Post on 25-Dec-2015
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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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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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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