Interpreting the Trispectrum as the Cross-Spectrum of the Wigner–Ville Distribution

Abstract

The fourth-order time-invariant spectrum, or trispectrum, has a simple derivation as the cross-spectrum among frequency bands in the Wigner-Ville distribution (WVD). Viewed this way, the trispectrum gains intuitive meaning as a measure of the linear dependence of power across frequencies, which yields some insight into its structure and interpretation. We highlight, in particular, a two-dimensional subdomain as useful for identifying modulated oscillations when the modulating envelope is non-negative or lowpass. Spectral characteristics of the carrier and modulating signals are revealed along separate axes of a two-dimensional representation of this domain. The application of this framework, combined with a previously described additive decomposition technique for higher-order spectra, is demonstrated by the blind identification and separation of sleep spindles and beta bursts in EEG.

I. Introduction

Higher-order spectra (HOS) are the frequency domain representations of time-invariant signal moments of order greater than two. The second-order spectrum is the familiar power spectrum (cross-spectrum in the multivariate context), while the (cross-) spectra of order three and four are called, respectively, the (cross-) bispectrum and (cross-) trispectrum. Higher moments are useful for identifying non-Gaussian signal characteristics and have found applications in non-minimum-phase [1] and nonlinear system identification [2] and the analysis of biomedical signals [3]. Nevertheless, practical applications of HOS have been hampered by their reputed lack of any simple real-world meaning [4], [5].

The aim of this note is to develop the interpretation of the fourth-order spectrum by viewing it as a second-order moment of spectrally resolved power. This view grants some access to intuition [5], and clarifies the relevance of the trispectrum to the measurement of dynamic properties of second-order signal statistics, such as amplitude and frequency modulation. In many settings, power represents the quantity of interest, and the goal is to measure the evolution of power over time and frequency, often in relation to experimental variables [6]. In these cases, the trispectrum presents a natural tool to describe linear relationships of power over time and frequency, both within and between signals.

The idea of interpreting HOS as cross-spectra among time-varying lower-order statistics is discussed by Akaike [5] who in turn, credits Tick [7]. Akaike described what he called a “mixed spectrum” giving, in the case of the bispectrum, the cross-spectrum between the time-varying autocorrelation of a signal with the signal itself (or a different signal in the case of the cross-bispectrum), indexed by the autocorrelation lag. Akaike’s development extends naturally to HOS of any order.

A closely related idea is developed here by starting with the Wigner-Ville distribution (WVD), leading to a mathematically equivalent derivation under a Fourier transform over lag axes of Akaike’s mixed spectrum. Although formally equivalent to this and other preceding derivations of HOS, viewing the trispectrum as the second moment of the WVD leads to a particularly simple framework for interpretation. Representations of time-varying spectral power, in the form of time-frequency decompositions of various kinds, are far more commonly encountered than representations of time-varying autocorrelation. In addition to informing the interpretation of the trispectrum as a whole, this view offers insight into the relevance of subdomains within the trispectrum for specific problems. In particular, it will allow us to recognize a two-dimensional slice of the trispectrum as useful for blindly identifying a modulated carrier signal when the modulating envelope is lowpass, such that energy is preserved in the carrier’s original band.

II. Higher-Order Spectra

Higher-order spectra are defined as Fourier-domain representations of time-invariant higher-order autocorrelations. They generalize the familiar (second-order) autocorrelation as expectations of the product of a signal with itself at multiple lags:

$$\begin{gathered} m^K\left({\tau }_1,{\tau }_2,\dots ,{\tau }_{K-1}\right)= \\ \text{E}\left[x\left(t\right)x\left(t+{\tau }_1\right)x\left(t+{\tau }_2\right)\dots x\left(t+{\tau }_{K-1}\right)\right] \end{gathered}$$ (1)

where $m^K$ is assumed not to depend on $t$. The multi-dimensional Fourier transform of (1) yields an expectation of a product of $K$ terms in the spectrum of $x$:

$$M^K\left({\omega }_1,{\omega }_2,\dots ,{\omega }_{K-1}\right)=\text{E}\left[X\left({\omega }_1\right)X\left({\omega }_2\right)\dots X^{*}\left({\Sigma }_k^{K-1}{\omega }_k\right)\right]$$ (2)

$X$ denoting the spectral representation of $x$ and $X^{*}$ its complex conjugate. The trispectrum corresponds to $K=4$ and is therefore a function of 3 frequency arguments. Time invariance results from the cancellation of linear phase in $X$ from the product in (2). While the expectation in (2) must be understood as an ensemble average, for the purpose of estimation, stationarity and ergodicity are commonly assumed so that estimators may be obtained through time averaging over windowed segments of the signal, which introduces forms of broadband bias and spectral leakage familiar from power spectral estimation [4], [8].

Although the definition in (2) extends easily to multivariate and complex-valued signals, it will be assumed here that $x$ is univariate and real-valued, so that we need consider only autospectra and so that taking the complex conjugate of $X$ can be equated to reversing the sign of the frequency argument (i.e. $X^{*}\left(\omega \right)=X(-\omega )$). A more general treatment must otherwise include combinations of sign inversion and conjugation for complex-valued signals. Naturally, it will also be assumed that the relevant higher moments of $x$ exist and are bounded.

III. The Wigner-Ville Distribution

The WVD is a quadratic time-frequency distribution (TFD), from which many other more commonly encountered TFDs, including the squared magnitudes of the conventional short-time Fourier transform (STFT), band-limited Hilbert transform (HT), and continuous wavelet transforms (CWT), may be derived by way of smoothing [9], [10]. All quadratic TFDs decompose the total energy of a signal over time and/or frequency to yield power in units of energy over increments of time and/or frequency. For a time series, $x$, The Wigner-Ville pseudo-distribution (WVD) is defined as:

$$𝒲\left\{x\right\}(\omega ,t)=\int x\left(t+\frac{\tau }{2}\right)x^{*}\left(t-\frac{\tau }{2}\right)e^{-i\omega \tau }d\tau$$ (3)

The WVD is not strictly positive and exhibits so-called interference at points in time and frequency midway between peaks in the signal power. For these reasons, it is not the TFD of first choice in most applications.

The WVD is, however, a lossless TFD in that it retains complete information about the original signal [11], up to a constant multiplicative root of unity (e.g. sign), and is thus invertible [12], and it is free of bias introduced by the choice of smoothing kernel in standard TFDs. As a result, statistics computed directly on the WVD retain information about signal phase that cannot be extracted from statistics computed on band-limited power. This fact proves useful for recovering waveforms and patterns of modulation that may not be observable with standard TFDs [13], or which are observed only under a limited set of parameters, requiring information about the target signal that may not be available at the outset.

A. Cross-Spectrum of the WVD

The cross-spectrum of the WVD is computed in analogy to a multivariate time series where the frequency dimension takes the place of channel index, giving a function of three frequency variables, ${\omega }_1$ and ${\omega }_2$ for a pair of frequency bands in the WVD, and $\eta$ for the frequency of association. Starting with the Fourier transform of the WVD along the time dimension,

$$\begin{gathered} ℳ\left\{x\right\}\left(\omega ,\eta \right)=\int 𝒲\left\{x\right\}\left(\omega ,t\right)e^{-\mathit{\text{int}}}\mathit{\text{dt}} \\ =\int \int x\left(t+\frac{\tau }{2}\right)x^{*}\left(t-\frac{\tau }{2}\right)e^{-i\omega \tau -i\eta t}d\tau \mathit{\text{dt}} \\ =X\left(\omega +\frac{\eta }{2}\right)X^{*}\left(\omega -\frac{\eta }{2}\right) \end{gathered}$$ (4)

the ensemble-average cross-spectral density of the WVD is given as the expectation:

$$\begin{gathered} S_{\mathit{\text{WVD}}}\left({\omega }_1,{\omega }_2,\eta \right)=\text{E}\left[ℳ\left({\omega }_1,\eta \right){ℳ}^{*}\left({\omega }_2,\eta \right)\right]= \\ \text{E}\left[X\left({\omega }_1+\frac{\eta }{2}\right)X^{*}\left({\omega }_1-\frac{\eta }{2}\right)X^{*}\left({\omega }_2+\frac{\eta }{2}\right)X\left({\omega }_2-\frac{\eta }{2}\right)\right] \end{gathered}$$ (5)

Eq. (5) represents a fourth-order time-invariant moment of the original signal with three linearly independent axes in which linear phase cancels, just as in (2). The result is therefore the trispectrum under an alternate parameterization of the frequency axes. Though otherwise equivalent to (2), parameterizing the axes in terms of the frequency arguments of the WVD lends the trispectrum an interpretation as the linear association of power between bands ${\omega }_1$ and ${\omega }_2$ of the WVD at cross-spectral frequency $\eta$.

IV. Interpreting the Trispectrum

Under the parameterization in (5), certain aspects of the structure of the trispectrum come into better view. In particular, it becomes easy to recognize subdomains that are relevant to the problem of identifying modulation of a band-limited carrier by a lowpass envelope. But first, we turn down a cautionary blind alley.

