User:Burhem/Time-frequency signal processing

Introduction and Backkground to TFSP
The field of "time-frequency signal processing" (TFSP) is a sub-field of "signal processing" that has grown enormously since the 1980s; initially, it started as a desire to design spectral analysis methods that can deal with non-stationary signals. 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 displaying 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 can benefit from this approach, especially when signals are non-stationary. In terms of terminology and description of the field, TFSP includes not only methods of signal analysis in the time-frequency or (t,f) domain but also methods 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.

The development of the field was helped by simultaneous advances in theory, implementation and application to real-life applications such as a clearer interpretation and estimation of the concept of instantaneous frequency, the realization of the need to use the analytic signal in the formulation of the Wigner-Ville Distribution and other TFDs so as to benefit practical applications.

The initial focus of the use of time-frequency techniques was for signal analysis was followed by new advances in designing TFSP methods with greater accuracy, with data dependent formulations and new applications such as detection, classification, filtering; new theoretical developments also continued including defining TFDs for complex arguments. 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. 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.

TFSP concepts, methods and techniques
The field of TFSP can be described by listing and discussing the relevant concepts and techniques specific to the field as they provide additional information that helps illustrate the methods in terms of meaning, context and relevance. These concepts and techniques are ordered and grouped according to three main technical topics (signal processing, time-frequency methods and time-scale methods).

Related Signal Processing Concepts and Methods
Given that Signal processing is the broader field of which Time-Frequency Signal Processing is a sub-field, we need to note that Time series is essentially the same terminology, but used in the field of Mathematics, to design a signal, so that time series analyis means the same as signal analysis. The Time domain refers to the natural first particular axis of the time-frequency domain. The Fourier transform transforms the signal representation from time only domain to frequency only domain. The result is a complex frequency spectrum from which a Fourier analysis can extract information relevant to the application considered. This resulting Frequency domain defines the other second particular axis of the time-frequency domain that results from the Fourier transformation. A relatively novel and key concept, the 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 most important concepts of TFSP as the IF establishes a practical and conceptual link between the time domain and the frequency domain. The Instantaneous Phase is the integral of the signal instantaneous frequency. TheFrequency Modulation(FM) is an example of practical use of the IF in Radar and Telecommunications for Radio transmission where the FM is given by the IF of the signal. A chirp is a particular signal, either natural or man-made, that is used to test the performance of TFDs in terms of selected criteria such as concentration and resolution. The 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. The formulation of the Analytic signal is the key to the definition of the IF.

Specific Time-Frequency concepts and Methods
The problem of 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.

The Ambiguity function (AF) is a function of delay and doppler that represents a Correlation function that includes a frequency doppler as well as a time lag; the AF equals the two-dimensional Fourier transform (2DFT) of the Wigner-Ville Distribution; by filtering the Ambiguity function and taking an inverse 2DFT, we can define Reduced Interference TFDS and the class of QTFDs.

Related Time-Scale concepts and Methods
A Wavelet is a basic function satisfying certain properties that is used with other time and frequency shifted versions 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. Wavelets and Quadratic TFDs are linked through the Harmonic wavelet transform, the Continuous Wavelet transform,Gabor Transform and the S transform.

A: analog formulation
1)	Given a real signal s(t) that needs processing, we first form the analytic associate z(t) according to the procedure outlined in [ref] so as to obtain z(t) = s(t) + H[s(t)], where [.] is the Hilbert Transform. 2)	We form the WVD of z(t) ie Wz (t,f) = FT [z(t + tau/2)z*(t – tau/2)], where FT is the Fourier transform. 3)	We form RO z (t,f)  = Wz (t,f) conv-2d gamma (t,f) where gamma is a kernel filter with characteristics adapted to the signal s(t).

B: digital formulation
1)	Same as in section A, but use DFT and other digital equivalents

C: example of methodologies
1)	We know s(t); find gamma (t,f) so as to get an estimate s^(t) of s(t) with improved SNR. 2)	T-F filtering of s(t) to remove some undesirable components 3)	We know RO z (t,f); find the optimal s(t) for which this RO is the TFD. 4)	S(t) is known; we want to estimate its components (eg multicomponent IF estimation) 5)	T-F detection 6)	T-F classification

Example of Applications
Almost any application in the wider fields of engineering and science and beyond can benefit from the use of TFSP techniques especially when data show signs of non-stationarities ie variations of the spectral content versus time. There is an increasing list of TFSP methods specially adapted to deal with specific practical applications ranging from Telecommunications to Machine condition monitoring and including Biomedical applications such as Newborn seizure detection.