IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 2, FEBRUARY 2001 381 Fast Computation of the Ambiguity Function and the Wigner Distribution on Arbitrary Line Segments Ahmet Kemal Özdemir, Student Member, IEEE, and Orhan Arıkan, Member, IEEE Abstract—By using the fractional Fourier transformation of the time-domain signals, closed-form expressions for the projections of their auto or cross ambiguity functions are derived. Based on a sim- ilar formulation for the projections of the auto and cross Wigner distributions and the well known two-dimensional (2-D) Fourier transformation relationship between the ambiguity and Wigner domains, closed-form expressions are obtained for the slices of both the Wigner distribution and the ambiguity function. By using dis- cretization of the obtained analytical expressions, efficient algo- rithms are proposed to compute uniformly spaced samples of the Wigner distribution and the ambiguity function located on arbi- trary line segments. With repeated use of the proposed algorithms, samples in the Wigner or ambiguity domains can be computed on non-Cartesian sampling grids, such as polar grids. Index Terms—Ambiguity function, fast computation, fractional Fourier transformation, Wigner distribution. I. INTRODUCTION T IME-FREQUENCY signal processing is one of the fun- damental research areas in signal processing. The Wigner distribution (WD) plays a central role in the theory and prac- tice of time-frequency signal processing [1]–[10]. Likewise, the ambiguity function (AF), which is the two-dimensional (2-D) Fourier transform of the Wigner distribution, plays a central role in time-frequency signal analysis [11]–[13] and radar and sonar signal processing [14]–[17]. Because of the availability of efficient computational algo- rithms, both the WD and AF are usually computed on Cartesian grids [1], [18], [19]. In this paper, by using the fractional Fourier transformation of the time-domain signals, closed-form expressions for the projections of their auto or cross ambiguity functions are derived. Based on a similar formulation for the projections of the auto and cross Wigner distributions [20] and the well-known 2-D Fourier transformation relationship between the ambiguity and Wigner domains, novel closed-form expressions are obtained for the slices of both the WD and the AF. By using discretization of the obtained analytical expres- sions, efficient algorithms are proposed to compute uniformly spaced samples of the WD and the AF located on arbitrary line segments. With repeated use of these algorithms, it is possible to obtain samples of the WD and AF on non-Cartesian grids, such as polar grids that are the natural sampling grids Manuscript received March 29, 1999; revised October 3, 2000. The associate editor coordinating the review of this paper and approving it for publication was Prof. Gregori Vazquez. The authors are with the Department of Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey (e-mail: kozdemir@ee.bilkent .edu.tr;oarikan@ee.bilkent.edu.tr). Publisher Item Identifier S 1053-587X(01)00620-1. of chirp-like signals. The ability of obtaining WD and AF samples over polar grids is potentially very useful in various important application areas including time-frequency domain kernel design, multicomponent signal analysis, time-frequency domain signal detection, and particle location analysis in Fresnel holograms [21]–[28]. The organization of the paper is in accordance with the dual nature of the ambiguity function and Wigner distribution. We first provide some preliminaries on these important concepts. In Section III, by using the Radon-Wigner transformation, an- alytical expressions are derived for the slices of the auto am- biguity functions. Then, by discretizing the obtained analytical expressions, efficient algorithms are presented for the computa- tion of slices of the ambiguity function. In Section IV, we follow a similar development, leading to novel closed-form expressions for the Radon-ambiguity function, and present efficient algo- rithms for the computation of slices of the Wigner distribution. In Section V, both the analytical and computational results are extended to the cross AF and WD. In Section VI, we provide results of simulated applications of the proposed algorithms. Fi- nally, the paper is concluded in Section VII. II. PRELIMINARIES ON THE WIGNER DISTRIBUTION AND THE AMBIGUITY FUNCTION Discrete time-frequency analysis is the primary investiga- tion tool in the synthesis, characterization, and filtering of time-varying signals. Among the alternative time-frequency analysis algorithms, those belonging to the Cohen’s class are the most commonly utilized ones. In this class, the time-fre- quency distributions of a signal are given by 1 TF (1) where the function is called the kernel [4], [29]. Recent research on the time–frequency signal analysis has revealed that signal dependent choice of the kernel helps in localization of the time-frequency components of the signals [11], [21]–[24]. By choosing , the most commonly used member of the Cohen’s class (the Wigner distribution) is obtained (2) Because of its nice energy localization properties, the WD has found important application areas. Definition (2) has been gen- 1 All integrals are from to unless otherwise stated. 1053–587X/01$10.00 © 2001 IEEE