Detection of Underwater Transient Acoustic Signals using. Time-Frequency Distributions and Higher-Order Spectra. Gordon J. Frazer and Boualem Boashash.
Detection of Underwater Transient Acoustic Signals using Time-Frequency Distributions and Higher-Order Spectra Gordon J. Frazer and Boualem Boashash Centre for Signal Processing Research, Queensland University of Technology, Brisbane, QLD. 4001, AUSTRALIA. Abstract I n this paper we demonstrate the application of the Wigner- Ville time-frequency distribution, the bispectrum, the time-varying bispectrum and Gerr’s third order Wigner distribution, to some underwater acoustic data and demonstrate the merit of including higherorder spectral information when signaturing underwater acoustic sources.
1
Introduction
Underwater signals, both biological and man-made, are often characterised by their acoustic emissions. Analysis in both the time domain and the frequency domain has been a standard approach for signals of this type[l]. More recently, it has been recognised that the time and frequency analysis of signals should be viewed jointly[13 and practical time-frequency analysis tools have been developed and applied to underwater acoustic signals[l][2]. The analysis methods mentioned use either the time series data directly or second moment information such as the power spectrum, the autocorrelation or the time-frequency distribution. These analysis tools do not make use of signaturing features introduced by non-Gaussian signals and nonlinearities inherent in the underwater source or introduced by the propagation medium. A supplementary analysis technique is to consider higher order cumulant sequences of the data and their corresponding spectra[6]. The most well known higher order spectra are the bispectrum and the trispectrum[lO]. With these spectra, non-Gaussianity and non-linearities inherent in the data can be identified and used to augment the signature of the acoustic source derived using the more traditional second order
methods. Existing higher-order spectral analysis techniques assume that the signal is stationary and that the ergodic property can be used to produce consistent spectral estimates. Often the signal to be analysed is not stationary and attempts have been made to define a time-varying higher-order spectra [7] [la] [3] [ll] [5]. Many analysis methods in current use are optimal for Gaussian and linear models but are suboptimal when the problem is one of analysing non-stationary, non-linear and non-Gaussian signals. The use of timevarying higher-order spectra represents one approach for problems of this type [5].
2
We now review some of the signal analysis techniques which we applied to underwater acoustic data. We assume that the signal { z ( t ) }is a random process which can be fully described by a probability density function (pdf), p ( z ) . If we impose the restriction that the pdf of the process, z ( t ) , is not changing with time, then for analysing the signal, one uses the power spectrum of the process defined as;
s(f)=
/
CO
R,(T)e-jz*fTdT
(1)
-03
Where R,(T) = E { z ( t ) z ( t+ T ) } is the autocorrelation of z ( t ) , and dependent only on the lag variable T . Power spectral analysis reveals the presence of periodicities in the signal, some of which may not have been obvious in the time domain. The assumption of stationarity is often not valid. When the signal is nonstationary, i.e. R,(t, T ) is time dependent, the power spectrum becomes “smeared”
I103 1058-6393/91$1.00 Q 1991 IEEE
Signal Analysis: A Review of Recent Developments
and much of the important information contained in the signal is lost. For this reason, the notion of a time-frequency distribution has arisen[ 11. This type of distribution attempts to localise the energy, or energy density, in time and frequency. A class of timefrequency distributions defined below has proven very useful.
for a symmetric pdf, the bispectrum and higher-order spectra are identically zero. In addition, the cumulant of the sum of two random variables is the sum of the cumulants of each random variable. For zero mean signals we use the bispectral definition derived from the moment sequence; 0 3 0 0
roo
coo
Wll f 2 )
roo
=
J_,J_, t( t
where % ( U ) is the analytic signal derived from t(n), and the function g(v, T) determines the particular time-frequency distribution. A more general class of time-frequency distributions which is optimal for any time-frequency distribution has recently been proposed in [4] and [5]. Two well known members of this class are the Wigner-Ville distribution and the spectrogram. The power spectrum and the time-frequency distribution present second moment properties of the signal. Signals with pdfs other than Gaussian contain information that cannot be analysed using second moment tools, but require higher-order moments.
2.1
Higher-Order Spectral Analysis
For a sequence of real random variables the pdf can be expanded in a power series with the terms in the series, the cumulants, describing the pdf. The rth order cumulants of a sequence (z1 , . . . , 2,) are defined to be;
+ Tl)
E{z(t)z(t
+ 72)) e- j
+ +
where r = IC1 . . . IC, and O ( f 1 , . . . , f n ) first characteristic function given by; @(fl, f 2 , .
