outline transmitters (chapters 3 and 4, source coding and modulation) (week 1 and 2) receivers...

Outline• Transmitters (Chapters 3 and 4, Source Coding and

Modulation) (week 1 and 2)• Receivers (Chapter 5) (week 3 and 4) • Received Signal Synchronization

(Chapter 6) (week 5)

• Channel Capacity (Chapter 7) (week 6)• Error Correction Codes (Chapter 8) (week 7 and 8)• Equalization (Bandwidth Constrained Channels) (Chapter

10) (week 9)• Adaptive Equalization (Chapter 11) (week 10 and 11)• Spread Spectrum (Chapter 13) (week 12)• Fading and multi path (Chapter 14) (week 12)

Channel Capacity (Chapter 7) (week 6)

• Discrete Memoryless Channels

• Random Codes

• Block Codes

• Trellis Codes

Channel Models

• Discrete Memoryless Channel– Discrete-discrete

• Binary channel, M-ary channel

– Discrete-continuous• M-ary channel with soft-decision (analog)

– Continuous-continuous• Modulated waveform channels (QAM)

Discrete Memoryless Channel

• Discrete-discrete– Binary channel, M-ary channel

11

11

...

....

.)|(..

...)|(

Qq

jiji

p

pxyP

xXyYP

P

Probability transition matrix


• Discrete-continuous– M-ary channel with soft-decision (analog)

outputx0

x1

x2

.

.

.xq-1

y

)|(

)|( 1

kxXyp

xXyp

P

22 2/)(

2

1)|(

kxy

k exyp

AWGN


• Continuous-continuous– Modulated waveform channels (QAM)– Assume Band limited waveforms, bandwidth = W

• Sampling at Nyquist = 2W sample/s

– Then over interval of N = 2WT samples use an orthogonal function expansion:

)(

)(

1

tfx

txN

iii

)(

)(

1

tfn

tnN

iii

N

iii tfy

ty

1

)(

)(


• Continuous-continuous– Using orthogonal function expansion:

)(

)(

1

tfx

txN

iii

)(

)(

1

tfn

tnN

iii

N

iiii

N

ii

T

i

N

ii

T

i

N

iii

tfnx

tfdttftntx

tfdttfty

tfy

ty

1

10

*

10

*

1

)(

)()()()(

)()()(

)(

)(


• Continuous-continuous– Using orthogonal function expansion get an

equivalent discrete time channel:

Nx

x

.

.

.

.1

Ny

y

y

.

.

.2

1

22 2/)(

2

1)|( iii xy

i

ii exyp

111 nxy Gaussian noise

Capacity of binary symmetric channel

• BSC

pp

pp

1

1P

}1,0{ }1,0{ YX

0 0

11

X Yp1

p1

p p


• Average Mutual Information 0 0

11

X Yp1

p1

p p

)1()1()0(

1log)1)(1(

)1()0()1(log)1(

)1()1()0(log)0(

)1()0()1(

1log)1)(0(

)1(

)1|1(log)1|1()1(

)0(

)1|0(log)1|0()1(

)1(

)0|1(log)0|1()0(

)0(

)0|0(log)0|0()0();(

XPpXpP

ppXP

XpPXPp

ppXP

XPpXpP

ppXP

XpPXPp

ppXP

YP

XYPXYPXP

YP

XYPXYPXP

YP

XYPXYPXP

YP

XYPXYPXPYXI


• Channel Capacity is Maximum Information– earlier showed:

0 0

11

X Yp1

p1

p p

pppp

XPpXpP

ppXP

XpPXPp

ppXP

XPpXpP

ppXP

XpPXPp

ppXPYXI

XPXP

2log)1(2log)1(

)1()1()0(

1log)1)(1(

)1()0()1(log)1(

)1()1()0(log)0(

)1()0()1(

1log)1)(0());(max(C

2

1)0()1(

2

1)0()1());(max( XPXPYXI

Capacity of binary symmetric channel• Channel Capacity

– When p=1 bits are inverted but information is perfect if invert them back!

0 0

11

X Yp1

p1

p p

pppp 2log)1(2log)1(C 22

Capacity of binary symmetric channel• Effect of SNR on Capacity

– Binary PAM signal (digital signal amplitude 2A)

0 0

11

X Yp1

p1

p p

)(tg

)(tg)(tg )(tg

)(tgA

A

02

02

/)(

0

2

/)(

0

1

1)|(

1)|(

Nr

Nr

b

b

eN

srp

eN

srp

AGWN

Capacity of binary symmetric channel• Effect of SNR


0 0

11

X Yp1

p1

p p

)(tg

)(tg)(tg )(tg

)(tgA

A

)|(2

1)|(

20

0 /)(

0

10

2

sePN

Q

dreN

seP

b

Nr b

0

1s



0 0

11

X Yp1

p1

p p

0

0

22

112

1

2

12

22

2

)|()|(

NAQ

QSNRQ

NQ

sePsePP

bb

b

b



0 0

11

X Yp1

p1

p p

noise rms

22

4

22

2

12

2 noise rms

2

1

2

1

0

0

0

AQ

AQ

NAQ

NAQP

N

b

Not sure about thisDoes it depend on bandwidth?



