Digital Communications 3units

8/9/2019 Digital Communications 3units

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Sri Sai Aditya Institute of Science & Technology ECE Department

Prepared by Venkata Satish N Page 1

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY KAKINADA

III Year B. Tech. Electronics and Communication Engineering – I Sem.

DIGITAL COMMUNICATIONS

UNIT I-PULSE DIGITAL MODULATION:

Elements of digital communication systems, advantages of digital communication systems,Elements of PCM: Sampling, Quantization & Coding, Quantization error, Compading in PCMsystems. Differential PCM systems (DPCM).

UNIT II-DELTA MODULATION :

Delta modulation, its draw backs, adaptive delta modulation, comparison of PCM and DM

systems, noise in PCM and DM systems.

UNIT III-DIGITAL MODULATION TECHNIQUES :

Introduction, ASK, FSK, PSK, DPSK, DEPSK, QPSK, M-ary PSK, ASK, FSK, similarity of BFSK and

BPSK.

UNIT IV-DATA TRANSMISSION :

Base band signal receiver, probability of error, the optimum filter, matched filter, probabilityof error using matched filter, coherent reception, non-coherent detection of FSK, calculation

of error probability of ASK, BPSK, BFSK,QPSK.

UNIT V-INFORMATION THEORY :

Discrete messages, concept of amount of information and its properties. Average information,

Entropy and its properties. Information rate, Mutual information and its properties,

UNIT VI-SOURCE CODING :

Introductions, Advantages, Shannon’s theorem, Shanon-Fano coding, Huffman coding,

efficiency calculations, channel capacity of discrete and analog Channels, capacity of a

Gaussian channel, bandwidth –S/N trade off.

UNIT VII-LINEAR BLOCK CODES :

Introduction, Matrix description of Linear Block codes, Error detection and error correction

capabilities of Linear block codes, Hamming codes, Binary cyclic codes, Algebraic structure,encoding, syndrome calculation, BCH Codes.

UNIT VIII-CONVOLUTION CODES :

Introduction, encoding of convolution codes, time domain approach, transform domainapproach. Graphical approach: state, tree and trellis diagram decoding using Viterbi

algorithm.

TEXT BOOKS :

1. Digital communications - Simon Haykin, John Wiley, 2005

2. Principles of Communication Systems – H. Taub and D. Schilling, TMH, 2003

REFERENCES :

1. Digital and Analog Communication Systems - Sam Shanmugam, John Wiley, 2005.

2. Digital Communications – John Proakis, TMH, 1983. Communication Systems Analog &

Digital – Singh & Sapre, TMH, 2004.

3. Modern Analog and Digital Communication – B.P.Lathi, Oxford reprint, 3rd edition,2004.


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UNIT I-PULSE DIGITAL MODULATION

ELEMENTS OF A DIGITAL COMMUNICATION SYSTEM

The analysis and design of digital communication systems Involves the

transmission of information in digital form from a source that generates the information

to one or more destinations.

The source output may be either an analog signal, such as an audio or video signal, or

a discrete signal, such as the output of a teletype machine, that is discrete in time and has

a finite number of output characters.

In a digital communication system, the messages produced by the source are

converted into a sequence of binary digits. The process of efficiently converting the

output of either an analog or discrete source into a sequence of binary digits is called

source encoding or data compression.

The sequence of binary digits from the source encoder, which we call the information

sequence, is passed to the channel encoder

FIGURE 1. Basic elements of a digital communication system.

. The purpose of the channel encoder is to introduce, in a controlled manner, some

redundancy in the binary information sequence that can be used at the receiver to


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overcome the effects of noise and interference encountered in the transmission of the

signal through the channel. This increases the reliability of the received data and

improves the fidelity of the received signal.

The binary sequence at the output of the channel encoder is passed to the digital

modulator, which serves as the interface to the communication channel. Since nearly all

the communication channels encountered in practice are capable of transmitting electrical

signals (waveforms), the primary purpose of the digital modulator is to map the binary

information sequence into signal waveforms.

To elaborate on this point, let us suppose that the coded information sequence is to be

transmitted one bit at a time at some uniform rate R bits per second (bits/s). The digital

modulator may simply map the binary digit 0 into a waveform So(t) and the binary digit 1

into a waveform S1(t). In this manner, each bit from the channel encoder is transmitted

separately. We call this binary modulation.

The communication channel is the physical medium that is used to send the signal

from the transmitter to the receiver. In wireless transmission, the channel may be the

atmosphere (free space). On the other hand, telephone channels usually employ a variety

of physical media, including wire lines, optical fiber cables, and wireless (microwave

radio).

Whatever the physical medium used for transmission of' the information, the essential

feature is that the transmitted signal is corrupted in a random manner by a variety of

possible mechanisms, such as additive thermal noise generated by electronic devices;

man-made noise, e.g., automobile ignition noise; and atmospheric noise,

e.g., electrical lightning discharges during thunderstorms.

At the receiving end of a digital communication system, the digital demodulator

processes the channel-corrupted transmitted waveform and reduces the waveforms to a

sequence of numbers that represent estimates of the transmitted data symbols. This

sequence of numbers is passed to the channel decoder, which attempts to reconstruct the

original information sequence from knowledge of the code used by the channel encoder

and the redundancy contained in the received data.

A measure of' how well the demodulator and decoder perform is the frequency with

which errors occur in the decoded sequence. More precisely, the average probability of a


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bit-error at the output of the decoder is a measure of the performance of the demodulator

decoder combination.

In general, the probability of error is a function of the code characteristics, the types of

waveforms used to transmit the information over the channel, the transmitter power, the

characteristics of the channel, and the method of' demodulation and decoding.

The source decoder accepts the output sequence from the channel decoder and, from

knowledge of the source encoding method used attempts to reconstruct the original

signal.

Because of channel decoding errors and possible distortion introduced by the source

encoder, and perhaps, the source decoder, the signal at the output of the source decoder is

an approximation to the original source output. The difference or some function of the

difference between the original signal and the reconstructed signal is a measure of the

distortion introduced by the digital communication system.

The points worth noting are:

The source coding algorithm plays important role in higher code rate

The channel encoder introduced redundancy in data

The modulation scheme plays important role in deciding the data rate and immunity ofsignal towards the errors introduced by the channel

Channel introduced many types of errors like multi path, errors due to thermal noise etc.


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ADVANTAGES OF DIGITAL COMMUNICATION OVER ANALOG MODULATION:

There are many advantages of using Digital Communication over analog

communication. Some of them are listed as below:

1. The Digital communication has mostly common structure of encoding a signal so

devices used are mostly similar.

2. The Digital Communication's main advantage is that it provides us added security to

our information signal.

3. The Digital Communication system has more immunity to noise and external

interference.

4. Digital information can be saved and retrieved when necessary while it is not possible

in analog.

5. Digital Communication system is cheaper than Analog Communication.

6. The configuring process of digital communication system is simple as compared to

analog communication system.

7. In Digital Communication System, the error correction and detection techniques can be

implemented easily.

8. Digital hardware implementation is flexible & permits the use of microprocessors,

digital switching elements & layer scale.

9. Digital systems are relatively less expensive than analog systems

10. Transmission rate can be changed easily.

11. Easy for processing and applying multiplexing techniques.


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ANALOG-TO-DIGITAL CONVERSION

A digital signal is superior to an analog signal because it is more robust to noise

and can easily be recovered, corrected and amplified. For this reason, the tendency today

is to change an analog signal to digital data. In this section we describe two techniques,

pulse code modulation and delta modulation.

PULSE CODE MODULATION (PCM)

Definition: Pulse code modulation (PCM) is essentially analog-to-digital conversion of a

special type where the information contained in the instantaneous samples of an analog

signal is represented by digital words in a serial bit stream.

