Detection of Surface Ships From Interception of ... - IEEE Xplore

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IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 37, NO. 3, JULY 2012

Detection of Surface Ships From Interception of Cyclostationary Signature With the Cyclic Modulation Coherence Jerome Antoni and David Hanson

Abstract—This paper addresses the problem of passive detection of surface ships from radiated propeller noise in the far field. This is traditionally accomplished by means of envelope analysis, a largely empirical technique. The first contribution of the paper is to give a formal justification why this approach is sound based on the cyclostationary modeling of propeller noise. Pursuing along similar lines, the paper proposes a new statistical test based on the recently introduced cyclic modulation coherence (CMC), which is close to optimal in detecting cyclostationary noise, while at the same time being easy and extremely fast to compute. Detection capability is thoroughly investigated as a function of the blade pass frequency and other parameter settings. Finally, the cyclostationarity framework makes possible the proposal of a closed-form statistical threshold that opens the way to automatic detection. Index Terms—Ship detection, propeller noise, cyclostationary signal, cyclic spectral analysis, Detection of Envelope Modulation On Noise (DEMON) processing. Fig. 1. Power spectral density of a hydrophone signal measured in the North Sea nearby a coaster: (a) [0; 5 kHz] range; (b) [0; 200 Hz] range. (Frequency 0.33 Hz.) resolution

I. INTRODUCTION

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F key interest to submariners is the ability to detect the presence of surface ships while remaining undetected themselves. To this end, passive detection techniques have been developed whereby the surface ship is detected, and in some cases classified, based on its noise emissions which are recorded by hydrophones on the submarine. Detection is impeded when the signal from the ship is lost in the noise, which can occur when the marine environment is particularly noisy and/or when the ship is distant from the observer. To maintain contact with a target vessel, the detection must rely on some form of signal processing to enhance the acoustic signature of the ship and attenuate the extraneous components in the sonar signal. Another reason that jeopardizes detection is the inherent broadband nature of propeller noise. The principal source of acoustic energy in the propeller signal is provided by cavitation [1]. Cavitation is a process whereby bubbles are drawn out of Manuscript received November 20, 2010; revised November 23, 2011; accepted April 11, 2012. Date of publication June 12, 2012; date of current version July 10, 2012. Associate Editor: W. Xu. J. Antoni is with the UMR CNRS 6253, University of Technology of Compiegne, Compiegne 60200, France and also with Vibrations and Acoustic Laboratory, University of Lyon (INSA), Villeurbanne CEDEX F-69621, France ([email protected]). D. Hanson is with the Dynamics Group, Sinclair Knight Merz, Sydney, N.S.W. 2065, Australia. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JOE.2012.2195852

the water by pressure gradients on the blade surface and edges. These bubbles are unstable, and it is their collapse that produces the noise. The spectral content of propeller cavitation noise is quite broadband, with significant energy out to at least 100 kHz. The degree of cavitation is related to the water pressure which varies with depth and nonuniformity in the wake field. Therefore, the cavitation noise will modulate as the propeller blade rotates through varying water depth. The propeller signal thereby comprises amplitude-modulated cavitation components, with a modulation period akin to the propeller frequency. Expressed another way, the propeller signal can be seen to be made up of a broadband carrier component modulated by the periodic blade rotation. The fact that the propeller signal is not exclusively composed of a purely periodic component but rather and mainly of a broadband carrier invalidates the use of a large number of signal processing techniques dedicated to the detection of hidden harmonics in noise—see, e.g., [2] and [3]. This is clearly illustrated in the example of Fig. 1, where the spectral analysis of the propeller noise emitted by a nearby coaster in the North Sea is unable to reveal the shaft rotation (expected at 5.75 Hz) and blade pass frequency (expected at 23 0.33 Hz) despite the very fine frequency resolution ( Hz) and the very long time recording (172 periods of the shaft). As a matter of fact, envelope detection has largely prevailed so far as the conventional way to process the propeller signal. It basically consists of three steps dedicated to extracting a periodic envelope carried by a broadband component. The first

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ANTONI AND HANSON: DETECTION OF SURFACE SHIPS FROM INTERCEPTION OF CYCLOSTATIONARY SIGNATURE

Fig. 2. Periodicity in the autocovariance of a second-order cyclostationary signal; a section of a burst random signal (top) and its autocovariance (bottom) . at time lag

step is to perform a bandpass filtration in a band where cavitation noise is assumed to dominate. Next, the signal is rectified and lowpass filtered so as to extract its envelope. Finally, the spectrum of the envelope is computed to check for the presence of periodic components. Detection of Envelope Modulation On Noise (DEMON) processing is one implementation of that technique employed by submariners to detect the presence of propeller craft [4]. However, this technique appears to be largely empirical, with nontrivial tunings and few papers related to it in the literature. Recent extensions to the technique were made by Li and Yang [5] who employed higher order statistics to suppress Gaussian noise, and Kummert [6] who proposed to replace the human operator intervention by fuzzy logic, but without amendment to the basic envelope processing technique. The main objective of this paper is to revisit the empirical envelope detection technique and to optimize it in light of recent results on cyclostationary signal processing [7], [8]. The term “cyclostationary” refers to a special class of nonstationary signals which are random in nature, but exhibit periodicity in their statistics. A first-order cyclostationary signal (CS1) will exhibit periodicity in its first-order statistics, i.e., its ensemble average will be periodic; at the second order, its autocovariance [9]. By way of an example, consider a burst random signal as represented in Fig. 2: the first-order statistics of the signal, i.e., the ensemble average over one ON/OFF cycle, is zero and so not periodic; however, the autocovariance of the signal can be seen to exhibit periodicity in time . Therefore, this signal may be described as second-order cyclostationary (CS2). Of particular importance to this work is the role played by cavitation of propeller blades as they pass through the water. As previously explained, cavitation may be considered as a broadband and random phenomenon, but occurring in a periodic fashion related to the shaft rotation, thus producing cyclostationarity. As far as the authors know, the cyclostationary property of propeller signals has rarely been recognized in the past. In a related field of applications, Gaonkar and Hohenemser [10], [11] and later George et al. [12] modeled the rotor blade flapping response to atmospheric turbulence as cyclostationary. In [13], Jha et al. pointed out that cyclostationary models are almost unknown to researchers in mechanics (vibroacoustics of bladed rotors). It is not until recently that the same authors gave a detailed model of the cyclostationary vibration of a ship propeller [14]. Similarly, Jurdic et al. [15] recently showed the benefits of a cyclostationary approach in analyzing rotor turbulence wakes. Indeed, as soon as the cyclostationary property of

