Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains

Abstract

The optical and digital implementations of general linear systems are costly. Through several examples we show that either exact realizations or useful approximations of these systems may be implemented in the form of repeated-filtering operations in consecutive fractional Fourier domains. These implementations are much cheaper than direct implementations of general linear systems. Thus we may significantly decrease the implementation costs of general linear systems with little or no decrease in performance by synthesizing them with the proposed repeated-filtering method.

The discrete fractional Fourier transform

“…For example, in the linear-system-approximation application in [17] and [18], we can implement the fractional Fourier transform stages with this fast algorithm. However, it would be preferable to have a fast algorithm that exactly com- putes the product of the fractional Fourier transform matrix defined here, with an arbitrary vector.…”

“…The transform has become popular in the optics and signal processing communities following the works of Ozaktas and Mendlovic [6]- [8], Lohmann [9] and Almeida [12]. Some of the applications explored include optimal filtering in fractional Fourier domains [13]- [16], cost-efficient linear system synthesis and filtering [17]- [21], time-frequency analysis [11], [12], [22], [23], and Fourier optics and optical information processing [24]- [26]. Additional recent publications include [27]- [33].…”

Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration

“…(A single-stage linear canonical transform filtering was treated in [12].) In [15], we have formulated the repeated filtering problem in terms of linear canonical transformations and showed that it can be reduced into an equivalent but simpler form involving only ordinary Fourier transforms (See Fig. 4 for the simple canonical form of the configuration in Fig.…”

“…There, the discrete version of this problem has also been provided. Again, in [15], the repeated filtering problem has been proposed to synthesize a desired linear transformation. In this correspondence, we will discuss this problem in terms of signal restoration.…”

“…Here, we generalize its form to repeated filtering in consecutive fractional Fourier domains. As we have shown, the equivalence of repeated filtering in consecutive domains with repeated filtering in consecutive ordinary time and frequency domains in [15] The form of (6) has (M + 1) 2 N degrees of freedom that is less than the N 2 degrees of freedom thatĜ opt has in (4). (Here, we are restricting ourselves to moderate values of M < N.) The estimator Gopt is well known.…”