A Unified Approach to the STFT, TFDs and Instantaneous Frequency

Spectral analysis of time varying signals is traditionally performed with the short time Fourier transformation (STFT). In the last few years, many authors have advocated the use of time frequency distributions for this task. This paper has 2 main aims. The first is to reformulate Cohen-class of time frequency representations (TFRs) into discrete-time, discrete-frequency, computer-implemented form. The second aim is to show how, in this form, many of the properties of the continuous-time, continuous-frequency formulation are either lost or altered. Intuitions applicable in the continuous-time case examined here. The properties of the discrete variable formulation examined are the presence and form of cross-terms, instantaneous frequency (IF) estimation and relations between Cohen's class TFRs. We define a parameterized class of distributions which is a blending between the STFT and wigner ville distribution (WVD). The two main conclusions to be drawn are that all TFRs of Cohen's class implementable in the form (which includes all commonly used TFRs) posses cross terms and that IF estimation using periodic moments of these TFRs is purposeless, since simpler methods obtain the same results

A Review of the Frequency Estimation and Tracking Problems

This report presents a concise review of some frequency estimation and frequency tracking problems. In particular, the report focusses on aspects of these problems which have been addressed by members of the Frequency Tracking and Estimation project of the Centre for Robust and Adaptive Systems. The report is divided into four parts: problem specification and discussion, associated problems, frequency estimation algorithms and frequency tracking algorithms. Part I begins with a definition of the various frequency estimation and tracking problems. Practical examples of where each problem may arise are given. A comparison is made between the frequency estimation and tracking problems. In Part II, block frequency estimation algorithms, fast block frequency estimation algorithms and notch filtering techniques for frequency estimation are dealt with. Frequency tracking algorithms are examined in Part III. Part IV of this report examines various problems associated with frequency estimation. Associated problems include Cramer-Rao lower bounds, theoretical algorithm performance, frequency resolution, use of the analytic signal and model order selection

On time-frequence analysis of heart rate variability

The aim of this research is to develop a time-frequency method suitable to study HRV in greater detail. The following approach was used: • two known time-frequency representations were applied to HRV to understand its advantages and disadvantages in describing HRV in frequency and in amplitude, over time; • a new method was developed that describes the time-varying fluctuations in the characteristic frequency bands of HRV by means of the instantaneous frequency and the instantaneous amplitude with an optimal time-resolution; • an index was developed to separate the oscillatory from the irregular periods in the instantaneous frequency; • from the instantaneous amplitude and frequency, we derived summarizing parameters which we applied to describe the changes in the instantaneous amplitude and frequency over time for the oscillatory and irregular periods separatel

