Higher-Order Spectra
p	Fact: the information contained in the power spectrum of a signal is essentially that which is contained in the autocorrelation sequence
p		This is sufficient for a signal with Gaussian statistics (why?), but...
p		Premise: there is useful information in higher-order moments of the signal (typically third and fourth-order moments or cumulants)

Historical Perspective:
p	The power spectrum is a member of the class of higher order spectra, and can be referred to as a second-order spectra:
p	Such transformations can suppress additive colored Gaussian noise of unknown power spectra
p	Identify and/or reconstruct non-minimum phase signals
p	Characterize deviations from Gaussian statistics
p	Identify nonlinear properties in signals
Mathematical Definitions
The nth-order moment is defined as:

The nth-order cumulant is defined as:

where  is the nth-order moment of an equivalent Gaussian signal with the same mean and autocorrelation as .

We can expand these functions as follows:
First-order moment:
	(mean value)
Second-order moment:
	(covariance)
Third-order moment:

If the mean of the signal is zero, then the second and third-order cumulants are equivalent to the moments, and the fourth-order cumulant is given by:

Mathematical Properties of Deterministic Signals

· Spectrum
Energy spectrum:	
Bispectrum:	
Trispectrum:	

· Quality Measures (zero mean)
Variance:	
Skewness:	
Kurtosis:	
Normalized Kurtosis:	

· Coherency
Biocoherency:	
Tricoherency:	


The latter measures are useful in discriminating linear processes from nonlinear ones. A signal is said to be a linear non-Gaussian process of order  if the magnitude of the nth-order coherency function is constant for all frequencies.

An Example: Non-Minimum Phase Systems
p	Fact: the information contained in the power spectrum of a signal is essentially that which is contained in the autocorrelation sequence
p		This is sufficient for a signal with Gaussian statistics (why?), but...
p		Premise: there is useful information in higher-order moments of the signal (typically third and fourth-order moments or cumulants)

Historical Perspective:
p	The power spectrum is a member of the class of higher order spectra, and can be referred to as a second-order spectra:
Autocorrelation
Third-Order
Statistics
Fourth-Order
Statistics
nth-Order
Statistics