Digital Communication I: Modulation and Coding Course-Lecture 13 ppt

Goals in designing a DCS Goals:  Maximizing the transmission bit rate  Minimizing probability of bit error  Minimizing the required power  Minimizing required system bandwidth  Ma

Digital Communications I:

Modulation and Coding Course

Term 3 - 2008 Catharina Logothetis

Lecture 13

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.

Today, we are going to talk about:

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

Error probability plane

(example for coherent MPSK and MFSK)‏

k=4 k=5

k=5

k=4 k=2 k=1

bandwidth-efficient power-efficient

Nyquist minimum bandwidth requirement

 The theoretical minimum bandwidth

needed for baseband transmission of Rs

symbols per second is Rs/2 hertz.

1 1

) ( t t T

Shannon limit

 Channel capacity: The maximum data rate at

which error-free communication over the channel is performed.

 Channel capacity of AWGV channel Hartley capacity theorem)‏:

(Shannon-] [bits/s 1

C

power noise

Average

: [Watt]

power signal

received Average

: ] Watt [

Bandwidth

: ] Hz [

0W N N

C E S

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.

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

E W

0

2 1 log

C

b

0

2 1 log

[dB]

6.1693

0log

1

:get we,

0

or

As

2 0

E

W

C W

-1.6 [dB]

Bandwidth efficiency plane

M=16

M=64 M=256

M=2 M=4

M=8 M=16

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:

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

b s

b

WT WT

M W

1 T s R s

] [bits/s/Hz

log / W 2 M

Rb 

] [bits/s/Hz

/ log

/ W 2 M M

Rb 

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-ary modulator

M-ary demodulator

] [symbols/s

log2 M

R

R s  [bits/s]

R

s s b

N

E R

N

E N

P

0 0

(

0

M P g P N

E f M

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

channelAWGN

r

C

P

R N

P

W

5 6

2

5 0

0

2 0

2 0

2

1010

3

7log

)(

102

.2)

/sin(

/22

)8(

67.62

1)

(log)

(log

[Hz]

4000[sym/s]

32003

/9600log

/8

modulationMPSK

channellimited

Band

M N

E Q

M

P

R N

P M N

E M N

E

W M

R R

M

W R

E B

s E

b

r b

s

C b

s

C b



 Choose a modulation scheme that meets the following

system requirements:

5 0

10

[bits/s]

9600

[dB.Hz]

48

[kHz]

45 with

channelAWGN

r

C

P

R N

P

W

5 6

1 5

0

0

2 0

2 0

2 0

0 0

10 10

3 7 )

( 1 2

2 10

4

1 2

exp 2

1 )

16 (

44 26

1 )

(log )

(log

[kHz]

45

[ksym/s]

4 38 4

/ 9600 16

) /(log

16

MFSK channel

limited -

power /

small relatively

and

[dB]

2 8 61 6 1

P N

E M

M

P

R N

P M N

E M N

E

W M

MR MR

W M

N E W

R

R N

P N

E

E k

k B

s E

b

r b

s

C b

s

b C

b

b

r b

Design example of uncoded systems …

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-ary modulator

M-ary demodulator

] [symbols/s

log2 M

R

R s  [bits/s]

R

s c

B f p

P 

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

channelAWGN

r

C

P

R N

P

W

systemlimited

power :

-enoughlow

Not 10

103

7log

)(

40003200

3/9600log

/8

modulationMPSK

channellimited

Band

-9 6

M R

R M

W R

E B

b s

C b

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 16

N E P

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

(

0 0

b

N

E N

E G

25 1

4000 3

9600 log

n W

M

R k

n M

R

s

Design example of coded systems …

 Bandwidth compatible BCH codes

0 4 1

3 3

106 127

4 3 6

2 2

113 127

2 2 7

1 1

120 127

2 3 6

2 2

51 63

2 2 8

1 1

57 63

0 2 8

1 1

26 31

k n

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:

[Hz]

4000 [sym/s]

3953 3

9600 51

n R

9 10

1

5 4

2

4 0

0 0

1010

2.1)

1(1

10

43

102

1log

)(

102

.1sin

22

)(

47.501

t j B

E c

s E

s

r s

p

p j

n j n

P

M

M P p

M N

E Q

M

P R

N

P N

Effects of error-correcting codes on error

performance

 Error-correcting codes at fixed SNR influence

the error performance in two ways:

 The degrading effect vanishes for non-real time

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

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

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

Block diagram of a DCS

Format encode Source Channel encode modulate Pulse Bandpass modulate

Format decode Source Channel

decode

Demod Sample Detect

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

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

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)‏

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

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

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)‏

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

Information about the exam: