Polyspectra, also known as higher-order spectra, are frequency-domain statistical quantities that generalize the spectral density (power spectrum) to higher orders. Polyspectra of a given signal can reveal non-Gaussian behavior, time-inversion asymmetries, and phase correlations between different frequency contributions to the signal. While a standard power spectrum quantifies the distribution of intensity across frequencies, polyspectra characterize higher-order phase and amplitude correlations between frequency components. Higher-order spectra include the third-order bispectrum, the fourth-order trispectrum, and their multivariate generalizations.
Definition In his original publications, Brillinger considers a stationary process (the signal) z ( t ) {\displaystyle z(t)} and its multi-time cumulant.
f ( τ 1 , ⋯ , τ n − 1 ) = C n ( z ( t ) , z ( t + τ 1 ) , ⋯ , z ( t + τ n − 1 ) ) . {\displaystyle f(\tau _{1},\cdots ,\tau _{n-1})=C_{n}(z(t),z(t+\tau _{1}),\cdots ,z(t+\tau _{n-1})).}
The cumulant does not depend on t {\displaystyle t} since z ( t ) {\displaystyle z(t)} is stationary. This allows for a definition of n {\displaystyle n} -th order polyspectra S z ( n ) {\displaystyle S_{z}^{(n)}} of a stationary signal z ( t ) {\displaystyle z(t)} by
C n ( z ( ω 1 ) , ⋯ , z ( ω n ) ) = 2 π δ ( ω 1 + ⋯ + ω n ) S z ( n ) ( ω 1 , ⋯ , ω n − 1 ) {\displaystyle C_{n}{\big (}z(\omega _{1}),\cdots ,z(\omega _{n}){\big )}=2\pi \delta (\omega _{1}+\cdots +\omega _{n})S_{z}^{(n)}(\omega _{1},\cdots ,\omega _{n-1})}
with the Fourier transform
z ( ω ) = ∫ − ∞ + ∞ e i ω τ z ( τ ) d τ , {\displaystyle z(\omega )=\int _{-\infty }^{+\infty }e^{i\omega \tau }z(\tau )d\tau ,}
and the Dirac delta function δ {\displaystyle \delta } which appears due to the independence of f {\displaystyle f} on t {\displaystyle t} .
The second order spectrum can be rewritten as
S z ( 2 ) ( ω ) = ∫ − ∞ + ∞ e i ω τ C 2 ( z ( t + τ ) , z ( t ) ) d τ , {\displaystyle S_{z}^{(2)}(\omega )=\int _{-\infty }^{+\infty }e^{i\omega \tau }C_{2}(z(t+\tau ),z(t))d\tau ,}
where C 2 ( x , y ) = ⟨ x y ⟩ − ⟨ x ⟩ ⟨ y ⟩ {\displaystyle C_{2}(x,y)=\langle xy\rangle -\langle x\rangle \langle y\rangle } is the covariance. The representation of S z ( 2 ) {\displaystyle S_{z}^{(2)}} is equivalent to the definition of the spectral density of z {\displaystyle z} (power spectrum) as Fourier transform of the autocorrelation function of z ( t ) {\displaystyle z(t)} .
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