Source Coding:Part 1-Formatting

Topics covered fromChapter 2 (Digital Communications-Bernard Sklar)Chapter 3 (Communication Systems-Simon Haykin)

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Layering of Source Coding

•Source coding includes▫ Formatting (input data) Sampling Quantization Symbols to bits (Encoding)

▫ Compression•Decoding includes

▫ Decompression▫ Formatting (output) Bits to symbols Symbols to sequence of numbers Sequence to waveform (Reconstruction)

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Layering of Source Coding4

Formatting

•The first important step in any DCS:

▫Transforming the information source to a form compatible with a digital system

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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

Formatting of Textual Data (Character Codes)

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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

Character Coding (Textual Information)

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Formatting of Analog Data

•To transform an analog waveform into a form that is compatible with a digital communication, the following steps are taken:1.Sampling2.Quantization and Encoding3.Base-band transmission (PCM)

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SamplingStrictly band limitedBand unlimited

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Sampling in Frequency Domain

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Sampling Theorem▫ The sampling theorem for strictly band-limited

signals of finite energy in two equivalent parts Analysis : A band-limited signal of finite energy that has

no frequency components higher than W hertz is completely described by specifying the values of the signal at instants of time separated by 1/2W seconds.

Synthesis : A band-limited signal of finite energy that has no frequency components higher than W hertz is completely recovered form knowledge of its samples taken at the rate of 2W samples per second. (using a low pass filter of cutoff freq. W)

▫ Nyquist rate (fs) The sampling rate of 2W samples per second for a signal

bandwidth of W hertz▫ Nyquist interval (Ts) 1/2W (measured in seconds)

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Type of Sampling

•Ideal•Natural•Practical•Sample and Hold (Flat-top)

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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

Consider the instantaneous sampling of the analog signal x(t)

Train of impulse functions select sample values at regular intervals

x(t)

x(t)x(t)

Ts

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Ideal Sampling15

Practical Sampling

In practice we cannot perform ideal sampling It is not practically possible 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:

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Natural SamplingIf 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

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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; here, the pulse is held to a constant height for the

whole sample period 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

Natural Sampling

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Time Domain

Frequency Domain

Flat-Top Sampling19

Flat-Top Sampling

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Aliasing• Aliasing Phenomenon

▫ The phenomenon of a high-frequency component in the spectrum of the signal seemingly taking on the identify of a lower frequency in the spectrum of its sampled version.

▫ To combat the effects of aliasing in practices Prior to sampling : a low-pass anti-alias filter is used to

attenuate those high-frequency components of a message signal that are not essential to the information being conveyed by the signal

The filtered signal is sampled at a rate slightly higher than the Nyquist rate.

▫ Physically realizable reconstruction filter The reconstruction filter is of a low-pass kind with a passband

extending from –W to W The filter has a non-zero transition band extending form W to

fstop-W Thus use Engr. Nyquist formula

Fig. a

Fig. b

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Fig. a Under-sampled Signal

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Fig. b Over-sampled Signal

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Pulse-Amplitude Modulation (PAM)•Output of Sampling (natural/S&H) is

known as PAM•Pulse-Amplitude Modulation (PAM)

▫ The amplitude of regularly spaced pulses are varied in proportion to the corresponding sample values of a continuous message signal.

▫ Two operations involved in the generation of the PAM signal Instantaneous sampling of the message signal m(t)

every Ts seconds, Lengthening the duration of each sample, so that it

occupies some finite value T.

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Other forms of Pulse Modulations

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Other forms of Pulse Modulations

• PDM (Pulse-duration modulation)▫Pulse-width or Pulse-length modulation.▫The samples of the message signal are used

to vary the duration of the individual pulses.▫PDM is wasteful of power

• PPM (Pulse-position modulation)▫The position of a pulse relative to its un-

modulated time of occurrence is varied in accordance with the message signal.

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Other forms of Pulse Modulations

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Quantization

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Quantization•Amplitude quantizing: Mapping samples of a

continuous amplitude waveform to a finite set of amplitudes.

In

Out

Quantized

values

Average quantization noise power

Signal peak power

Signal power to average quantization noise power

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Qunatization example

t

Ts: sampling time

x(nTs): sampled valuesxq(nTs): quantized values

boundaries

Quant. levels

111 3.1867

110 2.2762

101 1.3657

100 0.4552

011 -0.4552

010 -1.3657

001 -2.2762

000 -3.1867

PCMcodeword 110 110 111 110 100 010 011 100 100 011 PCM sequence

amplitudex(t)

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Quantization Effect• 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.

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Non-uniform 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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M any signals such as speech have a nonuniform distribution

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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Non-uniform Quantization

Non-uniform quantization is achieved by, first passing the input signal through a “compressor”. The output of the compressor is then passed through a uniform quantizer.

