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Quadratic time-frequency distributions
6 Quadratic time-frequency distributions Contents 6.1 Introduction ........................................................................................................ 255 6.2 Finite-energy signals: correlations and spectra ......................................................... 256 6.3 Spectrogram ........................................................................................................ 256 6.3.1 Expected value of spectrogram of ACS signals .................................................. 257 6.4 Quadratic TFDs .................................................................................................... 258 6.4.1 Expected values of quadratic TFDs of ACS signals.............................................. 259 6.5 Filtered quadratic TFDs ......................................................................................... 260 6.5.1 Kernels ....................................................................................................... 260 6.6 Filtered quadratic TFDs of ACS signals..................................................................... 261 6.6.1 Expected values of filtered quadratic TFDs of ACS signals .................................. 263 6.7 Cohen’s class of bilinear (or quadratic) time-frequency distributions ........................... 264 6.8 Proofs ................................................................................................................. 265
Abstract Quadratic time-frequency distributions for almost-cyclostationary signals are linked to the second-order cyclic statistics in time and frequency domains. Moreover, a link is established with the estimators of these statistics. Keywords Time-frequency distribution, Wigner-Ville distribution, Spectrogram, Kernels, Cohen class
6.1 Introduction In this chapter, results for second-order ACS signals are reinterpreted in terms of quadratic time-frequency distributions (TFDs). Time-frequency distributions have been initially studied for finite-energy deterministic signals (Boashash, 2015, Chaps. 1–3). Definitions and properties can be extended to finite-power signals considering finite-time windowing of these signals. In addition, if signals are modeled as stochastic processes, definitions originally given for deterministic signals also include an ensemble average to remove randomness. For cyclostationary Cyclostationary Processes and Time Series. https://doi.org/10.1016/B978-0-08-102708-0.00017-0 Copyright © 2020 Elsevier Ltd. All rights reserved.
255
256
Cyclostationary Processes and Time Series
and almost-cyclostationary signals the ensemble average can be replaced by the almostperiodic component extraction operator (Definition 2.14) if the analysis is made in the functional or fraction-of-time approach. The presence of time windowing naturally leads to establish relationships between quadratic time-frequency distributions of (windowed) ACS signals and estimators of second-order cyclic statistics (Section 5.2). An alternative approach consists in using for finite-power signals the same definitions adopted for finite-energy signals. In such a case, Dirac deltas appear in the expressions of time-frequency distributions. Time-frequency distributions for GACS signals are derived in (Izzo and Napolitano, 2005, Sec. 5) in the fraction-of-time approach. Moments of a suitable definition of instan˘ taneous frequency for ACS processes are derived in (Zivanovi´ c, 1991).
6.2 Finite-energy signals: correlations and spectra Let xi (t) ∈ L2 (R), i = 1, 2 complex-valued signals. Their asymmetric cross-ambiguity function is defined as Ax1 x2 (ν, τ ) x1 (t + τ ) x2 (t) e−j 2πνt dt (6.1a) R = X1 (f ) X2 (ν − f ) ej 2πf τ df (6.1b) R
where Xi (f ) is the Fourier transform of xi (t) and the second equality is proved in Section 6.8. If x2 (t) is replaced by x2∗ (t), then X2 (f ) must be replaced by X2∗ (−f ). The spectral cross-moment is defined as (6.2) Ex1 x2 (ν, f ) Ax1 x2 (ν, τ ) e−j 2πf τ dτ = X1 (f ) X2 (ν − f ) . R
The symmetric cross-ambiguity function is defined as x1 (t + τ/2) x2 (t − τ/2) e−j 2πνt dt A¯ x1 x2 (ν, τ ) R x1 (t + τ ) x2 (t ) e−j 2πν(t +τ/2) dt =
(6.3a)
R
= Ax1 x2 (ν, τ ) e−j πντ
(6.3b)
where the variable change t + τ/2 = t + τ is used.
6.3 Spectrogram Definition (1.31) of short-time Fourier transform (STFT) can be generalized by considering a data-tapering window bT (t) with support [−T /2, T /2]. That is, XT (t, f ) x(u) bT (u − t) e−j 2πf u du (6.4a) R
Chapter 6 • Quadratic time-frequency distributions =
R
x(t + s) bT (s) e−j 2πf (t+s) ds
257
(6.4b)
where in the second equality the variable change s = u − t is used. The (conjugate) spectrogram of the signal x(t) is defined as (∗) Sx (t, f ) XT (t, f ) XT (t, f )
(6.5)
where subscript x ≡ xx (∗) . If the complex conjugation is present, we have the spectrogram 2 Sxx ∗ (t, f ) |XT (t, f )|
(6.6)
.
6.3.1 Expected value of spectrogram of ACS signals 6.3.1.1 Stochastic approach Let x(t) be an ACS process. The expected value of the (conjugate) spectrogram is given by (Section 6.8) κf −α E {Sx (t, f )} = Rxα (τ ) rb (τ ) e−j 2πf τ dτ ej 2π(α−κf )t (6.7a) α∈A R
=
α∈A
Sxα (f ) ⊗ f
(∗) B 1 (f ) B 1 ((−)(β − f ))
where κ 1 + (−)1, β rb (τ )
T
T
ej 2π(α−κf )t β=κf −α
bT (s + τ ) bT (s) e−j 2πβs ds (∗)
R
and
B 1 (f )
R
T
(6.7b)
(6.8)
bT (t) e−j 2πf t dt .
(6.9)
If the complex conjugation is present, one has
E |XT (t, f )|
2
= =
Rxα (τ ) rb−α (τ ) e−j 2πf τ dτ ej 2παt
(6.10a)
Sxα (f ) ⊗ B 1 (f ) B ∗1 (α + f ) ej 2παt
(6.10b)
α∈A R
f
α∈A
T
T
The window bT (t) has duration T . Thus, its Fourier transform B 1 (f ) has approximate T bandwidth 1/T . Therefore, for |α| > 1/T the product B 1 (f ) B ∗1 (α + f ) is negligible. If T T
T
is large, then large cycle frequencies do not give significant contribution in (6.10b). If T is small, then the convolution by B 1 (f ) B ∗1 (α + f ) significantly modifies the shape of Sxα (f ). T
T
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Cyclostationary Processes and Time Series
6.3.1.2 Fraction-of-time approach Let us consider the lag product decomposition (Section 2.3.1.6) x(t + s1 ) x (∗) (t + s2 ) = E{α} x(t + s1 ) x (∗) (t + s2 ) + x (t; s1 , s2 )
(6.11)
with E{α} {·} almost-periodic component extraction operator (Definition 2.14) and x (t; s1 , s2 ) residual term non containing any finite-strength additive sine-wave component
x (t; s1 , s2 ) e−j 2παt = 0 ∀α ∈ R (6.12) t
From (6.11), and accounting for the FOT counterparts of (6.48) and (6.50), one obtains the (conjugate) spectrogram decomposition {α} {Sx (t, f )} + ηx (t, f ) Sx (t, f ) = E
= =
κf −α
α∈A R
Rxα (τ ) rb
(∗) Sxα (f ) ⊗ B 1 (f ) B 1 ((−)(β − f )) f
α∈A
where
ηx (t, f )
R2
T
T
β=κf −α
(6.13a)
ej 2π(α−κf )t + ηx (t, f )
x (t; s1 , s2 ) bT (s1 ) bT(∗) (s2 ) e−j 2πf s1 e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t
with
(τ ) e−j 2πf τ dτ ej 2π(α−κf )t + ηx (t, f )
ηx (t, f ) e−j 2παt
=
x (t; s1 , s2 ) e−j 2π(α+κf )t
t R2 (∗) bT (s1 ) bT (s2 ) e−j 2πf s1 e−(−)j 2πf s2
(6.13b)
(6.14)
t
ds1 ds2 = 0
∀α ∈ R
(6.15)
The Fourier coefficients of the almost-periodic component of Sx (t, f ) are given by ⎧ ⎪ ⎨ S α (f ) ⊗ B 1 (f ) B (∗) ((−)(β − f ))
γ ∈A 1 x −j 2π(γ −κf )t f T (6.16) Sx (t, f ) e = T β=κf −α ⎪ t ⎩ 0 γ ∈ A
6.4 Quadratic TFDs Let x(t) be a complex-valued continuous-time finite-power deterministic signal. For complex-valued signals, both lag-product waveforms with conjugated and not conjugated second term should be considered for a complete characterization (Section 1.3), (Schreier and Scharf, 2003b), (Schreier and Scharf, 2010), (Adali et al., 2011). In the following, only the case of conjugated second term of the lag product is considered. The case of not conjugated second term can be easily derived accounting for the guidelines in Section 1.3.