0 0

11

X Yp1

p1

p p

noise rms 2

Amplitudeerfc

2

1

noise rms

Amplitude

2

1

2

1QpPb

pppp 2log)1(2log)1(C



0 0

11

X Yp1

p1

p p

SNR Pb1 0.3085382 0.1586553 0.0668074 0.022755 0.006216 0.001357 0.0002338 3.17E-059 3.4E-06

10 2.87E-0711 1.9E-0812 9.87E-1013 4.02E-1114 1.28E-1215 3.19E-1416 6.11E-16

Pb (BER) vs SNR for binary channel

1.40E+01, 1.28E-12

1.20E+01, 9.87E-10

1E-16

1E-15

1E-14

1E-13

1E-12

1E-11

1E-10

1E-09

1E-08

1E-07

1E-06

1E-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

SNR = A/rms noise

BE

R

noise rms 2

Amplitudeerfc

2

12

1bP



0 0

11

X Yp1

p1

p p

pppp 2log)1(2log)1(C 22 SNR Pb C

0 0.5 01 0.308538 0.1085222 0.158655 0.3689173 0.066807 0.6461064 0.02275 0.8433855 0.00621 0.9455446 0.00135 0.9851857 0.000233 0.9968578 3.17E-05 0.9994819 3.4E-06 0.999933

10 2.87E-07 0.99999311 1.9E-08 0.99999912 9.87E-10 113 4.02E-11 114 1.28E-12 115 3.19E-14 116 6.11E-16 1

Capacity © and BER vs SNR for binary channel

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

SNR = A/rms noise

BE

R

noise rms 2

Amplitudeerfc

2

12

1bP

At capacity SNR = 7, so waste lots of SNR to get low BER!!!

Capacity of binary symmetric channel• Effect of SNRb


0 0

11

X Yp1

p1

p p

pppp 2log)1(2log)1(C 22 SNRb (dB)Pb C

-20 0.443769 0.009143-18 0.429346 0.014452-16 0.411325 0.022809-14 0.388906 0.03591-12 0.361207 0.056319-10 0.32736 0.087793-8 0.286715 0.135561-6 0.239229 0.206245-4 0.186114 0.306729-2 0.130645 0.4407970 0.07865 0.6025972 0.037506 0.7692614 0.012501 0.903056 0.002388 0.9757578 0.000191 0.997366

10 3.87E-06 0.99992512 9.01E-09 114 6.81E-13 1

Capacity C and BER vs SNR for binary channel

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-20 -12 -4 4 12

SNR per bit (dB)

Ca

pa

cit

y C

an

d B

ER

b

bb QpP

erfc2

1

2

Channel Capacity of Discrete Memoryless Channel

• Discrete-discrete– Binary channel, M-ary channel

11

11

...

....

.)|(..

...)|(

Qq

jiji

p

pxyP

xXyYP

P

Probability transition matrix


Average Mutual Information

1

0

1

1

1

0

1

1

)(

)|(log)|()(

)(

)|(log)|()();(

q

j

Q

i j

jijij

q

j

Q

i j

jijij

yP

xyPxyPxP

yYP

xXyYPxXyYPxXPYXI


Channel Capacity is Maximum InformationOccurs for only if Otherwise must work out max

1)( ,0)(

1

0

1

1)(

1

0

1

1)()(

1

0

)(

)|(log)|()(max

)(

)|(log)|()(max);(max

q

jjj

j

jj

xPxP

q

j

Q

i j

jijij

xP

q

j

Q

i j

jijij

xPxP

yP

xyPxyPxP

yYP

xXyYPxXyYPxXPYXIC

jpxP j allfor ,)( symmetric P

)( jxP

Channel Capacity Discrete Memoryless Channel

• Discrete-continuous

• Channel Capacity

x0

x1

x2

.

.

.xq-1

y

)|(

)|( 1

kxXyp

xXyp

P

1

0

1

0)()(

)|()()(

where

)(

)|(log)|()(max);(max

q

iii

q

i

iii

xPxP

xXyYpxXPyYp

dyyYp

xXyYpxXyYpxXPYXIC

ii


• Discrete-continuous

• Channel Capacity with AWGN

x0

x1

x2

.

