application of combined fourier series transform (sampling...
TRANSCRIPT
![Page 1: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/1.jpg)
Department of Electrical and Computer Engineering
Application of Combined Fourier Series Transform
(Sampling Theorem)
t
k
skTttx )()(
sT
t
)(tm t
)(tms
][kX
)( fM
)( fM s
s
sT
f1
Sampling Frequency
![Page 2: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/2.jpg)
Department of Electrical and Computer Engineering
tsT
][kX
k
skTttx )()( We need Fourier Series
))2sin(][)2cos(][(]0[)(1
tkfkXtkfkXXtx sss
k
k
c
s
T
s
Tt
ts Tdtt
Tdttx
TX
ss 1)(
1)(
1]0[
0
0
0
0)2sin()(2
][0
0
dttkftxT
kXsTt
t
s
s
s
![Page 3: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/3.jpg)
Department of Electrical and Computer Engineering
dttkftxT
kXsTt
t
s
s
c
0
0
)2cos()(2
][
ss
T
s
s TTdttkft
T
s 2)0cos(
2)2cos()(
2
0
))2sin(][)2cos(][(]0[)(1
tkfkXtkfkXXtx sss
k
k
c
Let’s go back to Fourier Series representation
k
k
s
ss
tkfTT
tx1
)2cos(21
)(
)2cos(][]0[1
tkfkXX s
k
k
c
![Page 4: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/4.jpg)
Department of Electrical and Computer Engineering
t
)(tm
f
)( fM
BB
Fourier Transform
B is the Bandwidth of m(t)
What about m(t)? – need Fourier Transform
![Page 5: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/5.jpg)
Department of Electrical and Computer Engineering
k
k
s
ss
tkfTT
tx1
)2cos(21
)(
k
k
s
ss
tkftmT
tmT
txtm1
)2cos()(2
)(1
)()(
We now have x(t) as frequency representation via Fourier Series
And m(t) as Frequency representation via Fourier Transform
)()( fMtm
Let’s multiply m(t) with x(t)
What is the frequency representation of m(t)x(t)?
![Page 6: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/6.jpg)
Department of Electrical and Computer Engineering
k
k
s
ss
tkftmT
tmT
txtm1
)2cos()(2
)(1
)()(
k
k
s
ss
tkftmT
tmT
FtxtmF1
)]2cos()(2
)(1
[)]()([
We need to take Fourier Transform
))]()((2
1[
2)(
1)(
1
s
k
k
s
ss
s kffMkffMT
fMT
fM
)]()([1
)(1
1
s
k
k
s
ss
kffMkffMT
fMT
![Page 7: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/7.jpg)
Department of Electrical and Computer Engineering
f
)( fM
BB
)]()([1
)(1
)()]()([1
s
k
k
s
ss
s kffMkffMT
fMT
fMtxtmF
f
sT
1
BB
)( fM s
sf sf2 sf3 sf4sf
sf2sf3sf4
![Page 8: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/8.jpg)
Department of Electrical and Computer Engineering
f
sT
1
BB
)( fM s
sf sf2 sf3 sf4sf
sf2sf3sf4
f
sT
1
BBsf sf2
sf3 sf4
sfsf2
sf3
sf4
f
sT
1
BBsf sf2 sf3 sf4
sfsf2sf3
sf4
Bfs 2
Bfs 2
Bfs 2
![Page 9: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/9.jpg)
Department of Electrical and Computer Engineering
f
sT
1
BB
)( fM s
sf sf2 sf3 sf4sf
sf2sf3sf4
f
sT
1
BBsf sf2
sf3 sf4
sfsf2
sf3
sf4
f
sT
1
BBsf sf2 sf3 sf4
sfsf2sf3
sf4
Bfs 2
Bfs 2
Bfs 2
Sampling Theorem
![Page 10: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/10.jpg)
Department of Electrical and Computer Engineering
Sampling frequency should be at least equal to or greater
than twice the bandwidth of the message signal for
successful recovery of the signal from it’s samples
Bfs 2
Sampling Theorem
f
sT
1
BBsf sf2 sf3 sf4
sfsf2sf3
sf4
Lowpass Filter
bandwidth of lowpass filter = B Hz
![Page 11: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/11.jpg)
Department of Electrical and Computer Engineering
A practical problem with Sampling
t
k
skTttx )()(
sT
t
)(tm t
)(tms
][kX
)( fM
)( fM s
s
sT
f1
Sampling Frequency
Not practical to generate!
