Filter Design for Constrained Signal Reconstruction in Linear Canonical Transform Domain

Mathematical Problems in Engineering

et al. 2022

The short-time linear canonical transform (STLCT) is a novel time-frequency analysis tool, which has attracted some attention recently. However, its applications in signal processing are limited because the time-frequency properties of the STLCT are still little known. Most existing studies focus on mathematical properties rather than time-frequency properties in signal processing. To handle this problem, first, we investigate some basic time-frequency properties such as 2-D resolution of the time-frequency plane, the STLCT domain support, computation of the STLCT, etc by generalizing the short-time Fourier transform to the STLCT. Then, based on these derived properties, we find that the Gaussian window is optimal window of the STLCT. Signal separation verified the results.

Section: Introductionmentioning

confidence: 99%

Time-Frequency Properties of the Short-Time Linear Canonical Transform and Its Application

Huang

Mathematical Problems in Engineering

et al. 2022

“…Many known integral transforms, including the Fourier transform, the fractional Fourier transform, and the Fresnel transform, are special cases of LCT. It is useful in optical signal processing, 1 radar system analysis, 2 filter design, 3 and many other fields. Recently, various aspects of LCT have been studied, which include the pseudo-differential operator related to LCT, 4,5 sampling theory, [6][7][8][9][10][11] uncertainty principle, [12][13][14][15] discrete LCT, 16,17 and fast algorithms.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Inversion of the windowed linear canonical transform with Riemann sums

Han

Math Methods in App Sciences

2022

In this paper, we study the inversion of the windowed linear canonical transform. We show that for certain window functions, a function can be recovered from its windowed linear canonical transform with a discrete series. We give necessary and sufficient conditions for the convergence of the series. Both the convergence in norm and the pointwise convergence are studied. Moreover, we get an inversion formula for integrable functions with the Cesàro summability.

“…e LCT's generality enables it to be capable of solving many mathematical, physical, and engineering problems that other conventional transformations fail to solve. For a signal f(t), its LCT associated with the parameter matrix A � (a, b; c, d) is given by [29][30][31][32][33][34][35]…”

Section: Lctmentioning

confidence: 99%

A Computationally Efficient Optimal Wigner Distribution in LCT Domains for Detecting Noisy LFM Signals

Shi

Mathematical Problems in Engineering

et al. 2022

Recently, Wigner distribution (WD) associated with linear canonical transforms (LCTs) is quickly becoming a promising technique for detecting linear frequency-modulated (LFM) signals corrupted with noises by establishing output signal-to-noise ratio (SNR) inequality model or optimization model. Particularly, the closed-form instantaneous cross-correlation function type of WD (CICFWD), a unified linear canonical Wigner distribution, has shown to be competitive in detecting noisy LFM signals under an extremely low SNR. However, the CICFWD has up to nine LCT free parameters so that it requires a heavy computational load. To improve the efficiency of real-time processing, this paper focuses on the instantaneous cross-correlation function type of WD (ICFWD), which has only six LCT free parameters but is not a special case of the CICFWD. The main advantage of ICFWD is that it could be expected to reduce the computational complexity while maintaining detection performance. This paper first proposes an optimization model to the ICFWD’s output SNR with respect to deterministic signals embedded in additive zero-mean noises. It then deduces the model’s solution to a single component LFM signal added with white noise, leading to the optimal selection strategy on LCT free parameters. Simulation results demonstrate that the ICFWD improves almost a doubling of computing speed in comparison with the CICFWD while sharing the same level of detection performance. To be specific, the computing time of ICFWD in sampling frequencies 5 Hz, 10 Hz, 15 Hz, and 20 Hz is about 0.048 s, 0.111 s, 0.226 s, and 0.392 s, respectively, while 0.075 s, 0.233 s, 0.478 s, and 0.821 s for the computing time of CICFWD; the ICFWD and CICFWD have nearly the same output SNR higher than that of the WD.