TABLE OF CONTENTS The operations of convolution, correlation and Fourier analysis play an important role in data analysis and experiment simulations. These operations on digitally sampled data are efficiently carried out with the Discrete Fourier Transform (DFT), and in particular with a fast version of the DFT called the Fast Fourier Transform (FFT). In this section, we introduce the DFT and FFT and explore their relationship to the analytic Fourier transform and Fourier series. We investigate Fourier deconvolution of a noisy signal with an optimal Weiner filter. We also learn how to minimally sample data without losing any information (Nyquist theorem) and about the aliasing that occurs when a waveform is sampled at an insufficient rate. Since the DFT is an approximation to the analytic Fourier transform, we learn how to zero pad a time series to obtain accurate Fourier amplitudes, and to remove spurious end effects in discrete convolution. Finally, we explore two commonly used approaches to spectral analysis and how to reduce the variance of spectral density estimators.
Before we consider the problem of representing a function in terms of a sum of orthogonal basis functions, we review the more familiar problem of representing a vector in terms of a set of orthogonal unit vectors. The vector F can be represented as the vector sum
| (B.1) |  |
where the 
s are unit vectors along 3 mutually perpendicular axes. Because the unit vectors satisfy the relation 
they are said to be an