PCM consists of three steps to digitize an analog signal:

1.

Sampling

2. Quantization

3. Binary encoding

Before we sample, we have to filter the signal to limit the maximum frequency of the

signal as it affects the sampling rate.

Filtering should ensure that we do not distort the signal, ie remove high frequency

components that affect the signal shape.

PCM Transmitter


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SAMPLING

Analog signal is sampled every TS secs. Ts is referred to as the sampling interval.

f s = 1/Ts is called the sampling rate or sampling frequency. According to the Nyquist

theorem, the sampling rate must be at least 2 times the highest frequency contained in the

signal. There are 3 sampling methods:

Ideal - an impulse at each sampling instant

Natural - a pulse of short width with varying amplitude

Flat top - a pulse of short width with constant amplitude

Usually Flat top sampled signal is generated by sampler.

Three different sampling methods for PCM

Nyquist sampling rate for low-pass and bandpass signals


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QUANTIZATION

Sampling results in a series of pulses of varying amplitude values ranging between

two limits: a min and a max. The amplitude values are infinite (or many) between the two

limits. We need to map the infinite amplitude values onto a finite set of known values.

This is achieved by dividing the distance between min and max into q zones, each of

height

rangeofinputsignal

noofQuantiztionlevels

max minv v

q

The midpoint of each zone is assigned a value from 0 to q-1 (resulting in q values). Each

sample falling in a zone is then approximated to the value of the midpoint. That is

quantization is a process of rounding-off each sampled value to the nearest value.

The reason for approximating to the mid point is that minimizes the maximum

quantization error.

Example:

Assume we have a voltage signal with amplitudes Vmin= -20V and Vmax=+20V.

We want to use q=8 quantization levels. Then zone width = (20 - -20)/8 = 5

The 8 zones are: -20 to -15, -15 to -10, -10 to -5, -5 to 0, 0 to +5, +5 to +10, +10 to +15,

+15 to +20

The midpoints are: -17.5, -12.5, -7.5, -2.5, 2.5, 7.5, 12.5, 17.5

Each zone is then assigned a binary code.

The number of bits required to encode the zones, or the number of bits per sample as it is

commonly referred to, is obtained as follows: v = log2 q

Hence no of bits required to represent each sample are v = 3

The 8 zone (or level) codes are therefore: 000, 001, 010, 011, 100, 101, 110, and 111

Assigning codes to zones: 000 will refer to zone -20 to -15; 001 to zone -15 to -10, etc


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Note: If suppose quantization levels are 16 (2v), no of bits required to represent each

sample are 4 (v bits).

If no of quantization levels are not in the power of 2, for example to distinguish 10 > 23

(

> 2

v

) quantization levels, 4 bits (v+1) are required. Possible no of 4 bit code words are16, use any 10 code words out of 16 for representing samples.

Quantization error is defined as the difference between actual sample and quantized

sample. i.e ( ) ( ) s q s x nT x nT


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TYPES OF QUANTIZERS

1. Uniform Quantizers

Types: a) Symmetrical type of mid rise quantizer

b) Symmetrical type of mid tread quantizer

2.

Non uniform Quantizers

Uniform Quantization

•

Most ADC‟s use uniform quantizers.

• The quantization levels of a uniform quantizer are equally spaced apart.

• Uniform quantizers are optimal when the input distribution is uniform ie when all

values within the Dynamic Range of the quantizer are equally likely.

Symmetrical type of mid rise quantizer

a) Symmetrical type of mid rise Quantizer b) Quantization error


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Origin lies in the middle of a rising part of the staircase graph like. Note that in mid rise

type, any input value in between 0 to Δ is mapped to an output value of Δ/2, any input

value between Δ to 2Δ is mapped to an output value of 3Δ/2 and so on.

Mid rise characteristic is desirable because of symmetry and because it uses the 2v levels

of a v bit coder efficiently. A disadvantage of this mid rise characteristic is that it cannot

represent a zero output level.

Symmetrical type of mid tread quantizer

a) Symmetrical type of mid tread Quantizer b) Quantization error

Origin lies in the middle of a tread of a staircase like graph. Note that in mid tread type,

any input value in between -Δ/2 to + Δ/2 is mapped to an output value of zero, any inputvalue between + Δ/2 to 3 Δ/2 is mapped to an output value of Δ and so on.

Unfortunately, this characteristic has an odd number of levels (if it is symmetric) or it

must be non symmetric about zero. Therefore it does not use the 2 v possible levels of a v

bit coder efficiently.


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Illustration of Quantization process for an analog signal & discrete time signal and

error signal in the approximations

Fig.: (a) An analog signal and its quantized version (b) The error signal

Fig: (c) Equispaced samples of m(t ) (d) Quantized sample sequence


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NOISE IN PCM SYSTEMS

The performance of a PCM system is influenced by two major sources of noise.

1. TRANSMISSION NOISE: It is introduced anywhere between the transmitter output

and the receiver input. The effect of transmission noise is to introduce bit errors into

the received PCM wave, with the result that, in case of a binary system, a symbol 1

occasionally is mistaken for a symbol 0, or vice versa. Clearly, the more frequently

such errors occur, the more dissimilar the receiver output becomes compared with the

original message signal.

2. DISTORTION DUE TO QUANTIZING

There are two types of distortions associated with a quantizer:

1. Overload or clipping distortion: Overload distortion occurs when the input signal

exceeds the quantizer's input range, then output will remain at its maximum (or

minimum) value until the input falls within the quantizer's input range. Overload

distortion results in a clipped output signal. To avoid clipping, a quantizer is

matched to the input signal.

2. Quantization distortion: Figure below shows the error signal introduced by the

quantizer. From this figure, it can be seen that quantization error occurs when the

input signal is within the input range of the quantizer. It arises because of the

difference between the input amplitude and the quantized sampled amplitude and

because of the limited sampling rate. The quantization error signal produces

quantization noise or distortion in the reconstructed message signal. Its frequency

spectrum covers a large bandwidth. Low-pass filtering which is used to smooth the

waveform will remove most of the quantization error above its cutoff frequency.

However, some of the quantization error is in the signal band, and that cannot be

removed by the low-pass filter. This will produce a gritty sound at the output of a

PCM system called quantization noise.

Figure: Characteristic of quantization and overload errors.


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Quantization noise is the result of the quantization process. Since the quantization

process adjusts the height of each sample, the original waveform cannot be exactly

reconstructed using a low-pass filter as is the case with PAM signals and the classical

sampling theorem. The sampling rate will also affect the quantization noise since the

quantization error will become larger as the sampling rate decreases.

Figure below shows an analog input signal and its quantized waveform. Shown

below this is the resulting quantization error signal. The maximum amplitude of this error

signal is half a quantization interval. The overall amplitude variation is from half a

quantization interval to minus half a quantization interval. During a period of small

intervals, the error signal appears to be a sawtooth wave.

Figure: Analog input signal, quantized waveform, and quantization error waveform.

Quantization error is another reason for using compressed encoding for digitizing

a voice signal. Compressed encoding allows a higher signal-to-quantization-noise ratio

(SNQR) than linear encoding. This ratio defined as where S is the voice signal level and

NQ is noise due to the quantization error. Clearly, keeping the quantization error small is

key to keeping a high SNQR. As signal amplitude gets smaller, NQ must get smaller to

keep SNQR from dropping. Compression accomplishes this by forcing quantization error

magnitude to decrease with lower amplitudes.


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Illustration of how quantization error is reduced by increasing quantization levels

When a signal is quantized, we introduce an error since the coded signal is an

approximation of the actual amplitude value. The difference between actual and coded

value (midpoint) is referred to as the quantization error.

