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Advanced Digital Signal Processing. Prof. Nizamettin AYDIN naydin @ yildiz .edu.tr http:// www . yildiz .edu.tr/~naydin. An example: Processing Complex Quadrature Signals. Quadrature Signals. Quadrature signals are based on the notion of complex numbers - PowerPoint PPT Presentation
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Prof. Nizamettin AYDIN
http://www.yildiz.edu.tr/~naydin
Advanced Digital Signal Processing
1
Amplitude ModulationAmplitude Modulation
2
• Review of FT properties– Convolution <--> multiplication– Frequency shifting
• Sinewave Amplitude Modulation– AM radio
• Frequency-division multiplexing– FDM
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
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
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
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 )
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 ))
Ideal Lowpass Filter
Hlp( j )
co co
y(t) x(t) if 0 co
y(t) 0 if 0 co
Ideal LPF: Fourier Series
y(t) 4
sin 50t 4
3sin 150t
fco "cutoff freq."
H( j ) 1 co
0 co
The way communication systems work
How do we sharebandwidth ?
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
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
)()()()()()( 21 jPjXjYtptxty
)()()()(
)cos()()(
21
cc
c
jXjY
ttxty
)()()(
)cos()(
cc
c
jP
ttp
))(())(()( 21
21
cc jXjXjY
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 ))
))(())(()(
)cos()()(
21
21
cc
c
jXjXjY
ttxty
)(
))sin((
)(
))sin(()(
)cos()()(
c
c
c
c
c
TTjY
ttxty
)(
)sin(2)(
0
1)(
T
jXTt
Tttx
x(t)
c c
))((21
cjX ))((21
cjX
))(())(()(
)cos()()(
21
21
cc
c
jXjXjY
ttxty
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 ))
DSBAM Waveform
• In the time-domain, the “envelope” of sine-wave peaks follows |x(t)|
Double Sideband AM (DSBAM)
“Typical” bandlimitedinput signal
Frequency-shiftedcopies Upper sideband
Lower sideband
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)
DSBAM Demodulation
V ( j) H( j)W( j ) X( j) if b co 2c b
H( j ) 2 | |co
0 | |co
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)
FDM Block Diagram (Xmitter)
cos(c1t)
cos(c2t)
c1 c2
Spectrum of inputsmust be bandlimited
Need c2 c1 2b
Frequency-Division De-Mux
cos(c1t)
cos(c2t)
c1 c2
Bandpass Filters for De-Mux
Pop Quiz: FT thru LPF
k
kjXtx )30(4)()(Input
cofor a value find then,2)( isoutput theIf ty
1
coco
)(LP jH
Sampling and ReconstructionSampling and Reconstruction
(Fourier View)(Fourier View)
28
• 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
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 )
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
DSBAM: Frequency-Domain
“Typical” bandlimitedinput signal
Frequency-shiftedcopies
))((21
cj jXe ))((2
1c
j jXe
Upper sidebandLower sideband
)( jX
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
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
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
Quadrature Modulation: 4 sigs
8700 Hz
3600 Hz
Ideal C-to-D Converter
• Mathematical Model for A-to-D
x[n] x(nTs )
FOURIERTRANSFORMof xs(t) ???
Periodic Impulse Train
s 2Ts
k
tjkk
ns
seanTttp )()(
s
T
T
tjk
sk T
dtetT
as
s
s1
)(1
2/
2/
Fourier Series
FT of Impulse Train
k
ssn
s kT
jPnTttp )(2
)()()(
ss T
2
Impulse Train Sampling
xs (t) x(t) (t nTs )n
x(t) (t nTs )
n
xs (t) x(nTs ) (t nTs )n
Illustration of Samplingx(t)
x[n] x(nTs )
n
sss nTtnTxtx )()()(
n
t
Sampling: Freq. Domain
EXPECTFREQUENCYSHIFTING !!!
k
tjkk
ns
seanTttp )()(
k
tjkk
sea
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
Frequency-Domain Representation of Sampling
Xs ( j) 1
TsX( j(
k
ks ))
“Typical”bandlimited signal
Aliasing Distortion
• If s < 2b , the copies of X(j) overlap, and we have aliasing distortion.
“Typical”bandlimited signal
Reconstruction of x(t)
xs (t) x(nTs ) (t nTs )n
Xs ( j ) 1
TsX( j(
k
ks ))
Xr ( j) Hr ( j)Xs ( j )
Reconstruction: Frequency-Domain
)()()(so overlap,not do )(of copies the,2 If
jXjHjXjX
srr
bs
Hr ( j)
Ideal Reconstruction Filter
hr (t) sin
Tst
Tst
Hr ( j) Ts
Ts
0 Ts
hr (0) 1
hr (nTs ) 0, n1,2,
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
Shannon Sampling Theorem
• “SINC” Interpolation is the ideal– PERFECT RECONSTRUCTION– of BANDLIMITED SIGNALS
Reconstruction in Time-Domain
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 ))