SIViP DOI 10.1007/s11760-013-0427-4 ORIGINAL PAPER Paley–Wiener criterion in linear canonical transform domains K. K. Sharma · Lokesh Sharma · Shobha Sharma Received: 9 October 2012 / Revised: 2 January 2013 / Accepted: 15 January 2013 © Springer-Verlag London 2013 Abstract The Paley–Wiener criterion involving the issue of the physical realizability of any linear time-invariant (LTI) system / filter is well known in the literature. This criterion gives us the frequency domain condition on the magnitude of the transfer function of the LTI system. In this letter, we extend this criterion to linear canonical transform domains. Keywords Fourier transform · Fractional Fourier transform · Linear canonical transforms · Paley–Wiener criterion · Physical realizability 1 Introduction The Paley–Wiener criterion, involving the physical realiz- ability of a linear time-invariant (LTI) system, gives us the necessary condition to be satisfied by the absolute value of its transfer function F (ω) in frequency domain [1, 2]. To be more precise, the Paley–Wiener criterion is given by [1] ∞  -∞ |log | F (ω)|| 1 + ω 2 dω< ∞, (1) K. K. Sharma (B ) Department of Electronics and Communication Engineering, Malaviya National Institute of Technology, Jaipur, India e-mail: kksharma_mrec@yahoo.com L. Sharma Department of Electronics and Communication Engineering, LNM IIT, Jaipur, India e-mail: Sharma.123lokesh@gmail.com S. Sharma Department of Electronics and Communication Engineering, Apex Institute of Engg. and Tech, Jaipur, India e-mail: sharma.shobha90@gmail.com where F (ω) denotes the conventional Fourier transform (CFT) of the impulse response f (t ) of the LTI system. The corresponding time domain condition on the impulse response of the system is also known [2]. Recently, it has been shown that the time and frequency domains are only two special domains out of a continuum of infinite number of domains called fractional Fourier domains (FFDs) [3, 4]. These FFDs have been further generalized to the linear canonical transform (LCT) domains [3]. It may be mentioned here that the equivalence of the LCT domains to the FFDs has already been shown in [8]. The filtering in these new domains has also been considered in the literature [5–7]. It may be mentioned here that given the absolute value of the LCT of the impulse response f (t ) of the LTI system, one cannot use (1) to find the causality of the system. In other words, one cannot obtain the absolute value of F (ω), the CFT of the impulse response f (t ) of the LTI system, from the absolute value of the LCT of the impulse response. Therefore, it is necessary to extend (1) to the FFDs or LCT domains and find a corresponding condition for realiz- ability/causality of a filter. In this paper, we extend the Paley–Wiener criterion to the LCT domains which can be used for investigation of the issue of the physical realizability of a LTI system given the absolute value of the LCT of the impulse response f (t ) of the LTI system. 2 Main result The LCT of a signal f (t ) with parameter matrix L = [ A, B ; C, D], denoted as F L (u L ), is given by [3] F L (u L ) = ⎧ ⎨ ⎩ ∞  -∞  1 j 2π B f (t ) K L (u L , t )dt , B  = 0 √ D exp( jCDu 2 M /2) f ( Du M ), B = 0 , (2) 123