4.4 The Bifrequency Cross-Periodogram
In this section, for jointly SC processes, the bifrequency cross-periodogram is defined and its expected value (Lemma 3.1) and covariance (Lemma 3.2) are derived. Moreover, the bifrequency cross-periodogram is shown to be an estimator of the bifrequency spectral cross-correlation density function 
asymptotically biased (Theorem 3.1) and with nonzero asymptotic variance (Theorem 3.2).
Definition 4.4.1 Given two stochastic processes 
and 
, their bifrequency cross-periodogram is defined as
where 
and 
are the short-time Fourier transforms (STFTs) of y(t) and x(t), respectively, defined according to
with Z(f) denoting the (generalized) Fourier transform of z(t) and 
a (1/Δf)-duration data-tapering window whose Fourier transform BΔf(f) has bandwidth Δf.
Assuming 
be an even function, the STFT (4.94a) is a low-pass filtered version of a frequency-shifted version of z(t) with the bandwidth Δf of the low-pass filter approximately equal to the reciprocal of the width of the data-tapering window. Thus, the function of t, 
is the superposition of the spectral components of z(t) within the spectral interval [f − Δf/2, f + Δf/2] (see (4.94b)).
In (Hurd 1973), it is shown that if z(t) is harmonizable, its spectral covariance is of bounded variation (see (4.1)), and 
is a Fourier-Stiltjes transform, then the STFT 
is harmonizable.
In the following, some assumptions are made that allow to interchange the order of expectations, sum, and integral operations in the derivations of the expressions for expected value and covariance of the bifrequency cross-periodogram, the time-smoothed bifrequency cross-periodogram, the frequency-smoothed periodogram, and their asymptotic expressions as the data-record length approaches infinity and the spectral resolution approaches zero.
Assumption 4.4.2 SC Statistics.
and 
are singularly and jointly (second-order) spectrally correlated, that is, for any choice of z1 and z2 in {x, x*, y, y*} it results that
is a countable set.
can be expressed as (Section 4.2.3)
is a countable set and the cumulant of complex random variables is defined according to (Spooner and Gardner 1994; Napolitano 2007a) (see also Section 1.4.2).
The functions 
and 
in (4.95) can always be chosen such that, for m ≠ n, 
at most in a set of zero Lebesgue measure in 
. Analogously, the functions 
and 
in (4.96) can always be chosen such that, for m ≠ n, 
at most in a set of zero Lebesgue measure in 
.
Assumption 4.4.3 Series Regularity.
in (4.95) are almost everywhere (a.e.) continuous, in 
and, moreover, such that
is the essential supremum of S(f) (Champeney 1990).
in 
and, moreover, such that
.
Note that, since the Fourier transform of a summable function is continuous, bounded, and infinitesimal at infinity (Champeney 1990), a sufficient condition assuring Assumption 4.2 a holds, is
where 
is the inverse Fourier transform of 
. Furthermore, a sufficient condition assuring Assumption 4.2 b holds, is
where 
is the inverse Fourier transform of 
.
In the special case of ACS processes, sufficient conditions assuring Assumption 4.2 a holds are derived in (Alekseev 1988). Moreover, (4.99) and (4.100) reduce to the well-known summability conditions on the second-and fourth-order cumulants as in (Hurd 1989a), (Hurd 1991), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Sadler and Dandawatacute; 1998).
Assumption 4.4.4 Support-Curve Regularity (I). For any choice of z1 and z2 in {x, x*, y, y*}, the functions 
in (4.95) are a.e. derivable with a.e. continuous derivatives.
Assumption 4.4.5 Data-Tapering Window Regularity. 
is a (1/Δf)-duration data-tapering window with Fourier transform BΔf(f) such that
(4.101) 
where WB(f), the Fourier transform of 
, is a.e. continuous and regular as |f|→ ∞, 
, and 
. From (4.102), it follows that
(4.104) 
(4.105) 
where δ(f) is Dirac delta and δf is Kronecker delta, (that is, δf = 1 if f = 0 and δf = 0 if f ≠ 0), and the first limit should be intended in the sense of distributions (generalized functions) (Zemanian 1987).
The general case of complex-valued data-tapering windows is considered. In fact, in (Politis 2005) it is shown that, by the adoption of appropriate complex-valued tapers, the bias of spectral estimators can be reduced by orders of magnitude. Moreover, in (Lahiri 2003) it is shown that different asymptotic independence properties of the Discrete Fourier Transform (DFT), and hence of the periodogram, can be obtained with different data-tapers.
