Advanced Digital Signal Processing
description
Transcript of Advanced Digital Signal Processing
![Page 1: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/1.jpg)
Prof. Nizamettin AYDIN
http://www.yildiz.edu.tr/~naydin
Advanced Digital Signal Processing
1
![Page 2: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/2.jpg)
Amplitude ModulationAmplitude Modulation
2
![Page 3: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/3.jpg)
• Review of FT properties– Convolution <--> multiplication– Frequency shifting
• Sinewave Amplitude Modulation– AM radio
• Frequency-division multiplexing– FDM
![Page 4: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/4.jpg)
Table of Easy FT Properties
ax1(t) bx2 (t) aX1( j ) bX2 ( j )
x(t td ) e jtd X( j )
x(t)e j0t X( j( 0 ))
Delay Property
Frequency Shifting
Linearity Property
x(at) 1|a | X( j(
a ))Scaling
![Page 5: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/5.jpg)
Table of FT Properties
x(t)h(t) H( j )X( j )
x(t)e j0t X( j( 0 ))
x(t)p(t) 1
2X( j )P( j )
dx(t)
dt ( j)X( j)
Differentiation Property
![Page 6: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/6.jpg)
Frequency Shifting Property
e j0 t x(t )e j tdt
x(t)e j ( 0 )t dt
X( j( 0))
x(t)e j0t X( j( 0 ))
y(t) sin 7t
te j 0 t Y ( j )
1 0 7 07
0 elsewhere
![Page 7: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/7.jpg)
Convolution Property
• Convolution in the time-domain
corresponds to MULTIPLICATIONMULTIPLICATION in the frequency-
domain
y(t) h(t) x(t) h( )
x(t )d
Y( j ) H( j )X( j )
y(t) h(t) x(t)x(t)
Y( j ) H( j )X( j )X( j )
![Page 8: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/8.jpg)
Cosine Input to LTI System
Y ( j) H( j )X( j)
H( j )[( 0 ) ( 0)]
H( j0 ) ( 0 ) H( j0 ) ( 0 )
y(t) H (j0 ) 12 e
j0t H( j0 ) 12 e
j 0t
H( j0 ) 12 e
j0t H *( j 0)12 e
j0t
H( j0 ) cos( 0t H( j0 ))
![Page 9: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/9.jpg)
Ideal Lowpass Filter
Hlp( j )
co co
y(t) x(t) if 0 co
y(t) 0 if 0 co
![Page 10: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/10.jpg)
Ideal LPF: Fourier Series
y(t) 4
sin 50t 4
3sin 150t
fco "cutoff freq."
H( j ) 1 co
0 co
![Page 11: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/11.jpg)
The way communication systems work
How do we sharebandwidth ?
![Page 12: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/12.jpg)
Table of FT Properties
x(t)h(t) H( j )X( j )
x(t)e j0t X( j( 0 ))
x(t)p(t) 1
2X( j )P( j )
dx(t)
dt ( j)X( j)
Differentiation Property
![Page 13: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/13.jpg)
Signal Multiplier (Modulator)
• Multiplication in the time-domain corresponds to convolution in the frequency-domain.
Y( j ) 1
2X( j )P( j )
y(t) p(t)x(t)
X( j)
x(t)
p(t)
Y( j ) 1
2X( j )
P( j( ))d
![Page 14: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/14.jpg)
)()()()()()( 21 jPjXjYtptxty
)()()()(
)cos()()(
21
cc
c
jXjY
ttxty
)()()(
)cos()(
cc
c
jP
ttp
))(())(()( 21
21
cc jXjXjY
![Page 15: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/15.jpg)
Amplitude Modulator
• x(t) modulates the amplitude of the cosine wave. The result in the frequency-domain is two shifted copies of X(j).
y(t) x(t)cos(ct)
X( j)
x(t)
cos(ct)Y( j ) 1
2X( j( c ))
12X( j( c ))
![Page 16: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/16.jpg)
))(())(()(
)cos()()(
21
21
cc
c
jXjXjY
ttxty
)(
))sin((
)(
))sin(()(
)cos()()(
c
c
c
c
c
TTjY
ttxty
)(
)sin(2)(
0
1)(
T
jXTt
Tttx
![Page 17: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/17.jpg)
x(t)
c c
))((21
cjX ))((21
cjX
))(())(()(
)cos()()(
21
21
cc
c
jXjXjY
ttxty
![Page 18: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/18.jpg)
DSBAM Modulator
• If X(j)=0 for ||>b and c >b,the result in the frequency-domain is two shifted and scaled exact copies of X(j).