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propeller signals is recognized, a strong theoretical framework together with an entire toolbox is then available to tackle the detection problem on rigorous grounds. Reference [16] seems to be one of the first works in this direction, yet where ambient noise rather than the signal of interest was considered cyclostationary. Degoul [17] explicitly exploited the cyclostationarity property of propeller noise to devise new detection statistics based on the spectral correlation density (the Fourier transform of the aforementioned autocovariance function in both time and time-lag ; see Section III-B). Computational aspects of the same method are investigated in [18]. Amindavar and Moghaddam [19] proposed similar detection statistics that proved significantly superior to classical harmonic detection. In view of the rather indigent state of the art, the main contributions of the paper are the following. 1) Section II: The cyclostationary property of propeller noise is demonstrated from the acoustic equations, thus explaining why second-order cyclostationarity associated with periodically modulated broadband noise may dominate over first-order cyclostationarity associated with purely periodic components (at least at high frequencies) or, stated differently, why propeller noise may bear a continuous rather than a discrete spectrum as exemplified in Fig. 1(b). 2) Section III: The cyclostationary property of propeller noise is exploited to devise the optimal strategies that should be followed for detection. 3) Section IV: A simple detection statistic is proposed which amends traditional envelope detection (DEMON) so as to make it nearly optimal, together with a closed-form expression for a statistical threshold that allows for automatic detection. II. CYCLOSTATIONARY MODELING OF PROPELLER NOISE The accurate modeling of the sound field produced by a marine propeller is a difficult problem of hydroacoustics that requires advanced analytical and numerical tools [20]–[22]. The object of this section is not to pursue this direction, but rather to give a phenomenological model that can explain, from a qualitative point of view, why the sound radiated by the rotation of a propeller in water becomes predominantly broadband in high frequencies and how it still conveys information on hidden periodicities (second-order cyclostationarity) as exemplified by actual measurements. It is emphasized that the objective is qualitative and by no means quantitative, the purpose being detection and not prediction. To do so, a number of simplifying assumptions are allowed. To start with, only the sound field produced by the rotation of a single blade will be considered, that resulting from a marine propeller being simply a summation over several blades in virtue of the principle of superposition of linear acoustics. This will not account for the interaction between adjacent blades (passage of the next blade in the wake of the previous one), yet the radiation in the far field of blade–vortex interaction noise may be reasonably negligible as compared to the other sources introduced hereafter. Next, sound waves will be assumed to propagate in an unbounded homogeneous medium. This implies that wave reflections on the sea surface and bottom

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Fig. 3. Sound radiation from a blade rotating in the the combination of a moving monopole with strength . with moment


-plane represented as and a moving dipole

will not be accounted for, without direct consequence for our purpose (secondary paths could be introduced with the image source method). The homogeneity assumption, although simplistic, will not jeopardize the statistical properties of the sound to be proved in this section, as explained in Section II-B. Finally and as is commonplace, propeller noise will be assumed to originate from a compact source of noise. Although such an assumption is unable to capture the subtle physics of local phenomena around the propeller blade, it is sufficient to provide good prediction in the far field.

A. Sound Field Radiated by Rotating Point Sources Based on the above assumptions, this section derives an expression of the propeller noise radiated in the far field. The basic idea is to decompose the noise source into a monopole and a dipole that accounts for thickness (cavitation) and loading noise, respectively, as proposed, for instance, in [25] and [26]. A major difference with the existing literature, however, is the consideration that the monopole strength and dipole moment are partly random functions of time. Cavitation bubbles produced by the rotation of the blade constitute a major source of propeller noise—so much indeed as to provide a characteristic feature of the acoustical signature of surface ships. This is conveniently modeled as a rotating monopole concentrated at some distance from the hub. Cavitation being mainly a random phenomenon in nature, it is sensible to model its effective source strength as the combination of a constant component and a stationary randomly fluctuating component , such that . In addition, since the rotation of the blade takes place in a spatially nonuniform flow owing to circumferential variation in the wake field and to the vertical gradient of the hydrostatic pressure, a periodic modulation is further imposed so that the monopole strength finally reads where is a periodic function of the propeller rotation angle with the shaft rotation speed. Another source of noise originates from the action of the propeller blade on the fluid volume. This is conveniently modeled as a condensed rotating force (thrust plus drag components) located at a distance from the hub that generates a rotating dipole field. As before, it is sensible to decompose into a constant term that accounts for the average loading and a stationary randomly fluctuating component that reflects “turbulent loading” of the fluid due to cavitation, inflow

turbulence, and various blade edge effects, and which does not contribute directly to the net force of motion but considerably to the radiated noise. For the same reasons as advocated above, the loading force is further modulated by a periodic function of angle so that, finally, the dipole moment reads . The sound field produced at time at the observation position by the moving cavitation source strength and point force is then returned by [27], [28]

(1) where

is the speed of sound in the medium, is the distance from the observer to the moving force (see Fig. 3), is the Mach number in the direction of the observer (which depends on through and ), is the component of the rotating force in the radiation direction, and the square brackets around the whole expression indicate that the latter is to be evaluated at the retarded time . Note that the underwater ambient noise is modeled in (1) as an additive stationary random process whose power spectral density is typically a decreasing function of frequency. The assumption of stationarity is obviously simplistic, but without serious implication as explained in Section III-B. Many insights can now be gained from (1), as is detailed in Sections III-A and III-B. B. Tonal and Broadband Noise in the Far Field Collecting deterministic and random terms apart, (1) shows that the overall radiated noise comprises a tonal part

(2) where the approximation leading to the second line is based on the assumption of a small modulation depth, i.e., and (it is reminded that modulation


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effects are on the second order with respect to static loading), and a broadband part

Fig. 4. Schematic propeller noise spectrum showing the relative contributions of tonal noise, broadband noise, and ambient noise in the far field.

