通訊原理

303
Chap. 2 1 Signal Models Deterministic signals vs. Random signals Deterministic Signals Completely being specified by functions of time. Predictable CHAP. 2 Signals & Linear Systems 0 , no ascertainty. e.g. cos , 1 1, 2 0, otherwise. Random signals take on random values at any given time instance mus xt A t t t t t be modeled probabilistically.

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Communication principle

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Page 1: 通訊原理

Chap. 2 11

Signal Models

Deterministic signals vs. Random signals Deterministic Signals

Completely being specified by functions of time. Predictable

CHAP. 2 Signals & Linear Systems 0

, no ascertainty. e.g. cos ,

11, 2

0, otherwise. Random signals

take on random values at any given time instance mus

x t A t t

tt

t be modeled probabilistically.

Page 2: 通訊原理

Chap. 2 22

0

0

Periodic vs. Aperiodic Peridic signals: A signal is periodic iff. there

exist a fundamental period , such that , , and

Aperiodic signals: Sig

x t

Tx t nT x t t n

0 0 00 0

nals excluding the periodic ones.

e.g.

2 2 cos cos , x t A t A t T

Page 3: 通訊原理

Chap. 2 33

0 0

Phasor signals and their Spectra

A special peridic function

,~ complex sinusoidal fn.

rotating phasor;

0 : magnitude : phasor with

j t j tj

j

x t Ae Ae e

x t

AAe

0*

0 0

, Note: : phase

More on the phasor signal

1. Given a fixed frequency or , information can be contained in and . 2.The related real sinusoidal fu

j tx t Ae

fA

*0

1

nction:

Re 2

x t Acos t x t x t x t

Page 4: 通訊原理

Chap. 2 44

3. Graphical Representation: Two ways of relating a phasor signal to a real sinusoidal signal

a projection of a rotating phasor onto the real axis.

b Addition of complex conjugate rotating phasors. 0j tx t Ae 12

x t 12

x t

Page 5: 通訊原理

Chap. 2 55

0 4. Frequency domain representation: cos 2A f t single sided spectra double sided spectra

Page 6: 通訊原理

Chap. 2 66

0 0

0 0

0

0

Singular Functions.singularity; 0

Unit Impulse function , 0 ; otherwise

1. Defined by

0 0 ;

1.

2. Sh

tt

x t t dt x t t dt x t dt x

t dt

0 0

ifting property:

x t t t dt x t 0 t

t0 t

0t t 0t

Page 7: 通訊原理

Chap. 2 77

2

1

2

1

0 1 0 2

0 0 0

0

0

3. Other properties:

1 1

2 even function

, 3

0 , otherwise

4

5 1

t

n

t

t

t n n

at ta

t t

x t t t tx t t dt

x t t t x t t t

x t t t dt x t

t

0 1 0 2,

where n

n

t t t

dt tdt

Page 8: 通訊原理

Chap. 2 88

2 2

2

1, 0 Unit step function,

0, 0

.

Signal Classifications:

Energy: lim Jouels

1 Power: lim Watts2

o

,

:

r

T

t

TT

T

TT

tu t

t

E x t dt x t

du tu t d t

dt

dt

P x t dtT

x t

Energy signal: iff. 0 , 0

Power signal: iff. 0 ,

E P

P E

Page 9: 通訊原理

Chap. 2 99

0

1

2

0

Examples: , 0, Energy signal

, power signal

, power signal

Periodic signals, Power signals

cos , power signal

t

j tj

p

x t Ae u t

x t Au t

x t Ae e

x t A t

Page 10: 通訊原理

Chap. 2 1010

0 0

0

0 0 0 00

0

periodic sig

Fourier series Synthesis & Analysis

nal

1

Eular's Formula: cos sin1 Given a , let , 2 ,

, where

j

jn tn n

n

jn t

T

e j

x t x t T f fT

X xx t eT

e dtt X

0

0

.

Frequency components Double-sided amplitude and phase line spectra. ,~ a complex number

: D.C. component: Amplitude of :

:: phase of .

n

n

j Xn n

n nn

nn n

X X e

XX XX

XX X harmonics

Page 11: 通訊原理

Chap. 2 1111

0

0

2

0

Power Spectral Density : , Define the time-average :

1lim , if is aperiodic2 1 , if is periodic

Parseval's theorem:

n

T

TT

T

jn tn

n

X n

v t dt v tTv tv t dt v t

T

x t X e

0

2 2 2

0

Ex. Please prove the Parseva

1

l's Thm.

nTn

P x t x t dt XT

Page 12: 通訊原理

Chap. 2 1212

Page 13: 通訊原理

Chap. 2 1313

*

Symmetric properties of Fourier coefficients

1. If , then its magnitude spectrum is even symmetric

and its phase spectrum is odd symmetric.

is real

, , nn n

x t

X Xx t X X even

odd

0

2. is a real, and ,

3. is a real, and , is imaginary,

1 4. If , halfwave symmetry, eg. square wa2

n n

n n

n

n

n

n

x t x t x t X X

x t x t x t X X X

x t x

X

T

X

t

ve

0, 0, 2, 4,...nX n 0

2T 0T

A

A

Page 14: 通訊原理

Chap. 2 1414

0

2T 0T

A

A 0

00

, 02. The square wave ,

, 2

Find the Fourier series expansion of .

TA tEx x t

TA t T

x t

Page 15: 通訊原理

Chap. 2 1515

Page 16: 通訊原理

Chap. 2 1616

0

12

1

1 harmonic:j nf t

n

t

n

s

X e 0

32

3

Sum up to the 3 harmonic

j nf tn

n

rd

X e 0

72

7

Sum up to the 7 harmonic

j nf tn

n

th

X e 0

112

11

Sum up to the 11 harmonic

j nf t

th

nn

X e 02j nf tn

n

x t X e

Page 17: 通訊原理

Chap. 2 1717

21 2

The Fourier transform Analysis & Synthesis: ~ an function

~ Fourier transform

aperiodi

cj ft

j ft

x t

x t X f x t e dt

X f x t X f e df

F

F ~ Inverse Fourier transform

Amplitude and phase spectra

, 0,

If is real,

j fX f X f e X f f X f

X f X fx t

f f

Page 18: 通訊原理

Chap. 2 1818

2

2 2

Symmetry Properties

1 and , and

is imaginary 2 and ,

Energy Spectral Density:

Energy

Rayleigh's energy

x t x t x t X f X f X f

X fx t x t x t

X f X f

E x t dt X f df G f

G f

df

X f

1 2 1 2 1 2

thm. Parseval's thm for Fourier transforms Convolution:

x t x t x x t d x t x d

Page 19: 通訊原理

Chap. 2 1919

0

1 1 2 2 1 1 2 2

20

Transformation theorems

Superposition:

Time-delay:

1 Scale-change:

Duality: ,

Frequency

j ft

a x t a x t a X f a X f

x t t X f e

fx at Xa a

x t X f X t x f

F

F

F

F F 020Translation:

The sinc functionsin

sin: c

j f t

z

x t e X f f

zz

F

Page 20: 通訊原理

Chap. 2 2020

-1 0 1-1012

Time (ms)-5 0 5

0

1

2

Frequency (k Hz)

-1 0 1-1012

Time (ms)-5 0 5

0

1

2

Frequency (k Hz)

-1 0 1-1012

Time (ms)-5 0 5

0

1

2

Frequency (k Hz)

t 1X f

1 2

0.5 1x t

1 2tx 1 2x t

1 fx at Xa a

F

Page 21: 通訊原理

Chap. 2 2121

0 0 0

1 2 1

lim 0

2

1 1 Modulation: cos 22 2

Differentiation: 2

1 Integration: 02 2

Convolution:

Multiplic

t

nn

xn

t

t

x t f t X f f X f f

d x tj f X f

dt

X fx d X f

j f

x t x t X f X f

F

F

F

F 1 2 1 2ation: x t x t X f X fF Fourier transform pairs see Appendix G.5 P.724

Page 22: 通訊原理

Chap. 2 2222

1 2 1 2

. Prove the following 1 a. scaling theorem:

b. Convolution theorem:

Exfx at X

a a

x t x t X f X fF

Page 23: 通訊原理

Chap. 2 2323

Page 24: 通訊原理

Chap. 2 2424

22

2

Example: 40sinc 20 , ?

1, ;2

0, o.w.

sin sinc2

Duality: ,

40sinc 2

s c

0

n

i

j f j fj ft

x t t X f

tt

t e e fe dt fj f f

x t X f

t

t

f

F

F

F 2

2

2

220

Energy Spectra Density: 420

Energy: 1600sinc 20 4 80 .20

x t

f

fG f X f

fE t dt G f df df J

Page 25: 通訊原理

Chap. 2 2525

Fourier transforms of periodic signal

Strictly speaking, the Fourier transform of a periodic signal does not exist since a periodic signal is a power signal with infinit

e energy.

By taking advantage of the Fourier series expansion, and incorporating with the use of delta function, the Fourier transform of a periodic signal can be expressed as

0

0

0

0

2

20

20

below ,~ periodic

, Fourier series expansion

,

jn f

jn f tn

n

jn f tn n

n n

te f n

x t x t T

x t X e

X f X e X f nf

fF

F

Page 26: 通訊原理

Chap. 2 2626

2

20

§ Power Spectral Density and Correlation Power Spectral Density PSD :

,

is a real, even-symmetric and non-negative function of .

Given , j nfn

S f

P S f df x t

S f f

x t x t T x t X e 0

20

0

2 2

,

so that is satisfied.

Example:

cos( ) ,2 2

4 4

,

o o

t

n

j t j t

o o

nn

A Ax t

S f X f nf

A t e e

A AS f f f f f

Page 27: 通訊原理

Chap. 2 2727

1 1 *

*

2

Time-average autocorrelation function for

,

For a real , ,

i

l

0

m

T

TT

G f X f X f

X f xx x x

X f x

x

x t x t dt x t x t dt

energy signals

F F t dt E

Page 28: 通訊原理

Chap. 2 2828

00

2

Time-average autocorrelation function for

1

1lim2

If is periodic,

0 , ~ PS

Wiener-Khinchine theor

D

e

T

T

TTR x t x t

x t R x t x t dtT

R x t P S f df S

x t x t dtT

f

power signals ★ 2

1 2

m:

= ,

=

j f

j f

R S f

S f R R e d

R S f S f e df

F

F

F

Page 29: 通訊原理

Chap. 2 2929

2

2

0 0

Properties of :

1. 0 , .

2. . even symmetric to

3. lim , if contains no periodic components.

4.

5. 0, .

R

R x t R

R x t x t R

R x t x t

x t x t T R R T

S f R f

F

Page 30: 通訊原理

Chap. 2 3030

0

2 2

0 0

21

0

2

0

Example: cos , ?

Sol.

1. By the Wiener-Khinchine thm.

,4 4

cos2

2. By direct calculation, 1 cos

x t A t R

A AS f f f f f

AR S f

R AT

F 0

0 0

0 0 0

2

0 0 00

2

0

cos

cos cos 2 22

cos2

T

T T

t t dt

A dt t dtT

A

Page 31: 通訊原理

Chap. 2 3131

1 1 2 2 1 1 2 2

Linear-Time-Invariant Systems

1.

Linear and Time-Invariant LTI systems: Linear Systems:

Satisfy the superposition principle:

y t H x t

y t H x t x t H x t H x t § 1 1 2 2

0 0

Time-Invariant Systems:

Delayed inputs produce the delayed outputs

,

y t y t

y t H x t H x t t y t t

x t y t H x t H

Page 32: 通訊原理

Chap. 2 3232

0

0

LTI systems:

Impulse response:

lim

lim

n

n

h t H t

x t x t t x t d

x n t n

y t H x t H x n t n

time-invariant, ( )

0linear, = lim

=

n

h t n

x n H t n

x h t d x t h t

t h tLTI x t y t x t h t LTI

Page 33: 通訊原理

Chap. 2 3333

Convolution theorem : For a LTI system, *

, ~ Transfer function.

BIBO Stability :

BIBO : Bounded Input Bounded Output

A LTI sys

y t x t h t Y f X f H f

H f h t

F

F Bounded Input

finite

tem is BIBO ( ) < ,

h t dt

y t x h t d

x h t d x d h t d

Page 34: 通訊原理

Chap. 2 3434

0

Causality

1. A system is causal if current output does not depend on future input. 2. For a causal system, 0, for 0

h t t

y t h x t d

h x t d

Symmetry properties of

: amplitude-response 1. ,

: phase-response

2. is real,

j H f

H f

H fH f H f e

H f

H f H fh t

H f H f

t h t

t

t

x t

0 t h tLTI

Page 35: 通訊原理

Chap. 2 3535

. 1. Find the transfer function of the following RC circuit.

12 2. Let the input ( ) , ?

Ex H f

t Tx t A y t

T

Page 36: 通訊原理

Chap. 2 3636

R

C x t y t

i t

Page 37: 通訊原理

Chap. 2 3737

Page 38: 通訊原理

Chap. 2 3838

0 T A x

0 t T 1 u t

0t 1 u t t T 1 u t

Page 39: 通訊原理

Chap. 2 3939

2

* *

2 2

Input-Output Relations for Spectra

1. ,

y

Y f H f X f

G f Y f Y f Y f

H f X f H f X f

H f X f

2

2

2. A similar relationship hold for power signal and spectra :

x

y x

H f G f

S f H f S f

Page 40: 通訊原理

Chap. 2 4040

0

0

0

0 0

2

2

20

20

Reponse to Periodic Inputs

1. For a complex exponential sinusoidal input =

. . is a scaled and

j f t

j f tss

j f t

j f t H f

ss

x t Ae

y t h Ae d

H f Ae

H f Ae

i e y t

0

0

0

0

2

2

2

amplitude scaling

20 0

phase-shifted copy of the input .

