from fourier series to fourier transformihayee/teaching/ee2111/ece... · some properties of fourier...
TRANSCRIPT
![Page 1: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/1.jpg)
Department of Electrical and Computer Engineering
From Fourier Series to Fourier Transform
tjkk
k
FekXtx
][)(
dtetxT
kXF
F
Tt
t
tjk
F
0
0
)(1
][
dtetxT
kX
F
F
F
T
T
tjk
F
2
2
)(1
][
F
tjk
F
TwhendtetxT
kX F ,)(1
][
OR
![Page 2: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/2.jpg)
Department of Electrical and Computer Engineering
dtetxX tj )()(
Let’s Consider a function
We can express X[k] in terms of X(ω)
)(1
][ F
F
kXT
kX
F
tjk
F
TwhendtetxT
kX F ,)(1
][
![Page 3: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/3.jpg)
Department of Electrical and Computer Engineering
t
ω
FT
X )(][kX
)(tx
2
FT
2
FT
F F2 F30FF2F3
Ff Ff2 Ff30FfFf2Ff3
FxT1
![Page 4: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/4.jpg)
Department of Electrical and Computer Engineering
t
)(tx
2
FT
2
FT
ω
FT
X )(][kX
F F2 F30FF2F3
Ff Ff2 Ff30FfFf2Ff3
FxT2
![Page 5: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/5.jpg)
Department of Electrical and Computer Engineering
t
)(tx
2
FT
2
FT
FT
X )(][kX
ω
F F2 F30FF2F3
Ff Ff2 Ff30FfFf2Ff3
FxTx22
![Page 6: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/6.jpg)
Department of Electrical and Computer Engineering
t
FT
X )(
)(tx
FxT
TransformFourier
][kX
![Page 7: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/7.jpg)
Department of Electrical and Computer Engineering
dtetxX tj )()(
Fourier Transform of a Singal x(t)
)]([)( 1 XFtx
Now Let’s talk about the Inverse Fourier Transform
)]([)( txFX
![Page 8: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/8.jpg)
Department of Electrical and Computer Engineering
k
tjk FekXtx
][)(
k
tjk
F
F FeT
kXtx
)()(
k
tjk
F
eT
kXtx )()(
22
F
FT
k
tjkekX
tx
2
)()(
FTF
lim
![Page 9: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/9.jpg)
Department of Electrical and Computer Engineering
k
tjkekXtx
)(2
1)(
k
tjkekXtx )(2
1)(
deXtx tj
)(2
1)(
)]([)( 1 XFtx Inverse Fourier Transform
22
F
FT
k
tjkekX
tx
2
)()(
![Page 10: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/10.jpg)
Department of Electrical and Computer Engineering
)]([)( 1 XFtx
deXtx tj
)(2
1)(
Inverse Fourier Transform
dtetxX tj )()(
Fourier Transform of a Signal x(t) )]([)( txFX
)()( Xtx
![Page 11: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/11.jpg)
Department of Electrical and Computer Engineering
)]([)( 1 XFtx
deXtx tj
)(2
1)(
Inverse Fourier Transform
dtetxX tj )()(
Fourier Transform of a Signal x(t)
)]([)( txFX
OR
dtetxfX ftj 2)()(
)]([)( txFfX
)]([)( 1 fXFtx
dfefXtx ftj
2)()(
OR
![Page 12: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/12.jpg)
Department of Electrical and Computer Engineering
)]([)( 1 XFtx
deXtx tj
)(2
1)(
Inverse Fourier Transform
dedt
dX
dt
tdx tj
)(2
1)(
deXjdt
tdx tj
)(2
1)(
)]([)( 1 XjF
dt
tdx
)()(
Xjdt
tdx
![Page 13: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/13.jpg)
Department of Electrical and Computer Engineering
00)( ttDefinition:
1)(
t
Fourier Transform of Impulse Function
t
)(t
dtetxX tj )()(
Fourier Transform of a Singal x(t)
1)()( 0
edtetX tj
![Page 14: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/14.jpg)
Department of Electrical and Computer Engineering
Let’s revisit LTI system
)(tx C
R
)(ty
)(ti
)()()(
txtydt
tdyRC
)()()( XYYRCj
)(]1)[( XRCjY
1
1
)(
)(
RCjX
Y
![Page 15: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/15.jpg)
Department of Electrical and Computer Engineering
)(tx C
R
)(ty
)(ti
1
1
)(
)(
RCjX
Y
RCjX
Y
1
1
)(
)(
RCj
H
1
1
1
)(
When the input, x(t) is unit impulse function, the output is h(t)
RCjH
1
1)(
![Page 16: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/16.jpg)
Department of Electrical and Computer Engineering
1. An impulse function
