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UNIT – I

ELEMENTS OF DIGITAL COMMUNICATION SYSTEMS

The purpose of a Communication System is to transport an information bearing signal from a

source to a user destination via a communication channel.

MODEL OF A COMMUNICATION SYSTEM

Fig. 1.1: Block diagram of Communication System

The three basic elements of every communication systems are

1. Transmitter,

2. Receiver and

3. Channel.

The Overall purpose of this system is to transfer information from one point (called Source)

to another point, the user destination.

The message produced by a source, normally, is not electrical. Hence an input transducer is

used for converting the message to a time – varying electrical quantity called message signal.

Similarly, at the destination point, another transducer converts the electrical waveform to the

appropriate message.

The transmitter is located at one point in space, the receiver is located at some other point

separate from the transmitter, and the channel is the medium that provides the electrical

connection between them.

The purpose of the transmitter is to transform the message signal produced by the source of

information into a form suitable for transmission over the channel.

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The received signal is normally corrupted version of the transmitted signal, which is due to

channel imperfections, noise and interference from other sources.The receiver has the task of

operating on the received signal so as to reconstruct a recognizable form of the original

message signal and to deliver it to the user destination.

Communication Systems are divided into 3 categories:

1. Analog Communication Systems are designed to transmit analog information using

analog modulation methods.

2. Digital Communication Systems are designed for transmitting digital information using

digital modulation schemes, and

3. Hybrid Systems that use digital modulation schemes for transmitting sampled and

quantized values of an analog message signal.

ELEMENTS OF DIGITAL COMMUNICATION SYSTEMS:

The figure 1.2 shows the functional elements of a digital communication system.

Source of Information: 1. Analog Information Sources.

2. Digital Information Sources.

Analog Information Sources → Microphone actuated by a speech, TV Camera

scanning a scene, continuous amplitude signals.

Digital Information Sources → These are teletype or the numerical output of computer

which consists of a sequence of discrete symbols or letters.

An Analog information is transformed into a discrete information through the process of

sampling and quantizing.

Digital Communication System

Fig. 1.2: Block Diagram of Digital Communication System

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SOURCE ENCODER / DECODER:

The Source encoder ( or Source coder) converts the input i.e. symbol sequence into a

binary sequence of 0s and 1s by assigning code words to the symbols in the input

sequence. For eg. :-If a source set is having hundred symbols, then the number of

bits used to represent each symbol will be 7 because 27=128 unique combinations are

available. The important parameters of a source encoder are block size, code

word lengths, average data rate and theefficiency of the coder (i.e. Actual output

data rate compared to the minimum achievable rate)

At the receiver, the source decoder converts the binary output of the channel

decoder into a symbol sequence. The decoder for a system using fixed – length code

words is quite simple, but the decoder for a system using variable – length code

words will be very complex.

Aim of the source coding is to remove the redundancy in the

transmitting information, so that bandwidth required for transmission is

minimized. Based on the probability of the symbol code word is assigned. Higher the

probability, shorter is the codeword.

Ex: Huffman coding.

CHANNEL ENCODER / DECODER:

Error control is accomplished by the channel coding operation that consists of

systematically adding extra bits to the output of the source coder.

These extra bits do not convey any information but helps the receiver to detect and /

or correct some of the errors in the information bearing bits. There are two methods

of channel coding:

1. Block Coding: The encoder takes a block of k information bits from

the source encoder and adds r error control bits, where r is dependent on k and

error control capabilities desired.

2. Convolution Coding: The information bearing message stream is encoded in

a continuous fashion by continuously interleaving information bits and error

control bits.

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The Channel decoder recovers the information bearing bits from the coded binary

stream. Error detection and possible correction is also performed by the channel

decoder.

The important parameters of coder / decoder are: Method of coding, efficiency,

error control capabilities and complexity of the circuit.

MODULATOR:

The Modulator converts the input bit stream into an electrical waveform

suitable for transmission over the communication channel. Modulator can be

effectively used to minimize the effects of channel noise, tomatch the frequency

spectrum of transmitted signal with channel characteristics, to provide the

capability to multiplex many signals.