(3) On the one hand, the tonal part in (2) radiates energy at multiples of the rotation speed due to the periodicity of the modulating functions , and . However, because these are smooth functions, the extent of the frequency spectrum is typically limited to only a few significant harmonics. On the other hand, the broadband noise radiates energy over a continuum of frequencies and in a much wider range due to its amplification in high frequencies by the time derivative of the random fluctuations and . This is confirmed by experimental spectra of cavitation noise which typically evidence a “white noise” behavior that extends far beyond the audible range [21]. It is seen in (2) and (3) that the amplitude of both tonal and broadband noises decreases like in the far field. Since tonal noise typically occupies the same frequency range as underwater ambient noise, its signal-to-noise ratio (SNR) is likely to decrease to so small a level when it arrives at the hydrophone in the far field that its detection by classical spectral analysis is jeopardized. This is schematically illustrated in Fig. 4, and was experimentally verified in the introductory example of Fig. 1. Broadband noise, on the other hand, has its energy spread high enough in frequency to maintain a good SNR against low-frequency ambient noise even in the far field. This explains why the spectrum of propeller noise, as measured in the far field, may turn out completely broadband, without any remaining discrete frequency component, somehow in contradiction with one’s intuition that tonal noise should have radiated more efficiently due to lesser attenuation with distance of low-frequency components. Although this situation is surely not typical [23], [24], the present attempt to explain it will shortly establish that, on a more general basis: 1) ship signature also shows periodic modulations of broadband noise in a high-frequency range; and, consequently, 2) this can be used for detection whenever the low-frequency (discrete) spectral components are buried into ocean noise. Finally, note that the more realistic scenario where propagation takes place in a nonhomogeneous medium (e.g., with sound-speed variations) will even strengthen the random nature of the received sound in the far field (due to refraction and phase alteration) as compared to the simplistic homogenous medium considered so far. It now remains to be seen in which way broadband propeller noise can transmit the information on the propeller frequencies (rotational speed and blade passage frequency) without necessarily evidencing any discrete spectral component.

C. Broadband Noise Is Cyclostationary Section II-B has given some insights into the reasons why propeller noise is essentially broadband in the far field. However, broadband noise still contains the information on hidden periodicities in the form of periodic modulations due to the Doppler effect and the rotation of blades in a nonuniform flow. Equation (3) is explicit in this regard. First, it involves an amplitude modulation through multiplication by the periodic functions , and . Second, its evaluation at the retarded time [ varies periodically with ] introduces a periodic phase modulation. These are two mechanisms that make the radiated broadband noise nonstationary with periodic modulations, that is, cyclostationary. Cyclostationarity is the key property by which a broadband (highfrequency) wave can carry a discrete (lowpass) information. III. THE SEARCH FOR HIDDEN PERIODICITY This section discusses how the cyclostationary property of propeller noise can be exploited to reveal the existence of hidden periodicities related to the rotation frequency and its harmonics, even in the presence of extremely low SNRs. To start with, let us then consider the scenario where the signal measured at the hydrophone is purely broadband, i.e., . From the above discussion, if only amplitude modulation is considered, then where is a periodic function of period [for instance, , and in (3)] and is a stationary random carrier1 [for instance, and in (3)]. Similarly, a phase-modulated signal may be seen as a summation of many amplitude-modulated narrowband processes, such that each frequency component in the signal appears intermittently in time. Thus, where again each is of period and where ’s are narrowband stationary random carriers. Upon expanding each in Fourier series , one arrives at (4) where . It is noteworthy that (4) is simply a Fourier series whose coefficients are mutually correlated stationary random functions of time. Beyond the physical reasons which have motivated its derivation, this equation happens to 1Referring to [29], the “carrier signal” has its amplitude multiplied by the information-bearing signal—typically referred to as the “modulating signal.”

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be fully general to describe a cyclostationary signal [29]. Not only does it explicitly highlight the hidden periodicity inherent to a cyclostationary signal, but it also helps in understanding the major statistical properties of such a signal, as briefly outlined in Section III-A.

where , , and are the filtered versions of the fluctuating monopole strength , dipole component , and additive noise , respectively. Direct application of (6) then yields

A. Time Statistics The energy conveyed by a cyclostationary signal as a function of time, say , may be conveniently computed from (4) as (5) [the bracket operator dewhere fines the stationary time-average ] is the periodic time-averaging operator that extracts all periodic components, (with “*” the complex conjugate symbol) is the cross correlation between the Fourier coefficients and , and is the variance of the stationary noise. This is clearly a periodic function of time with period , meaning that the energy of a cyclostationary process is flowing periodically. The same analysis applies per frequency bands. Letting denote the filtered signal in a narrowband of width centered on frequency , it readily follows that

(10) and the power spectral densities of the with monopole and dipole sources whose fluctuations have been assumed mutually uncorrelated for simplicity (taking correlation into account would be an easy matter, yet it would not provide any deeper insight into the results of the present demonstration). Clearly, the periodicity of is inherited from the periodic modulations and . Equation (10) indicates in passing that the dipole source essentially radiates cyclostationarity along the propeller axis where the force component bears most of its intensity (thrust component), whereas the monopole source radiates cyclostationarity uniformly in all directions. B. Frequency Statistics As soon as the frequency domain is of interest, another important quantity is returned by the spectral correlation density (11)