The more zones, the smaller which results in smaller errors. But, the more zones, the

more bits required to encode the samples which leads higher bit rate.

Example:

In the above example, increasing the no of quantization levels from 5 to 10, decreases the

step size by 2, there by decreases the quantization error.


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NON UNIFORM QUANTIZING

In this step size of the quantizer is not fixed over entire input range and it varies

according to the input signal. i.e step size of the quantizer is reduced at low levels and

increased at high levels.

Non uniform Quantizer of 8 levels

Importance of Non uniform Quantization: Voice signals are more likely to have

amplitudes near zero than at extreme peaks.. Signals with lower amplitude values will

suffer more from quantization error as the error range: /2, is fixed for all signal levels.

Non linear quantization is used to alleviate this problem. The Goal is to keep SNQR fixed

for all sample values.

Two approaches for obtaining Non uniform Quantization:

Direct approach:

The quantization levels follow a logarithmic curve. Smaller ‟s at lower amplitudes and

larger ‟s at higher amplitudes. But this process of varying directly is very difficult.


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Indirect approach:

An Effect of non linear quantizing can be can be obtained by first passing the

sample values through a compressor at the sender, then through a uniform quantizer. This

technique increase amplitudes near zero. To compensate the effects happened at the

sender, pass the sample values through an expander at the receiver. The process of

compression, uniform quantization and expansion is called Companding.

A-law and µ -law Companding• These two are standard companding methods.

• µ -Law is used in North America and Japan

• A-Law is used elsewhere to compress digital telephone signals


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Two types according to compression filter

– -law : used in US

– A-law : used in Europe

A-law & µ-law compression curve

Similarities between A−law and µ −law

Both are linear approximations of logarithmic input/output relationship.

Both are implemented using eight−bit code words (256 levels, one for each

quantization interval).

Eight−bit code words allow for a bit rate of 64 kilobits per second (kbps). This is

calculated by multiplying the sampling rate (twice the input frequency) by the size

of the code word (2 x 4 kHz x 8bits = 64 kbps).

Both break a dynamic range into a total of 16 segments

ln(1 )sgn( )

ln(1 )

x y x

1sgn( ), 01 ln

1 ln( )1sgn( ), 1

1 ln

A x x x

A A y

A x x x

A A


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Differences Between A−law and µ −law

Different linear approximations lead to different lengths and slopes.·

A−law provides a greater dynamic range than u−law.

u−law provides better signal/distortion performance for low level signals than

A−law.

A−law requires 13−bits for a uniform PCM equivalent. u−law requires 14−bits for

a uniform PCM equivalent

SNR of Compander

Example: µ-law Companding


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ENCODING

The output of the quantizer is one of “q” possible signal levels. If we want to use

a binary transmission system, then we need to map each quantized sample into a “v” bit

binary word.

Encoding is the process of representing each quantized sample by an bit code

word. The mapping is one-to-one so there is no distortion introduced by encoding. Some

mappings are better than others.

A Gray code gives the best end-to-end performance.

With gray codes adjacent samples differ only in one bit position.

The weakness of Gray codes is poor performance when the sign bit (MSB) is

received in error.

Example (3 bit quantization):

With this gray code, a single bit error will result in an amplitude error of only 2.

Unless the MSB is in error.

There are several ways by which binary symbols 1 and 0 can be represented by electrical

signals:

Unipolar NRZ (on-off signaling): Symbol 1 is represented by transmitting a pulse of

constant amplitude for the duration of symbol, and symbol 0 is represented by switching

off the pulse. This type of signal is referred to as an on-off signaling or Unipolar non

return to zero.

Polar NRZ: Symbols 1 and 0 are represented by pulses of equal positive and negative

amplitudes. This type of signal is referred to as a polar Non Return to Zero signal.


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Unipolar RZ: A rectangular pulse (half symbol wide) is used for a 1 and no pulse for a

0. This type of signal is called Unipolar Return to zero.

Bipolar RZ: Positive and negative pulses are used alternatively for symbol 1, and no

pulse for symbol 0. This type of signal is called a bipolar signal.

Manchester or Split-phase code: Symbol 1 is represented by a positive pulse followed

by a negative pulse, with both pulses being of equal amplitude and half-symbol wide; for

symbol 0, the polarities of these pulses are reversed. This type of signal is called a split

phase or Manchester code.

Electrical representations of binary data

Reasons and advantages of different encodings will be discussed in UNIT 4


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Bit rate of PCM

The bit rate of a PCM signal can be calculated form the number of bits per sample x the

sampling rate. i.e. Bit rate = v x f s

Bandwidth requirements of PCM

The bandwidth of (serial) binary PCM waveforms depends on the bit rate R and the

waveform pulse shape used to represent the data.

For no aliasing case (f s≥ 2B), the MINIMUM Bandwidth of PCM is:

B pcm(Min) = R/2 = vf s//2.

The Minimum Bandwidth of vf s//2 is obtained only when sin(x)/x pulse is used to

generate the PCM waveform.

For PCM waveform generated by rectangular pulses, the First-null Bandwidth is:

B pcm = R = nf s

A digitized signal will always need more bandwidth than the original analog signal. Price

we pay for robustness and other features of digital transmission.

EXAMPLE: DESIGN OF A PCM SIGNAL FOR TELEPHONE SYSTEMS

Assume that an analog audio voice-frequency (VF) telephone signal occupies a band

from 300 to 3,400Hz. The signal is to be converted to a PCM signal for transmission over

a digital telephone system. The minimum sampling frequency is 2x3.4 = 6.8 ksample/ sec.

To be able to use of a low-cost low-pass anti aliasing filter, the VF signal is oversampled

with a sampling frequency of 8ksamples/sec. This is the standard adopted by the Unites

States telephone industry. Assume that each sample values is represented by 8 bits; then

the bit rate of the binary PCM signal is Bit rate = v x f s = 8 x 8k = 64k bit/sec

This 64-kbit/s signal is called a DS-0 signal (digital signal, type zero).

The minimum absolute bandwidth of the binary PCM signal when sin(x)/x pulse is

used to generate is B pcm(Min) = R/2 = vf s//2 = 32k bit/sec

If we use a rectangular pulse for sampling the first null bandwidth is given by

B pcm(Min) = R = vf s = 64k bit/sec

We require a bandwidth of 64 kHz to transmit this digital voice PCM signal, whereas the

bandwidth of the original analog voice signal was, at most, 4 kHz.


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APPLICATIONS OF PCM

With the advent of fibre optic cables, PCM is used in telephony.

In space communication, space craft transmits signal to earth. Here the ransmitted

power is quite small and the distances are very large.Hence due to high noise

immunity, only pcm systems can be used in such applications.

ADVANTAGES OF PCM

• Relatively inexpensive digital circuitry may be used extensively.

• PCM signals derived from all types of analog sources may be merged with

data signals and transmitted over a common high-speed digital

communication system.

• In long-distance digital telephone systems requiring repeaters, a clean

PCM waveform can be regenerated at the output of each repeater, where

the input consists of a noisy PCM waveform.

• The noise performance of a digital system can be superior to that of an

analog system.

• The probability of error for the system output can be reduced even further

by the use of appropriate coding techniques.

DRAW BACKS OF PCM

Encoding, Decoding and quantizing circuitry of PCM is complex

PCM requires a large bandwidth as compared to other systems


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QUANTIZATION ERROR/NOISE IN PCM



If the step size of the quantizer (mid rise or mid tread) is , then maximum quantization

errormax is

2

and the range of quantization error is ,

2 2

.