![Page 12: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/12.jpg)
Department of Electrical and Computer Engineering
Let’s Try Square Wave
t
)(tx
sT
t
)(tm t
)(tms
][kX
)( fM
)( fM s
s
sT
f1
Sampling Frequency
Problem Resolved1
![Page 13: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/13.jpg)
Department of Electrical and Computer Engineering
t
)(tx
sT
][kX We need Fourier Series
)2
sin(4
][k
kkX c
2
1]0[ X
1
0][ kX s
We know
))2sin(][)2cos(][(]0[)(1
tkfkXtkfkXXtx sss
k
k
c
and
)2cos()2
sin(4
2
1)(
1
tkfk
ktx s
k
k
![Page 14: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/14.jpg)
Department of Electrical and Computer Engineering
k
k
stkftmk
ktmtxtm
1
)2cos()()2
sin(4
)(2
1)()(
We now have x(t) as frequency representation via Fourier Series
And m(t) as Frequency representation via Fourier Transform
)()( fMtm
Let’s multiply m(t) with x(t)
What is the frequency representation of m(t)x(t)?
)2cos()2
sin(4
2
1)(
1
tkfk
ktx s
k
k
![Page 15: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/15.jpg)
Department of Electrical and Computer Engineering
k
k
stkftmk
ktmtxtm
1
)2cos()()2
sin(4
)(2
1)()(
We need to take Fourier Transform
])2cos()()2
sin(4
)(2
1[)]()([
1
k
k
stkftmk
ktmFtxtmF
))]()((2
1)[
2sin(
4)(
2
1)(
1
s
k
k
ss kffMkffMk
kfMfM
))]()()[(2
sin(2
)(2
1
1
s
k
k
s kffMkffMk
kfM
![Page 16: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/16.jpg)
Department of Electrical and Computer Engineering
f
)( fM
BB
)]()(sin([4
)(2
1)()]()([
1
s
k
k
ss kffMkffMk
fMfMtxtmF
f
2
1
BB
)( fM s
sf sf2 sf3 sf4sf
sf2sf3sf4
![Page 17: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/17.jpg)
Department of Electrical and Computer Engineering
f
)( fM
BB
)]()(sin([4
)(2
1)()]()([
1
s
k
k
ss kffMkffMk
fMfMtxtmF
f
2
1
BB
)( fM s
sf sf2 sf3 sf4sf
sf2sf3sf4
Lowpass Filter
Sampling Theorem still works with practical samples
![Page 18: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/18.jpg)
Department of Electrical and Computer Engineering
Distortion and Linear systems
)(*)()( thtxty
)(ty)(tx )(th
)(th - Impulse Response of a Linear Time Invariant System
)()()( HXY )()( Hth
![Page 19: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/19.jpg)
Department of Electrical and Computer Engineering
Distortionless System
)()( dttkxty
dtjekXY
)()(
dtjkeH
)(
)()()( HXY we know
kH )(
dtH )(
![Page 20: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/20.jpg)
Department of Electrical and Computer Engineering
Filters
centerfcenterf
Bandpass Filter
f
)( fH
offcutf offcutf
Lowpass Filter
f
offcutf offcutf
Highpass Filter
f
Cut-off Frequency
Cut-off Frequency
Center Frequency
Bandwidth
B
)( fH
)( fH
![Page 21: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/21.jpg)
Department of Electrical and Computer Engineering
Lowpass and Highpass combined to generate Bandpass Filter
centerfcenterf
Highpass Filter
f
)( fH
LPcf ,
Lowpass Filter
f
HPcLPc ffB ,,
)( fH
HPcf ,LPcf ,HPcf ,
B 2
,, HPcLPc
center
fff
![Page 22: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/22.jpg)
Department of Electrical and Computer Engineering
