Notes Digital Communication Lecture 2

8/2/2019 Notes Digital Communication Lecture 2

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Digital Communication Systems

Lecture-2, Prof. Dr. Habibullah

Jamal

Under Graduate, Spring 2008


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Formatting


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Example 1:

In ASCII alphabets, numbers, and symbols are encoded using a 7-bit code

A total of 27= 128different characters can be represented using

a 7-bit unique ASCII code (see ASCII Table, Fig. 2.3)


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Formatting Transmit and Receive Formatting

Transition from information source digital symbolsinformation sink


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Character Coding (Textual Information)

A textual information is a sequence of alphanumeric characters Alphanumeric and symbolic information are encoded into digital bits

using one of several standard formats, e.g, ASCII, EBCDIC


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Transmission of Analog Signals Structure of Digital Communication Transmitter

Analog to Digital Conversion


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Sampling Sampling is the processes of converting continuous-time analog

signal, xa

(t), into a discrete-time signal by taking the samples atdiscrete-time intervals

Sampling analog signals makes them discrete in time but stillcontinuous valued

If done properly (Nyquist theoremis satisfied), sampling does notintroduce distortion

Sampled values: The value of the function at the sampling points

Sampling interval:

The time that separates sampling points (interval b/w samples), Ts If the signal is slowly varying, then fewer samples per second will

be required than if the waveform is rapidly varying So, the optimum sampling rate depends on the maximum

frequencycomponent present in the signal


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Analog-to-digital conversion is (basically) a 2 step process:

Sampling Convert from continuous-time analog signal xa(t)to discrete-

time continuous value signal x(n)

Is obtained by taking the samples ofxa(t)at discrete-timeintervals, Ts

Quantization

Convert from discrete-time continuous valued signal to discretetime discrete valued signal


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Sampling Sampling Rate (or sampling frequency fs):

The rate at which the signal is sampled, expressed as thenumber of samples per second (reciprocal of the samplinginterval), 1/Ts= fs

Nyquist Sampling Theorem (or Nyquist Criterion):

If the sampling is performed at a proper rate, no info is lost aboutthe original signal and it can be properly reconstructed later on

Statement:

If a signal is sampled at a rate at least, but not exactly equal totwice the max frequency component of the waveform, then the

waveform can be exactly reconstructed from the sampleswithout any distortion

max2sf f


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Ideal Sampling ( or Impulse Sampling) Therefore, we have:

Take Fourier Transform (frequency convolution)

1( ) ( ) e s

jn t

s

ns

x t x t

T

1 1

( ) ( ) * ( ) *s s jn t jn t

s

n ns s

X f X f e X f e

T T

1( ) ( ) * ( ),

2

ss s s

ns

X f X f f nf f T

1 1( ) ( ) ( )s s

n ns s s

n X f X f nf X f

T T T


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Sampling If Rs< 2B, aliasing(overlapping of the spectra) results

If signal is not strictly bandlimited, then it must be passed throughLow Pass Filter(LPF) before sampling

Fundamental Rule of Sampling (Nyquist Criterion)

The value of the sampling frequency fs must be greater than twicethe highest signal frequency fmax of the signal

Types of sampling

Ideal Sampling

Natural Sampling

Flat-Top Sampling


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Ideal Sampling ( or Impulse Sampling) Is accomplished by the multiplication of the signal x(t)by the uniform

train of impulses (comb function) Consider the instantaneous sampling of the analog signal x(t)

Train of impulse functions select sample values at regular intervals

Fourier Series representation:

( ) ( ) ( )s s

n

x t x t t nT

1 2( ) ,s

jn t

s s

n ns s

t nT eT T


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Ideal Sampling ( or Impulse Sampling)This shows that the Fourier Transform of the sampled signal is the

Fourier Transform of the original signal at rate of 1/Ts


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Ideal Sampling ( or Impulse Sampling) As long as fs> 2fm,no overlap of repeated replicas X(f - n/Ts)will

occur in Xs(f) Minimum Sampling Condition:

Sampling Theorem: A finite energy function x(t)can be completelyreconstructedfrom its sampled value x(nTs)with

provided that =>

2s m m s m f f f f f

2 ( )sin

2( ) ( )

( )

s

s

s s

n s

f t nT

T x t T x nT

t nT

( ) sin (2 ( ))s s s sn

T x nT c f t nT

1 12

s

s m

Tf f


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Ideal Sampling ( or Impulse Sampling)This means that the output is simply the replication of the original

signal at discrete intervals, e.g


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Tsis called the Nyquist interval:It is the longest time interval that canbe used for sampling a bandlimited signal and still allowreconstruction of the signal at the receiver without distortion


