lecture 7 channel capacity

8/8/2019 Lecture 7 Channel Capacity

1/35

Outline

Transmitters (Chapters 3 and 4, Source Coding andModulation) (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)


2/35

Channel Capacity (Chapter 7) (week 6)

Discrete Memoryless ChannelsRandom Codes

Block CodesTrellis Codes


3/35

Channel Models

Discrete Memoryless Channel Discrete-discrete

Binary channel, M-ary channel

Discrete-continuousM-ary channel with soft-decision (analog)

Continuous-continuous

Modulated waveform channels (QAM)


4/35

Discrete Memoryless Channel

Discrete-discrete Binary channel, M-ary channel

-

!!!

!

11

11

...

....

.)|(..

...)|(

Qq

ji ji

p

p x y P

x X yY P

P

Probability transition matrix


5/35


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

output x

0 x1 x2..

. xq-1

y

-

)(

)( 1

k x y p

x y pP

22 2/)(

2

1)|( W

TW

k x yk

e x y p !AWGN


6/35


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 anorthogonal function expansion:

)(

)(

1

t f x

t x N

iii

!

!

)(

)(

1

t f n

t n N

i

ii!

!

!

! N

iii t f y

t y

1

)()(


7/35


Continuous-continuous Using orthogonal function expansion:

)(

)(

1

t f x

t x N

iii

!

!

)(

)(

1

t f n

t n N

iii

!! _ a

? A_ a

? A

!

!

!

!

!

!

!

!

N

iiii

N

ii

T

i

N

ii

T

i

N

iii

t f n x

t f dt t f t nt x

t f dt t f t y

t f y

t y

1

10

*

10

*

1

)(

)()()()(

)()()(

)(

)(


8/35


Continuous-continuous Using orthogonal function expansion get an

equivalent discrete time channel:

N x

x

.

.

.

.1

N y

y

y

.

.

.2

1

22 2/)(

2

1)( iii x y

i

iie x y p

W

TW !

111 n x y ! Gaussian noise


9/35

Capacity of binary symmetric channel

BSC

-!

p p

p p

1

1P

}1,0{ }1,0{ !! Y

0 0

11

X Y p1

p1

p p


10/35


Average Mutual Information 0 0

11

X Y p1

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()11(

log)11()1(

)0()10(

log)10()1()1(

)01(log)01()0(

)0()00(

log)00()0();(

!!!

!!!

!!!

!!!!

!!!!!!

!!!!!!

!

!!!!!

!!!!!!!

X P p X p P

p p X P

X p P X P p p

p X P

X P p X p P p

p X P

X p P X P p

p p X P

Y P X Y P X Y P X P

Y P X Y P X Y P X P

Y P

X Y P X Y P X P

Y P X Y P

X Y P X P Y X I

.

.

.

.

.

.


11/35


Channel Capacity is Maximum Information earlier showed:

0 0

11

X Y p1

p1

p p

p p p p

X P p X p P p

p X P

X p P X P p p

p X P

X P p X p P p

p X P

X p P X P p p p X P Y X I

X P X P

2log)1(2log)1(

)1()1()0(1

log)1)(1(

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

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

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

21)0()1(

!

!!!

!!!

!!!

- !!!!!

!!!!

.

.

.

21

)0()1());(max( !!!! X P X P Y X I


12/35


13/35


Effect of SNR on Capacity Binary PAM signal (digital signal amplitude 2 A )

0 0

11

X Y p1

p1

p p

)(t g

)(t g )(t g )(t g

)(t g A

A

02

02

/)(

02

/)(

01

1)|(

1)|(

N r

N r

b

b

e N

sr p

e N

sr p

I

I

T

T

!

!AGWN


14/35


Effect of SNR Binary PAM signal (digital signal amplitude 2 A )

0 0

11

X Y p1

p1

p p

)(t g

)(t g )(t g )(t g

)(t g A

A

)(2

1)(

20

0 /)(

01

02

se P N

dr e N

se P

b

N r b

!

!

! gI

I

T

0

1 s


15/35



0 0

11

X Y p1

p1

p p

!

!!

!

!

0

0

221

121

21

2

22

2

)|()|(

N AQ

QS NRQ

N Q

se P se P P

bb

b

b

K

I


16/35



0 0

11

X Y p1

p1

p p

!