2. For a period

eigen-signal of LTI systems

ic input

.

,

n

j f tn

n

j nf t X H nj f tn n

n

j f t

x t

x t X e

y t X H nf e

Ae

X H nf e 0

phaseshift

f

n

H f

02= j f t

x t

Ae 0

ssy t

H f x t

Page 41: 通訊原理

Chap. 2 4141

/10

0

: Consider a filter having the transfer functon

2 ,42

what is its reponse to a unit-amplitude triangular signal with period 0.1 sec. and amplitude 1.

j ffH f e

TA

Example

2 2

the Fourier coefficients of the 4 ,

triangular signal . 0,

n

A n oddX n

n even 2121 2

f

H f

2121 H f

f

A0

2T0

2T x t

tA

Page 42: 通訊原理

Chap. 2 4242

2121 2

f

H f

2121 H f

f

Page 43: 通訊原理

Chap. 2 4343

0

020

20 0 0

1. Distionless system:

( )

1. The amplitude response has to be a constant for all frequecies.

j

j

f

ft

ty t H x t t Y f H X f e H f X f

H f H e

Distortionless Transmission

F 2. The phase response is a linear function of , for all frequencies. f

f

H f

0H

f

02H f t f 02slope t

Page 44: 通訊原理

Chap. 2 4444

2. Three types of distortions in communication channel:

1. Amplitude distortion: Linear system, but the amplitude

response is not a constant .

2. Phase delay distortion : Linear system, but the phase

shift is not a linear function of frequency.

3. Nonlinear distortion : nonlinear system

Page 45: 通訊原理

Chap. 2 4545

1

2

3

4

1. cos 10 cos 12

2. cos 10 cos 26

3. cos 26 cos 34

4. cos 32 cos 34

x t t t

x t t t

x t t t

x t t t

Δ Example H f iy t ix t

( )H f H f f ( )f Hz ( )f Hz

1515 00

2

2

21

201020 10( )

20 40f fH f , 15

30

, 152

, 15 2

f f

fH f

f

Page 46: 通訊原理

Chap. 2 4646

1

1

2

1 1 1. 2cos 10 2cos 126 5

1 1 1 2cos 10 2cos 12 2 , distortionless60 60 60

1 13 2. 2cos 10 cos 266 30

y t t t

t t x t

y t t t

3

1 1 2cos 10 cos 26 , amplitude distortion60 60

13 3. cos 26 cos 3430 2

1 1 cos 26 cos 3460 68

t t

y t t t

t t

4

, phase distortion

1 1 4. cos 32 cos 342 2

1 1 cos 32 cos 34 , phase distortion64 68

y t t t

t t

Page 47: 通訊原理

Chap. 2 4747

21 2

1 1 2 2

- Consider a nonlinear system with the input-output relation If the input signal is cos cos then the resultant output

y t a x t a

x

x

t A t A t

t Non linear Distortion 1 2

1 1 1 2 2

2 2 2 22 1 2 2 1 2

2 1 2 1 2 1 2

cos cos

1 1 cos 2 cos 22 2

cos cos

The system has produced frequencies in the output

y t a A t A t

a A A a A t A t

a A A t t

other than the original ones.

Page 48: 通訊原理

Chap. 2 4848

0

0

20

0 0

20

1. Ideal Low-pass Filter:

2

2 sinc 2 ,

non-causal

2. Ideal High-pass Filter

12

j ftLP

LP

j ftHP

fH f H eB

h t BH B t t

fH f H eB

• Filters 0 0 0 2 sinc 2HPh t H t t B B t t f

fBB LPH f LPH ff

fBB HPH f HPH f

Page 49: 通訊原理

Chap. 2 4949

021 0 1 0

1 0

0 0 0 0

3. Ideal Band-pass Filter

where

2 sinc cos 2

j ftBP

BP

H f H f f H f f e

fH f HB

h t H B B t t f t t

f

fBB

BPH f BPH f0f 0f

Page 50: 通訊原理

Chap. 2 5050

Page 51: 通訊原理

Chap. 2 5151

Page 52: 通訊原理

Chap. 2 5252

SAMPLING THEORY 1. Ideal instantaneous sampling waveform

s s sn n

s s s sn n

s t t nT S f f f nf

x t x t s t x t t nT x nT t nT

F x t sx t s

n

s t t nT LPH f 1 2LP

s

fH ff B

, if

R

s

x t x tW B f W

x t

t sx t

tsT

Page 53: 通訊原理

Chap. 2 5353

,

Uniform sampling theorem for Low-pass signals

A low-pass signal with finite bandwidth can be co

mple

tel

s

s

s sn

s sn

x t x t s t

X f X f S f

X f f f nf

x

f X f n

t W

f ★ y

1 described by , if .2

The original signal can be exactly reconstructed by passing through an ideal lowpass filter with bandwidth satisf

1 2 Nyquist sampling frequency

yi

s s

s

ss

x t TW

x tB

f WT

ng

sW B f W

Page 54: 通訊原理

Chap. 2 5454

BB s sX f x tWW (0)sf X

sf W sfsf Wsf W

sf W sf X f x tWW 0X

f LPH f1sf R RX f x t

WW 0X

f

f

f

( )x t( )sx t s

n

s t t nT LPH f 12LP

s

fH ff B

, if

R

s

x t x tW B f W

,

2s

s

W B f Wf W

Page 55: 通訊原理

Chap. 2 5555

Aliasing: If is not band-limited on 2 , distortion aliasing of overlapped spectra is inevitable.

sx t f W 1 Increase the sampling freq.

2 Pre-filter the signal with a low-pass filter anti-aliasing filter

solution :solution :

sf

0 fsf s s sn

X f f X f nf

Page 56: 通訊原理

Chap. 2 5656

1, 2 2

0, o.w.

1 , . Please verify 0, o.w.

1

tt

tt tEx

t t

Page 57: 通訊原理

Chap. 2 5757

t 2

21

1

2t

2t

2 2

2

t 2

t t

Page 58: 通訊原理

Chap. 2 5858

2 2

2

t 2

t t 2

2

2t 2

t t

Page 59: 通訊原理

Chap. 2 5959

t t

2

2

2t

2t t

2

sinc

sinc

t f

t f

F

F

Page 60: 通訊原理

Chap. 2 6060

Property:

0, only in the interval , ,

0, only in the interval ,

0, only in ,

x t t a b

y t t c d

x t y t t a c b d

Page 61: 通訊原理

Chap. 3 11

CHAPTER 3 BASIC MODULATION TECHNIQUES Analog modulation Continuous-wave CW modulation: Linear: AM Exponential angle : FM, PM

Pulse modulation sampled data 3.1 Linear Modulation

G

§

eneral form: cos

: one-to-one correspondence with message : fixed cos : carrier

c c

c

c

x t A t t

A t

t

Page 62: 通訊原理

Chap. 3 22

m t cx t rx t d t

cosc cA t 2cos ct Low pass filter

Dy t

Modulator

Demodulator

Double Side-band Modulation Suppressed Carrier : DSB or DSBSC

1.

cos

, 2 2 2

c c c

c c cc c c c

x t A m t t

A AX f M f f M f f f

F

M f

W Wf cM f f cM f f cX f

cf

0

fcf 0

Page 63: 通訊原理

Chap. 3 33

2

the phase and the fre2 Coherent synchronous Demodulator detector :

The receiver exactly knows of the

carrier signal.

2 cos cos

1 cos 2

quenc

2

y

cos 2

c c c

cc c c

d t A m t t t

A m t t A m t

同調 2 = cos 2

message pass the LPF; high-freq. term rejected

. : Upper sideband and lower sideband:

c c c

A B

t

A m t A

A

m t t

Def

B

0

cX f

cff

cf cf Wcf W: upper sideband

: lower sideband

rx t d t

2cos ct Low pass filter

Dy t

Demodulator

Page 64: 通訊原理

Chap. 3 44

M f

W Wf cM f f cM f f cX f

cf D f

2 cf2 cf 0

f

f

cf 0

0

m t cx t rx t d t

cosc cA t 2cos ct Low pass filter

Dy t

Modulator

Demodulator

Page 65: 通訊原理

Chap. 3 55

3 What if the receiver reference is ? A phase error occurs:

cos 2cos

cos + c

non-coherent

os 2

cos , 1 cos 1

is mult

c c c

c c c

D

d t A m t t t t

A m t t A m t t t

y t m t t t

m t

iplicatively distorted by a time-varying

factor cos .t rx t d t Low pass filter 2cos ct t Dy t

Page 66: 通訊原理

Chap. 3 66

2 2 2 2

2 2 2 2

4 Carrier recovery: regenerate the carrier from the receive signal at receiver cos

1 1 cos 2 2 2

r c c

c c c

x t A m t t

A m t A m t t

rx t cos ct cos 2 ct 2 Narrow-band

BPF at 2 cf2f H f

2 cf2 cf

Page 67: 通訊原理

Chap. 3 77

Amplitude Modulation AM :

Also known as DSB with carrier, DSB-Large Carrier (DSB-LC) 1

,~ original message,

,~

co

normalized messmin

s 1 cos ;

c c n c c c

n

x t A m t

m t

m tm t

m t

A t am t A t A AA age;

. min 1, if min 0

min,~ modulation index

nEx m t m t

m ta

A

cx t m t A m tcos cA t

,~ D.C. biasA 1, min ;

over-modulation.

a m t A

Page 68: 通訊原理

Chap. 3 88

2 AM waveforms and spectra

1 cosc n c cx t am t A t M f

f0

cfcf cX f

f0

2 2c c

AA Af f M f f cx t m t A m tcos cA t

,~ D.C. biasA

Page 69: 通訊原理

Chap. 3 99

3 AM demodulation a. Coherent detection: precise but requires carrier recovery circuit very complex . b. Incoherent detection: envelope detection. Simple receiver LPF , but requires sufficient carrier power and

1 . Envelope detection

ca f W

Page 70: 通訊原理

Chap. 3 1010

Modulated carrier and envelope detector outputs for various values of the modulation index

0.5a 1.0a 1.5a

Page 71: 通訊原理

Chap. 3 1111

22 2 2

4 Efficiencypower of the information-bearing signal Efficiency

total power of the transmitted signal For AM: Total power:

1 cos

=

ff

c n c c

E

P x t am t A t 22 2

2 2

22 2

2

1 2 1 cos 22

, 1 2Assume

Assume 02

,

, 1

2c

cn n

cn

nn

c

nc

Aam t

f W

m AP a m

a m t t

AP a m t a m t

t t

Page 72: 通訊原理

Chap. 3 1212

22 2

22 2

max

2 m x

2

a

2

2 22 100% =

12

If max min and 1, then 50%.

Ex. square-wave, 1, = 50%, for 1

If is sinus

10

oi

0%1

d

cn

ffc

n

ff

n ff

n

n

Aa m tE

Aa m t

m t m t a E

m t m t E a

a m t

a t

m

m

t

2 max

2 2 2 2

2 2 2

22

1al, , 33.3%, for 1.2

For DSB-SC: cos

1 1 cos 2 , ,2

, 100% 2

n ff

c c c

c c c c

cff

m t E a

x t A m t t

x t A m t t assume f W

A m t E

Page 73: 通訊原理

Chap. 3 1313

Example:

Modulation index 0.5 Carrier power = 50 Watt.

4cos 2 2sin 49

Find 1. The efficiency =?, given that min 4.364 occurs

at 0.435 by numeric

m m

ff t

m

a

m t f t f t

E m t

f t

al method .

2. The output spectrum for the AM modulation.

Page 74: 通訊原理

Chap. 3 1414

Page 75: 通訊原理

Chap. 3 1515

Page 76: 通訊原理

Chap. 3 1616

Page 77: 通訊原理

Chap. 3 1717

Single-Sideband Modulation SSB

1. For DSB modulation, two sidebands contain the same information. Therefore, it is a good idea to convey message by only a single LSB or USB

sideband for saving power and the transmission bandwidth. 2. Generation of SSB signals

Method 1: sideband filteringcosc cA t Sideband

filter c SSBx t x t DSBx t m t a

0

DSBX f

cff

cf USB

LSB2W2W

cM f f cM f f M f

f02W

Page 78: 通訊原理

Chap. 3 1818

0

0

f

fcf

cf sgn cf f sgn cf f 1

1 cf cf

12

f

1

1

1 M f

f

DSBX f cX f

LH f

W W

f

f

cM f f cM f f 1, 0 For a lower-sideband SSB signal, define sgn

1, 01 Sideband filter: sgn sgn2

1 1 2 2

L c c

DSB c c c c

ff

f

H f f f f f

X f A M f f A M f f

Page 79: 通訊原理

Chap. 3 1919

Lower-sideband SSB signal

1 sgn sgn4

sgn sgn

c

c

c L DSB

c c c c c

c c

M f f

M f

c

f

c

X f H f X f

A M f f f f M f f f f

M f f f f M f f f f

.

b.

1 = +41 sgn sgn4

?

c c c

c c c c c

c c

a

A M f f M f f

A M f f f f M f f f f

X f x t

F

0 fcf sgn cf f 1

1 cM f fcf

Page 80: 通訊原理

Chap. 3 2020

Method 2: Phase-shift modulator: Generate directly from its time-domain representation.