Chapter 6 • Quadratic time-frequency distributions
Kx (t, τ )
t ↔ν Ax (ν, τ ) τ ↔f
259
τ ↔f
Wx (t, f )
t ↔ν
kx (ν, f )
FIGURE 6.1 Quadratic TFDs.
According to the definitions given in (Boashash, 2015, Sec. 3.2.1) for finite-energy signals and those given in (Boashash, 2015, Sec. 9.4) for finite-power stochastic processes, the following functions can be defined, where Fourier transforms must be intended in the sense of distributions (generalized functions) (Gel’fand and Vilenkin, 1964, Chap. 3), (Henniger, 1970). Their mutual relationships are illustrated in Fig. 6.1. Symmetric second-order lag-product or instantaneous autocorrelation function Kx (t, τ ) x(t + τ/2) x ∗ (t − τ/2)
(6.17)
Symmetric ambiguity function Ax (ν, τ )
R
Kx (t, τ ) e−j 2πνt dt
(6.18)
Kx (t, τ ) e−j 2πf τ dτ
(6.19)
(Symmetric) Wigner distribution Wx (t, f )
R
Symmetric spectral autocorrelation function kx (ν, f ) Kx (t, τ ) e−j 2πνt e−j 2πf τ dt dτ R
(6.20)
6.4.1 Expected values of quadratic TFDs of ACS signals Let x(t) be a complex-valued continuous-time ACS stochastic process. In such a case, the quadratic TFDs defined in Section 6.4 are random functions. In the following, their expected values are expressed in terms of cyclic autocorrelation functions and cyclic spectra. Symmetric autocorrelation function Rx (t, τ ) E {Kx (t, τ )} = Rxα (τ ) e−j πατ ej 2παt α∈A
(6.21a) (6.21b)
260
Cyclostationary Processes and Time Series
where Rxα (τ ) e−j πατ
1 = lim T →∞ T
T /2 −T /2
Rx (t, τ ) e−j 2παt dt
(6.22)
is the symmetric cyclic autocorrelation function (1.34). Expected value of the symmetric ambiguity function (Napolitano, 2015b, Eq. (9.6.15)) E {Ax (ν, τ )} Rx (t, τ ) e−j 2πνt dt (6.23a) R = Rxα (τ ) e−j πατ δ(ν − α) (6.23b) α∈A
Expected value of the Wigner distribution (Martin and Flandrin, 1985), (Amin, 1992), (Flandrin, 1999), (Antoni, 2007a), (Napolitano, 2015b, Eq. (9.6.14)) E {Wx (t, f )} Rx (t, τ ) e−j 2πf τ dτ (6.24a) R Sxα (f + α/2) ej 2παt (6.24b) = α∈A
where Sxα (f + α/2) is the symmetric cyclic spectrum (1.35) which is the Fourier transform of the symmetric cyclic autocorrelation function (6.22). Spectral autocorrelation function or rotated Loève bifrequency spectrum (Section 6.8) {k E x (ν, f )} Rx (t, τ ) e−j 2πνt e−j 2πf τ dt dτ (6.25a) 2 R (6.25b) = E X(f + ν/2) X ∗ (f − ν/2) = Sxα (f + α/2) δ(ν − α) (6.25c) α∈A
6.5 Filtered quadratic TFDs 6.5.1 Kernels In this section, the finite-time counterpart of the almost-periodic component extraction operator (Definition 2.14) is considered as time-lag kernel for a given ACS signal x(t). Then, the other kernels are derived by Fourier transformations. Their mutual relationships are illustrated in Fig. 6.2. Let v(τ ) a unit-width lag window with Fourier transform V (f ) such that v(0) = R V (f )df = 1. Thus, lim f →0 V (f/ f )/ f = δ(f ). For example, v(τ ) = rect(τ ) ↔ V (f ) = sinc(f ). Let A be the countable set of second-order cycle frequencies of x(t).
Chapter 6 • Quadratic time-frequency distributions
t ↔ν g(ν, τ ) τ ↔f
G(t, τ )
261
τ ↔f
γ (t, f )
t ↔ν
G(ν, f )
FIGURE 6.2 Kernels.
The following kernels can be defined. Time-lag kernel
t 1 ej 2παt v( f τ ) rect T T
G(t, τ ) =
(6.26)
α∈A
Doppler-lag kernel
G(t, τ ) e−j 2πνt dt R sinc((ν − α)T ) v( f τ ) =
g(ν, τ )
(6.27a) (6.27b)
α∈A
Time-frequency kernel
G(t, τ ) e−j 2πf τ dτ 1 t j 2παt 1 f e = rect V T T f f
γ (t, f )
R
(6.28a) (6.28b)
α∈A
Doppler-frequency kernel
G(t, τ ) e−j 2πνt e−j 2πf τ dt dτ 1 f sinc((ν − α)T ) = V f f
G(ν, f )
R2
(6.29a) (6.29b)
α∈A
The presence of the window v( f τ ) is not crucial in the definitions of kernels G(t, τ ) x (ν, τ ) (see Section 6.6) at values of |τ | larger x (t, τ ) and A and g(ν, τ ). It will set to zero K than 1/2 f since for these values of |τ | the variance of the estimators is large. In contrast, x (t, f ) and V (f/ f )/ f is important to get consistency of estimators W kx (ν, f ).
6.6 Filtered quadratic TFDs of ACS signals Filtered quadratic TFDs for generic stochastic processes are defined according to (Boashash, 2015, Secs. 3.2.2–3.2.4). Their mutual relationships are illustrated in Fig. 6.3. In the following, their expressions for ACS processes are presented.
