There are many useful computations such as correlations and convolutions that can be implemented using FTs. In fact, taking advantage of computationally efficient DFT techniques such as the FFT often executes much faster than more straightforward implementations. Subsequent chapters reuse these tools in an optical context. For example, convolution is used in Ch. 5 to simulate the effects of diffraction and aberrations on image quality, and structure functions are used in Ch. 9 to validate the statistics of turbulent phase screens.

Three of these tools, namely convolution, correlation, and structure functions, are closely related and have similar mathematical definitions. Furthermore, they are all written in terms of FTs in this chapter. However, their uses are quite different, and each common use is explained in the upcoming sections. These different uses cause the implementations of each to be quite different. For example, correlations and structure functions are usually performed on data that pass through an aperture. Consequently, their computations are modified to remove the effects of the aperture.

The last computation discussed in this chapter is the derivative. Like the other computations in this chapter, the method presented is based on FTs to allow for efficient computation. The method is then generalized to computing gradients of two-dimensional functions. While derivatives and gradients are not used again in later chapters, derivatives are discussed because some readers might want to compute derivatives for topics related to optical turbulence, like simulating the operation of wavefront sensors.

3.1 Convolution

We begin this discussion of FT-based computations with convolution for a couple of reasons. First, convolution plays a central role in linear-systems theory.¹⁴ The output of a linear system is the convolution of the input signal with the system's impulse response. In the context of simulating optical wave propagation, the linear-systems formalism applies to coherent and incoherent imaging, analog optical image processing, and free-space propagation.