A. Does Power Have a Power Spectrum?

To quantify the dynamic properties of power within isolated bands of the WVD, it may seem natural to consider the diagonal of (5), ${\omega }_1={\omega }_2$, which ought to give the power spectrum of a single band of the WVD in isolation. Setting ${\omega }_1={\omega }_2=\omega$, gives the squared magnitude of $ℳ$:

$$\begin{gathered} S_{\mathit{\text{WVD}}}(\omega ,\omega ,\eta )=\text{E}\left[|ℳ(\omega ,\eta ){|}^2\right] \\ =\text{E}\left[{\left|X\left(\omega +\frac{\eta }{2}\right)\right|}^2{\left|X\left(\omega -\frac{\eta }{2}\right)\right|}^2\right] \end{gathered}$$ (6)

This quantity and its equivalents have been suggested for identifying modulation in the context of cyclostationary [14] and audio signals [15], [16]. There is however a difficulty with its interpretation as the power spectrum of time-varying power in the WVD. First, it can be observed that, as a result of the symmetry of the trispectrum, an equivalent quantity is obtained by setting $\eta =0$ in (5):

$$S_{\mathit{\text{WVD}}}\left({\omega }_1,{\omega }_2,0\right)=\text{E}\left[{\left|X\left({\omega }_1\right)\right|}^2{\left|X\left({\omega }_2\right)\right|}^2\right]$$ (7)

which seems to give the association of the 0 Hz component of power between two bands (after subtracting the product of expected power). Is (6) to be interpreted as the non-dynamic association of power in two bands on either side of $\omega$ or as the power spectrum of time-varying power within a single band of the WVD?

The contradiction raised by this question arises from a short-coming in the analogy between frequency bands of the WVD and the components of a multivariate time series: power cannot both vary dynamically and be independent across bands of the WVD. Modulation of power introduces correlations across bands, while the relevant correlation for any dynamic property of the signal cannot reside within the 0 Hz part of the cross-spectrum. The subdomain of (6) is therefore the wrong place to identify modulation. We must instead consider the dynamic part of the association across bands, as described next.

B. The Diagonal Slice

Because the spectrum of a modulated carrier is the convolution between the separate spectra of the carrier and modulating signal, the relevant correlations for a real-valued modulating signal appear between sidebands offset symmetrically on either side of the carrier band. A reparameterization of (5) better describes this situation in terms of a carrier frequency, $\omega$, and the offset to the sidebands, $\beta$:

$$\begin{gathered} S\left(\omega ,\beta ,\eta \right)=S_{\mathit{\text{WVD}}}\left(\omega +\beta /2,\omega -\beta /2,\eta \right)= \\ =\text{E}\left[X\left(\omega +\frac{\eta -\beta }{2}\right)X^{*}\left(\omega -\frac{\eta +\beta }{2}\right)\right. \\ \left.\times X^{*}\left(\omega +\frac{\eta +\beta }{2}\right)X\left(\omega -\frac{\eta -\beta }{2}\right)\right] \end{gathered}$$ (8)

Identifying an arbitrary carrier modulated by an arbitrary envelope requires the entire trispectrum, but when the modulating signal is non-negative or lowpass, so that the envelope has a non-vanishing mean, energy from the carrier is preserved in the signal, and the relevant association is a captured within the two-dimensional subdomain (exemplified in Fig. 1(a)) that conveys the three-way association of the carrier and the symmetrically offset sidebands, obtained by setting $\beta =\eta$.

$$\begin{gathered} D\left(\omega ,\eta \right)=S\left(\omega ,\eta ,\eta \right)= \\ =\text{E}\left[X^2\left(\omega \right)X^{*}\left(\omega +\eta \right)X^{*}\left(\omega -\eta \right)\right] \end{gathered}$$ (9)

This slice, which we refer to as the diagonal slice, is the basis of what we have called the modulogram, a device for representing spectral properties of carrier and modulating signals along separate dimensions of a two-dimensional plot (example in Fig. 1(b)). In addition to giving the correct measure of association for modulation of power, this slice lies within the cumulant subdomain of the moment HOS for $\omega ,\eta \ne 0$ and $\left|\omega \right|-\left|\eta \right|\ne 0$, so that lower-order corrections [17] do not have to be introduced to obtain a measure of association, in contrast to (6).

Fig. 1.

Application of tricoherence towards identifying modulated oscillations associated with sleep in in human EEG data. (a) Full tricoherence computed over 7 hours of EEG data recorded from the left frontal channel (F3) exhibits activity around the diagonal slice. (b) Diagonal slice represented as the modulogram reveals modulation of carrier oscillations within distinct bands of 10–15 Hz (magenta box) and 15–30 Hz (yellow boxes), modulated at scales of 1 s and 0.25 s, respectively. (c) Three oscillatory features recovered by HOSD applied to the trispectral diagonal slice correspond to sleep spindles (magenta) and beta bursts (yellow) of differing frequency. (d) The first component signal (magenta) is associated with non-REM sleep (N2 and N3), while the second (yellow) is associated with REM sleep (R) and awake (W).