DEMODULATOR:

The extraction of the message from the information bearing waveform produced by

the modulation is accomplished by the demodulator. The output of the demodulator

is bit stream. The important parameter is the method of demodulation.

CHANNEL:

The Channel provides the electrical connection between the source and destination.

The different channels are: Pair of wires, Coaxial cable, Optical fibre, Radio

channel, Satellite channel or combination of any of these.

The communication channels have only finite Bandwidth, non-ideal frequency

response, the signal often suffers amplitude and phase distortion as it travels over the

channel. Also, the signal power decreases due to the attenuation of the channel.

The signal iscorrupted by unwanted, unpredictable electrical signals referred to as

noise.

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The important parameters of the channel are Signal to Noise power Ratio(SNR),

usable bandwidth, amplitude and phase response and the statistical properties of

noise.

Certain Issues in Digital Transmission

Sometimes, a comparison between two digital transmission systems is needed. There

are many parameters that can be used to compare between digital transmission

systems, but some of the most important parameters of a digital transmission system

are:

• Transmission Rate (measured in bits per second): This is a measure of the

number of bits that can be transmitted over the communication channel per unit time.

• Bandwidth Requirements (measured in Hz): This is a measure of the

spectrum that the communication system requires to transmit the information at the

desired transmission rate.

• Error probability (measured in percentages): This represents the percentage of

bits that are in error relative to the overall number of bits that are transmitted by the

communication system.

• Transmission Power (or Bit Energy) (measured in Watts (or Jules/bit)): This

represents the amount of power of the transmitted signal that would be required to

achieve a particular desired error probability.

• System Complexity (measured in cost of building or operating the system):

This represents that amount of money that a manufacturer will have to spend to build

the system and the amount of money that a user will have to pay to use the system.

The performance of a digital transmission system is a function of the

following factors:

1. Amount of energy in each digital bit (or pulse): Generally, the more energy a

digital bit (or pulse) has, the better the performance that the system will have.

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2. The distance between the transmitter and receiver: Because energy is spread

or attenuated as it travels over the channel and more noise is added due to the

existence of more noise sources over long channels, generally the longer the path that

the digital transmitted signal has to travel, the worse the performance that the system

will have. However, you do not always have control over the distance between the

transmitted and receiver.

3. Amount of noise that is added to the signal: Certainly, the less the noise that

is added to the transmitted signal, the better the performance of the communication

system. We usually have limited control over the added noise.

4. Bandwidth of the transmission channel: By using larger bandwidth, we can

either transmit at a higher transmission bit rate while keeping the same probability of

bit error, or we can transmit at the same transmission bit rate but reduce the

probability of bit error. Generally, the larger the bandwidth allocated to a

communication system, the better the performance it will have.

Advantages of Digital Communication

1. The effect of distortion, noise and interference is less in a digital communication

system. This is because the disturbance must be large enough to change the pulse

from one state to the other.

2. Regenerative repeaters can be used at fixed distance along the link, to identify and

regenerate a pulse before it is degraded to an ambiguous state.

3. Digital circuits are more reliable and cheaper compared to analog circuits.

4. The Hardware implementation is more flexible than analog hardware because of

the use of microprocessors, VLSI chips etc.

5. Signal processing functions like encryption, compression can be employed to

maintain the secrecy of the information.

6. Error detecting and Error correcting codes improve the system performance by

reducing the probability of error.

7. Combining digital signals using TDM is simpler than combining analog signals

using FDM. The different types of signals such as data, telephone, TV can be treated

as identical signals in transmission and switching in a digital communication system.

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8. We can avoid signal jamming using spread spectrum technique.

Disadvantages of Digital Communication:

1. Large System Bandwidth:- Digital transmission requires a large system

bandwidth to communicate the same information in a digital format as compared to

analog format.

2. System Synchronization:- Digital detection requires system synchronization

whereas the analog signals generally have no such requirement.

Channels for Digital Communications

The modulation and coding used in a digital communication system depend on the

characteristics of the channel. The two main characteristics of the channel are

BANDWIDTH and POWER. In addition the other characteristics are whether the

channel is linear or nonlinear, and how free the channel is free from the external

interference.

