kyunghee university chapter 2 digital communication 1 chapter 2: formatting and baseband modulation

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Chapter 2Chapter 2 Digital Communication 1

Chapter 2: Formatting and Baseband

Modulation

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Contents

Formatting

Sampling Theorem

Pulse Code Modulation (PCM)

Quantization

Baseband Modulation

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Introduction

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IntroductionFormatting

To insure that the message is compatible with digital signalSampling : Continuous time signal x(t) Discrete time pulse signal x[n]Quantization : Continuous amplitude Discrete amplitudePulse coding : Map the quantized signal to binary digits

When data compression is employed in addition to formatting, the process is termed as a source coding.

Baseband Signaling: Pulse modulationConvert binary digits to pulse waveformsThese waveforms can be transmitted over cable.

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Baseband Systems

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Formatting Textual Data

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Formatting Textual Data (Cont.)

A system using a symbol set with a size of M is referred to as an M-ary system.

For k=1, the system is termed binary.For k=2, the system is termed quaternary or 4-ary. M=2k

The value of k or M represents an important initial choice in the design of any digital communication system.

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Formatting Analog Information

Baseband analog signal

Continuous waveform of which the spectrum extends from dc to some finite value (e.g. few MHz)

Analog waveform Sampled version

PAM Analog waveform

)(tx

t

)( fX

f

mfmf

Sampling process ((e.g.) sample-and-hold)

Low pass filtering

FT

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Sampling TheoremUniform sampling theorem A bandlimited signal having no spectral components above fm hertz can be determined uniquely by values sampled at uniform intervals of

sec2

1

mS fT

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Nyquist CriterionA theoretically sufficient condition to allow an analog signal to be reconstructed completely from a set of uniformly spaced discrete-time samples

Nyquist rate

mS ff 2

mS ff 2e.g.) speech 8kHz audio 44.1kHz

fm fs

Speech 3.2kHz( 사람의 한계 )

8 kHz ( 휴대폰 )

Audio 20kHz ( 가청주파수 )

44.1kHz (mp3)

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Impulse SamplingIn the time domain,

In the frequency domain,

n

SSS nTtnTxtxtxtx )()()()()(

n

SS

S nffXT

fXfXfX )(1

)()()(

n

SnTttx )()(

1( ) ( )S

ns

X f f nfT

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Impulse Sampling (Cont.)

FT

FT

FT

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Impulse Sampling (Cont.)The analog waveform can theoretically be completely recovered from the samples by the use of filtering (see next figure).Aliasing If , some information will be lost.

Cf) Practical considerationPerfectly bandlimited signals do not occur in nature. A bandwidth can be determined beyond which the spectral components are attenuated to a level that is considered negligible.

mS ff 2

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Impulse Sampling (Cont.)

2sf W

2sf W

2sf W

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Aliasing Due to undersampling Appear in the frequency band between

mms fff and )(

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Sampling and aliasingHelicopter 100Hz

Sampling at 220Hz 100Hz

Sampling at 80Hz 20Hz

How about sampling at 120Hz?

f0 100-100

f0 100-100

320120

-120

-320

540340

-340

f100-100

20

-20-180 -60

60 -140

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Aliasing (Cont.)Effect in the time domain

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Natural SamplingMore practical method

A replication of X(f), periodically repeated in frequency every fs Hz. Weighted by the Fourier series coefficients of the pulse train, compared with a constant value in the impulse-sampled case.

n

tnfjnpS

Sectxtxtxtx 2)()()()(

n

Snn

tnfjnS nffXcectxFfX S )(})({)( 2

)sinc(1

ssn T

nT

Tc

otherwise ,0

2/2/,/1)(

TnTtTnTTtx ss

p

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Natural Sampling (Cont.)

1/T-1/T

1/T-1/T

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Sample-and-HoldThe simplest and most popular sampling method

Significant attenuation of the higher-frequency spectral replicatesThe effect of the nonuniform spectral gain P(f) applied to the desired baseband spectrum can be reduced by postfiltering operation.

n

SSS nTtnTxtptxtxtptx )()()()()()()(

n

SS

S nffXT

fPfXfXfPfX )(1

)()()()()(

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Oversamplingthe most economic solution Analog processing is much more costly than digital one.

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QuantizationAnalog quantized pulse with “Quantization noise”L-level uniform quantizer for a signal with a peak-to-peak range of Vpp= Vp-(-Vp) =2Vp

The quantization step

The sample value on is approximate to

The quantization error is

LVq pp /

)2

,2

[

ii i

2/2/ e

o

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Quantization Errors

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Quantization Errors (Cont’d)SNR due to quantization errors

2/ 2 / 22 2 2 2

/ 2 / 2

1[ ] ( )

12e E e e p e de e de

2 2

2

2 2pp

p

V LV

2 2 22

2 2

/ 43

/12p

q e

VS LL

N

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Pulse Code Modulation (PCM)Each quantization level is expressed as a codeword L-level quantizer consists of L codewordsA codeword of L-level quantizer is represented by l-bit binary digits.

PCMEncoding of each quantized sample into a digital word

Ll 2log

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Pulse Code Modulation (PCM)

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PCM Word SizeHow many bits shall we assign to each analog sample?The choice of the number of levels, or bits per sample, depends on how much quantization distortion we are willing to tolerate with the PCM format.

