿ⁾ are known) of the bifrequency spectral cross-correlation density function. For ACS processes it is well known that a frequency-smoothed version of the cross-periodogram provides a consistent estimator of the spectral cross-correlation density function, provided that the cycle frequencies are perfectly known (Gardner 1986a), (Gardner 1987d), (Hurd 1989a), (Hurd and Lekow 1992a), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Gerr and Allen 1994), (Dehay and Lekow 1996), (Sadler and Dandawatacute; 1998). Such a result cannot be extended, in general, to SC processes that are not ACS. In (Allen and Hobbs 1992) and (Lii and Rosenblatt 2002), it is shown that smoothing the cross-periodogram in the frequency domain provides a consistent estimator of the spectral cross-correlation density function if the support curves f₂ = Ψ⁽ⁿ⁾(f₁) are lines (with not necessarily unit slopes). Furthermore, in (Allen and Hobbs 1992) and (Lii and Rosenblatt 2002) it is shown that the proposed frequency-smoothing techniques are effective only if the location of the support lines is known. That is, in general, frequency-smoothing techniques to reduce the variance of the cross-periodogram cannot be adopted if the location of the support curves of the Loève bifrequency spectrum is unknown. This result is in accordance with the more general result of (Gardner 1988a), (Gardner 1991b), where it is shown that reliable (low bias and variance) estimates of statistical functions of nonstationary processes can be obtained only if the nonstationarity is of almost-periodic nature (almost-cyclostationarity) or of known form.

Motivated by the interpretation of the time-smoothed bifrequency cross-periodogram as time-averaged spectral cross-correlation and by the asymptotic equivalence of time-and frequency-smoothing the cross-periodogram in the case of ACS processes (Gardner 1987d), the function is proposed as an estimator of the bifrequency spectral cross-correlation density function for SC processes when the location of the support curves is unknown. Moreover, its asymptotic bias and variance are discussed and a constraint on the reliability of the estimate is found.

Assumption 4.5 Time-Smoothing Window Regularity. a_T(t) is a T-duration time-smoothing window that can be expressed as

(4.124)

with and .

Assumption 4.5.3 Lack of Support-Curve Clusters (I). There is no cluster of support curves. That is, let

(4.125)

then for every η₀ and f₁₀, the set is finite (or empty) and for any no curve can be arbitrarily close to the value η₀ for f₁ = f₁₀. That is, for any η and f₁ it results in

(4.126)

In the special case of ACS processes, Assumption 4.5.3 means that there is no cluster point of second-order cycle frequencies (Dehay and Hurd 1994).

Assumption 4.5.4 Support-Curve Regularity (II). The second-order derivatives of the functions Ψ⁽ⁿ⁾, , exist a.e. and are uniformly bounded.

Lemma 4.5.5 Expected Value of the Time-Smoothed Bifrequency Cross-Periodogram (Napolitano 2003, Lemma 4.1). Let and be second-order harmonizable jointly SC stochastic processes with bifrequency cross-spectrum (4.15). Under Assumptions 4.4.3a (series regularity), 4.4.5 (data-tapering window regularity), and 4.5.2 (time-smoothing window regularity), the expected value of the time-smoothed cross-periodogram (4.123) is given by

(4.127)

where is the Fourier transform of a_T(t).

Proof: See Section 5.3.

Lemma 4.5.6 Covariance of the Time-Smoothed Bifrequency Cross-Periodogram (Napolitano 2003, Lemma 4.2). Let and be second-order harmonizable zero-mean singularly and jointly SC stochastic processes with bifrequency cross-spectrum (4.15). Under Assumptions 4.4.2 (SC statistics), 4.4.3 (series regularity), 4.4.5 (data-tapering window regularity), and 4.5.2 (time-smoothing window regularity), the covariance of the time-smoothed cross-periodogram (4.123) is given by

(4.128)

where

(4.129)

(4.130)

(4.131)

where for notation simplicity.

Proof: See Section 5.3.