2. A constant function (via inverse transform)
3. Complex exponential function (via inverse transform)
4. Sinosoidal Function
5. Gate Function
Some Examples of Fourier Transform
![Page 17: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/17.jpg)
Department of Electrical and Computer Engineering
Fourier Transform of a Constant Function
t
1
dtetxX tj )()(
Fourier Transform of a Singal x(t)
dte tj
)(tx
0
j
e tj
Let’s try indirectly – let’s find the Inverse Fourier Transform of ()
deXXFtx tj
)(2
1)]([)( 1
2
1
2
1)(
2
1)]([ 01
edeF tj
)(]2
1[
F )(2]1[ F
)(2
![Page 18: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/18.jpg)
Department of Electrical and Computer Engineering
Fourier Transform of a Complex Exponential
Let’s find the Inverse Fourier Transform of (0)
deXXFtx tj
)(2
1)]([)( 1
tjtj edeF 0
2
1)(
2
1)]([ 00
1
)(]2
[ 0
0
tj
eF
)(2][ 00
tj
eF
)( 0
0
![Page 19: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/19.jpg)
Department of Electrical and Computer Engineering
Fourier Transform of a Sinusoidal Signal
)( 0
0
2)cos()(
00
0
tjtjee
ttx
)()(]2
[)][cos( 000
00
tjtj
eeFtF
j
eettx
tjtj
2)sin()(
00
0
)()(]2
[)][sin( 000
00
jjj
eeFtF
tjtj
)]()([ 00 j
)]()([ 00
)( 0
0
![Page 20: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/20.jpg)
Department of Electrical and Computer Engineering
Fourier Transform of a Rectangular Function
2/10
2/12/1
2/11
)(
t
t
t
trect
t
1
2
1
2
1
dtetxXtxF tj )()()]([
2/1
2/1
)]([ dtetrectF tj
j
e
j
e
j
e jjtj
2/2/2/1
2/1
j
ee
j
e
j
e jjjj 2/2/2/2/
![Page 21: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/21.jpg)
Department of Electrical and Computer Engineering
j
eetrectF
jj 2/2/
)]([
j
ee jj
2
][2 2/2/
)2/(
)2/sin()2/sin(
2
)2/(sin c
t
ttc
)sin()(sin
Remember?
)sin()(sin c
or
![Page 22: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/22.jpg)
Department of Electrical and Computer Engineering
1. Linearity
2. Symmetry
3. Scaling
4. Time Shifting
5. Frequency Shifting
6. Time Differentiation
7. Convolution
Some Properties of Fourier Transform
![Page 23: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/23.jpg)
Department of Electrical and Computer Engineering
Linearity
)()()()( 22112211 XkXktxktxk
dtetxXtxF tj )()()]([
dtetkxtkxF tj)()]([
dtetxk tj)(
)(kX
![Page 24: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/24.jpg)
Department of Electrical and Computer Engineering
Symmetry)()( Xtx
)(2)( xtX
deXtx tj
)(2
1)(
deXtx tj
)(2
1)(
deXtx tj
)()(2
deXtx tj
)()(2
deXx j
)()(2
dtetXx jt
)()(2
dtetXx tj
)()(2
)(2)( xtX
![Page 25: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/25.jpg)
Department of Electrical and Computer Engineering
Scaling)()( Xtx
)(1
)(a
Xa
atx
dteatxatxF tj
)()]([
dexa
atxF aj
/)(1
)]([
dadtat Let’s make a variable change
)(1
)(1 )/(
aX
adex
a
aj
Let’s assume ‘a’ is positive
![Page 26: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/26.jpg)
Department of Electrical and Computer Engineering
Time Shifting – linear phase shift
0)()( 0
tjeXttx
dtettxttxF tj
)()]([ 00
ddttt 0Let’s make a variable change
dexttxF
tj
)(
00)()]([
deextjj
0)(
)()( 00 Xedexe
tjjtj
![Page 27: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/27.jpg)
Department of Electrical and Computer Engineering
)()( 00
Xetxtj
Frequency Shifting - modulation
dteetxetxF tjtjtj
00 )(])([
dtetxtjtj
0)(
dtetxtj
)( 0)(
)( 0 X
![Page 28: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/28.jpg)
Department of Electrical and Computer Engineering
Amplitude Modulation
)()( Mtm
)]()([2
1)cos()()(mod ccc MMttmts
)(tm
)cos( tc
)cos()( ttm c
Modulating signalModulated Signal
Modulating carrier
)(mod fS
)( fM
B2B2
)()( modmod Sts
cc
![Page 29: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/29.jpg)
Department of Electrical and Computer Engineering
Amplitude Modulation – Another Method
Can we do it with square wave? If so, what else do we need?