y(t) x(t)cos(ct)
X( j)
x(t)
cos(ct)Y( j ) 1
2X( j( c ))
12X( j( c ))
![Page 19: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/19.jpg)
DSBAM Waveform
• In the time-domain, the “envelope” of sine-wave peaks follows |x(t)|
![Page 20: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/20.jpg)
Double Sideband AM (DSBAM)
“Typical” bandlimitedinput signal
Frequency-shiftedcopies Upper sideband
Lower sideband
![Page 21: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/21.jpg)
DSBAM DEmodulator
w(t) x(t)[cos(ct)]2 1
2x(t) 1
2x(t)cos(2ct)
W( j ) 1
2X( j) 1
4X( j( 2c )) 1
4X( j( 2c ))
V ( j) H( j)W( j )
w(t) v(t)x(t)
cos(ct) cos(ct)
y(t) x(t)cos(ct)
![Page 22: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/22.jpg)
DSBAM Demodulation
V ( j) H( j)W( j ) X( j) if b co 2c b
H( j ) 2 | |co
0 | |co
![Page 23: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/23.jpg)
Frequency-Division Multiplexing (FDM)
• Shifting spectrum of signal to higher frequency:– Permits transmission of low-frequency signals with
high-frequency EM waves– By allocating a frequency band to each signal
multiple bandlimited signals can share the same channel
– AM radio: 530-1620 kHz (10 kHz bands)– FM radio: 88.1-107.9 MHz (200 kHz bands)
![Page 24: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/24.jpg)
FDM Block Diagram (Xmitter)
cos(c1t)
cos(c2t)
c1 c2
Spectrum of inputsmust be bandlimited
Need c2 c1 2b
![Page 25: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/25.jpg)
Frequency-Division De-Mux
cos(c1t)
cos(c2t)
c1 c2
![Page 26: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/26.jpg)
Bandpass Filters for De-Mux
![Page 27: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/27.jpg)
Pop Quiz: FT thru LPF
k
kjXtx )30(4)()(Input
cofor a value find then,2)( isoutput theIf ty
1
coco
)(LP jH
![Page 28: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/28.jpg)
Sampling and ReconstructionSampling and Reconstruction
(Fourier View)(Fourier View)
28
![Page 29: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/29.jpg)
• Sampling Theorem Revisited– GENERAL: in the FREQUENCY DOMAIN
– Fourier transform of sampled signal
– Reconstruction from samples
• Review of FT properties– Convolution multiplication– Frequency shifting
– Review of AM
![Page 30: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/30.jpg)
Table of FT Properties
x(t td ) e jtd X( j )
x(t)e j0t X( j( 0 ))
Delay Property
Frequency Shifting
x(at) 1|a | X( j(
a ))Scaling
x(t)h(t) H( j )X( j )
![Page 31: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/31.jpg)
Amplitude Modulator
• x(t) modulates the amplitude of the cosine wave. The result in the frequency-domain is two SHIFTED copies of X(j).
y(t) x(t)cos(ct )
X( j)
x(t)
cos(ct )
Y (j) 12 e
jX( j( c))
12 e
jX( j( c))Phase
![Page 32: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/32.jpg)
DSBAM: Frequency-Domain
“Typical” bandlimitedinput signal
Frequency-shiftedcopies
))((21
cj jXe ))((2
1c
j jXe
Upper sidebandLower sideband
)( jX
![Page 33: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/33.jpg)
DSBAM Demod Phase Synch
w(t) v(t)x(t)
cos(ct) )cos( tc
)cos()()( ttxty c
))2(())2((
)()()(
41
41
41
41
cj
cj
jj
jXejXe
jXejXejW
? ifwhat )()cos()( 21
21 jXjV
![Page 34: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/34.jpg)
Quadrature Modulator
TWO signals on ONE channel: “out of phase” Can you “separate” them in the demodulator ?