As the error is equally likely in the range ,2 2

, it is better to assume error as uniform

random variable. The probability density function of this error is given by

1 1( )

2 2

f

Mean Square value of this quantization error (Noise power) is given by

22 22 2 2

2 2

1[ ] ( )

12 E f d d

SIGNAL TO QUANTIZATION NOISE RATIO IN PCM

Case 1: input signal is sinusoidal signal ( ) sinm m x t A t

SNR = Signal Power (rms) / Quantization noise power =

2

22

12

m A

Where2 2

2

m m

v

A Arangeofinputsignal

noofQuantiztionlevels q

SNR = Signal Power / Quantization noise power

=

2

2

2

12

m A

=

2

2

2

32

222

2

12

m

v

mv

A

A

or23

22

m

BWpcm f

SNR in decibels = 2103

10 log ( 2 ) 1.76 62

v v


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Case 2: Input signal is a DC signal ranging between – Am to +Am

SNR= Signal Power / Quantization noise power =2

2

12

m A

Where2 2

2

m m

v

A Arangeofinputsignal

noofQuantiztionlevels q

SNR =2

2

12

m A

=2

2

2 3* 2

2

2

12

vm

m

v

A

A

SNR in decibels = 21010 log (3* 2 ) 4.76 6v v .

Note: Signal to noise ratio of PCM system improved by 6db for every one bit increase.

PCM TRANSMISSION PATH & REGENERATION

The path between the PCM transmitter and PCM receiver over which the PCM signal

travel, is called as PCM transmission path. The most important feature of PCM system

lies in its ability to control the effects of distortion and noise when the PCM signal travels

on the channel. This capability is accomplished by reconstructing the PCM wave by

means of a chain of regenerative repeaters located at sufficiently close spacing along the

transmission route.

There are three basic functions are performed by a regenerative repeater, namely

1.

Equalization

2. Timing

3. Decision Making

The equalizer shapes the received pulses so as to compensate for the effects of amplitude

and phase distortions produced by the transmission characteristics of the channel.

The timing circuitry provides a periodic pulse train, derived from the received pulses, for

sampling the equalized pulses at the instants of time where the signal to noise ratio is a

maximum.


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The decision making device makes a decision in the favor of 1, if the equalized pulse plus

noise is above the threshold level and it makes a decision in the favor of 0 if the equalized

pulse plus noise is below the threshold level.

PCM RECEIVER

The first operation in the receiver is to regenerate the received pulses. These clean

pulses are then regrouped into code words and decoded into a quantized PAM signal. The

decoding process involves generating a pulse the amplitude of which is the linear sum of

all the pulses in the code word, with each pulse weighted by its place value in the code.

The final operation in the receiver is to recover the signal wave by passing the

decoder output through a low-pass reconstruction filter whose cutoff frequency is equal

to the message bandwidth W. Assuming that the transmission path is error free, the

recovered signal includes no noise with the exception of the initial distortion introduced

by the quantization process


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TIME DIVISION MULTIPLEXING PAM SYSTEM

Normally, in PAM system, the duration of the pulse is much less than the time

period of pulses Ts. Thus no information is being transmitted through the system for most

of the time. The time space Ts – can be utilized to transmit information from other

signals. The signal numbers 2, 3 and 4 are transmitting information with the help of

samples numbered 2, 3 and 4 respectively. This is along with the samples numbered 1 of

the signal number 1.

The time period Ts is equally divided between the four signals, thus allocating a

time slot of

4to each signal. Thus the duration of time slot is such that

4 > . Thus,

there is a guard time

4- between all successive sampling pulses, ensuring that there is

less cross talk between signals. The arrangement by which the information from morethan one signal is transmitted in this manner is known as time division multiplexing.

A TDM PAM system is shown in figure below, which transmits information from

n signals. The switch 1 and switch 2 respectively known as commutator and

decommutator are synchronized electronic switches which rotate at the same speed of 2f M

rotations per second. The commutator samples and combines the samples, while the

decommutator seperates the samples belonging to individual signals.


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Synchronization is the most crucial in TDM system. Thus, for example, if the

commutator is at position 2, the decommutator must also be in position 2. To provide

synchronization, a synchronizing puse is transmitted in every frame (time interval

between two successive samples of the same signal, i.e Ts).

Thus to multiplex n channels, n+1 time slots are provided in a frame; n for

channels and 1 for the synchronizing pulse. The synchronizing pulse is chosen in such a

way that it is easily distinguishable. For this purpose, one of its properties is adjusted in

such a way that it is never attained by the other pulses. For example, in case of PAM, its

amplitude is made larger than the amplitudes of all the other pulses.


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DIFFERENTIAL PULSE CODE MODULATION

When a voice or video signal is sampled at a rate slightly higher than the nyquist

rate, the resulting sampled signal is found to exhibit a high correlation between adjacent

samples. The meaning of this high correlation is that, in an average sense, the signal does

not change rapidly from one sample to next with the result that the difference between

adjacent samples has a variance that is smaller than the variance of the signal itself.

When these highly correlated samples are encoded, as in standard PCM system,

the resulting encoded signal contains redundant information. By removing this

redundancy before encoding, we obtain a more efficient coded signal.

For example, we can observe that the samples taken at 4Ts, 5Ts and 6Ts are

encoded to same value of 110. This information can be carried only by one sample. But

three samples are carrying the same information means that it is redundant. Cosider

another example of samples taken at 9Ts and 10Ts. The difference between these samples

only due to last bit and first two bits are redundant, since they do nit change.

If this redundancy is reduced, then overall bit rate will decrease and number of

bits will decrease and number of bits required to transmit one sample will also be

reduced.

Redundant information in PCM


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DPCM TRANSMITTER

DPCM can be treated as a variation of PCM; it also involves the three basic steps

of PCM, namely, sampling, quantization and coding. But, in the case of DPCM, what is

quantized is the difference between the actual sample and its predicted value, as

explained below.

Let x(t) represent the analog signal that is to be DPCM coded, and let it be

sampled with a period Ts. The sampling frequency fs = 1/Ts is such that there is no

aliasing in the sampling process. Let x(nTs) = m(t) at t= nTs. Quite a few real world

signals such as speech signals, biomedical signals (ECG, EEG, etc.), telemetry signals

(temperature inside a space craft, atmospheric pressure, etc.) do exhibit sample-to-sample

correlation. This implies that x(n) and x(n + 1) (or x( (n) and x (n − 1)) do not differ

significantly. In fact, given a set of previous M samples, say x (n − 1), x (n − 2),…… x (n

− M ) , it may be possible for us to predict (or estimate) x (n) to within a small percentage

error.

DPCM transmitter

Let ( ) s x nT denote the predicted value of x(nTs) and let( ) ( ) ( )

s s se nT x nT x nT

Which is the difference between the unquantized input sample m(nTs) and a prediction

of it, denoted by ( ) s x nT . This predicted value is produced bu using a prediction filter

whose input, as we will see, consists of a quantized version of the input signal x (nTs).


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The difference signal ( ) se nT is called a prediction error, since it is the amount by which

the prediction filter fails to predict the input exactly.

In DPCM, error sequence is quantized, coded and obtained a variation of PCM,

which is known as differential pulse code modulation.

The quantizer output may be expressed as ( ) ( ) ( )q s s e se nT e nT q nT , where

( )e sq nT is the quantization error.

According to fig, the quantizer output ( )q se nT is added to the predicted value

( ) s x nT to produce the prediction filter input ( ) ( ) ( )q s s q s x nT x nT e nT .

( ) ( ) ( ) ( )q s s s e s x nT x nT e nT q nT

( ) ( ) ( )q s s e s x nT x nT q nT That is irrespective of the properties of the prediction filter, the quantized signal

( )q s x nT at the prediction filter input differs from the original input signal ( ) s x nT by the

quantizing error ( )e sq nT . Accordingly if prediction is good, the variance of the prediction

error ( )q se nT will be smaller than the variance of ( ) s x nT .

