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Digital Communications

Prof H Xu


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Text Book

Principles of Communication Systems - Taub &

Schilling 2nd Edition

Communication Systems Haykin Principles of Communications Ziemer &

Tranter

Communication Systems Engineering Proakis& Salehi


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Copyright 2001, S. K. Mitra

Cheating

No late tutorial/practical will be accepted.

Cheating will be not be tolerated and it will be

strongly dealt with. This includes: Copying someone elses Prac codes and report.

Copying someones tutorial/exam answers.

etc.

DP requirements: faculty rule


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Contents

Part 1: waveform coding

Part 2: source coding

Part 2: channel coding

Part 2: digital modulation


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Part 1 waveform coding


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Introduction

Communication systems are used to transport informationbearing signal from source to destination via a channel.

The information bearing signal can be:

(b) Digital : digital communication system

(a) Analog : analog communication system;

Digital communication is expanding because:

(a) The impact of the computer;

(b) flexibility and compatibility;

(c) possible to improve reliability;

(d) integrated solid-stateelectronic technology

(d) availability of wideband channels


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Basic communication system

Introduction


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Introduction

Information Source

(a) Generates the message(s) . Examples are voice,

television picture, computer key board, etc..

(b) If the message is not electrical, a transducer is used to

convert it into an electrical signal.(c) Source can be analog or digital.

(d) Source can have memory or memoryless.


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Sourceencoder/decoder

Introduction

(a) The sourceencoder maps the signal produced by the

source into a digital form (for both analog and digital).

(b) The mapping is done so as to remove redundancy in the

output signal and also to represent the original signal asefficiency as possible (using as few bits as possible).

(c) The mapping must be such that an inverse operation

(source decoding) can beeasily done.

(d) Primary objective of sourceencoding/decoding is to reduce

bandwidth, while maintaining adequate signal fidelity.


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Introduction

Channel encoder/decoder

(a) Maps the input digital signal into another digital signal in

such a way that the noise will be minimized.

(b) Channel coding thus provides for reliable communication

over a noisy channel.

(c) Redundancy is introduced at the channel encoder and

exploited at the decoder to correct errors.

Modulator

(a) Modulation provides for efficient transmission of the

signal over channel.(b) Most modulation schemes impress the information on

either the amplitude, phase or frequency of a sinusoid.

(c) Modulation and demodulation is done such that

Bit error rate is minimized and Bandwidth is conserved.


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Introduction

Channel

Characteristics of channel are

(a) Bandwidth

(b) Power

(c) Amplitude and phase variations(d) Linearity, etc..

Typical channel models are Additive White Gaussian Channel

and Rayleigh fading channel;


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Waveform coding

-- introduction

Techniques for converting analog signal into a digital bit stream

fall into the broad category waveform coding. Example are:

(a) Pulse code modulation (PCM)

(b) Differential PCM

(c) Delta modulation

(d) Linear prediction coding (LPC)

(e) Subband coding

The basic operations in most waveform codes are:

(a) Sampling

(b) Quantization

(c) Encoding


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Waveform coding

-- introduction

Example: PCM

Lower Pass Filter (LPF) at transmitter is used to attenuate

high frequency components Sampling operation is performed in accordance with the

sampling theorem a band limited signal of finiteenergy, with

no frequency components higher than is completely

described by samples taken at a rate .Mf

Mf2


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Waveform coding

-- introduction

Aliasing results if sampling frequency .Ms ff 2

Quantization produces a discrete amplitude, discrete-time

signal from the discrete time, continuous amplitude signal.

Encoding assigns binary codewords into thequantized signal.


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Waveform coding

-- Quantization

Classification ofquantization

Uniform quantization

Mid-rise type Mid-tread type

nonuniform quantization

law A lawQ


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Waveform coding

-- uniform quantization

A uniform Quantizing is the type in which the 'step size'remains same throughout the input range.

No assumption about amplitude statistics and correlation

properties of the input.

Mid-treadZero is one of the output

levels M is odd

Mid-rise

Zero is not one of the

output levels M is even


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Quantizing error consists of the difference between the input and

output signal of thequantizer.

02/2/ !(e( outputinput

(!(e( outputinput 2/32/

Waveform coding

-- uniform quantizing noise


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Maximum instantaneous value ofquantization error is2

(

Waveform coding

-- uniform quantizing noise


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Waveform coding

-- performance of a uniform quantizer

The performance of a quantizer is measured in terms of the

signal to quantizing error ratio:

_ anoisequatizingsquaremean

kTmESQER s

)(2

!

For a signal with distribution , the signal power is)(mp

_ a !

(

(

!Q

i

m

mis

i

i

dmmpmkTmE1

2

2

22 )()(


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Waveform coding

-- Sampling, quantization and coding

For example: Q=16 quantization steps; ;vsizestep 5.0!!(M

sf

T2

1!

Output coding : natural binary


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Waveform coding


For example: Q=16 quantization steps; ;vsizestep 5.0!!(M

sf

T2

1!

Bit rate= MfQ 2log2

Output coding: natural binary


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Waveform coding


!

!Q

i

iiPMSQEMSQE

1

Where is the probability that signal falls in the ith interval.

is the mean squarequantization error in the ith

interval.

iP

i

MSQE

(

(!

2/

2/

2 )|()(i

i

m

mii dmimpmmMSQE

ii P

mp

P

impimp

)(),()|( !!

(

(!

2/

2/ )(

i

i

m

mi dmmpP

So !

(

(

- !

Q

i

i

m

mi PdmimpmmMSQE

i

i1

2/

2/

2 )|()(


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Waveform coding


!

