Discrete Fourier Transform

Abstract

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. <strong>12.1 Discrete Fourier Series</strong> If the continuous signal <i>f</i>(<i>x</i>) is replaced by <i>g</i>(<i>x</i>) and the radial frequency &#x03C9;₀ by its spatial counterpart <i>u</i>₀(&#x03C9;₀ = 2&#x03C0;<i>u</i>₀), and subscript <i>p</i> is added to mark the periodicity over (0, <i>l</i>), 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.