ii. modulation & coding. © tallal elshabrawy design goals of communication systems 1.maximize...

II. Modulation & Coding

© Tallal Elshabrawy

Design Goals of Communication Systems

1. Maximize transmission bit rate

2. Minimize bit error probability

3. Minimize required transmission power

4. Minimize required system bandwidth

5. Minimize system complexity, computational load & system cost

6. Maximize system utilization

2


Some Tradeoffs in M-PSK Modulaion

1 Trades off BER and Energy per Bit2 Trades off BER and Normalized Rate in b/s/Hz3 Trades off Normalized Rate in b/s/Hz and Energy per Bit

3

0 2 4 6 8 10 12 14 16 18

10-4

10-3

10-2

10-1

100

Eb/N0

Pb

BPSK,QPSK8 PSK16 PSK

1

2

3

m=4

m=3

m=1, 2


Shannon-Hartley Capacity Theorem

C: System Capacity (bits/s)

W: Bandwidth of Communication (Hz)

S: Signal Power (Watt)

N: Noise Power (Watt)

4

2

SC W log 1

N

System Capacity for communication over of an AWGN Channel is given by:


Shannon-Hartley Capacity Theorem

5

-10 0 10 20 30 40 50

1/8

1/4

1

2

4

8

16

SNR

Nor

mal

ized

Cha

nnel

Cap

acity

C/W

(b/

s/H

z)

Practical Systems

UnattainableRegion


Shannon Capacity in terms of Eb/N0

Consider transmission of a symbol over an AWGN channel

6

CWlog21E

SR

S

N0W

SES

TS

ESR

S

NN0W

ESR

SmE

bR

SE

bC

C

Wlog

21 E

b

N0

C

W


Shannon Limit

7

C

Wlog

21 E

b

N0

C

W

x

Eb

N0

log21 x

1Eb

N0

1

xlog

21 x

1Eb

N0

log21 x

1

x

x 0 1

x1 x e

Eb

N0

1

loge0.693 1.6dB

xEb

N0

C

W

Let


Shannon Limit

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

1/4

1/2

1

2

4

8

16

1/8

1/16

Eb/N0

Mor

mal

ized

Cha

nnel

Cap

acity

b/s

/Hz

Shannon Limit=-1.6 dB


Shannon Limit

No matter how much/how smart you decrease the rate by using channel coding, it is impossible to achieve communications with very low bit error rate if Eb/N0 falls below -1.6 dB


-4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

1/16

1/8

1/4

1/2

1

2

4

8

16

Shannon Limit


BPSK UncodedPb = 10-5

QPSK Uncoded Pb = 10-5

8 PSK UncodedPb=10-5


Room for improvement by channel coding

Norm

aliz

ed

Ch

an

nel C

ap

aci

ty b

/s/H

z

Eb/N0


1/3 Repetition Code BPSK

Is this really purely a gain?No! We have lost one third of the information transmitted rate

11

0 1 2 3 4 5 6 7 8 9 1010

-6

10-5

10-4

10-3

10-2

10-1

Eb/N0

Pb

BPSK UncodedBPSK 1/3Repetition Code

Coding Gain= 3.2 dB


0 1 2 3 4 5 6 7 8 9 1010

-6

10-5

10-4

10-3

10-2

10-1

Eb/N0

Pb

BPSK Uncoded8 PSK 1/3 Repitition Code

1/3 Repetition Code 8 PSK

12

Coding Gain= -0.5 dB

When we don’t sacrifice information rate 1/3 repetition codes did not help us


The waveform generator converts binary data to voltage levels (1 V., -1 V.) The channel has an effect of altering the voltage that was transmitted Waveform detection performs a HARD DECISION by mapping received

voltage back to binary values based on decision zones

Channel

e

rv

Channel

Encoder

Waveform

Generator

Waveform

Detection

Channel

Decoder

Channel

v rx y

v = [v1 v2 … vi … vn]e = [e1 e2 … ei … en]r = [r1 r2 … ri … rn]x = [x1 x2 … xi … xn]y = [y1 y2 … yi … yn]

0 T

0 T

+1 V.

-1 V.

vivi=1

vi=0

xi

0yi>0

yi<0

ri=1

ri=0

ri+

zi]-∞, ∞[

yi

Hard Decision Decoding


The waveform generator converts binary data to voltage levels (1 V., -1 V.) The channel has an effect of altering the voltage that was transmitted The input to the channel decoder is a vector of voltages rather than a vector

of binary values

Channel

e

rv

Channel

Encoder

Waveform

Generator

Channel

Decoder

Channel

v rx

v = [v1 v2 … vi … vn]e = [e1 e2 … ei … en]r = [r1 r2 … ri … rn]x = [x1 x2 … xi … xn]

0 T

0 T

+1 V.