)(tm
)cos( tc
)cos()( ttm c
Modulating signalModulated Signal
Modulating carrier
Square Wave
![Page 30: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/30.jpg)
Department of Electrical and Computer Engineering
Amplitude Modulation – Another Method
Can we do it with square wave? If so, what else do we need?
)(tm )cos()( ttm c
Modulating signalModulated Signal
Square Wave
BPF
)(tm
)2cos( tfc
)2cos()( tftm c
Modulating signalModulated Signal
Modulating carrier
Square Wave
What is the advantage of doing so?
![Page 31: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/31.jpg)
Department of Electrical and Computer Engineering
)( fM
2
1
B2B2
)(mod S
c2 c3 c4cc2c3c4
Bandpass Filter
Advantage: A higher multiple of carrier frequency can be chosen
)(mod fS
B2B2
B2B2c2 c3 c4cc2c3c4 c
c
![Page 32: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/32.jpg)
Department of Electrical and Computer Engineering
Amplitude Demodulation – Synchronous
Demodulation: )(cos)()( 2 ttmtS cdem
)]2cos()()([2
1ttmtm c
)]2()2([4
1)(
2
1)( ccdem MMMS
)cos( tc
)cos()( ttm c
Modulated signalDemodulated Signal
Demodulating carrier
)(tSdem
Low Pass Filter
![Page 33: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/33.jpg)
Department of Electrical and Computer Engineering
Amplitude Demodulation – Envelop Detection
+
-
1vCharges Discharge
CR
Time Constant = RC
c
RC
1
BRC
2
1
2v 3v
C should decay slow enoughC should charge quick enough
![Page 34: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/34.jpg)
Department of Electrical and Computer Engineering
deXtx tj
)(2
1)(
Time Differentiation
dedt
dX
dt
tdx tj
)(2
1)(
deXjdt
tdx tj
)(2
1)(
)]([)( 1 XjF
dt
tdx
)()(
Xjdt
tdx
![Page 35: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/35.jpg)
Department of Electrical and Computer Engineering
Convolution
dtyxtytx )()()(*)(
Then
)()( Xtx )()( Yty If and
)()()(*)( YXtytx
dtdtyxetytxF tj ])()([)](*)([
ddttyex tj ])([)(
deYx j ])([)(
![Page 36: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/36.jpg)
Department of Electrical and Computer Engineering
deYxtytxF j ])([)()](*)([
dexY j
)()(
)()( XY
)()()(*)( YXtytx
![Page 37: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/37.jpg)
Department of Electrical and Computer Engineering
Let’s revisit LTI system - again
)(tx C
R
)(ty
)(ti
)()()(
txtydt
tdyRC
)()()( XYYRCj
)(1
1
)(
)(
H
RCjX
Y
)()()( HXY
)(*)()( thtxtyknowwebut
![Page 38: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/38.jpg)
Department of Electrical and Computer Engineering
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
![Page 39: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/39.jpg)
Department of Electrical and Computer Engineering
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
22 )
2(sin)()2( jj eceXtx
![Page 40: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/40.jpg)
Department of Electrical and Computer Engineering
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
22 )
2(sin)()2( jj eceXtx
)2
(sin2)(2)(2
cXtx
![Page 41: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/41.jpg)
Department of Electrical and Computer Engineering
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
