Digital Communications I:Modulation and Coding Course

Term 3 - 2008Catharina Logothetis

Lecture 13

Lecture 13 2

Last time, we talked about:

The properties of Convolutional codes. We introduced interleaving as a means

to combat bursty errors by making the channel seem uncorrelated.

We also studied “Concatenated codes” that simply consist of inner and outer codes. They can provide the required performance at a lower complexity.

Lecture 13 3

Today, we are going to talk about:

Shannon limit Comparison of different modulation

schemes Trade-off between modulation and

coding

Lecture 13 4

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

Lecture 13 5

Error probability plane(example for coherent MPSK and MFSK)

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

Bit

erro

r pro

babi

lity

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

Lecture 13 6

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)

Lecture 13 7

Nyquist minimum bandwidth requirement

The theoretical minimum bandwidth needed for baseband transmission of Rs symbols per second is Rs/2 hertz.

T21

T21−

T

)( fH

f t

)/sinc()( Ttth =1

0 T T2T−T2−0

Lecture 13 8

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

+=

NSWC

power noise Average :[Watt]power signal received Average :]Watt[

Bandwidth :]Hz[

0WNNCES

W

b

==

Lecture 13 9

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 >

Lecture 13 10

Shannon limit …C/W [bits/s/Hz]

SNR [bits/s/Hz]

Practical region

Unattainableregion

Lecture 13 11

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.

+=

WC

NE

WC b

02 1log

==

+=

WNNCES

NSWC

b

0

2 1log

[dB] 6.1693.0log

1

:get we,0or As

20

−≈=→

→∞→

eNE

WCW

b

0/ NEb

Shannon limit

Lecture 13 12

Shannon limit …

[dB] / 0NEb

W/C [Hz/bits/s]

Practical region

Unattainableregion

-1.6 [dB]

Lecture 13 13

Bandwidth efficiency plane

R<CPractical region

R>CUnattainable region

R/W [bits/s/Hz]

Bandwidth limited

Power limited

R=C

Shannon limit MPSKMQAMMFSK

510−=BP

M=2

M=4

M=8M=16

M=64

M=256

M=2M=4M=8

M=16

[dB] / 0NEb

Lecture 13 14

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)

Lecture 13 15

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

WTWTM

WR ==

[Hz] /1 ss RTW ==

][bits/s/Hz log/ 2 MWRb =

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

Lecture 13 16

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.

M-arymodulator

M-arydemodulator

][symbols/s log2 MRRs =[bits/s] R

ssbr RNER

NE

NP

000

==[ ])( ,)(0

MPgPNEfMP EBs

E =

=

Input

Output

Lecture 13 17

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

PRNP

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/8modulationMPSK channel limited-Band

−−

−

<×=≈

×=≈=

===

=<===⇒=⇒⇒>

MMPP

MNEQMP

RNPM

NEM

NE

WMRRMWR

EB

sE

b

rbs

Cbs

Cb

π

Lecture 13 18

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

PRNP

W

561

5

0

02

02

0

2

0

00

10103.7)(12

2104.12

exp2

1)16(

44.261)(log)(log

[kHz] 45 [ksym/s] 4.384/960016)/(log16MFSK channel limited-power / small relatively and

[dB] 2.861.61

−−−

− <×=−

≈⇒×=

−−≤=

===

=<=×===⇒=⇒⇒<

===

MPPNEMMP

RNPM

NEM

NE

WMMRMRWMNEWR

RNP

NE

Ek

k

Bs

E

b

rbs

Cbs

bCb

b

rb

Design example of uncoded systems …

Lecture 13 19

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 MRRs =[bits/s] R

ss

ccbr R

NER

NER

NE

NP

0000

===[ ])( ,)(0

MPgpNEfMP Ecs

E =

=

Input

Output

Encoder

Decoder

[bits/s] RknRc

=

)( cB pfP =

Lecture 13 20

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

PRNP

W

system limited-power :enough lowNot 10103.7log

)(400032003/9600log/8

modulationMPSK channel limited-Band

96

2

2

⇒>×=≈

<===⇒=⇒⇒>

−−

MMPP

MRRMWR

EB

bs

Cb

Lecture 13 21

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

Lecture 13 22

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

NE

NEG

25.140003

9600loglog 22

≤⇒≤

⇒≤

==

kn

knW

MR

kn

MRR C

bs

Lecture 13 23

Design example of coded systems … Bandwidth compatible BCH codes

0.41.331061274.36.221131272.27.111201272.36.2251632.28.1157630.28.1126311010 95 −− == BB PPtkn

Coding gain in dB with MPSK

Lecture 13 24

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

96005163

log2

=<=

=

= C

bs W

MR

knR

910

1

54

2

4

000

10102.1)1(1

1043102.1

log)(

102.1sin22)( 47.501

−−−

+=

−−

−

<×≈−

≈

×=×=≈

×=

≈⇒==

∑ jnc

jc

n

tjB

Ec

sE

s

rs

ppjn

jn

P

MMPp

MNEQMP

RNP

NE π

Lecture 13 25

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.

Lecture 13 26

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

Lecture 13 27

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

Lecture 13 28

Block diagram of a DCS

FormatSourceencode

Channelencode

Pulsemodulate

Bandpassmodulate

FormatSourcedecode

Channeldecode

Demod. SampleDetect

Channel

Digital modulation

Digital demodulation

Lecture 13 29

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

Lecture 13 30

Course summary – cont’d

1. 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

Lecture 13 31

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)

Lecture 13 32

Course summary – cont’d

1. 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

Lecture 13 33

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

Lecture 13 34

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)

Lecture 13 35

Course summary – cont’d

1. 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

Lecture 13 36

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.