ELECTRIC CIRCUITS EIGHTH EDITION JAMES W. NILSSON & SUSAN A. RIEDEL

CHAPTER 17 THE FOURIER TRANSFORM © 2008 Pearson Education

CONTENTS 17.1 The Derivation of the Fourier Transform 17.2 The Convergence of the Fourier Integral 17.3 Using Laplace Transforms to Find Fourier Transforms 17.4 Fourier Transforms in the Limit © 2008 Pearson Education

CONTENTS 17.5 Some Mathematical Properties 17.6 Operational Transforms 17.7 Circuit Applications 17.8 Parseval’s Theorem © 2008 Pearson Education

17.1 The Derivation of the Fourier Transform   The Fourier transform gives a frequency-domain description of an aperiodic time-domain function. © 2008 Pearson Education F

  Depending on the nature of the time- domain signal, one of three approaches to finding its Fourier transform may be used: 1) 1)If the time-domain signal is a well- behaved pulse of finite duration, the integral that defines the Fourier transform is used. © 2008 Pearson Education 17.1 The Derivation of the Fourier Transform

2) 2)If the one-sided Laplace transform of f(t) exists and all the poles of F(s) lies in the left half of the s plane, F(s) may be used to find F(  ). 3) 3)If f(t) is a constant, a signum function, a step function, or a sinusoidal function, the Fourier transform is found by using a limit process. © 2008 Pearson Education 17.1 The Derivation of the Fourier Transform

© 2008 Pearson Education 17.1 The Derivation of the Fourier Transform Inverse Fourier transform

© 2008 Pearson Education 17.1 The Derivation of the Fourier Transform A voltage pulse

C n versus nω 0, T/  =5 © 2008 Pearson Education 17.1 The Derivation of the Fourier Transform  Transition of the amplitude spectrum as f(t) goes from periodic to aperiodic.

C n versus nω 0, T/  =10 © 2008 Pearson Education 17.1 The Derivation of the Fourier Transform

V(ω) versus ω © 2008 Pearson Education 17.1 The Derivation of the Fourier Transform

17.2 The Convergence of the Fourier Integral The decaying exponential function Ke -at u(t) © 2008 Pearson Education

17.2 The Convergence of the Fourier Integral The approximation of a constant with an exponential function © 2008 Pearson Education

17.3 Using Laplace Transforms to Find Fourier Transforms The reflection of a negative-time function over to the positive-time domain © 2008 Pearson Education

17.4 Fourier Transforms in the Limit The signum function © 2008 Pearson Education

17.4 Fourier Transforms in the Limit A function that approaches sgn(t) asεapproaches zero © 2008 Pearson Education

17.4 Fourier Transforms in the Limit Fourier transforms of elementary functions © 2008 Pearson Education

17.5 Some Mathematical Properties From the defining integral, © 2008 Pearson Education

17.5 Some Mathematical Properties

17.6 Operational Transforms © 2008 Pearson Education  The Fourier transform of a response signal y(t) is  where X(  ) is the Fourier transform of the input signal x(t), and H(  ) is the transfer function H(s) evaluated at s = j .

17.6 Operational Transforms © 2008 Pearson Education

17.7 Circuit Application   The Laplace transform is used more widely to find the response of a circuit than is the Fourier transform, for two reasons: 1) 1)The Laplace transform integral converges for a wider range of driving functions. 2) 2)The Laplace transform accommodates initial conditions. © 2008 Pearson Education

17.7 Circuit Application © 2008 Pearson Education Example: Using the Fourier Transform to Find the Transient Response Use the Fourier transform to find i o (t) in the circuit shown below. The current source i g (t) is the signum function 20sgn(t) A.

17.8 Parseval’s Theorem   The magnitude of the Fourier transform squared is a measure of the energy density (joules per hertz) in the frequency domain (Parseval’s theorem).   The Fourier transform permits us to associate a fraction of the total energy contained in f(t) with a specified band of frequencies. © 2008 Pearson Education

17.8 Parseval’s Theorem Example: Applying Parseval’s Theorem The current in a 40Ω resistor is © 2008 Pearson Education What percentage of the total energy dissipated in the resistor can be associated with the frequency band 0≤  ≤2√3 rad/s? A

17.8 Parseval’s Theorem Example: Applying Parseval’s Theorem to a Low- Pass Filter Parseval’s theorem makes it possible to calculate the energy available at the output of the filter even if we don’t know the time-domain expression for v 0 (t). Suppose the input voltage to the low-pass RC filter circuit shown below is © 2008 Pearson Education V

17.8 Parseval’s Theorem Example: Applying Parseval’s Theorem to a Low- Pass Filter © 2008 Pearson Education a) What percentage of the 1Ω energy available in the input signal is available in the output signal? b) What percentage of the output energy is associated with the frequency range 0 ≤  ≤ 10 rad/s?

17.8 Parseval’s Theorem © 2008 Pearson Education The rectangular voltage pulse

17.8 Parseval’s Theorem © 2008 Pearson Education The Fourier transform of v (t)

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