The combined effect of the compressor and the uniform quantizer is that of a non-uniform quantizer.

At the receiver the voice signal is restored to its original form by using an expander.

This complete process of Compressing and Expanding the signal before and after uniform quantization is called Companding.

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Non-uniform Quantization (Companding)

Expander

)(ˆ tm

Compressor

)(tm

Uniform Quantizer

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Non-uniform Quantization (Companding)

)(tm

Expander

)(ˆ tm

Compressor

)(tm

Uniform Quantizer

The 3 stages combine to give the characteristics of a Non-uniform quantizer.

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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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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 CO M pressing 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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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

Pulse Code Modulation (PCM)

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Pulse Code Modulation (PCM)

Pulse Code Modulation refers to a digital baseband

signal that is generated directly from the quantizer and

encoder output

Sometimes the term PCM is used interchangeably with

quantization

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Figure 3.13(Communication System-Simon Haykin)The basic elements of a PCM system. (Topic 3.7)

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Pulse-Code Modulation

• PCM (Pulse-Code Modulation)▫ A message signal is represented by a sequence of coded pulses,

which is accomplished by representing the signal in discrete form in both time and amplitude

▫ The basic operation Transmitter : sampling, quantization, encoding Receiver : regeneration, decoding, reconstruction

• Operation in the Transmitter1. Sampling

1.The incoming message signal is sampled with a train of rectangular pulses

2.The reduction of the continuously varying message signal to a limited number of discrete values per second

2. Nonuniform Quantization1.The step size increases as the separation from the origin of the input-

output amplitude characteristic is increased, the large end-step of the quantizer can take care of possible excursions of the voice signal into the large amplitude ranges that occur relatively infrequently.

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3. Encoding1.To translate the discrete set of sample vales

to a more appropriate form of signal2.A binary code The maximum advantage over the effects of

noise in a transmission medium is obtained by using a binary code, because a binary symbol withstands a relatively high level of noise. The binary code is easy to generate and

regenerate

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• Regeneration Along the Transmission Path▫ The ability to control the effects of distortion and noise

produced by transmitting a PCM signal over a channel▫ Equalizer

Shapes the received pulses so as to compensate for the effects of amplitude and phase distortions produced by the transmission

▫ Timing circuitry Provides a periodic pulse train, derived from the received pulses Renewed sampling of the equalized pulses

▫ Decision-making device The sample so extracted is compared o a predetermined threshold

▫ ideally, except for delay, the regenerated signal is exactly the same as the information-bearing signal 1.The unavoidable presence of channel noise and interference causes

the repeater to make wrong decisions occasionally, thereby introducing bit errors into the regenerated signal

2. If the spacing between received pulses deviates from its assigned value, a jitter is introduced into the regenerated pulse position, thereby causing distortion.

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Fig.5.13

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Receiver• Operations in the Receivers

1. Decoding and expanding1.Decoding : regenerating a pulse whose amplitude

is the linear sum of all the pulses in the code word2.Expander : a subsystem in the receiver with a

characteristic complementary to the compressor1.The combination of a compressor and an expander is a

compander

2. Reconstruction1.Recover the message signal : passing the expander

output through a low-pass reconstruction filter

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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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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Summary of Major Line Codes

• Categories of Line Codes▫ Polar - Send pulse or negative of pulse▫ Uni-polar - 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

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 (on-off Signaling)

• 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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Figure 3.15Line codes for the electrical representations of binary data. (a) Unipolar NRZ signaling. (b) Polar NRZ signaling. (c) Unipolar RZ signaling. (d) Bipolar RZ signaling. (e) Split-phase or Manchester code.

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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 Uni-polar 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 recovery1. 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 easy2.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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Differential Encoding

(a) Original binary data. (b) Differentially encoded data, assuming reference bit 1. (c) Waveform of differentially encoded data using unipolar NRZ signaling.

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Differential Coding•Encoding

▫encoded(k) = encoded(k – 1) XOR original(k)▫where k starts from 0 ▫Encoded(-1) is called the reference bit which

can be either 1 or 0•Decoding

▫original(k) = encoded (k – 1) XOR encoded(k)▫where k starts from 0▫Reference bit remains same for both

encoding and decoding process

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69Sources of Corruption in the sampled, quantized and transmitted pulses• Channel Effects

▫ Channel Noise (AWGN, White Noise, Thermal etc)▫ Intersymbol Interference (ISI)

• Sampling and Quantization Effects▫Quantization (Granularity) Noise▫Quantizer Saturation or Overload Noise▫Timing Jitter

• 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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Note:Topics Covered

• Digital Communications-Bernard Sklar▫ Chapter 2

• Communication System-Simon Haykin 4th Ed.▫ Chapter 3 3.1-3.8

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