(6) is the noise power spectral density and where the where summation is now limited to those harmonics that are included in the frequency band of interest, i.e., such that . This shows that the energy of a cyclostationary signal is also flowing periodically through a frequency band . One limit of this approach is that it requires at least two harmonics to be included in the band of interest for the complex exponential not to vanish on the right-hand side of (6), that is, the condition (7) must hold. This places a frequency resolution bound which, if not respected, will jeopardize the detection of the presence of cyclostationarity in a signal. Note that in the case of blades, the detection of the blade pass frequency will require (8) The energy flow radiated by a propeller in a high-frequency narrowband may be readily established from (6) under the assumption of a small Mach number at the blade tip and of negligible phase modulation in the retarded time . From (3), the filtered broadband noise behaves in the higher frequency range as

(9)

As indicated by its name, measures the statistical link between two frequency components at frequencies and . From (4), the spectral correlation density of a cyclostationary signal is (see Appendix A) (12) stands for the cross-spectral density between where Fourier coefficients and and where is the Kronecker symbol equal to one for and zero otherwise. From (9) and (11), the spectral correlation density of the propeller broadband noise radiated in the high-frequency range is

(13) and are where the Fourier coefficients of the modulating functions and , respectively. Several important remarks arise in view of this result. 1) The classical power spectral density is seen as a particular case of the spectral correlation density when , that is, . Because it involves a summation of continuous spectra, the power spectral density is thus itself a continuous spectrum, the reason why it may not show


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frequency domain approach, as is customary in acoustics. The optimal approach in that case is probably based on the spectral coherence density, which is briefly resumed in Section IV-A. An alternative suboptimal test based on the recently proposed cyclic modulation spectrum [38] is then proposed, which is a much more attractive tool from the computational point of view while achieving a nearly identical detection capability. A. Detection Based on the Spectral Coherence Density It has then been established in the literature that the (squared) spectral coherence density (14)

Fig. 5. Schematic spectral correlation density showing a typical cyclostationary signature continuous in the spectral frequency and discrete in the cyclic frequency .

any discrete frequency in spite of the presence of periodic modulations. 2) The spectral correlation density is zero almost everywhere except at values of which coincides with a multiple of the shaft rotation speed . In other words, the spectral correlation density is a discrete function of the so-called “cyclic frequency” (although continuous in the “spectral frequency” ). This is the “cyclostationary signature” of the blade, as illustrated in Fig. 5, which will be intensively exploited in Section IV. 3) The effect of stationary underwater noise is entirely concentrated at the zero cyclic frequency (independently of its color), thus leaving theoretically unaffected the detection of discrete components along the cyclic frequency axis even under extremely noisy environments. This property remains robust even in the more realistic case of nonstationary noise (indeed as long as the noise itself is not cyclostationary) because the spectral correlation density would then spread near , but without seriously interfering with the discrete pattern of propeller noise in [30] (this is further discussed in Section V and illustrated in Fig. 11 therein). 4) Since the spectral correlation density contains a factor that acts like a highpass filter, its examination in a high-frequency range will somewhat compensate for its decrease as , in particular when noisy measurements in the far field are of concern. IV. DETECTION OF CYCLOSTATIONARITY IN THE FAR FIELD The central idea of the paper is to propose a statistical test for detecting marine propeller noise based on the cyclostationary signature produced in the far field. It is now clear that such an approach is expected to perform well even in the case of broadband noise and in unfavorable SNRs. The detection of the presence of cyclostationarity has been addressed in a number of papers, either from the time domain or from the frequency domain [31]–[35]. This paper favors the

is both the uniformly most powerful and the likelihood ratio test statistic under the Gaussian assumption, two properties which make it optimal in the sense of being invariant under any linear prefiltering of the signal and of maximizing the probability of correct detection given a fixed rate of false alarm, respectively [36], [37], [39]. In this respect, it should be preferred to other detection statistics such as those based on the spectral correlation, as formerly proposed in [17]–[19] for propeller noise detection. Given an estimate of the spectral coherence density from a finite-length record , the statistical test reads: Reject the null hypothesis H0 “there is no presence of cyclostationarity at cyclic frequency and spectral frequency ” at the level of significance if (15) where is the percentile of the chi-square law with two degrees of freedom and is the variance reduction factor of the estimator [39]. The fact that the spectral coherence density is scaled between 0 and 1 makes it ideally suited to comparisons on a normalized basis, its interpretation being that of a measure of cyclostationary “intensity.” To see this, let us reasonably assume the propeller noise spectrum is smooth enough, so that, for a low cyclic frequency (a small multiple of the shaft rotation speed ), . Thus SNR

(16)

where SNR

is the SNR and where (from Schwarz’s inequality) is clearly a relative measure of cyclostationarity for a given sound pressure level. In addition to its optimality, another advantage of the detection test (15) is surely its versatility; however, its main drawback is its high computational burden which makes it hardly suited to real-time implementation unless restricted to narrow frequency ranges in and , but this being at the cost of increased intervention of a human operator. The object of Sections IV-B and IV-C is to propose an alternative detection statistic with nearly equal performance, yet one which is ideally suited to real-time processing. B. Definition of the Cyclic Modulation Coherence The cyclic modulation coherence (CMC) was recently introduced in [38] as a power-normalized version of the energy flow

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defined in (6). The CMC intends to approximate the spectral coherence density while offering a much faster way of computation by making a systematic use of the discrete Fourier transform (DFT) and its related fast Fourier transform (FFT) algorithm. Let , , be the -long discrete signal to be analyzed (note the change from continuous time to discrete time , with the sampling frequency). An -long analysis window —typically a Hanning window—with unit norm is introduced and shifted along the signal by steps of samples, such that (with ) smoothly cuts off a block of the signal (a “snapshot”) around time samples . Note that there are overlapping snapshots, with the fraction of overlap. The next step is to compute the short-time DFT (17) which provides samples at times of the filtered signal in a narrow frequency band of width centered on frequency . As explained before, this is not enough to detect a cyclostationary signal which, in general, is not periodic in its waveform but allows a random carrier to transmit energy with periodic bursts in some frequency band. The detection of such a behavior is achieved by means of a second DFT on the squared magnitude of that transforms time into its dual-frequency variable (18) The quantity was coined the cyclic modulation spectrum in [38] because of its ability to detect periodic modulation of random noise. In short, it is simply the DFT of the so-called spectrogram. Note that, in the special case where boils down to the averaged-periodogram estimate of the power spectral density (PSD) of the signal. The CMC is then defined as CMC

