We apply the continuous wavelet transform to dispersive pulse propagation and obtain an approximation that is easily applied. The approximation shows that one can evolve the wavelet transform of the pulse in a simple manner, by calculating the wavelet transform at time zero and making a simple algebraic substitution.
Instantaneous frequency is an important characteristic of time-varying or nonstationary signals. The definition and interpretation of instantaneous frequency have been the subject of discussion and debate for decades. The most common approach is due to Gabor, whereby a specific complex signal, called the analytic signal, is associated with a given real signal by inverting the spectrum of the real signal over only the positive frequency axis; the instantaneous frequency is then taken to be the derivative of the phase. Other approaches for associating a particular complex signal to a given real signal, and hence obtaining different instantaneous frequencies, have also been proposed. One way to define the associated complex signal / instantaneous frequency is by imposing physical constraints, which we discuss. We also discuss the common interpretation of instantaneous frequency as the average frequency at each time, and point out when this interpretation holds, which is not usually the case. This leads to the question of what is the “average frequency at each time?” The answer, coupled with physical constraints on the complex signal representation, leads to a quadrature-AM / FM signal model. Finally, we consider methods that manipulate the poles and zeros of the signal to obtain a complex representation.
In active sensing such as in sonar and radar, target recognition is adversely impacted by target-like returns from non-target objects (i.e. clutter). Because the target and clutter returns are in general nonstationary, the application of linear time-varying (LTV) pre-filters has been suggested to enhance target classification. We apply a minimum probability of error (MPE) classifier with and without LTV filters to distinguish targets from clutter in active sonar data. Classification performance was improved with LTV filtering.
KEYWORDS: Signal analyzers, Time-frequency analysis, Signal generators, Fourier transforms, FM band, RF communications, Automatic target recognition, Mahalanobis distance, Current controlled current source, Mathematics
Determining the amplitude and phase of a signal is important in many areas of science and engineering. The derivative of the phase is typically called the "instantaneous frequency," which in principle mathematically describes (and ideally coincides with) the common physical experiences of variable-frequency phenomena, such as a siren. However, there is an infinite number of different amplitude-phase pairs that will all generate the same real signal, and hence there is an unlimited number of "instantaneous frequencies" for a given real signal. Gabor gave a procedure for associating a specific complex signal to a given real signal, from which a unique definition of the amplitude and phase, and consequently the instantaneous frequency, of the real signal is obtained. This complex signal, called the analytic signal, is obtained by inverting the Fourier spectrum of the real signal over the positive frequency range only. We introduce a new complex signal representation by applying Gabor's idea to the Wigner time-frequency distribution. The resulting complex signal, which we call the Wigner-Gabor signal, has a number of interesting properties that we discuss and compare with the analytic signal. In general the Wigner-Gabor signal is not the analytic signal, although for a pure tone A cos(ω0t) the Wigner-Gabor and analytic
signals both equal A exp(jω0t). Also, for a time-limited signal s(t) = 0, |t| > T, the
analytic signal is not time-limited, but the Wigner-Gabor signal is time-limited.
Classifying underwater targets from their sonar backscatter is often complicated by induced or self-noise (i.e. clutter, reverberation) arising from the scattering of the sonar pulse from non-target objects. Because clutter is inherently nonstationary, and because the propagation environment can induce nonstationarities as well, in addition to any nonstationarities / time-varying spectral components of the target echo itself, a joint phase space approach to target classification has been explored. In this paper, we apply a previously developed minimum mean square time-frequency spectral estimation method to design a bank of time-frequency filters from training data to distinguish targets from clutter. The method is implemented in the ambiguity domain in order to reduce computational requirements. In this domain, the optimal filter (more commonly called a “kernel” in the time-frequency literature) multiples the ambiguity function of the received signal, and then the mean squared distance to each target class is computed. Simulations demonstrate that the class-specific optimal kernel better separates each target from the clutter and other targets, compared to a simple mean-squared distance measure with no kernel processing.
In previous work, we have given a method for obtaining propagation invariant features for classification of underwater objects from their sonar backscatter in dispersive but rangG'-independent environments. In this paper we consider the derivation of invariant features for classification in range dependent environments, based on the parabolic equation.
We have previously developed a feature extraction process for propagation-invariant
classification of a target from its propagated sonar backscatter. The features are invariant to the
frequency dependent propagation effects of absorption and dispersion, for range-independent channels.
