A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics

Abstract

The Linear Canonical Transform (LCT) is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform (FT), the FRactional Fourier Transform (FRFT), and the FreSnel Transform (FST) can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.

Multiplicative filtering in the linear canonical transform domain

“…In the derived operation, it is found that the result does not meet the requirement of symmetric definition of FT. Further, the derived convolution/correlation operation contains two and ten chirp multiplications on the LHS and RHS, respectively. In 2013, Goel and Singh [29] derived the convolution theorem with different approach. The convolution operation defined in time domain is dependent on time variable only.…”

“…The origin of LCT is in quantum mechanics and a brief overview may be found in [14]. As a generalisation of the FT, FRFT and FST, the basic theories of the LCT have been developed including sampling the signals [15][16][17][18][19][20][21][22][23][24], convolution and correlation operations, discrete approximations to the transforms [25][26][27][28][29][30][31][32][33][34][35][36][37][38]a n ds oo n ,w h i c h augment the theoretical system of the LCT.…”

“…From simulation, it is found that mean square error (MSE) is minimum for different values of signal-to-noise ratio (SNR) in LCT domain filtering as compared with frequency domain filtering and fractional domain filtering. Moreover, the authors have calculated the computational complexity of the designed filter and made available to the literature [16,26,28,29].…”

A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform

“…It can be regarded as a generalization of many mathematical transforms such as the Fourier transform, Laplace transform, the fractional Fourier transform, and the Fresnel transform. Many fundamental properties of this extended transform are already known, including shift, modulation, convolution, and correlation and uncertainty principle, for example, in [1][2][3][4][5][6].…”

“…The quaternion algebra over R, denoted by H, is an associative noncommutative four-dimensional algebra: (1) which obeys the following multiplication rules: ij = −ji = k, jk = −kj = i, ki = −ik = j, i 2 = j 2 = k 2 = ijk = −1.…”