DPCM RECEIVER

The receiver for reconstructing the quantized version of the input is shown in figure. It

consists of a decoder to reconstruct the quantized error signal. The quantized version of

the original input is reconstructed from the decoder output using the same prediction

filter as used in the transmitter.

DPCM receiver


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Output Signal to Noise ration of the DPCM system:

By definition,

(SNR)o =Variance of the input signal/Variance of the quantized noise =2

2

X

Q

=2 2

2 2* X E

E Q

Where 2 E

is the variance of the prediction error.

(SNR)o =2 2

2 2* X E

E Q

= G p * prediction error to quantization noise ratio.

Where Gp is the predictive gain. This prediction gain must be high as possible. This Gp is

maximized by minimizing the variance

2

E of the prediction error.

THE PREDICTION FILTER

The predicted value ( ) s x nT is modeled as a linear combination of past values of the

quantized input as shown below

1

( ) ( ) p

s k q s s

k

x nT w x nT kT

Where the tapped delay line weights w1, w2, w3………w p define the desired predictionfilter coefficients and p is order of the prediction filter.

Tapped delay line filter used as prediction filter


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The prediction error1

( ) ( ) ( ) p

s s k q s s

k

e nT x nT w x nT kT

The variance of the prediction error is therefore 2 E

= 2[ ( )] s E e nT

2 2

1

[ ( )] [{ ( ) ( )} ]

p

s s k q s s

k

E e nT E x nT w x nT kT

In order to choose a set of weights that minimize the variance 2 E

, we must differentiate

2

E with respect to each weight and then put the resulting derivatives equal to zero.

ADVANTAGES OF DPCM

1. As the difference is being encoded and transmitted by the DPCM technique, a

small difference voltage is to be quantized and encoded.

2. This will require less number of quantization levels and hence less number of bits

to represent them

3. Thus signaling rate and bandwidth of a DPCM system will be less than that of

DPCM.

COMPARISON BETWEEN PCM AND DPCM

Parameter of

comparison

Pulse code modulation Differential Pulse Code

Modulation

Number of bits It can use 4, 8 or 16 bits per sample Bits can be more than one but

are less than PCM

Quantization error Quantization error depends on

number of levels used.

Quantization error is present

Transmission

Bandwidth

Highest bandwidth is required since

number of bits are high

Bandwidth requires is lower

than PCM

Feed back There is no feedback in transmitter

and receiver

Here, feedback exists

Complexity of

implementation

System Complex Simple

Signal to noise ratio Good Fair

Applications Audio and video telephony Speech and video


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Communication Systems- Simon haykin 4th edition Exercise Problems

Problem1: A speech signal has a total duration of 10 s. It is sampled at the rate of 8 kHz

and the encoded. The signal to Quantization noise ratio is required to be 40db. Calculate

the minimum storage capacity needed to accommodate this digitized speech signal.

Solution: The minimum number of bits per sample is 7 for a signal to quantization noise

ratio of 40 dB. Hence

The number of samples in a duration of 10 s = 8000*10 = 8*104 samples.

The minimum storage is therefore = 7*8*104= 560 Kbits

Problem 2 : A PCM system uses a uniform quantizer followed by a 7 bit binary encoder.

The bit rate of the system is equal to 650 10 /b s .

(a) What is the maximum message bandwidth for which the system operates

satisfactorily?

(b) Determine the output signal to quantization noise ratio when a full- load

sinusoidal modulating wave of frequency 1 MHz is applied to the input.

Solution: (a)

Bit rate of the PCM system is given by s R vf

For the system to operate satisfactorily, sampling rate must be atleast equal to the

nyquist rate. Hence max2 s R vf v f

650 10 /b s = max7 2 f

max f = 3.57*106 Hz

(c) The output signal to Quantizing noise ratio is given by SNR in dB= 1.8 + 6v=

43.8 dB


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Problem 3: (a) A sinusoidal signal, with an amplitude of 3.25 volts, is applied to a

uniform quantizer of the mid rise type whose output takes on the values 0, ±1, ±2,±3

volts. Sketch the waveform of the resulting quantizer output for one complete cycle of the

input.

(b) Repeat this evaluation for the case when the quantizer is of the midrise type

whose output takes on the values 0.5, ±1.5, ±2.5, ±3.5 volts.


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Problem 4: The signal m (t) = 6sin (2πt) volts is transmitted using a 4 bit binary PCM

system. The quantizer is of the midrise type, with a step size of 1 volt. Sketch the

resulting PCM wave for one complete cycle of the input. Assume a sampling per second,

with samples taken at t = ±18, ±3/8, ±5/8,……, seconds.

Solution:


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UNIT2- DELTA MODULATION

Delta modulation, like DPCM is a predictive waveform coding technique and can

be considered as a special case of DPCM. It uses the simplest possible quantizer, namely

a two level (one bit) quantizer. The price paid for achieving the simplicity of the

quantizer is the increased sampling rate (much higher than the Nyquist rate) and the

possibility of slope-overload distortion in the waveform reconstruction, as explained in

greater detail later on in this section. In DM, the analog signal is highly over-sampled in

order to increase the adjacent sample correlation. The implication of this is that there is

very little change in two adjacent samples, thereby enabling us to use a simple one bit

quantizer, which like in DPCM, acts on the difference (prediction error) signals. In its

original form, the DM coder approximates an input time function by a series of linear

segments of constant slope. Such a coder is therefore referred to as a Linear (or non-

adaptive) Delta Modulator (LDM). Subsequent developments have resulted in delta

modulators where the slope of the approximating function is a variable. Such coders are

generally classified under Adaptive Delta Modulation (ADM) schemes. We use DM to

indicate either of the linear or adaptive variety.

LINEAR DELTA MODULATION

PRINCIPAL OF WORKING

The principle of operation of an LDM system can be explained with the help of

Fig 2.1 below. The signal x (t), band limited to W Hz is sampled at the rate 2 s f W .

If x(nTs) denote the sample of x(t) at t= nTs. The staircase approximation to x(t), denote

by ( ) s x nT is arrived as follows. One notes, at t=nT s, the polarity of the difference

between x(nTs) and the latest approximation to it; that is ( ) s x nT at t= nTs.

The difference between the input and the previous approximation is quantized

into only two levels, namely, , corresponding to positive and negative differences,

respectively. Thus, if the approximation falls below the signal at any sampling epoch, it is

increased by . If on the other hand, the approximation lies above the signal, it is

diminished by . Provided that the signal does not change too rapidly from sample to

sample, we find that the staircase approximation remains within of the input signal.


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Figure 2.1 Illustration of Delta Modulation

If ( ) 0 se nT then ( )q se nT & ( ) 1 sb nT

If ( ) 0 se nT then ( )q se nT & ( ) 0 sb nT

DELTA MODULATION TRANSMITTER

The principal virtue of delta modulation is its simplicity. It may be generated by applying

the sampled version of the incoming baseband signal to a modulator that involves a

summer, quantizer and accumulator interconnected as shown in figure 2.2.

Fig.2.2 Delta Modulation Transmitter


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Denoting the input signal as x(t) and the staircase approximation as xq(t), the basic

principal of delta modulation may be formalized in the following set of discrete-time

relations.

( ) ( ) ( ) 1

( ) sgn( ( )) 2

s s q s s

q s s

e nT x nT x nT T eq

e nT e nT eq

& ( ) ( ) ( ) 3q s q s s q s x nT x nT T e nT eq

where Ts is the sampling period; e(nTs) is an error signal representing the difference

between the present sample value x(nTs) of the input signal and the latest approximation

to it. Namely, ( ) ( ) s q s s x nT x nT T ; and ( )q se nT is the quantized version of e(nTS).