By taking the statistical expectation of the bifrequency cross-periodogram (4.93) with the expressions of the STFTs (4.94b) of x(t) and y(t) substituted into, one obtains the following result.
Lemma 4.4.6 Expected Value of the Bifrequency Cross-Periodogram (Napolitano 2003, Lemma 3.1). Let 
and 
be second-order harmonizable jointly SC stochastic processes with bifrequency cross-spectrum (4.15). Under Assumptions 4.2 a (series regularity) and 4.4 (data-tapering window regularity), the expected value of the bifrequency cross-periodogram (4.93) is given by
where
Proof: See Section 5.2.
By expressing the covariance of the bifrequency cross-periodogram in terms of second-order moments and a fourth-order cumulant of the STFTs of x(t) and y(t), the following result is proved.
Lemma 4.4.7 Covariance of the Bifrequency Cross-Periodogram (Napolitano 2003, Lemma 3.2). Let 
and 
be second-order harmonizable zero-mean singularly and jointly SC stochastic processes with bifrequency spectra and cross-spectra (4.95). Under Assumptions 4.1 (SC statistics), 4.2 (series regularity), and 4.4 (data-tapering window regularity), the covariance of the bifrequency cross-periodogram (4.93) is given by
with
where, for notation simplicity, 
, 
, 
, and 
.
Proof: See Section 5.2.
In the following, Theorems 4.4.8 and 4.4.9 provide the asymptotic expected value and covariance, respectively, of the cross-periodogram (4.93) when the spectral resolution Δf → 0, that is, when the data-record length 1/Δf → ∞.
Theorem 4.4.8 Asymptotic Expected Value of the Bifrequency Cross-Periodogram (Napolitano 2003, Theorem 3.1). Let 
and 
be second-order harmonizable jointly SC stochastic processes with bifrequency cross-spectrum (4.15). Under Assumptions 4.4.2a (SC statistics), 4.4.3a (series regularity), 4.4.4 (support-curve regularity (I)), and 4.4 (data-tapering window regularity), the asymptotic (Δf → 0) expected value (4.106) of the bifrequency cross-periodogram (4.93) is given by
where
Proof: See Section 5.2.
The presence of the multiplicative term E(n)(f1) in (4.112) implies that, in general, the cross-periodogram of jointly SC processes is an asymptotically biased estimator of the bifrequency spectral cross-correlation density function 
(see 4.19). However, if the functions 
are known, the bias is known.
In the special case of jointly ACS processes, there is a one-to-one correspondence between indices 
and cycle frequencies 
, 
, 
, and (4.106) reduces to the well known result (Gardner 1987d), (Hurd 1989a), (Hurd and Le
kow 1992a), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Dehay and Lekow 1996), (Sadler and Dandawatacute; 1998)
(4.114) 
where 
is the set of the (conjugate) cycle frequencies of the (conjugate) statistical cross-correlation function of y(t) and x(t). Therefore, the cyclic cross-periodogram 
is an asymptotically unbiased (but for a known scaling factor) estimator of the (conjugate) cross cyclic spectrum 
, provided that the cycle frequency α is perfectly known.
Theorem 4.4.9 Asymptotic Covariance of the Bifrequency Cross-Periodogram (Napolitano 2003, Theorem 3.2). Let 
and be second-order harmonizable zero-mean singularly and jointly SC stochastic processes with bifrequency cross-spectrum (4.15). Under Assumptions 4.1 (SC statistics), 4.2 (series regularity), 4.3 (support-curve regularity (I)), 4.4 (data-tapering window regularity), the asymptotic (Δf → 0) covariance (4.108) of the cross-periodogram (4.93) is given by
where
Proof: See Section 5.2.
From Theorems 4.4.8 and 4.4.9 it follows that the bifrequency cross-periodogram of SC processes is an inconsistent estimator of the bifrequency spectral cross-correlation density function 
.
In the special case of jointly ACS processes, accounting for (1.101)) we have
Then, by substituting (4.118)–(4.121) into (4.116) and (4.117) and the result into (4.115), it follows that (corrected version of (Napolitano 2003, eq. (57)))
in accordance with the results of (Hurd 1989a), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Dehay and Lekow 1996), (Sadler and Dandawatacute; 1998, eq. (13)). In (4.122), α1 and α2
, and E1 = E2 and 
are defined by substituting (4.118)–(4.121) into the expressions of 
.