Five channels are considered in the digital communication, namely: telephone

channels, coaxial cables, optical fibers, microwave radio, and satellite channels.

Telephone channel: It is designed to provide voice grade communication. Also

good for data communication over long distances. The channel has a band-pass

characteristic occupying the frequency range 300Hz to 3400hz, a high SNR of

about 30db, and approximately linear response.

For the transmission of voice signals the channel provides flat amplitude

response. But for thetransmission of data and image transmissions, since the phase

delay variations are important an equalizer is used to maintain the flat amplitude

response and a linear phase response over the required frequency band.

Transmission rates upto16.8 kilobits per second have been achieved over the

telephone lines.

Coaxial Cable: The coaxial cable consists of a single wire conductor centered

inside an outer conductor, which is insulated from each other by a dielectric. The

main advantages of the coaxial cable are wide bandwidth and low external

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interference. But closely spaced repeaters are required. With repeaters spaced at 1km

intervals the data rates of 274 megabits per second have been achieved.

Optical Fibers: An optical fiber consists of a very fine inner core made of silica

glass, surrounded by a concentric layer called cladding that is also made of glass. The

refractive index of the glass in the core is slightly higher than refractive index of the

glass in the cladding. Hence if a ray of light is launched into an optical fiber at the

right oblique acceptance angle, it is continually refracted into the core by the

cladding. That means the difference between the refractive indices of the core and

cladding helps guide the propagation of the ray of light inside the core of the fiber

from one end to the other.

Compared to coaxial cables, optical fibers are smaller in size and they offer

higher transmission bandwidths and longer repeater separations.

Microwave radio: A microwave radio, operating on the line-of-sight link, consists

basically of a transmitter and a receiver that are equipped with antennas. The

antennas are placed on towers at sufficient height to have the transmitter and

receiver in line-of-sight of each other. The operating frequencies range from 1 to

30 GHz.

Under normal atmospheric conditions, a microwave radio channel is very

reliable and provides path for high-speed digital transmission. But during

meteorological variations, a severe degradation occurs in the system performance.

Satellite Channel: A Satellite channel consists of a satellite in geostationary orbit, an

uplink from ground station, and a down link to another ground station. Both link

operate at microwave frequencies, with uplink the uplink frequency higher than the

down link frequency. In general, Satellite can be viewed as repeater in the sky. It

permits communication over long distances at higher bandwidths and relatively low

cost.

Bandwidth:

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Bandwidth is simply a measure of frequency range. The range of frequencies

contained in a composite signal is its bandwidth. The bandwidth is normally a

difference between two numbers. For example, if a composite signal contains

frequencies between 1000 and 5000, its bandwidth is 5000 - or 4000. If a range of

2.40 GHz to 2.48 GHz is used by a device, then the bandwidth would be 0.08 GHz

(or more commonly stated as 80MHz).It is easy to see that the bandwidth we define

here is closely related to the amount of data you can transmit within it - the more

room in frequency space, the more data you can fit in at a given moment. The term

bandwidth is often used for something we should rather call a data rate, as in “my

Internet connection has 1 Mbps of bandwidth”, meaning it can transmit data at 1

megabit per second.

Sampling

A message signal may originate from a digital or analog source. If the

message signal is analog in nature, then it has to be converted into digital form before

it can transmitted by digital means. The process by which the continuous-time signal

is converted into a discrete–time signal is called Sampling.

Sampling operation is performed in accordance with the sampling theorem.

Sampling Theorem for Lowpass Signals

Part - I If a signal x(t) does not contain any frequency component beyond W Hz, then

the signal is completely described by its instantaneous uniform samples with

sampling interval (or period ) of Ts< 1/(2W) sec.

Part – II The signal x(t) can be accurately reconstructed (recovered) from the set of

uniform instantaneous samples by passing the samples sequentially through an ideal

(brick-wall) lowpass filter with bandwidth B, where W ≤ B < fs – W and fs = 1/(Ts).

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As the samples are generated at equal (same) interval (Ts) of time, the process of

sampling is called uniform sampling. Uniform sampling, as compared to any non-

uniform sampling, is more extensively used in time-invariant systems as the theory of

uniform sampling (either instantaneous or otherwise) is well developed and the

techniques are easier to implement in practical systems.