The quantized output ( )q se nT is finally coded to produce the desired DM wave.

Figure 2.1 illustrates the way in which the staircase approximation ( )q x t follows

variations in the input signal x (t) in accordance with above equations and it also displays

the corresponding binary sequence at the delta modulator output

Working of Accumulator (Stair case wave form generator)

1. In particular, quantizer consists of a hard limiter with input and output relation

defined by eq2 which is depicted in fig 2.2.1. The quantizer output is applied to an

accumulator, producing the result1 1

( ) sgn( ( )) ( )n n

q s s q s

i i

x nT e iT e iT

.

Fig 2.2.1: Input – output characteristic of quantizer for DM system

2. Thus at the sampling instant nTs, the accumulator increments the approximation

by a step in a positive or negative direction, depending upon the algebraic sign

of error signal e(nTs).


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3. If the input signal x(nTs) is greater than the most recent approximation ( ) s x nT , a

positive increment + is applied to the approximation.

4. If on other hand, the input signal is smaller, a negative increment - is applied to

the approximation.5. In this way the accumulator does the best it can to track the input samples by one

step at a time.

DELTA MODULATION RECEIVER

In the receiver, the staircase approximation xq(t) is reconstructed by passing the sequence

of positive and negative pulses, produced at the decoder output, through an accumulator

in a manner similar to that used in the transmitter. Then pass this staircase waveform

through a low pass filter (with a bandwidth equal to Original signal bandwidth) to recover

the original signal.

Fig.2.3 Delta Modulation Receiver

In comparing the DPCM and DM networks, we note that they are basically similar,

except for two important differences, namely, the use of a one-bit quantizer in delta

modulator and the replacement of the prediction filter by a single delay element.


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QUANTIZING NOISE

Delta modulation systems are subject to two types of quantizing error.

(1) Slope overload distortion

(2) Granular Noise

SLOPE OVERLOAD DISTORTION:

This distortion arises because of large dynamic range of the input signal. The rate of

rise of input signal x(t) is so high that the staircase signal cannot approximate it. The

slope overload is said to occur when the step size „Δ‟ is too small to follow steep

segment of the input waveform x(t). To reduce this error, the step size must be

increased when slope of the signal x(t) is high. Since the step size of delta modulator

remains fixed, its maximum or minimum slopes occur along straight lines. Therefore

this modulator is also known as Linear Delta Modulator.

Quantiztion errors in delta modulation for an arbitrary input

To reduce this slope overload distortion, the slope of the quantizer must be greater

than the maximum slope of the input signal.

Ie.max

( )

imum s

dm t

T dt

GRANULAR NOISE (IDLE NOISE):

Granularity, on other hand refers to a situation where the stair case function ( ) s x nT

hunts around a relatively flat segment of the input function, with a step size that is too

large relative to local slope characteristic of the input. This means that for very small


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DRAWBACKS OF DELTA MODULATION

The Delta Modulation has two major drawbacks as under;

(i) Slope overload distortion

(ii)

Granular Noise

SIMON HAYKIN Problem1: Given a sine wave of frequency f m and amplitude Am

applied to a delta modulator having step size Δ. Show that the slope overload will

occur if2

m

m s

A f T

here Ts is the sampling period.

Solution: Let us consider that the sine wave is represented as ( ) sin(2 )m m x t A f t

Maximum slope of delta modulator is given as sT

.

We know that, the slope overload distortion will take place if slope of the sine wave

is greater than slope of delta modulator i.e.,( )

max dx t

dt >

sT

sin(2 )max m m

s

dA f t

dt T

max 2 cos(2 )m m m s

f A f t T

2 m m f A > sT

2m

m s

A f T

Note:

To avoid slope overload distortion, the condition that must be satisfied is2

m

m s

A f T


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QUANTIZATION ERROR/NOISE IN DELTA MODULATION



If the step size of the quantizer is , then maximum quantization error max is and the

range of quantization error is , .

As the error is equally likely in the range , , it is better to assume error as uniform

random variable. The probability density function of this error is given by

1 1

( )2

f

Mean Square value of this quantization error (Noise power) is given by2

2 2 2 1[ ] ( )3

E f d d

SIGNAL TO QUANTIZATION NOISE RATIO IN DELTA MODULATION

Case 1: input signal is sinusoidal signal ( ) sinm m x t A t

SNR = Signal Power (rms) / Quantization noise power =

2

2

2

3

m A

No slope overload distortion occurs, for2

m

m s

A f T

, then substituting into the above

equation gives SNR =

2

2

2

3

m A

=

2

2

1

2 2

3

m s f T

This noise power2

3

is uniformly distributed over the frequency band upto s f (which is

more than m f ). Then the output quantization power within the bandwidth BWLPF f is given

by2

'

3

BWLPF q

s

f N

f


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In the receiver, at the output of Low pass filter of Bandwidth BWLPF f

SNR =

2

3

2 2 2

1

2 2 3

8

3

m s s

BWLPF BWLPF m

s

f T f

f f f

f

ADAPTIVE DELTA MODULATION

To reduce slope overload distortion, a large step size is required to accommodate

wide dynamic range of the input signal and small steps are required to reduce granular

noise. In fact, adaptive delta modulation is the modification to overcome these errors.

Finally, we should mention that a delta modulator may also be made adaptive,

wherein the variable step size increases during a steep segment of the input signal and

decreases when the modulator is quantizing an input signal with a slowly varying

segment. In this way the step size is adapted to the level of the input signal. The resulting

system is called an adaptive delta modulator.

The problem in adaptive delta modulation, of course, is to specify suitable rules

for step size variation. Figure below illustrates the operation of ADM.

Waveforms illustrative of ADM operation


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ADAPTIVE DELTA MODULATION TRANSMITTER:

The logic for step size control is added in the diagram. The step size increases or

decreases according to a specified rule depending on one bit quantizer output. As an

example, if one bit quantizer output is high (i.e. 1), then the step size may be doubled for

next sample. If one bit quantizer output is low, then step size may be reduced by one step.

ADAPTIVE DELTA MODULATION RECEIVER:

In the receiver of Adaptive delta modulator shown in figure, there are two portions. The

first portion produces the step size from each incoming bit. Exactly the same process is

followed as that in transmitter. The previous input and present input decides the step size.


48/108



It is then applied to an accumulator which builds up staircase waveform. The low pass

filter then smoothens out the staircase waveform to reconstruct the original signal.

ADVANTAGES OF ADAPTIVE DELTA MODULATION

1.

Signal to Noise ratio becomes better than ordinary delta modulation because of

the reduction in slope overload distortion and idle noise.

2. Because of the variable step size, the dynamic range of ADm is wider than simple

DM.

3. Bandwidth required for the transmission through channel is also less.

Results have been reported in the literature which compares the (SNR)o performance of

μ -law PCM and the ADM scheme discussed above. One such result is shown in Fig.

below for the case of band pass filtered (200-3200 Hz) speech. For PCM telephony, thesampling frequency used is 8 kHz. As can be seen from the figure, the SNR comparison

between ADM and PCM is dependent on the bit rate. An interesting consequence of this

is, below 50 kbps, ADM which was originally conceived for its simplicity, out-performs

the logarithmic PCM, which is now well established commercially all over the world. A

60 channel ADM (continuous adaptation) requiring a bandwidth of 2.048 MHz (the same

as used by the 30 channel PCM system) was in commercial use in France for sometime.

French authorities have also used DM equipment for airborne radio communication and

air traffic control over Atlantic via satellite. However, DM has not found wide-spread

commercial usage simply because PCM was already there first!

