Time-Frequency Analysis has previously been successfully applied to characterize and quantify a variety of acoustic signals, including marine mammal sounds. In this research, Time-Frequency analysis is applied to human speech signals in an effort to reveal signal structure salient to the biometric speaker verification challenge. Prior approaches to speaker verification have relied upon signal processing analysis such as linear prediction or weighted Cepstrum spectral representations of segments of speech and classification techniques based on stochastic pattern matching. The authors believe that the classification of identity of a speaker based on time-frequency representation of short time events occurring in speech could have substantial advantages. Using these ideas, a speaker verification algorithm was developed1 and has been refined over the past several years. In this presentation, the authors describe the testing of the algorithm using a large speech database, the results obtained, and recommendations for further improvements.
Highly sophisticated methods for detection and classification of signals and images are available. However, most of these methods are not robust to nonstationary variations such as imposed by Doppler effects or other forms of warping. Fourier methods handle time-shift or frequency shift variations in signals or spatial shifts in images. A number of methods have been developed to overcome these problems. In this paper we discuss some specific approaches that have been motivated by time-frequency analysis. Methodologies developed for images can often be profitably used for time-frequency analysis as well, since these representations are essentially images. The scale transform introduced by Cohen can join Fourier transforms in providing robust representations. Scale changes are common in many signal and image scenarios. We call the representation which results from appropriate transformations of the object of interest the Scale and Translation Invariant Representation or STIR. The STIR method is summarized and results from machine diagnosis, radar, marine mammal sounds, TMJ sounds, speech and word spotting are discussed. Some of the limitations and variations of the method are discussed to provide a rationale for selection of particular elements of the method.
Time-frequency distributions (TFDs) of Cohen's class often
dramatically reveal complex structures that are not
evident in the raw signal. Standard linear filters are often not
able to separate the underlying signal from background clutter and noise. The essense of the signal can often be extracted from the TFD by evaluating strategic slices through the TFD for a series of frequencies. However, TFDs are often computationally intense compared
to other methods. This paper demonstrates that quadratic filters may
be designed to capture the same information as is available in the specific slices through the TFD at a considerably lower computational cost. The outputs of these filters can be combined to provide a robust impulse-like response to the chosen signal. This is particularly useful when the exact time series representation of the signal is unknown, due to variations and background clutter
and noise. It is also noted that Teager's method is closely related to TFDs and are an example of a quadratic filter. Results using an ideal matched filter and the TFD motivated quadratic
filter are compared to give insight into their relative responses.
Reliable monitoring methods are essential for maintaining a high level of quality control in laser welding. In industrial processes, monitoring systems allow for quick decisions on the quality of the weld, allowing for high productions rates and reducing overall cost due to scrap. A monitoring system using infrared, ultraviolet, audible sound, and acoustic emission was implemented for monitoring CO2 laser welds in real-time. The signals were analyzed using time-frequency analysis techniques. The time-frequency distribution using the Choi-Williams kernel was calculated, and the resulting distributions were analyzed using the Renyi information distribution. Results for porosity monitoring showed that an acoustic emission sensor held the most promise with 100% classification in two weld studies. These encouraging results led to a second study for monitoring of weld penetration and in the second case, infrared, ultraviolet, and audible sound showed the most promise with 100% classification for both laboratory and industrial data.
KEYWORDS: Time-frequency analysis, Denoising, Interference (communication), Signal to noise ratio, Wavelets, Smoothing, Probability theory, Image information entropy, Fourier transforms, Visualization
Signals used in time-frequency analysis are usually corrupted by noise. Therefore, denoising the time-frequency representation is a necessity for producing readable time-frequency images. Denoising is defined as the operation of smoothing a noisy signal or image for producing a noise free representation. Linear smoothing of time-frequency distributions (TFDs) suppresses noise at the expense of considerable smearing of the signal components. For this reason, nonlinear denoising has been preferred. A common example to nonlinear denoising methods is the wavelet thresholding. In this paper, we introduce an entropy based approach to denoising time-frequency distributions. This new approach uses the spectrogram decomposition of time-frequency kernels proposed by Cunningham and Williams.In order to denoise the time-frequency distribution, we combine those spectrograms with smallest entropy values, thus ensuring that each spectrogram is well concentrated on the time-frequency plane and contains as little noise as possible. Renyi entropy is used as the measure to quantify the complexity of each spectrogram. The threshold for the number of spectrograms to combine is chosen adaptively based on the tradeoff between entropy and variance. The denoised time-frequency distributions for several signals are shown to demonstrate the effectiveness of the method. The improvement in performance is quantitatively evaluated.
