User:Burhem/Time-frequency signal processing
Introduction to TFSP
The field of "time-frequency signal processing" (TFSP) is a sub-field of "signal processing" that has grown enormously since the 1980s; the field TFSP includes other sub-fields such as time-frequency representation, time-frequency analysis, time-frequency detection, time-frequency filteriing and others. It is concerned with the representation, analysis and processing of signals whose spectral characteristics are time-varying. TFSP represents signals (or time-series) in a joint time and frequency domain, a key difference from the traditional signal representations that are either time domain representations or a frequency domain representation. Such a representation uses Time-Frequency Distributions that are mathematical formulations for showing the distribution of signal energy in the time-frequency plane. Such a TFSP approach allows us to take into account information not accessible through trational signal processing methods so that this new information allows a more accurate representation, analysis and processing; all traditional applications of signal processing would benefit from this approach. In terms of terminology, TFSP is now more appropriate than TFSA as TFDS are now used not only for signal analysis in the time-frequency (t,f) domain but also for signal processing in the (t,f) domain. Such processing includes (t,f) filtering, parameter estimation, signal detection and classification, feature extraction and many others.
Background of TFSP
A major historical development occurred in the late 1970s and early 80s simulaneously and independently in France and the Netherlands, facilitated by new progress in computing, that led to what are known as the earliest applications of TFSP to real-life problems. In France, this development took place in elf-aquitaine (now Total) and was led by B. Bouachache[1] (re-spelled "B. Boashash" in 1984 when he settled in Australia); in the Netherlands, the development took place in Philips and was led by Claasen and Mecklenbrauker[2]. Independently of each other, both B. Boashash and Claasen and Mecklenbrauker applied TFSP to a practical application relevant to the company employing them; in France, the application was to use the Wigner-Ville Distribution and the instantaneous frequency for the estimation of absorption in vertical seismic prospecting for the purpose of a more accurate determination of oil reservoirs contours. In the Netherlands, the application related to the design of loudspeakers. Both teams independently designed an algorithm for implementing the Wigner Distribution and applied it to their engineering problem. After 1984, the effort startcontinued in Australia and focussed on making further progress in applying the TFSP concepts and methods to new applications. For this purpose, the first TFSA package was set-up in Fortran, then C and then Matlab in the 1980s, and the first conference on Time-Frequency Signal Analysis took place in 1990 in Brisbane Australia. In parallel, an SPIE special session on time-frequency signal analysis was organised by the same team leader between 1985 and 1995. During this time, some other auhors became aware of these developments and realized that the formulations used to define TFDs were the same as the ones used in Quantum Mechanics. They then joined the effort of the Australian team and went on to publish a review of the results facilitated by the TFSA software that was then available from the Australian team at the University of Queensland. The Australian team emphasized a holistic approach that included developments of the theory, its implementation and application to real-life applications with special focus on the interpretation and estimation of the concept of instantaneous frequency [3], the need to use the analytic signal in the formulation of the Wigner-Ville Distribution[4] and other TFDs, and the importance of progressing the design of efficient algorithms[5].
From "Time-Frequency Signal Analysis to "Time-Frequency Signal Processing"
From the 1990s, more researchers became involved in the development of TFSP; these efforts led to new advances in designing TFSP methods with greater accuracy, with data dependent formulations[6] /refer/baraniuk and jones/ and new applications to detection[7], classification, filtering and others /refer/???. These developments allowed to not just analyze but also process signals in a joint time-frequency domain, allowing the possibility of more real-life applications as reported in the most comprehensive treatment of these questions found in [8]. The development of new efficient algorithms and improvements in computer technology combined to allow more users to adapt these time-frequency methods to more and more applications in a wider range of fields.
Links to other TFSP relevant articles
We list below and give context to some relevant links to existing articles that provide additional information to complement this article. We also provide a detailed comment for the materiall in each link so as to clarify the meaning and context of these articles and the relevance to this material. To make it easier and more useful to the reader, these links are ordered and grouped according to three major technical topics (signal processing, time-frequency methods and time-scale methods).
Signal Processing Concepts and Methods
Signal processing is the broader field of which Time-Frequency Signal Processing is a sub-field.
Time series is the terminology used in the field of Mathematics to design a signal, so that time series analyis means is the same as signal analysis.