Performance of PCM and ADM versus bit rate


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sT

> Aβ sech2(βt)

> AβTs since the maximum value of sech(βt) is 1 at t=0

Problem4: Consider a DM system designed to accommodate analog message signals

limited to bandwidth W= 5kHz. A sinusoidal test signal of amplitude A=1 volt and

frequency f m = 1 kHz is applied to the system. The sampling rate of the system is 50 kHz

(a) Calculate the step size required to minimize slope overload distortion.

(b) Calculate signal to Quantization noise ratio of the system for the specified

sinusoidal test signal.

Solution: (a) To avoid slope overload distortion s

T > 2 m m f A

Therefore = 2 m m s f A T = 2 m m

s

f A

f

=

2 1 1

2 50

k v

k

=0.126 v

(c) Signal to Quantization noise ratio

SNR

3

2 2

3

8

s

BWLPF m

f

f f =

3

2 2

3 (50 )

8 5 (1 )

k

k k

= 475

SNR in db = 10 47510log = 26.8 db


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Problem 5: Consider a low pass signal with a bandwidth of 3 kHz. A linear delta

modulation system with step size Δ=0.1 v is used to process this signal at a sampling

rate ten times the nyquist rate.

(a) For linear delta modulation, the maximum amplitude of a sinusoidal test signal of

frequency 1 kHz which can be processed by the system without slope-overload

distortion.

(b)For the specifications given in part a, evaluate the output signal to noise ratio

under (i) prefilterd and (ii) postfiltered conditions

Solution: (a) For linear delta modulation, the maximum amplitude of a sinusoidal test

signal that can be used without slope overload distortion is

2m

m s

A f T

=

2

s

m

f

f

=

0.1 10 2 3

2 1

k

k

=0.95 v

(b) (i) Under the pre-filtered condition, it is reasonable to assume that the granular

quantization noise is uniformly distributed between – Δ and +Δ. Hence the

variance of the quantization noise is

22 2 2 1[ ] ( )

3 E f d d

When input signal is sinusoidal signal ( ) sinm m x t A t

SNR =

2

22

3

m A

=

2

2

0.95

20.1

3

= 135= 21.3 db

(iii) The signal to noise ratio under the post filtered condition is

SNR

3

2 2

3

8

s

BWLPF m

f

f f

3

2 2

3 (60 )

8 3 (1 )

k

k k

=1367== 31.3 db

The filtering gain in signal to noise ratio due to the use of a reconstruction

filter at the demodulator output is therefore 31.3 db- 21.3 db= 10db.


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Problem 6: A linear delta modulator has a step size of 100 mV and the minimum output

amplitude is + 50 mV. A signal s(t ) = 0.5 u(t ) is applied to the input of the delta

modulator. Show how the modulator tracks the input indicating the distortions in the

waveform. Sketch the waveform for 12 clock cycles, beginning at least 2 clock cycles

before t = 0. Also, sketch the output waveform in NRZ format.

Solution:

Figure (a) below shows the sketch of the delta modulator input and the tracking

distortions.

The input is a step signal of amplitude 0.5 volts beginning at t = 0 as shown by the heavy

line. The input for t < 0 is 0 volts. Initial amplitude of the DM predictor, at clock instant

1, is assumed to be + 50 mV. The clock instants are shown in (b).

At the clock instant 2 the predictor output is higher than the input (0 V) and hence, a

negative step (-100mV) is added to the predictor output. At clock instant 3 the predictor

output is lower (- 50 mV) than the input (0.5 V) and hence, a positive step is added to the


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predictor. At clock instant 4 the predictor output (+50 mV) is still lower than the input.

Hence, a 100 mV step is added. At clock instants 4, 5, 6, 7, and 8 the predictor output is

lower than input and at each instant a 100 mV step is added to the previous predictor out.

At clock instant 9 the predictor output (550 mV) is found higher than the input. Hence, a

100 mV step is subtracted from the predictor output. At clock instant 10 a 100 mV step is

added. The DM output waveform is shown in figure (c)

Problem 7: A segment of a delta modulated data stream is a sequence given below.

0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1

This sequence is applied to a linear delta demodulator having a step size of 100 mV.

Assuming initial output of the demodulator is 0 V, show the output sample voltages at

each input bit and sketch the waveform.

Solution:

The output of demodulator is obtained by adding the step voltage for input 1 and

subtracting the step voltage for input 0. The output is shown in the table below for the

input bits given above.

Input 0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1

Output -0.1 0 -0.1 0 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.3 0.4 0.3 0.2 0.1 0.2

The waveform of the output is shown in figure below.


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Problem 8:

An instantaneously companded delta modulator employs the following step

size adaptation algorithm.

where Sk and Sk-1 are the current and previous step sizes, Bk and Bk-1 are the

current and previous output bits, Bk and 1k B have opposite polarity. The

minimum step size is 100 mV, so the amplitude of the steps when the input

is zero is ±50 mV. If a step input x(t ) = 1.2 V is applied to the modulator at

t =0 show how the predictor output tracks the input by sketching the

waveform. Sketch the binary output waveform of the delta modulator.

Solution:

We can show the step size Sk , predictor output Pk and modulator outputs for some clock

cycles in a tabular form as below. The input of 1.2 V is applied to the modulator at t = 0

and we start at t = -2.

The waveforms of the predictor output and the modulator output are plotted in the figure

below.


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Problem 9: A linear delta modulator is used to digitize speech signal band limited to 3.4

kHz. An output filter with 4 kHz cutoff frequency is used. Find the sampling frequency

required to get a performance equivalent to that of a 6-bit linear PCM coder. Compare the

information rates for PC and DM outputs.

Solution:The S/N obtained from a linear PCM coder is

max

6 1.8 6 6 1.8 37.8S

v db N

The S/N obtained from a linear DM coder is SNR

3

2 2

3

8

s

BWLPF m

f

f f

In db we can write the above equation as SNR dB = 10 log

3

2 2

3

8

s

BWLPF m

f

f f

To get DM performance equivalent to PCM technique SNR of DM must be equal to SNR

of PCM.


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37.8 =2

310 log 30 log 10 log 20 log

8 s BWLPF m f f f

37.8 = 3 323

10 log 30 log 10 log 4 10 20 log 3.4 108

s f

f s = 191 kb/s

Assuming 8kHz sampling the information rate for PCM data is R PCM=8*6=48kb/s

For DM the information rate is same as the sampling rate, hence R DM = 191 kb/s

Thus, DM requires approximately four times the data rate compared to 6-bit PCM for

similar performance.

Problem 10: A stereo music signal is sampled at 44.1 kHz and digitized with 16 bits for

recording on CD. If the CD stores 80 minutes of music find the total capacity of the CD

in bytes. What is the quality of the music if it has an RMS value 15 dB below the peak

value of the quantizer?

Solution:

We have fs = 44.1 kHz, n = 16 and number of channels is 2.

Hence, the bit rate

R = 2.n. fs = 2 x 16 x 44.1 x 103 = 1411.2 kb/s

The capacity of the CD is

C = 80 x 60 x 1411.2 kbits

= 6773760 kbits

= 846.72 Mbytes

The signal to noise ratio is

S / Nq = 6.n +1.8 -15 = 6 x16 +1.8 -15 = 82.8 dB


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Important Questions

1. a) Consider a test signal m(t)= A tanh(ßt) defined by a hyperbolic tangent function.

Where A and ß are constants. Determine the minimum step size ∆ for delta modulation of

this signal, which is required to avoid slope overload.

b) Comparison between PCM and DM.

2. a) A signal to transmitted is of the form S(t)=10COS 1000πt+5COS1500πt.

i) Choose an appropriate f and step size for delta modulator.

ii) Find the SNR for your design.

b) Draw the block diagram for Adaptive delta modulation system and explain each

block?