Theorem 4.5.7 Asymptotic Expected Value of the Time-Smoothed Bifrequency Cross-Periodogram (Napolitano 2003, Theorem 4.1). Let and be second-order harmonizable jointly SC stochastic processes with bifrequency cross-spectrum (4.15). Under Assumptions 4.4.3a (series regularity), 4.4.4 (support-curve regularity (I)), 4.4.5 (data-tapering window regularity), and 4.5.2 (time-smoothing window regularity), the asymptotic (T→ ∞, Δf → 0, with TΔf→ ∞) expected value (4.127) of the time-smoothed cross-periodogram (4.123) is given by

(4.132)

where if g(f) = 0 in a neighborhood of f and otherwise ( also if g(ν) = 0 for ν = f but g(ν) ≠ 0 for ν ≠ f). [In (Napolitano 2003, Theorem 4.1), is erroneously indicated by δ].

Proof: See Section 5.3.

In (4.132) the order of the two limits cannot be interchanged. In fact, only for T→ ∞ and Δf → 0 with TΔf→ ∞ the asymptotic variance vanishes (Theorem 4.5.9).

From Theorem 4.5.7 it follows that the asymptotic expectation of the time-smoothed cross-periodogram can be nonzero only for those (f₁, f₂) such that f₂ = Ψ⁽ⁿ⁾(f₁) with Ψ⁽ⁿ⁾ having unit slope in a neighborhood of f₁ for some . Thus, in the case of SC processes that are not ACS, the time-smoothed cross-periodogram is an asymptotically biased estimator of the bifrequency spectral cross-correlation density function (4.19). Such a result is in accordance with that derived in (Lii and Rosenblatt 2002) for the Daniell-like estimate in the special case of spectral support constituted by lines.

For jointly ACS processes, if (*) is present and otherwise. Thus, the second Kronecker delta in (4.132) is always unity and the time-smoothed cross-periodogram is an asymptotically unbiased estimator of (but for a known multiplicative factor depending on the data-tapering and time-smoothing windows). More specifically, for Ψ⁽ⁿ⁾(f₁) = (−)(α − f₁) and , (4.132) reduces to the well-known result (Gardner 1987d), (Hurd 1989a), (Hurd and Lekow 1992a), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Sadler and Dandawatacute; 1998)

(4.133)

In the following, the bias of the time-smoothed bifrequency cross-periodogram is analyzed in points belonging to the neighborhoods of the support curves where the slope of the curve is not too far from unity and, hence, accounting for Theorem 4.5.7, a small bias could be expected for finite T and Δf, provided that, however, T is sufficiently large and Δf sufficiently small.

Corollary 4.5.8 (Napolitano 2003, Corollary 4.1). Let and be second-order harmonizable jointly SC stochastic processes with bifrequency cross-spectrum (4.15). Under Assumptions 4.4.3a (series regularity), 4.4.4 (support-curve regularity (I)), 4.4.5 (data-tapering window regularity), 4.5.2 (time-smoothing window regularity), 4.5.3 (lack of support-curve clusters (I)), and 4.5.4 (support-curve regularity (II)), if at the point (f₁, f₂) of the bifrequency plane it results that

(4.134)

and

(4.135)

for , then

(4.136)

Proof: See Section 5.3.

Corollary 4.5.8 means that nonzero contribution to the expected value of the time-smoothed bifrequency cross-periodogram in the point (f₁, f₂) is given only by spectral densities S⁽ⁿ⁾(f₁) corresponding to support curves such that the value of Ψ⁽ⁿ⁾(f₁) is close to f₂ (in the sense of (4.134)) and the slope in (f₁, f₂) is close to ±1 (in the sense of (4.135)).

Theorem 4.5.9 Asymptotic Covariance of the Time-Smoothed Bifrequency Cross-Periodogram (Napolitano 2003, Theorem 4.2). Let and be two second-order harmonizable zero-mean singularly and jointly SC stochastic processes with bifrequency cross-spectrum (4.15). Under Assumptions 4.4.2 (SC statistics), 4.4.3 (series regularity), 4.4.4 (support-curve regularity (I)), 4.4.5 (data-tapering window regularity), and 4.5.2 (time-smoothing window regularity), the asymptotic (T→ ∞, Δf → 0, with TΔf→ ∞) covariance (4.128) of the time-smoothed cross-periodogram (4.123) is such that

(4.137)

where

(4.138)

(4.139)

with

(4.140)

(4.141)

(4.142)

Proof: See Section 5.3.

In the special case of singularly and jointly ACS processes, by substituting (4.118)–(4.121) into (4.129)–(4.131), one obtains the known result (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Sadler and Dandawatacute; 1998, eq. (23)) (corrected version of (Napolitano 2003, eq. (76)))

(4.143)

where α₁ and α₂,

with and given by (4.141) and (4.142), respectively, with (4.118)–(4.121) substituted into.

