class_10_fouriertransform.pdf
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Chengbin Ma UM-SJTU Joint Institute
Class#10
- Continuous-time periodic signals: the Fourier series (3.5)-An application
- Continuous-time nonperiodic signals: the Fourier transform (3.7)
Slide 1
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Chengbin Ma UM-SJTU Joint Institute
Preview of Previous Lecture
Natural response and forced response
natural response transient response
(homogeneous solution), and forced response
steady-state response (particular solution)
Fourier Series (FS)
DC component, fundamental frequency, Nth
harmonic component
FS coefficient
Dirichlet conditions
Trigonometric FS
Slide 2
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Chengbin Ma UM-SJTU Joint Institute
This Lecture
A real application of FS in analyzing a power
electronic circuit
Extend FS to analyze nonperiodic signals, i.e.,
Fourier transform (FT).
Slide 3
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Chengbin Ma UM-SJTU Joint Institute
Application: DC-AC Converter
Slide 4
Example 3.16, p227
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Chengbin Ma UM-SJTU Joint Institute
DC/AC Conversion Efficiency
Example 3.16, p227
Slide 5
0
0
( ) [ ]cos( )k
x t B k k t
0[0] 2 /B T T
02sin( 2 / )[ ] , 0k T T
B k kk
-
Chengbin Ma UM-SJTU Joint Institute
Class#10
- Continuous-time periodic signals: the Fourier series (3.5)-An application
- Continuous-time nonperiodic signals: the Fourier transform (3.7)
Slide 6
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Chengbin Ma UM-SJTU Joint Institute
The Fourier transform
The Fourier transform (FT) is used to represent
a continuous-time nonperiodic signal as a
superposition of complex sinusoids.
1( ) ( ) (IFT)
2
( ) ( ) (FT)
( ) ( )
j t
j t
FT
x t X j e d
X j x t e dt
x t X j
T
tjkdtetx
TkX
0
0)(1
][
k
tjkekXtx 0][)(
Frequency-domain representation
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Chengbin Ma UM-SJTU Joint Institute
Basic Idea (1)
Development of the Fourier transform representation of a continuous-time nonperiodic signal - we think of a nonperiodic signal as the limit of a periodic signal as the period becomes arbitrarily large, and we examine the limiting behavior of the Fourier series representation for this signal.
dejXtx tj)(2
1)(
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Chengbin Ma UM-SJTU Joint Institute
Basic Idea (2)
Points: T is infinitely large; thus 0 is infinitely
small and k0 is actually continuous.
1. Represent w(t) using IFS
2. Solve W[k] using FS
3. Further represent w(t) using k
Eventually
Slide 9
T
tjk
k
tjk
dtetxT
kX
ekXtx
0
0
)(1
][
][)(
dejXtx tj)(2
1)(
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Chengbin Ma UM-SJTU Joint Institute
Derivations (1) Reference
0
0 0
0
0
/ 2 / 2
/ 2 / 2
0
( ) [ ]
1 1[ ] ( ) ( )
1( )
, [ ] 0.
, [ ] ( ) ( ) does not
necessarily approach 0, where ( ) ( )
jk t
k
T T
jk t jk t
T T
jk t
jk t
j t
w t W k e
W k w t e dt x t e dtT T
x t e dtT
T W k
T W k T x t e dt X jk
X j x t e dt
is
the envelope function of [ ] .W k T
=
=
T
tjk
k
tjk
dtetxT
kX
ekXtx
0
0
)(1
][
][)(
In single period
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Chengbin Ma UM-SJTU Joint Institute
Derivations (2) Reference
We can express the periodic signal in terms of the envelope function as
0
0 0
0
0
0
0 0
( ) [ ]
1 1[ ] ( )
1( )
2
1( )
2
jk t
k
jk t jk t
k k
jk t
k
jk t
k
w t W k e
W k Te X jk eT T
X jk e
X jk e
0
0
2
)(][
][)( 0
T
jkXTkW
ekXtxk
tjk
Because T is infinite, 0 is infinitely small!
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Chengbin Ma UM-SJTU Joint Institute
Derivations (3) Reference
We have
dejX
dejX
ejkX
twtwtx
tj
tj
k
tjk
T
)(2
1
)(2
1
)(2
1lim
)(lim)(lim)(
0
00
2T
Inverse FT
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Chengbin Ma UM-SJTU Joint Institute
IFT and FT
The Fourier transform pair
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
T
tjk
k
tjk
dtetxT
kX
ekXtx
0
0
)(1
][
][)(
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Chengbin Ma UM-SJTU Joint Institute
Dirichlet conditions
A continuous-time signal x(t) is Fourier transformable
if it satisfies the Dirichlet conditions:
x(t) is absolutely integrable.
x(t) has finite number of maxima and minima in any finite
interval.
x(t) has finite number of discontinuities within any finite
interval. Furthermore, each of these discontinuities must be
finite.