1 . 4ˆ ˆ . sgn , sgn

1ˆ , Hilber

cos 22

t

c

c

cc c c

x t

a A M f f M f f

b m t j f M f jm t f M f

m t m

A m

tt

t f t

F

F F 2 2

transform of

1 sgn sgn4

1 ˆ ˆ 41 ˆ 2 sin 24

1 ˆ sin 22

c c

c c c c c

j f t j f tc

c c c cA m t f t

m t

A M f f f f M f f f f

A jm t e jm t e

A jm t j f t

-1F

Page 81: 通訊原理

Chap. 3 2121

Similarly , for a upper-sideband SSB USB/SSB

ˆ LSB/SSB: c

ˆUSB/SSB

os 2 sin 2

: cos 2 sin 22

:

2

cc c c

cc c c

Ax t m

Ax t m t f t m t f

t f t m t

t

t f

Page 82: 通訊原理

Chap. 3 2222

0

2 2 2

2 2 20

Hilbert transform:

1, 0 , lim sgn

1, 01 1 4 2 2 4

4 sgn lim4

Duality theorem of Fourie

1ˆ ˆs

r

gn

t t tx t e u t e u t x t t

tj fX f

M f j f M f m t m

j f j f fj f jt

f f

tt

-1F

F Transform: sgn ,

1 sgn sgn , sgn

jtf

j f f j ft t

F

F F

tt

x t1

10 sgn t

Page 83: 通訊原理

Chap. 3 2323

1ˆ The Hilbert transform of :

1ˆ ˆ sgn

Ex. cos 21 1 2 2

1 1ˆ sgn2 2

1 2

ˆ 2

o

o o

o o

o o

j

x t x t x tt

X f x t x t j f X ft

x t f t

X f f f f f

X f j f f f f f

j f f j f f

jx t e

F F F 2 2 sin 2 o of t j f toe f t f

0f0f sgn f

Page 84: 通訊原理

Chap. 3 2424

ˆEx. sin 2 , ?ox t f t x t

Page 85: 通訊原理

Chap. 3 2525

Page 86: 通訊原理

Chap. 3 2626

2 2 2

2

ˆ

Properties of Hilbert Transformˆ 1. Energy or power of Energy or power of

Pf.

ˆ sgn

ˆ

ˆr 2. If is , teal n andhe

XX

x t x t

X f j f X f X f

E X f d E

x t x t

f , i.e.,

ˆ 0, for energy signals , or 1 ˆlim 0, for power signals

2 P

are orthogonal

lease prove it.

T

TT

x t x t dt

x t x t

x t

T

Page 87: 通訊原理

Chap. 3 2727

Page 88: 通訊原理

Chap. 3 2828

1 1

lowpass highpassnon-overlapping sp

3. Let and being a and a signal. If the two signals have , then

ˆ

pf.

ectra

lm t c t

m t c t m t c t

m t c t M f C f

M

F F 0 02 20

2

2

2

: sgn

, please verify i

sgn

tj f t j f t

j f f t

j f f t

j f f t

e j f e

f C f e dfdf

m t c t M f C f e dfdf

M f C f j f f e dfdf

Page 89: 通訊原理

Chap. 3 2929

region or

0, for lowpass Non-overlapping spectral: assume

0, for highpass

is nonzero only for and .

M f f W

C f f W

M f C f f W f W

A B

A B

2( ) ( ) sgn j f f tm t c t M f C f j f f e dfdf f WW

WW f

B

AW

W

WWf

f region of 'f f A

region of 'f f Bsgn( ) 1f f sgn( ) 1f f

Page 90: 通訊原理

Chap. 3 3030

2

2 2

By observing the function behavior in region and ,, .

sgn

sgn

in this case sgn sgnj f f t

j ft j f t

m t c t M f C f j f e dfdf

M f e df j f C

f f f

f e df

A B

A B 0 0

0 0

0

Example For a lowpass signal with 0 for , with , we conclude

ˆ

cos 2 sin 2

sin 2 cos 2

m t c t

m t f t m t f t

m t f t m t f

m t M f Wf

t

fW

Page 91: 通訊原理

Chap. 3 3131

*

ˆ Analytic signals: , is real

ˆ

ˆ ,1 Re2

sgn sgn

2 , 0

0 , 0ˆ

p

n

p

p p p

p

p

n

x t x t jx tx t

x t x t jx t

x t x t jx t

x t x t x t x t

X f X f j j f X f X f f X f

X f fX f

f

x t x t j

,

0 , 0 1 sgn

2 , 0n

x t

fX f X f f

X f f

Page 92: 通訊原理

Chap. 3 3232

0

0 0 0 0

21 0 1 0

1 0

0 0 0 0 0

Example Ideal bandpass filter with bandwidth ,

2 sinc cos

,

where

, 2 sinc sin

BP

j ftBP

BP

P

B

h t H B B t t t t

H f H f f H f f e

fH f HB

B f h t H B B t t t t

x t h

2 20 0

ˆ 2 sinc .

BP BP

P BP BP

t jh t

x t h t h t H B B t t

Page 93: 通訊原理

Chap. 3 3333

f0 0f0f0 0f

f0

B

B

X f pX f X f B

A

2A

2A

f

complex baseband

Complex Envelope Representation of Bandpass Signal For a bandpass signal x t X f ˆ

sgn

p

p

x t x t jx t

X f X f f X f

02

0

,~ complex baseband

j f tp

p

x t x t e

X f X f f

Page 94: 通訊原理

Chap. 3 3434

0

0

2

2

real

Comple

bandpass

x envelope of :

,

Given the signal , the

signal can be expressed as

Re Re

complex base

Re

b

cos

and

2

j f tP

R

I

I

j f tP R

x t

x t x t e x t j

x t

x t

x t x t x t e

x t jx t

x

f

t 0 0

0 0

sin 2

,~ the band-pass signal real

Bandpass Systems: Consider a bandpass system with impulse respose . T

cos 2 s

he impulse response can be repr

in

en

2

es

R Ix t f t

t

x t f

j f t

t

h t

02

ted in

terms of a complex envelop as

Re ,

where

j f t

R I

h t

h t h t e

h t h t jh t

h t y t x t H f

f0 0f0ff0

B

B

X f X f A

2Acomplex baseband

Page 95: 通訊原理

Chap. 3 3535

0

0

0

0

2

2

2

2 *

The response of the system to a bandpass signal is

Re

Re

Re

1 2

j f t

j f t

j f t

j f t

x t

Y f X f H f

x t x t e

h t h t e

H f h t h t e

h t e h t e

F F

F 02

*0 0

*0 0

1 2

also1 2

j f t

H f f H f f

X f X f f X f f

h t y t x t

Page 96: 通訊原理

Chap. 3 3636

0 0 0 0

0 0

non-overlapping spectra

non-overlapping sp

0

e

0ctra

1 4

Y f H f X f

X f f H f f X f f H f f

X f f H f f

0 0

0 0 0 01 4

X f f H f f

X f f H f f X f f H f f

f0f0f 0H f f 0X f f

Page 97: 通訊原理

Chap. 3 3737

0

0

0

0 0

*2

2

2

1 ,4

1

Define

1 R

4

e2

j f

j t

t

f t j f

Y f H f X f h t x t

y

Y f Y f f

t e

Y f f

y t y t e y t e Ff0 0f0ff0

X f X f A

2Acomplex baseband

f0 0f0ff0

H f H fcomplex baseband

Page 98: 通訊原理

Chap. 3 3838

0

0

cos 2

cos 2

?

t

tx t f t

h t e u t f t

y t

Example x t y t h t

0

0

0

2

2

2

12

Re Re

Re

j f t

j f t

j f t

y t x t h tx t x t ey t y t e

h t h t e

Page 99: 通訊原理

Chap. 3 3939

y t

y tt

t

t1 e 22

2

Page 100: 通訊原理

Chap. 3 4040

4. Demodulation of SSB signals Coherent detection with possibly phase error

+ : LSB-SSB1 1 ˆ cos sin ,: USB-SSB2 2c c c c cx t A m t t A m t t

LPF cx t Dy t d t cos , phase error

let 4 cK t t t

K

Page 101: 通訊原理

Chap. 3 4141

1 1 ˆcos sin 4cos2 2

2 cos cos

ˆ 2 sin cos

cos cos 2

c c c c c

c c c

c c c

c c c

d t A m t t A m t t t t

A m t t t t

A m t t t t

A m t t A m t t t

message crosstalk

ˆ ˆ sin 2 sin

After lowpass filtering and amplitude scalingˆ cos sin

If 0, .

c c c

D

D

A m t t t A m t t

y t m t t m t t

t y t m t

LPF cx t Dy t d t 4cos ct t

Page 102: 通訊原理

Chap. 3 4242

2 2

1

cos

1 1 ˆ cos sin2 2

What is the output of envelope detector ?

cos sin

Let tan

r c

c c c c

c c

b ta t

e t x t K t

A m t K t A m t t

e t a t t b t t

R t a t b t

bt

Carrier re-insertion : cos, ;

sin

a t R t tt b t R t t

a t

EnvelopeDetector

rx t Dy t e t cos cK t in pahsedirect

quadrature-axis R t t a t

b t

Page 103: 通訊原理

Chap. 3 4343

envelope

2 2

cos cos sin sin

cos

Assumming that the bandwidth of is much smaller than

1 1 2 2

c c

c

c

D c c

e t R t t t R t t t

R t t t

m t f

y t R t A m t K A m t

2

2 2

1 ˆ1 21 122

1 1 If is chosen large enough such that , 2 2

1 1 then , providing min2 2

c

c

c

c c

D c c

A m tA m t K

A m t K

K A m t K A m t

y t A m t K K A m t

Page 104: 通訊原理

Chap. 3 4444

1 1 1

1 1 1

2 2 1

Example

cos 0.4cos 2 0.9cos 3

ˆ sin 0.4sin 2 0.9sin 3 For SSB:

ˆ cos sin cos2

where

ˆˆ , and tan

2

cc c c c

c

m t t t t

m t t t t

Ax t m t t m t t R t t t

m tAR t m t m t tm t

Page 105: 通訊原理

Chap. 3 4545

1 Let 1Hz, 50Hz, 2. The time-domain waveform of ,

ˆ , , USB in d and LSB in e are shown below:c c

c

f f A m t

m t R t x t

Page 106: 通訊原理

Chap. 3 4646

Problems of SSB: 1 Need nearly perfect sideband filter with sharp cut-off frequency response.

2 Realistically, to avoid baseband interferen

Vestigial - Sideband Modulation

殘邊帶調變 ce due to the transition region at the cut off frequency of the sideband filter, the message signal is prohibited to have low frequency components. 3 Cannot transmit carrier to reduce the complexity . Allowing a vestige of the unwanted sideband to appear at the output of the modulator. VSB

Modification

H f

f

Page 107: 通訊原理

Chap. 3 4747

Idea:

DSB signaling

fcf cf0

cM f f cM f f H ff

cM f f H f cM f f H f cX f

0 f

demodulation cos ct rx t d tLPF

Dy t DY f

f DY f Assume an ideal channel:

r c c cX f X f M f f M f f H f cfcf f0

H f

Page 108: 通訊原理

Chap. 3 4848

D

]

c c c c

c c c c

c c

c c

D f d t

M f f M f H f f M f M f f H f f

M f H f f H f f M f f H f f

M f f H f f

Y f M f H f f H f f

1 1

2 22 2

1 12

2 2

12

2

1

2

F D ,

if constant at

the vestige band .c c

y t m t

H f f H f f

f cf2

cf2

cH f f cH f f 0 f

f0

Page 109: 通訊原理

Chap. 3 4949

D

Solution Let be an lowpass anti-symmetric filter, i.e.,

and for .

, Set ,

,

,

c c

c c

U f H f U f H f

c c

H f

H f f

U f f H f f for fH f

U f f H f f for f

H f

H f H f

f H f f

Y f

0

0

0

1 M f1

212

12

H ff cfcf cf cf 1

12

f0

H f

Page 110: 通訊原理

Chap. 3 5050

Example: cos cos

cos

cos

cos

cos

cos

DSB c

c

c

c

c

m t A t B t

x t m t t

A t

A t

B t

B t

1 2

1

1

2

2

1

2

1

2

1

2

1

2

Page 111: 通訊原理

Chap. 3 5151

The VSB signal Tx. :

cos cos

cos

The demoulated signal Rx. : , ?

cos

cos

cos co

s

c DSB

c c c

c

D

c c

c

x t X f H f

x t A t A t

B t

d t y t

d t x

t

t t

A tA A t

1 1

2

11 1

1 11

2 2

4

2 1

1

2 cos coscos

cos cos co

.

sc c

D

B t

A t A t B t

A t B t

y t m t

1 22

1 1 21

1 2 2 cos ct4

cx t d t LPF Dy t

Page 112: 通訊原理

Chap. 3 5252

cosdx t

m t t 1

e t ( )cosx t m t t 2

Local Oscillator

BPF cos[ ]t 1 22

2Center Freq. : Mixer The carrier freq. of the band-pass signal is now translated to .dx t f

2

cos cos , +: high-side t

uning

: low-sid

cos cos

e tuning

dx t

e t m t t t

m t t m t t

1 1 2

2 1 2

2

2

Frequency Translation and Mixing

Page 113: 通訊原理

Chap. 3 5353

Problem: signals at , for example

cos , will appear at the output of BPF.

cos cos

cos cos

. ., When pa

imagx t

k t t

k t t t

k t t k t t

image frequenc s

e

e

i

i 1 2 1

2

1

1

2

2 2

2 2

3

2

2 imag

ssing through the mixer, the desired signals

and the image signal will be coupled to each other at mixer output.

dx t

x t

Page 114: 通訊原理

Chap. 3 5454

IF 2

IF 2

C 1

Illustration of the image signals high-side tuning

LO 1 2 1 22

c IF 1 22 2 1 22 3

Desired signal

Localoscillator

Imagesignal

Imagesignal atmixer output

Signal atmixer output

Passbandof the IF filter

f

f

f

f

f

dX f E f e t imagX f

Page 115: 通訊原理

Chap. 3 5555

Super-heterodyne Receiver A super-heterodyne receiver has two "amplification-and- filtering" sections prior to demodulation.

