Theory of radar imaging using time-frequency distribution

The concept of radar was developed for the estimation of the distance (range) and velocity of a target from a receiver. The distance measurement is obtained by measuring the time taken for the transmitted signal to propagate to the target and return to the receiver. The target's velocity is determined by measuring the Doppler induced frequency shift of the returned signal caused by the rate of change of the time- delay from the target. As researchers further developed conventional radar systems it become apparent that additional information was contained in the backscattered signal and that this information could in fact be used to describe the shape of the target itself. It is due to the fact that a target can be considered to be a collection of individual point scatterers, each of which has its own velocity and time- delay. DelayDoppler parameter estimation of each of these point scatterers thus corresponds to a mapping of the target's range and cross range, thus producing an image of the target. Much research has been done in this area since the early radar imaging work of the 1960s. At present there are two main categories into which radar imaging falls. The first of these is related to the case where the backscattered signal is considered to be deterministic. The second is related to the case where the backscattered signal is of a stochastic nature. In both cases the information which describes the target's scattering function is extracted by the use of the ambiguity function, a function which correlates the backscattered signal in time and frequency with the transmitted signal. In practical situations, it is often necessary to have the transmitter and the receiver of the radar system sited at different locations. The problem in these situations is 'that a reference signal must then be present in order to calculate the ambiguity function. This causes an additional problem in that detailed phase information about the transmitted signal is then required at the receiver. It is this latter problem which has led to the investigation of radar imaging using time- frequency distributions. As will be shown in this thesis, the phase information about the transmitted signal can be extracted from the backscattered signal using time- frequency distributions. The principle aim of this thesis was in the development, and subsequent discussion into the theory of radar imaging, using time- frequency distributions. Consideration is first given to the case where the target is diffuse, ie. where the backscattered signal has temporal stationarity and a spatially white power spectral density. The complementary situation is also investigated, ie. where the target is no longer diffuse, but some degree of correlation exists between the time- frequency points. Computer simulations are presented to demonstrate the concepts and theories developed in the thesis. For the proposed radar system to be practically realisable, both the time- frequency distributions and the associated algorithms developed must be able to be implemented in a timely manner. For this reason an optical architecture is proposed. This architecture is specifically designed to obtain the required time and frequency resolution when using laser radar imaging. The complex light amplitude distributions produced by this architecture have been computer simulated using an optical compiler.

Time-Frequency Distribution Moments of Heart Rate Variability for Neonatal Seizure Detection

In this paper, we propose features extracted from the heart rate variability (HRV) based on the first and second conditional moments of time-frequency distribution (TFD) as an additional guide for seizure detection in newborn. The features of HRV in the low frequency band (LF: 0-0.07 Hz), mid frequency band (MF: 0.07-0.15 Hz), and high frequency band (HF: 0.15-0.6 Hz) have been obtained by means of the time-frequency analysis using the modified-B distribution (MBD). Results of ongoing time-frequency research are presented. Based on our preliminary results, the first conditional moment of HRV which is also known as the mean/central frequency in the LF band and the second conditional moment of HRV which is also known as the variance/instantaneous bandwidth (IB) in the HF band can be used as a good feature to discriminate the newborn seizure from the non-seizure

Identification of Nonlinear Systems Through Time-Frequency Filtering Technique

In the previous paper, a class of nonlinear system is mapped to a so-called skeleton linear model (SLM) based on the joint time-frequency analysis method. Behavior of the nonlinear system may be indicated quantitatively by the variance of the coefficients of SLM versus its response. Using this model we propose an identification method for nonlinear systems based on nonstationary vibration data in this paper. The key technique in the identification procedure is a time-frequency filtering method by which solution of the SLM is extracted from the response data of the corresponding nonlinear system. Two time-frequency filtering methods are discussed here. One is based on the quadratic time-frequency distribution and its inverse transform, the other is based on the quadratic time-frequency distribution and the wavelet transform. Both numerical examples and an experimental application are given to illustrate the validity of the technique.

Heart rate variability characterization using a time-frequency based instantaneous frequency estimation technique

In this paper, a new method for characterizing the newborn heart rate variability (HRV) is proposed. The central of the method is the newly proposed technique for instantaneous frequency (IF) estimation specifically designed for nonstationary multicomponen signals such as HRV. The new method attempts to characterize the newborn HRV using features extracted from the time–frequency (TF) domain of the signal. These features comprise the IF, the instantaneous bandwidth (IB) and instantaneous energy (IE) of the different TF components of the HRV. Applied to the HRV of both normal and seizure suffering newborns, this method clearly reveals the locations of the spectral peaks and their time-varying nature. The total energy of HRV components, ET and ratio of energy concentrated in the low-frequency (LF) to that in high frequency (HF) components have been shown to be significant features in identifying the HRV of newborn with seizures.