Simulations have shown that these features lose their effectiveness when applied to waves propagating
in a range-dependent environment. In this paper we extend our previous approach to obtain invariant
features for classification in range-dependent environments. Numerical simulations are presented for
the classification of two shells from their acoustic backscatter propagating in an ideal wedge.
Propagation effects, such as dispersion, absorption and multi-path, can adversely
impact classification of underwater objects from their sonar backscatter. One approach to handling
this problem is to extract features from the wave that are minimally affected by propagation effects, if
possible. In previous work, a signal processing and feature extraction method was developed to obtain
moment-like features that are invariant to dispersion and absorption. The method was developed based
on linear wave propagation in range- independent environments. However, most ocean environments,
especially littoral environments, exhibit range dependence. Deriving propagation invariant features for
such environments remains an especially challenging task. In this paper, we explore the classification
utility of the previously developed range-independent features in a range-dependent environment, via
simulation of the propagation of the backscatter from two different cylinders in an ideal wedge. Our
simulation results show that, while performance does drop off for increasing distances in a range dependent
environment, the previously developed invariant moment features do provide better classification
performance than ordinary temporal moments.
We extend a recent method by Kay that maximizes the probability of detecting an elastic object in the presence
of Gaussian reverberation and additive Gaussian interference. Kay's solution specifies the spectral magnitude
for the optimal transmit waveform, and hence there is an unlimited number of "optimal" waveforms that can
be transmitted, all with the same spectral magnitude but differing in terms of time domain characteristics
such as duration and peak power. We extend Kay's approach in order to obtain a unique optimal waveform
by incorporating time-domain constraints, via two optimization problem formulations. One approach yields a
waveform that preserves the optimal spectral magnitude while achieving the minimum temporal duration. The
second complementary approach considers temporal concentration rather than duration, and yields a waveform
that, depending on the degree of concentration imposed, achieves the optimal the spectral magnitude to varying
degrees.
Some marine mammals as well as bats are known to emit sophisticated waveforms while searching for objects or hunting prey. Some dolphins have been observed to change their sonar pulse depending on the environment. Incorporating these strategies into sonar waveform and receiver design has become an active area of research. In this paper, we explore the application of an optimal waveform design scheme recently given by Kay, to the detection of elastic objects. We examine the benefits of optimal waveform design versus transmitting a linear FM waveform, as well as performance loss suffered by assuming a point target. The optimization approach designs the magnitude spectrum of the transmit waveform and, accordingly, there is an unlimited number of "optimal" transmit waveforms with the same magnitude spectrum. We propose a time domain optimization criterion to obtain the transmit waveform with the optimal magnitude spectrum and the smallest possible duration, as well as the waveform with the optimal magnitude spectrum and the longest possible duration. The former waveform allows for higher ping rates, but necessarily has higher time domain peak power, while the latter waveform has lower time domain peak power and lower ping rates. A method to obtain waveforms that are a blend of these two extremes is also presented, allowing a smooth trade-off between ping rate and peak power.
In underwater automatic target recognition via active sonar, the transmitted sonar pulse and the returning target
backscatter can undergo significant distortion due to channel effects, such as frequency-dependent attenuation
(damping) and dispersion, as well as random effects due to noise and other channel variability. These propagationinduced
effects can be detrimental to classification because the observed backscatter depends not only on the
target but also on the propagation environment and how far the wave has traveled, resulting in increased
variability in the received sonar signals. Using a recently developed phase space approximation for dispersive
propagation, we present a method for analyzing these effects on temporal and spectral moment features of the
propagating signal, including uncertainty in certain channel parameters, in particular target distance.
We give a brief review of ideas and insights derived from the study and application
of positive time-frequency distributions, in the nearly 25 years since their formulation
by Cohen, Posch and Zaparovanny. Associated topics discussed include instantaneous
frequency and conditional moments, the "time varying spectrum" and joint versus conditional
distributions, the uncertainty principle, kernel design, cross terms, AM-FM signal
decompositions, among others. Many of the conventionally held ideas in time-frequency
analysis are challenged by results from positive time-frequency distributions.