The concept of ‘instantaneous’ sampling is more of a mathematical abstraction as no

practical sampling device can actually generate truly instantaneous samples (a

sampling pulse should have non-zero energy). However, this is not a deterrent in

using the theory of instantaneous sampling, as a fairly close approximation of

instantaneous sampling is sufficient for most practical systems. To contain our

discussion on Nyquist’s theorems, we will introduce some mathematical expressions.

If x(t) represents a continuous-time signal, the equivalent set of instantaneous

uniform samples {x(nTs)} may be represented as,

{x(nTs)}≡ xs(t) = Σ x(t).δ(t- nTs) (1)

where x(nTs) = x(t) t =nTs , δ(t) is a unit pulse singularity function and ‘n’ is an

integer

Conceptually, one may think that the continuous-time signal x(t) is multiplied by an

(ideal) impulse train to obtain {x(nTs)} as in equation(1) can be rewritten as,

xs(t) = x(t).Σ δ(t- nTs) …. (2)

Now, let X(f) denote the Fourier Transform F(T) of x(t), i.e.

+∞

X ( f )= ∫ x (t ).exp( − j 2π ft )dt …. (3)

−∞

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Now, from the theory of Fourier Transform, we know that the F.T of Σ δ(t- nTs), the

impulse train in time domain, is an impulse train in frequency domain:

F{Σ δ(t- nTs)} = (1/Ts).Σ δ(f- n/Ts) = fs.Σ δ(f- nfs) … (4)

If Xs(f) denotes the Fourier transform of the energy signal xs(t), we can write using

Eq. (1.2.4) and the convolution property:

Xs(f) = X(f)* F{Σ δ(t- nTs)}

= X(f)*[fs.Σ δ(f- nfs)]

= fs.X(f)*Σ δ(f- nfs)

= fs. ∫ X(λ). Σ δ(f − nfs − λ)dλ +∞

−∞ = fs. Σ ∫ X(λ).δ(f- nfs-λ)dλ = fs.Σ X(f- nfs) ….

(5)

[By sifting property of δ(t) and considering δ(f) as an even function, i.e. δ(f) = δ(-

f)]

This equation, when interpreted appropriately, gives an intuitive proof to Nyquist’s

theorems as stated above and also helps to appreciate their practical implications.

Let us note that while writing Eq.(5), we assumed that x(t) is an energy signal so that

its Fourier transform exists.

Further, the impulse train in time domain may be viewed as a periodic singularity

function with almost zero (but finite) energy such that its Fourier Transform [i.e. a

train of impulses in frequency domain] exists.

With this setting, if we assume that x(t) has no appreciable frequency component

greater than W Hz and if fs > 2W, then Eq.(1.2.5) implies that Xs(f), the Fourier

Transform of the sampled signal xs(t) consists of infinite number of replicas of X(f),

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centered at discrete frequencies n.fs, -∞ < n < ∞ and scaled by a constant fs= 1/Ts

(Fig. 1.3).

Fig. 1.3 Spectra of (a) an analog signal x(t) and (b) its sampled version

Fig. 1.3 indicates that the bandwidth of this instantaneously sampled wave

xs(t) is infinite while the spectrum of x(t) appears in a periodic manner, centered at

discrete frequency values n.fs.

Now, Part – I of the sampling theorem is about the condition fs > 2.W i.e. (fs – W) >

W and (– fs + W) < – W. As seen from Fig. 1.3, when this condition is satisfied, the

spectra of xs(t), centered at f = 0 and f = ± fs do not overlap and hence, the spectrum

of x(t) is present in xs(t) without any distortion. This implies that xs(t), the

appropriately sampled version of x(t), contains all information about x(t) and thus

represents x(t).

The second part of Nyquist’s theorem suggests a method of recovering x(t) from its

sampled version xs(t) by using an ideal lowpass filter. As indicated by dotted lines in

Fig. 1.3, an ideal lowpass filter (with brick-wall type response) with a bandwidth W

≤ B < (fs – W), when fed with xs(t), will allow the portion of Xs(f), centered at f = 0

and will reject all its replicas at f = n fs, for n ≠ 0. This implies that the shape of the

continuous-time signal xs(t), will be retained at the output of the ideal filter.