(19)

that is the cyclic modulation spectrum normalized by the estimated PSD so as to eliminate all scaling effects. Therefore, the capability of detecting the presence of cyclostationarity in some frequency band will not depend on the actual energy level in that band, but only on whether energy fluctuates periodically therein. Note that the normalization is also equivalent to computing the

CMC

cyclic modulation spectrum of the whitened signal, which is a customary preprocessing step in most detection tests. C. The CMC in the Presence of Cyclostationarity To understand the behavior of the CMC in the presence of cyclostationarity (and in particular how it relates to the spectral coherence density) let us investigate its expression in the simple case where signal exhibits cyclostationarity at a single and arbitrary cyclic frequency (note this is without loss of generality, since the result will be duplicated accordingly in the case of multiple cyclic frequencies). In that case, according to the definition of cyclostationarity, one looks for the presence of a sinusoidal component in the squared-magnitude energy flow (see Fig. 2), i.e., (20) is an arbitrary phase, and where is a stationary “noise” component that contains a high enough direct current (dc) level such that . Under some mild assumptions about the smoothness of the PSD of , it can be shown that (see Appendix B) CMC (21) where

1)

the

Dirichlet

is the only function of , and therefore, it controls the cyclic frequency resolution, i.e., , as returned by the width of its main lobe; 2) the “cyclostationary” SNR returns the strength of the cyclostationary component relatively to the signal power at frequency ; and 3) the residual estimation noise has magnitude that vanishes like in probability, with the number of snapshots. Therefore, because rapidly tends to a train of discrete delta functions with period , we have (22), shown at the bottom of the page. This is fine enough to detect the presence of cyclostationarity at , but not fully satisfying since the CMC also returns nonzero values at all other cyclic frequencies , thus erroneously indicating other cyclostationary components where they do not exist. The reason stems from undersampling (seen as a function of time ) by factor in (17) which entails frequency aliasing in . To gain more insight into this issue and see how to solve it, it must be realized that in (20) is related to the spectral correlation density as (see Appendix C) (23)

for any interger elsewhere.

kernel,

(22)


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where

with . In (23), is seen to act as a lowpass filter in which gradually brings down to zero as increases. For instance, for an -long Hanning window, and 30 dB for [this upper bound is set voluntarily large to be conservative with the stronger condition required for aliasing to disappear in (49) of Appendix D]. The reason for this is clear: the narrower the frequency band , the slower the variations of the energy flow through it, with cutoff frequency2 (24) The existence of such a cutoff frequency is actually a chance to reject the undesirable aliased cyclic frequencies in (22). This is achieved provided that , i.e., with a Hanning window, meaning that at least 75% overlap should be set when computing the short-time DFT in (17). The CMC is then nonzero at and only, thus detecting cyclostationarity at the correct location. A final remark concerns the magnitude of the CMC at the cyclic frequency , which, according to (22) and (23), is asymptotically

Fig. 6. Schematic illustration of the CMC in the presence of cyclostationarity at cyclic frequency . Length of the signal controls the cyclic frequency reso; length of the analysis window controls lution . The shaded area the cyclic frequency bandwidth indicates regions of theoretical zero values but affected by estimation noise on . the order of

CMC (25) In short, the CMC is a scaled approximation of the spectral coherence density introduced in Section IV-A, and therefore, it inherits all its useful properties. As explained above, the scaling factor places a limit beyond which the CMC virtually vanishes. This is the most serious drawback of the CMC as compared to the spectral coherence density which does not suffer from such a limitation. Fig. 6 illustrates the main features of the CMC discussed in this paragraph. D. The CMC in the Presence of Tonal Noise The CMC has been designed to detect the presence of periodically modulated (i.e., second-order cyclostationary) broadband noise. A natural question arises as whether it can also detect the presence of tonal noise. The issue is particularly relevant as Fig. 4 exhibits a frequency range where tonal and broadband noises are likely to coexist (the extent of that range depends on such factors as the propeller design, dimensions, speed, surface roughness, etc.). Even more dramatically, there may be temptation to use the CMC to detect pure tonal noise as it appears in the low-frequency range of Fig. 4. To answer this question, let us consider without loss of generality to our purpose the case where the signal contains a pure tone in stationary additive noise, i.e., (26) 2This limit can also be proved by a direct application of the uncertainty principle in (7), the difference in factor 4 coming from the assumption of an ideal bandpass filter in Section III-A instead of the DFT filter used in this section.

Fig. 7. Example of a simulated cyclostationary signal ( 0 dB). SNR

20 Hz,

Then, the energy flow in a narrowband subject to condition (24) (under this condition, the negative and positive frequency components of the cosine are both passed through the band centered on ) reads

(27) which, in turn, implies that CMC

(28) This result is found quite similar to (21) except that 1) the tone frequency is detected at instead of ; and 2) the CMC is weighted by factor . Since the width of is on the order of , the CMC is significantly nonzero in the frequency band only. This proves the CMC is able to detect the presence of tonal noise and to recognize it as such (for it is confined below contrary to broadband noise, which