OutputDemo-

dulator

Localoscillator

R.F. filter andAmplifier

I.F. filter andAmplifier

Mixer

near-by imagdx t x t x t

Page 116: 通訊原理

Chap. 3 5656

near-by imag

RF amplifier generally has limited amplification gain. Therefore, receive signal amplification is mainly done at the IF-amplifier.

At the antenna:

: td

d

x t x t x t

x t

imag

near-by

near-by

he desired signal at

: image signal at

: other unwanted signals in the channel near by .

The carrier frequency of does not confuse the IF m

c

c IF

c

x t

x t

x t

2 near-by

near-by

ixer.

After the RF amplifier: .

The RF filter is a wideband tunable filter. Although

can pass the RF-filter also, they will be removed by the succ

dx t x t

x t

eeding IF mixer

After the mixer: dx t

Page 117: 通訊原理

Chap. 3 5757

IF

2

c 1

c IF 2

Received signal

RF-filteroutput

Signal atIF filter output

f

dX f

Illustration of the superheterodyne receiver high-side tuning

i c IF 2

imagX f near-byX f

c 1

f

f IF IF

IF

IF

is . 1 AM I.F.: 455KHz; IF-BW ; RF: 540K~1.6MHz

2 FM I.F.: 10.7MHz; IF-BW ; RF: 88~108MHz High-side tuning

Low-side tuning

fixed

c

c

LO c IF

LO c IF

ff KHz f

f KHz f

2

10

200

Page 118: 通訊原理

Chap. 3 5858

LORelation between and in terms of :c if f f

IFf2

LOf

f

i LO IFf f f i LO IFf f f Desire signal

1 c

2 IF

Desire signal: , d

f fX f

f f

Image signal

Image signal: imagX f

c LO IFf f f c LO IFf f f a low-side tuning b high-side tuning

IFf2

f

LOf

Page 119: 通訊原理

Chap. 3 5959

Which one is better? High-side tuning vs. Low-side tuning For AM receiver, RF frequency range: 540 KHz 1600 KHz

I.F. 455 KHz ~ fixed

1 Low-side tuning: Local oscillator freque

L F

c

O c If f f

f ncy range: KHz KHz

KHz KHz min : max : .

2 High-side tuning: Local oscillator frequency range: KHz KHz

LO

LO L

LO

LO

O

c

O

IF

L

f

f

f

f

f f

f

f

540 455 1600 455

85 1145 1 13 47

540 455 1600 455 KHz KHz min : max : . Smaller local oscillator frequency range is preferred easy to implemen High-side tuning t , is better

LO LO LOf f f 995 2055 1 2 07

Page 120: 通訊原理

Chap. 3 6060

§ Angle Modulation

General form: cos

Instantaneous phase:

Instantaneous frequency:

Phase deviation:

Frequency deviation:

i

c c c

i c

ic

t

i

x t A t t

t t t

d t d tt

dt dtt

d

Phase modulation PM

; Phase deviation message.

: deviation constant, : message,

cos

p

c c c p

p

t k m t

x t A t

tdt

k

m

t

k t

m

Page 121: 通訊原理

Chap. 3 6161

0

0

0

0

0

Frequency modulation (FM)

; Frequency deviation message

2

= : Frequency deviation constant2

EX.

1

c

F r

os 2

o

t t

f

t

c c c

f dt t

fd

d t

d tk m t

dt

x t A t

t k m d f m d

kf

f m d

, , ;2

PM: cos2

FM: cos

a phase discontinuity occurs at 0 for the PM signal

p f d

c c c

c c c d

m t u t k k

x t A t u t

x t A u t t

t

Page 122: 通訊原理

Chap. 3 6262

PM: cos cos2 For cos , ;

FM: cos sin

c c c p m

m fc c c m

m

x t A t k tm t t k

x t A t t

Page 123: 通訊原理

Chap. 3 6363

2

carrier s

Narrowband Angle Modulation

cos

Re Re 12!

If ( ) 1, then

Re cos sin

c c

c

c

c

c c c

j tj t j tc c

j t j tc c c c c c c

m t A

x t A t t

tA e e A e j t

t

x t A e A t je A t A t t

( ) 1

( ) 1

in

narrowband angle modulation AM-like modulation

BW of , BW of 2W.

c

t

t c

t

t W x t

Page 124: 通訊原理

Chap. 3 6464

0 0

. A FM system with cos , 2

Let 0, cos sin

cos sin

m f d

td

f m mm

dc

t

c c mm

Ex m t A t k fAft t k A d tf

Afx t A t tf narFIG rowURE 3. band a24 Generatio ngle modulatn of ion: m t

dtdf2

pk

FM

PM

tsin ct

cos ct cA cx t

carrieroscillator

o90 phase shifter

Page 125: 通訊原理

Chap. 3 6565

2 2 2

cos sin sin

cos cos cos2

1, si

Re 12

n 1,

c m m

dc c c c m c

m

c dc c c m c m

m

j f t j f t j f td

d dm

m

m

c

m

Afx t A t A t tf

A

Af Aft

AfA t t tf

AfA e e ef

tf f NB-FM

NB-FM

AM

2

Compare to the AM signal with modulation index ,

1 cos 2 cos 2

cos 2 cos 2 cos 22

Re 12

c

c

c

c c m c

cc c c m c m

j f t jc

x t

x t a

x t A a f t f t

AA f t a f f t f f t

aA e e 2 2m mf t j f te

Page 126: 通訊原理

Chap. 3 6666

Spectrum of an Angle-Modulated Signal

wideband spectrum of single-tone message

Assume a single-tone message,

sin

: modulation index controls the maximum phase deviation

mt t si

in

n

s cos sin Re

Since is a with period , it can be

expressed by its Fourier series representation with coefficients

periodic function

mc

m

j tj tc

j

c

t

c

m

c mx t A t t A e e

e

2 / / sinsin

/

si

/

n

Let ,

m m m mm m

m m

j t n tj t jn tm mn

mj x nx

n n

C e e dt e dt

x Jt C e dx

2 2

1

2

Page 127: 通訊原理

Chap. 3 6767

sin

is the Bessel-function of the first-kind of orde

c

r and argument .

,

Re ,

os ,

m m

c m

n

j t jn tn n n

c n c

n

j t jn t

n

cn

m

c n

J n

C J e J

A J n

e

x t A e J e

t

An angle-modulated signal of a single-tone message has infinite number of line spectra, each spaced with , centered at the carrier frequency .

n

m

c

J

ff

Page 128: 通訊原理

Chap. 3 6868

Amplitude

cA phase rad

f

J 4

J 3

J 2

J 1 J 0

J 1 J 2 J 3 J 4

cm

ff4

cm

ff2 cf

cm

ff2

cm

ff4

cm

ff3

cm

ff1 cf cm

ff

cm

ff3 single-sided amplitude spectrum

single-sided phase spectrum

f

Page 129: 通訊原理

Chap. 3 6969

sin

Properties of :

1 is real-valued.

2

3

4 Recursive relation:

Given and , we can re

j x nxn

n

nn n

nn n

n n n

J e dx

J

J J

J J

nJ J J

J J

1 1

0 1

1

2

1

1

2 cursively find

all the for .nJ n 2

Page 130: 通訊原理

Chap. 3 7070

5 When is small eg., narrow-band FM, PM ,

!

, , for .

6 When is real and fixed, lim

7 , for all

Ex. Prove 1 and 7

n

n n

n

nn

nn

Jn

J J J n

J

J

0 1

2

2

1 0 12

0

1

Page 131: 通訊原理

Chap. 3 7171

Page 132: 通訊原理

Chap. 3 7272

, , , the narrow band angle-modulated signals:

cos

cos cos

n

c c n c mn

c c c c m

J J J n

x t A J n t

A J t A J t

0 1

0 1

1 2

Page 133: 通訊原理

Chap. 3 7373

Page 134: 通訊原理

Chap. 3 7474

Carrier nulls: The values for which

, . n

n

J

n J J

0

0

0 0 0 0 1

Page 135: 通訊原理

Chap. 3 7575

The above analysis is based on the single-tone modulation case in which sin , For PM, this means sin

sin sin

= or

Fo

m

m

p p m m

p p

t t

m t A t

t k m t k A t t

k A kA

r FM, this means cos

sin sin

m

td

d m mm

m t A tf At f m d t tf

2

Page 136: 通訊原理

Chap. 3 7676

= : given a fixed , is reverse propotional to

; The amplitude spectrum of a FM signal increase with a decreasing .

dd m

m

m m

m

f A f ff

f f

f

f

f

f

f

f

cf

. 1 0

. 0 5

. 2 0

. 5 0

. 10 0

mf

mf

mf

Page 137: 通訊原理

Chap. 3 7777

Power in an Angle-Modulated signal

single-tone case: cos

cos cos

c c n c mn

c c n k c

k n

m c mn k

c nn

x t A J n t

x t A J J n t k t

A J

2 2

1

2

1

2

2

1

2

1

2 frequencydeviation

.

general cases: cos ,

cos

, cos , .

Power of the angle modulated signal is constant

c c c

c

c

cc

c c

c c

x t A

A

t tAx t t t

d t t t xd

tt

A

2

2

22

2

1 2 2

02

2

2 2 . indep. of t

Page 138: 通訊原理

Chap. 3 7878

Bandwith of angle-modulated signals

. Single-tone t :

series expansion of : see appendix G3

, !

for moderate .

For

n

n

n n

m

J

Jn n n n

2 4

2 4

1

12 2 1 2 2 1 2 fixed large , , lim

! Strictly speaking, the bandwidth of an angle-modulated signal is infinitely large. However, the bandwidth can be approxiamtely specified

n

n nn nn J J

n 0

2

by the frequency band of the major power distribution.

Page 139: 通訊原理

Chap. 3 7979

20

single-to

Power ratio:1

component power 2 = 1total power2

for a angle modulated signal,

. single-underscored terms i

ne

n Table 3.2 :

c n k

r nn

n k

m

r

k

c

BW

A JkP J J

k

A

P

f

k

2 2

2

2 1

2

2

0 7 . double-underscored terms in Table 3.2 For 0.989, by observing Table 3.2,

integer part of +1 , i.e., +1 ,

+1 m

r

r

PP

k k

BW f 2

0 98

Page 140: 通訊原理

Chap. 3 8080

FM case

.) Arbitrary :

Define: Frequency deviation ratio

maxpeak frequency deviation max

bandwidth of

,

1

Carson's rule

,

td

m t

d tdt fD m t

m t W W

D

B W

W

D

B

W

2

1

2

1

2 1

2 narrow-band angle-modulated signal

2 , max

wide-band angle-modulated signal

d

W

D BW DW f m t

1 2 2

Page 141: 通訊原理

Chap. 3 8181

Ex. Consider a FM modulation system below with

cos

, cos , ,

BPF:

please determine the p

cc

c

d m

BP

Ax t t t

f

f m t t A f

fH f

100100 2 1000

1000

8 5 16 5 8

1000

56

ower at the filter output. m t cx t output

outx t FMModulator

BPH f

,c df f 1000 8

Page 142: 通訊原理

Chap. 3 8282

Page 143: 通訊原理

Chap. 3 8383

f

cX f

13.126.1

36.5

4.7

32.817.8

32.8

4.7

36.539.1

26.113.1

952 96

096

897

698

499

210

0010

0810

1610

2410

3210

4010

48

1000972

1

1028 f

39.1 BPH f

c nA J

Page 144: 通訊原理

Chap. 3 8484

sin sin

Ex. Given cos cos ,

sin sin , For FM mod.,

,

cos sin sin

Re

c

d d

c c c

j t j tj tc

m t A t B t

t t tAf Bff f

x t A t t t

A e e e 1 1 2 2

1 2

1 1 2 2

1 2

1 2

1 1 2 2 sin sin , and

Re

= cos

c

j t jn t j t jm tn m

n m

jn t jn tj tc c n m

n m

c n m cn m

e J e e J e

x t A e J e J e

A J J n m t

1 1 1 2 2 2

1 2

1 2

1 2

1 2 1 2

Page 145: 通訊原理

Chap. 3 8585

Narrowbandfrequency modulator

Localoscillator

Bandpass filter

Mixer

1

0

1

Narrowband FM signal : Carrier frequencyPeak frequency deviation :

Deviation ratio

maxd df f m t

f

D

0

2 1

2 1

Wideband FM signal :Carrier frequency Peak frequency deviation :Deviation ratio :

d d

nff nf

D nD

cx t Frequency Multiplier n Narrowband-to-wideband Conversion Indirect FM

(i.e., Figure 3.24)

FIGURE 3.31 Frequency modulation utilizing narrowband-to-wideband conversion

LOe t x t y t

y t m t

Page 146: 通訊原理

Chap. 3 8686

0

0

LO LO

LO

2 Contains two stages: 1 Narrowband FM, Frequency multiplier

cos narrowband FM output

After the freq. multiplier,

cos

local oscillator: 2cos

c

c

x t A t t

y t A n t n t

e t t

y t y t e t

0 LO 0 LO

c 0 LO c 0 LO

c

cos cos

The BPF may choose to select or

as its passband center frequency.

= cos wideband FM output

c c

c c

A n t n t A n t n t

n n

x t A t n t

x t n

z t

x t y t

Frequency multiplier is often a memoryless nonliner device followed by a

BPF.