262
Cyclostationary Processes and Time Series
x (t, τ ) = Kx (t, τ ) ⊗ G(t, τ ) K t ↔ν x (ν, τ ) = Ax (ν, τ ) g(ν, τ ) A τ ↔f
t
τ ↔f x (t, f ) = Wx (t, f ) ⊗⊗ γ (t, f ) W t f
t ↔ν
kx (ν, f ) = kx (ν, f ) ⊗ G(ν, f ) f
FIGURE 6.3 Filtered quadratic TFDs.
The smoothed 2nd-order lag product or smoothed instantaneous autocorrelation function is an estimator of the symmetric autocorrelation function Rx (t, τ ). For ACS processes it is given by (Napolitano, 2015b, Eq. (9.6.17)) (Section 6.8) x (t, τ ) Kx (t, τ ) ⊗ G(t, τ ) K t (T ) = Rx (α, τ ; t) ej 2παt v( f τ )
(6.30a) (6.30b)
α∈A
where Rx(T ) (α, τ ; t)
1 T
t+T /2
x(u + τ/2) x ∗ (u − τ/2) e−j 2παu du
(6.31)
t−T /2
denotes here the symmetric cyclic correlogram of x(t). Under mild regularity assumptions on the finite or practically finite memory of x(t) expressed in terms of summability of cumulants, the cyclic correlogram is a mean-square consistent (as T → ∞) estimator of the √ cyclic autocorrelation function. Moreover, T [Rx(T ) (α, τ ; t) − Rxα (τ )] is zero-mean asymptotically complex normal (Section 5.2.1). The symmetric generalized ambiguity function is given by x (ν, τ ) = Ax (ν, τ ) g(ν, τ ) A = x(t + τ/2) x ∗ (t − τ/2) e−j 2πνt dt sinc((ν − α)T ) v( f τ ) R
(6.32a) (6.32b)
α∈A
The smoothed Wigner distribution is an estimator of the Wigner distribution Wx (t, f ). For ACS processes it is given by (Napolitano, 2015b, Eq. (9.6.18)) x (t, τ ) e−j 2πf τ dτ (6.33a) Wx (t, f ) K R 1 f (6.33b) V ej 2παt = Ix(T ) (α, f ; t) ⊗ f f f α∈A
where Ix(T ) (α, f ; t)
1 XT (t, f + α/2) XT∗ (t, f − α/2) T
(6.34)
Chapter 6 • Quadratic time-frequency distributions
is the symmetric cyclic periodogram of x(t) with t+T /2 XT (t, f ) x(u) e−j 2πf u du
263
(6.35)
t−T /2
short-time Fourier transform (STFT) of x(t). It results (Lemma 2.39) Ix(T ) (α, f ; t) = Rx(T ) (α, τ ; t) e−j 2πf τ dτ . R
(6.36)
Expression (6.33b) can be equivalently obtained as x (t, f ) = Wx (t, f ) ⊗⊗ γ (t, f ) . W t f
(6.37)
(T )
The cyclic periodogram Ix (α, f ; t) is an asymptotically (as T → ∞) unbiased but not consistent estimator of Sxα (f + α/2) (Section 5.2.2). The right-hand side of (6.33b) is the superposition of frequency-smoothed (conjugate) cyclic periodograms. It is a mean-square consistent estimator of the Wigner distribution of ACS signals provided that T → ∞ and f → 0 with T f → ∞ (Section 5.2.3) and the sums over α ∈ A of biases and variances of the terms in (6.33b) are absolutely convergent. The smoothed spectral autocorrelation function is given by kx (ν, f ) = kx (ν, f ) ⊗ G(ν, f ) f x (t, f ) e−j 2πνt dt = W R 1 f (T ) = V e−j 2π(ν−α)t dt . Ix (α, f ; t) ⊗ f f f R
(6.38a) (6.38b) (6.38c)
α∈A
6.6.1 Expected values of filtered quadratic TFDs of ACS signals In this section, the expected values of the filtered quadratic TFDs of ACS signals are expressed in terms of cyclic autocorrelation functions and cyclic spectra. Proofs are in Section 6.8. Smoothed autocorrelation function x (t, τ ) = E {Kx (t, τ )} ⊗ G(t, τ ) E K t (T ) E Rx (α, τ ; t) ej 2παt v( f τ ) =
(6.39a) (6.39b)
α∈A
= E Kx (t, τ ) v( f τ ) β Rx (τ ) e−j πβτ sinc(T (β − α)) ej 2πβt v( f τ ) + α∈A β∈A β=α
(6.39c)
264
Cyclostationary Processes and Time Series
with (Theorem 5.13)
E Rx(T ) (α, τ ; t) = Rxα (τ ) + O(T −r )
(6.40)
where r = γ in Theorem 5.13. Filtered generalized ambiguity function x (t, τ ) e−j 2πνt dt E Ax (ν, τ ) E K R = E Ax (ν, τ ) v( f τ ) β + Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν − β) v( f τ )
(6.41a)
(6.41b)
α∈A β∈A β=α
Filtered Wigner distribution x (t, f ) = E Wx (t, f ) ⊗ 1 V f E W f f f β 1 f V sinc(T (β − α)) ej 2πβt Sx (f + β/2) ⊗ + f f f
(6.42)
α∈A β∈A β=α
Filtered rotated Loéve bifrequency spectrum 1 f E kx (ν, f ) = E kx (ν, f ) ⊗ V f f f β 1 f + V sinc(T (β − α)) δ(ν − β) Sx (f + β/2) ⊗ f f f
(6.43)
α∈A β∈A β=α
6.7 Cohen’s class of bilinear (or quadratic) time-frequency distributions The bilinear or quadratic time-frequency distributions of the Cohen’s class can be expressed as (Cohen, 1989), (Cohen, 1995) Cx (t, f ) = Ax (ν, τ ) (ν, τ ) ej 2π(νt−f τ ) dν dτ (6.44a) R2 Wx (θ, λ) (t − θ, f − λ) dθ dλ (6.44b) = R2
where the kernels are linked by
(ν, τ ) =
R2
(t, f ) e−j 2π(νt−f τ ) dt df .
(6.45)
Chapter 6 • Quadratic time-frequency distributions
265
In the special case of ACS processes, the expected value of Cx (t, f ) is obtained by using (6.23b) into the expected value of (6.44a) or, equivalently, using (6.24b) into the expected value of (6.44b). The case of GACS signals is considered in (Izzo and Napolitano, 2005, Sec. 5) in the fraction-of-time approach.
6.8 Proofs In this Section, proofs are reported for results of the present chapter. Unless otherwise specified, only formal manipulations are reported, without explicitly justifying inversion of linear operators such as time averages, infinite sums, integrals, and ensemble average (expectation). In general, these manipulations are justified at least in the sense of generalized functions (distributions) (Champeney, 1990), (Zemanian, 1987).