22 )
2(sin)()2( jj eceXtx
)2
(sin2)(2)(2
cXtx
)4
(sin2
1)
2(
2
1)2(
cXtx
![Page 42: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/42.jpg)
Department of Electrical and Computer Engineering
How about FT of
)22(2 tx
jec )4
(sin
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
22 )
2(sin)()2( jj eceXtx
)2
(sin2)(2)(2
cXtx
)4
(sin2
1)
2(
2
1)2(
cXtx
![Page 43: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/43.jpg)
Department of Electrical and Computer Engineering
Time For Discrete Time Fourier Series
CT Exponential Fourier Series
tkfjk
k
FekXtx)(2
][)(
where
dtetxT
kXF
F
Tt
t
tkfj
F
0
0
)(2)(
1][
DT Exponential Fourier Series
nkFjk
k
FekXnx)(2
][][
nkFjNnn
nnF
F
F
enxN
kX)(2
10
0
][1
][
F
FT
f1
whereF
FN
F1
![Page 44: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/44.jpg)
Department of Electrical and Computer Engineering
One Important Difference
CT Exponential Fourier Series
tkfjk
k
FekXtx)(2
][)(
where
DT Exponential Fourier Series
nkFjk
k
FekXnx)(2
][][
F
FT
f1
whereF
FN
F1
nkFj Fe)(2
nFNjnkFjnFNkj FFFFF eee)(2)(2)(2
njnkFjee F 2)(2
nkFj Fe)(2
Basis functiontkfj Fe)(2
Basis function
tfTjtkfjtfTkj FFFFF eee)(2)(2)(2
tjtkfjee F 2)(2
)]2sin()2[cos()(2
njnenkFj F
)]2sin()2[cos()(2
tjtetkfj F
for all ntkfj Fe
)(2 for all t
![Page 45: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/45.jpg)
Department of Electrical and Computer Engineering
One Important Difference … cont.
CT Exponential Fourier Series
tkfjk
k
FekXtx)(2
][)(
where
DT Exponential Fourier Series
nkFjk
k
FekXnx)(2
][][
F
FT
f1
whereF
FN
F1
nkFj Fe)(2
Basis functiontkfj Fe)(2
Basis function
Still need infinite number
of exponential functions
Need only NF exponential
Functions
nkFjNkk
kk
F
F
ekXnx)(2
10
0
][][
tkfjk
k
FekXtx)(2
][)(
![Page 46: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/46.jpg)
Department of Electrical and Computer Engineering
DTFS vs. CTFS
![Page 47: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/47.jpg)
Department of Electrical and Computer Engineering
Transition from DT Fourier Series to DT Fourier Transform
nkFjNnn
nnF
F
F
enxN
kX)(2
10
0
][1
][
nkFjNn
NnF
F
F
F
enxN
kX)(2
2/)1(
2/
][1
][
nkFjn
nF
FenxN
kX)(2
][1
][
FN
Fnjn
n
enxFX 2][)(
Let’s define a function
When
)(1
][ F
F
kFXN
kX
![Page 48: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/48.jpg)
Department of Electrical and Computer Engineering
Transition from DT Fourier Series to DT Fourier Transform
nkFjNnn
nnF
F
F
enxN
kX)(2
10
0
][1
][
nkFjNn
NnF
F
F
F
enxN
kX)(2
2/)1(
2/)1(
][1
][
nkFjn
nF
FenxN
kX)(2
][1
][
FN
Fnjn
n
enxFX 2][)(
Let’s define a function
When
)(1
][ F
F
kFXN
kX
DTFT
![Page 49: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/49.jpg)
Department of Electrical and Computer Engineering
Inverse Discrete Fourier Transform
nkFjNk
k
F
F
ekXnx)(2
1
0
][][
nkFjNkk
kk
F
F
ekXnx)(2
10
0
][][
nkFjNk
k F
F F
F
eN
kFXnx
)(2)1
0
)(][
FNWhennFkj
Nk
k
eF
FkXnx
F
)(21
0 /1
)(][
![Page 50: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/50.jpg)
Department of Electrical and Computer Engineering
FNWhennFkj
Nk
k
eF
FkXnx
F
)(21
0 /1
)(][
nFkjNk
k
eFFkXnxF
)(21
0
)(][
dFeFXnx Fnj 2
1