Time-frequency analysis of human motion during rhythmic exercises

Biomechanical signals due to human movements during exercise are represented in time-frequency domain using Wigner Distribution Function (WDF). Analysis based on WDF reveals instantaneous spectral and power changes during a rhythmic exercise. Investigations were carried out on 11 healthy subjects who performed 5 cycles of sun salutation, with a body-mounted Inertial Measurement Unit (IMU) as a motion sensor. Variance of Instantaneous Frequency (I.F) and Instantaneous Power (I.P) for performance analysis of the subject is estimated using one-way ANOVA model. Results reveal that joint Time-Frequency analysis of biomechanical signals during motion facilitates a better understanding of grace and consistency during rhythmic exercise.

Time-frequency spectral estimation of speech

In recent years there has been a growing interest amongst the speech research community into the use of spectral estimators which circumvent the traditional quasi-stationary assumption and provide greater time-frequency (t-f) resolution than conventional spectral estimators, such as the short time Fourier power spectrum (STFPS). One distribution in particular, the Wigner distribution (WD), has attracted considerable interest. However, experimental studies have indicated that, despite its improved t-f resolution, employing the WD as the front end of speech recognition system actually reduces recognition performance; only by explicitly re-introducing t-f smoothing into the WD are recognition rates improved. In this paper we provide an explanation for these findings. By treating the spectral estimation problem as one of optimization of a bias variance trade off, we show why additional t-f smoothing improves recognition rates, despite reducing the t-f resolution of the spectral estimator. A practical adaptive smoothing algorithm is presented, whicy attempts to match the degree of smoothing introduced into the WD with the time varying quasi-stationary regions within the speech waveform. The recognition performance of the resulting adaptively smoothed estimator is found to be comparable to that of conventional filterbank estimators, yet the average temporal sampling rate of the resulting spectral vectors is reduced by around a factor of 10. © 1992.

Time-Frequency Representations for Speech Signals

This work addresses two related questions. The first question is what joint time-frequency energy representations are most appropriate for auditory signals, in particular, for speech signals in sonorant regions. The quadratic transforms of the signal are examined, a large class that includes, for example, the spectrograms and the Wigner distribution. Quasi-stationarity is not assumed, since this would neglect dynamic regions. A set of desired properties is proposed for the representation: (1) shift-invariance, (2) positivity, (3) superposition, (4) locality, and (5) smoothness. Several relations among these properties are proved: shift-invariance and positivity imply the transform is a superposition of spectrograms; positivity and superposition are equivalent conditions when the transform is real; positivity limits the simultaneous time and frequency resolution (locality) possible for the transform, defining an uncertainty relation for joint time-frequency energy representations; and locality and smoothness tradeoff by the 2-D generalization of the classical uncertainty relation. The transform that best meets these criteria is derived, which consists of two-dimensionally smoothed Wigner distributions with (possibly oriented) 2-D guassian kernels. These transforms are then related to time-frequency filtering, a method for estimating the time-varying 'transfer function' of the vocal tract, which is somewhat analogous to ceptstral filtering generalized to the time-varying case. Natural speech examples are provided. The second question addressed is how to obtain a rich, symbolic description of the phonetically relevant features in these time-frequency energy surfaces, the so-called schematic spectrogram. Time-frequency ridges, the 2-D analog of spectral peaks, are one feature that is proposed. If non-oriented kernels are used for the energy representation, then the ridge tops can be identified, with zero-crossings in the inner product of the gradient vector and the direction of greatest downward curvature. If oriented kernels are used, the method can be generalized to give better orientation selectivity (e.g., at intersecting ridges) at the cost of poorer time-frequency locality. Many speech examples are given showing the performance for some traditionally difficult cases: semi-vowels and glides, nasalized vowels, consonant-vowel transitions, female speech, and imperfect transmission channels.

A time frequency approach to blind deconvolution in multipath underwater channels

Blind deconvolution is studied in the underwater acoustic channel context, by time-frequency (TF) processing. The acoustic propagation environment is modelled by ray tracing and mathematically described by a multipath propagation channel. Representation of the received signal by means of a signal-dependent TF distribution (radially Gaussian kernel distribution) allowed to visualize the resolved replicas of the emitted signal, while signi cantly attenuating the inherent interferences of classic quadratic TF distributions. The source signal instantaneous frequency estimation was the starting point for both source and channel estimation. Source signature estimation was performed by either TF inversion, based on the Wigner-Ville distribution of the received signal, or a subspace- -based method. The channel estimate was obtained either via a TF formulation of the conventional matched- lter, or via matched- - ltering with the previously obtained source estimate. A shallow water realistic scenario is considered, comprising a 135-m depth water column and an acoustic source located at 90-m depth and 5.6-km range from the receiver. For the corresponding noiseless simulated data, the quality of the best estimates was 0.856 for the source signal, and 0.9664 and 0.9996 for the amplitudes and time-delays of the impulse response, respectively. Application of the proposed deconvolution method to real data of the INTIMATE '96 sea trial conduced to source and channel estimates with the quality of 0.530 and 0.843, respectively. TF processing has proved to remove the typical ill-conditioning of single sensor deterministic deconvolution techniques.