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Fig. 8. CMC in the case of (a) a stationary signal; (b) a cyclostationary signal with 20 Hz and SNR 20 Hz and SNR 10 dB; and (d) a cyclostationary signal with 70 Hz and SNR 0 dB. (

is allowed to occupy the full frequency spectrum). However, (28) being essentially the Fourier transform of the filtered and squared signal, it is generally not as optimal as the Fourier transform of the raw signal to detect a tone [3] (squaring will worsen poor SNRs lower than unity). In other words, the CMC and the power spectrum complement each other: on the one hand, the Fourier transform is optimal to detect tonal noise, but it cannot detect second-order cyclostationary noise; on the other hand, the CMC is optimal to detect second-order cyclostationary noise, but it is not optimal for tonal noise even though it can actually detect it. Indeed, the combination of the two tools provides a simple and efficient solution to discriminate the nature of propeller noise received in the far field. E. Statistical Test for the Detection of Propeller Noise The capability of the CMC to mimic the spectral coherence density at a marginal computational cost makes it an ideal candidate to detect propeller noise. However, its usage is limited to a restrained cyclic frequency range. In most sonar applications, this will not be a problem, since the cyclic frequency of interest (typically a few tens of hertz) will usually be much smaller than the coarser allowable spectral resolution, i.e., , which is consistent with the requirement in (24). Such a setting may easily be fixed once and for all to allow fully automatic processing. Although the CMC seems to follow in essence similar processing lines as DEMON (i.e., a bandpass filter and an envelope detection), the main differences are the following. • The CMC was derived in a deductive way to approach the optimality of the spectral coherence density; as such, it is expected to perform better than the empirical DEMON approach [e.g., some technical differences involve a squaring of the signal envelope instead of a rectification (so as to properly define an energy flow), a fixed bank of narrowband filters instead of a user-defined bandpass filter, and a normalization of the signal energy flow by its power spectral density so as to gain immunity against scaling effects].

0 dB; (c) a cyclostationary signal with 0.3 Hz and 39 Hz.)

• The CMC is an offspring of the theory of cyclostationary processes, which makes possible the assessment of all its statistical properties and in particular the construction of a statistical threshold for automatic detection as shown below. The CMC being a complex quantity in general, its squared-magnitude will be used without loss of information as far as detection is of concern. The image formed by CMC as a function of and will then provide a good and fast visual test to check for the presence of cyclostationarity in the signal. In view of automatic detection, a better statistical test is given by the integrated squared-magnitude CMC (ICMC) over a frequency band where cavitation is known to predominate, namely: Reject the null hypothesis H0 “there is no presence of cyclostationarity at cyclic frequency (i.e., )” at the level of significance if CMC

(29)

where with and the DFT bins corresponding to and , and is a statistical threshold to be determined. By allowing the user to select a relevant frequency band (either online or a priori) where the cyclostationary SNR is high, the proposed statistical test will be all the more efficient. In addition, because of the integration over frequency , the test will amount to comparing a function of only against the threshold the ICMC should not exceed with probability when the signal is stationary. Stated differently, the threshold should envelop 100(1-p)% of the values of the squared-magnitude estimation noise in (20). It is shown in Appendix D that (30)


487

Fig. 9. Integrated squared-magnitude cyclic modulation coherence (ICMC) in band [1000; 3000] Hz in the four cases of Fig. 8 together with the statistical threshold (dotted line) at the 0.5% level of significance.

• Disregard the first cyclic frequency which always returns CMC and may then absorb most of the dynamical range of the CMC. • Set at least 75% overlap when computing the short-time DFT with a Hanning window (the question as which optimal window to be used for minimizing cyclic leakage is currently under investigation, although of marginal importance). • On the CMC displayed as an image in the -plane, select a frequency band where the cyclostationary SNR is maximized. Set if detection of possible tonal noise is requested and otherwise. • Compute the ICMC in that frequency band and compare the result against the statistical threshold (30) for a given level of significance. • Eventually, compute the “detection SNR” Fig. 10. ICMC in the case of a cyclostationary signal with several harmonics of 15 Hz (SNR 0 dB) together with the statistical the cyclic frequency threshold (dotted line) at the 0.5% level of significance.

with with

the 100(1-p) percentile of the degrees of freedom and

distribution

(31) It is emphasized that this result applies in the most general case as a function of the number of integrated frequency bins, the number of snapshots, their shift , their length , and the type of the analysis window . Practical recommendations for an optimal use of the proposed statistical test are resumed hereafter.

CMC

(32)

a useful indicator for appraising the significance of the detection. As a final remark, it should be noted that the proposed statistical test does not strictly test for the presence of cyclostationarity, but rather for the absence of stationarity (this is a common feature of all similar tests that have been proposed in the literature). Consequently, the ICMC is likely to exceed the statistical threshold whenever the signal exhibits nonstationarity, but not necessarily cyclostationarity (this is further discussed in Section V and illustrated in Fig. 11 therein). However, it has been proved in [6] that cyclostationarity only can produce significantly high magnitudes of the spectral coherence density, and therefore of the CMC. In that sense, the detection test should prove quite robust.

