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Use of the Kurtosis Statistic in the Frequency Domain As an Aid In lEEE JOURNALlEEE OF OCEANICENGINEERING, VOL. OE-9, NO. 2, APRIL 1984 85 Use of the Kurtosis Statistic in the FrequencyDomain as an Aid in Detecting Random Signals Absmact-Power spectral density estimation is often employed as a couldbe utilized in signal processing. The objective ofthis method for signal ,detection. For signals which occur randomly, a paper is to compare the PSD technique for signal processing frequency domain kurtosis estimate supplements the power spectral witha new methodwhich computes the frequency domain density estimate and, in some cases, can be.employed to detect their presence. This has been verified from experiments vith real data of kurtosis (FDK) [2] forthe real and imaginary parts of the randomly occurring signals. In order to better understand the detec- complex frequency components. Kurtosis is defined as a ratio tion of randomlyoccurring signals, sinusoidal and narrow-band of a fourth-order central moment to the square of a second- Gaussian signals are considered, which when modeled to represent a order central moment. fading or multipath environment, are received as nowGaussian in Using theNeyman-Pearson theory in thetime domain, terms of a frequency domain kurtosis estimate. Several fading and multipath propagation probability density distributions of practical Ferguson [3] , has shown that kurtosis is a locally optimum interestare considered, including Rayleigh and log-normal. The detectionstatistic under certain conditions. The reader is model is generalized to handle transient and frequency modulated referred to Ferguson'swork for the details; however, it can signals by taking into account the probability of the signal being in a be simply said thatit is concernedwith detecting outliers specific frequency range over the total data interval. It is shown that from an otherwise Gaussian sample. The outliers are equivalent this model produces kurtosis values consistent with real data meas- urements. tothe randomly occurring signal that is to be detected. By The abilityof the power spectral density estimate and the frequencyextending this idea to thefrequency domain and based on domain kurtosis estimate to detect randomly occurring signals, gen- analyses of real underwateracoustics data, we have found erated from the model,is compared using the deflection criterion. isIt conditionsunder which the FDK indicates the presence of shown, for the cases considered, that over a large rangeof conditions, randomly occurring signals [2] , [4]. Both time and frequency the power spectral density estimate is a better statistic based on the deflection criterion. However, there is a small range of conditions domainanalyses of the real data havebeen performed. By over which it appears that the frequency domain kurtosis estimate settinghas the frequency parameter equal to zero in a DFT, it an advantage. The real data that initiated this analytical investigationcan be shown that the time domain is a specialcase of the are also presented. frequency domain. Analogous results should also hold in the spatial domain; however, we will only consider the frequency I. INTRODUCTION domainhere. In addition, the results are applicable to both N MAYY IMPORTANTsignal processing applications, active and passive sonar, although we will concentrate on the I includingunderwater acoustics: an estimateof the power latter application rather than the former. The objective of this spectral density (PSD) of the received data is often employed paper is to analytically determine the potential for exploiting for signal detection.The data arefirst transformed into the kurtosisestimation in thefrequency domain to indicate the freguency domain by utilizing the discrete Fourier transform presence of randomlyoccurring signals. To accomplishthis, (DFT),which can be efficientlyexecuted by analgorithm we introducea model for the receiveddata which contains called thefast Fourier transform (FFT). At this point, the the effects of amplitude and phase fluctuation of the signal. data areconsidered to be inthe frequency domain and an Inaddition, to be morerealistic, transient and frequency estimate of the PSD can be easily obtairied. Often, this esti- modulation effects of the signal are also incorporated into the mateconsists ofaveraging together asufficient number of model. References which support this model will be cited in individual FFTspectrums or periodogamsto ensure con- the text. To justify the results presented here, the PSD and sistent results [ I ] . FDK estimates will be compared in the last section using the The PSD is essentially a sum of the estimatesof the second- real underwater acoustic data that initiated this work. How- order moments for both the real and imaginary parts of each ever, subsequent data have also supported the analytical work frequencycomponent in thefrequency domain. If thefre- presented here. quency domainsignals arerandomly occurring and not Gaussian Theseresults should also apply in other fields wherethe distributed, then higherorder moments of the complex fre- detection of arandomly occurring signal is important.For quency components may contain additional information that example,the detection ofvariable stars in astronomymay benefit from this approach. Manuscriptreceived July 18, 1983; February 2, 1984. Thiswork was supported by the Office of Naval Research (Probability and Sta- 11. FREQUENCY DOMAIN KURTOSIS tistics Program). Theauthor withis the Naval UnderwaterSystemsCenter, New Lon- Let 4) = -k (4 - 'P')''] ' = '7 3 ".: - 7 don, CT 06320. q = 1, 2?.