.

.xq-1

y

22 2/)(

2

1)|(

kxy

k exyp

1

01

0

2/)(

2/)(

2/)(

)( 22

22

22

2

1)(

2

1

log2

1)(max

q

iq

i

xyi

xy

xyi

xPdy

exXP

eexXPC

i

i

i

i


• Binary Symmetric PAM-continuous

• Maximum Information when:

x0

x1

x2

.

.

.xq-1

y

2

1)()( AXPAXP

dyeee

e

dyeee

eC

y

AA

A

y

AA

A

22

2222

22

22

2222

22

2/

2/2/

2/2

2/

2/2/

2/2

2

1

2log

2log

2

1




dyeee

e

dyeee

eC

y

AA

A

y

AA

A

22

2222

22

22

2222

22

2/

2/2/

2/2

2/

2/2/

2/2

2

1

2log

2log

2

1



• Versus Binary Symmetric discrete SNRb (dB)Pb C-20 0.443769 0.009143-18 0.429346 0.014452-16 0.411325 0.022809-14 0.388906 0.03591-12 0.361207 0.056319-10 0.32736 0.087793-8 0.286715 0.135561-6 0.239229 0.206245-4 0.186114 0.306729-2 0.130645 0.4407970 0.07865 0.6025972 0.037506 0.7692614 0.012501 0.903056 0.002388 0.9757578 0.000191 0.997366

10 3.87E-06 0.99992512 9.01E-09 114 6.81E-13 1

Capacity C and BER vs SNR for binary channel

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-20 -12 -4 4 12

SNR per bit (dB)

Ca

pa

cit

y C

an

d B

ER


• Continuous-continuous– Modulated waveform channels (QAM)– Assume Band limited waveforms, bandwidth = W

• Sampling at Nyquist = 2W sample/s

– Then over interval of N = 2WT samples use an orthogonal function expansion:

)(

)(

1

tfx

txN

iii

)(

)(

1

tfn

tnN

iii

N

iii tfy

ty

1

)(

)(


• Continuous-continuous– Using orthogonal function expansion get an

equivalent discrete time channel:

Nx

x

.

.

.

.1

Ny

y

y

.

.

.2

1

22 2/)(

2

1)|( iii xy

i

ii exyp

111 nxy Gaussian noise


• Continuous-continuous

• Capacity is (Shannon))(

)(

1

tfx

txM

iii

)(

)(

1

tfn

tnM

iii

M

iii tfy

ty

1

)(

)(

);(1

maxlim)(

YXIT

CxpT

ii

N

i i

iijii

NNN

NNNNNNN

dxydyp

xypxpxyp

ddp

pppI

WTN

NN

1 )(

)|(log)()|(

)(

)|(log)()|();(

2

XY

yxy

xyxxyYX

22 2/)(

2

1)|( iii xy

i

ii exyp




0

2

0

2

1 0

2

2

1

)(

21log

21log

2

1

21log);(max

NWT

NN

NI

x

x

N

i

xNN

xp

YX

22 2/

2

1)( xix

x

i exp

Statistically independent

zero mean Gaussian inputs

then



• Constrain average power in x(t):

22

1

2

0

2

2

)(2

1

)]([1

xx

N

ii

T

av

WT

N

xE

dttxET

P



• Thus Capacity is:)(

)(

1

tfx

txM

iii

)(

)(

1

tfn

tnM

iii

M

iii tfy

ty

1

)(

)(

0

0

2

)(

1log

21loglim

);(1

maxlim

WN

PW

NW

IT

C

av

x

T

NNxpT

YX

22 2/)(

2

1)|( iii xy

i

ii exyp



• Thus Normalized Capacity is:)(

)(

1

tfx

txM

iii

)(

)(

1

tfn

tnM

iii

M

iii tfy

ty

1

)(

)(

WCN

WN

C

CPWN

P

W

C

WCb

b

bavav

/

12

1log

but ,1log

/

0

02

02

22 2/)(

2

1)|( iii xy

i

ii exyp

etab/No (dB)C/W-1.44036 0.1-1.36402 0.15-1.24869 0.225-1.07386 0.3375-0.8075 0.50625

-0.39875 0.7593750.234937 1.1390631.230848 1.7085942.822545 2.5628915.41099 3.844336

9.669259 5.76650416.65749 8.64975627.92605 12.9746345.69444 19.4619573.22669 29.19293115.4055 43.78939179.5542 65.68408 0.1

1

10

-10 0 10 20 30

outline transmitters (chapters 3 and 4, source coding and modulation) (week 1 and 2) receivers...

Documents

channel capacity chapter

capacity snr

y slide

shannon slide

agwn slide

max slide

normalized capacity

y awgn slide