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Hartley Shannon Law

The theory behind designing and analyzing channel codes is called Shannon’s noisy

channel coding theorem. It puts an upper limit on the amount of information you can

send in a noisy channel using a perfect channel code. This is given by the following

equation:

where C is the upper bound on the capacity of the channel (bit/s), B is the bandwidth

of the channel (Hz) and SNR is the Signal-to-N ise ratio (unitless).

Bandwidth-S/N Tradeoff

The expression of the channel capacity of the Gaussian channel makes intuitive

sense:

1. As the bandwidth of the channel increases, it is possible to make faster

changes in the information signal, thereby increasing the information rate.

2. As S/N increases, one can increase the information rate while stillpreventing

errors due to noise.

3. For no noise, S/N tends to infinity and an infinite information rate is

possibleirrespective of bandwidth.

Thus we may trade off bandwidth for SNR. For example, if S/N = 7 and B = 4kHz,

then the channel capacity is C = 12 ×103 bits/s. If the SNR increases to S/N = 15 and

B is decreased to 3kHz, the channel capacity remains the same. However, as B tends

to 1, the channel capacity does not become infinite since, with an increase in

bandwidth, the noise power also increases. If the noise power spectral density is ɳ/2,

then the total noise power is N = ɳB, so the Shannon-Hartley law becomes

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Pulse Code Modulation

Introduction

In the simplest model of a telephone speech communication there is a direct,

dedicated, physical connection between the two participants in the conversation, and

this link is held for the duration of the conversation. The analogue electrical signal

produced by the telephone at either end is sent on to connection without

modification.

In Pulse Amplitude Modulation (PAM), the unmodified electrical signal is not sent

on to the connection. Instead, short samples of the signal are taken at regular

intervals, and these samples are sent on to the connection. The amplitude of each

sample is identical to the signal voltage at the time when the sample was taken.

Typically, 8,000 samples are taken per second, so that the interval between samples

is 125s, and the duration of each sample is approximately 4s.

Because each sample is very short (~4s) there is a lot of time between samples

(~121s). Samples from other conversations are put into this “spare time”. Usually

the samples from 32 separate conversations are put on to a single line. This process is

called Time Division Multiplexing (TDM).

Each sample is very short, and will be distorted as it travels across a communications

network. In order to reconstruct the original analogue signal the only information the

receiver needs to have about a sample is its amplitude, but if this is distorted then all

information about the sample has been lost. To overcome this problem, the pulse is

not transmitted directly, instead its amplitude is measured and converted into an 8

binary number - a sequence of 1s and 0s. At the receiver end, the receiver merely

needs to detect if a 1 or a 0 has been received so that it can still recover the amplitude

of a PAM pulse even if the 1s and 0s used to describe it have been distorted.

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The process of converting the amplitude of each pulse into a stream of 1s and 0s is

called Pulse Code Modulation (PCM)

Note that the process of PAM and PCM (but without the use of TDM) is essentially

used to store music and speech on CDs, but with a higher sample rate, more bits per

sample and complex error correction mechanisms.

Some terms are:

Sampling The process of measuring the amplitude of a continuous-time signal at discrete instants. It converts a continuous-time signal to a discrete-time signal.

Quantizing Representing the sampled values of the amplitude by a finite set of levels. It converts a continuous-amplitude sample to a discrete-amplitude sample.

Encoding Designating each quantized level by a (binary) code.

Sampling and quantizing operations transform an analogue signal to a digital signal.

Use of quantizing and encoding distinguishes PCM from analogue pulse modulation methods.

The quantizing and encoding operations are usually performed in the same circuit at the transmitter, which is called an Analogue to Digital Converter (ADC). At the receiver end the decoding operation converts the (8 bit) binary representation of the pulse back into an analogue voltage in a Digital to Analogue Converter (DAC)

Pulse Code Modulation

Pulse Code Modulation (PCM) is an extension of PAM wherein each analogue sample value is quantized into a discrete value for representation as a digital code word.