488


V. NUMERICAL AND EXPERIMENTAL RESULTS The application of the statistical test described in Section IV-D is now demonstrated on some numerical examples and on the actual North Sea data introduced in Section I. The first set of numerical data consists of four signals of 32 768 samples each with sampling frequency 10 kHz: 1) a stationary signal; 2) a cyclostationary signal with 20 Hz and SNR 0 dB; 3) a cyclostationary signal with 20 Hz and SNR 10 dB; and 4) a cyclostationary signal with 70 Hz and SNR 0 dB. The cyclostationary signals 2)–4) were synthesized by modulating a filtered Gaussian noise in band [1000; 3000] Hz with a squared sine wave—producing a burst-random behavior as shown in Fig. 2(a)—and further adding stationary Gaussian noise. For instance, Fig. 7 displays signal 2). The CMC was computed with a 256-long Hanning window and 75% overlap to achieve a frequency resolution 40 Hz and a cyclic frequency resolution 0.3 Hz. It is displayed in Fig. 8 as a gray-level image for each of the four signals. The presence of cyclostationarity in cases 2)–4) is seen to produce a vertical line in the frequency band [1000; 3000] Hz as predicted by (22) and as schemed in Fig. 6, the intensity of which is proportional to the SNR as indicated by (16). The effect of the lowpass filter is particularly notable, which makes the detection of cyclostationarity at 70 Hz in case 4) more critical than at 20 Hz in case 2) for an identical SNR. The ICMC of each of signals was then computed by integrating over the frequency band [1000; 3000] Hz. Results are shown in Fig. 9 together with the statistical threshold at the 0.5% level of significance and the detection SNR defined in (32). This provides a fully automatic detection test with remarkable performance: the presence of cyclostationarity is clearly detected in cases 2)–4), even under very low SNR. It is also noteworthy that the statistical threshold returned by (30) (dotted line) closely envelops the noise baseline as a function of the cyclic frequency in each case. However, it should be emphasized that this is generally so only under hypothesis H0 of stationarity, and not necessarily in the presence of cyclostationarity. This is illustrated in Fig. 10 in the case of a cyclostationary signal whose multiple harmonics of the cyclic frequency all contribute to push up the noise baseline above the predicted threshold (yet without harm on the detection capability since this happens under hypothesis H1). The robustness of the stationary assumption of ambient ocean noise was further investigated on actual measurements. A time record ( 11 kHz) of ocean noise purposely selected for its nonstationarity is displayed in the excerpt of Fig. 11(a). After integration of the CMC over the full frequency range [0; 5500] Hz, this produces a baseline in the ICMC that significantly stands above the H0 statistical threshold. However, this effect: 1) is a continuous function of the cyclic frequency; 2) is confined to the lower cyclic frequency range; and 3) gradually vanishes at higher frequencies (as typical of a Markov-type evolution of the noise envelope ) in contrast with a cyclostationary behavior that would have instead produced one or several peaks.

Moreover, the ocean noise nonstationarity is almost completely removed when the ICMC computation is restricted to the frequency range [1000; 3000] Hz as demonstrated in Fig. 11(b). As a final remark on nonstationary noise, it is worth mentioning that in some oceanic environments, particularly littorals, the low-frequency ambient noise background can be dominated by the so-called “distant shipping noise,” generally composed of an incoherent summation of noises from multiple, distant ships scrambled by environmental propagation effects. Yet, if modulations remain more or less periodic, this type of noise would also be cyclostationary and would be detected as such by the proposed approach. Note that in circumstances where it is essential that nonstationary noise does not produce excessive false alarms (in particular, in automatic detection scenarios) this can be easily fixed by setting an alarm threshold , with a “security factor” that explicitly accounts for that nonstationarity. An intensive investigation of the detection capability of the ICMC was further carried out by simulating cyclostationary signals for various values of their cyclic frequency and SNR. All other parameters (cyclic and spectral resolution, fraction of overlap, integration range) were set as above. Results are reported in terms of the detection SNR of (32) and of the probability of detection for a given probability of false alarm of 5% in Fig. 12(a) and (b), respectively, and are systematically compared with the detection capability of the spectral coherence density (integrated squared magnitude over the same frequency range as the ICMC) which serves as a reference point due to its optimality. It is seen in Fig. 12 that the detection capability of the ICMC strongly depends on the cyclic frequency contrary to the spectral coherence density: • the two statistics roughly return the same detection SNR up to and an excellent probability of detection (close to one) up to for SNR 10 dB; • the performance of the ICMC rapidly decreases afterwards to the ultimate limit after which no more detection is possible even in the absence of additive noise (infinite SNR). These results are found perfectly in line with the upper limit established in (24) on different grounds. It is emphasized once again that using at least 75% overlap is essential to maximize the available cyclic frequency range and that any smaller percentage of overlap would result in a loss of detection capability—notwithstanding the additional aliasing errors that would also result as explained in Section IV-C and in Appendix D. The proposed detection test was finally applied to the North Sea data that served as an introductory example in Section I. A 512 sample-long Hanning window was used with 75% overlap to achieve a frequency resolution 86 Hz and a cyclic frequency resolution 0.04 Hz. The CMC is displayed in Fig. 13 as a gray-level image, and excerpts where propeller harmonics are detected are displayed in Fig. 14(a). This suggests integrating the CMC over the [5; 15]-kHz frequency range where the SNR seems to stand out, although any other frequency range below 15 kHz (cutoff frequency of the instrumentation)


489

Fig. 11. ICMC of an actual nonstationary ocean noise together with the statistical threshold at the 0.5% level (dotted line) and 50% level (continuous line): (a) -kHz band; (b) integration in the [1; 3]-kHz band. integration in the full

would work just as well in this example. The resulting ICMC is shown in Fig. 14(b). It clearly exhibits a family of harmonics at the shaft rotation speed of 5.75 Hz and a higher fourth harmonic at 23 Hz—the blade pass frequency—which is the signature of a four-blade propeller. The detection SNR reaches as high a value as 3087, which strikingly contrasts with the inefficiency of the classical spectral analysis of Fig. 1. Note that the statistical threshold is slightly below the noise baseline, either because of the reason discussed in Fig. 9 or of the actual nonstationarity of ocean noise,3 or both. The cyclic frequency range where these harmonics are detected is well below 86 Hz, which means, according to the analysis in Fig. 12, that the ICMC virtually returns the same result as the spectral coherence density in that band. From a computational point of view, the ICMC took 1.14 central processing unit (CPU) time on a laptop computer with a 1662-MHz processor against 12 600 for the spectral coherence density estimated from the averaged cyclic periodogram method [39]; this is four orders of magnitude faster for achieving virtually the same detection result. VI. CONCLUSION This paper presented a new method for detecting secondorder cyclostationary signals (as produced by propeller craft) in significant extraneous noise from sonar signals. The method is based on the recently proposed CMC and returns nearly identical results as the optimal spectral coherence density in a cyclic frequency range not greater than the order of the spectral resolution , yet at a considerably lower cost. It is closely related in principle to the DEMON processing technique, but enjoys several improvements over the latter thanks to the cyclostationary framework which underpins it. The detection capability of the method was demonstrated on both simulated and measured signals and provides, at this stage, a visual aid for sonar operators 3As previously explained, that situation could easily be fixed by using a security factor (e.g., of 2 or 3) on the statistical threshold.

to assist in detecting surface ships. The availability of a statistical threshold as a function of cyclic frequency also makes it eligible for automatic detection.