C. Physical Interpretations

Applications of HOS are often stymied by questions of interpretation [4], [5], which naturally depend on the system under study. The problem is therefore not to attach any universal meaning to HOS but to develop frameworks that facilitate the assignment of meaning. Second-order measures of association (e.g covariance and cross-spectra) underlie a host of standard techniques of linear analysis, whose familiarity and conceptual simplicity ease the way to interpretation. Understanding third- and fourth-order spectra as linear measures of dynamic association among first and second-order statistics offers a way to recursively bootstrap insight into their meaning [5], [18]. Additive decompositions of second-order statistics, as with principal component analysis, are another indispensable tool of linear analysis, used to identify a small number of, often meaningful, dimensions along which data covary [19]. Additive decompositions of HOS may likewise guide interpretation by relating HOS to concrete features within the signal [20]. Interpretation of resulting components is most natural in the context of mixtures of independent non-Gaussian linear processes; but even for nonlinear systems, the reduction of high-dimensional data to a small number of features that characterize the dynamics of the system remains a broadly useful step towards interpretation [21].

V. Identifying Modulated Oscillations in EEG

Transiently modulated oscillations are a prominent feature of human electroencephalographic (EEG) recordings correlated with arousal and sleep, cognitive and motor function, and neuropathology [22]. Among the best characterized of these are sleep spindles, which are aperiodically modulated oscillations of 11 – 16 Hz, with a duration of 0.5 – 2 s, visible in unaveraged traces [23], [24]. Beta bursts represent another class of transiently modulated oscillations in the frequency range of 12 – 35 Hz, associated with task-dependent motor and cognitive regulation [25]. Automated methods to identify oscillatory bursting typically rely on filtering and amplitude thresholding [25], both of which impose potentially arbitrary assumptions about the relevant frequency band and threshold of significance.

We demonstrate that the modulogram reveals the presence and characteristics of modulated oscillations in EEG during sleep with few starting assumptions. The example in Fig. 1 shows the blind identification and separation of sleep spindles and beta bursts within a single night’s recording in one frontal EEG channel (F3). Tricoherence within the diagonal slice, estimated over the whole duration of the recording, shows evidence of slow modulation within distinct bands between 10 Hz and 30 Hz. Decomposing the modulogram using the method of Kovach and Howard (HOSD) [20] recovered 3 features associated with peaks in the modulogram. Characteristic spectral and temporal features of modulation for each component are revealed by the associated feature waveform. In each case, modulation was characterized by a transient oscillatory burst with a descending frequency ramp (Fig. 1(c), bottom panel). The first component (Fig. 1(c), magenta trace) appears within the 10 – 15 Hz band with modulation on the scale of 1 s. The second and third (Fig. 1(c), yellow traces) appear within bands of 13 – 21 Hz and 21 – 28 Hz, respectively, both modulated on the scale of 0.25 s. RMS amplitude of the first component is strongly correlated with non-REM N2 and N3 sleep and anti-correlated with REM and awake stages (Table I). Components 2 and 3 exhibit the opposite association, being suppressed in N2 and N3 sleep and elevated in REM (R) and awake (W) stages. Characteristics of component 1 closely match those of sleep spindles [24], while in their higher frequency and shorter duration, components 2 and 3 resemble beta bursts [25].

TABLE I.

Correlation Between Component RMS Amplitude and Sleep Stage

Component	NREM (N2+N3)	REM	W
1	0.48	−0.41	−0.53
2	−0.26	0.22	0.33
3	−0.24	0.19	0.22

Although the modulogram provides direct evidence of modulation, it does not by itself reveal unambiguously whether one or a number of components are present. To address component identification and separation, we have applied a previously described additive decomposition of HOS (HOSD) which yields a filter for each component [13], [20] and a reconstructed component signal (Fig. 1(d)). Because HOS decompose the static cumulant of a given order, HOSD further provides a principled statistical criterion for thresholding, based on zeroing out the static cumulant of the component filter output.

Methods:

To demonstrate the use of the modulogram, a representative, fully deidentified data set containing 16 channels of continuous EEG data was obtained with permission from the Human Sleep Project database (sub-S0001122033808/ses-1). along with sleep staging performed by a certified polysomnographer [26]. Results are shown for channel F3. Following low-cut filtering at 0.5 Hz and line-noise removal [27], outlier rejection was performed through iterative thresholding such that all data samples fell within 5 standard deviations of the mean. Tricoherence within the diagonal slice was estimated using the direct method as described in [20] and [13] with overlapping 6-s analysis windows for frequency combinations in the range 0.3 Hz to 30 Hz, excluding windows containing any rejected samples. Feature waveforms and component signals were recovered for 3 components using the HOSD method described in [20] and [13]. Root-mean-square (RMS) amplitude was computed within successive 10 s windows for respective component signals and correlated with sleep stage indicator functions to quantify the associations with sleep stage.