Lowpass and Highpass combined to generate Bandstop Filter
stopbandcenterf ,stopbandcenterf ,
Highpass Filter
f
)( fH
LPcf ,
Lowpass Filter
f
LPcHPcstopband ffB ,,
)( fH
HPcf ,LPcf ,HPcf ,
stopbandB 2
,,
,
HPcLPc
stopbandcenter
fff
![Page 23: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/23.jpg)
Department of Electrical and Computer Engineering
How a signal should look like after passing through a filter
![Page 24: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/24.jpg)
Department of Electrical and Computer Engineering
Practical Lowpass Filter
)(tx C
R
)(ty
)(ti
RCjH
1
1)(
Basic Lowpass Filter
![Page 25: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/25.jpg)
Department of Electrical and Computer Engineering
R=1, 000 Ohm
R=10,000 Ohm
R=1, 000 Ohm
R=10,000 Ohm
C=1 mF
C=1 mF
Ma
gn
itu
de
Ph
ase
(ra
d)
Frequency in rads/sec
sec1mRC
sec10mRC
![Page 26: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/26.jpg)
Department of Electrical and Computer Engineering
)(tx C
R
)(ty
)(ti
RCjH
1
1)(
Single Stage Lowpass Filter
C
R
)(ti
C
R
)(ti
Two Stage Lowpass Filter
)1
1)(
1
1()(
RCjRCjH
![Page 27: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/27.jpg)
Department of Electrical and Computer Engineering
R=1000 Ohm
C=1 mF
R=1000 Ohm
C=1 mF
Single Stage
Dual Stage
Single Stage
Dual Stage
Ma
gn
itu
de
Ph
ase
(ra
d)
Frequency in rads/sec
![Page 28: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/28.jpg)
Department of Electrical and Computer Engineering
Practical Bandpass Filter
)(tx C
R
)(ty
)(ti
)/1()/()(
/)(
2 LCRCjj
RCjH
Basic Bandpass Filter
L
)/1(0 LC
![Page 29: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/29.jpg)
Department of Electrical and Computer Engineering
Magnitude and Phase Response of BPF
![Page 30: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/30.jpg)
Department of Electrical and Computer Engineering
D=12
1
][nx ][ny
][]1[2
1][ nxnyny
Discrete Time Filters
][2
1][ nunh
n
Impulse Response
Fe
FH2
2
11
1)(
Transfer Function by taking DTFT of h[n]
![Page 31: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/31.jpg)
Department of Electrical and Computer Engineering
CT vs. DT Lowpass Filters
)(tx C
R
)(ty
)()()(
txtydt
tdyRC
Continuous Time
)(ti
)()()(
tthdt
tdhRC
D=12
1
][nx ][ny
][]1[2
1][ nxnyny
Discrete Time
][]1[2
1][ nnhnh
)()( / tueth RCt ][2
1][ nunh
n
![Page 32: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/32.jpg)
Department of Electrical and Computer Engineering
][2
1][ nunh
n
)()( / tueth RCt
t n
F
)(FH
11f
)( fH
CT vs. DT Lowpass Filters … cont.
Time Response
Frequency Response
![Page 33: Application of Combined Fourier Series Transform (Sampling ...ihayee/Teaching/ee2111/ece2111_chapters_6_7.pdf · Department of Electrical and Computer Engineering Application of Combined](https://reader031.vdocuments.site/reader031/viewer/2022041403/5e17b4f1a963cf2d8c1f1d54/html5/thumbnails/33.jpg)
Department of Electrical and Computer Engineering
F
)(FH
11f
)( fH
F
)(FH
11f
)( fH
CT vs. DT Lowpass Filters … cont.
Magnitude Response
Phase Response