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 ω 0 by its spatial counterpart u 0 (ω 0 = 2π u 0 ), 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.