Proof of (6.1b) Let Xi (f ) be the Fourier transform of xi (t) (i = 1, 2). From definition (6.1a), we have j 2πf1 (t+τ ) Ax1 x2 (ν, τ ) = X1 (f1 ) e df1 X2 (f2 ) ej 2πf2 t df2 e−j 2πνt dt R R R j 2πf1 τ X1 (f1 ) X2 (f2 ) e ej 2π(f1 +f2 −ν)t dt df1 df2 = R R R j 2πf1 τ = X1 (f1 ) X2 (f2 ) e δ(f1 + f2 − ν) df1 df2 R R = X1 (f1 ) X2 (ν − f1 ) ej 2πf1 τ df1 . (6.46) R
Proof of (6.7a) and (6.7b) Using expression (6.4b) for the STFT we have Sx (t, f ) (∗)
XT (t, f ) XT (t, f ) (∗) x (∗) (t + s2 ) bT (s2 ) e−(−)j 2πf (t+s2 ) ds2 = x(t + s1 ) bT (s1 ) e−j 2πf (t+s1 ) ds1 R R (∗) = x(t + s1 ) x (∗) (t + s2 ) bT (s1 ) bT (s2 ) e−j 2πf s1 e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t R2
(6.47)
This expression allows one, in the following, to interchange the order of expectation op erator E{·} and R (·) ds1 ds2 even if E{·} is replaced by the almost-periodic component extraction operator E{α} {·} (Definition 2.14) for the analysis the fraction-of-time (FOT) approach. In contrast, using (6.4a) for the STFT, leads to an expression where the or der of E{α} {·} and R (·) ds1 ds2 cannot be interchanged in the FOT approach (see also Remark 2.30).
266
Cyclostationary Processes and Time Series
E {Sx (t, f )} (∗) E XT (t, f ) XT (t, f ) = E x(t + s1 ) x (∗) (t + s2 ) bT (s1 ) bT(∗) (s2 ) e−j 2πf s1 e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t 2 R (∗) = Rxα (s1 − s2 ) ej 2πα(t+s2 ) bT (s1 ) bT (s2 ) e−j 2πf s1 e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t R2 α∈A
= = =
α∈A R
α∈A R
Rxα (τ ) ej 2πα(t+s2 ) bT (s2 + τ ) bT (s2 ) e−j 2πf (s2 +τ ) e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t (∗)
2 α∈A R
Rxα (τ ) e−j 2πf τ (κf −α)
Rxα (τ ) rb
R
bT (s2 + τ ) bT (s2 ) e−j 2π(κf −α)s2 ds2 dτ ej 2π(α−κf )t (∗)
(τ ) e−j 2πf τ dτ ej 2π(α−κf )t
(6.48) β
where in the fourth equality the variable change τ = s1 − s2 is made and rb (τ ) is defined in (6.8). Using the Fourier transform pair (3.254) F β (∗) (∗) (6.49) rb (τ ) bT (s + τ ) bT (s) e−j 2πβs ds ←→ B 1 (f ) B 1 ((−)(β − f )) R
T
we have E {Sx (t, f )} = =
α∈A R
α∈A
=
β Rxα (τ ) rb (τ ) e−j 2πf τ dτ
β=κf −α
T
ej 2π(α−κf )t
(∗) Sxα (f ) ⊗ B 1 (f ) B 1 ((−)(β − f )) f
T
T
β=κf −α
(∗) Sxα (f ) ⊗ B 1 (f ) B 1 (f − (−)α) ej 2π(α−κf )t f
α∈A
T
ej 2π(α−κf )t (6.50)
T
where the second equality holds provided that B 1 (f ) is even (and hence bT (t) is even). T In fact, (−)(κf − α − f ) = (−)(f + (−)f − α − f ) = f − (−)α and, since B 1 (f ) is even, the T replacement β = κf − α can be made also before doing the convolution over f .
Proof of (6.25b) and (6.25c) Substituting (6.17) and (6.21a) into (6.25a) and making the variable change t1 = t + τ/2, t2 = t − τ/2 (τ = t1 − t2 , t = (t1 + t2 )/2), we have E x(t + τ/2) x ∗ (t − τ/2) e−j 2πf τ e−j 2πνt dτ dt E {kx (ν, f )} = 2 R E x(t1 ) x ∗ (t2 ) e−j 2πf (t1 −t2 ) e−j 2πν(t1 +t2 )/2 dt1 dt2 = 2 R = E x(t1 ) x ∗ (t2 ) e−j 2π[(f +ν/2)t1 −(f −ν/2)t2 ] dt1 dt2 R2
Chapter 6 • Quadratic time-frequency distributions = E X(f + ν/2) X ∗ (f − ν/2)
267
(6.51)
which proves (6.25b). Expression (6.25c) for ACS signals follows accounting for (1.29).
Proof of (6.30b)
1 t+T /2 x(u + τ/2) x ∗ (u − τ/2) e−j 2παu du ej 2παt v( f τ ) T t−T /2 α∈A u − t 1 x(u + τ/2) x ∗ (u − τ/2)rect ej 2πα(t−u) du v( f τ ) = T R T α∈A 1 t ej 2παt v( f τ ) rect = Kx (t, τ ) ⊗ t T T
(T )
Et,A {Kx (t, τ )}
α∈A
= Kx (t, τ ) ⊗ G(t, τ ) t
x (t, τ ) =K
(6.52)
where (6.30a) and the fact that rect(·) is an even function have been accounted for.
Expected value of the cyclic correlogram
E Rx(T ) (α, τ ; t) v( f τ ) 1 t+T /2 = E x(u + τ/2) x ∗ (u − τ/2) e−j 2παu du v( f τ ) T t−T /2 1 u − t E Kx (u, τ ) = rect ej 2πα(t−u) du e−j 2παt v( f τ ) T T R 1 t ej 2παt e−j 2παt v( f τ ) = E Kx (t, τ ) ⊗ rect t T T t ↔ν ←→ E {Ax (ν, τ )} sinc(T (ν − α)) ⊗ δ(ν + α) v( f τ ) ν = =
β Rx (τ ) e−j πβτ δ(ν − β) sinc(T (ν − α)) ⊗ δ(ν + α) v( f τ ) ν
β∈A
β Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν − β) ⊗ δ(ν + α) v( f τ ) ν
β∈A
=
Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν + α − β) v( f τ ) β
β∈A
= Rxα (τ ) e−j πατ δ(ν) v( f τ ) β + Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν + α − β) v( f τ ) β∈A β=α
where the last term is due to cycle leakage and becomes negligible as T → ∞.