))(())((
))(())(()(
22121
22121
cj
c
cj
c
jXjX
jXjXjY
![Page 35: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/35.jpg)
Demod: Quadrature System
)cos( tc
)()(
)()()(
22/
41
141
22/
41
141
jXeejXe
jXeejXejVjjj
jjj
0 if )()( 1 txtv
2/ if )()( 2 txtv
))(())((
))(())(()(
22121
22121
cj
c
cj
c
jXjX
jXjXjY
![Page 36: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/36.jpg)
Quadrature Modulation: 4 sigs
8700 Hz
3600 Hz
![Page 37: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/37.jpg)
Ideal C-to-D Converter
• Mathematical Model for A-to-D
x[n] x(nTs )
FOURIERTRANSFORMof xs(t) ???
![Page 38: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/38.jpg)
Periodic Impulse Train
s 2Ts
k
tjkk
ns
seanTttp )()(
s
T
T
tjk
sk T
dtetT
as
s
s1
)(1
2/
2/
Fourier Series
![Page 39: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/39.jpg)
FT of Impulse Train
k
ssn
s kT
jPnTttp )(2
)()()(
ss T
2
![Page 40: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/40.jpg)
Impulse Train Sampling
xs (t) x(t) (t nTs )n
x(t) (t nTs )
n
xs (t) x(nTs ) (t nTs )n
![Page 41: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/41.jpg)
Illustration of Samplingx(t)
x[n] x(nTs )
n
sss nTtnTxtx )()()(
n
t
![Page 42: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/42.jpg)
Sampling: Freq. Domain
EXPECTFREQUENCYSHIFTING !!!
k
tjkk
ns
seanTttp )()(
k
tjkk
sea
![Page 43: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/43.jpg)
Frequency-Domain Analysis
xs (t) x(t) (t nTs )n
x(nTs ) (t nTs )
n
xs (t) x(t) 1Tsk
e jkst 1
Tsx(t)
k
e jkst
Xs ( j) 1
TsX( j(
k
ks ))
s 2Ts
![Page 44: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/44.jpg)
Frequency-Domain Representation of Sampling
Xs ( j) 1
TsX( j(
k
ks ))
“Typical”bandlimited signal
![Page 45: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/45.jpg)
Aliasing Distortion
• If s < 2b , the copies of X(j) overlap, and we have aliasing distortion.
“Typical”bandlimited signal
![Page 46: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/46.jpg)
Reconstruction of x(t)
xs (t) x(nTs ) (t nTs )n
Xs ( j ) 1
TsX( j(
k
ks ))
Xr ( j) Hr ( j)Xs ( j )
![Page 47: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/47.jpg)
Reconstruction: Frequency-Domain
)()()(so overlap,not do )(of copies the,2 If
jXjHjXjX
srr
bs
Hr ( j)
![Page 48: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/48.jpg)
Ideal Reconstruction Filter
hr (t) sin
Tst
Tst
Hr ( j) Ts
Ts
0 Ts
hr (0) 1
hr (nTs ) 0, n1,2,
![Page 49: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/49.jpg)
Signal Reconstruction
xr (t) hr (t) xs (t) hr (t) x(nTs ) (t nTs )n
xr (t) x(nTs )sin
Ts(t nTs )
Ts
(t nTs )n
Ideal bandlimited interpolation formula
xr (t) x(nTs )hr (t nTs )n
![Page 50: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/50.jpg)
Shannon Sampling Theorem
• “SINC” Interpolation is the ideal– PERFECT RECONSTRUCTION– of BANDLIMITED SIGNALS
![Page 51: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/51.jpg)
Reconstruction in Time-Domain
![Page 52: Advanced Digital Signal Processing](https://reader036.fdocuments.us/reader036/viewer/2022062322/56814dbf550346895dbb1888/html5/thumbnails/52.jpg)
Ideal C-to-D and D-to-C
x[n] x(nTs )xr (t) x[n]
sin Ts
(t nTs )Ts
(t nTs )n
Ideal Sampler Ideal bandlimited interpolator
Xr ( j) Hr ( j)Xs ( j )Xs ( j) 1
TsX( j(
k
ks ))