Page 147: 通訊原理

Chap. 3 8787

0 0

1

Ex. A narrowband-to-wideband converter as in Fig. 3.31 Narrowband FM :

cos , 2 100,000

peak frequency deviation of : 50

bandwidth of : 500 Wideband

c

d

x t A t t Hz

t f Hz

t W Hz

2

output : carrier frequency 85 deviation ratio 5 Determine 1. ? ,

2. the two possible local oscillator frequencies,

3. the center frequency and

cf MHzD

n

bandwidth of the BPF .

cx t Freq.Mul.

n LOe t x t y t

y t m t

Page 148: 通訊原理

Chap. 3 8888

Page 149: 通訊原理

Chap. 3 8989

0

output the frequency-deviation of

Demodulation of Angle-Modulated s

the input signal.

ignals

Frequency discriminator :

Received signal

2 , for FM; cos ;

t

dr c

f m dx t A t t t

, for PMpk m t

Dy t

input frequencycf

slope DK Hzf

FrequencyDiscriminator

rx t D D

d ty t K

dt

Page 150: 通訊原理

Chap. 3 9090

Discriminator output:

, : discriminator constant.

For FM,

For PM,

2

2

1D

D D d

D p

D D

D

K

y t

d t

K f m tK k dy t m t

d

d

t

y t Kt

FrequencyDiscriminator :r

FMx t D D dy t K f m t

FrequencyDiscriminator :r

PMx t

dt 2D pK k

y t m t Dy t

Page 151: 通訊原理

Chap. 3 9191

Dy t e t ddt

EnvelopeDetector

rx t Envelope

Approximation to Ideal Discriminator

cos

sin

Envelope of

If for all , 0

r c c

c c c

c c

c c

x t A t t

d te t A t t

dt

d te t y t A

dt

d t d tt

dt dt

2 , for FM 2D c c d D c

d ty t A A f m t k A

dt

Page 152: 通訊原理

Chap. 3 9292

Due to interference perturbation and channel noise, may sometimes be time-varying , use bandpass limiter.

c

c

AA t

Limiter

Bandpass Filter

EnvelopeDetector

rx t

Band pass limiter

Dy t ddt

K

K limiter BPF

Page 153: 通訊原理

Chap. 3 9393

ddt

Discriminator implementation: Ideal differentiator : x t y t

H f

f0 2 , High-pass filter.H f j f 0 0

For small

Implementation: 1 Time-delay:

,

lim lim

, .

r r

r r

r r

e t x t x t

e t x t x t

dx t dx te t

dt dt

rx t e t

Time-delay

Page 154: 通訊原理

Chap. 3 9494

2 RC network:2 ;1 1 2

21 , 2 differentiator with gain .

2

R j fRCH fj fRCR

j fC

f H f j fRC RCRC

C

R

H f

f0

10.707

cf1

2 RC

Page 155: 通訊原理

Chap. 3 9595

The resultant Discriminator: 2D CK A RCDifferentiator Envelope detector

D D dy t K f m t :r

FMx t

1Disadvantage : 2

If freq. 10.7 for commercial FM,

, ,D D

fRC

MHzRC K y t

Page 156: 通訊原理

Chap. 3 9696

3 BPF-based Differentiator linear region differentiator Disadvantages : 1. small linear region 2. dc bias

Linearregion

H f

f 4 Balanced Discriminator Advantages : 1. wider linear region 2. no dc bias BPF at

Cf fBPF at

Cf f Envelopedetector

Envelopedetector

Dy t rx t

Page 157: 通訊原理

Chap. 3 9797

Page 158: 通訊原理

Chap. 3 9898

desired signal undesired interfere

Interference Interference in Linear Modulation

eg., DSB, AM, SSB

Interference : cos

cos cos cos cos

i c i

r c c m m c i c i

A t

x t A t A t t A t

§ message

nce

Coherent demodulation cos cos

Envelope detection Nonlinear detector phasor diagram :

cos cos cos cos

D m m i i

r c c m m c i c i

m t

y t A t A t

x t A t A t t A t

Dy tLPF 2cos ct rx t

Page 159: 通訊原理

Chap. 3 9999

Re2 2

Re2 2

c c m c m c i

c m m i

j t j t j t j t j t j t j tm mr c i

j t j t j t j tm mc i

j tR t e

A Ax t A e e e e e Ae e

A Ae A e e Ae

Page 160: 通訊原理

Chap. 3 100100

Envelo

Envelope detector output: 1 :

cos cos cos

+ cos cos sin sin

cos cos

c i

r c c m m c

i i c i c

c m m i i

A A

x t A t A t t

A t t t t

A A t A t

pe

cos

sin sin

If , with a DC-blocker to remove term cos cos , same as the

c

i i c

c i

A t

c

D m m i i

B t

t

A t t

A A Ay t A t A t

coherent demod.

12

02 2

cos sin cos ,

, A t B tA t

z t A t t B t t R t t t

R t A t B t R t A t

Page 161: 通訊原理

Chap. 3 101101

2 If

cos cos cos cos let

cos cos

cos cos

cos cos si

c i

r c c m m c i c i

c c i i i c i

m m c i i

c c i

c i

i

c i

A A

x t A t A t t A t

A t t A t

A t t t

A t t

Envelope

n sin

cos cos cos cos

sin sin

cos cos cos cos

c i i

i c i m m c i i

c i i

i c i m m i

A

c i

c

t

t t

A t A t t t

t t

A A t A t t t

A

sin cos sin sinB t

i m m i c it A t t t

Page 162: 通訊原理

Chap. 3 102102

cos cos2

If , then ,

cos + cos cos , The message is lost

Interference in Angle Modulation

Unmodulated carrier interference at

m i m i

c i

D c i m m i

t t

c i

A A R t A t

y t A t A t t

cos cos

cos cos cos sin sin

cos

r c c i c i

c c i i c i i c

c

x t A t A t

A t A t t A t t

R t t t

Page 163: 通訊原理

Chap. 3 103103

2 2

1

cos sin sin

tancos

cos if , ;

sin

1 cos cos sin

c tA

c i i i i

i i

c i i

c i i

c i ii

c

i ir c i c i

c c

R t A A t A t

A tt

A A t

R t A A tA A At t

A

A Ax t A t t tA A

1 3 5

1

Taylor's series:1 1tan3 5

1, tan

u u u u

u u u

Page 164: 通訊原理

Chap. 3 104104

Ideal discriminator output:

sin , for PM

cos , for FM2

Since th interference output at the FM discriminator is

proportio

e

nal to

i

D D ic

DiD

D ic

i

AK t K tA

y td t AK K t

df

t A

, for a small , the interference affects

less on the FM system than the PM system, and vice ver

sa.i if f

Dy t ddt

EnvelopeDetector

rx t dt Dy t

PM

FM

Page 165: 通訊原理

Chap. 3 105105

The interference analysis of angle modulation becomes difficult, if the preceding assumption is not hold. However, an approximation can be made through the use of the

phasor diagram. pleas

c iA A

interference impacting on the FM system is

proportioal to t

e refer to the textbook Generally, the

. One approach for interference

alleviation is to use the pre-emphas

he

is

frequenc

and the e

y

de-if

mphasis filters

in FM systems. pH f

Pre-emphasis Filter

Mod. cosi c iA t m t rx t dH fDe-emphasis Filter

Dy t

m tDiscriminator.

ddt

Page 166: 通訊原理

Chap. 3 106106

32

3

3

Deemphasis filter: 1st-order RC LPF1 1 , set

21

1 , ,

,

assume in the case of

While suppressing the interference wi

d

d

iD D

c

c i

H f f WRCf

f

f f H ff

Ay t KA

A A

th large , the de-emphasis filter distort the message signal as well, so we need to deploy a

1 pre-emphasis filter with at the transmitter.

i

pd

f

H fH f

dH f

fif3f

0.7071

R

C

Page 167: 通訊原理

Chap. 3 107107

Page 168: 通訊原理

Chap. 3 108108

§ Feedback DemodulatorsPhase-Lock Loops PLL for FM demodulation

Main buliding blocks: 1. Phase Dector PD

2. Loop Filter

3. Loop Amplifier

4. Voltage-Controlled Oscillator VCO

rx t Phasedetector

Loopfilter

Loopamplifier

VCO

de t 0e t e tDemodulated output

gain =

Page 169: 通訊原理

Chap. 3 109109

0

Phase Detector PD : Assume the input of PD:

cos ,

sin

, : constant

the corresponding output is a funct

1 si

ion of phase-dif

n2

ferenc

d

r c c

d

v c

c de t

x t A t t

e t A

A A K t t

t t

K e of the inputs

Ex. An implementation of PD

LPF rx t de t 0e t

1 rx tPD de t 0e t

Page 170: 通訊原理

Chap. 3 110110

phase di

VCO : Frequency-deviation of VCO output is proportional to the magnitude of its input.

rad sec ; : VCO constant

.

1 Since sin2

v v v

t

v v

d c d

d tK e t K

dt

t K e d

e t A A K t t fference only

,

it follows that the PLL can be modeled without taking the carrieN

r terms. onlinear PLL model. VCO e t 0

sinv c

e t

A t t

Page 171: 通訊原理

Chap. 3 111111

t tK dt t e t

Demodulated output

Loopfilter

Amplifiergain

: Nonilnear PLL modelFigure 3.46

sin 12 c dA A K

Phase Detector

de t Furthermore, when the VCO operating in lock, ,

sin

Nonlinear PLL model Linear PLL m, odelt t

t t

t t t t

Page 172: 通訊原理

Chap. 3 112112

: Linear PLL modelFigure 3.47

t tK dt t e t

Demodulated output

Loopfilter

Amplifiergain

Phase Detector

de t 12 c dA A K The loop filter can be

assumed a short-circuitfor simplification.

for FM

VCO input demodulated mes

When the PLL is

sage

in lock,

,

1 ;

t t

d t d tm t K e t

dt dt

e t m tK

Page 173: 通訊原理

Chap. 3 113113

,Phase Error Loop Gain

1VCO: sin2

sin

nonlinea

sin

r 1st-or

,

v v c v d v

t

t

ttK

d tK e t A A K K t t

dt

d t d t d tK t

dtd t d t

K tdt d

dt

t

dt

der differential equation

t tK dt t e t

Demodulated output

Loopfilter

Amplifiergain

: Nonilnear PLL modelFigure 3.46

sin 12 c dA A K

Phase Detector

de t

Page 174: 通訊原理

Chap. 3 114114

Assume that the input to a FM modulator:

s sin , 0

si

in ,

n , 0

tt

t

d tu t

dtd t

K t d K tdt

d K tdt

ordt

0d t

dt 0

d tdt increase

decrease ,

sint

dydt

y K

ss

ss

Under the condition , 0, and the trajectory of

staggers around ; 0,t

t t t

K

t

Page 175: 通訊原理

Chap. 3 115115

For 0, is monotonically ascending

going right on the -axis, so that the opearting point

will move from left to right along the "sin" curve as .

For 0,

d tt

dt

td t

dt

being monotonically descending,

the operating point will move in opposite direction. Point is a locally stable point.Note:

1 Steady-state error: At point , 0, no frequency erro

t

A

dAdt

1ss

r

but also at point , sin 0, phase error existstA K

Page 176: 通訊原理

Chap. 3 116116

is the lock

2 Lock range: The system converge to point if .

If , sin 0, the operating

curve will not intersect range

with -axis

t t

t

t

A

K K t

K

K

ddt tK Ex.: 1

ss

ss

Phase error: sin ,

.t

t

K

K

Page 177: 通訊原理

Chap. 3 117117

1st order linear model PD: Assuming that t is very small eg. ,

sin ,2

VCO: 1st-order linear model

Using the Laplace tr

t

c v dv

t

t t KA A Kt t t t e t t t

d tK t t

dt

,

ans

,

form,

,

,

.

t

t

t

t

t

t

K tt

s KH ss s K

h t K e u

Ks ss K

K h t t

t t h t t

t

signal

t

FM

. ., output of the VCO

t

i e

tK td t t

t

d tK t t

dt

Page 178: 通訊原理

Chap. 3 118118

Ex. The output to a FM modulator is ,

cos , 0cos

cos , 0

By using a first-order linear PLL, determine the demodulated output.

t c c

c c c fc c f

m t Au t

A t tx t A t K A u d

A K A t t

Sol.

Page 179: 通訊原理

Chap. 3 119119

1st order approximated PLL RC-LPF with a 3-dB bandwidth tK

Page 180: 通訊原理

Chap. 3 120120

ss

Summary of the 1st order approximated PLL:1. limited lock range:

.12. phase error: 0

3. The loop gain,1

drawbacks

2

is a

of

function

1st orde

of t

r PLLt

t

t c v d v

K

K

K A A K K

he amplitude of input signal. 4. 1st-order approximated PLL is equivalent to a RC LPF with 3-dB bandwidth . 5. As , 1. However, it is impractical.

c

t

t

A

KK H s

Page 181: 通訊原理

Chap. 3 121121

The two drawbacks of 1st order PLL can be solved by the 2nd order PLL: Perfect 2nd order PLL:

Linear model is assumed. i.e., is small

Loop filter transfer function:

.

s aF ss

t t For 1st order PLL: 1F s s1VCO: s Demodulated output

Loop Filter F s

: Linear PLL model freq. domain representationFigure 3.46 s s Loop gain tK

Page 182: 通訊原理

Chap. 3 122122

2

2 2

2 2 2

standard form of 2nd- order s

Linear model:

,

Phase error: 1 ,

12

n

t

t t

t t t

t t n

s aF s

sK F s

s s sss K F s K s a

H ss s K F s s K s K a

s s s s H s

s s sH ss s K s K a s s ystems

1 damping factor, 02 where

natural frequency

t

n t

Ka

K a

Page 183: 通訊原理

Chap. 3 123123

2

2

2 2 2

2 2

For step frequency input FM signal ,

,

Phase error: 2

2

n n

n n

d tu t s s s

dt s s

sss s s

s s

s

22 2

2

2

ss

1

Using inverse Laplace transform,

si

, 0, no steaty-state

n 1 , 1

phase e rror

1n

n n

tn

n

t e t u t f

t t

or 2 2

Laplace-transform pair: sin

bte at u t

as b a

Page 184: 通訊原理

Chap. 3 124124

Ex.: Some computer simulation results 0.707, 10 , 88.9, 44.4n tf Hz K a t d t

dt

ss 8 four cycle phase slipped ss 6 three cycle phase slipped

Page 185: 通訊原理

Chap. 3 125125

20f Hz 35f Hz 40f Hz 45f Hz

ss 6 3 cycles ss 8

4 cycles Features of the 2nd order PLL:

1. lock range ,

lim 0,

for arbitrary 2. Has cycle-slipping.

t

d tdt

Page 186: 通訊原理

Chap. 3 126126

Analog Pulse Modulation Pulse modulation is used to represent uniformly sampled signals Analog Pulse Modulation: A pulse train is used as the carrier wave in which amplitud

§ , or of each pulse can be used to represent the sampled message.

