ah e n n l

ti noise.3 For one-dimensional ~1-D! signals it is pos- We develop an optical system to obtain an approx-sible to obtain a single two-dimensional display that contains a continuous representation of the RWT for all possible projection angles. This display is known as the Radon–Wigner spectrum4 or the Radon– Wigner display.5 A relation between the RWT and another new important transformation, the frac-tional Fourier transform ~FRT!,6 was found by Loh-mann and Soffer.7 They demonstrated that the RWT is the square modulus of the FRT, in which the fractional order p is related to the projection angle f of the RWT through the relationship p 5 fy90°. This finding suggests that the optical computation of the RWT is possible directly from the input function, omitting the passage through its Wigner distribution function. imated Radon–Wigner display, in which the frac-tional order varies continuously. The setup behaves like a multichannel parallel Radon–Wigner trans-former and makes use of only two refractive ele-ments: a cylindrical lens and a varifocal lens. In Section 2 we review some aspects of the optical pro-duction of the FRT, and we use them in the develop-ment of the optical system. The constraints for the involved parameters are also analyzed to achieve the optimum design. In Section 3 the performance of our proposal is tested experimentally with an oph-thalmic progressive addition lens as a varifocal lens. These results are compared with computer simula-tions. 2. Basic Theory Our approach for obtaining the Radon–Wigner dis-play is through the FRT formalism with the use of the theorem demonstrated in Ref. 7. This approach is divided into two steps: first we review the optical production of the FRT in terms of Fresnel diffraction; next, we obtain in parallel the Radon–Wigner display for a 1-D input, by imaging simultaneously at th