Thus, as shown in Fig. 2.1 a PAM system can be converted into a PCM system by adding a suitable analogue-to-digital (A/D) converter at the source and a digital-to-analogue (D/A) converter at the destination.

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Fig. 2.1

PCM is a true digital process as compared to PAM. In PCM the speech signal is

converted from analogue to digital form.

PCM is standardised for telephony by the ITU-T (International Telecommunications

Union - Telecoms, a branch of the UN), in a series of recommendations called the G

series. For example the ITU-T recommendations for out-of-band signal rejection in

PCM voice coders require that 14 dB of attenuation is provided at 4 kHz. Also, the

ITU-T transmission quality specification for telephony terminals require that the

frequency response of the handset microphone has a sharp roll-off from 3.4 kHz.

In quantization the levels are assigned a binary codeword. All sample values falling

between two quantization levels are considered to be located at the centre of the

quantization interval. In this manner the quantization process introduces a certain

amount of error or distortion into the signal samples. This error known as

quantization noise, is minimised by establishing a large number of small quantization

intervals. Of course, as the number ofquantization intervals increase, so must the

number or bits increase to uniquely identify the quantization intervals. For example,

if an analogue voltage level is to be converted to a digital system with 8 discrete

levels or quantization steps three bits are required. In the ITU-T version there are 256

quantization steps, 128 positive and 128 negative, requiring 8 bits. A positive level is

represented by having bit 8 (MSB) at 0, and for a negative level the MSB is 1.

A to D

C o n v e r te r

B in a ry

C o d e r

P a ra lle l

to S e r ia l

C o n v e r te r

D ig ita l

P u lse

G e n e ra to r

S e r ia l to

P a ra lle l

C o n v e r te r

D to A

C o n v e r te rL P F

S a m p le r

A n a lo g u e

In p u t

P C M

O u tp u t

M o d u la to r

P C M

In p u t

A n a lo g u e

O u tp u t

D e m o d u la to r

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Quantization

The process of quantizing a signal is the first part of converting an sequence of

analog samples to a PCM code. In quantization, an analog sample with an amplitude

that may take value in a specific range is converted to a digital sample with an

amplitude that takes one of a specific pre–defined set of quantization values. This is

performed by dividing the range of possible values of the analog samples into L

different levels, and assigning the center value of each level to any sample that falls

in that quantization interval. The problem with this process is that it approximates the

value of an analog sample with the nearest of the quantization values. So, for almost

all samples, the quantized samples will differ from the original samples by a small

amount. This amount is called the quantization error. To get some idea on the effect

of this quantization error, quantizing audio signals results in a hissing noise similar to

what you would hear when play a random signal.

Assume that a signal with power Psis to be quantized using a quantizer with L = 2n

levels ranging in voltage from –mp tomp as shown in the fig. 2.2

t4TsTs 3Ts 5Ts2Ts0

mp

–mp

L = 2n

L levels

n bits0

v

Q uantizer O utput Sam ples q

x

Q uantizer Input Sam ples x

A quantization interval Corresponding quantization value

Fig. 2.2

We can define the variable v to be the height of the each of the L levels of the

quantizer as shown above. This gives a value of v equal to

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2

pm

vL

.

Therefore, for a set of quantizers with the same mp, the larger the number of levels of

a quantizer, the smaller the size of each quantization interval, and for a set of

quantizers with the same number of quantization intervals, the larger mp is the larger

the quantization interval length to accommodate all the quantization range.

Now if we look at the input output characteristics of the quantizer, it will be similar

to the red line in the following figure. Note that as long as the input is within the

quantization range of the quantizer, the output of the quantizer represented by the red

line follows the input of the quantizer. When the input of the quantizer exceeds the

range of –mp tomp, the output of the quantizer starts to deviate from the input and

the quantization error (difference between an input and the corresponding output

sample) increases significantly.

v v v vvvvv

v/2

v/2

v/2

v/2

v/2

v/2

v/2

v/2

Quantizer

Input x

Quantizer

Output xq

qx

x

mp

Fig. .2.3

Now let us define the quantization error represented by the difference between the

input sample and the corresponding output sample to be q, or

q

q x x .