Pulse Amplitude Modulation PAM

Pulse Width Modulation PWM

Pulse Posit

ewidth p

ion Mod

os

ul

ition

ation PP

M

sT sT2 sT3

amplitude

width

positiont

Page 187: 通訊原理

Chap. 3 127127

for zero-value samplessT2

Page 188: 通訊原理

Chap. 3 128128

impulse-train s

holding networkampling

=

Pulse Amplitude Modulation PAM

Generation of PAM waveform :

*

c

s s c

tn th

c

m t

m t m t m nT t nT m t

m t m t h t

m nT

2 *

=

s sn

s sn

t nT t

m nT t nT

2

2

h t sn

t nT m t m t cm t = th t

2

Page 189: 通訊原理

Chap. 3 129129

, and sinc ,

sinc

j fs

n s

j fc s

n s

nm t f M f H f f eT

nM f f M f f eT

F

Page 190: 通訊原理

Chap. 3 130130

Pulse Width Modulation PWM

proportional to the sample values of a message signal.

Nonlinear modulation. Complicated spectrum.

Used extensively for DC motor con

Pulse

trol.

widt

The pulse

h

amplitude is a constant. Therefore, the pulse area of a PWM

waveform is proportional to the mes

Lowpass filters or integrators can be used as a demodulato

sage value.

Pulse Position M u

r.

od l

ation PPM

Pulse position proportional to the sample values of a message signal.

The spectrum is very similar to that of a PWM signal.

Page 191: 通訊原理

Chap. 3 131131

Delta Modulation and PCM Delta Modulation DM

The message signal is en- coded into a sequenc

p

e of binary symbols represented by the fun

olaritctions

y of impulse.

Operations:

§ 0

0

0

;

sgn

,

,

sd t m t m t

t d t

d t

d t

0

0

Page 192: 通訊原理

Chap. 3 132132

a stairstep approximation of .

For a specific time interval

c s s sn n

t

s s sn

s s

s s skn

x t t t nT nT t nT

m t nT nT d

m t

k T t kT

m t nT nT d

1 .

At the time instance ,

, if

, if

s

t

T

s s

s

s s ss s

s s s

m k T

t kT

m k T kTm kT

m k T kT

1

0 0

0 0

1

1

1

Page 193: 通訊原理

Chap. 3 133133

Page 194: 通訊原理

Chap. 3 134134

0

0

Slope overload:

Occurs when the message signal has a slope greater than that can be followed by the stairstep approximation . . : If step size , Maximum slo

s

s

d m tdt T

m t

m tEx

0

0

pe:

in the case , can exactly catch up the

variation of .

The slope overload implies a bandwidth limitation on .

m s

ss

S Td m t m tdt T

m t

m t

Page 195: 通訊原理

Chap. 3 135135

Digital Pulse Modulation: Signal samples are represented by discrete values (a countable set of values). Pulse-Code Modulation PCM

The message is

first sampled analog amplitude and then quantized into discrete values. The quantized values are encoded into a binary sequence.

011 ,111 ,110 ,011 , 010

10

Page 196: 通訊原理

Chap. 3 136136

A rough discussion of bandwidth Quantization levels: use bits to represents a quantized value Bits per sample: log Message bandwidth: Minimum sampling rate: Nyquist

nq nn q

WW

2

2

2 max

sampling rate. 2 pulse/sec. or bits/sec. Each pulse carries 1 bit message.

Maximum pulse width minimum bandwidth :

recalling sinc

PCM Bandwidt

nW

nWt f

1

2

F

max

h:

Major error quantization error.

quantization error

'

q

B k knW

n B

1 1 1 2 2

sinc ff

Page 197: 通訊原理

Chap. 3 137137

frequency, time, spatial,...

3.7 Multiplexing By using the variety of of communication channel,

multiple number of messages can be transmitted in a common

nature features

commu

§ ncation link. Frequency-Division Multiplexing FDM Several message signals are first translated, using modulation, to different spectral locations and then added to form a base- ban

d signal. Bandwidth ,

: bandwidth of the message

: gard-band of the message.

i i

thi i

thi

W G

W i m t

G i

1G1W 2G2W

frequency

Page 198: 通訊原理

Chap. 3 138138

: DSB

: SSB

: Angle modulationN

m t

m t

m t12

Page 199: 通訊原理

Chap. 3 139139

cos v.s. sin

Quadrature Multiplexing QM - are used for frequency translations.

- Two messages are transmitted in same frequency band.

- Strictly speaking,

Quadra

QM

ture carri

is

es

c ct t not a frequency-division technique. - Modulation:

co s n s ic c c cx t A m t t m t t 1 2

sinc cA t sin ct 2

m t1 m t2

cosc cA t cos ct 2 ..... cx t rx t

DDy t DQy tLowpass filter

QDSB modulator QDSB demodulator

Lowpass filter

Dx t Qx t

Page 200: 通訊原理

Chap. 3 140140

local osc.

- Demodulation: Assume ,

Local oscillator has a phase error

cos

cos cos sin cos

cos co

r c

D r c

c c c c c

c

x t x t

x t x t t

A m t t t m t t t

A m t m t

1 2

1 1

2

2 s

sin sin

cos sin

Phase error on local oscillator causes both attenuation and crosstalk at

c

c

DD c

t

m t t m t

y t A m t m t

2 2

1 2

2

2 detection output. , DD cy t A m t 10

sin ct 2

cos ct 2 rx t

DDy t DQy tLowpass filter

Lowpass filter

Dx t Qx t

Page 201: 通訊原理

Chap. 3 141141

Time-divission Multiplexing TDM - Each message signal occupies a time-slot in every data frame of second. - Suitable for sampled digital signals.

T 分時多工

BW W 1

Nii

B W 1

BW W 2

NBW W s2

s1

Ns

Page 202: 通訊原理

Chap. 3 142142

- The minimum bandwidth of a TDM baseband signal: The message BW. Assuming Nyquist rate sampling, the total number of samples

for the TDM baseband signal in seconds is

thi

s

i W

T n

Assume the TDM baseband signal is a lowpass signal with bandwidth .

,

N

ii

N

s ii

N

ii

WT

B

n BT WT B W

1

1 1

2

2 2

Page 203: 通訊原理

Chap. 3 143143

- A TDM example: The Digital Telephone System T1-line PCM TDM Bandwidth of human voice : 4 voice channel: 8000 samples sec, 8 bits sample data rate

W KHz

bps

8000 8 64000 sec. .

A T1 frame contains 24 8

data bits plus

24 voice channels

one frame synchronous bit in 0.125 ms interval.

data rate

yield one T1 ca

of

rrier.

T ms sampling period 10 125

8000

T1-line . Mbps.

bitsms

24 8 11 544

0 125

10011100

time

1channel11110010

2channel00010010

24channel

Page 204: 通訊原理

Chap. 3 144144

Page 205: 通訊原理

Chap. 5 1

Chap5: Review of Probabilities and Random Variables Set Theory: Set collection of elements

Ex. All possible outcomes of tossing a dice. 1,2,3,4,5,6 Subsets:

S

B

, Venn diagram: always be used to express the relationships between sets.

Ex. : 1 10 , .

: 1 10 , , and is even

A x B x A

A X X X Z

B X X X Z X

B A 1, 3, 5, 7, 9,

2, 4, 6, 8, 10

A

B

Page 206: 通訊原理

Chap. 5 2

1. Intersection (production)2. Union (Addition)

Set operations : 3. Complement4. Difference

1. Intersection: , or

and .

2. Union : or

or

A B A B

A B x x A x B

A B A B

A B x x A x

.

3. Complement :

; and

4. Difference: and

B

A

A S A S A x x S x A

A B x x A x A B

Page 207: 通訊原理

Chap. 5 3

, De Morgan s Law :

Ex. , 2,4,9

5,6,7,8

1,3

5,6,7,8 1,3

A B A B

A B A B

A B S A B

A S A

A B

A B

A B

2 4 9

S

5 6

7 8

BA1 3

Page 208: 通訊原理

Chap. 5 4

1 2

1

1

1

Mutually Exclusive or disjoint , and are mutually exclusive. Exhaustive:

, , .

Partition: ; 1, , ;

; ,

, ,

N

N ii

i

i j

N

ii

A B A B

A A A A S

A S i NA A i j

A S

A A

form a partition of .N S

1A 2A

3ANA

S

Page 209: 通訊原理

Chap. 5 5

Review of Probabilities and Random Variables Given a chance experient,

: Total number of equally likely and mutually exclusive outcomes : number of outcomes included in an event .

A

NN A

2

: headEx. Tossing a fair coin twice

: tail

,

two tossing with the same outcome1 4 , =1, ( ) ,4

1 , , , , .2B

A

A

N

NP AN

HT

A H T

B

N N A P A

B H H T T P B ,T H

,H H ,T T

,H TA

B

Page 210: 通訊原理

Chap. 5 6

Probability : Relative frequency of occurance

lim .

Sample space collection of all possible

outcomesof a chance experiment

Axioms of Proba

A

N

NP AN bilities:

. 0 ; , ~ non-negativity

. 1,~ Normalization

. ,

i P A A

ii P

iii A B P A B P A P B

Page 211: 通訊原理

Chap. 5 7

Ex. Tossing a fair dice 1,2,3,4,5,6

event 1 ,

even points ,

odd points1 61 2

1 1 2 , 6 2 31 , , 2

A

B

C

P A

P B P C

P A B A B

P A C A C P A C P A P C

1P B C P A B

C

Page 212: 通訊原理

Chap. 5 8

Some useful probability relationships:

1 , 1

Conditional Probability :

Probability of , given that event has been known.

, or

P A B P A P B P A B

P A A P P A P A P A P A

P A B

P A B A B

P ABP A B P AB P A

P B

B P B

Page 213: 通訊原理

Chap. 5 9

Ex. Tossing a fair dice: 1 , odd points , even points , 2

116 ; 1 3

20

, , are inde

012

16 1.

penden

16

t.

A B C D

P A BP A B

P B

P A B P

P A CP A C

P C

P A BP B A

P A

A A B

A

BC

D

Page 214: 通訊原理

Chap. 5 10

,T H ,T T

A

B

, ,H T ,H H

if , are independent, ,

.

1 tossing is Ex. Tossing a fair coin twice, define ,

2 tossing is

Are A, B inde

st

nd

P ABA B P A B P A

P B

P AB P A P B

A H

B T

p.?

Page 215: 通訊原理

Chap. 5 11

1

1

1

Bay's theorem:

; is a partition of , i.e., ,

event ,

.

i jN

Ni i

ii

jj

j jN

i ii

A A i jA

A

B

P A BP A B

P B

P B A P A

P B A P A

B

1A

2A

iA

NA

Page 216: 通訊原理

Chap. 5 12

Ex. A binary symmetric channel BSC is shown,

0 0.3

given 0 0 0.8 .

1 1 0.8

Find 1 , 1 1 , 1 ,

and error transmission ?

S

R S

R S

S S R R

P

P

P

P P P

P

1S

0S 0R

1R

(0 0 ) 0.8R SP (1 1 ) 0.8R SP

Page 217: 通訊原理

Chap. 5 13

Page 218: 通訊原理

Chap. 5 14

Random Variables r.v. :

A ramdom variable is a mapping or function which assigns a real number to each outcome of a chance experiment.

: , X

12

q X

Page 219: 通訊原理

Chap. 5 15

Ex. Flipping a fair coin twice , you would possiblely win $20, if you have , , you would loose $5 if you have , or

, ; otherwise, you would loose $10. Let being the . which represents the

H H H T

T H X r v money you win from the game, please find 5 ?, 10 ?, 20 ?.

.1 5 , 4

3 10 , , , 4 20 1

P X P X P X

Sol

P X P T T

P X P H T T H T T

P X P

,H T ,T H ,T T ,H H

20

510

Page 220: 通訊原理

Chap. 5 16

1

1. Discrete r.v., exists only on discrete points on the

real lin

Probability mass fu

e, ,

.

2. Continuous r.v. could be any values on th r.

nc

v.:

tion X i

qi

i

i

X

X x

X

P x P X x

Cumulative distribution function . . :

Probability density function . . :

e real line.

,

= .

X

X X

c d f

F x P X x

p d fdfx

X

x F xd

Page 221: 通訊原理

Chap. 5 17

1 For discrete r.v.:

Cumulative distribution function . . .