. .,f n ) = E
{
e-j2*(flz1+,..+fnln)
2.2
Time-Varying Higher-Order Spectral Analysis
There is currently no single unified approach to the analysis of time-varying higher-order spectra. Several extentions to time-varying second moment analysis methods have been proposed [7] [ll] [5]. Thatcher and Amin extended the third moment stationary bispectrum, introducing the "running bispectrum" [12] by analogy with the spectrogram. In this paper this is referred to as the time-varying bispectrum. There are no reported applications of these methods to real data and no reports of how the distributions are best interpreted. The definition for the time-varying bispectrum is a generalisation of the definition of the spectrogram.
B ( ~ , f l , f i )=
l-A/2
t(t
}
+
A class of spectra can be defined from the higher order cumulant sequences. For zero mean random variables the moment and cumulant sequences are identical for orders one, two and three. For higher orders, however, they are not the same. Two useful higherorder spectra are the bispectrum and the trispectrum which are the Fourier transforms of C ~ ( T I72) , and cq(~1,~~ 2 ,3 ) the , third and fourth order cumulant sequences, respectively. Higher-order spectra based on cumulant sequences have the desirable property that
t+A/2
z(t)
l-A/2
Ul)Z(t
e - j 2nf 1
(4)
~ 2
Clearly, the bispectrum and other higher-order spectra present extra information to that contained in the power spectrum, since they use additional terms from the series expansion of the pdf of the process. Naturally the series expansion of the pdf may itself be time-varying. In this case the definitions of higherorder spectra presented so far will smear the spectra and information will be lost.
t+A/2
(3) is the joint
dqd
a
(5)
+
(6)
212)
~ - j1 z*fZu2
dUldu2
Gerr introduced the third-order Wigner distribution, which he called the Wigner bispectrum. He derived it from the Wigner distribution by retaining the lag centering property of the Wigner-Ville distribution and requiring the following property to be met; if the signal is third order stationary, then the expectation of the new definition is the same as the traditional bispectrum. Gerr's Wigner bispectrum, for a signal z ( t ) , is;
LL 0
WX(t, f l , f2)
=
3
0
3
2
1
3
3
t ( t - -211 - -u2)
(7)
spectrum (see figure 2) indicates a concentration of energy around the frequency range It is not clear from the power spectrum whether there is any relationship between the frequency components visible. Indeed, the power spectrum plot shows little other than the frequency extent of the signal. For example, it cannot indicate whether all frequencies are present all the time, or whether they begin and end at different times. The Wigner-Ville distribution (WVD) (see figure 4) shows clearly a dominant spectral line starting at about $ t j and increasing from approximately to approximately where it remains before terminating at about z t f . Additional lower level signal energy is also visible, some of which will be artifacts introduced by the unsmoothed bilinear term in the definition of the WVD. Nevertheless, this distribution shows considerably more information than either the time signal plot or the power spectrum and clearly indicates that the signal is non-stationary. The spectrogram (see figure 3) is a smoothed version of the WVD. While it removes the artifacts visible in the WVD, it also smoothes useful signal information and blurs the beginning and end of the signal transients. It shows the dominant time-varying frequencies well. Once again, this distribution is a substantial improvement on the signal plot and the power spectrum. The bispectrum, as defined, assumes stationarity in the same manner as the power spectrum. Consequently the bispectral plot shown (see figure 5) is the average, or smeared, bispectrum over the complete signal. The critical features visible (all 12 regions of symmetry are shown) are the major peak at approximately , $ f ), indicating phase coupling between these frequencies and the minor peak at approximately f f ; ) which indicates the generation of harmonics. Most definitely the signal should be considered to be generated by some non-linear process, and this knowledge may help detection, analysis, classification and interpretation of signals of this type. Note that this information is not available from any of the previous plots. The normalised power time-varying bispectrum (see figure 6) shows the mean of the magnitude of the time-varying bispectrum, for successive frames, normalised by the power in the frame of signal for which the bispectrum has been calculated. This gives an indication as to the presence of non-Gaussian and nonlinear behaviour throughout the duration of the signal. The single time slice of the time-varying bispectrum (see figure 7) indicates harmonic behaviour but does
kif,$if.
Swami generalised the Wigner bispectrum by retaining the same conditions as Gerr but allowing a wider class of centered lags. Boashash and O’Shea derived a class of higherorder spectra by recognising that the bilinear product in the Wigner-Ville distribution is the first order central finite difference phase estimator. They selected higher order phase estimators and derived the appropriate distributions. Interestingly, they select a subspace of the higher-order distribution and show it to be the optimal time-frequency distribution for signals with polynomial phase law. Their distribution is of the form;
Pi.
(tl
f) =
Jp, Jp, J_m_ ej2 * v ( A
4(VI
-t)dvdXe-j
+-;.(A
2*f
T)
(8)
Tdr
where; at2
I
