A general framework for sampling and reconstruction in function spaces associated with fractional Fourier transform

The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.

Some important fractional transformations for signal processing

Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain

The linear canonical transform (LCT) has been shown to be useful and powerful in signal processing, optics, etc. Many results of this transform are already known, including sampling theory. Most existing sampling theories of the LCT consider the class of bandlimited signals. However, in the real world, many analog signals arising in engineering applications are non-bandlimited. In this correspondence, we propose a sampling and reconstruction strategy for a class of function spaces associated with the LCT, which can provide a suitable and realistic model for real applications. First, we introduce definitions of semi- and fully-discrete convolutions for the LCT. Then, we derive necessary and sufficient conditions pertaining to the LCT, under which integer shifts of a chirp-modulated function generate a Riesz basis for the function spaces. By applying the results, we present a more comprehensive sampling theory for the LCT in the function spaces, and further, a sampling theorem which recovers a signal from its own samples in the function spaces is established. Moreover, some sampling theorems for shift-invariant spaces and some existing sampling theories for bandlimited signals associated with the Fourier transform (FT), the fractional FT, or the LCT are noted as special cases of the derived results. Finally, some potential applications of the derived theory are presented.

The fractional Fourier transform (FRFT) has the mixed time and frequency characteristics of signals. And FRFT is a powerful and effective tool for time-varying and nonstationary signal processing. To address face recognition problem, an approach based on 2D discrete fractional Fourier transform (2D-DFRFT) is proposed in this paper. First of all, the 2D-DFRFT of a facial image is computed. Then Principal Component Analysis (PCA) and Fisher Linear Discriminant Analysis (FLDA) are combined to perform discriminative feature extraction. Then the nearest neighbor (NN) classifier is applied to perform face recognition, based on Euclidean distance. Experimental results on ORL face database indicate that the proposed approach obtains good recognition effects.

Difference operators and generalized discrete fractional transforms in signal and image processing

The traditional signal processing method of chirp sub-bottom profiling is a type of matched filtering and is an optimal detector when there is no Doppler shift. Fractional Fourier transform (FrFT) is a linear transformation whose orthogonal basis is a linear frequency modulation signal, so there is no cross term interference when processing multiple or multipath chirp signals. The central characteristic of FrFT is that it has a similar theoretical basis to pulse compression, which makes it possible to use FrFT in the signal processing of chirp sub-bottom profiling. In this paper, the basic principle and theory of chirp sub-bottom profiler signal processing based on FrFT is introduced in detail. Firstly, orthogonal transform is performed to obtain the corresponding analytical form of the echo signal, then the time domain signal is transformed to a fractional Fourier domain (u domain) by optimal FrFT, and the signal enhancement is done using the u domain shading method. Finally, time dimensional transform is used to get the time domain envelope signal of the enhanced sediment impulse response. Time dimensional transform is realized by sample sequence mapping, time offset compensation, extraction and low pass filtering. This process can transform a u domain signal to the time domain directly without complex calculations. Performance comparison of this method to matched filtering is given too, and recommendations for future work are presented.

The fractional Fourier transform (FRFT) has been used to detect and estimate the parameters of linear frequency-modulated continuous-wave (LFMCW) in low probability of intercept radar waveforms. The FRFT, which is optimal for single linear frequency-modulated (LFM) signals, becomes sub-optimal when applied to LFMCW signals because the observed waveform of this type of signal is composed of concatenated LFM pulses. A new signal processing method, called the periodic FRFT (PFRFT), is proposed for the detection of LFMCW signals. First, the discrete PFRFT is studied and the signal processing gain of this transform for LFMCW signals is analyzed. Second, an adaptive threshold detection and estimation algorithm for LFMCW signals is formulated after analysis of the test statistics of the squared modulus of LFMCW signals when using the probability density function in the presence of Gaussian white noise. It is then proved that PFRFT-based estimation is equivalent to maximum likelihood estimation in the detection and estimation of LFMCW signals. Finally, the results of both the theoretical analysis and verification simulations show that the PFRFT significantly outperforms the FRFT for LFMCW signals.