-., n representthe real discretewheredata lz is the U.S. Government work not protected by U.S. copyright .... .I . I ~,. .. .. ~. r ~ . .. .. ." 86 JOURNALIEEE OF OCEANIC ENGINEERING, VOL. OE-9, NO. 2, APRIL 1984 intervalbetween successive observations of the process. We in theestimate [8]. For randomlyoccurring signals that will use the samedefinition for the DFT, asgiven in [5] . producenoncaussian distributions, the kurtosis estimate The DFT is defined as can be less than 3 or it can have a value much greater. Several cases are examined in the paper to demonstrate the range of 1M- 1 kurtosis values for various situations. x(q,F~) = Tqix(i, 4)exp (-jFpi) (1) Techniquesfor optimally processingsignals contaminated i= 0 by under-iceambient noise aswell as other noiseenviron- ments are presented in [9] and . where the symbols are identified as [IO] The FDK is defined by taking the expected valueof the j=a. fourth-order central moment and the square of the expected value of thesecond-order central moment separately, and then forming the ratio. The resultof this operation is Fp = 27rfph is thepth radian frequencycomponent, p = 0, 1, .*a, M - 1, and fp = p/Mh Hz. For simplicity, we shall resume the window weights equal one, Le., Wi= 1, for all i, K(Fp)=E{[X(q,F,)14}/IE[(X(q,F,))21)2. (3) andh=l. The power spectrum estimate is defined as Beforeproceeding further, we need to definea model for the received data. Our goal is to compare the FDK and n PSD estimatesunder some conditions which are known to Vp)= (1/n> X(q,Fp)x*(q, Fp) (2) q=1 occur in underwateracoustic detection problems, but have not been explicitly evaluated in this way before. The model where the asterisk represents complex conjugate. The variance we employ assumes that the transmitted or radiated signal is actedupon multiplicatively by the medium which causes of the periodogram does not go to zero as A4 + -, and there- fore the periodogram is not a consistent estimate of the PSD amplitudemodulation and frequency spreading tooccur. 151. In the PSDestimate considered here, n nonoverlapped This is obviously notthe most general modelpossible, but DFT segmentsare averaged to ensurethat each frequency it is adequateto answersome importantsonar design ques- componentrepresents a consistent PSD estimate [5] . Thus tions. (2) is anasymptotically unbiased estimate of the power Theinput, x(i, q), will be azero-mean process which is spectral density [5] . composed of an additive mixture of signal and noise of the The FDK estimate wasdiscussed in terms of a detection form problem in [6] . The main concern here is to determine the sensitivity of the PSD and FDK estimates to randomly occur- x(i, 4) = w,4) + m(i, qMi, 4) (4) ring signals, assuming a sufficient number of DFT segments is available. The FDK represents a measure for the probability where m(i, q) modulates the signal and will either represent distribution over a time interval consisting of many DFT seg- the effects of the propagation medium or reflect a physical ments. Many of the arctic segments, which will be discussed in characteristic of thetransmitted or radiated signal s(i, q). the last section, showed highly dynamic frequency components.The components N(i, qj and s(i, q) are zero-mean stationary This was dueto the highlydynamic nature of ice sounds. processes and N(i, q), m(i, q) and s(i, q),will be assumed to Thesedynamic components were easily descernedusing a be mutually independent from each other. For the particular spectrogram.The advantages of the DFK for estimating the choicesof m(i, q) and s(i, q) givenin the text, x(i, q) will statistical beha\lor of ice sounds over the spectro, Gram are be stationary in the wide sense. that an operator is not needed and that it produces a quantita- Ourmodel for the fading receivedsignal m(i, q)s(i, q) tive measure of the probabiIity distribution. This measure can assumes that the total effects of theamplitude fluctuations also be used to distinguish stationary sinusoids and Gaussian due tomultipath interference or to nonstationarities of the signals from ice sounds. source, receiver, or of themedium canbe simply included It should alsobe pointedout that overlapped segments in the multiplicativefunction m(i, q). Onthe other hand, havealso been studiedto reduce the variancein the PSD the phase fluctuations of the signal will be contained in the estimatefor another application [7], but ths technique function s(i, q) itself. This approach appliesto sound propagat- will not be treated here. ing in the ocean [l 11 and to electromagnetic communication The FDK is defined separately for the real and imaginary systems [12] . Later, we will generalize this model to include parts of (1).
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