Properties of :

1. 0 1, and

0

. righ

q

X X i ii

X i X i X i

X

XX

X

X

f x P x x x

P x F x F x

c d f

F x

Fi F x

F

F

i

x

i

x P X

0 0

2 1 2 1

t continuous :

. non-decreasing: ,X X

X X

F x F x

iii x x F x F x

Page 222: 通訊原理

Chap. 5 18

Ex. Find the of the previous coin-tossing gamble.XF x ,H T ,T H ,T T ,H H

20

510

Page 223: 通訊原理

Chap. 5 19

20510 x14

12

14

1 1 110 5 204 2 4XF x u x u x u x

Page 224: 通訊原理

Chap. 5 20

1 2 2 1

0 0 0

0

0

=

Probability density function . . .

lim

lim

Proper

X X

X X

X XX X x

x

P x X x F x F x

P X x F x F x

p d f

F x x F xdf x F xdx x

P x X x xx

2

1

1 2 2 1

.

. ties of :

. 1

x

X X

X X

xXXx

X X

a F x f d

b P x X x F x F xf x

f d

c f x dx F

Page 225: 通訊原理

Chap. 5 21

0

2

0

Ex. Rotating a spinning pointer, denotes the angle readings1 , 0 2

~ 0,2 , ~ uniform distribution2 0 , o.w.

0, 0 01 , 0 2 =

21 , 2

2

f

F d

d

, 0

, 0 22 1 , 2

F 2 1

f 2 1

2

Page 226: 通訊原理

Chap. 5 22

Ex. For the previous flipping coin example 1 1 1 10 5 204 2 4

1 1 1 10 5 204 2 4

Joint ' and ' In the sense of event probability,

X

X X

F x u x u x u x

df x F x x x xdx

cdf s pdf s

2

event and both occur Similarily, in . . , and ,~ joint . . . and

, , , ~ joint . . .

XY

XYXY

P A B P A Br v s

F x y P X x Y y c d f

F x yf x y p d f

x y

Page 227: 通訊原理

Chap. 5 23

2 2

1 11 2 1 2 , ,

, , ( ) 1

, ,

, ,

, , ,

,~,

,

XY Y

XY X

y x

XYy x

XY

XY

XY X

XY Y

P x X x y Y y f x y dxdy

F P X y P

f x y dxdy P x dx X x y dy Y y

F x P X x Y P X x F x

F y P X Y y P Y y F y

f x y dx f y

f x y dy f x

Marginal densities.

Page 228: 通訊原理

Chap. 5 24

Independence: , r.v.s

, or

and are independent , or for discrete r.v.s

,

Conditional density function:

,, ,

XY X Y

XY X Y

XY X Y

XY YX YXY

X YY

X Y

f x y f x f y

X Y F x y F x F y

P x y P x P y

f x yf x y

f yf x y f x y f y

, are indep , . . XX YX Y f x y f x

Page 229: 通訊原理

Chap. 5 25

(2 )

Ex. Given two . . , with joint .

, 0, 0 ,

0 , o.w.

. ? , . ?, . ?, . Are , indep.?

. ?, . ?, and . ?

x y

XY

X Y

X Y X Y

r v s X Y pdf

Ae x yf x y

i A ii f x iii f y iv X Y

v F x vi F y vii f x y

Page 230: 通訊原理

Chap. 5 26

Page 231: 通訊原理

Chap. 5 27

0

2

1 , 0, 0.

0 , 0 0 , 0

. , indep., 2

I. 1 rv. to 1 rv. Transformation of random variables:

II. 2 rvs. to 2 rv

y yy YY Y

xXX Y

e yf d yvi F y f d

yy

vii X Y f x y f x e u x

1

1

s.

I. 1 rv. to 1 rv. transformation:

Given ~ , let , ?

1. From

Steps:

, Find ( ), 1,

,

2.

i

NX i

Y

X

i i

Y

Y g X x g y i

X f x Y g X

N

f y

f xf y

g x

Page 232: 通訊原理

Chap. 5 28

1

1 ;0 2

1 , Ex. Given ~ , 2

0 , o.w.

1, ?

Sol.

1. 1, ,

1 , 0 2, 1=

1 , 0 2 2. 2

0,

Y

Yy

y

f

Y f y

g y

g y y y

dg gd

f yf y

g . . o w

Page 233: 通訊原理

Chap. 5 29

1

2

12

2

1 2

21 2

1 1 2

Ex.

~ 0.5

, ?

Sol.

1. , ,

2 , and 2

2. 2

0

xX

Y

x y

yX i X X

Yi i

X f x e

Y X f y

x yy x

x y

dyg x y g x ydx

f x f x f x ef y u yg x g xg x y

y

Page 234: 通訊原理

Chap. 5 30

?

II. Two r.v.s. to two r.v.s. transformation:

, , , , , . . ; , ,

,

Steps: 2.

, ,1. From , find , 1, ,

, ,

Jacobian:

XY ZW XY

i

i

i

Z g X YX Y f x y r v s f z w f x y

W h X Y

z g x y x M z wi n

w h x y y N z w

g gx y

Jhx

,,

1

, , ,3. ,

ii

x M z wy N z w

ni iXY

ZWi i

hy

f M z w N z wf z w

J

Page 235: 通訊原理

Chap. 5 31

2 2

1

2 2 2 2

2 2 2 2cos cossin sin

Ex. , , , , , ?Tan

Sol.

cos 1. ,

sin

2.

XY R

x r x ry r y r

R X YX Y f x y f rY

X

x ry r

x yR Rx y x y x yJ

y xx y x y x y cos sin

1 sin cos

3. , cos , sinR XY

rr r

f r r f r r

Page 236: 通訊原理

Chap. 5 32

2 22

2

1 1 12 22 2 2 2 2

2 2 22 2

0,. , independent; ;

0,

. , ?, . ?, . ?, . , indep.?

Sol.

. , ;

,1 1 1 ,

,1 1

ZW Z W

Z X YX NEx X Y YY N W

Xa f z w b f z c f w d Z W

ya w y wxx

z wzx y wxw wz x y x wz wzx y wxw w

Page 237: 通訊原理

Chap. 5 33

Page 238: 通訊原理

Chap. 5 34

Page 239: 通訊原理

Chap. 5 35

2 2

, independent Gaussian r.v.s.,

distributed

distributed

, independent

X Y

Z X Y RayleighYW CauchyX

Z W

Page 240: 通訊原理

Chap. 5 36

Statistical Averages Mean or statistical average

~ ,

,~ mean value of

For diserete . . ,

X

X

X X i ii

X i X ii

X f x

E X X x f x dx X

r v s f x P x x x

E X x f x dx x P x

Y

~ . .

, , , ~ , ,

, ,

Y

Y

XY

X

X

g X r v

E Y E g X g x f x dx y f y dy

Z g X Y X Y

E Z g x y f x y dxd

f y

y

x

Page 241: 通訊原理

Chap. 5 37

2

2

1 ; Ex. ~ , cos 2

0 ; o.w

02 .

1 cos 02

Ex.1 1 1,2,4 , 1 , 2 4 ; log2 4

1 1 12 4 22 4 4 log

X X X

f X

E f d d

E X d

X P P P Y X

E X

E Y

2 21 1 1 31 log 2 log 4 .2 4 4 4

Page 242: 通訊原理

Chap. 5 38

1 1

,

pf. ,

= , ,

, , ~ . . , , , ,~ cons

X Y

XY

XY XY

f x f y

N N

Z X Y E Z E X E Y

E Z x y f x y dxdy

x f x y dydx y f x y dxdy

E X E Y

X X r v s a a

1 1

, are indep

tants

, . ., ,

.

N N

i i i ii i

XY X Yi e f x y f

E a X a E X

X Y

E XY E X

x y

E

f

Y

Page 243: 通訊原理

Chap. 5 39

2

2 2

2 2 2

1

Variance: ~ ,

,~ Variance of

{ 2 }

{ }, are randomstatistically independent

X x

X

x x

X X

N

X f x E X

Var X E X X

E X X

E XX X

1 2 1 2

1

2

1

1

2

1

variables,

. ., ( , , ) ,

and are constants;

N i

N

X X X N X

N N

i ii

i

ii

i

N

iV

i e f x x x f x

a

a

ar a X

a

Page 244: 通訊原理

Chap. 5 40

1 1

2

1 1 1

2

1

pf. ,

N N

i i i i i ii i

N N N

i i i i i ii i i

N

i i ii

E a X a E X

Var a X E a X a

E a X

2

0,

, 1

, 1

2 2

,

1

, 1

i

N

i j i i j ji j

N

i j i i j ji j

N N

i j i j i j i

i ji j

ii j i

E a a X X

a a E X X

a a E X X a

Page 245: 通訊原理

Chap. 5 41

Conditional mean:

The expected value of given ,

is a function of sti,

ll

a

r.v.

X Y

Y

YX Y

E X Y

X Y y

E X Y y x f x y dx

E X Y y Y

E E X Y y E X Y y f y dy

x f x y f y

,

For discrete r.v.s: i

i

XY

i i i iX Yx

i Y iy

dxdy

x f x y dxdy

E X

E X Y y x P x Y y

E X Y y P y E X

Page 246: 通訊原理

Chap. 5 42

2 2

Covariance and the Correlation Coefficients , ~ , ,

, , , Covariance Coefficient:

~ A measure of the similarity be

XY

X X Y Y

XY X Y X Y

X Y f x y

E X Var X E Y Var Y

E X Y E XY

tween and . , 0 , are uncorrelated.

, are indep.,

X Y XY

X YE XY

X Y

X Y

, are uncorrelated.X Y

Page 247: 通訊原理

Chap. 5 43

Correlation coefficiant :

, are uncorrelated, 0

Ex. Two random variables are related as , where is a constant. Please determine ?

XYXY

X Y

XY

XY

X Y

Y X

Page 248: 通訊原理

Chap. 5 44

Page 249: 通訊原理

Chap. 5 45

1 21 2 1

Bernoulli Trial: 0,1 , 0 , 1 1 Binomial distribution:

Consider a sequence of independently and identically distributed i.i.d. Bernoulli trials,

, , , , ,

n

X X

N X X X

X P p P p q

n

X X X P x

21

1 2

0

0 0

, ,

Let number of 0's in , , ,! ,

! !

. 1

n

N X ii

n

n k n k k n kY k

n

Y Yk

n nnn k n k

Y Y kk k

x x P x

Y X X XnP Y k P k C p q p q

k n k

f y P k y k

i f y dy P k C p q p q

Page 250: 通訊原理

Chap. 5 46

2

2

0

11

1

2

1

2

.

1 !

1 ! !

1 ! .

1 ! !

1 Gaussian distribution: ~ , , ( )= ,2

n

Yk

nk n k

k

nnk n k

k

x

X

ii E Y kP k

n np q

k n k

nnp p q np p q np

k n k

X N f x e

E X

2

2

2

2

2

2

2

, .

1 2

1 12

x

X X

X

Var X

F x P X x e d

F e d

Page 251: 通訊原理

Chap. 5 47

2

2

2

2

2

2

1-function: 2

Properties :

1 . 12

1 . 1 , 2

. 1

x

tx

Q Q x e d

i Q e d

ii e dt Q x

iii Q x Q x

-4 -3 -2 -1 0 1 2 3 4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

zf x(z

) Q x

Page 252: 通訊原理

Chap. 5 48

2

2

2

2

2

2

2

24

50

25 2

2 2

2

2

1 Ex. ? 5 2

2 1 2 Sol. let , 1 .5 52

1 ~ , , ,2

1 ( )2

let ,

1 12

x

y

x

X

x

X

x y

X

e dx

xy e dy Q

X N f x e

F x e d

F x e dy

y

xQ

Page 253: 通訊原理

Chap. 5 49

2

2

2

2

2

Ex. ~ ,

, where is a const., ?

, 1 ;

1 2

~ ,

Y

yX

Y X

X N

Y X f y

g x x g x

f yf y f y e

g x

Y N

Page 254: 通訊原理

Chap. 5 50

Page 255: 通訊原理

Chap. 6 1

Random Processes r.p. :

For a r.v., : maps each in to a real number.

For r.p., : , , maps each in to a time fn..

,Chap 6:

X X

X t X t

t

,X t ,X t ,X t

,X tX

1t 2t 1 1, ~ r.v.X t X 2 2,

~ r.v.X t X

Page 256: 通訊原理

Chap. 6 2

1 1

for a fixed time , , ~ . .

a sequence of time instances , , , , ~ a sequence of r.v.s. for a fixed state

i i i

XN N

t t X t X r v

t t X X

1 1 1 1

, ( , ) ~ a sample fn.

, = collection of all possible sample fns, , ;

~ Ensemble , , ~ r.v., ( , )

k k

i

X

X t

X t X t i

t t X t F x t P X x

~ order . . .

( , ) is time-varying.( , ) ( , ) , ~ order . . .

X

XX

lst c d fF x t

F x tf x t lst p d fx

Page 257: 通訊原理

Chap. 6 3

1 1Ex. Tossing a fair coin with , 2 2, sin

, ,

Find ( , ) and ( , )1 for 0, , and 12

X X

P H P T

X H t tX t

X T t t

F x t f x t

t

H

T

,X tt

112

1 ,X T t ,X H t

0

Page 258: 通訊原理

Chap. 6 4

Page 259: 通訊原理

Chap. 6 5

1 1

2 2

, ,~ . .

, , ~ . .

, , ~ . . let

, , ~ . .

First order c.d.f:

, , 1, ,

, , , ~ 1 order

i

i

i

n n

X i i i i

X i i stX i i

i

X t R P

X t X r v

X t X r v

X t X r v

F x t P X t x i n

F x tf x t pd

x

★ 1 2

1 2

1 2

1 2 1 2 1 1 2 2

21 2 1 2

1 2 1 21 2

.

Second order c.d.f: , ; , and

, ; , , ; , ,~ 2 order .