APPENDIX A PROOF OF (12) Using (4) in conjunction with the shifting property of the Fourier transform , the spectral correlation density of marine propeller noise becomes

(33) Now using the property that for any pair of jointly and stationary signals if and 0 otherwise, (12) immediately follows.

APPENDIX B PROOF OF (21) From (20), it readily follows that

(34)

490

with As long as


the Dirichlet kernel as defined below (21). (35)

(i.e., is greater than the width of the main lobe of the Dirichlet kernel), the contribution of centered on can be neglected when only positive cyclic frequencies are considered. In addition, under the assumption of strong mixing of the additive noise (ergodicity)

where is defined below (23). In addition, since assumed with unit norm, . Therefore

(36) was

CMS (37) Condition (35) implies that (21) with .

, which finally yields

APPENDIX C PROOF OF (23) Upon invoking the property of cycloergodicity (i.e., convergence of the below Fourier coefficient to its expected value) [9]

(38) Using Cramer’s spectral decomposition of the short-time DFT (39) the right-hand-side term in (38) becomes

(40) If cyclostationarity is assumed on a set of cyclic frequencies

Fig. 12. Comparison of the detection capability of the ICMC (continuous lines) with that of the spectral coherence density (dotted lines): (a) detection SNR of ; (b) probability of detection versus cyclic (32) versus cyclic frequency ; and (c) versus SNR for a given false alarm rate of 5% (10 000 frequency Monte Carlo runs).

When cyclostationarity is present at in (20), then

only, as indicted

(41) CMC

and thus

(42)

(43)


Fig. 13. CMC of the North Sea data (

0.04 Hz and

491

86 Hz).

where it has been used that: 1) the contribution of can be neglected over positive cyclic frequencies because of condition (35); 2) for the same reason; and 3) the spectral correlation density is smooth enough as compared to the bandwidth of so that it can be factored out of the integrals. Setting , (23) immediately follows from identifying (43) with (21).

now remains to be found, which, according to Cramer’s decomposition (39), expands as

(46) Now, under H0 [40]

APPENDIX D PROOF OF (30) Under hypothesis H0, the numerator of the CMC is, in probability, orders of magnitude smaller than its denominator; therefore, the CMC probability distribution is well approximated by that of . Since involves a DFT, it tends asymptotically to a Gaussian complex variable [40]. Therefore, under H0, the CMC is asymptotically complex Gaussian with zero mean and variance (which will be derived shortly hereafter) and CMC

(44)

under H0, where symbol means “distributed as.” As a consequence, the ICMC turns out to be a sum of complex Chi2 random variables with two degrees of freedom. Since these are asymptotically independent in virtue of a property of the DFT [40], then CMC

(47) yielding after some calculations

(48) Using the fact that for large , one finds

behaves like

(45)

This is the distribution of a scaled Chi2 variable with degrees of freedom from which the statistical threshold of (30) immediately follows. A closed-form expression for the variance of the CMC under H0 now remains to be found. Let us first note that , where , with for under H0 and , as given by (42) for . Moreover, condition (24) implies that .

(49)

492


Fig. 14. (a) Excerpts of Fig. 13 zooming around harmonics of the shaft speed and showing frequency range of interest (delimited by horizontal dotted lines). (b) ICMC of the North Sea data with statistical threshold (dotted line) at the 0.5% level of significance.

The last step is to recognize that: 1) can be factored out of the integrals as long as the power spectral density is smooth enough as compared to the bandwidth; 2) aliasing between terms shifted by does not occur provided that condition (24) is met; and 3) under the same condition. Therefore

(50) The first integral on the right-hand side of (50) is negligible provided that , with the bandwidth of , and the second one is trivially shown to be the Fourier transform of the square of the autocorrelation of the analysis window . This finishes the proof of (30). ACKNOWLEDGMENT The authors would like to thank Prof. R. Coates of Seiche Pty Ltd, and the Australian Defence Science and Technology

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Jerome Antoni received the M.S. degrees in mechanical engineering and in electrical engineering from the University of Technology of Compiegne, Compiegne, France, in 1995 and 1996, respectively, and the Ph.D. degree in signal processing from the Grenoble Institute of Technology, Grenoble, France, in 2000. Currently, he is a full time Professor at the University of Lyon, Villeurbanne CEDEX, France. He has spent several periods at the University of New South Wales, Sydney, N.S.W., Australia, and at the Vrije Universiteit Brussel, Brussels, Belgium. The main direction of his research activity is the diagnostics of mechanical systems from vibration and acoustical measurements. His current interests deal with cyclostationary modeling of rotating machine signals, blind source separation, and (non)parametric identification of mechanical systems and material properties. Prof. Antoni is a member of the Editorial Boards of Mechanical System and Signal Processing, International Journal of Condition Monitoring, and Diagnostika.

David Hanson received the B.S. degree in mechanical engineering from the University of Newcastle, Callaghan, N.S.W., Australia, in 2004 and the Ph.D. degree from the University of New South Wales, Sydney, N.S.W., Australia, in 2007, where he examined the application of cyclostationary signal processing for operational modal analysis. He is a Technical Specialist in Noise and Vibration in the Dynamics Group, Sinclair Knight Merz, Sydney, N.S.W, Australia. His research interests include rail noise mitigation and prediction, particularly curve squeal, and operational modal analysis. The application of traditionally mechanical signal processing techniques to ship detection arose from discussions with Prof. R. Coates regarding DEMON processing.