# CH8 - CHAPTER 8 The Discrete Fourier Transform Fourier...

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141 CHAPTER 8 The Discrete Fourier Transform Fourier analysis is a family of mathematical techniques, all based on decomposing signals into sinusoids. The discrete Fourier transform (DFT) is the family member used with digitized signals. This is the first of four chapters on the real DFT , a version of the discrete Fourier transform that uses real numbers to represent the input and output signals. The complex DFT , a more advanced technique that uses complex numbers, will be discussed in Chapter 31. In this chapter we look at the mathematics and algorithms of the Fourier decomposition, the heart of the DFT. The Family of Fourier Transform Fourier analysis is named after Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist. (Fourier is pronounced: , and is for @ ¯e @ ¯a always capitalized). While many contributed to the field, Fourier is honored for his mathematical discoveries and insight into the practical usefulness of the techniques. Fourier was interested in heat propagation, and presented a paper in 1807 to the Institut de France on the use of sinusoids to represent temperature distributions. The paper contained the controversial claim that any continuous periodic signal could be represented as the sum of properly chosen sinusoidal waves. Among the reviewers were two of history's most famous mathematicians, Joseph Louis Lagrange (1736-1813), and Pierre Simon de Laplace (1749-1827). While Laplace and the other reviewers voted to publish the paper, Lagrange adamantly protested. For nearly 50 years, Lagrange had insisted that such an approach could not be used to represent signals with corners , i.e., discontinuous slopes, such as in square waves. The Institut de France bowed to the prestige of Lagrange, and rejected Fourier's work. It was only after Lagrange died that the paper was finally published, some 15 years later. Luckily, Fourier had other things to keep him busy, political activities, expeditions to Egypt with Napoleon, and trying to avoid the guillotine after the French Revolution (literally!).

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The Scientist and Engineer's Guide to Digital Signal Processing 142 Sample number 0 4 8 12 16 -40 -20 0 20 40 60 80 DECOMPOSE SYNTHESIZE FIGURE 8-1a (see facing page) Amplitude Who was right? It's a split decision. Lagrange was correct in his assertion that a summation of sinusoids cannot form a signal with a corner. However, you can get very close. So close that the difference between the two has zero energy . In this sense, Fourier was right, although 18th century science knew little about the concept of energy. This phenomenon now goes by the name: Gibbs Effect , and will be discussed in Chapter 11. Figure 8-1 illustrates how a signal can be decomposed into sine and cosine waves. Figure (a) shows an example signal, 16 points long, running from sample number 0 to 15. Figure (b) shows the Fourier decomposition of this signal, nine cosine waves and nine sine waves, each with a different frequency and amplitude. Although far from obvious, these 18 sinusoids add to produce the waveform in (a). It should be noted that the objection
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CH8 - CHAPTER 8 The Discrete Fourier Transform Fourier...

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