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

If T  infinite, fn. never repeats itself  aperiodic.

: See :

17.1 The Derivation of the Fourier Transform(abbr.)   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

Transform of const :

This fn.  impulse fn. at Verification :

The following rules apply : (ex) Then,

(ex) Then,  When f(t) exist for,FT is sum of 2 transforms. Thus if we let Then,

(ex) f(t) =. That is, Then,

 Now, Thus,

Where, Then,

(ex)

17.5 Some Mathematical Properties (abbr.) From the defining integral, © 2008 Pearson Education

17.5 Some Mathematical Properties  For even fn.,

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 .

 

 See :

 

© 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.

 See :



f(t) : vtg. or ct. in Then, “Parseval’s theorem” Derivation :

By the way,

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

See : Then,

See : 

Total dissipated energy :



See :

 대입

Therefore,

EE14154 Home work Prob 제출기한 : - 다음 요일 수업시간 까지 - 제출기일을 지키지않는 레포트는 사정에서 제외함

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