from fourier series to fourier transforms. recall that where now let t become large... and so ω...
TRANSCRIPT
![Page 1: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/1.jpg)
From Fourier Series to
Fourier Transforms
![Page 2: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/2.jpg)
Recall that
n
Txninecxf /2)(
where ,...2,1,0,1 /2
2/
2/
ndxexf
Tc Txni
T
T
n
Now let T become large ...
andT
nπω
2
so ω becomes small ...
deFxf xi
2
1)(
dxexfF xi )(
Fourier Transform
of f(x)
Inverse Fourier Transform of F(ω).
![Page 3: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/3.jpg)
Example 1Determine the Fourier Transform of
113 tututf
![Page 4: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/4.jpg)
113 tututf
1
1
3
tiei
dtetfF tiωω )(
1
1
3 dteF ti
sin6
ii eei
3
ii eei
3
sin23
ii
sinc6)( F)(sinc
sin
![Page 5: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/5.jpg)
113 tututf ω
ωω
sin6F
Note: F(ω) is REAL in this example.
These are the graphs of f(t) and F(ω):
![Page 6: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/6.jpg)
Example 2Determine the Fourier Transform of
tetutf 2
![Page 7: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/7.jpg)
i
2
1
dtetfF tiωω )( tetutf 2
0
2 dteeF tit
0
2 dte ti
0
2
2
1 tiei
102
1
i
![Page 8: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/8.jpg)
Note: F(ω) is COMPLEX in this example.
Draw the graph of the modulus of F(ω) (the amplitude spectrum).
tetutf 2 ω
ωi
F
2
1
24
1
2
1
2
1
iiF
tetutf 2 F
![Page 9: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/9.jpg)
Even Functions
If f is an even function, then tftf
0
cos2)( dtttfdtetfF ti ωω ω
This result arises because cosine is even ...
... and so is even ... ttf ωcos
![Page 10: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/10.jpg)
Example 3Determine the Fourier Transform of
1111 tututtututtf
Even function!
![Page 11: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/11.jpg)
1
0
cos12 dtttF
1cos22
1
0
1
0
sin12
sin12 dt
ttt
1
0
sin2
dtt
002
1
0
cos2
t AAA 22 sincos2cos
A2sin21AA 2sin22cos1
2
2)(
F 2/sin2 2 2/sin
)2/(
1 22
2/sinc2
![Page 12: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/12.jpg)
Odd Functions
If f is an odd function, then tftf
0
sin2)( dtttfidtetfF ti ωω ω
This result arises because sine is odd ...
... and so is even ... ttf ωsin
![Page 13: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/13.jpg)
Example 4Determine the Fourier Transform of
2424 tututututf
Odd function!
![Page 14: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/14.jpg)
2
0
sin42 dttiF
2424 tututututf
0
sin2 dtttfiF ωω
2sin16i
2
0
cos8
t
i
12cos8
i
2sin28
i
FIm
![Page 15: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/15.jpg)
Summary:
Example f(t) F(ω)
1 Even Real
2 Neither odd nor even
Complex
3 Even Real
4 Odd Imaginary
113 tutuω
ωsin6
tetu 2
ωi2
1
...11 tutut
2sinc2 ω
...24 tutuω
ω2sin16i
![Page 16: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/16.jpg)
Special Case 0,2
aetf at
Use this known result:
πdxe x2
Substitute yax
πdyea ay2
Now use:a
ity
2
ω
atiat
a
itaay
42
22
22
a
dte atiat πωω
4
22
Hence:
atiat ea
dte 4
2
2ω
ω πor
set at iofTransformFourier2 ae
a4
2
![Page 17: From Fourier Series to Fourier Transforms. Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier](https://reader036.vdocuments.us/reader036/viewer/2022062804/56649e995503460f94b9ca9a/html5/thumbnails/17.jpg)
Now look at Tutorial 1