From an analytical perspective, the Fourier series represents a periodic signal as an infinite sum of multiples of the fundamental frequencies, while the Fourier transform permits an aperiodic waveform to be described as an integral sum over a continuous range of frequencies. Despite this separation by series and integral representations, in mathematical terms the Fourier series is regarded as a special case of the Fourier transform. Some of the basic definitions associated with the continuous Fourier series and transform are given in Appendix A; these definitions are extended to discrete signal samples in this chapter. The derivations here provide a conceptual framework for DFT algorithms and the associated parameters frequently quoted in the description of FFT software, and provide the background for frequency-based filtering developed in Chapter 13.

12.1 Discrete Fourier Series

If the continuous signal f(x) is replaced by g(x) and the radial frequency ω₀ by its spatial counterpart u₀(ω₀ = 2πu₀), and subscript p is added to mark the periodicity over (0, l), the derivations in Appendix A, Sec. A.1 lead to the following Fourier series:

(12.1)

with the coefficients given in Eqs. (A.5) and (A.6) in Appendix A.