Introduction of Renyi information to time-frequency analysis occurred in 1991, by Williams et al at SPIE. The Renyi measure provides a single objective indication of the complexity of a signal as reflected in its time-frequency representation. The Gabor logon is the minimum complexity signal and its informational value is zero bits. All other signals exhibit increased Renyi information. Certain time-frequency distributions are information invariant, meaning that their Renyi information does not change under time-shift, frequency shift and scale changes. The Reduced Interference Distributions are information invariant. Thus a given signal within that class will always have the same Renyi result. This can be used to survey large data sequences in order to isolate certain types of signals. One application is to extract instances of such a signal from a streaming RID representation. Examples for temporomandibular joint clicks are provided.
KEYWORDS: Fourier transforms, Transform theory, Signal processing, Probability theory, Numerical analysis, Solids, Signal generators, System identification, Optimization (mathematics), Information theory
We have recently introduced the class of generalized scale transforms and its subclass of warped Fourier transforms. Members in each class are defined by continuous time warping functions. While the two transforms admit a mathematically elegant analysis of warp-shift invariant systems it is still unclear how to design warping functions that deliver optimal representations for a given class of signals or systems. In many cases we can obtain an optimal choice for the warping function via a closed form analysis of the system that generates the signal of interest. In cases in which a closed form analysis is not possible we have to rely on a warp function estimation method. The approach we are taking in this paper is founded in information theory. We consider the observed signal as a random process. A power estimate of the warped Fourier transform parameterized by an underlying warping function is obtained from a finite number of realizations. We treat the power estimate as a probability density in warp-frequency and minimize its differential entropy over the space of admissible warping functions. We use an iterative numerical method for the minimization process. A proper formulation of a discrete time warped Fourier transform is employed as a foundation for the numerical analysis. Applications of the proposed algorithm can be found in detection, system identification, and data-compression.
KEYWORDS: Time-frequency analysis, Wavelets, Fourier transforms, Electrical engineering, Computer science, Signal analyzers, Signal processing, Space operations, Matrices, Lead
Previous work has shown that time-frequency distributions (TFDs) belonging to Cohen's class can be represented as a sum of weighted spectrograms. This representation offers the means of reducing the computational complexity of TFDs. The windows in the spectrogram representation may either be the eigenfunctions obtained from an eigen decomposition of the kernel or any complete set of orthonormal basis functions. The efficiency of the computation can further be increased by using a set of scaled and shifted functions like wavelets. In this paper, the concept of scaling is considered in discrete-time domain. The scale operator in the frequency domain is formulated and the vectors which correspond to the solutions of this eigenvalue problem in discrete-time are derived. These new eigenvectors are very similar in structure to the eigenvectors obtained from eigensystem decomposition of reduced interference distribution (RID) kernels. The relationship between these two sets of window functions is illustrated and a new efficient way of decomposing time-frequency kernels is introduced. The results are compared to the previous decomposition methods. Finally, some possible applications of these discrete scale functions in obtaining new time-frequency distributions are discussed.
KEYWORDS: Microwave radiation, Time-frequency analysis, High power microwaves, Fourier transforms, Electron beams, Frequency modulation, Magnetism, Waveguides, Dispersion, Pulsed power
Research is being conducted on high power microwave devices (e.g., gyrotrons) at the University of Michigan. Of utmost concern is the phenomenon of pulse shortening, that is, the duration of the microwave pulse is shorter than the duration of the cathode voltage. For years researchers have applied the Fourier transform to the heterodyned microwave signals. The problem with this technique is that a signal with multiple frequency components has the same spectrum as that of a signal with frequency components emitted at different times. Time-frequency analysis (TFA) using Reduced Interference Distributions provided an entirely different outlook in the community when it was recently applied to heterodyned microwave signals. Results show, with unprecedented clarity, mode hopping, mode competition, and frequency modulation due to electron beam voltage fluctuations. The various processes that lead to pulse shortening may finally be identified. Time resolved maximum intensity of the TFA has produced results very similar to the microwave power signal, verifying the utility of TFA in the analysis of the temporal evolution of power in each mode.
