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Digital Communication SystemsLecture-2

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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 = 128 different 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 symbols information 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” at discrete-time intervals Sampling analog signals makes them discrete in time but still

continuous valued If done properly (Nyquist theorem is satisfied), sampling does not

introduce 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 frequency component 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” of xa(t) at discrete-time intervals, Ts

Quantization Convert from discrete-time continuous valued signal to discrete

time discrete valued signal

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Sampling

Sampling Rate (or sampling frequency fs): The rate at which the signal is sampled, expressed as the

number of samples per second (reciprocal of the sampling interval), 1/Ts = fs

Nyquist Sampling Theorem (or Nyquist Criterion): If the sampling is performed at a proper rate, no info is lost about

the 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 to twice the max frequency component of the waveform, then the waveform can be exactly reconstructed from the samples without any distortion”

max2sf f

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Ideal Sampling ( or Impulse Sampling)

Therefore, we have:

Take Fourier Transform (frequency convolution)

1( ) ( ) e sjn t

sns

x t x tT

1 1( ) ( )* ( )*s sjn t jn t

sn ns s

X f X f e X f eT T

1( ) ( )* ( ),

2s

s s sns

X f X f f nf fT

1 1( ) ( ) ( )s s

n ns s s

nX 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 through

Low Pass Filter (LPF) before sampling Fundamental Rule of Sampling (Nyquist Criterion)

The value of the sampling frequency fs must be greater than twice the 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 sn

x t x t t nT

1 2( ) ,sjn t

s sn 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 completely reconstructed from its sampled value x(nTs) with

provided that =>

2s m m s mf f f f f

2 ( )sin

2( ) ( )

( )

s

ss s

n s

f t nT

Tx t T x nT

t nT

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

T x nT c f t nT

1 1

2ss 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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Ts is called the Nyquist interval: It is the longest time interval that can be used for sampling a bandlimited signal and still allow reconstruction 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 thin

rectangular 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 Sampling

If we multiply x(t) by a train of rectangular pulses xp(t), we obtain a gated waveform that approximates the ideal sampled waveform, known as natural sampling or gating (see Figure 2.8)

( ) ( ) ( )s px t x t x t2( ) sj nf t

nn

x t c e

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

2[ ( ) ]sj nf tn

n

c x t e

[ ]n s

n

c X f nf

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

The top of each pulse follows the variation of the signal being sampled

Xs (f) is the replication of X(f) periodically every fs Hz

Xs (f) is weighted by Cn Fourier 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 sampling is used to alleviate

this problem

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Flat-Top Sampling

Here, the pulse is held to a constant height for the whole sample period

Flat top sampling is obtained by the convolution of the signal obtained after ideal sampling with a unity amplitude rectangular 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 the value 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

( ) '( )* ( )sx t x t p t

( )* ( ) ( ) ( )* ( ) ( )sn

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 sinc function

( ) [ ( )]s sX f x t

( ) ( ) ( )sn

P f x t t nT

1( ) ( )* ( )s

ns

P f X f f nfT

1( ) ( )s

ns

P f X f nfT

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Flat top sampling (Frequency Domain)

Flat top sampling becomes identical to ideal sampling as the width 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 > 2B then we recover x(t) exactly

Else we run into some problems and signal is not fully recovered

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

spectral overlap in 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, an irreversible overlap of the spectral replicas is produced

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This could be due to:

1. x(t) containing higher frequency than were expected2. 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 above

half the sampling frequency (fs/2), aliasing will occur Aliasing can be prevented by first passing the analog signal

through an anti-aliasing filter (also called a prefilter) before sampling is performed

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

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

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Solution 2: Over Sampling and Filtering in the Digital Domain 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 high

performance digital filter and down sample the resulting signal.

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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 overlapping known as aliasing will occur

( ) ( ) ( ) ( ) ( )

( ) ( )

s sn

s sn

x t x t x t x t t nT

x nT t nT

2( ) ( ) ( ) ( ) sj nf ts p n

n

x t x t x t x t c e

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

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=5000 samples/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 cos12000ax 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 8000 samples/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 at least 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 is generated 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 different

sources can be transmitted over a common digital channel (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 when

quantization levels are not finely spaced apart enough to accurately approximate input signal resulting in truncation or rounding error.