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Practical Sampling In practice we cannot perform ideal sampling

It is practically difficult to create a train of impulses Thus a non-ideal approach to sampling must be used

We can approximate a train of impulses using a train of very thinrectangular pulses:

Note:

Fourier Transform of impulse train is another impulse train

Convolution with an impulse train is a shifting operation

( ) sp

n

t nTx t


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Natural SamplingIf we multiply x(t)by a trainof rectangular pulses xp(t),

we obtain a gated waveformthat approximates the idealsampled waveform, knownas natural sampling orgating (see Figure 2.8)

( ) ( ) ( )s px t x t x t

2( ) s

j nf t

n

n

x t c e

( ) [ ( ) ( )]s pX f x t x t

2[ ( ) ]s

j nf t

n

n

c x t e

[ ]n sn

c X f nf


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Each pulse in xp(t)has width Tsand amplitude 1/Ts

The top of each pulse follows the variation of the signal beingsampled

Xs(f)is the replication of X(f)periodically every fsHz

Xs(f)is weighted by CnFourier Series Coeffiecient

The problem with a natural sampled waveform is that the tops of the

sample pulses are not flat It is not compatible with a digital system since the amplitude of each

sample has infinite number of possible values

Another technique known as flat top samplingis used to alleviatethis problem


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Flat-Top Sampling Here, the pulse is held to a constant height for the wholesample period

Flat top sampling is obtained by the convolution of the signalobtained after ideal sampling with a unity amplituderectangular pulse,p(t)

This technique is used to realize Sample-and-Hold(S/H)operation

In S/H, input signal is continuously sampled and then thevalue is held for as long as it takes to for the A/D to acquire

its value


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Flat top sampling (Time Domain)

'( ) ( ) ( ) x t x t t

( ) '( ) * ( )s

x t x t p t

( ) * ( ) ( ) ( ) * ( ) ( )s

n

p t x t t p t x t t nT


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Taking the Fourier Transform will result to

where P(f)is a sincfunction

( ) [ ( )]s s X f x t

( ) ( ) ( )sn

P f x t t nT

1( ) ( ) * ( )

s

ns

P f X f f nf T

1( ) ( )

s

ns

P f X f nf

T


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Flat top sampling (Frequency Domain)

Flat top sampling becomes identical to ideal sampling as thewidth of the pulses become shorter


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Recovering the Analog Signal One way of recovering the original signal from sampled signal Xs(f)

is to pass it through a Low Pass Filter (LPF) as shown below

If fs> 2Bthen we recover x(t)exactly

Else we run into some problems and signalis not fully recovered


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Undersampling and Aliasing

If the waveform is undersampled(i.e. fs < 2B) then there will be

spectral overlapin the sampled signal

The signal at the output of the filter will be

different from the original signal spectrum

This is the outcome of aliasing!

This implies that whenever the sampling condition is not met, anirreversible overlap of the spectral replicas is produced


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This could be due to:

1. x(t)containing higher frequency than wereexpected

2. An error in calculating the sampling rate Under normal conditions, undersampling of signals causing

aliasing is not recommended


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Solution 1: Anti-Aliasing Analog Filter

All physically realizable signals are not completely bandlimited

If there is a significant amount of energy in frequencies abovehalf the sampling frequency (fs/2), aliasing will occur

Aliasing can be prevented by first passing the analog signalthrough an anti-aliasingfilter (also called a prefilter) beforesampling is performed

The anti-aliasing filter is simply a LPF with cutoff frequencyequal to half the sample rate


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Aliasing is prevented by forcing the bandwidth of the sampledsignal to satisfy the requirement of the Sampling Theorem


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Solution 2: Over Sampling and Filtering in the DigitalDomain

The signal is passed through a low performance (less costly)analog low-pass filter to limit the bandwidth.

Sample the resulting signal at a high sampling frequency.

The digital samples are then processed by a highperformance digital filter and down sample the resultingsignal.


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Summary Of Sampling

Ideal Sampling

(or Impulse Sampling)

Natural Sampling

(or Gating)

Flat-Top Sampling

For all sampling techniques

If fs > 2B then we can recover x(t)exactly

If fs < 2B) spectral overlappingknown as aliasingwill occur

( ) ( ) ( ) ( ) ( )

( ) ( )

s s

n

s s

n

x t x t x t x t t nT

x nT t nT

2

( ) ( ) ( ) ( )

s j nf t

s p nnx t x t x t x t c e

( ) '( ) * ( ) ( ) ( ) * ( )s s

n

x t x t p t x t t nT p t


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Example 1: Consider the analog signal x(t)given by

What is the Nyquist rate for this signal?

Example 2: Consider the analog signal xa(t)given by

What is the Nyquist rate for this signal?

What is the discrete time signal obtained after sampling, if

fs=5000samples/s. What is the analog signal x(t)that can be reconstructed from the

sampled values?

( ) 3cos(50 ) 100sin(300 ) cos(100 ) x t t t t

( ) 3cos 2000 5sin 6000 cos12000a x t t t t


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Practical Sampling Rates Speech

- Telephone quality speech has a bandwidth of 4 kHz(actually 300 to 3300Hz)

- Most digital telephone systems are sampled at 8000samples/sec

Audio:- The highest frequency the human ear can hear is

approximately 15kHz

- CD quality audio are sampled at rate of 44,000

samples/sec Video

- The human eye requires samples at a rate of atleast 20 frames/sec to achieve smooth motion


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Pulse Code Modulation (PCM)

Pulse Code Modulation refers to a digital baseband signal that isgenerated directly from the quantizer output

Sometimes the term PCM is used interchangeably with quantization


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See Figure 2.16 (Page 80)


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Advantages of PCM:

Relatively inexpensive

Easily multiplexed: PCM waveforms from differentsources can be transmitted over a common digitalchannel (TDM)

Easily regenerated: useful for long-distance

communication, e.g. telephone Better noise performance than analog system

Signals may be stored and time-scaled efficiently (e.g.,satellite communication)

Efficient codes are readily available

Disadvantage:

Requires wider bandwidth than analog signals


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2.5 Sources of Corruption in the sampled,quantized and transmitted pulses Sampling and Quantization Effects

Quantization (Granularity) Noise: Results whenquantization levels are not finely spaced apart enoughto accurately approximate input signal resulting intruncation or rounding error.

Quantizer Saturation or Overload Noise: Results wheninput signal is larger in magnitude than highestquantization level resulting in clipping of the signal.

Timing Jitter: Error caused by a shift in the sampler

position. Can be isolated with stable clock reference. Channel Effects

Channel Noise

Intersymbol Interference (ISI)


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The level of quantization noise is dependent on how close anyparticular sample is to one of the L levels in the converter

For a speech input, this quantization error resembles a noise-like disturbance at the output of a DAC converter

Signal to Quantization Noise Ratio


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Uniform Quantization A quantizer with equal quantization level is a Uniform Quantizer

Each sample is approximated within a quantile interval Uniform quantizers are optimal when the input distribution is

uniform

i.e. when all values within the range are equally

likely

Most ADCs are implemented using uniform quantizers

Error of a uniform quantizer is bounded by 2 2

q qe


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The mean-squared value (noise variance) of the quantization error isgiven by:

/ 2 / 2 / 22 2 2

/ 2 / 2 / 2

1 1( )2

q q q

q q q

e p e de e de e deq q

3/ 2

/ 2

213 12

q

q

qeq

Signal to Quantization Noise Ratio


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The peak power of the analog signal (normalized to 1Ohms )can beexpressed as:

Therefore the Signal to Quatization Noise Ratio is given by:

22 2 2

2 41

pppVV L q

P

2 2

2

/ 4

/12

23q

L q

qSNR L


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where L = 2nis the number of quantization levels for the converter.(n is the number of bits).

Since L = 2n, SNR = 22nor in decibels

ppV

Lq

210log (2 ) 6

10

nS n dBN

dB

If qis the step size, then the maximum quantization error that canoccur in the sampled output of an A/D converter is q


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Nonuniform Quantization Nonuniform quantizershave unequally spaced levels

The spacing can be chosen to optimize the Signal-to-Noise Ratio

for a particular type of signal It is characterized by:

Variable step size

Quantizer size depend on signal size


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Many signals such as speech have a nonuniform distribution

See Figure on next page (Fig. 2.17)

Basic principleis to use more levels at regions with large probabilitydensity function (pdf)

Concentrate quantization levels in areas of largest pdf

Or use fine quantization (small step size) for weak signals andcoarse quantization (large step size) for strong signals


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Statistics of speech Signal Amplitudes

Figure 2.17: Statistical distribution of single talker speech signal

magnitudes (Page 81)


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Nonuniform quantization using companding Companding is a method of reducing the number of bits required in

ADC while achieving an equivalent dynamic range or SQNR

In order to improve the resolution of weak signals within a converter,and hence enhance the SQNR, the weak signalsneed to beenlarged, or the quantization step size decreased, but only for theweak signals

But strong signalscan potentially be reducedwithout significantly

degrading the SQNR or alternatively increasing quantization step size The compression process at the transmitter must be matched with an

equivalent expansion process at the receiver


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The signal below shows the effect of compression, where theamplitude of one of the signals is compressed

