Prof. Nizamettin AYDIN

naydin@yildiz.edu.tr

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 ))