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Chapter 2:
Sampled Functions and the Discrete Fourier Transform
Abstract

When implementing Fourier optics simulations on the computer it is necessary to represent functions by discrete arrays of sampled values and apply transform and processing methods designed for these discrete signals. To come as close as possible to simulating continuous space, it would be great to model the physical elements with a gazillion samples. However, computer memory and execution time limitations won't allow this. Thus, devising practical Fourier optics simulations becomes an act of balancing acceptable sampling artifacts and available computer resources. This chapter begins to address this matter with discussions of the sampling of continuous functions, the Shannon-Nyquist sampling theorem, and the concept of effective bandwidth. The remainder of the chapter concerns the discrete Fourier transform (DFT), the workhorse tool for computational Fourier analysis. We look at its relationship with the analytic transform, describe implementation details, and discuss how DFT results differ from the analog world.

2.1 Sampling and the Shannon-Nyquist Sampling Theorem

Consider the two-dimensional (2D) analytic function g(x,y) and suppose it is sampled in a uniform manner (Fig. 2.1) in the x and y directions, which is indicated by

(2.1)

where the sample interval is Δx in the x direction and Δy in the y direction, and m and n are integer-valued indices of the samples. The respective sample rates are 1/Δx and 1/Δy. In practice, the sampled space is finite and, assuming it is composed of MN samples in the x and y directions, respectively, m and n are often defined with the following values:

(2.2)