After compression, input to the quantizer will have a more uniformdistribution after sampling

At the receiver, the signal is

expanded by an inverseoperation

The process of COMpressingand exPANDING the signal iscalled companding

Companding is a techniqueused to reduce the number of bitsrequired in ADC or DAC whileachieving comparable SQNR


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Basically, companding introduces a nonlinearity into the signal

This maps a nonuniform distribution into something that moreclosely resembles a uniform distribution

A standard ADC with uniform spacing between levels can be usedafter the compandor (or compander)

The companding operation is inverted at the receiver

There are in fact two standard logarithm based compandingtechniques

US standard called -law companding

European standard called A-law companding


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Input/Output Relationship of Compander

Logarithmic expression Y = log Xis the most commonly

used compander Thisreduces the dynamic range of Y


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Types of Companding -Law Companding Standard (North & SouthAmerica, and Japan)

where

x and y represent the input and output voltages

is a constant number determined by experiment

In the U.S., telephone lines uses companding with = 255

Samples 4 kHz speech waveform at 8,000 sample/sec Encodes each sample with 8 bits, L = 256quantizer levels

Hence data rate R = 64kbit/sec

= 0corresponds to uniform quantization

maxmax

log 1 (| | / sgn( )

log (1 )

e

e

x x y y x

A L C di S d d (E Chi R i A i


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A-Law Companding Standard (Europe, China, Russia, Asia,

Africa)

where

x and y represent the input and output voltages A = 87.6

A is a constant number determined by experiment

maxmax

max

max

max

max

| |

| | 1sgn( ), 0

(1 )( )

| |

1 log 1 | |sgn( ), 1

(1 log )

e

e

x

Ax xy x

A x Ay x

x

Ax xy x

A A x


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Pulse Modulation Recall that analog signals can be represented by a sequence of discrete

samples (output of sampler)

Pulse Modulation results when some characteristic of the pulse (amplitude,width or position) is varied in correspondence with the data signal

Two Types:

Pulse Amplitude Modulation (PAM)

The amplitude of the periodic pulse train is varied in proportion to thesample values of the analog signal

Pulse Time Modulation

Encodes the sample values into the time axis of the digital signal

Pulse Width Modulation (PWM)

Constant amplitude, width varied in proportion to the signal Pulse Duration Modulation (PDM)

sample values of the analog waveform are used in determining thewidth of the pulse signal


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There are many types of waveforms. Why? performance criteria!

Each line code type have merits and demerits The choice of waveform depends on operating characteristics of a

system such as:

Modulation-demodulation requirements

Bandwidth requirement

Synchronization requirement

Receiver complexity, etc.,


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Goals of Line Coding (qualities to look for)

A line code is designed to meet one or more of the following goals:

Self-synchronization

The ability to recover timing from the signal itself

That is, self-clocking (self-synchronization) - ease of clock lockor signal recovery for symbol synchronization

Long series of ones and zeros could cause a problem

Low probability of bit error Receiver needs to be able to distinguish the waveform associated

with a markfrom the waveform associated with a space

BER performance

relative immunity to noise

Error detection capability

enhances low probability of error


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Spectrum Suitable for the channel

Spectrum matching of the channel

e.g. presence or absence of DC level

In some cases DC components should be avoided

The transmission bandwidth should be minimized

Power Spectral Density

Particularly its value at zero

PSD of code should be negligible at the frequency near zero Transmission Bandwidth

Should be as small as possible

Transparency

The property that any arbitrary symbol or bit pattern can be

transmitted and received, i.e., all possible data sequence shouldbe faithfully reproducible


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Line Coder The input to the line encoder is

the output of the A/D converter

or a sequence of values anthatis a function of the data bit

The output of the line encoderis a waveform:

where f(t)is the pulse shape and Tbis the bit period (Tb=Ts/nfor nbit quantizer)

This means that each line code is described by a symbol mappingfunction anand pulse shape f(t)

Details of this operation are set by the type of line code that isbeing used

( ) ( )n bn

s t a f t nT


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Summary of Major Line Codes Categories of Line Codes

Polar - Send pulse or negative of pulse

Unipolar - Send pulse or a 0 Bipolar (a.k.a. alternate mark inversion, pseudoternary)

Represent 1 by alternating signed pulses

Generalized Pulse Shapes

NRZ -Pulse lasts entire bit period

Polar NRZ

Bipolar NRZ

RZ - Return to Zero - pulse lasts just half of bit period

Polar RZ

Bipolar RZ

Manchester Line Code

Send a 2- pulse for either 1 (high low) or 0 (low high)

Includes rising and falling edge in each pulse

No DC component


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When the category and the generalized shapes are combined, we havethe following:

Polar NRZ:

Wireless, radio, and satellite applications primarily use PolarNRZ because bandwidth is precious

Unipolar NRZ

Turn the pulse ON for a 1, leave the pulse OFF for a 0

Useful for noncoherentcommunication where receiver cant

decide the sign of a pulse fiber optic communication often use this signaling format

Unipolar RZ

RZ signaling has both a rising and falling edge of the pulse

This can be useful for timing and synchronization purposes

Bi l RZ


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Bipolar RZ

A unipolar line code, except now we alternatebetween positive and negative pulses to send a 1

Alternating like this eliminates the DC component This is desirable for many channels that cannot

transmit the DC components

Generalized Grouping

Non-Return-to-Zero: NRZ-L, NRZ-M NRZ-S Return-to-Zero: Unipolar, Bipolar, AMI

Phase-Coded: bi-f-L, bi-f-M, bi-f-S, Miller, DelayModulation

Multilevel Binary: dicode, doubinary

Note:There are many other variations of line codes (see Fig. 2.22,page 80 for more)


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Commonly Used Line Codes Polar line codes use the antipodal mapping

Polar NRZ uses NRZ pulse shape

Polar RZ uses RZ pulse shape

, 1

, 0

n

n

n

A when X a A when X


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Unipolar NRZ Line Code

Unipolar non-return-to-zero (NRZ) line code is defined byunipolar mapping

In addition, the pulse shape for unipolar NRZ is:

where Tbis the bit period

, 10, 0

n

n

n

A when X awhen X

Where Xn is the nth data bit

( ) , NRZ Pulse Shape

b

tf t

T


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Manchester Line Codes


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Manchester Line Codes

Manchester line codesuse the antipodal mappingand the following split-phasepulse shape:

4 4( )

2 2

b b

b b

T Tt t

f tT T


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Summary of Line Codes


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Comparison of Line Codes Self-synchronization

Manchester codes have built in timing information because theyalways have a zero crossing in the center of the pulse

Polar RZ codes tend to be good because the signal level alwaysgoes to zero for the second half of the pulse

NRZ signals are not good for self-synchronization

Error probability

Polar codes perform better (are more energy efficient) thanUnipolar or Bipolar codes

Channel characteristics

We need to find the power spectral density (PSD) of the linecodes to compare the line codes in terms of the channelcharacteristics


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Comparisons of Line Codes Different pulse shapes are used

to control the spectrum of the transmitted signal (no DC value,

bandwidth, etc.) guarantee transitions every symbol interval to assist in symbol timing

recovery

1. Power Spectral Density of Line Codes (see Fig. 2.23, Page 90)

After line coding, the pulses may be filtered or shaped to further

improve there properties such as Spectral efficiency

Immunity to Intersymbol Interference

Distinction between Line Coding and Pulse Shaping is not easy

2. DC Component and Bandwidth

DC Components

Unipolar NRZ, polar NRZ, and unipolar RZ all have DC components

Bipolar RZ and Manchester NRZ do not have DC components

First Null Bandwidth


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First Null Bandwidth

Unipolar NRZ, polar NRZ, and bipolar all have 1st null bandwidthsof Rb = 1/Tb

Unipolar RZ has 1st null BW of 2Rb

Manchester NRZ also has 1st null BW of 2Rb, although thespectrum becomes very low at 1.6Rb


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Generation of Line Codes

The FIR filter realizes the different pulse shapes

Baseband modulation with arbitrary pulse shapes can bedetected by

correlation detector

matched filter detector (this is the most common detector)


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Section 2.8.4: Bits per PCM Word and Bits per Symbol L=2l

Section 2.8.5: M-ary Pulse Modulation Waveforms M = 2k

Problem 2.14: The information in an analog waveform, whosemaximum frequency fm=4000Hz, is to be transmitted using a 16-levelPAM system. The quantization must not exceed1% of the peak-to-peak analog signal.

(a) What is the minimum number of bits per sample or bits per PCMword that should be used in this system?

(b) What is the minimum required sampling rate, and what is theresulting bit rate?

(c) What is the 16-ary PAM symbol Transmission rate?

Bits per PCM word and M-ary Modulation


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2 2

2

2

| | | |2

12

2

1log log (50) 62

8000 48000 16

48000 12000 / seclog ( ) 4

pp

pp l

pp

qe pV e

VV Lq q L

L p

l lp

fs Rs M

R R symbolsM

Solution to Problem 2.14

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Notes Digital Communication Lecture 2