Resolution enhancement of low resolution wavefields with POCS algorithm - Electronics Letters

Introduction: The fractional Fourier transform (FRT) has received significant interest since the early 1990s [1, 2]. The ath order FRT operation corresponds to the ath power of the ordinary Fourier transform operation. The zeroth-order FRT of a function is the function itself and the first-order FRT is equal to the ordinary Fourier transform. For additional properties and references, see [3]. In this Letter, an iterative algorithm for signal interpolation or extrapolation from partial FRT information is developed. More specifically, it is assumed that partial sets of samples are available for several a values. The problem is to interpolate or extrapolate the signal from this information. The reconstruction algorithm is globally convergent and it is based on the method of projections onto convex sets (POCS), a classical numerical technique [4–6]. The convergence of this algorithm can be proved easily in both continuous and discrete FRT domains because low resolution fractional Fourier measurements in both continuous and discrete domains correspond to closed and convex sets in L or ‘, respectively. The FRT has been shown to describe the evolution of waves and beams as they propagate in space, in the Fresnel approximation [3]. Wavefields or beam profiles undergo continual fractional Fourier transformation as they propagate. Increasing values of a correspond to different measurement planes further along the direction of propagation. Therefore the problem solved here corresponds to the problem of reconstructing wavefields or beam profiles from partial measurements at more than one plane. This covers a wide range of applications where it is possible to make measurements in more than one plane, but not at every point or over the complete interval in any given plane. The availability of information from several planes is used to compensate the missing information in each of them. Let us denote the ath order FRT operator as F . The ath FRT xa1⁄4F x of a function x is given by xaðuÞ 1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi