Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part36

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part36

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 1 Copyright © Orchard Publications Chapter 8 The Fourier Transform his chapter introduces the Fourier Transform, also known as the Fourier Integral. The defi- nition, theorems, and properties are presented and proved. The Fourier transforms of the most common functions are derived, the system function is defined, and several examples are provided to illustrate its application in circuit analysis. 8.1 Definition and Special Forms We recall that the Fourier series for periodic functions of time, such as those we discussed on the previous chapter, produce discrete line spectra with non-zero values only at specific frequencies referred to as harmonics. However, other functions of interest such as the unit step, the unit impulse, the unit ramp, and a single rectangular pulse are not periodic functions. The frequency spectra for these functions are continuous as we will see later in this chapter. We may think of a non-periodic signal as one arising from a periodic signal in which the period extents from . Then, for a signal that is a function of time with period from , we form the integral (8.1) and assuming that it exists for every value of the radian frequency , we call the function the Fou- rier transform or the Fourier integral. The Fourier transform is, in general, a complex function. We can express it as the sum of its real and imaginary components, or in exponential form, that is, as (8.2) The Inverse Fourier transform is defined as (8.3) We will often use the following notation to express the Fourier transform and its inverse. (8.4) and (8.5) T to + + F ω () ft e j ω t t d = ω F ω F ω Re F ω {} jIm F ω + F ω e j ϕω == 1 2 π ------ F ω e j ω t ω d = F F ω = F 1 F ω =
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Chapter 8 The Fourier Transform 8 2 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications 8.2 Special Forms of the Fourier Transform The time function is, in general, complex, and thus we can express it as the sum of its real and imaginary parts, that is, as (8.6) The subscripts and will be used often to denote the real and imaginary parts respectively. These notations have the same meaning as and . By substitution of (8.6) into the Fourier integral of (8.1), we obtain (8.7) and by Euler’s identity (8.8) From (8.8), we see that the real and imaginary parts of are (8.9) and (8.10) We can derive similar forms for the Inverse Fourier transform as follows: Substitution of (8.2) into (8.3) yields (8.11) and by Euler’s identity, (8.12) Therefore, the real and imaginary parts of in terms of the Inverse Fourier transform are (8.13) and ft () f Re t jf Im t + = Re f t {} Im f t F ω f t e j ω t tj f t e j ω t t d +
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This note was uploaded on 11/20/2009 for the course EE EE 102 taught by Professor Bar during the Fall '09 term at UCLA.

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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part36

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