Download - 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
KyungHeeUniversity
Chapter 2Chapter 2 Digital Communication 1
Contents
Formatting
Sampling Theorem
Pulse Code Modulation (PCM)
Quantization
Baseband Modulation
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Chapter 2Chapter 2 Digital Communication 1
Introduction
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Baseband Systems
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Chapter 2Chapter 2 Digital Communication 1
Formatting Textual Data
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Impulse Sampling (Cont.)
FT
FT
FT
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Impulse Sampling (Cont.)
2sf W
2sf W
2sf W
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Chapter 2Chapter 2 Digital Communication 1
Aliasing Due to undersampling Appear in the frequency band between
mms fff and )(
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Aliasing (Cont.)Effect in the time domain
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Natural Sampling (Cont.)
1/T-1/T
1/T-1/T
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Oversamplingthe most economic solution Analog processing is much more costly than digital one.
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Quantization Errors
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Pulse Code Modulation (PCM)
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Uniform Quantization (Cont.)
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Nonuniform Quantization (Cont.)
Ξ
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
PCM Waveforms (Fig. 2.22)
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
Rectangular function Rect(t)
0 | |( ) 2
0
m
TV t
x totherwise
0
sin( ) mV fT
X ff
FT
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Chapter 2Chapter 2 Digital Communication 1
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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Chapter 2Chapter 2 Digital Communication 1
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