EE 3220: Digital Communication

Dr Hassan Yousif 1

Dr. Hassan Yousif AhmedDepartment of Electrical EngineeringCollege of Engineering at Wadi AldwasserSlman bin Abdulaziz University

Lec-13: Shannon and modulation schemes

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

Dr Hassan Yousif 2

Error probability plane(example for coherent MPSK and MFSK)

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[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

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)

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Nyquist minimum bandwidth requirement

• The theoretical minimum bandwidth needed for baseband transmission of R

s symbols per

second is Rs/2 hertz.

Dr Hassan Yousif 5T2

1

T2

1

T

)( fH

f t

)/sinc()( Ttth 1

0 T T2TT20

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):

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][bits/s1log2

N

SWC

power noise Average :[Watt]

power signal received Average :]Watt[

Bandwidth :]Hz[

0WNN

CES

W

b

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.

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CRb bR

CRb

Shannon limit …

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C/W [bits/s/Hz]

SNR [bits/s/Hz]

Practical region

Unattainableregion

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.

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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

Shannon limit …

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[dB] / 0NEb

W/C [Hz/bits/s]

Practical region

Unattainableregion

-1.6 [dB]

Bandwidth efficiency plane

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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

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)

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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.

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][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

Design example of uncoded systems

• Design goals:1. The bit error probability at the modulator output must meet the

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

bandwidth.

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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

Design example of uncoded systems …

• Choose a modulation scheme that meets the following system requirements:

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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

Design example of uncoded systems …

• Choose a modulation scheme that meets the following system requirements:

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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 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.

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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

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.

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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

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.

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dB 16100

9

uncoded

bB N

EP

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.

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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

Design example of coded systems …• Bandwidth compatible BCH codes

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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

Design example of coded systems …• Examine that the combination of 8-PSK and (63,51) BCH

codes meets the requirements:

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[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

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.

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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

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Course summary• In a big picture, we studied:– Fundamentals issues in designing a digital

communication system (DSC)• Basic techniques: formatting, coding, modulation• Design goals:

– Probability of error and delay constraints

• Trade-off between parameters:– Bandwidth and power limited systems– Trading power with bandwidth and vise versa

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Block diagram of a DCS

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FormatSourceencode

Channelencode

Pulsemodulate

Bandpassmodulate

FormatSourcedecode

Channeldecode

Demod. Sample

Detect

Channel

Digital modulation

Digital demodulation

Course summary – cont’d• In details, we studies:

1. Basic definitions and concepts• Signals classification and linear systems• Random processes and their statistics

– WSS, cyclostationary and ergodic processes– Autocorrelation and power spectral density

• Power and energy spectral density• Noise in communication systems (AWGN)• Bandwidth of signal

2. Formatting• Continuous sources

– Nyquist sampling theorem and aliasing– Uniform and non-uniform quantization

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Course summary – cont’d1. Channel coding

• Linear block codes (cyclic codes and Hamming codes)– Encoding and decoding structure

» Generator and parity-check matrices (or polynomials), syndrome, standard array

– Codes properties:» Linear property of the code, Hamming distance,

minimum distance, error-correction capability, coding gain, bandwidth expansion due to redundant bits, systematic codes

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Course summary – cont’d• Convolutional codes

– Encoder and decoder structure» Encoder as a finite state machine, state diagram,

trellis, transfer function– Minimum free distance, catastrophic codes, systematic

codes– Maximum likelihood decoding:

» Viterbi decoding algorithm with soft and hard decisions

– Coding gain, Hamming distance, Euclidean distance, affects of free distance, code rate and encoder memory on the performance (probability of error and bandwidth)

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Course summary – cont’d1. Modulation

• Baseband modulation– Signal space, Euclidean distance– Orthogonal basic function– Matched filter to reduce ISI– Equalization to reduce channel induced ISI – Pulse shaping to reduce ISI due to filtering at the

transmitter and receiver» Minimum Nyquist bandwidth, ideal Nyquist pulse

shapes, raise cosine pulse shape

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Course summary – cont’d• Baseband detection

– Structure of optimum receiver– Optimum receiver structure

» Optimum detection (MAP)» Maximum likelihood detection for equally likely symbols

– Average bit error probability– Union bound on error probability– Upper bound on error probability based on minimum distance

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Course summary – cont’d• Passband modulation

– Modulation schemes» One dimensional waveforms (ASK, M-PAM)» Two dimensional waveforms (M-PSK, M-QAM)» Multidimensional waveforms (M-FSK)

– Coherent and non-coherent detection– Average symbol and bit error probabilities– Average symbol energy, symbol rate, bandwidth– Comparison of modulation schemes in terms of error

performance and bandwidth occupation (power and bandwidth)

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Course summary – cont’d1. Trade-off between modulation and coding

• Channel models – Discrete inputs, discrete outputs

» Memoryless channels : BSC » Channels with memory

– Discrete input, continuous output» AWGN channels

• Shannon limits for information transmission rate• Comparison between different modulation and coding

schemes– Probability of error, required bandwidth, delay

• Trade-offs between power and bandwidth– Uncoded and coded systems

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Information about the exam:

• Exam date: – 8th of March 2008 (Saturday)

• Allowed material:– Any calculator (no computers)– Mathematics handbook– Swedish-English dictionary

• A list of formulae that will be available with the exam.

Dr Hassan Yousif 34