(

(

- !

Q

i

i

m

mi PdmimpmmMSQE

i

i1

2/

2/

2)|()(

iP

mpimp

)()|( !

!

(

(!

Q

i

m

mi

i

i

dmmpmmMSQE1

2/

2/

2 )()(


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Waveform coding

-- Linear quantizer with larger Q

If the number ofquantizing steps is larger, then can be

considered constant in a quantization interval.

)(mp

So !

(

(!

Q

i

m

mi

i

i

dmmpmmMSQE

1

2/

2/

2 )()(

larger Q, can be considered constant)(mp

!

(

(!

Q

i

m

mii

i

i

dmmmmpMSQE1

2/

2/

2)()(

!!

(

(

(vv!!

Q

i

i

Q

i

i mpdxxmpMSQE1

3

1

2/

2/

2

22

31)()(

!

!(Q

i

imp1

1)(

12

2(!MSQE


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Waveform coding

-- Nonuniform quantizing

Problems with uniform quantization

Only optimal for uniformly distributed signal

Real audio signals (speech and music) are more concentrated

near zeros

Human ear is more sensitive to quantization errors at smallvalues

Solution: use non-uniform quantization

quantization interval is smaller near zero


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Waveform coding


uses variable steps;

small steps in regions where the signal has a higher

probability;

thequantizer steps ( ) and the levels ( ) are chosen tomaximize the SQER.

i( im

in practice, a nonuniform quantizer is realized by signal

compression followed by uniform quatizer )(mgy !

at the receiver an expander is used to produce the inverseoperation )(1 ygm !

the compressor and expander taken together

constitute a compander.


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Waveform coding


Two common laws are the law and the A law.Q

Q law

A law

ee

ee!

1||1

,log1

|)|log(1

1||0,log1

||

xAA

xA

AxA

xA

y

)()1log(

)||

1log(

maxmax xsignx

x

xy Q

Q

!


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Waveform coding



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Waveform coding

-- implementation of -law

(1) Transform the signal using -law

)()1log(

)||

1log(

)( maxmax xsignx

x

xxFyQ

Q

!!

(2) Quantize the transformed value using a uniform

quantizer(3) Transform thequantized value back using inverse -

law

)(110)(||

)1log(

max1 max ysignx

yFxy

x

!!

Q

Q


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Waveform coding

-- performance of a nonuniform quantizer

Recall :

The performance of a quantizer is measured in terms of the

signal to quantizing error ratio:

_ a )()(2

MSQEnoisequatizingsquaremeankTmESQER s!

For a signal with distribution , the signal power is)(mp

_ a !(

(!

Q

i

m

mis

i

i

dmmpmkTmE1

2

2

22 )()(

!

(

(!

Q

i

m

mi

i

i

dmmpmmMSQE1

2/

2/

2 )()(

For nonuniform quantization system, it is very difficult to

calculate MSQE.


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Waveform coding

-- performance of a nonuniform quantizer

Normally we use mean squareerror (MSE) between original and

quantized samples or signal to noise ratio (SNR) to evaluate the

performance of nonuniform quantization system.

! !N

k kxkxNMSE 1

2

)()(

1

where N is the number of samples in the sequence.

MSESNR x

2W

!

2

xWwhere is the variance of the original signal


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Waveform coding

-- Differential PCM

speech and many signals contain enough structure such that

there is correlation among adjacent samples.

mostly evident when sampled at higher than Nyquist.

if samples are , the first difference

.

...),3(),2(),(sss

TmTmTm

))1(()( ssr TrmrTmD !

_ a _ a _ a _ a))1(()(2))1(()( 222 ssssr TrmrTmETrmErTmEDE !For a zero mean stationary process,

_ a _ a 222 ))1(()( wss TrmErTmE W!!_ a )())1(()( 2 smmmss TRTrmrTmE !! VW

where are correlation coefficients.V


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Waveform coding

-- Differential PCM

_ a _ a _ a _ a )1(2))1(()(2))1(()( 2222 VW !! mssssr TrmrTmETrmErTmEDE

For2

1"V then _ a 22 mrDE W

That means that the variance of is

less than the variance of sampled signal.

))1(()(ssr

TrmrTmD !

So a given number ofquantization steps, better performance

can be obtained by quantizing rather

than the samples.

))1(()( ssr TrmrTmD !

( Differential PCM; PCM)2

1"V

2

1V


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the procedure is to encoder the

difference ])1[()()( sss TrmrTmrTe !

Waveform coding

-- Differential PCM

Where is predicted by using previous values of

unquantized output.

])1[( sTrm

+

+

+

-

Quantiser7

Predictor

encoder

7

Transmitter

S(i) e( i) e(i) + q(i) DPCM

S (i)^


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Waveform coding

-- Delta Modulation

uses single bit quantization.

possible with oversampling to increase correlation

between adjacent samples.

its a 1-bit version of DPCM

uses a staircase approximation to the oversampled signal


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Bit Rate of a Digital Sequence

Nyquist sampling rate mS ff 2u

Quantization resolution: B bit/sample

Bit rate: bit/secBfR S v!

For example:

Speech signal sampled at 8 KHz, quantized to 8 bit/sample,

Then kbits/sec6488 !v!v! BfR S


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Summary of waveform coding

Understand the general concept ofquantization

Can perform uniform quantization on a given signal and

calculate the SQER

Understand the principle of non-uniform quantization, and canperform mu-law quantization and calculate SQER

Can calculate bit rate given sampling rate and quantization

Levels

Know advantages of digital representation

understand the difference between DPCM and PCM.

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