1

Today, we are going to talk about:

Shannon limit Comparison of different modulation

schemes Trade-off between modulation and

coding

2

Goals in designing a DCS

Goals: Maximizing the transmission bit rate Minimizing probability of bit error Minimizing the required power Minimizing required system bandwidth Maximizing system utilization Minimize system complexity

3

Error probability plane(example for coherent MPSK and MFSK)

[dB] / 0NEb [dB] / 0NEb

Bit

erro

r pr

obab

ility

M-PSK M-FSK

k=1,2

k=3

k=4

k=5

k=5

k=4

k=2

k=1

bandwidth-efficient power-efficient

4

Limitations in designing a DCS

Limitations: The Nyquist theoretical minimum

bandwidth requirement The Shannon-Hartley capacity theorem

(and the Shannon limit) Government regulations Technological limitations Other system requirements (e.g

satellite orbits)

5

Nyquist minimum bandwidth requirement

The theoretical minimum bandwidth needed for baseband transmission of R

s symbols per

second is Rs/2 hertz.

T2

1

T2

1

T

)( fH

f t

)/sinc()( Ttth 1

0 T T2TT20

6

Shannon limit

Channel capacity: The maximum data rate at which error-free communication over the channel is performed.

Channel capacity of AWGV channel (Shannon-Hartley capacity theorem):

][bits/s1log2

N

SWC

power noise Average :[Watt]

power signal received Average :]Watt[

Bandwidth :]Hz[

0WNN

CES

W

b

7

Shannon limit … The Shannon theorem puts a limit

on the transmission data rate, not on the error probability: Theoretically possible to transmit

information at any rate , with an arbitrary small error probability by using a sufficiently complicated coding scheme

For an information rate , it is not possible to find a code that can achieve an arbitrary small error probability.

CRb bR

CRb

8

Shannon limit …C/W [bits/s/Hz]

SNR [bits/s/Hz]

Practical region

Unattainableregion

9

Shannon limit …

There exists a limiting value of below which there can be no error-free communication at any information rate.

By increasing the bandwidth alone, the capacity can not be increased to any desired value.

W

C

N

E

W

C b

02 1log

WNN

CES

N

SWC

b

0

2 1log

[dB] 6.1693.0log

1

:get we,0or As

20

eN

EW

CW

b

0/ NEb

Shannon limit

10

Shannon limit …

[dB] / 0NEb

W/C [Hz/bits/s]

Practical region

Unattainableregion

-1.6 [dB]

11

Bandwidth efficiency plane

R<C

Practical region

R>CUnattainable region

R/W [bits/s/Hz]

Bandwidth limited

Power limited

R=C

Shannon limit MPSKMQAMMFSK

510BP

M=2

M=4

M=8M=16

M=64

M=256

M=2M=4M=8

M=16

[dB] / 0NEb

12

Power and bandwidth limited systems

Two major communication resources: Transmit power and channel bandwidth

In many communication systems, one of these resources is more precious than the other. Hence, systems can be classified as: Power-limited systems:

save power at the expense of bandwidth (for example by using coding schemes)

Bandwidth-limited systems: save bandwidth at the expense of power (for

example by using spectrally efficient modulation schemes)

13

M-ary signaling

Bandwidth efficiency:

Assuming Nyquist (ideal rectangular) filtering at baseband, the required passband bandwidth is:

M-PSK and M-QAM (bandwidth-limited systems)

Bandwidth efficiency increases as M increases.

MFSK (power-limited systems)

Bandwidth efficiency decreases as M increases.

][bits/s/Hz 1log2

bs

b

WTWT

M

W

R

[Hz] /1 ss RTW

][bits/s/Hz log/ 2 MWRb

][bits/s/Hz /log/ 2 MMWRb

Lecture 13 14

Design example of uncoded systems

Design goals:1. The bit error probability at the demodulator output must

meet the system error requirement.2. The transmission bandwidth must not exceed the

available channel bandwidth.