The quantization distortion error p = 1/(2 # of levels)

bits 2

1log2 p

l pppVe ||

23LN

S

q

bits 3/log2 l

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Uniform QuantizationThe steps are uniform in size.The quantization noise is the same for all signal magnitudes.SNR is worse for low-level signals than for high-level signals.

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Uniform Quantization (Cont.)

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Nonuniform QuantizationFor speech signals, many of the quantizing steps are rarely used.Provide fine quantization to the weak signals and coarse quantization to the strong signals. Improve the overall SNR by

The more frequent the more accurateThe less frequent the less accurate

But, we only have uniform quantizer.

Volt

Probability

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Nonuniform Quantization (Cont.)

Companding = compress + expandAchieved by first distorting the original signal with a logarithmic compression characteristic and then using a uniform quantizer.At the receiver, an inverse compression characteristic, called expansion, is applied so that the overall transmission is not distorted.

Compression Quantization Expansionx[n] x[n]’

Volt

Probability

Volt

Probability

oo

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Nonuniform Quantization (Cont.)

Ξ

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Baseband TransmissionDigits are just abstractions: a way to describe the message information.

Need something physical that will represent or carry the digits.

Represent the binary digits with electrical pulses in order to transmit them through a (baseband) channel.Binary pulse modulation: PCM waveforms

Ex) RZ, NRZ, Phase-encoded, Multi-level binary

M-ary pulse modulation : M possible symbolsEx) PAM, PPM (Pulse position modulation), PDM (pulse duration modulation)

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PCM Waveform TypesNonreturn-to-zero (NRZ)

NRZ-L, NRZ-M (differential encoding), NRZ-S

Return-to-zero (RZ)Unipolar-RZ, bipolar-RZ, RZ-AMI

Phase encodedbi--L (Manchester coding), bi--M, bi--S

Multilevel binaryMany binary waveforms use three levels, instead of two.

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PCM Waveforms (Fig. 2.22)

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Choosing a PCM WaveformDC component

Magnetic recording prefers no dc (Delay, Manchester..)

Self-clockingManchester

Error detectionDuobinary (dicode)

Bandwidth compressionDuobinary

Differential encodingNoise immunity

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Spectral Densities of PCM Waveforms

Normalized bandwidthCan be expressed as W/RS [Hz/(symbol/s)] or WT

Describes how efficiently the transmission bandwidth is being utilizedAny waveform type that requires less than 1.0Hz for sending 1 symbol/s is relatively bandwidth efficient.

Bandwidth efficiency

R/W [bits/s/Hz]

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Spectral Densities of PCM Waveforms Mean = 0

DC=0Transition period↑ W ↓ R/W [bits/s/Hz]

Bandwidth efficiency

FT

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Autocorrelation of Bipolar

x(t)

x(t+Ʈ)

Rx(Ʈ)

xi xi+1xi-1

xi xi+1xi-1

-Ʈ To-Ʈ 2To-Ʈ

0 To 2To

xi+2

xi+2

0 To 2To

-Ʈ To-Ʈ 2To-Ʈ

A2 A2

Note that where 1

0i

ii

A volts for bx

A volts for b

2[ ] [ ]i jE x x A i j

0

1) 0 ( ) [ ( ) ( )] (1 )xi T R E x t x t

T

1[ ]i iE x x [ ]i iE x x 1[ ]i iE x x 1 1[ ]i iE x x

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Autocorrelation of Bipolar

x(t)

x(t+Ʈ)

Rx(Ʈ)

xixi+1xi-1

xi

xi+1xi-1

-Ʈ To-Ʈ 2To-Ʈ

0 To2To

xi+2

xi+2

0 To 2To

-Ʈ To-Ʈ 2To-Ʈ

A2 A2 A2

Note that where 1

0i

ii

A volts for bx

A volts for b

2[ ] [ ]i jE x x A i j

0

1) 0 ( ) [ ( ) ( )] (1 )xii T R E x t x t

T

1[ ]i iE x x [ ]i iE x x 1[ ]i iE x x 1 1[ ]i iE x x

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Autocorrelation and power spectrum density of Bipolar

0

1(1 | |) 0 | |

( )

0ox

TTR

otherwise

0

0 0

2

0

0

sin( )x

fTG f

fT

FT

Gx(f) 1

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Rectangular function Rect(t)

0 | |( ) 2

0

m

TV t

x totherwise

0

sin( ) mV fT

X ff

FT

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M-ary Pulse-ModulationM=2k-levels : k data bits per symbol

longer transition period narrower W more channel

Kinds of M-ary pulse modulationPulse-amplitude modulation (PAM)Pulse-position modulation (PPM)Pulse-duration modulation (PDM)

M-ary versus BinaryBandwidth and transmission delay tradeoffPerformance and complexity tradeoff

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Example. Quantization Levels and Multilevel Signaling

Analog waveform fm=3kHz, 16-ary PAM, quantization distortion 1%(a) Minimum number of bits/sample?

|e|= 1/2L < 0.01 L>50 L=64 6bits/sample(b) Minimum required sampling rate?

fs > 2*3kHz = 6kHz

(c) Symbol transmission rate? 9103symbols/sec 6*6 kbps = 36kbps 36[kbps]/4[bits/symbol]

(d) If W=12kHz, the bandwidth efficiency? R/W = 36kbps / 12kHz = 3bps/Hz

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HW#3P2.2P2.8P2.9P2.14P2.18

kyunghee university chapter 2 digital communication 1 chapter 2: formatting and baseband modulation

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