!

!

!

!!

noiserms22

422

21

2

2 noiserms

21

21

0

0

0

AQ

AQ

N AQ

N AQ P

N

b

W

W Not sure about thisDoes it depend on bandwidth?


17/35



0 0

11

X Y p1

p1

p p

!

!!

noiserms2

mplitudeer c

21

noiserms

mplitude

21

21Q p P b

p p p p 2log)1(2log)1(!


18/35



0 0

11

X Y p1

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-131E-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 1 0 11 12 13 14 15 16 17

SNR = A/rms noise

B E R

!

noiserms2

mplitudeer c

21

21

b P


19/35



0 0

11

X Y p1

p1

p p

p p p p 2log)1(2log)1(C 22!S

Pb C0 0.5 01 0.308538 0.1085222 0.158655 0.368917

3 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 1

15 3.19E-14 116 6.11E-16 1

C apacity and E vs S 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

S

/r s noise

E

!

noiserms2

mplitudeer c

21

21

b P

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


20/35


Effect of SNR b Binary PAM signal (digital signal amplitude 2 A )

0 0

11

X Y p1

p1

p p

p p p p 2log)1(2log)1(C 22!S NR (dB)

-

! "

"

#

- # $

" !

#

%

- # !

# #

%

$ "

- #

$ $ " !

%

"

#

- #

!

#

% !

#

"

- #

!

$

"

-$

$ !

#

%

#

% % !

#

-!

" "

! %

-

#

$ !

# #

! "

-

#

! %

"

$ ! %

! % "

% !

! " !

#

#

%

#

" %

!

$ $

" % %

$

#

"

#

" "

! !

#

$

& -!

" " " " %

#

"

# & -"

#

#

!

$

# & - #

#

Ca p aci t ' C a nd B ( R vsSNR fo ) b in ary c h a nn 0 1

- 2 3 4

2

2 3 4

2 3

5

2 3 6

2 3 7

8

8 3 4

- 4 2 - 8 4 -5 5

8 4

S NR p 9 r b it (dB)

C a p a c

i t y

C a

n d B

@

bbb Q p P

K

K

erfc21

2

!!!


21/35

Channel Capacity of DiscreteMemoryless Channel

Discrete-discrete Binary channel, M-ary channel

-

!

!!

!

11

11

...

....

.)|(..

...)|(

Qq

ji ji

p

p x y P

x X yY P

P

Probability transition matrix


22/35

Channel Capacity of DiscreteMemoryless Channel

Average Mutual Information

! !

! !

|

!

!!!!!!

1

0

1

1

1

0

1

1

)(

)|(log)|()(

)(

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

q

j

Q

i j

ji ji j

q

j

Q

i j

ji

ji j

y P

x y P x y P x P

yY P

x X yY P x X yY P x X P Y X I


23/35


24/35


25/35

Channel Capacity DiscreteMemoryless Channel

Discrete-continuousChannel Capacity with AWGN

x0 x1 x2... xq-1

y

22

2/)(

21)|( W TW

k x yk e x y p !

!!!

!g

g

!

1

01

0

2/)(

2/)(

2/)(

)( 22

22

22

21)(

21

log2

1)(max

q

iq

i

x yi

x y

x yi

x P

d ye x X P

e

e x X P i

i

i

i W

W

W

TW

TW

TW


26/35


Binary Symmetric PAM-continuousMaximum Information when:

x0 x1 x2... xq-1

y

21

)()( !!!! A X P A X P

-

!

A

A

g

g

d yeee

e

d yeee

eC

y

A A

A

y

A A

A

22

2222

22

22

2222

22

2/

2/2/

2/2

2/

2/2/

2/2

21

2log

2log

2

1

W

W W

W

W

W W

W

TW


27/35


Binary Symmetric PAM-continuousMaximum Information when:

-

!