(6.53)
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Cyclostationary Processes and Time Series
Proofs for Section 6.6.1 Accounting for (6.53) it results: (T ) x (t, τ ) = E K E Rx (α, τ ; t) ej 2παt v( f τ )
t ↔ν
α∈A
α x (ν, τ ) = Rx (τ ) e−j πατ δ(ν) E A α∈A
+
β Rx (τ ) e−j πβτ sinc(T (β
− α)) δ(ν + α − β) ⊗ δ(ν − α) v( f τ ) ν
β∈A β=α
=
Rxα (τ ) e−j πατ δ(ν − α) v( f τ )
α∈A
E{Ax (ν, τ )} β Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν − β) v( f τ ) +
(6.54)
α∈A β∈A β=α
τ ↔f
1 f V E kx (ν, f ) = E {kx (ν, f )} ⊗ f f f +
β Sx (f
α∈A β∈A β=α
t ↔ν
1 f + β/2) sinc(T (β − α)) δ(ν − β) ⊗ V f f f
1 f V E Wx (t, f ) = E {Wx (t, f )} ⊗ f f f
(6.55)
+
τ ↔f
α∈A β∈A β=α
β Sx (f
+ β/2) sinc(T (β − α)) e
j 2πβt
1 f ⊗ V f f f
x (t, τ ) = E Kx (t, τ ) v( f τ ) E K β Rx (τ ) e−j πβτ sinc(T (β − α)) ej 2πβt v( f τ ) + α∈A β∈A β=α
(6.56)
(6.57)
Abstract Quadratic time-frequency distributions for almost-cyclostationary signals are linked to the second-order cyclic statistics in time and frequency domains. Moreover, a link is established with the estimators of these statistics. Keywords Time-frequency distribution, Wigner-Ville distribution, Spectrogram, Kernels, Cohen class
6.1 Introduction In this chapter, results for second-order ACS signals are reinterpreted in terms of quadratic time-frequency distributions (TFDs). Time-frequency distributions have been initially studied for finite-energy deterministic signals (Boashash, 2015, Chaps. 1–3). Definitions and properties can be extended to finite-power signals considering finite-time windowing of these signals. In addition, if signals are modeled as stochastic processes, definitions originally given for deterministic signals also include an ensemble average to remove randomness. For cyclostationary Cyclostationary Processes and Time Series. https://doi.org/10.1016/B978-0-08-102708-0.00017-0 Copyright © 2020 Elsevier Ltd. All rights reserved.
255
256
Cyclostationary Processes and Time Series
and almost-cyclostationary signals the ensemble average can be replaced by the almostperiodic component extraction operator (Definition 2.14) if the analysis is made in the functional or fraction-of-time approach. The presence of time windowing naturally leads to establish relationships between quadratic time-frequency distributions of (windowed) ACS signals and estimators of second-order cyclic statistics (Section 5.2). An alternative approach consists in using for finite-power signals the same definitions adopted for finite-energy signals. In such a case, Dirac deltas appear in the expressions of time-frequency distributions. Time-frequency distributions for GACS signals are derived in (Izzo and Napolitano, 2005, Sec. 5) in the fraction-of-time approach. Moments of a suitable definition of instan˘ taneous frequency for ACS processes are derived in (Zivanovi´ c, 1991).
6.2 Finite-energy signals: correlations and spectra Let xi (t) ∈ L2 (R), i = 1, 2 complex-valued signals. Their asymmetric cross-ambiguity function is defined as Ax1 x2 (ν, τ ) x1 (t + τ ) x2 (t) e−j 2πνt dt (6.1a) R = X1 (f ) X2 (ν − f ) ej 2πf τ df (6.1b) R
where Xi (f ) is the Fourier transform of xi (t) and the second equality is proved in Section 6.8. If x2 (t) is replaced by x2∗ (t), then X2 (f ) must be replaced by X2∗ (−f ). The spectral cross-moment is defined as (6.2) Ex1 x2 (ν, f ) Ax1 x2 (ν, τ ) e−j 2πf τ dτ = X1 (f ) X2 (ν − f ) . R
The symmetric cross-ambiguity function is defined as x1 (t + τ/2) x2 (t − τ/2) e−j 2πνt dt A¯ x1 x2 (ν, τ ) R x1 (t + τ ) x2 (t ) e−j 2πν(t +τ/2) dt =
(6.3a)
R
= Ax1 x2 (ν, τ ) e−j πντ
(6.3b)
where the variable change t + τ/2 = t + τ is used.
6.3 Spectrogram Definition (1.31) of short-time Fourier transform (STFT) can be generalized by considering a data-tapering window bT (t) with support [−T /2, T /2]. That is, XT (t, f ) x(u) bT (u − t) e−j 2πf u du (6.4a) R
Chapter 6 • Quadratic time-frequency distributions =
R
x(t + s) bT (s) e−j 2πf (t+s) ds
257
(6.4b)
where in the second equality the variable change s = u − t is used. The (conjugate) spectrogram of the signal x(t) is defined as (∗) Sx (t, f ) XT (t, f ) XT (t, f )
(6.5)
where subscript x ≡ xx (∗) . If the complex conjugation is present, we have the spectrogram 2 Sxx ∗ (t, f ) |XT (t, f )|
(6.6)
.
6.3.1 Expected value of spectrogram of ACS signals 6.3.1.1 Stochastic approach Let x(t) be an ACS process. The expected value of the (conjugate) spectrogram is given by (Section 6.8) κf −α E {Sx (t, f )} = Rxα (τ ) rb (τ ) e−j 2πf τ dτ ej 2π(α−κf )t (6.7a) α∈A R
=
α∈A
Sxα (f ) ⊗ f
(∗) B 1 (f ) B 1 ((−)(β − f ))
where κ 1 + (−)1, β rb (τ )
T
T
ej 2π(α−κf )t β=κf −α
bT (s + τ ) bT (s) e−j 2πβs ds (∗)
R
and
B 1 (f )
R
T
(6.7b)
(6.8)
bT (t) e−j 2πf t dt .
(6.9)
If the complex conjugation is present, one has
E |XT (t, f )|
2
= =
Rxα (τ ) rb−α (τ ) e−j 2πf τ dτ ej 2παt
(6.10a)
Sxα (f ) ⊗ B 1 (f ) B ∗1 (α + f ) ej 2παt
(6.10b)
α∈A R
f
α∈A
T
T
The window bT (t) has duration T . Thus, its Fourier transform B 1 (f ) has approximate T bandwidth 1/T . Therefore, for |α| > 1/T the product B 1 (f ) B ∗1 (α + f ) is negligible. If T T
T
is large, then large cycle frequencies do not give significant contribution in (6.10b). If T is small, then the convolution by B 1 (f ) B ∗1 (α + f ) significantly modifies the shape of Sxα (f ). T
T
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Cyclostationary Processes and Time Series
6.3.1.2 Fraction-of-time approach Let us consider the lag product decomposition (Section 2.3.1.6) x(t + s1 ) x (∗) (t + s2 ) = E{α} x(t + s1 ) x (∗) (t + s2 ) + x (t; s1 , s2 )
(6.11)
with E{α} {·} almost-periodic component extraction operator (Definition 2.14) and x (t; s1 , s2 ) residual term non containing any finite-strength additive sine-wave component
x (t; s1 , s2 ) e−j 2παt = 0 ∀α ∈ R (6.12) t
From (6.11), and accounting for the FOT counterparts of (6.48) and (6.50), one obtains the (conjugate) spectrogram decomposition {α} {Sx (t, f )} + ηx (t, f ) Sx (t, f ) = E
= =
κf −α
α∈A R
Rxα (τ ) rb
(∗) Sxα (f ) ⊗ B 1 (f ) B 1 ((−)(β − f )) f
α∈A
where
ηx (t, f )
R2
T
T
β=κf −α
(6.13a)
ej 2π(α−κf )t + ηx (t, f )
x (t; s1 , s2 ) bT (s1 ) bT(∗) (s2 ) e−j 2πf s1 e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t
with
(τ ) e−j 2πf τ dτ ej 2π(α−κf )t + ηx (t, f )
ηx (t, f ) e−j 2παt
=
x (t; s1 , s2 ) e−j 2π(α+κf )t
t R2 (∗) bT (s1 ) bT (s2 ) e−j 2πf s1 e−(−)j 2πf s2
(6.13b)
(6.14)
t
ds1 ds2 = 0
∀α ∈ R
(6.15)
The Fourier coefficients of the almost-periodic component of Sx (t, f ) are given by ⎧ ⎪ ⎨ S α (f ) ⊗ B 1 (f ) B (∗) ((−)(β − f ))
γ ∈A 1 x −j 2π(γ −κf )t f T (6.16) Sx (t, f ) e = T β=κf −α ⎪ t ⎩ 0 γ ∈ A
6.4 Quadratic TFDs Let x(t) be a complex-valued continuous-time finite-power deterministic signal. For complex-valued signals, both lag-product waveforms with conjugated and not conjugated second term should be considered for a complete characterization (Section 1.3), (Schreier and Scharf, 2003b), (Schreier and Scharf, 2010), (Adali et al., 2011). In the following, only the case of conjugated second term of the lag product is considered. The case of not conjugated second term can be easily derived accounting for the guidelines in Section 1.3.
