6/1/2018 © 2003, JH McClellan & RW Schafer 1
Signal Processing First
Chapter 11Continuous-TimeFourier Transform
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6/1/2018 © 2003, JH McClellan & RW Schafer 3
LECTURE OBJECTIVES Review Frequency Response Fourier Series
Definition of Fourier transform
Relation to Fourier Series Examples of Fourier transform pairs
dtetxjX tj )()(
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11.1 Definition of Fourier Transform
Forward Fourier transform
Inverse Fourier transform
dtetxjX tj )()(
dejXtx tj)()( 21
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11.2 Fourier Transform and the Spectrum
Everything = Sum of sinusoids (?) One Square Pulse = Sum of Sinusoids (?)
Finite Length Not Periodic Limit of Square Wave as Period infinity
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Fourier Series: Periodic x(t)x(t) x(t T0 )
T02T0 T0 2T00 t
x(t) akej 0k t
k
ak 1T0
x(t)e j0ktdtT0/ 2
T0 / 2
Fundamental Freq.0 2 / T0 2f0
Fourier Synthesis
Fourier Analysis
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Square wave signal
ak e j0kt
j0kT0 T0 /4
T0 / 4
e jk / 2 ejk / 2
j2k
sin(k / 2)k
x(t) x(t T0 )
T02T0 T0 2T00 t
4/
4/0
0
0
0)1(1 T
T
tkjk dte
Ta
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Spectrum from Fourier Series
,4,20
,3,1,00)2/sin(k
k
kkak
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What if x(t) is not periodic?
Sum of Sinusoids? Non-harmonically related sinusoids Would not be periodic, but would probably be
non-zero for all t. Fourier transform gives a “sum” (actually an integral) that
involves ALL frequencies
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Limiting Behavior of Spectrum
T0=2T
T0=4T
T0=8T
)(Plot
0 kaT
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Fourier Transform Defined
For non-periodic signalsFourier Synthesis
Fourier Analysis
dtetxjX tj )()(
dejXtx tj)()( 21
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11.4 Examples of Fourier Transform Pairs
X( j) 1a j
x(t) eatu(t)
Example 1:
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Example 1: x(t) eatu(t)
X( j) 1a j
X( j ) eat
0
e j tdt 0
e (a j )tdt
X( j ) eate j t
a j 0
1
a j
a 0
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Frequency Response
Fourier Transform of h(t) is the Frequency Response
jjHtueth t
11)()()(
)()( tueth t
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Magnitude and Phase Plots
jajH
1)(
ajH 1tan)(
22
11
aja
)()( jHjH
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X( j) sin(T / 2) / 2
Example 2: x(t) 1 t T / 20 t T / 2
X( j ) e j t
j T / 2
T /2
e jT / 2 e jT /2
j
X( j) (1)e jtdtT / 2
T /2 e jtdt
T / 2
T /2
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Example 3:
b
bjX
0
1)(
tttx b
)sin()(
b
b
dedejXtx tjtj
121)(
21)(
jtee
jtetx
tjtjtj bbb
b
21
21)(
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Example 4:
X( j ) (t)e jtdt
1
Shifting Property of the Impulse
)()( 0tttx
0)()( 0tjtj edtettjX
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Example 5: X( j ) 2 ( 0 )
x(t) 12
2 ( 0 )e jtd
e j0t
x(t) 1 X( j ) 2 ()
x(t) e j0 t X( j) 2 ( 0 )
x(t) cos(0t) X( j) ( 0) ( 0)