Fast Discrete Linear Canonical Transform Based on CM-CC-CM Decomposition and FFT

Abstract

In this paper, a discrete LCT (DLCT) irrelevant to the sampling periods and without oversampling operation is developed. This DLCT is based on the well-known CM-CC-CM decomposition, that is, implemented by two discrete chirp multiplications (CMs) and one discrete chirp convolution (CC). This decomposition doesn't use any scaling operation which will change the sampling period or cause the interpolation error. Compared with previous works, DLCT calculated by direct summation and DLCT based on center discrete dilated Hermite functions (CDDHFs), the proposed method implemented by FFTs has much lower computational complexity. The relation between the proposed DLCT and the continuous LCT is also derived to approximate the samples of the continuous LCT. Simulation results show that the proposed method somewhat outperforms the CDDHFs-based method in the approximation accuracy. Besides, the proposed method has approximate additivity property with error as small as the CDDHFs-based method. Most importantly, the proposed method has perfect reversibility, which doesn't hold in many existing DLCTs. With this property, it is unnecessary to develop the inverse DLCT additionally because it can be replaced by the forward DLCT.

Two-dimensional nonseparable discrete linear canonical transform based on CM-CC-CM-CC decomposition

“…In this section, the proposed 2D HA-NsDLCT and 2D LC-NsDLCT will be compared with Koç's method [27] and Ding's 2D method [28] in computational complexity, accuracy, additivity property and reversibility property. The computational complexity has been analyzed in (41), (44), (62) and (67), and is summarized in TABLE 1.…”

Fast and accurate computation of normalized Bargmann transform

“…We can find out that the above inverse transform is simpler than that in (13). For digital computation, we only require a 2D pointwise product for e jxy , a 2D summation along x axis , and a 1D FFT along y axis.…”

“…Since the Bargmann transform is a special case of the complex LCT, one can use the existing algorithms of the complex LCT to compute the Bargmann transform, and then normalize it to obtain the normalized Bargmann transform. Digital computation of the real LCT has been widely discussed in [13,[31][32][33][34][35][36]. Koç et al extended the computation of the real LCT to the complex LCT [18].…”

“…Thus, the normalized Bargmann transform is a unitary transform. Besides implementing the normalized Bargmann transform and its inverse directly by (12) and (13), some other computational methods are proposed in the following sections. We don't pay much attention to the conventional (i.e.…”

Computation of the Short‐Time Linear Canonical Transform with Dual Window

“…When we select Gaussian window, the STFT becomes the Gabor transform (GT). In recent years, with the development of nonstationary signal processing technology, the linear canonical transform (LCT) was developed by many scholars [2][3][4][5][6][7][8][9]. It is a generalized form of the FT and the Fractional Fourier transform (FRFT) and has been considered to be a powerful analyzing tool in signal processing and optics [10][11][12][13][14][15].…”