Quantizer Saturation or Overload Noise: Results when input signal is larger in magnitude than highest quantization 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 any particular 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 ADC’s 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 is given 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 be expressed as:

Therefore the Signal to Quatization Noise Ratio is given by:

22 2 2

2 41ppp VV L qP

2 2

2

/ 4

/1223

qL q

qSNR L

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

Since L = 2n, SNR = 22n or in decibels

ppV

Lq

210log (2 ) 610

nS n dBN dB

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

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Nonuniform Quantization Nonuniform quantizers have 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 principle is to use more levels at regions with large probability density function (pdf)

Concentrate quantization levels in areas of largest pdf

Or use fine quantization (small step size) for weak signals and coarse 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 signals need to be enlarged, or the quantization step size decreased, but only for the weak signals

But strong signals can potentially be reduced without 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 the amplitude of one of the signals is compressed

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

At the receiver, the signal is expanded by an inverse operation

The process of COMpressing and exPANDING the signal is called companding

Companding is a technique used to reduce the number of bits required in ADC or DAC while achieving comparable SQNR

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Basically, companding introduces a nonlinearity into the signal This maps a nonuniform distribution into something that more

closely resembles a uniform distribution A standard ADC with uniform spacing between levels can be used

after the compandor (or compander) The companding operation is inverted at the receiver

There are in fact two standard logarithm based companding techniques US standard called µ-law companding European standard called A-law companding

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Input/Output Relationship of Compander

Logarithmic expression Y = log X is the most commonly used compander

This reduces the dynamic range of Y

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Types of Companding -Law Companding Standard (North & South America, 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 = 256 quantizer levels Hence data rate R = 64 kbit/sec

= 0 corresponds to uniform quantization

maxmax

log 1 (| | /sgn( )

log (1 )e

e

x xy y x

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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

maxmax

max

| |

| | 1sgn( ), 0

(1 )( )

| |1 log

1 | |sgn( ), 1

(1 log )

e

e

xA

x xy x

A x Ay x

xA

x 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 the sample 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 the width of the pulse signal

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PCM Waveform Types The output of the A/D converter is a set of binary bits But binary bits are just abstract entities that have no physical definition We use pulses to convey a bit of information, e.g.,

In order to transmit the bits over a physical channel they must be transformed into a physical waveform

A line coder or baseband binary transmitter transforms a stream of bits into a physical waveform suitable for transmission over a channel

Line coders use the terminology mark for “1” and space to mean “0” In baseband systems, binary data can be transmitted using many kinds of

pulses

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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 lock or 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 mark from 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 should be 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 an that is a function of the data bit

The output of the line encoder is a waveform:

where f(t) is the pulse shape and Tb is the bit period (Tb=Ts/n for n bit quantizer)

This means that each line code is described by a symbol mapping function an and pulse shape f(t)

Details of this operation are set by the type of line code that is being 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 have the following:

Polar NRZ: Wireless, radio, and satellite applications primarily use Polar

NRZ because bandwidth is precious Unipolar NRZ

Turn the pulse ON for a ‘1’, leave the pulse OFF for a ‘0’ Useful for noncoherent communication where receiver can’t

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

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Bipolar RZ A unipolar line code, except now we alternate

between 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, Delay

Modulation 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

, 0n

nn

A when Xa

A when X

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Unipolar NRZ Line Code Unipolar non-return-to-zero (NRZ) line code is defined by

unipolar mapping

In addition, the pulse shape for unipolar NRZ is:

where Tb is the bit period

, 1

0, 0n

nn

A when Xa

when X

Where Xn is the nth data bit

( ) , NRZ Pulse Shapeb

tf t

T

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Bipolar Line Codes With bipolar line codes a space is mapped to zero and a

mark is alternately mapped to -A and +A

It is also called pseudoternary signaling or alternate mark inversion (AMI)

Either RZ or NRZ pulse shape can be used

, when 1 and last mark

, when 1 and last mark

0, when 0

n

n n

n

A X A

a A X A

X

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Manchester Line Codes Manchester line codes use the antipodal mapping and

the following split-phase pulse 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 they

always have a zero crossing in the center of the pulse Polar RZ codes tend to be good because the signal level always

goes 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) than

Unipolar or Bipolar codes Channel characteristics

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

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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

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First Null Bandwidth Unipolar NRZ, polar NRZ, and bipolar all have 1st null bandwidths of

Rb = 1/Tb Unipolar RZ has 1st null BW of 2Rb Manchester NRZ also has 1st null BW of 2Rb, although the

spectrum 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 be

detected 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, whose maximum frequency fm=4000Hz, is to be transmitted using a 16-level PAM system. The quantization must not exceed ±1% of the peak-to-peak analog signal.(a) What is the minimum number of bits per sample or bits per PCM word that should be used in this system?(b) What is the minimum required sampling rate, and what is the resulting bit rate?(c) What is the 16-ary PAM symbol Transmission rate?

Bits per PCM word and M-ary Modulation

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max

2 2

22

| | | |2

12

2

1log log (50) 6

2

8000 48000 16

4800012000 / sec

log ( ) 4

pp

pp lpp

qe pV e

VV Lq q L

L p

l lp

fs Rs M

RR symbols

M

Solution to Problem 2.14