When a wave propagates in a medium with dispersion and damping, different frequencies
propagate at different velocities and are attenuated at different rates. Accordingly, the wave
changes as it propagates. These propagation effects can negatively impact automatic classification,
since what is observed changes from location to location. We examine various moments of a wave,
such as duration and bandwidth, which are often used as features for classification, and quantify the
effects of dispersion and damping on these moments. We also identify moment-like features that are
invariant to dispersion and damping, and thus may offer advantages over ordinary moments as features
for classification.
Given the moments of a time-frequency distribution, one can, in principle,
construct the characteristic function from which one then obtains the distribution by
Fourier transformation. However, often one can not find a closed form for the characteristic
function and hence one can not obtain the distribution in a direct manner. We formulate
the problem of constructing time-frequency representations from moments without first
constructing the characteristic function. Our method is based on expanding the distribution
in terms of a complete set of functions where the expansion coeficients are dependent
directly on the moments. We apply the method to a case where the even moments are
manifestly positive which is a necessary condition for obtaining a proper time-frequency
representation.
We show how to construct distributions from moments directly, that is,
without first calculating the characteristic function. We apply the method to compute the
Wigner distribution from its conditional moments.
In active sonar or radar, if the channel is spatially- and/or temporally-varying,
then the target echo can change with propagation, such that echoes from identical targets
may not be identified as such at the receiver. Two common propagation effects that induce
changes in the signal are dispersion and dissipation (or damping), which give rise
to frequency-dependent velocity of propagation and frequency-dependent attenuation, respectively.
We have previously developed a feature extraction process for target echoes in
dispersive channels, to obtain moment-like features that are invariant to dispersion, per
mode. Accordingly, even though the target echo can change with propagation in a dispersive
channel, the "dispersion-invariant moment" features do not. However, these moment
features are affected by damping. In this paper, we consider the case of a channel with
dispersion and damping, and derive features that are invariant to both phenomena, for
any dispersion relation and exponential or power-law damping. Results are presented from
classification simulations to demonstrate the utility of these features.
Signals with time-varying spectral content arise in a number of situations, such as in shallow water sound propagation, biomedical signals, machine and structural vibrations, and seismic signals, among others. The Wigner distribution and its generalization have become standard methods for analyzing such time-varying signals. We derive approximations of the Wigner distribution that can be applied to gain insights into the effects of filtering, amplitude modulation,
frequency modulation, and dispersive propagation on the time-varying spectral content of signals.
In active sonar or radar, the received signal can often be modeled as a convolution of the transmitted signal with the channel impulse response and the target impulse response. Because the received signal may have a time-varying spectrum, due for example to target motion or to changes in the channel impulse response, time-frequency methods have been used to characterize propagation effects and target effects, and to extract features for classification. In this paper, we consider the time-varying spectrum, in particular the Wigner time-frequency representation, of a received signal modeled as the convolution of the transmitted signal with the channel and target responses. We derive a simple but insightful approximation that shows the effects of the magnitude and phase of the frequency response of the target and of the channel on the Wigner representation of the transmitted signal. We also consider time-varying effects on the Wigner representation, such as changes in reflected energy, which we model by amplitude modulation.
The vibrations produced by objects, for example by a plate or cylinder insonified by a sonar wave, exhibit characteristics unique to the particular structure, which
can be used to distinguish among different objects. The situation is complicated, however,
by many factors, a particularly important one being propagation through media. As a vibration
propagates, its characteristics can change simply due to the propagation channel;
for example, in a dispersive channel, the duration of the vibration will increase with propagation
distance. These channel effects are clearly detrimental to automatic recognition
because they do not represent the object of interest and they increase the variability of
the measured responses, especially if measurements are obtained from targets at different
locations. Our principal aim is to identify characteristics of propagating vibrations and
waves that may be used as features for classification. We discuss various moment-like
features of a propagating vibration. In the first set of moments, namely temporal moments
such as mean and duration at a given location, we give explicit formulations that
quantify the effects of dispersion. Accordingly, one can then compensate for the effects
of dispersion on these moments. We then consider another new class of moments, which
are invariant to dispersion and hence may be useful as features for dispersive propagation.
We present classification results comparing these invariant features to related non-invariant
features, for classification of simulated backscatter from different steel shells in a dispersive
environment.
We consider dispersive propagation with damping in phase-space, and derive the Wigner distribution in terms of the initial wave and the dispersion relation. The case for no damping, that is, lossless propagation, has been previously considered by Cohen, and is a special case of the more general result presented here. Simple and physically revealing approximations of the Wigner distribution in terms of the initial Wigner distribution are presented. Also, exact low-order conditional moments are given and their interpretation is discussed.