This research program investigates high power microwave generation utilizing a microsecond electron beam accelerator to study means of eliminating microwave pulse-shortening. The particular device under study is the coaxial gyrotron oscillator in the S-band frequency range. Experiments have concentrated on three types of gyrotron cavities: (1) coaxial, unslotted, (2) coaxial, slotted, and (3) noncoaxial, unslotted. The first major result is that the coaxial rod raises the limiting current in the e-beam diode, permitting reliable, higher current extraction into the microwave tube. The second major finding is that the slotted cavity gives the highest peak powers (approximately 90 MW) but very short pulselengths (approximately 10 - 20 ns). The unslotted coaxial gyrotron emits power levels of 20 - 40 MW with longer pulselengths (up to 40 ns). The noncoaxial gyrotron radiates lower peak power levels (approximately 20 MW). All of the gyrotron types exhibited signs of the pulse shortening mechanisms of mode hopping and mode competition as diagnosed by time-frequency-analysis (TFA). TFA also shows that lower power microwave oscillation is maintained over some 300 ns, but the power level may be modulated due to e-beam voltage fluctuations.
We are presenting a new class of transforms which facilitates the processing of signals that are nonlinearly stretched or compressed in time. We refer to nonlinear stretching and compression as warping. While the magnitude of the Fourier transform is invariant under time shift operations, and the magnitude of the scale transform is invariant under (linear) scaling operations, the new class of transforms is magnitude invariant under warping operations. The new class contains the Fourier transform and the scale transform as special cases. Important theorems, like the convolution theorem for Fourier transforms, are generalized into theorems that apply to arbitrary members of the transform class. Cohen's class of time-frequency distributions is generalized to joint representations in time and arbitrary warping variables. Special attention is paid to a modification of the new class of transforms that maps an arbitrary time-frequency contour into an impulse in the transforms that maps an arbitrary time-frequency contour into an impulse in the transform domain. A chirp transform is derived as an example.
This paper outlines means of using special sets of orthonormally related windows to realize Cohen's class of time-frequency distributions (TFDs). This is accomplished by decomposing the kernel of the distribution in terms of the set of analysis windows to obtain short time Fourier transforms (STFTs). The STFTs obtained using these analysis windows are used to form spectrograms which are then linearly combined with proper weights to form the desired TFD. A set of orthogonal analysis windows which also have the scaling property proves to be very effective, requiring only 1 + log2(N - 1) distinct windows for an overall analysis of N + 1 points, where N equals 2n, with n a positive integer. Application of this theory offers very fast computation of TFDs, since very few analysis windows needed and fast, recursive STFT algorithms can be used. Additionally, it is shown that a minimal set of specially derived orthonormal windows can represent most TFDs, including Reduced Interference Distributions (RIDs) with only three distinct windows plus an impulse window. Finally, the Minimal Window RID (MW-RID) which achieves RID properties with only one distinct window and an impulse window is presented.
Scale as a physical quantity is a recently developed concept. The scale transform can be viewed as a special case of the more general Mellin transform and its mathematical properties are very applicable in the analysis and interpretation of the signals subject to scale changes. A number of single-dimensional applications of scale concept have been made in speech analysis, processing of biological signals, machine vibration analysis and other areas. Recently, the scale transform was also applied in multi-dimensional signal processing and used for image filtering and denoising. Discrete implementation of the scale transform can be carried out using logarithmic sampling and the well-known fast Fourier transform. Nevertheless, in the case of the uniformly sampled signals, this implementation involves resampling. An algorithm not involving resampling of the uniformly sampled signals has been derived too. In this paper, a modification of the later algorithm for discrete implementation of the direct scale transform is presented. In addition, similar concept was used to improve a recently introduced discrete implementation of the inverse scale transform. Estimation of the absolute discretization errors showed that the modified algorithms have a desirable property of yielding a smaller region of possible error magnitudes. Experimental results are obtained using artificial signals as well as signals evoked from the temporomandibular joint. In addition, discrete implementations for the separable two-dimensional direct and inverse scale transforms are derived. Experiments with image restoration and scaling through two-dimensional scale domain using the novel implementation of the separable two-dimensional scale transform pair are presented.
It is mathematically convenient to consider both positive nd negative frequencies in signal representation. This idea is critically important to time-frequency analysis. Usually, however one is presented with real signals. It is also well known that the analytic signal is formed using the Hilbert transform. Some interesting and potentially useful relationships are developed for a signal and its Hilbert transform. Some interesting and potentially useful relationships are developed for a signal and its Hilbert transform in this paper. Cross-Hilbert time-frequency distributions (TFDs) between a signal and its Hilbert transform. The relationships between TFDs of signals and cross-Hilbert TFDs of signals are examined. It is shown how interactions between these TFDs yield results which are confined to either the positive or negative frequency planes. Some results which may seem counterintuitive are pointed out. Finally, some interesting results which use the concepts developed for separation of mixed signals by manipulating the associated TFDs are presented.