Chapter 6 • Quadratic time-frequency distributions
Kx (t, τ )
t ↔ν Ax (ν, τ ) τ ↔f
259
τ ↔f
Wx (t, f )
t ↔ν
kx (ν, f )
FIGURE 6.1 Quadratic TFDs.
According to the definitions given in (Boashash, 2015, Sec. 3.2.1) for finite-energy signals and those given in (Boashash, 2015, Sec. 9.4) for finite-power stochastic processes, the following functions can be defined, where Fourier transforms must be intended in the sense of distributions (generalized functions) (Gel’fand and Vilenkin, 1964, Chap. 3), (Henniger, 1970). Their mutual relationships are illustrated in Fig. 6.1. Symmetric second-order lag-product or instantaneous autocorrelation function Kx (t, τ ) x(t + τ/2) x ∗ (t − τ/2)
(6.17)
Symmetric ambiguity function Ax (ν, τ )
R
Kx (t, τ ) e−j 2πνt dt
(6.18)
Kx (t, τ ) e−j 2πf τ dτ
(6.19)
(Symmetric) Wigner distribution Wx (t, f )
R
Symmetric spectral autocorrelation function kx (ν, f ) Kx (t, τ ) e−j 2πνt e−j 2πf τ dt dτ R
(6.20)
6.4.1 Expected values of quadratic TFDs of ACS signals Let x(t) be a complex-valued continuous-time ACS stochastic process. In such a case, the quadratic TFDs defined in Section 6.4 are random functions. In the following, their expected values are expressed in terms of cyclic autocorrelation functions and cyclic spectra. Symmetric autocorrelation function Rx (t, τ ) E {Kx (t, τ )} = Rxα (τ ) e−j πατ ej 2παt α∈A
(6.21a) (6.21b)
260
Cyclostationary Processes and Time Series
where Rxα (τ ) e−j πατ
1 = lim T →∞ T
T /2 −T /2
Rx (t, τ ) e−j 2παt dt
(6.22)
is the symmetric cyclic autocorrelation function (1.34). Expected value of the symmetric ambiguity function (Napolitano, 2015b, Eq. (9.6.15)) E {Ax (ν, τ )} Rx (t, τ ) e−j 2πνt dt (6.23a) R = Rxα (τ ) e−j πατ δ(ν − α) (6.23b) α∈A
Expected value of the Wigner distribution (Martin and Flandrin, 1985), (Amin, 1992), (Flandrin, 1999), (Antoni, 2007a), (Napolitano, 2015b, Eq. (9.6.14)) E {Wx (t, f )} Rx (t, τ ) e−j 2πf τ dτ (6.24a) R Sxα (f + α/2) ej 2παt (6.24b) = α∈A
where Sxα (f + α/2) is the symmetric cyclic spectrum (1.35) which is the Fourier transform of the symmetric cyclic autocorrelation function (6.22). Spectral autocorrelation function or rotated Loève bifrequency spectrum (Section 6.8) {k E x (ν, f )} Rx (t, τ ) e−j 2πνt e−j 2πf τ dt dτ (6.25a) 2 R (6.25b) = E X(f + ν/2) X ∗ (f − ν/2) = Sxα (f + α/2) δ(ν − α) (6.25c) α∈A
6.5 Filtered quadratic TFDs 6.5.1 Kernels In this section, the finite-time counterpart of the almost-periodic component extraction operator (Definition 2.14) is considered as time-lag kernel for a given ACS signal x(t). Then, the other kernels are derived by Fourier transformations. Their mutual relationships are illustrated in Fig. 6.2. Let v(τ ) a unit-width lag window with Fourier transform V (f ) such that v(0) = R V (f )df = 1. Thus, lim f →0 V (f/ f )/ f = δ(f ). For example, v(τ ) = rect(τ ) ↔ V (f ) = sinc(f ). Let A be the countable set of second-order cycle frequencies of x(t).
Chapter 6 • Quadratic time-frequency distributions
t ↔ν g(ν, τ ) τ ↔f
G(t, τ )
261
τ ↔f
γ (t, f )
t ↔ν
G(ν, f )
FIGURE 6.2 Kernels.
The following kernels can be defined. Time-lag kernel
t 1 ej 2παt v( f τ ) rect T T
G(t, τ ) =
(6.26)
α∈A
Doppler-lag kernel
G(t, τ ) e−j 2πνt dt R sinc((ν − α)T ) v( f τ ) =
g(ν, τ )
(6.27a) (6.27b)
α∈A
Time-frequency kernel
G(t, τ ) e−j 2πf τ dτ 1 t j 2παt 1 f e = rect V T T f f
γ (t, f )
R
(6.28a) (6.28b)
α∈A
Doppler-frequency kernel
G(t, τ ) e−j 2πνt e−j 2πf τ dt dτ 1 f sinc((ν − α)T ) = V f f
G(ν, f )
R2
(6.29a) (6.29b)
α∈A
The presence of the window v( f τ ) is not crucial in the definitions of kernels G(t, τ ) x (ν, τ ) (see Section 6.6) at values of |τ | larger x (t, τ ) and A and g(ν, τ ). It will set to zero K than 1/2 f since for these values of |τ | the variance of the estimators is large. In contrast, x (t, f ) and V (f/ f )/ f is important to get consistency of estimators W kx (ν, f ).
6.6 Filtered quadratic TFDs of ACS signals Filtered quadratic TFDs for generic stochastic processes are defined according to (Boashash, 2015, Secs. 3.2.2–3.2.4). Their mutual relationships are illustrated in Fig. 6.3. In the following, their expressions for ACS processes are presented.
262
Cyclostationary Processes and Time Series
x (t, τ ) = Kx (t, τ ) ⊗ G(t, τ ) K t ↔ν x (ν, τ ) = Ax (ν, τ ) g(ν, τ ) A τ ↔f
t
τ ↔f x (t, f ) = Wx (t, f ) ⊗⊗ γ (t, f ) W t f
t ↔ν
kx (ν, f ) = kx (ν, f ) ⊗ G(ν, f ) f
FIGURE 6.3 Filtered quadratic TFDs.
