1 chapter 5: fourier transform. fourier transform: 2 definition of the fourier transforms definition...
TRANSCRIPT
![Page 1: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/1.jpg)
1
Chapter 5: Chapter 5:
Fourier Fourier TransformTransform
![Page 2: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/2.jpg)
FOURIER TRANSFORM:FOURIER TRANSFORM:
2
Definition of the Fourier transformsDefinition of the Fourier transformsRelationship between Laplace Transforms and
Fourier TransformsFourier transforms in the limitProperties of the Fourier TransformsCircuit applications using Fourier TransformsParseval’s theoremEnergy calculation in magnitude spectrum
![Page 3: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/3.jpg)
Definition of Fourier Definition of Fourier TransformsTransforms
3
dtetf
tfFF
tj
)(
)()(
Fourier Transforms:
![Page 4: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/4.jpg)
Inverse Fourier Transforms:
4
dteF
FFtf
tj
)(2
1
)()( 1
![Page 5: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/5.jpg)
Example 1:Obtain the Fourier Transform for thefunction below:
0
1
f ( t)
t
ate
5
![Page 6: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/6.jpg)
Solution:
Given function is:
00
0)(
t
tetf
at
6
![Page 7: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/7.jpg)
Fourier Transforms:
7
ja
eja
dtedtee
dtetfF
tja
tjatjat
tj
1
1
)(
)()(
0
)(
0
)(
0
![Page 8: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/8.jpg)
FOURIER TRANSFORM:FOURIER TRANSFORM:
8
Definition of the Fourier transformsRelationship between Laplace Transforms and Relationship between Laplace Transforms and
Fourier TransformsFourier TransformsFourier transforms in the limitProperties of the Fourier TransformsCircuit applications using Fourier TransformsParseval’s theoremEnergy calculation in magnitude spectrum
![Page 9: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/9.jpg)
Relationship between Fourier Transforms and Laplace Transforms
9
There are 3 rules apply to the use of Laplace transforms to find Fourier Transforms of such functions.
![Page 10: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/10.jpg)
Rule 1:If f(t)=0 for t<=0-
Replace s=jω
jstfLtfF )()(
10
![Page 11: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/11.jpg)
Example:
11
0cos
00)(
tte
ttf
oat
![Page 12: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/12.jpg)
Replace s=jω
12
22
22
)(
)()(
o
jso
aj
aj
as
astfF
![Page 13: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/13.jpg)
Rule 2: Inverse negative function
13
jstfLtfF )()(
![Page 14: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/14.jpg)
Example:
14
0cos
00)(
tte
ttf
oat
0cos
00)(
tte
ttf
oat
Negative
![Page 15: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/15.jpg)
Fourier Transforms
15
22
22
)(
)(
)()(
o
jso
js
aj
aj
as
as
tfLtfF
![Page 16: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/16.jpg)
Rule 3:Add the positive and negative function
16
0)()(
0)()(
ttftf
ttftf
)()()( tftftf
![Page 17: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/17.jpg)
Thus,
17
jsjs tfLtfL
tfFtfFtfF
)()(
)()()(
![Page 18: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/18.jpg)
Example 1:
18
at
at
etf
etf
)(
)(
as
tfL
astfL
1)(
1)(
![Page 19: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/19.jpg)
Fourier transforms:
19
22
2
11
11)(
a
a
ajaj
asastfF
jsjs
![Page 20: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/20.jpg)
Example 2:
Obtain the Fourier Transforms for the function below:
0sin
00)(
tte
ttf
oat
20
![Page 21: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/21.jpg)
Solution:
21
22
22
)(
)(
)()(
o
o
jso
o
js
aj
as
sFF
![Page 22: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/22.jpg)
Example 3:
22
0
00)(
tte
ttf
at
![Page 23: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/23.jpg)
Solution:
23
2
2
)(
1
)(
1
)()(
aj
as
tfLF
js
js
![Page 24: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/24.jpg)
Example 4:
24
0
0)(
tte
ttetf
at
