Extraction of higher-order coupling feature using three and one half dimension spectrum

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Abstract

In this paper, three and a half dimension spectrum is defined. Also, the frequency coupling properties of the spectrum are investigated by using different definitions of fifth-order cumulants of real or complex sinusoidal signals. It is shown that the three and a half dimension spectrum has ability to entirely suppress symmetrical distributed noise and no frequency coupling components, that employing three and a half dimension spectrum of complex signals can extract the attended quartic or quadratic-to-cubic frequency coupling components or procreant frequency components for coupling, and that the original frequency components and the procreant frequency components for coupling can be synchronously extracted using the three and a half dimension spectrum of real signals. Computer simulations with the underwater moving target-radiated data are used to demonstrate the performance of three and a half dimension spectrum of real or complex signals under a wide range of conditions.

Introduction

When harmonic signals radiated by underwater moving target (UMT) propagate in nonlinearity underwater acoustic channel [1], [2], [3], the outputs of this channel are likely to include the original frequency components and the procreant frequency components duo to nonlinearity coupling such as the pairing frequency coupling components, the quadratic frequency coupling components, the cubic frequency coupling components [4], and other formal frequency coupling components. These frequency coupling components of the harmonic signals are the embodiment of important features of the UMT. Recently, higher-order statistics (HOS) have moved to the forefront among many researchers [5], [6], [7], [8], [9]. Applying HOS in extracting the frequency coupling components are very useful in dealing with nonlinearity problem. The quadratic frequency coupling component was extracted by using one and a half dimension spectrum or bispectrum [10], [11], [12], [13], [14], [15], [16], [17], [20] and cubic or pairing frequency coupling components were drawn out from all frequency components by using two and a half dimension spectrum or trispectrum [4], [14], [21], [23]. Yet, it is not reported how to extract higher-order frequency coupling components via employing higher-order statistics or other methods in the given literatures.

In this paper, in order to explore the method for extracting higher-order frequency coupling components of the UMT-radiated noise, the features and properties of fifth-order cumulant spectrum are studied and analyzed in theory and simulation tests.

The organization of this paper is as follows. In Section 2, the definitions of fifth-order cumulant spectrum are given and the frequency coupling properties of the spectrum studied. Section 3 describes the estimation method of fifth-order cumulant spectrum and shows the extensive simulation results. Finally, some conclusions are drawn in Section 4.

Section snippets

Three and one half dimension spectrum

If a signal x(t) at instantaneous time t is a real-valued, zero-mean, stationary random process, its fifth-order cumulants can be consistently estimated using [18], [19], [22]C5x(τ1,τ2,τ3,τ4)cum{x(t),x(t+τ1),x(t+τ2),x(t+τ3),x(t+τ4)}=E[x(t)x(t+τ1)x(t+τ2)x(t+τ3)x(t+τ4)]-E[x(t)x(t+τ1)]E[x(t+τ2)x(t+τ3)x(t+τ4)]-E[x(t)x(t+τ2)]E[x(t+τ1)x(t+τ3)x(t+τ4)]-E[x(t)x(t+τ3)]E[x(t+τ1)x(t+τ2)x(t+τ4)]-E[x(t)x(t+τ4)]E[x(t+τ1)x(t+τ2)x(t+τ3)]-E[x(t+τ1)x(t+τ2)]E[x(t)x(t+τ3)x(t+τ4)]-E[x(t+τ1)x(t+τ3)]E[x(t)x(t+τ2)x(t+τ

Estimation of three and one half dimension spectrum

The three and a half dimension spectrum of random signal x(k) at discrete time k may be estimated by the following algorithm:

  • Step (1)

    The measured data {x1, x2,  , xL} are divided into K groups, each group has M point data, and L = K × M, as well as L, K and M are integers. Each datum minus the mean value of this group measured data.

  • Step (2)

    Assume that {x(i)(k), k = 1, 2,  , M  1} is the measured data set of the ith group, and i = 1, 2,  , P. We transform the measured data {x(i)(k)} into complex counterpart xˆ(i)(k), i.e.,xˆ(i)(k

Test results

We present some numerical examples to analyze the performance of the three and a half dimension spectrum in this section. In these examples, the measured data x(k)(k represents discrete time) is the underwater moving target (UMT)-radiated data, sampling frequency fs, data record K, data length of each record N, and data total length L are 28 kHz, 7 × 4096, and 7 × 4096, respectively. The computational steps are as follows:

Conclusions

Higher-order frequency coupling components is one of important features of complex nonlinear vibration system. It is very necessary and valid to employ higher-order spectrum for analyzing these nonlinear features. According to our research results, we can come to following conclusions:

  • (1)

    Three and a half dimension spectrum is a new conception and different from one and a half dimension spectrum, two and a half dimension spectrum.

  • (2)

    Employing the three and a half dimension spectrum of complex signals

Acknowledgements

The work was supported by the Science Foundation of Anhui Province, China (No. 050420304), the Science Fund of Educational Office, Anhui Province, China (No. 2005KJ008ZD) and the Doctor fund of Anhui university of science and technology, China (No. 2004YB05).

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