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##### From Signals and Systems with MATLAB Computing and Simulink Modeling, Third Edition

This chapter begins with the actual computation of frequency spectra for discrete time systems. For brevity, we will use the acronyms DFT for the Discrete Fourier Transform and FFT for Fast Fourier Transform algorithm respectively. The definition, theorems, and properties are also discussed, and several examples are presented to illustrate their uses.

## 10.1 The Discrete Fourier Transform (DFT)

In the Fourier series topic, Chapter 7, we learned that a periodic and continuous time function, results in a non periodic and discrete frequency function. Next, in the Fourier transform topic, Chapter 8, we saw that a non-periodic and continuous time function, produces a non-periodic and continuous frequency function. In Chapter 9 we learned that the and Inverse transforms produce a periodic and continuous frequency function, since these transforms are evaluated on the unit circle. This is because the frequency spectrum of a discrete time sequence f [n] is obtained from its transform by the substitution of z=e sT=e j?T as we saw from the mapping of the s-plane to the z-plane in Chapter 9, Page 9 23. It is continuous because there is an infinite number of points in the interval 0 to 2 ?, although, in practice, we compute only a finite number of equally spaced points.

In this chapter we will see that a periodic and discrete time function results in a periodic and discrete frequency function. For convenience, we summarize these facts in Table 10.1.

Table 10.1: Characteristics of Fourier and transforms

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