g

g

g

g

d yeee

e

d yeee

eC

y

A A

A

y

A A

A

22

2222

22

22

2222

22

2/

2/2/

2/2

2/

2/2/

2/2

21

2log

2log

2

1

W

W W

W

W

W W

W

TW


28/35


Binary Symmetric PAM-continuousVersus Binary Symmetric discreteS NR B (dB) C B D

-E F F

G

H H I P

Q R F

G

F F R

S

H I

- S T F

G

H E R I H Q F

G

F

S

H H U E

- S Q F

G

H

S S

I E U

F

G

F E E T F R

- S H F

G

I T T R F Q F

G

F I U

R

S

- S E F

G

I Q

S

E F P F

G

F U Q I

S

R

- S F F

G

I E P

I Q F

G

F T P P

R I

-T F

G

E T Q P

S

U F

G S

I U U Q

S

-Q F

G

E I R E E R F

G

E F Q E H U

-H F

G S

T Q

S S

H F

G

I F Q P E R

-E F

G S

I F Q H U

F

G

H H F P

R P

F F

G

F P T Q U F

G

Q F E U R P

E F

G

F I P U F Q F

G

P Q R E Q

S

H F

G

F

S

E U

F

S

F

G

R F I F U

Q F

G

F F E I T T F

G

R P U P U P

T F

G

F F F

S

R

S

F

G

R R P

I Q Q

S

F I

G

T P

V -F Q F

G

R R R R E U

S

E R

G

F

S V -F R

S

S

H Q

G

T

S V - S I

S

Ca p aci ty C a nd B W R vsSNR fo r b in ary c h a nn X Y

- ` a b

`

` a b

` a

c

` a d

` a e

f

f a b

- b ` - f b -c c

f b

S NR p g r b it (dB)

C a p a c

i t y

C a

n d B

h


29/35


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 anorthogonal function expansion:

)(

)(

1

t f x

t x N

iii

!

!

)(

)(

1

t f n

t n N

iii

!!

!

! N

iii t f y

t y

1

)()(


30/35


Continuous-continuous Using orthogonal function expansion get an

equivalent discrete time channel:

N x

x

.

.

.

.1

N y

y

y

.

.

.2

1

22 2/)(

2

1)( iii x y

i

iie x y p

W

TW !

111 n x y ! Gaussian noise


31/35


Continuous-continuousCapacity is (Shannon)

)(

)(

1

t f x

t xM

iii

!

!

)(

)(

1

t f n

t n M

iii

!!

!

! M

iii t f y

t y

1

)(

)(

);(1maxlim)(

Y X I T

C x pT gp

!

ii

N

i i

ii jii

N N N

N N N N N N N

d x yd y p

x y p x p x y p

d d p

p p p I

WT N

N N

!

g

g

g

g!

!

!

1 )()(

log)()(

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

2

XY

yxy

xyxxyYX ..

22 2/)(

21)|( iii x y

i

iie x y p

W

TW

!


32/35


Continuous-continuousMaximum Information when:

!

!

!

!

0

2

0

2

1 0

2

21

)(

21log

21log

21

21log);(max

N W T

N N

N I

x

x

N

i

x N N

x p

W

W

W YX

222/

21)( xi x

x

i e x pW

TW ! Statistically independentzero mean Gaussian inputs

then


33/35


34/35


Continuous-continuousThus Capacity is:

)(

)(

1

t f x

t xM

iii

!

!

)(

)(

1

t f n

t n M

iii

!!

!

! M

iii t f y

t y

1

)(

)(

!

!

!

gp

gp

0

0

2

)(

1log

21loglim

);(1

maxlim

W N

P W

N

W

I T

C

a v

x

T

N N x pT

W

YX

22 2/)(

21)|( iii x y

i

iie x y p

W

TW

!


35/35


Continuous-continuousThus Normalized Capacity is:

)(

)(

1

t f x

t xM

iii

!

!

)(

)(

1

t f n

t n M

iii

!!

!

! M

iii t f y

t y

1

)(

)(

W C

N

WN

C

C P WN P

W

C

W C b

b

bavav

/

12

1log

but,1log

/

0

0

2

02

!

!

!

!

I

I I

22 2/)(

21)|( iii x y

i

iie x y p

W

TW

!

etab/No (d C/W-1.44036 0.1-1. 36402 0.15-1.24869 0.225-1.07386 0.3375

-0.8075 0.50625-0.39875 0.759375

0.234937 1.1390631.230848 1.7085942.822545 2.562891

5.41099 3.8443369.669259 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

lecture 7 channel capacity

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