The smoothed 2nd-order lag product or smoothed instantaneous autocorrelation function is an estimator of the symmetric autocorrelation function Rx (t, τ ). For ACS processes it is given by (Napolitano, 2015b, Eq. (9.6.17)) (Section 6.8) x (t, τ ) Kx (t, τ ) ⊗ G(t, τ ) K t (T ) = Rx (α, τ ; t) ej 2παt v( f τ )
(6.30a) (6.30b)
α∈A
where Rx(T ) (α, τ ; t)
1 T
t+T /2
x(u + τ/2) x ∗ (u − τ/2) e−j 2παu du
(6.31)
t−T /2
denotes here the symmetric cyclic correlogram of x(t). Under mild regularity assumptions on the finite or practically finite memory of x(t) expressed in terms of summability of cumulants, the cyclic correlogram is a mean-square consistent (as T → ∞) estimator of the √ cyclic autocorrelation function. Moreover, T [Rx(T ) (α, τ ; t) − Rxα (τ )] is zero-mean asymptotically complex normal (Section 5.2.1). The symmetric generalized ambiguity function is given by x (ν, τ ) = Ax (ν, τ ) g(ν, τ ) A = x(t + τ/2) x ∗ (t − τ/2) e−j 2πνt dt sinc((ν − α)T ) v( f τ ) R
(6.32a) (6.32b)
α∈A
The smoothed Wigner distribution is an estimator of the Wigner distribution Wx (t, f ). For ACS processes it is given by (Napolitano, 2015b, Eq. (9.6.18)) x (t, τ ) e−j 2πf τ dτ (6.33a) Wx (t, f ) K R 1 f (6.33b) V ej 2παt = Ix(T ) (α, f ; t) ⊗ f f f α∈A
where Ix(T ) (α, f ; t)
1 XT (t, f + α/2) XT∗ (t, f − α/2) T
(6.34)
Chapter 6 • Quadratic time-frequency distributions
is the symmetric cyclic periodogram of x(t) with t+T /2 XT (t, f ) x(u) e−j 2πf u du
263
(6.35)
t−T /2
short-time Fourier transform (STFT) of x(t). It results (Lemma 2.39) Ix(T ) (α, f ; t) = Rx(T ) (α, τ ; t) e−j 2πf τ dτ . R
(6.36)
Expression (6.33b) can be equivalently obtained as x (t, f ) = Wx (t, f ) ⊗⊗ γ (t, f ) . W t f
(6.37)
(T )
The cyclic periodogram Ix (α, f ; t) is an asymptotically (as T → ∞) unbiased but not consistent estimator of Sxα (f + α/2) (Section 5.2.2). The right-hand side of (6.33b) is the superposition of frequency-smoothed (conjugate) cyclic periodograms. It is a mean-square consistent estimator of the Wigner distribution of ACS signals provided that T → ∞ and f → 0 with T f → ∞ (Section 5.2.3) and the sums over α ∈ A of biases and variances of the terms in (6.33b) are absolutely convergent. The smoothed spectral autocorrelation function is given by kx (ν, f ) = kx (ν, f ) ⊗ G(ν, f ) f x (t, f ) e−j 2πνt dt = W R 1 f (T ) = V e−j 2π(ν−α)t dt . Ix (α, f ; t) ⊗ f f f R
(6.38a) (6.38b) (6.38c)
α∈A
6.6.1 Expected values of filtered quadratic TFDs of ACS signals In this section, the expected values of the filtered quadratic TFDs of ACS signals are expressed in terms of cyclic autocorrelation functions and cyclic spectra. Proofs are in Section 6.8. Smoothed autocorrelation function x (t, τ ) = E {Kx (t, τ )} ⊗ G(t, τ ) E K t (T ) E Rx (α, τ ; t) ej 2παt v( f τ ) =
(6.39a) (6.39b)
α∈A
= E Kx (t, τ ) v( f τ ) β Rx (τ ) e−j πβτ sinc(T (β − α)) ej 2πβt v( f τ ) + α∈A β∈A β=α
(6.39c)
264
Cyclostationary Processes and Time Series
with (Theorem 5.13)
E Rx(T ) (α, τ ; t) = Rxα (τ ) + O(T −r )
(6.40)
where r = γ in Theorem 5.13. Filtered generalized ambiguity function x (t, τ ) e−j 2πνt dt E Ax (ν, τ ) E K R = E Ax (ν, τ ) v( f τ ) β + Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν − β) v( f τ )
(6.41a)
(6.41b)
α∈A β∈A β=α
Filtered Wigner distribution x (t, f ) = E Wx (t, f ) ⊗ 1 V f E W f f f β 1 f V sinc(T (β − α)) ej 2πβt Sx (f + β/2) ⊗ + f f f
(6.42)
α∈A β∈A β=α
Filtered rotated Loéve bifrequency spectrum 1 f E kx (ν, f ) = E kx (ν, f ) ⊗ V f f f β 1 f + V sinc(T (β − α)) δ(ν − β) Sx (f + β/2) ⊗ f f f
(6.43)
α∈A β∈A β=α
6.7 Cohen’s class of bilinear (or quadratic) time-frequency distributions The bilinear or quadratic time-frequency distributions of the Cohen’s class can be expressed as (Cohen, 1989), (Cohen, 1995) Cx (t, f ) = Ax (ν, τ ) (ν, τ ) ej 2π(νt−f τ ) dν dτ (6.44a) R2 Wx (θ, λ) (t − θ, f − λ) dθ dλ (6.44b) = R2
where the kernels are linked by
(ν, τ ) =
R2
(t, f ) e−j 2π(νt−f τ ) dt df .
(6.45)
Chapter 6 • Quadratic time-frequency distributions
265
In the special case of ACS processes, the expected value of Cx (t, f ) is obtained by using (6.23b) into the expected value of (6.44a) or, equivalently, using (6.24b) into the expected value of (6.44b). The case of GACS signals is considered in (Izzo and Napolitano, 2005, Sec. 5) in the fraction-of-time approach.
6.8 Proofs In this Section, proofs are reported for results of the present chapter. Unless otherwise specified, only formal manipulations are reported, without explicitly justifying inversion of linear operators such as time averages, infinite sums, integrals, and ensemble average (expectation). In general, these manipulations are justified at least in the sense of generalized functions (distributions) (Champeney, 1990), (Zemanian, 1987).
Proof of (6.1b) Let Xi (f ) be the Fourier transform of xi (t) (i = 1, 2). From definition (6.1a), we have j 2πf1 (t+τ ) Ax1 x2 (ν, τ ) = X1 (f1 ) e df1 X2 (f2 ) ej 2πf2 t df2 e−j 2πνt dt R R R j 2πf1 τ X1 (f1 ) X2 (f2 ) e ej 2π(f1 +f2 −ν)t dt df1 df2 = R R R j 2πf1 τ = X1 (f1 ) X2 (f2 ) e δ(f1 + f2 − ν) df1 df2 R R = X1 (f1 ) X2 (ν − f1 ) ej 2πf1 τ df1 . (6.46) R
Proof of (6.7a) and (6.7b) Using expression (6.4b) for the STFT we have Sx (t, f ) (∗)
XT (t, f ) XT (t, f ) (∗) x (∗) (t + s2 ) bT (s2 ) e−(−)j 2πf (t+s2 ) ds2 = x(t + s1 ) bT (s1 ) e−j 2πf (t+s1 ) ds1 R R (∗) = x(t + s1 ) x (∗) (t + s2 ) bT (s1 ) bT (s2 ) e−j 2πf s1 e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t R2
(6.47)
This expression allows one, in the following, to interchange the order of expectation op erator E{·} and R (·) ds1 ds2 even if E{·} is replaced by the almost-periodic component extraction operator E{α} {·} (Definition 2.14) for the analysis the fraction-of-time (FOT) approach. In contrast, using (6.4a) for the STFT, leads to an expression where the or der of E{α} {·} and R (·) ds1 ds2 cannot be interchanged in the FOT approach (see also Remark 2.30).