M-arymodulator

M-arydemodulator

][symbols/s log2 M

RRs [bits/s] R

ssbr RN

ER

N

E

N

P

000

)( ,)(0

MPgPN

EfMP EB

sE

Input

Output

15

Design example of uncoded systems …

Choose a modulation scheme that meets the following system requirements:

5

0

10 [bits/s] 9600 [dB.Hz] 53

[Hz] 4000 with channel AWGNAn

Bbr

C

PRN

P

W

56

2

50

02

02

0

2

10103.7log

)(

102.2)/sin(/22)8(

67.621

)(log)(log

[Hz] 4000 [sym/s] 32003/9600log/8

modulationMPSK channel limited-Band

M

MPP

MNEQMP

RN

PM

N

EM

N

E

WMRRM

WR

EB

sE

b

rbs

Cbs

Cb

16

Choose a modulation scheme that meets the following system requirements:

5

0

10 [bits/s] 9600 [dB.Hz] 48

[kHz] 45 with channel AWGNAn

Bbr

C

PRN

P

W

561

5

0

02

02

0

2

0

00

10103.7)(12

2104.1

2exp

2

1)16(

44.261

)(log)(log

[kHz] 45 [ksym/s] 4.384/960016)/(log16

MFSK channel limited-power / small relatively and

[dB] 2.861.61

MPPN

EMMP

RN

PM

N

EM

N

E

WMMRMRWM

NEWR

RN

P

N

E

Ek

k

Bs

E

b

rbs

Cbs

bCb

b

rb

Design example of uncoded systems …

17

Design example of coded systems Design goals:

1. The bit error probability at the decoder output must meet the system error requirement.

2. The rate of the code must not expand the required transmission bandwidth beyond the available channel bandwidth.

3. The code should be as simple as possible. Generally, the shorter the code, the simpler will be its implementation.

M-arymodulator

M-arydemodulator

][symbols/s log2 M

RRs

[bits/s] R

ss

ccbr R

N

ER

N

ER

N

E

N

P

0000

)( ,)(0

MPgpN

EfMP Ec

sE

Input

Output

Encoder

Decoder

[bits/s] Rk

nRc

)( cB pfP

18

Design example of coded systems …

Choose a modulation/coding scheme that meets the following system requirements:

The requirements are similar to the bandwidth-limited uncoded system, except that the target bit error probability is much lower.

9

0

10 [bits/s] 9600 [dB.Hz] 53

[Hz] 4000 with channel AWGNAn

Bbr

C

PRN

P

W

system limited-power :enough lowNot 10103.7log

)(

400032003/9600log/8

modulationMPSK channel limited-Band

96

2

2

M

MPP

MRRM

WR

EB

bs

Cb

19

Design example of coded systems

Using 8-PSK, satisfies the bandwidth constraint, but not the bit error probability constraint. Much higher power is required for uncoded 8-PSK.

The solution is to use channel coding (block codes or convolutional codes) to save the power at the expense of bandwidth while meeting the target bit error probability.

dB 16100

9

uncoded

bB N

EP

20

Design example of coded systems

For simplicity, we use BCH codes. The required coding gain is:

The maximum allowed bandwidth expansion due to coding is:

The current bandwidth of uncoded 8-PSK can be expanded by still 25% to remain below the channel bandwidth.

Among the BCH codes, we choose the one which provides the required coding gain and bandwidth expansion with minimum amount of redundancy.

dB 8.22.1316)dB()dB()dB(00

coded

c

uncoded

b

N

E

N

EG

25.140003

9600

loglog 22

k

n

k

nW

M

R

k

n

M

RR C

bs

21

Design example of coded systems …

Bandwidth compatible BCH codes

0.41.33106127

4.36.22113127

2.27.11120127

2.36.225163

2.28.115763

0.28.112631

1010 95 BB PPtkn

Coding gain in dB with MPSK

22

Design example of coded systems …

Examine that the combination of 8-PSK and (63,51) BCH codes meets the requirements:

[Hz] 4000[sym/s] 39533

9600

51

63

log2

C

bs W

M

R

k

nR

910

1

54

2

4

000

10102.1)1(1

1043

102.1

log

)(

102.1sin2

2)( 47.501

jnc

jc

n

tjB

Ec

sE

s

rs

ppj

nj

nP

M

MPp

MN

EQMP

RN

P

N

E

23

Effects of error-correcting codes on error performance

Error-correcting codes at fixed SNR influence the error performance in two ways:1. Improving effect:

The larger the redundancy, the greater the error-correction capability

2. Degrading effect: Energy reduction per channel symbol or coded bits

for real-time applications due to faster signaling. The degrading effect vanishes for non-real time

applications when delay is tolerable, since the channel symbol energy is not reduced.

24

Bandwidth efficient modulation schemes

Offset QPSK (OQPSK) and Minimum shift keying Bandwidth efficient and constant envelope

modulations, suitable for non-linear amplifier M-QAM

Bandwidth efficient modulation Trellis coded modulation (TCM)

Bandwidth efficient modulation which improves the performance without bandwidth expansion