X X

X X ndX X

f

F x x t t P X x X x

F x x t tf x x t t pdf

x x

t

,X t1t 2t 1 1,

~ r.v.X t X 2 2,

~ r.v.X t X

Page 260: 通訊原理

Chap. 6 6

1 2

222 2

1 2 1 2

1 2 1 2

Mean and Autocorrelation: ~ , Mean:

,

Variance:

Autocorrelation:

,

, ;

X

X X

X

X

X X

X t f x t

E X t xf x t dx m t X t

t E X t X t X t X t

R t t E X t X t

x x f x x

★ 1 2 1 2

2 1

,

,X

t t dx dx

R t t

Page 261: 通訊原理

Chap. 6 7

1 2 1 1 2 2

1 2 1 2

1 2 1 2

21 1 1

1 2 1 2

Autocovariance function

,

,

, Cross-correlation fn.:

~ . ,

,

X

X

X X

XY YX

C t t E X t X t X t X t

E X t X t X t X t

R t t X t X t

C t t t

X tr p

Y t

R t t E X t Y t R

★ 2 1

1 2

2 1

,

Wide Sense Stationary W.S.S : ,~ . .

1. constant If ,

2. ,

in which , then is W.S.S.X X

t t

X t r p

E X t

R t t R E X t X t

t t X t

t

,X t1t 2t 1 1X t X 2 2X t X

t

,Y t1t 2t 1 1Y t Y 2 2Y t Y

Page 262: 通訊原理

Chap. 6 8

2 2

Ergodic ~ . .

Define the time average of : 1 lim

2,

,

~ Ergodic

T

TT

X

X t r p

X t

X t X t dtT

E X t X t

E X t X t

R E X t X t X t X t

X t

Page 263: 通訊原理

Chap. 6 9

0Ex. cos 2

2 , ~ 4 4

0, o.w.

Please determine if is ergodic?

X t A f t

f

X t

Page 264: 通訊原理

Chap. 6 10

Page 265: 通訊原理

Chap. 6 11

0 0

Ex. cos 2 ; , ~ constant

1 , ~ ,2

0 , o.w.

Is an ergodic process?

n t A f t A f

f

n t

Page 266: 通訊原理

Chap. 6 12

Page 267: 通訊原理

Chap. 6 13

Page 268: 通訊原理

Chap. 6 14

2 2

2 2 2

22

2 2

,~ Ergodic Process

1. ~ D.C. component of

2. , ~ Total power of

3.

,~ A.C. power of

4. , ~ power of ( )

X X

X t

E X t X t X t

E X t X t X t

E X t m

X t X t X t

X t X t DC x t

Page 269: 通訊原理

Chap. 6 15

2

0

0

Properties of :

~ W.S.S.,

1. 0 ,

2. , ~ even symmetric

3. lim

4. If ,~ periodic

then

Please prove the above

X

X

X X

X X

X

X X

R

X t R E X t X t

R R

R R

R X t

X t X t T

R R T

properties.

Page 270: 通訊原理

Chap. 6 16

Page 271: 通訊原理

Chap. 6 17

Page 272: 通訊原理

Chap. 6 18

0,: ,

Ex. ,

, ,2 2

, and independent r.v.s; : deterministic pulse function

kk

k m m kk

k

k

X

k mk m

E aa

E a a RX t a p t kTT TU

a p t

R E X t X t

E a a p t kT p t mT

2

2

1

k mk m

T

Tm kk m

u

E a a E p t kT p t mT

R p t kT p t mT dT

Page 273: 通訊原理

Chap. 6 19

2

2

2

2

1

change variable, let ,

1

let , , ,

1

Tt kT

TX m k t kTk m

Tt kT

Tn

n

t kTk

t kT u

R R p u p u m k T duT

m k n m k n

R p u p u nT duT

,

1where

pulse correlation fn. of the deterministic pulse .

p u p u nT d

n

nn

u r nTT

R r nT

r p t p t dtT

p t

Page 274: 通訊原理

Chap. 6 20

. Binary random waveform:

,

1; , ;2

, are independent,

?

kk

k

k k

k n

X

Ex X t a p t kT

Aa

Atp t P a A P a A kT

a a k n

R T

2T3T

4T

A

A t

T 3Tt

Page 275: 通訊原理

Chap. 6 21

2 2sincXS f A T Tff

Page 276: 通訊原理

Chap. 6 22

2

2

Wiener-khinchine theorem ~ W.S.S

,

0, o.w.

,

~ random1 lim

2

T

T j ftT T T

T

X TT

X t

X t T t TX t

f X t X t e dt

f

S f E fT

F★ 2

Wiener-khinchine theore

~ Average power d

m:

ensity Spect

i.e.,

ra o

f

X X

j fX X X

R E X t X t S f

R R e d S

X

f

t F

F

Page 277: 通訊原理

Chap. 6 23

2

2

2

22 2

2 2

.

a real random process

, Define: , energy signal

0, o.w.

lim2

1 lim21 lim

2n

T

T

n T

T T j f tT TT TT

T TTR

pf

n t

n t t Tn t

E n tS f

T

E n t n e dtdT

E n t nT

F 2

21 lim2

T T j f t

T Tt

T T j f tnT TT

e dtd

R t e dtdT

Page 278: 通訊原理

Chap. 6 24

1 1, Jacobian: 1,

0 1

2 0, 0 2 ,0 : , 0 :

u uu t tlet J dudv dtd

v vv tt

T u u Tu u

T v u T u T v T

tT T

T

T tT T

T

T t t

0,,

0,u t u t

letv t u t

v

u2T 2T

T

T

Page 279: 通訊原理

Chap. 6 25

0 22 2

2 0

0 22 2

2 0

0 22 2

2 0

1 lim21 lim 2 2

21 lim 2 2

2

u T T Tj fu j fun n nT T u TT

Tj fu j fun nTT

Tj fu j fun nTT

S f R u e dv du R u e dv duT

T u R u e du T u R u e duT

T u R u e du T u R u e duT

2

1, a

2

2

2

s

lim 12

T j funTT

j f

T

un n n

uR u e du

T

R u e du R R

F F

Page 280: 通訊原理

Chap. 6 26

0

2

0

2 2

0 0

1 , Ex. cos 2 ; ~ 2

0, o.w.

cos22

4 4

Ex. The power spectral density . . of a bandlimited is defined as

X

X X

X t A f t f

AR f

A AS f R f f f f

P S D n t

F 0 0

10

, =2

2 2 0 , o.w.

sinc2

, : white noise.

n

n n

N B f B N fS fB

R S f N B B

B n t

F

Page 281: 通訊原理

Chap. 6 27

0

0

,2 white noise :

2

n

n

NS fn t

NR ★ nS f

0

2N

f

Page 282: 通訊原理

Chap. 6 28

Cross correlation function , . .

Define: Cross correlation function

A measure of the similarity between and in the time-domain.

pf.

XY

XY YX

X t Y t R P

R E X t Y t

X t Y t

R R

★ , let

. Q.E.D.

XY

YX

R E X t Y t t t

E X t Y t

R

Page 283: 通訊原理

Chap. 6 29

Cross power spectral density:

,

A measure of the similarity between the two random processes in the frequency-domain. 0, ; , orthogonal. In general, the orthogonality of

XY XY

XY

S f R

R X t Y t

F

two R.P. is nothing to do with the statistically independent.

Page 284: 通訊原理

Chap. 6 30

Let ,

Powe

n

X XY XY Y

n t X t Y t

R E n t n t

E X t Y t X t Y t

E X t X t E X t Y t

E Y t X t E Y t Y t

R R R R

2

r of :

0 0 2 0 0

where cross power

and are orthogonal, .

YX XY

n n X XY Y

PP P

XY

n X Y

n t

P E n t R R R R

P E X t Y t

X t Y t P P P

Page 285: 通訊原理

Chap. 6 31

Random Processes through a LTI system

, W.S.S.,

,

?

X

X

X

E X tX t

E X t X t R

Y t X t h t X t h d

h t H f

E Y t

R

★ ???

?

Y H fXX

YYX

Y

E X t

RR

S fS f

X t Y t h t

Page 286: 通訊原理

Chap. 6 32

2

0

Y x

XY XX

YY XX

Y X

H

R R h

R R h h

S f H f S f

X t Y t h t

Page 287: 通訊原理

Chap. 6 33

Page 288: 通訊原理

Chap. 6 34

Page 289: 通訊原理

Chap. 6 35

1

1

1 1

1

21

22

Gaussian . .

let , ~ r.v.s,

, , , , , ,

{ } ~ autocovariance matr

1 If , , wh2

ix

i

n

n

T

X i

n

TTn X X

T T

n

R P

X t XE X

X t

n f

X X

e

X

E

X X Xx

X

X

μ C x μ

X

X X

X

X μ

C X X μ

C

μ

x

★ 2

2

1 2

2 2

2

ere , , , ,

then is a Gau

, ,

1 , ~

ssian proc

,2

ess.

i Xi

i

i i

T

i i i

x

X i

n

i X

i

t t X t

x x x

X

X

f x e N

t x

Page 290: 通訊原理

Chap. 6 36

~Gaussian

pf.

~ Gaussian R.P. ~ Gaussian R.P. LTI

k

k

X

k

Y t X h t d

X k h t

X

k

t Y t ★ , 0

, linear combination of Gaussian . . .

~ Gaussian process

k kk

X r v s

Y t

h t X t Y t

Page 291: 通訊原理

Chap. 6 37

1 2

5

1 2 1 2

Ex. ~ Ergodic and Gaussian distributed.

3 ,

5 2 sinc2 , .2 2

1. find ; , 2. find ; ,

3. find , ; 1, 3

Sol.

XX

X Y

X X

X t

R e

fh t B Bt H f B HzB

f x t f y t

f x x t t

Page 292: 通訊原理

Chap. 6 38

Page 293: 通訊原理

Chap. 6 39

Page 294: 通訊原理

Chap. 6 40

Page 295: 通訊原理

Chap. 6 41

0

0 00

2

20 0 00

Noise equivalent bandwidth

: white noise,2

For an ideal bandpass filter:

Filter output:

2

o

n

N N

o

n n

N N

Nn t S f

f f f fH f HB B

n t

S f S f H f

N f f f fHB B

2 20 0output noise power: .

oo n NE n t S f df N H B n t ideal filterH f

f

0H

NB

on t

Page 296: 通訊原理

Chap. 6 42

2 2200 0 2

0

For a realistic filter : ?

Use an ideal filter to approximately evaluate the bandwidth of .Noise equivalent bandwidth:

1let , .2 2

r

r

r N N r

H f BW

H f

N H f df N H B B H f dfH

Page 297: 通訊原理

Chap. 6 43

Narrowband noise:Most communication signals are modulated within a frequency band ofwhich the bandwith is generally much smaller than the carrier frequency.Therefore, the in band noise can be expres

0 0

0 0 00

sed as a narrowband noisecos sin

where ; 2

, . .

0,2 ; is independent to

c s

n

c s

n t n t t n t t

N f f f fS f B fB B

n t n t R P

U n t

Page 298: 通訊原理

Chap. 6 44

2 202 2

cos

2 2

sin

To observe the realization of the narrowband noise, canreformulated as

cos

s

cc s

c sR t

t

s

c s

t

n t

n tn t n t n t t

n t n t

n t

n t n t

0

2 2

0 1

in

cos , c s

s

c

t

R t n t n tR t t t n t

t Tann t

Page 299: 通訊原理

Chap. 6 45

0 0cos sinc sn t n t t n t t

Page 300: 通訊原理

Chap. 6 46

1

0

1

0

1 0

0 0

0

0

cos 2 2

0

0

0

2 cos +

4 cos + cos +

2 cos

2 cos 2 2

2 cosnR E t

z

n

z

z t n t t

R E n t n t t t

E n t n t

E n t n t t

R

S

2

2 1

1

1

2

0 0

2 0 0

0 0

20 0

.

Similarily 2 si

Lp

, : Low-pass equivalent noises of the b

n + , 2 c

and

os

.

pa

c sn n z n n

z n n

z n

z z n

B

z

c s

f

n

S f S f S f H f S f f S f f

f F R S f f S f f

z t n t t R R

S f

n t n t

F R S f f S f f S f

ss .n t

Page 301: 通訊原理

Chap. 6 47

1 2

1 2

1 2

0 0

0

0 0

1

1

Cross-correlation:

4 cos + sin +

2 sin .

c s

z z

n

z z n n

n n c s

s

R E z t z t

E n t n t t t

R

S f j S f f S f f

R E n t n t

E z t h t n t

E z t

1

1

1

z n

s

s

s

s

n

R

z

h d n t

E z t n t h d

R h

Page 302: 通訊原理

Chap. 6 48

1 2

1

1 2

1 2

1

1 2

1 2

1 2

z

c s

s

zR

n n z z

z n s

z z

R E z t n t

E z t z t h t

E z t z t h d

E z t z t h d

R R

R

h

h

h

1 2

2

0 0

Lp

is imaginary,

must be an odd function of . Fourier property

It follows that if is continuo

c s

c s

c s

c s

n n z z

n n

n n

n n

n n

S f S f H f

j S f f S f f

S f

R

R

F us,

0 0 ~c sn n c sR E n t n t uncorrelated

Page 303: 通訊原理

Chap. 6 49

0

,

0 0, , if Lp 0

, if is a Gaussian R.P;

indep. Gaussian R.P.

. 0,

c s

c

s

s

cn n n n n n

c

s

c

s

LTI

n t n tuncorrelated

R S f S f f S f f

n tn t n t

n t

n t

n t

a E n t

E n

2 2

0

0 0

2

2 cos 0

. ,

Lp

=

1The joint pdf: , ; ,2

c

c s

c s

c s

c n n n

n s

n n

n n c s

t E n t t h t E n t

b Var n t N

Var n t S f df S f f S f f df

S f df N Var n t

f n n t t eN

N