Development of Time - Frequency Techniques for Sonar Applications

Sonar signal processing comprises of a large number of signal processing algorithms for implementing functions such as Target Detection, Localisation, Classification, Tracking and Parameter estimation. Current implementations of these functions rely on conventional techniques largely based on Fourier Techniques, primarily meant for stationary signals. Interestingly enough, the signals received by the sonar sensors are often non-stationary and hence processing methods capable of handling the non-stationarity will definitely fare better than Fourier transform based methods.Time-frequency methods(TFMs) are known as one of the best DSP tools for nonstationary signal processing, with which one can analyze signals in time and frequency domains simultaneously. But, other than STFT, TFMs have been largely limited to academic research because of the complexity of the algorithms and the limitations of computing power. With the availability of fast processors, many applications of TFMs have been reported in the fields of speech and image processing and biomedical applications, but not many in sonar processing. A structured effort, to fill these lacunae by exploring the potential of TFMs in sonar applications, is the net outcome of this thesis. To this end, four TFMs have been explored in detail viz. Wavelet Transform, Fractional Fourier Transfonn, Wigner Ville Distribution and Ambiguity Function and their potential in implementing five major sonar functions has been demonstrated with very promising results. What has been conclusively brought out in this thesis, is that there is no "one best TFM" for all applications, but there is "one best TFM" for each application. Accordingly, the TFM has to be adapted and tailored in many ways in order to develop specific algorithms for each of the applications.

Four-window technique for measuring optical-phase-space-time-frequency tomography

A new approach, the four-window technique, was developed to measure optical phase-space-time-frequency tomography (OPSTFT). The four-window technique is based on balanced heterodyne detection with two local oscillator (LO) fields. This technique can provide independent control of position, momentum, time and frequency resolution. The OPSTFT is a Wigner distribution function of two independent Fourier transform pairs, phase-space and time-frequency. The OPSTFT can be applied for early disease detection.

Size-frequency distribution of boulders ≥7 m on comet 67P/Churyumov-Gerasimenko

Aims. We derive for the first time the size-frequency distribution of boulders on a comet, 67P/Churyumov-Gerasimenko (67P), computed from the images taken by the Rosetta/OSIRIS imaging system. We highlight the possible physical processes that lead to these boulder size distributions. Methods. We used images acquired by the OSIRIS Narrow Angle Camera, NAC, on 5 and 6 August 2014. The scale of these images (2.44−2.03 m/px) is such that boulders ≥7 m can be identified and manually extracted from the datasets with the software ArcGIS. We derived both global and localized size-frequency distributions. The three-pixel sampling detection, coupled with the favorable shadowing of the surface (observation phase angle ranging from 48° to 53°), enables unequivocally detecting boulders scattered all over the illuminated side of 67P. Results. We identify 3546 boulders larger than 7 m on the imaged surface (36.4 km2), with a global number density of nearly 100/km2 and a cumulative size-frequency distribution represented by a power-law with index of −3.6 +0.2/−0.3. The two lobes of 67P appear to have slightly different distributions, with an index of −3.5 +0.2/−0.3 for the main lobe (body) and −4.0 +0.3/−0.2 for the small lobe (head). The steeper distribution of the small lobe might be due to a more pervasive fracturing. The difference of the distribution for the connecting region (neck) is much more significant, with an index value of −2.2 +0.2/−0.2. We propose that the boulder field located in the neck area is the result of blocks falling from the contiguous Hathor cliff. The lower slope of the size-frequency distribution we see today in the neck area might be due to the concurrent processes acting on the smallest boulders, such as i) disintegration or fragmentation and vanishing through sublimation; ii) uplifting by gas drag and consequent redistribution; and iii) burial beneath a debris blanket. We also derived the cumulative size-frequency distribution per km2 of localized areas on 67P. By comparing the cumulative size-frequency distributions of similar geomorphological settings, we derived similar power-law index values. This suggests that despite the selected locations on different and often opposite sides of the comet, similar sublimation or activity processes, pit formation or collapses, as well as thermal stresses or fracturing events occurred on multiple areas of the comet, shaping its surface into the appearance we see today.