at
![Page 25: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/25.jpg)
Solution:
25
at
at
tetf
tetf
)(
)(
2
2
1)(
1)(
astfL
astfL
![Page 26: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/26.jpg)
26
222
22
22
4
)(
1
)(
1
)(
1
)(
1)(
a
aj
jaaj
asastfF
jsjs
![Page 27: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/27.jpg)
FOURIER TRANSFORM:FOURIER TRANSFORM:
27
Definition of the Fourier transformsRelationship between Laplace Transforms and
Fourier TransformsFourier transforms in the limitFourier transforms in the limitProperties of the Fourier TransformsCircuit applications using Fourier TransformsParseval’s theoremEnergy calculation in magnitude spectrum
![Page 28: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/28.jpg)
Fourier Transforms in the limitFourier transform for signum function
(sgn(t))
1 .0
0
-1 .0
t
s g n ( t)
28
![Page 29: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/29.jpg)
)()()sgn( tutut 0)()(lim)sgn(0
tuetuet tt
1 .0
0
-1 .0
t
f ( t)
)(tue t
)( tue t
29
![Page 30: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/30.jpg)
30
22
2
11
11)(
j
jj
sstfF
jsjs
![Page 31: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/31.jpg)
assume ε→0,
31
j
tF2
)sgn(
![Page 32: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/32.jpg)
Fourier Transforms for step function:
)sgn(2
1
2
1)( ttu
j
tFFtuF
1
)sgn(2
1
2
1)(
32
![Page 33: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/33.jpg)
Fourier Transforms for cosine function
tjetf 0)(
)(2 00 tjeF
33
![Page 34: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/34.jpg)
22
2cos
00
00
0
tjtj
tjtj
ee
eet
34
![Page 35: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/35.jpg)
Thus,
00
00
0
222
12
1cos 00
tjtj eFeFtF
35
![Page 36: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/36.jpg)
FOURIER TRANSFORM:FOURIER TRANSFORM:
36
Definition of the Fourier transformsRelationship between Laplace Transforms and
Fourier TransformsFourier transforms in the limitProperties of the Fourier TransformsProperties of the Fourier TransformsCircuit applications using Fourier TransformsParseval’s theoremEnergy calculation in magnitude spectrum
![Page 37: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/37.jpg)
Properties of Fourier Transforms
Multiplication by a constant
)()( FtfF
KFtKfF )(37
![Page 38: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/38.jpg)
Addition and subtraction
33
22
11
)(
)(
)(
FtfF
FtfF
FtfF
)()()(
)()()(
321
321
FFF
tftftfF
38
![Page 39: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/39.jpg)
Differentiation
Fjdt
tfdF
Fjdt
tdfF
nn
n
)(
)(
39
![Page 40: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/40.jpg)
Integration
)0()(
)()(
Fj
FtgF
dxxftgt
40
![Page 41: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/41.jpg)
Scaling
01
)(
aa
Fa
atfF
41
![Page 42: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/42.jpg)
Time shift
)()( 0 FeatfF tj
42
![Page 43: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/43.jpg)
Frequency shift
)()( 00 FtfeF tj
43
![Page 44: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/44.jpg)
Modulation
00
0
2
1
2
1
)()cos(
FF
tftF
44
![Page 45: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/45.jpg)
Convolution in time domain
)()()()( 2121 FFtftf
45
![Page 46: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/46.jpg)
Convolution in frequency domain:
)()(2
1)()( 2121
FFtftf
46
![Page 47: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/47.jpg)
Example 1:
Determine the inverse Fourier Transforms for the function below:
86)(
410)(
2
jj
jF
47
![Page 48: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/48.jpg)
Solution:
48
24
)2)(4(
410
86)(
410)(
2
s
B
s
A
ss
s
ss
ssF
LAPLACELAPLACETRANSFORMSTRANSFORMS
![Page 49: 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace](https://reader035.vdocuments.site/reader035/viewer/2022082207/56649f055503460f94c1a666/html5/thumbnails/49.jpg)
A and B value: 818 BA
2
8
4
18
jjF
49
)()818()( 24 tueetf tt