-1 V.

vivi=1

vi=0

xi +

zi]-∞, ∞[

ri

Soft Decision Decoding


Hard Decision- Each received bit is detected individually- If the voltage is greater than 0 detected bit is 1- If the voltage is smaller than 0 detected bit is 0- Detection information of neighbor bits within the same codeword is

lost

Channel

e

rv

Channel

Encoder

Waveform

Generator

Waveform

Detection

Channel

Decoder

Channel

0 0 0

ry


0 -1 -1 -1 0.1 -0.9 0.1 1 0 1 1

Hard Decision: Example 1/3 Repetition Code BPSK


Soft Decision- If the accumulated voltage within the codeword is greater than 0

detected bit is 1- If the accumulated voltage within the codeword is smaller than 0

detected bit is 0- Information of neighbor bits within the same codeword contributes to

the channel decoding process

Channel

e

rv

Channel

Encoder

Waveform

Generator

Channel

Decoder

Channel

0 0 0

r


0 -1 -1 -1 0.1 -0.9 0.1 0

Accumulated Voltage = 0.1-0.9+0.1=-0.7<0

Soft Decision: Example 1/3 Repetition Code BPSK


1/3 Repetition Code BPSK Soft Decision

{ }1,0b∈Channel Coding

(1/3 Repetition Code)

c 000,111Waveform

Representation

b b b b b bs E , E , E , E , E , E

Channel

]n,n,n[=n 321

Soft DecisionDecoding

r*b

Uncoded0

b

0

bCode.pRe3/1

0

b

NE

=N3E3

=NE

Important Note


BER Performance Soft Decision 1/3 Repetition Code BPSK

Select b*=0 if

Note that r0 r1 and r2 are independent and identically distributed. In other words

Therefore

Similarly

f R b 0 f R b 1

0 1 2 0 1 2f r r r b 0 f r r r b 1

2i b

2

r E

2σi 2

1f r b 0 e

2πσ

2i b

2

3 r E2

2σi 2

i 0

1f r b 0 e

2πσ

2i b

2

3 r E2

2σi 2

i 0

1f r b 1 e

2πσ


Select b*=0 if

( ) ( )1=bRf>0=bRf

2 2

i b i b

2 2

3 3r E r E2 2

2σ 2σ

2 2i 0 i 0

1 1e e

2πσ 2πσ

2 22 2i b i b

2 2i 0 i 0

r + E r E

2σ 2σ

2 22 2

i b i bi 0 i 0

r + E r E

2 2

i b i b

2 2

r E r E2 2

2σ 2σ

i 0 i 0

ln e ln e

2

ii 0

r 0

BER Performance Soft Decision 1/3 Repetition Code BPSK


where

BER Performance Soft Decision 1/3 Repetition Code BPSK 2

ii 0

Pr error b = 0 = Pr r > 0 b = 0

2

b ii 0

Pr error b = 0 = Pr E n 0

bPr error b = 0 = Pr n 3 E

2

ii 0

n n

n is Gaussian distributed with mean 0 and variance 3N0/2

b

0

3 EPr error b = 0 = Q

3N / 2

b

0

3E1Pr error b 0 erfc

2 N

2

i bi 0

Pr error b = 0 = Pr n 3 E


Hard Vs Soft Decision: 1/3 Repetition Code BPSK

0 1 2 3 4 5 6 7 8 9 1010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0

Pb

BPSK UncodedBPSK 1/3 Repitition Code Hard DecisionBPSK 1/3 Repetition Code Soft Decision

Coding Gain= 4.7 dB


1/3 Repetition Code 8 PSK Hard Decision

22

0 1 2 3 4 5 6 7 8 9 1010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb /N0

Pb

BPSK Uncoded8PSK 1/3 Repetition Code Hard Decision8PSK 1/3 Repetition Code Soft Decision

Coding Gain= 1.5 dB


-4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

1/16

1/8

1/4

1/2

1

2

4

8

16

Shannon Limit and BER Performance

23


BPSK UncodedPb = 10-5

QPSK Uncoded Pb =

10-5



BPSK 1/3 Rep. Code Hard Decision

Pb = 10-5

BPSK 1/3 Rep. Code Sodt Decision

Pb = 10-5

Norm

aliz

ed

Ch

an

nel C

ap

aci

ty b

/s/H

z

Eb/N0

1/3

8PSK 1/3 Rep. Code Hard Decision

Pb = 10-5

8PSK 1/3 Rep. Code Soft DecisionPb = 10-5

ii. modulation & coding. © tallal elshabrawy design goals of communication systems 1.maximize...

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