Bispectral Methods of Signal Processing

Alexander V. Totsky, Alexander A. Zelensky, Victor F. Kravchenko

• Publisher: De Gruyter
• Year: 2014
• ISBN:
  9783110386066
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Signals & Signal Processing
• Index: Technology & Engineering

By studying applications in radar, telecommunications and digital image restoration, this monograph discusses signal processing techniques based on bispectral methods. Improved robustness against different forms of noise as well as preservation of phase information render this method a valuable alternative to common power-spectrum analysis used in radar object recognition, digital wireless communications, and jitter removal in images.

1 General properties of bispectrum-based digitalsignal processing

1.1 General properties of cumulant and moment functions

Common power or energy spectral density estimation is a well-known and widely spread tool for random signal analysis. Ensemble averaged Fourier magnitude spectrum density does not contain any information about behavior of a centered random process in the frequency domain since the spectral components are statistically independent in different observed realizations. In this case, the energy distribution of statistically independent spectral components must be estimated since the energy content does not depend on the phase relationships for separate frequencies. Indeed, for the processes containing independent spectral components, the energy spectrum estimate is the exhaustive characteristic conventionally used in spectral analysis of such processes.

In several practical applications of signal processing, an analyzed process can contain phase coupled spectral contributions. Study of these spectral correlation relationships can give us very useful and important information for correct understanding, analysis and description of physical effects that cause a given process. Note that such information about phase coupling is irretrievably lost in common energy spectrum estimates.

Cumulant function and cumulant spectrum estimation can serve as a very useful and promising tool for signal analysis and processing. Cumulant-based approach has several important and attractive benefits as compared with energy spectrum estimation. These benefits are listed and described below.

First, consider mathematical description of cumulant spectra for a real-valued stationary and discrete-valued process given by the time series as {x(i), i = 0, 1, 2, ...}. The joint cumulants

of r^th order can be defined as

(1.1.1)

where Θ(ω₁, ω₂, ..., ω_r) = 〈exp[j(ω₁x₁+ω₂x₂ +...+ω_rx_r)]〉_x is the multidimensional characteristic function; ω₁, ω₂, ..., ω_r are the angular frequencies;

; 〈...〉 _x denotes ensemble averaging procedure; and τ_1, τ₂, ..., τ_r—1 are the time shifts.

The cumulants (1.1.1) serve as the characteristic of the probability distribution and they can be represented by the following coefficients in Taylor series for the function In Θ(ω) in the neighborhood of the point of origin

(1.1.2)

The joint moments m_x(τ_1, τ_2, ..., τ_r—1), defined for a stationary process {x(i), i = 0, 1, 2, ...} differ from the cumulants (1.1.1) as follows

(1.1.3)

The joint moments (1.1.3) can be defined by the expansion coefficients of the characteristic function Θ(ω) in Taylor series in the neighborhood of the point of origin as

(1.1.4)

The relationships between the joint cumulants (1.1.1) and the joint moments (1.1.3) in the origin under assumption that τ₁ = τ₂ = ...τ_r—1 = 0 can be written by the following formulas

(1.1.5)

For the case of a zero-mean process, that is, for

, the formulas (1.1.5) transform to the following structure

(1.1.6)

where σ² is the variance of a process under consideration.

Let us consider a real-valued discrete and zero-mean process {x(i), i = 0, 1, 2, ..., I — 1}, (x(i)) = 0. The relationships between the moment and cumulant functions for this zero-mean process can be described by the following formulas

(1.1.7a)

(1.1.7b)

(1.1.7c)

where k, l and m are the shift indices.

The formula (1.1.7a) describes the relationship between the second-order statistics and it defines conventional autocorrelation ...