One of the key problems in high resolution, time-varying spectral analysis is the suppression of interference terms which can obscure the true location of auto components in the resulting time-frequency distribution (TFD). Commonly used reduced interference distributions tackle the problem with a properly chosen 2D low pass filter (kernel). A recently published novel approach uses alternative means to achieve the desired goal. The idea of the new method is to obtain an estimate of the cross terms form a given prior distribution based on the magnitude and location of its negative components. The estimate is constructed via an iterative projection method that guarantees that the resulting distribution is positive and satisfies the marginals. Even though the marginals are usually a desirable property of TFDs in general, they can impose an undesirably strong constraint on positive TFDs in particular. For these cases it is thus beneficial to relax the marginals-constraints. In this paper we present a new method that does not require the incorporation of this constraint and thus leads to positive TFDs with reduced interference terms but without the restrictions due to the marginals.
Recognition of specific patterns and signatures in images has long been of interest. Powerful techniques exist for detection and classification, but are defeated by straightforward changes and variations in the pattern. These variations include translation and scale changes. Translation and scaling are well understood in a mathematical sense and transformations exist such that, when applied to an image, the result is invariant to these disturbances. Hence, methods may be designed wherein these effects are absent in the resultant representations. This paper describes a pattern recognition procedure which uses scale and translation invariant representations (STIRs) as one step of the process. A novel feature extraction method then identifies features of the STIRs orthogonal to noise variation. This is followed by a detection approach which exploits these features to detect desired patterns in noise. By explicitly modeling the variation due to noninteger scaling factors and sub-pixel translation, strong discrimination between similar patterns is achieved. Using the orthogonal features of the invariant representations, several tests are shown to classify well. A two dimensional image is the basic starting point for the technique. This may be an actual image of an object or the two dimensional form of signal representation such as a time-frequency distribution. The example of keyword spotting in scanned documents serves to illustrate the pattern recognition method.
This paper outlines means of combining and reconciling concepts associated with Cohen's class of distributions and with the wavelet transform. Both have their assets and their liabilities. Previous work has shown that one can decompose any time-frequency distribution (TFD) in Cohen's class into a weighted sum of spectrograms. A set of orthogonal analysis windows which also have the scaling property in common with wavelets is proposed. Successful application of this theory offers very fast computation of TFDs, since very few analysis windows may be needed and fast algorithms can be used. In addition, the decomposition idea offers the possibility of shaping the analysis such that good local and global properties as well as a number of desirable TFD properties are retained. Finally, one may view the result in terms of conventional Cohen's class concepts or, alternatively, in terms of wavelet concepts and potentially combine powerful insights and concepts from both points of view. Preliminary results applied to radar backscatter are provided. Performance curves for several wavelet types are also provided.
A new method to estimate the energy distribution over the time-frequency plane of time-varying stochastic signals is presented. A state space modeling approach is used to represent the signal. A Kalman-smoothing algorithm is used to estimate the states from which the so-called `Kalman- smoothed time frequency distribution (KS-TFD)' is obtained. The KS-TFD estimate is positive, has good cross-term properties and high temporal resolution. The Kalman smoother-based estimates are optimal in the mean square sense and therefore the KS-TFD estimate has excellent noise performance. We demonstrate the `localizing' property of KS- TFD using deterministic signals such as impulses and Gabor logons. Minimum interference is seen with multi component signals. For Gabor logons buried in white noise at various signal-to-noise ratios, we show the excellent performance of the KS-TFD estimate in comparison to the non-causal spectrogram using quantitative performance indices.
Processing a noise signal using Cohen's class of transformation gives a time-frequency distribution with distinct characteristics. Understanding the effects of noise is essential for using such distributions for signal analysis. The variance of a generalized discrete time- frequency distribution is developed analytically and confirmed numerically by simulation for several different time-frequency kernels.
Images and signals can be characterized by representations invariant to time shifts, spatial shifts, frequency shifts, and scale changes as the situation dictates. Advances in time-frequency analysis and scale transform techniques have made this possible. The next step is to distinguish between invariant forms representing different classes of image or signal. Unfortunately, additional factors such as noise contamination and `style' differences complicate this. A ready example is found in text, where letters and words may vary in size and position within the image segment being examined. Examples of complicating variations include font used, corruption during fax transmission, and printer characteristics. The solution advanced in this paper is to cast the desired invariants into separate subspaces for each extraneous factor or group of factors. The first goal is to have minimal overlap between these subspaces and the second goal is to be able to identify each subspace accurately. Concepts borrowed from high-resolution spectral analysis, but adapted uniquely to this problem have been found to be useful in this context. Once the pertinent subspace is identified, the recognition of a particular invariant form within this subspace is relatively simple using well-known singular value decomposition techniques.