3. (a) Explain the noise effects in delta modulation

(b) A DM system is designed to operate at 3 times the Nyquist rate for a signal with a

3 KHz BW. The quantization step size is 250mv

(i) Determine the maximum amplitude of a 1 KHz sinusoid for which delta modulator

does not show slope overload.

(ii) Determine post filtered output SNR for the signal at part (i)

4. (a) Derive the condition for step size of the quantizer in a DM system to avoid slope

over load distortion for the message signal x(t) = A cos(wt)

(b) Explain the major drawback of the DM system with relevant waveforms.

5. A sinusoidal modulating signal is represented by m(t)= A cos (wmt) where wm=2πf m.

Derive the expression for the maximum output signal to quantization noise ratio in the

DM system system with no slope overload distortion and also determine maximum

output signal to quantization noise ratio at the post receiver?


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UNIT 3- DIGITAL MODULATION TECHNIQUES

Modulation is defined as the process by which some characteristics of a carrier is

varied in accordance with a modulating signal. In digital communications, the modulating

signal consists of binary data or an M-ary encoded version of it. The data is used to

modulate a carrier wave (usually sinusoidal) with fixed frequency.

The modulation process involves switching or keying the amplitude, frequency or

phase of the carrier in accordance with the input data.

Thus there are three basic modulation techniques for the transmission of digital

data. They are known as amplitude-shift keying (ASK), frequency shift keying (FSK) and

phase shift keying (PSK).

If the amplitude of the carrier is switched depending on the input digital signal,

then it is called amplitude shift keying (ASK). This process is quite similar to analog

amplitude modulation.

If the frequency of the sinusoidal carrier is switched depending upon the input

digital signal, then it is known as the frequency shift keying. This is very much similar to

the analog frequency modulation.

If the phase of the carrier is switched depending upon the input digital signal, then

it is called phase shift keying. This is similar to phase modulation.

ASK, PSK & FSK waveforms (with sine as carrier signal)


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Since the phase and frequency modulation has constant amplitude envelope,

therefore FSK and PSK, the effect of non-linearities, noise interference is minimum on

signal detection. However, these effects are more pronounced on ASK. Therefore FSK

and PSK are preferred over ASK.

Figure shows the waveforms for amplitude-shift keying, phase shift keying and

frequency shift keying. In these waveforms, a single feature of the carrier (i.e. amplitude,

phase or frequency) undergoes modulation.

In digital modulations, instead of transmitting one bit at a time, we transmit two

or more bits simultaneously. This is known as M-ary transmission. This type of

transmission results in reduced channel bandwidth.

However, sometimes, we use two quadrature carriers for modulation. This process is

known as Quadrature modulation.

Thus we see that there are a number of modulation schemes available to the designer of a

digital communication system required for data transmission over a bandpass channel.

Every scheme offers system trade-offs of its own. In particular choice is made in favour

of a scheme which possesses as many of the following design characteristics as possible:

(i) Maximum data rate

(ii) Minimum probability of symbol error

(iii) Minimum transmitted power

(iv) Minimum channel bandwidth

(v) Maximum resistance to interfering signals.

(vi) Minimum circuit complexity.

DEFINITIONS AND TERMINOLOGY

There are basically two types of transmission of digital signals

i) BASEBAND DATA TRANSMISSION:

The digital data is transmitted over the channel directly. There is no carrier or any

modulation. This is suitable for transmission over short distances.

A signal whose frequency content (i.e. its spectrum) is in the vicinity of zero (i.e.,

f = 0 or dc) is said to be a baseband signal.

Original source signal are sometimes said to be baseband. Baseband systems

transmit baseband signals.


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This is usually not an effective means of communication.

ii) PASSBAND (BAND PASS OR NARROW BAND) DATA TRANSMISSION: The digital data

modulates high frequency sinusoidal carrier. Hence it is also called digital CW

modulation. It is suitable for transmission over long distances.

Types of passband Modulation are ASK, PSK, FSK and etc.

Bandpass signal spectrum is nonzero in some band of frequency with BW = 2B

centered about f = ±fc, where fc >> 0.

Effective transmission of signal usually requires bandpass signal.

Bandpass transmission involves some translation of the baseband signal to some

band of frequency centered around fc.

Types of Reception for Passband transmission

There are two types of methods for detection of passband signals.

i) COHERENT (SYNCHRONOUS) DETECTION: In this method, the local carrier generated at

the receiver is phase locked with the carrier at the transmitter. Hence it is also called

synchronous detection.

ii) NON COHERENT (ENVELOPE) DETECTION: In this method, the receiver carrier no need

to be phase locked with transmitter carrier. Hence it is also called envelope detection.

Noncoherent detection is simple but it has higher probability of error.


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Types of Digital Modulation techniques (Classification based on envelope)

Types of Digital Modulation techniques (Classification based on coherence)


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DIGITAL MODULATION TECHNIQUES

As mentioned earlier, the binary (i.e. Digital) modulation has three basic forms

amplitude-shift keying(ASK), phase-shift keying(PSK) and frequency shift keying

(FSK).

BINARY AMPLITUDE SHIFT KEYING (ON-OFF KEYING)

Definition:

Amplitude shift keying (ASK) or ON-OFF keying (OOK) is the simplest digital

modulation technique. In this method, there is only one unit energy carrier and it is

switched on or off depending upon the input binary sequence.

a) Binary Modulating signal and b) BASK signal

a) Modulating signal b) spectrum of „a‟ c) Spectrum of BASK signal


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EXPRESSION:

( ) 2 cos(2 ) s c s t P f t To transmit symbol „1‟

( ) 0 s t To transmit symbol „0‟ i.e. no signal is transmitted.

Signal s(t) contains some complete cycles of carrier frequency „f c‟. Hence ASK waveform looks like an ON-OFF of the signal. Therefore it is also known as

the ON-OFF keying (OOK).

SIGNAL SPACE DIAGRAM (CONSTELLATION DIAGRAM) OF BASK

Study of signal spaces provides us with a geometrical method of conceptualizing the

modulation process.

The ASK waveform of equation for symbol 1 can be represented as,

1

2( ) cos(2 ) ( ) s b c s b

b

s t PT f t PT t T

This means that there is only one carrier function 1( )t which is a unit energy signal over

(0, T b). The signal space diagram will have two points on 1( )t . One will be at zero and

other will be at s b P T . The collection of all possible signal points is called the signal

constellation.

Thus, the distance between the two signal points is d= s b P T = b E

The decision boundary is determined by the threshold value λ. If x lies in the region Z 1,

then a decision of a “1” is made. If x lies in the region Z 2, then a decision of a “0” is

made.

One advantage in using the signal space representation is that it is much easier to identify

the “distance” between signal points. The distance between two signal points will be

increased which makes the received signal point less probable be located in the wrong

region.


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GENERATION OF BASK SIGNAL

ASK signal may be generated by simply applying the incoming binary data and the

sinusoidal carrier to the two inputs of a product modulator. The resulting output will be

the ASK waveform.

Generation of BASK signal

BASK RECEPTION:

COHERENT DETECTION OR DEMODULATION OF BINARY ASK SIGNAL

The demodulation of BASK waveform can be achieved with the help of coherent detector

as shown in figure.

It consists of a product modulator which is followed by an integrator and Decision

making device. The incoming ASK signal is applied to one input of the product

modulator. The other input of the product modulator is supplied with a sinusoidal carrier

which is generated with the help of a local oscillator.

The output of the product modulator goes to input of the integrator. The integrator

operates on the output of the multiplier for successive bit intervals and essentially

performs a low-pass filtering action. The output of the integrator goes to the input of a

decision making device.

.

Coherent detection of Binary ASK signal


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Now, the decision making de

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Digital Communications 3units