Abstract - When a continuous-time signal is sampled at a rate less than the Nyquist criterion, the signal is aliased. This distortion is usually irrecoverable. However, we show that for certain AM-FM signals, the distortion due to aliasing can be mitigated and an unaliased version of the signal can be recovered from its aliased samples. We present a method for determining whether or not a signal has potentially been distorted by aliasing, and an algorithm for recovering an unaliased version of the signal. The method is based on the manifestation of aliasing in the time-frequency plane, and estimating the instantaneous phase/frequency of the aliased signal.
As a wave propagates in a dispersive medium certain characteristics change and hence it may not be recognized as the same wave by
different observers. For lossless dispersive propagation, temporal moments such as the mean time and duration of the wave
change as a function of position, while frequency moments do not. We show that there are other moment-like temporal features of the wave
that are also invariant to dispersion. These moments may be useful
in automatic classification because indeed they are invariant to dispersive channel effects and hence do not depend on the position at which they are calculated, and they provide additional information beyond that given by frequency moments.
In dispersive wave propagation, the standard stationary phase approximation to the wave is accurate in the asymptotic regime.
Typically the calculation of the stationary points is
taken to depend only on the dispersion relation. We examine the effects of including the spatial phase of the initial wave as well in the calculation and show that doing so can improve the approximation.
We address the question of the importance of satisfying the marginal conditions in a time-frequency distribution and discuss in detail the fundamental and practical issues involved. We also examine the marginals of the spectrogram. The spectrogram often gives reasonable results even though both marginals can never be satisfied exactly, but sometimes it can give very unreasonable results. We show by examples that the spectrogram gives the clearest characterization of the time-frequency properties of a signal when the combined error in both of its marginals is as small as possible. In contrast, when the error in either one of the marginals of the spectrogram is large, the time-frequency characterization degrades. Some different error measures are described. We discuss the issues involved both theoretically and by way of numerical simulations. We also address the issue of marginals for the random case.
Underwater sound propagation is inherently nonstationary, particularly in shallow water where the ocean surface and bottom act like waveguide boundaries, giving rise to structural (or geometric) dispersion. The spectrogram has been a principal means to study the nonstationarities and dispersion characteristics of shallow-water sound propagation. In this paper, we give the low-order conditional time-frequency moments of a wave propagating in a waveguide.
Comparison of these results is made to spectrograms of explosive source sound propagation in the Yellow Sea.
We review recent work on defining the time-frequency moments of a signal. Expressions are given for moments of all orders, in terms of the amplitude and phase of the signal and spectrum. Knowing the time-frequency moments is of interest for a variety of reasons, including their potential utility as features for classification of nonstationary signals, and also because from the moments one can construct the time-varying spectral density, or approximate it using a few moments.
It has been shown that often the onset of developing faults in machines is clearly manifest in the time-frequency plane before any problems are noted by conventional methods such as the power spectrum. In this paper we explore a particular feature of some faulting machines, wherein a single vibration frequency briefly and intermittently appears as 'eyelets' in the time-frequency plane. We show that abrupt phase shifts in a tone, or equivalently sudden, rapid changes in the amplitude, cause a transient increase in the instantaneous spectral moments, particularly the instantaneous bandwidth and the instantaneous kurtosis, and cause eyelets in time-frequency similar to those seen in real machine vibrations.
KEYWORDS: Linear filtering, Digital filtering, Fermium, Frequency modulation, Signal to noise ratio, Amplitude modulation, Optical correlators, Filtering (signal processing), Modulation, Telecommunications
While spread spectrum systems are robust to many types of interference, performance can be significantly degraded if the interference is strong enough, particularly for wideband interferences. In these situations, various signal processing methods can be employed to remove, or excise the jammer prior to despreading the received signal, resulting in enhanced performance. We investigate the effects of amplitude and frequency modulated (AM-FM) jammers on the performance of direct sequence spread spectrum communication systems. We demonstrate that such jammers cause significant degradation in bit-error-rate with increasing AM on systems designed to excise FM jammers only (i.e., systems with fixed notch-width excision filters). We propose an adaptive technique that utilizes the instantaneous bandwidth of the jammer, in addition to its instantaneous frequency, to filter wideband AM- FM interference from the DSSS signal. We also investigate the effects of adapting the filter notch-depth as well, according to the instantaneous power of the jammer. Simulations demonstrate additional improvement in system performance for the proposed adaptive technique compared to fixed notch-width and fixed notch- depth excision filters.