Seismic characterisation based on time-frequency spectral analysis

We present high-resolution time-frequency spectral analysis schemes to better resolve seismic images for the purpose of seismic and petroleum reservoir characterisation. Seismic characterisation is based on the physical properties of the Earth's subsurface media, and these properties are represented implicitly by seismic attributes. Because seismic traces originally presented in the time domain are non-stationary signals, for which the properties vary with time, we characterise those signals by obtaining seismic attributes which are also varying with time. Among the widely used attributes are spectral attributes calculated through time-frequency decomposition. Time-frequency spectral decomposition methods are employed to capture variations of a signal within the time-frequency domain. These decomposition methods generate a frequency vector at each time sample, referred to as the spectral component. The computed spectral component enables us to explore the additional frequency dimension which exists jointly with the original time dimension enabling localisation and characterisation of patterns within the seismic section. Conventional time-frequency decomposition methods include the continuous wavelet transform and the Wigner-Ville distribution. These methods suffer from challenges that hinder accurate interpretation when used for seismic interpretation. Continuous wavelet transform aims to decompose signals on a basis of elementary signals which have to be localised in time and frequency, but this method suffers from resolution and localisation limitations in the time-frequency spectrum. In addition to smearing, it often emerges from ill-localisation. The Wigner-Ville distribution distributes the energy of the signal over the two variables time and frequency and results in highly localised signal components. Yet, the method suffers from spurious cross-term interference due to its quadratic nature. This interference is misleading when the spectrum is used for interpretation purposes. For the specific application on seismic data the interference obscures geological features and distorts geophysical details. This thesis focuses on developing high fidelity and high-resolution time-frequency spectral decomposition methods as an extension to the existing conventional methods. These methods are then adopted as means to resolve seismic images for petroleum reservoirs. These methods are validated in terms of physics, robustness, and accurate energy localisation, using an extensive set of synthetic and real data sets including both carbonate and clastic reservoir settings. The novel contributions achieved in this thesis include developing time-frequency analysis algorithms for seismic data, allowing improved interpretation and accurate characterisation of petroleum reservoirs. The first algorithm established in this thesis is the Wigner-Ville distribution (WVD) with an additional masking filter. The standard WVD spectrum has high resolution but suffers the cross-term interference caused by multiple components in the signal. To suppress the cross-term interference, I designed a masking filter based on the spectrum of the smoothed-pseudo WVD (SP-WVD). The original SP-WVD incorporates smoothing filters in both time and frequency directions to suppress the cross-term interference, which reduces the resolution of the time-frequency spectrum. In order to overcome this side-effect, I used the SP-WVD spectrum as a reference to design a masking filter, and apply it to the standard WVD spectrum. Therefore, the mask-filtered WVD (MF-WVD) can preserve the high-resolution feature of the standard WVD while suppressing the cross-term interference as effectively as the SP-WVD. The second developed algorithm in this thesis is the synchrosqueezing wavelet transform (SWT) equipped with a directional filter. A transformation algorithm such as the continuous wavelet transform (CWT) might cause smearing in the time-frequency spectrum, i.e. the lack of localisation. The SWT attempts to improve the localisation of the time-frequency spectrum generated by the CWT. The real part of the complex SWT spectrum, after directional filtering, is capable to resolve the stratigraphic boundaries of thin layers within target reservoirs. In terms of seismic characterisation, I tested the high-resolution spectral results on a complex clastic reservoir interbedded with coal seams from the Ordos basin, northern China. I used the spectral results generated using the MF-WVD method to facilitate the interpretation of the sand distribution within the dataset. In another implementation I used the SWT spectral data results and the original seismic data together as the input to a deep convolutional neural network (dCNN), to track the horizons within a 3D volume. Using these application-based procedures, I have effectively extracted the spatial variation and the thickness of thinly layered sandstone in a coal-bearing reservoir. I also test the algorithm on a carbonate reservoir from the Tarim basin, western China. I used the spectrum generated by the synchrosqueezing wavelet transform equipped with directional filtering to characterise faults, karsts, and direct hydrocarbon indicators within the reservoir. Finally, I investigated pore-pressure prediction in carbonate layers. Pore-pressure variation generates subtle changes in the P-wave velocity of carbonate rocks. This suggests that existing empirical relations capable of predicting pore-pressure in clastic rocks are unsuitable for the prediction in carbonate rocks. I implemented the prediction based on the P-wave velocity and the wavelet transform multi-resolution analysis (WT-MRA). The WT-MRA method can unfold information within the frequency domain via decomposing the P-wave velocity. This enables us to extract and amplify hidden information embedded in the signal. Using Biot's theory, WT-MRA decomposition results can be divided into contributions from the pore-fluid and the rock framework. Therefore, I proposed a pore-pressure prediction model which is based on the pore-fluid contribution, calculated through WT-MRA, to the P-wave velocity.Open Acces

The detection of unknown waveforms in ESM receivers: FFT-based real-time solutions

Radars and airborne electronic support measures (ESMs) systems are locked in a tactical battle to detect each other whilst remaining undetected. Traditionally, the ESM system has a range advantage. Low probability of intercept (LPI) waveform designers are, however, more heavily exploiting the matched filter radar advantage and hence degrading the range advantage. There have been literature and internal, SELEX Galileo proposals to regain some ESM processing gain of low probability of intercept (LPI) waveforms. This study, however, has sought digital signal processing (DSP) solutions which are: (1) computationally simple; (2) backward-compatible with existing SELEX Galileo digital receivers (DRxs) and (3) have low resource requirements. The two contributions are complementary and result in a detector which is suitable for detection of most radar waveforms. The first contribution is the application of spatially variant apodization (SVA) in a detection role. Compared to conventional window functions, SVA was found to be beneficial for the detection of sinusoidal radar waveforms as it surpassed the fixed window function detectors in all scenarios tested. The second contribution shows by simulation that simple spectral smoothing techniques improved DRx LPI detection capability to a level similar to more complicated non-parametric spectral estimators and far in excess of the conventional (modified) periodogram. The DSP algorithms were implemented using model-based design (MBD). The implication is that a detector with improved conventional and LPI waveform detection capability can be created from the intellectual property (IP). Estimates of the improvement in SELEX Galileo DRx system detection range are provided in the conclusion