266
Cyclostationary Processes and Time Series
E {Sx (t, f )} (∗) E XT (t, f ) XT (t, f ) = E x(t + s1 ) x (∗) (t + s2 ) bT (s1 ) bT(∗) (s2 ) e−j 2πf s1 e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t 2 R (∗) = Rxα (s1 − s2 ) ej 2πα(t+s2 ) bT (s1 ) bT (s2 ) e−j 2πf s1 e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t R2 α∈A
= = =
α∈A R
α∈A R
Rxα (τ ) ej 2πα(t+s2 ) bT (s2 + τ ) bT (s2 ) e−j 2πf (s2 +τ ) e−(−)j 2πf s2 ds1 ds2 e−j 2πκf t (∗)
2 α∈A R
Rxα (τ ) e−j 2πf τ (κf −α)
Rxα (τ ) rb
R
bT (s2 + τ ) bT (s2 ) e−j 2π(κf −α)s2 ds2 dτ ej 2π(α−κf )t (∗)
(τ ) e−j 2πf τ dτ ej 2π(α−κf )t
(6.48) β
where in the fourth equality the variable change τ = s1 − s2 is made and rb (τ ) is defined in (6.8). Using the Fourier transform pair (3.254) F β (∗) (∗) (6.49) rb (τ ) bT (s + τ ) bT (s) e−j 2πβs ds ←→ B 1 (f ) B 1 ((−)(β − f )) R
T
we have E {Sx (t, f )} = =
α∈A R
α∈A
=
β Rxα (τ ) rb (τ ) e−j 2πf τ dτ
β=κf −α
T
ej 2π(α−κf )t
(∗) Sxα (f ) ⊗ B 1 (f ) B 1 ((−)(β − f )) f
T
T
β=κf −α
(∗) Sxα (f ) ⊗ B 1 (f ) B 1 (f − (−)α) ej 2π(α−κf )t f
α∈A
T
ej 2π(α−κf )t (6.50)
T
where the second equality holds provided that B 1 (f ) is even (and hence bT (t) is even). T In fact, (−)(κf − α − f ) = (−)(f + (−)f − α − f ) = f − (−)α and, since B 1 (f ) is even, the T replacement β = κf − α can be made also before doing the convolution over f .
Proof of (6.25b) and (6.25c) Substituting (6.17) and (6.21a) into (6.25a) and making the variable change t1 = t + τ/2, t2 = t − τ/2 (τ = t1 − t2 , t = (t1 + t2 )/2), we have E x(t + τ/2) x ∗ (t − τ/2) e−j 2πf τ e−j 2πνt dτ dt E {kx (ν, f )} = 2 R E x(t1 ) x ∗ (t2 ) e−j 2πf (t1 −t2 ) e−j 2πν(t1 +t2 )/2 dt1 dt2 = 2 R = E x(t1 ) x ∗ (t2 ) e−j 2π[(f +ν/2)t1 −(f −ν/2)t2 ] dt1 dt2 R2
Chapter 6 • Quadratic time-frequency distributions = E X(f + ν/2) X ∗ (f − ν/2)
267
(6.51)
which proves (6.25b). Expression (6.25c) for ACS signals follows accounting for (1.29).
Proof of (6.30b)
1 t+T /2 x(u + τ/2) x ∗ (u − τ/2) e−j 2παu du ej 2παt v( f τ ) T t−T /2 α∈A u − t 1 x(u + τ/2) x ∗ (u − τ/2)rect ej 2πα(t−u) du v( f τ ) = T R T α∈A 1 t ej 2παt v( f τ ) rect = Kx (t, τ ) ⊗ t T T
(T )
Et,A {Kx (t, τ )}
α∈A
= Kx (t, τ ) ⊗ G(t, τ ) t
x (t, τ ) =K
(6.52)
where (6.30a) and the fact that rect(·) is an even function have been accounted for.
Expected value of the cyclic correlogram
E Rx(T ) (α, τ ; t) v( f τ ) 1 t+T /2 = E x(u + τ/2) x ∗ (u − τ/2) e−j 2παu du v( f τ ) T t−T /2 1 u − t E Kx (u, τ ) = rect ej 2πα(t−u) du e−j 2παt v( f τ ) T T R 1 t ej 2παt e−j 2παt v( f τ ) = E Kx (t, τ ) ⊗ rect t T T t ↔ν ←→ E {Ax (ν, τ )} sinc(T (ν − α)) ⊗ δ(ν + α) v( f τ ) ν = =
β Rx (τ ) e−j πβτ δ(ν − β) sinc(T (ν − α)) ⊗ δ(ν + α) v( f τ ) ν
β∈A
β Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν − β) ⊗ δ(ν + α) v( f τ ) ν
β∈A
=
Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν + α − β) v( f τ ) β
β∈A
= Rxα (τ ) e−j πατ δ(ν) v( f τ ) β + Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν + α − β) v( f τ ) β∈A β=α
where the last term is due to cycle leakage and becomes negligible as T → ∞.
(6.53)
268
Cyclostationary Processes and Time Series
Proofs for Section 6.6.1 Accounting for (6.53) it results: (T ) x (t, τ ) = E K E Rx (α, τ ; t) ej 2παt v( f τ )
t ↔ν
α∈A
α x (ν, τ ) = Rx (τ ) e−j πατ δ(ν) E A α∈A
+
β Rx (τ ) e−j πβτ sinc(T (β
− α)) δ(ν + α − β) ⊗ δ(ν − α) v( f τ ) ν
β∈A β=α
=
Rxα (τ ) e−j πατ δ(ν − α) v( f τ )
α∈A
E{Ax (ν, τ )} β Rx (τ ) e−j πβτ sinc(T (β − α)) δ(ν − β) v( f τ ) +
(6.54)
α∈A β∈A β=α
τ ↔f
1 f V E kx (ν, f ) = E {kx (ν, f )} ⊗ f f f +
β Sx (f
α∈A β∈A β=α
t ↔ν
1 f + β/2) sinc(T (β − α)) δ(ν − β) ⊗ V f f f
1 f V E Wx (t, f ) = E {Wx (t, f )} ⊗ f f f
(6.55)
+
τ ↔f
α∈A β∈A β=α
β Sx (f
+ β/2) sinc(T (β − α)) e
j 2πβt
1 f ⊗ V f f f
x (t, τ ) = E Kx (t, τ ) v( f τ ) E K β Rx (τ ) e−j πβτ sinc(T (β − α)) ej 2πβt v( f τ ) + α∈A β∈A β=α
(6.56)
(6.57)