Research On Some Theoretical Issues Of The Linear Canonical Transform

The linear canonical transform (LCT) is a new signal processing tool developed inrecent years. As a unified and multi-parameter class of linear integral transform, LCThas its unique advantages in non-stationary signal processing. Although it is not verymuch known, its special cases are widely used in various fields, often under differentnames, such as the Fourier transform, fractional Fourier transform (FRFT), Fresneltransform, time scaling, and others. Therefore, understanding the LCT may help to gainmore insights on its special cases and to carry over knowledge gained from one subjectto others. However, from the existing literature, the basic theoretical system of LCT isnot completed yet, some theories related the signal processing need to be furtherestablished or strengthened, such as the time-frequency analysis and sampling theory.Thus, carrying out the research of the LCT and some related theories is very significantin theory and practice.Based on above issues, this dissertation focuses on the study of time-frequencyanalysis, signal sampling and some related theories of LCT. The main contributions andinnovations of this dissertation are summarized as follows:1. Beginning with the relationship between the LCT and the Fourier transform,some basic theories associated with the LCT are investigated in this dissertation. Basedon the uniform sampling theorem of Fourier transform, the uniform sampling theoremand signal reconstruction formula associated with the LCT are derived; from thetraditional convolution theorem, the convolution theorem associated with the LCT isalso derived; by analyzing the classic Hilbert transform, the Hilbert transform associatedwith the LCT is proposed.The definition and properties of Hilbert transform of a signalin the LCT domain have been studied. Finally, the definition and properties of LCT fordiscrete time signal are deduced. It’s very important to establish the above basic theoriesassociated with the LCT.2. Time-frequency analysis theories associated with the LCT are studied in thisdissertation. Starting with the relationship between the linear canonical transform andthe Wigner distribution, the theory of time-frequency filtering about the linear canonical transform has been studied, a time-frequency filtering method based on the linearcanonical transform is proposed and the parameter selection of the filter is discussed indetail. Secondly, the relationship between the linear canonical transform and short-timeFourier transform (STFT) is analyzed. A time-frequency signal analysis method basedon LCT and STFT is proposed, which has no cross-terms problem. It can be used torealize interference suppression of chirp signals and separate components from atime-frequency signal. The simulation results illustrate the validity of the proposedmethod. Finally, the relationship between the linear canonical transform and theambiguity function is analyzed, which is verified by simulations. The above researchresults have laid a good theoretical basis in applied research of further development ofLCT to time-frequency analysis field.3. The uniform sapling theories for band-pass signal associated with the LCT arededuced in this dissertation. The sampling theories related to LCT have not beencompleted yet, so the sampling theorem needs to be restudied in the LCT domain.Firstly, the definition of band-limited signal in the LCT sense is introduced. Secondly,the band-pass signal sampling theorem and reconstruction formula associated with LCTare deduced. Finally, an example of sampling a chirp signal is provided to demonstratethe application of the sampling theorem. The band-pass signal sampling theoremassociated with the LCT is a generalization of the classical sampling theories and willenrich the theoretical system of the linear canonical transform.4. The convolution and multiplicative filtering theories associated with the offsetlinear canonical transform (OLCT) are investigated in this dissertation.The OLCT,which is a time-shifted and frequency-modulated version of the linear canonicaltransform, has been shown to be a powerful tool for signal processing and optics.However, some basic results for this transform, such as convolution and correlationtheorems, remain unknown. Based on a new convolution operation, the convolution andcorrelation theorems associated with the OLCT are derived. The sampling theorem forthe bandlimited signal in the OLCT domain is also derived. The formulas of uniformsampling and lowpass reconstruction related to the OLCT are obtained. The designmethod of the multiplicative filter in the OLCT domain is analyzed. Based on the modelof the multiplicative filter in the OLCT domain, a practical method to achievemultiplicative filtering through convolution in the time domain is proposed.These result further enrich the theoretical system of the linear canonical transform.5. The generalized sampling theories for bandlimited signal associated with theOLCT are deduced in this dissertation.The aim of the generalized sampling is thereconstruction of a bandlimited signal f (t), from the samples of the responses of Mlinear time invariant systems, each sampled by the1Mth Nyquist rate. Firstly, basedon Papoulis’generalized sampling model, the generalized sampling theorem forbandlimited signals in OLCT domains is proposed. Secondly, by designing differentOLCT domain filters, reconstruction formulae for uniform sampling from the signal,from the signal and its first derivative or its generalized Hilbert transform are obtainedbased on the generalized sampling theorem. Since recurrent nonuniform sampling forsignal has valuable applications, reconstruction expression for recurrent nonuniformsamples of signal bandlimited in the offset linear canonical transform domain is alsoobtained by using the generalized sampling theorem and the properties of the offsetlinear canonical transform. Finally, simulations to verify results have been presented.