15.6 Fast Fourier Transform
INTRODUCTION
Consider a function f that is defined and continuous on the interval [0, 2p]. If x0, x1, x2, … , xn, … are equally spaced points in the interval, then the corresponding function values f0, f1, f2, … , fn, … shown in FIGURE 15.6.1 are said to represent a discrete sampling of the function f. The notion of discrete samplings of a function is important in the analysis of continuous signals.
![A continuous function f is plotted on an x y coordinate plane on the interval [0, 2 p] wherein 2 p > 0. For equally spaced points on the x axis the corresponding values on the curve are marked with a dot and a dotted vertical line connects them to the x axis. Starting on the y axis, from left to right the dotted lines are labeled: f subscript 0, f subscript 1, f subscript 2,…, f subscript n. Two dotted lines after f subscript n are not labeled the second one being for x = 2 p. The distance between the vertical lines f subscript 1 and f subscript 2 is indicated by an accolade and labeled T. The curve starts from a point on the y axis which is above the origin rises then drops down then gradually rises again and drops down again. The curve is indicated by an arrow and labeled y = f(x). The corresponding point to the dotted line labeled f subscript n is indicated by an arrow and labeled f(n T).](/api/v2/epubs/9781284206258/files/images/ch15-9781284207972_CH15_FIGF0601.png)
FIGURE 15.6.1 Sampling of a continuous function
In this section, the complex or exponential form of a Fourier series plays an important role in the discussion. A review of Section 12.4 is recommended.
Discrete Fourier ...
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