)(][ DT Inverse Fourier Transform
10 fromchangesFk
![Page 51: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/51.jpg)
Department of Electrical and Computer Engineering
Fnjn
n
enxFX 2][)(
dFeFXnx Fnj 2
1
)(][
njn
n
enxX
][)(
deFXnx nj
2
)(2
1][
Discrete Time
Fourier Transform
Discrete Time
Inverse Fourier Transform
OR OR
F2Where
![Page 52: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/52.jpg)
Department of Electrical and Computer Engineering
1. Linearity
2. Scaling
3. Time Shifting
4. Frequency Shifting
5. Time Differencing
6. Convolution
Some Properties of Discrete Fourier Transform
![Page 53: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/53.jpg)
Department of Electrical and Computer Engineering
CTFS CTFT
DTFS DTFT
Continuous Time
Discrete Time
Continuous
Frequency
Discrete
Frequency
Relationship between Time and Frequency Domains
![Page 54: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/54.jpg)
Department of Electrical and Computer Engineering
Discrete Frequency Continuous Frequency
CT
DT
With Examples
![Page 55: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/55.jpg)
Department of Electrical and Computer Engineering
Parseval’s Theorem in Continuous Frequency
n
x nxE2
][
dttxEx
2)(
Energy of CT Signals Energy of DT Signals
dffX2
)(
dX2
)(2
1
1
2)( dFFX
2
2)(
2
1dX
![Page 56: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/56.jpg)
Department of Electrical and Computer Engineering
dttxEx
2)(
dttxtx )(*)(
dtdeXtx tj ])(*2
1)[(
ddtetxX tj
])()[(*2
1
dXdXX
2
)(2
1)()(*
2
1
![Page 57: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/57.jpg)
Department of Electrical and Computer Engineering
Parseval’s Theorem in Discrete Frequency
0
2
0
][1
Nn
x nxN
P
0
2
0
)(1
T
x dttxT
P
Power of Periodic CT Signals Power of Periodic DT Signals
k
kX2
][
0
2][
Nk
kX
![Page 58: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/58.jpg)
Department of Electrical and Computer Engineering
tkfjk
k
FekXtx)(2
][)(
0
0
2)(2
][,, ][T
tkfj
kXTx dtekXE F
0
2)(22
][T
tkfjdtekX F
Energy of each of the above frequency component over T0
0
22][][
0
TkXdtkXT
![Page 59: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/59.jpg)
Department of Electrical and Computer Engineering
...]]2[]1[
...]2[]1[]0[[
22
222
0, 0
XX
XXXTE Tx
...]2[]1[
...]2[]1[]0[
)2(2)(2
)2(2)(2
tfjtfj
tfjtfj
FF
FF
eXeX
eXeXX
tkfjk
k
FekXtx)(2
][)(
From conservation of energy principle
...]]2[]1[
...]2[]1[]0[[1
22
222
,
00
XX
XXXET
P Txx
![Page 60: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/60.jpg)
Department of Electrical and Computer Engineering
...]]2[]1[
...]2[]1[]0[[1
22
222
,
00
XX
XXXET
P Txx
k
x kXP2
][
![Page 61: From Fourier Series to Fourier Transformihayee/Teaching/ee2111/ece... · Some Properties of Fourier Transform. Department of Electrical and Computer Engineering Linearity k 1 x 1](https://reader034.vdocuments.us/reader034/viewer/2022052612/5f0f7e6d7e708231d4447099/html5/thumbnails/61.jpg)
Department of Electrical and Computer Engineering
Example
Determine the power of x(t) without performing any integration
)2cos()( tfAtx F
244
222 AAAPx
tkfjk
k
FekX)(2
][
tfjtfj FF eXeX)(2)(2
]1[]1[
tfjtfj FF eA
eA )(2)(2
22