As with the case of instantaneous frequency, it is often difficult to interpret the instantaneous bandwidth of most signals: both quantities typically range beyond the spectral support of the signal, yielding the paradox that the instantaneous bandwidth (and frequency) can be greater than the global bandwidth of the signal. A new definition of instantaneous frequency that does not suffer from this difficulty has recently been given, and we build on those results here to obtain a new definition of instantaneous bandwidth. Kernel constraints for a Cohen-class time-frequency distribution to yield these new results for its conditional moments are also given.
We present a number of methods that use image and signal processing techniques for removal of noise from a signal. The basic idea is to first construct a time-frequency density of the noisy signal. The time-frequency density, which is a function of two variables, can then be treated as an 'image,' thereby enabling use of image processing methods to remove noise and enhance the image. Having obtained an enhanced time-frequency density, one then reconstructs the signal. Various time frequency-densities are used and also a number of image processing methods are investigated. Examples of human speech and whale sounds are given. In addition, new methods are presented for estimation of signal parameters from the time- frequency density.
We apply time-frequency methods to automotive vibration signals for sound quality analysis. Our analysis indicates that time-frequency methods provide additional information beyond that provided by the spectral density and vibration time series that is relevant to the assessment of noise and sound quality.
The question of what the joint and conditional time-frequency moments are in terms of the signal and spectral amplitudes and phases is considered. From these moments, a time- frequency density can be constructed using the method of maximum entropy. This technique is used to assess the plausibility of expressions recently derived for low-order conditional and joint moments.ALso investigate is whether or not there is a lower bound on the local time- bandwidth product of a time-frequency density, as there is for the global case.
Many waves exhibit characteristics that depend on time and/or frequency. For example, the frequencies of a pulse propagating in a dispersive medium travel at different velocities. This kind of dependency gives rise to the need for joint time- frequency analysis and methods for describing the local temporal and spectral nature of waves, particularly pulses and transients. Useful concepts arising for this description from time-frequency theory are the average frequency at each time of a wave, and the spread about that average. These quantities are obtained as conditional moments of a time-frequency density (TFD). We explore the conditional variances of some common TFDs, and determine when these are positive (which isn't always the case). We also investigate local characteristics of a wave, in terms of these conditional moments, and show through experimental results that they robustly characterize the time-frequency behavior of transients and pulses.
We consider the definition and interpretation of instantaneous frequency and other time-varying frequencies of a signal, and related concepts of instantaneous amplitude, instantaneous bandwidth and the time-varying spectrum of a signal. A definition for the average frequency at each time is given, and we show that spectrograms and Cohen-Posch time-frequency distributions can yield this result for the first conditional moment in frequency. For some signals this result equals the instantaneous frequency, but generally instantaneous frequency is not the average frequency at each time in the signal. We discuss monocomponent versus multicomponent signals, and give an estimate of the time-varying spectrum given the instantaneous frequencies and bandwidths of the components. We also consider the role of the complex signal in defining instantaneous amplitude, frequency and bandwidth, and ways to obtain a complex signal satisfying certain physical properties, given a real signal (or its time-varying spectrum). Depending upon the physical properties desired (e.g., the instantaneous amplitude of a magnitude-bounded signal should itself be bounded), one obtains different complex representations -- and hence different instantaneous amplitudes, frequencies and bandwidths -- of the given signal.
We propose a time-frequency based pattern classification method which utilizes the joint moments of time-frequency distributions (TFDs) for features. The method is applied to a biomedical data set, and compared to a template matching scheme and to methods utilizing only temporal moments or spectral moments. Our results show that a classification algorithm which utilizes joint time-frequency information, as quantified by the joint moments of the TFD, can potentially improve performance over time or frequency-based methods alone, for classification of nonstationary time series.
In this paper, we show how quadratic time-frequency representations are a generalization of the spectrogram and we review our results for time-frequency analysis and display of chirps and speech. We then show comparative performance on phase-shifted keyed communication signals. The concept of quadratic filtering is then introduced and linked to Teager's energy detector and the resolution advantages over linear filtering are demonstrated.
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