From Theorem 4.5.9 it follows that, even if

is not regular as T→ ∞ and Δf → 0 with TΔf→ ∞, the result is that

(4.144)

where denotes the “big oh” Landau symbol. Therefore, for the variance of the estimate (4.123) is (1/TΔf). Moreover, from Corollary 4.5.8 one has that the bias at the point (f₁, f₂) with f₂ = Ψ⁽ⁿ⁾(f₁) is negligible, provided that the condition

(4.145)

is satisfied. Thus, in the case of departure of the nonstationarity from the almost-cyclostationarity, that is, different from 1 for some and some f₁, the variance cannot be made arbitrarily small (e.g., augmenting the collect time T) without obtaining a significant bias. On the contrary, when the stochastic processes x(t) and y(t) are jointly ACS, it results in ∀n and ∀f₁ and, hence, (4.145) is always satisfied allowing one to obtain an estimator of the bifrequency spectral cross-correlation density function which is mean-square consistent (that is, asymptotically unbiased and with vanishing asymptotic variance). Thus, there exists a trade-off between the departure of the nonstationarity from the almost-cyclostationarity and the estimate accuracy obtainable by single sample-path measurements. The larger is such a departure, that is, the larger is the left-hand side of (4.145) for some f₁, the smaller is the allowed value of TΔf and, hence, the larger is the estimate variance. Note that, such a trade-off is not the usual bias-variance trade-off as for (stationary and) ACS processes where, for a fixed data-record length, the variance of the cyclic spectrum estimate can be reduced by augmenting the bandwidth of the frequency-smoothing window paying the price of a bias increase. Moreover, unlike the case of ACS processes, for SC processes the estimator performance cannot be improved as wanted by increasing the data-record length and the spectral resolution.

It is worthwhile emphasizing that, in the special case where the Ψ⁽ⁿ⁾(f₁) are known linear functions, the frequency-smoothing based techniques presented in (Allen and Hobbs 1992) and (Lii and Rosenblatt 2002) provide consistent estimators of the spectral cross-correlation density functions. However, note that the proposed estimator (4.123), unlike the estimators in (Allen and Hobbs 1992) and (Lii and Rosenblatt 2002), can be adopted when the functions Ψ⁽ⁿ⁾(f₁) are not necessarily linear. Moreover, the estimator (4.123) does not require the a priori knowledge of the shape and location of the support curves f₂ = Ψ⁽ⁿ⁾(f₁). The paid price for such a lack of knowledge is the absence of asymptotic unbiasedness and consistency of the estimator. This is in accordance with the result stated in (Gardner 1988a), (Gardner 1991b), where it is shown that spectral correlation measurements can be reliable only when the nonstationarity is of almost-periodic nature (almost-cyclostationarity) or known form. The results of this section establish quantitatively how accurate can be spectral correlation measurements made by a single sample-path of a spectrally correlated stochastic process.

In the case of ACS processes, the lack of knowledge of the location of the support lines is equivalent to the fact that cycle frequencies are unknown. In such a case, asymptotically unbiased estimates of the cyclic spectra cannot be obtained. For ACS processes, however, from Theorems 4.5.7 and 4.5.9 it follows that the estimate of the whole bifrequency spectral cross-correlation density function performed by the time-smoothed cross-periodogram is mean-square consistent. In such a case, in the limit as T→ ∞ and Δf → 0 with TΔf→ ∞, the regions of the bifrequency plane where the time-smoothed cross-periodogram is significantly different from zero tend to the support lines and, hence, the unknown cycle frequencies can potentially be estimated. From Theorems 4.5.7 and 4.5.9 it follows that an analogous result does not hold for SC processes that are not ACS.

Finally, note that the theorems on the behavior of bias and variance of the estimators (4.93) and (4.123) can be extended to estimate the density of the impulsive term in the Loève bifrequency spectrum of processes exhibiting joint spectral correlation (Definition 4.2.5), provided that the continuous terms are bounded in for any choice of z₁ and z₂ in {x, x^*, y, y^*} and the continuous term in cum{Y(f₁), X^(*)(f₂), Y^*(f₃), X^(*)*(f₄)} is bounded in .