Transcript the fourier transform

JAMES W. NILSSON
&
SUSAN A. RIEDEL
ELECTRIC
CIRCUITS
EIGHTH EDITION
CHAPTER 17
THE FOURIER
TRANSFORM
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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
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CONTENTS
17.5 Some Mathematical Properties
17.6 Operational Transforms
17.7 Circuit Applications
17.8 Parseval’s Theorem
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17.1 The Derivation of the Fourier
Transform
 The
Fourier transform gives a
frequency-domain description of an
aperiodic time-domain function.
F ( )  F { f (t )}  


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f (t )e
 jt
dt
17.1 The Derivation of the
Fourier Transform
 Depending on the nature of the timedomain signal, one of three approaches to
finding its Fourier transform may be used:
1) If the time-domain signal is a wellbehaved pulse of finite duration, the
integral that defines the Fourier transform
is used.
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17.1 The Derivation of the
Fourier Transform
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) 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.
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17.1 The Derivation of the
Fourier Transform
1
f (t ) 
2



jt
F ( )e d
Inverse Fourier transform
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17.1 The Derivation of the
Fourier Transform
A voltage pulse
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17.1 The Derivation of the
Fourier Transform
Transition of the amplitude spectrum as
f(t) goes from periodic to aperiodic.
Cn versus nω0 , T/ =5
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17.1 The Derivation of the
Fourier Transform
Cn versus nω0 , T/ =10
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17.1 The Derivation of the
Fourier Transform
V(ω) versus ω
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17.2 The Convergence of the
Fourier Integral
The decaying exponential function Ke-at u(t)
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17.2 The Convergence of the
Fourier Integral
The approximation of a constant
with an exponential function
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17.3 Using Laplace Transforms to Find
Fourier Transforms
The reflection of a negative-time function
over to the positive-time domain
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17.4 Fourier Transforms in the Limit
The signum function
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17.4 Fourier Transforms in the Limit
A function that approaches sgn(t)
asεapproaches zero
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17.4 Fourier Transforms in the Limit
Fourier transforms of elementary functions
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17.5 Some Mathematical Properties
From the defining integral,
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17.5 Some Mathematical Properties
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17.6 Operational Transforms
The Fourier transform of a response
signal y(t) is
Y ( )  X ( ) H ( )
 where X() is the Fourier transform of
the input signal x(t), and H() is the
transfer function H(s) evaluated at s = j.
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17.6 Operational Transforms
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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) The Laplace transform integral
converges for a wider range of driving
functions.
2) The Laplace transform accommodates
initial conditions.
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17.7 Circuit Application
Example: Using the Fourier Transform to Find
the Transient Response
Use the Fourier transform to find io (t) in the
circuit shown below.
The current source ig (t) is the signum function
20sgn(t) A.
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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.
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17.8 Parseval’s Theorem
Example: Applying Parseval’s Theorem
The current in a 40Ω resistor is
2t
i  20e u (t ) A
What percentage of the total energy dissipated in the resistor
can be associated with the frequency band 0≤  ≤2√3 rad/s?
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17.8 Parseval’s Theorem
Example: Applying Parseval’s Theorem to a LowPass 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 v0(t).
Suppose the input voltage to the low-pass RC
filter circuit shown below is
5t
vi (t )  15e u (t ) V
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17.8 Parseval’s Theorem
Example: Applying Parseval’s Theorem to a LowPass Filter
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?
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17.8 Parseval’s Theorem
The rectangular voltage pulse
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17.8 Parseval’s Theorem
The Fourier transform of v(t)
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THE END
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