The decomposition of time-frequency representations (TFRs) in terms of weighted spectrograms has been recently proposed by several authors. Spectrogram decomposition concepts allow inner products to be used in the computations rather than the more cumbersome outer products usually associated with TFR computations. Kernels are decomposed in terms of eigenvectors in such a manner that a TFR may be represented by a truncated spectrogram series according to the strength of the eigenvalues associated with these eigenvectors. Many TFRs can be represented by relatively few spectrograms due to the small contributions of the remaining spectrograms with eigenvalues below some threshold value. The windows of the spectrograms forming the spectrogram series are the eigenvectors of the decomposition of the kernel of the particular representation. In the present paper it is shown how full TFR representations can be obtained by using a carefully chosen set of scaled cross spectrogram windows, thus avoiding the inherent approximations of the eigenvector approach. Much redundancy can be taken advantage of to permit computation of a small number of short-time Fourier transforms (STFTs). It is not practical to compute the Wigner distribution via the spectrogram decomposition approach due to the fact that the singular values of the decomposition are plus or minus one, precluding truncation of the spectrogram series. The new approach, on the other hand, can represent a Wigner distribution and other TFRs with a small number of STFTs. These STFTs can be used to compute a number of spectrograms and cross-spectrograms which, when appropriately weighted and summed, yield a given TFR, depending on the kernel used in the decomposition.
Studies of two orbiting spheres demonstrate the usefulness of time-frequency distributions for the analysis of radar signals. The spheres served as a simplified model for moving blades of an engine propeller or helicopter rotor. Illuminating the moving spheres with a continuous wave radar generated a backscatter signal which was difficult to interpret in either the time or the frequency domains. By applying the binomial distribution (a discrete time-frequency distribution) we could clearly associate each sphere with its corresponding doppler return. The binomial distribution provided a detailed view of the target dynamics, opening the way for target classification and identification. The structures and details available in the time- frequency domain were not readily exploited in the original signal representation.
A new class of time-frequency kernels is introduced. Members in this class satisfy the desired time-frequency distribution properties and simultaneously provide local autocorrelation functions (LAF) which are amenable to high resolution techniques over periods of stationarities. These high spectral resolution kernels map the sinusoids in time into damped/undamped sinusoidal bilinear data products over the LAF lag variable. The damped sinusoids represent cross-terms. Using SVD-based backward linear prediction techniques, the signal zeros, the cross-term zeros, and the extraneous zeros, respectively, lie on, outside, and inside the unit circle, providing a mechanism to distinguish between different types of components. It is shown that the binomial kernel introduced is a member of this class.
Cohen's class of time-frequency distributions has been recognized to have significant potential for the analysis of complicated signals. The spectrogram, though it offers comparatively lower time-frequency resolution than other, more recently investigated members of Cohen's class, is still the most broadly used TFD today. Packages are available which perform SP evaluation and offer signal synthesis from the closely related short-time Fourier transform. These packages allow the SP to be the widely accessible signal processing tool that it is. In this paper, we introduce two decompositions of TFDs, the SP decomposition and the weighted reversal correlator decomposition. The decompositions are useful for TFD interpretation, fast implementations using SP building blocks and related high-resolution, linear signal synthesis method using STFT signal synthesis building blocks. It is hoped that these decompositions will facilitate the development of TFD signal analysis/synthesis packages (using existing SP analysis/synthesis packages), and that these packages will make TFDs other than the SP accessible to a broad audience.
The well-known uncertain principle is often invoked in signal processing. It is also often considered to have the same implications in signal analysis as does the uncertainty principle in quantum mechanics. The uncertainty principle is often incorrectly interpreted to mean that one cannot locate the time-frequency coordinates of a signal with arbitrarily good precision, since, in quantum mechanics, one cannot determine the position and momentum of a particle with arbitrarily good precision. Renyi information of the third order is used to provide an information measure on time-frequency distributions. The results suggest that even though this new measure tracks time-bandwidth results for two Gabor log-ons separated in time and/or frequency, the information measure is more general and provides a quantitative assessment of the number of resolvable components in a time frequency representation. As such, the information measure may be useful as a tool in the design and evaluation of time-frequency distributions.
In this paper we introduce a new definition for the instantaneous frequency of a discrete-time analytic signal. Unlike the existing definition which uses only two data samples around a particular time this method utilizes all the data samples for estimating the instantaneous frequency. We prove that this quantity is identical to the average frequency evaluated at the particular time in the discrete-time TFD. This property is consistent with the analogous continuous-time property. We also derive requirements on the discrete-time kernel needed to satisfy this property. Using computer-generated signals and real data performance comparisons are made between the proposed approach and the existing one.
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