Time domain refers to the natural first particular axis of the time-frequency domain.
Fourier transform transforms the signal representation from time only domain to frequency only domain.
Frequency domain refers to the other second particular axis of the time-frequency domain that results from the Fourier transformation.
Time-Frequency concepts and Methods
Time–frequency representation is concerned with defining the best mathematical formulation to represent signals in a joint time-frequency domain.
The Spectrogram is the earliest TFD used; it is obtained by taking the Short-time Fourier transform (STFF) of the signal. An equivalent TFD is obtained with filter-banks whereby the frequency axis and time axis are in essence conceptually swapped.
The Wigner distribution function or Wigner-Ville Distribution is the core time-frequency representation used to represent signals in a joint time-frequency domain.
Bilinear time-frequency distribution is a class of TFDS that have specific properties that make them suitable to represent signals in a joint time-frequency domain. The Wigner-Ville Distribution is a particular member of this class.
Reduced Interference Distributions such as the Gaussian TFD (also called sometimes Choi-Williams Distribution) allow to obtain a trade-off between resolution in the (t,f) domain and the reduction of artifacts (also called interferences) created by the bilinear nature of these methods.
Transformation between distributions in time-frequency analysis is a useful way to move from one time-frequency representation to another way without having to recompute the whole TFD.
Time–frequency analysis refers to the use of TFDs to analyse a signal and find its characteristics in a joint time-frequency domain.
Instantaneous Frequency(IF) is a function of time which describes the variations of the signal spectral contents with time.Instantaneous Frequency is one of the key concepts of TFSP as the IF establishes a practical and conceptual link between the time domain and the frequency domain.
Frequency Modulation(FM) is an example of practical use of the IF in Telecommunications for Radio transmission where the FM is given by the IF of the signal.
Instantaneous Phase is the integral of the signal instantaneous frequency.
Analytic signal is a complex signal formed by adding an imaginary part to the real signal, where the imaginary part is the Hilbert Transform of the real part. This key concept defines the complex signal z(t) associated with the real signal s(t). Its use in the formulation of the WD results in the WVD which leads to the practical use of the WVD for time-frequency representation of signals.
Time-Scale concepts and Methods
Wavelet or in pluriel, wavelets, are basic functions satisfying certain properties that are used as a basis for signal decomposition; each wavelet occupies a certain time and frequency position in the time-frequency domain; the decomposition can be done with the wavelet transform. The relationship to TFSP is that in the case of signals that have discontinuities or are transients, a wavelet decomposition can provide a more accurate time-frequency representation than quadratic or bilinear time-frequency distributions.
Example of Applications of TFSP
A wide range of applications can be found in the reference tfsap
references
- ^ B. Bouachache, "Representation temps-frequence," Soc. Nat. ELF Aquitaine, Pau, France, Publ. Recherches, no. 373-378, 1978
- ^ T. A. C. M. Classen and W. F. G. Mecklenbrauker, “The Wigner distributiona tool for time-frequency signal analysis; Part I,” Philips J. Res., vol. 35, pp. 217–250, 1980.
- ^ B. Boashash, "Estimating and Interpreting the Instantaneous Frequency of a Signal-Part I: Fundamentals", Proceedings of the IEEE, Vol. 80, No. 4, pp. 519-538, April 1992
- ^ B. Boashash, "Note on the Use of the Wigner Distribution for Time Frequency Signal Analysis", IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 36, No. 9, pp. 1518–1521, Sept. 1988
- ^ B. Boashash and P. Black, "An Efficient Real-Time Implementation of the Wigner-Ville Distribution", IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 35, No. 11, pp. 1611-1618, November 1987.
- ^ R. G. Baraniuk and D. L. Jones, “A Signal-Dependent Time-Frequency Representation: Optimal Kernel
- ^ B. Yazici and G. B. Kliman, “An adaptive statistical time-frequency method for detection of broken bars and bearing faults in motors using stator current,” IEEE Trans. Ind. Applicat., vol. 35, pp. 442–452,Mar./Apr. 1999.
- ^ B. Boashash, editor, “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003; ISBN 0080443354
links
Most Comprehensive reference on time-frequency signal analysis and processing
tfsp toolbox; insert url + text