In previous papers, the Fourier transform (FT) has been generalized into the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the simplified fractional Fourier transform (SFRFT). Because the cosine, sine, and Hartley transforms are very similar to the FT, it is reasonable to think they can also be generalized by the similar way. We introduce several new transforms. They are all the generalization of the cosine, sine, or Hartley transform. We first derive the fractional cosine, sine, and Hartley transforms (FRCT/FRST/FRHT). They are analogous to the FRFT. Then, we derive the canonical cosine and sine transforms (CCT/CST). They are analogous to the LCT. We also derive the simplified fractional cosine, sine, and Hartley transforms (SFRCT/SFRST/SFRHT). They are analogous to the SFRFT and have the advantage of real-input-real-output. We also discuss the properties, digital implementation, and applications (e.g., the applications for filter design and space-variant pattern recognition) of these transforms. The transforms introduced in this paper are very efficient for digital implementation. We can just use one half or one fourth of the real multiplications required for the FRFT and LCT to implement them. When we want to process even, odd, or pure real/imaginary functions, we can use these transforms instead of the FRFT and LCT. Besides, we also show that the FRCT/FRST, CCT/CST, and SFRCT/SFRST are also useful for the one-sided (t /spl isin/ [0, /spl infin/]) signal processing.

Quaternionic one-dimensional linear canonical transform

Taking advantage of the high speed arithmetic offers by the parallelism of the digital signal processor DSP TMS320C6711 of TI, we implemented two algorithms to find the Fourier transform of fractional order of two dimensional signals ("The discrete fractional Fourier transform" and "Fractional Fourier Transform calculation through the fast-Fourier-transform algorithm"). As a result, we have two algorithms for calculate the fractional Fourier transform of images using a DSP, the images are captured using a camera CCD and the fractional Fourier transform is display in a monitor, both are connected to the DSP device directly. Finally, the work focuses in the evaluation of this algorithms, in order to find which is best performance related to four criteria determined from the properties of the Fourier transform of fractional order: The inverse transform, fractional Fourier transform of order one or standard Fourier transform, fractional Fourier transform of order zero or operator identity and order additivity.

This paper proposed an improved method based on fractional Fourier transform (FRFT) for multicarrier LFM signals of MIMO radar (MIMO-LFM) by analyzing its characteristics. First, the intercepted signal model and FRFT are analyzed. Then the signal detection and parameter estimation of MIMO-LFM signals based on FRFT autocorrelation envelope is derived after analyzing the superiority and shortage of conventional FRFT. After that, the detection performance of the proposed method is deduced, and the theoretical results show that this method outperformance the conventional time-frequency method for MIMO-LFM signals processing. Finally, the simulation results demonstrate the validity of the proposed method.

A distributional theory of fractional transformations is developed. A constructive approach, based on the eigenfunction expansion method pioneered by Zemanian, is used to produce an appropriate space of test functions and corresponding space of generalized functions. The fractional transformations that are defined are shown to form an equicontinuous group of operators on the space of test functions and a weak* continuous group on the space of generalized functions. Integral representations for the fractional transformations are also obtained under certain conditions. The fractional Fourier transformation is considered as a particular case of our general theory.

Given a signal, whether it is a discrete vector or a continuous function, one desires to write it in terms of simpler components. Typically, these components or “building blocks” form what is called a basis. A basis is an optimal set, containing the minimal number of elements needed to uniquely represent any signal in a given space. A frame can be thought of as a redundant basis, having more elements than needed. In fact, in any finite dimensional vector space every finite spanning set is a frame. The redundancy of a frame leads to a non-unique representation, however, this makes signal representation resilient to noise and robust to transmission losses. Frames are now standard tools in signal processing and are of great interest to mathematicians and engineers alike. This chapter presents a brief introduction to frames in finite dimensional spaces, and in particular discusses a highly desirable class of frames called equiangular tight frames. Possible research ideas suitable for an undergraduate curriculum are also discussed.

Radial acceleration estimation of multiple high maneuvering targets