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Chengbin Ma UM-SJTU Joint Institute
Example (1)
Example 3.24 FT of a real decaying exponential.
jaja
edte
dtetuedtetxjX
atuetx
tjatja
tjattj
at
1
)(
)()()(
0),()(
0
)(
0
)(
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
MichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Example (2)
Figure 3.39 (b) magnitude spectrum. (c) Phase spectrum.
Low-pass filter
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Chengbin Ma UM-SJTU Joint Institute
3.7 The Fourier Transform
Example 3.24 FT of a rectangular pulse.
0
0
00 0 0 0
0
0
00 0 0 0
0
1,( )
0, otherwise
( ) ( )
2
2
sin( )2sin( ) 2 2 ( ) ( )
T
j t j t
T
Tj T j T j T j Tj t
T
t Tx t
X j x t e dt e dt
e e e e e
j j j
TT T T Sa T Sa f
T
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
In textbook (page 223), Sa is named as sinc
sinc(u)
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Chengbin Ma UM-SJTU Joint Institute
Example (3)
Figure 3.40 (b) Spectrum of the rectangular pulse.
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Chengbin Ma UM-SJTU Joint Institute
Example (4)
FT of the unit impulse.
0
( ) 1
( ) ( )
( ) ( ) 1
FT
j t j t
t
t
x t t
X j t e dt e
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
MichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Example (5)
FT of 1.
0
1 2 ( )
( ) 2 ( )
1( ) 2 ( ) 1
2
FT
j t j t
X j
x t e dt e
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
MichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Example (6)
FT of the delayed unit impulse.
0
0
0
0
0
0
( )
( ) ( )
( ) ( )
FTj t
j tj t j t
t t
t t e
x t t t
X j t t e dt e e
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
MichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Example (7)
FT of the complex sinusoids.
0
0
0
0
0
0
2 ( )
( ) 2 ( )
1( ) 2 ( )
2
FTj t
j tj t j t
e
X j
x t e dt e e
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
MichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Example (8)
FT of the cosine.
0 0
0 0
0 0 0
0 0
0 00
cos( ) [ ( ) ( )]
2 ( ), 2 ( )
2 ( ) 2 ( )cos( )
2 2
FT
FT FTj t j t
j t j t FT
t
e e
e et
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
Note that the nature of integration is summation, and linearity will automatically follow.
MichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Example (9)
FT of the sine.
0 0
0 0
0 0 0
0 0
0 00
sin( ) [ ( ) ( )]
2 ( ), 2 ( )
2 ( ) 2 ( )sin( )
2 2
FT
FT FTj t j t
j t j t FT
tj
e e
e et
j j
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
MichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Example (10)
FT of the sinc function.
1,sin( )( ) ( )
0,2
( )2
1 1( )
2 2 2
1 1 1
2 2 2
FT
W
j t j t
W
Wj t jWt jWt jWt jWt
W
WWtu W u W
Wt W
X jW
x t e d e dW
e e e e e
jt jt t j
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
page 246
MichaelRectangleMichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Example (11)
FT of the signum function.
( )
1, 02
sign( ) 0, 0
1, 0
[PROOF]
sign( ) ( ) ( )
1( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 1( ) ( ) ( )
FT
j t
j t j
t
t tj
t
t u t u t
u t u t e dt U jj
u t u t e dt u e d U j
U jj j
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
)(1
)(
j
tu FT
Refer to the following slides
MichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Reference (1.1)
FT of the unit step function.
1( ) ( )
1 1 1( ) ( ) ( ) ( )
2
1 1 1( )
2 2
1 cos( ) sin( ) 1 1 sin( ) 1
2 2 2 2
1 si
21 sin( )
2
FT
j t
j t j t
u tj
X j x t e dj j
e d e dj
t j t td d
j
u d duu t du td d
u
td
n( ), 0
1 sin( ), 0
2
udu t
u
udu t
u
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
u is negative
cos(wt)/(jw) is odd function, therefore its integration from inf to inf is zero
MichaelRectangleMichaelRectangle -
Chengbin Ma UM-SJTU Joint Institute
Reference (1.2)
1, 1, 1sin( ) sin( )
0, 0, 12 2
sin( ) sin( ) 1 sin( ) 01
2 2
1, 0
sin( ) 1 sin( ) 2
12, 0
2
j t
WWt t
Wt W t
t t te dt dt dt
t t t
tu t
du du
t
x
1, 01 sin( ) 1( )
0, 02 2
ttt d
t
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
FT of the unit step function (Continue).
=0
let W = 1
This is exactly the definition of u(t)
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Chengbin Ma UM-SJTU Joint Institute
Homework
None
Read the PPTs and try to solve the examples
by yourself carefully!
Slide 29