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Plotting this quantization error versus the input signal of a quantizer is seen next.

Notice that the plot of the quantization error is obtained by taking the difference

between the blow and red lines in the above Fig. 2.3

v v v vvvvv

v/2 Quantizer

Input x

Quantization Error q

v/2

mp

Fig. 2.4

It is seen from the Fig 2.4 that the quantization error of any sample is restricted

between –v/2 andv/2 except when the input signal exceeds the range of

quantization of –mp to mp.

Uniform Quantization

We assume that the amplitude of the signal m(t) is

confined to the range (-mp, +mp ). This range (2mp) is

divided into L levels, each of step size , given by

= 2 mp / L

A sample amplitude value is approximated by the

midpoint of the interval in which it lies. The

input/output characteristics of a uniform quantizer is

shown in Fig. 2.5

Fig. 2.5

-mp

+ mp

In p u t

Ou

tpu

t

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Companding

-Companding is the process of compressing and then expanding

-High amplitude analog signals are compressed prior to txn. and then expanded in the

receiver

-Higher amplitude analog signals are compressed and Dynamic range is improved

-Early PCM systems used analog companding, where as modern systems use digital

companding.

Fig 2.6 Basic companding process

Analog companding

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2.7 PCM system with analog companding

--In the transmitter, the dynamic range of the analog signal is compressed, and then

converted o a linear PCM code.

--In the receiver, the PCM code is converted to a PAM signal, filtered, and then

expanded back to its original dynamic range.

-- There are two methods of analog companding currently being used that closely

approximate a logarithmic function and are often called log-PCM codes.

The two methods are 1) -law and

2) A-law

-law companding

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Fig. 2.8m-law companding

where Vmax = maximum uncompressed analog input amplitude(volts)

Vin = amplitude of the input signal at a particular instant of time (volts)

= parameter used tio define the amount of compression (unitless)

Vout = compressed output amplitude (volts)

A-law companding

--A-law is superior to -law in terms of small-signal quality

--The compression characteristic is given by

where y = Vout

x = Vin /

Vmax

m a x

m a x

ln 1

ln 1

in

o u t

VV

VV

1||1

,log1

|)|log(1

1||0,

log1

||

xAA

xA

Ax

A

xA

y

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Digital Companding:

--With digital companding, the analog signal is first sampled and converted to a

linear PCM code, and then the linear code is digitally compressed.

-- In the receiver, the compressed PCM code is expanded and then decoded back to

analog.

-- The most recent digitally compressed PCM systems use a 12- bit linear PCM code

and an 8-bit compressed PCM code.

Digital compression error

--To calculate the percentage error introduced by digital compression

%error=12-bit encoded voltage - 12-bit decoded voltage X 100

12-bit decoded voltage

1.3 PCM Encoding Process (HDB3)

The output from the analogue to digital converter (ADC) has n parallel bits. In the

case of telephony n = 8. The most significant bit is the signed bit. If the measured

sample is positive then the signed bit is 0. If the measured sample is negative then

the signed bit is 1. The remaining 7 bits are used to code the sample value. The ITU-

T define a look up table which allocates a particular binary code to each quantified

A-law value.

The line coding which is used assigns opposite polarities to successive “1”s. This

eliminates any DC voltage on the line, and reduces the inter symbol interference if

adjacent bits are “1”. If there is silence on the PCM channel then the measured

samples will be 0 Vrms and the output of the DAC will be 1000 0000. A stream of all

zeros is not desirable on an active channel because

all zeros could also be a fault condition and

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it is difficult to recover the clock signal from the incoming signal.

The coding system HDB3 is used and was developed to eliminate all zeros, and to

assign opposite polarities to successive “1”s.

This is a bipolar signalling technique (i.e. relies on the transmission of both positive

and negative pulses).

In AMI positive and negative pulses (of equal amplitude) are used for alternative

symbols 1. No pulse is used for symbol 0. In either case the pulse returns to 0 before

the end of the bit interval. This eliminates any DC on the line.

HDB3 encoding rules follow those for AMI, except that a sequence of four

consecutive 0's are encoded using a special "violation" bit. The 4th 0 bit is given the

same polarity as the last 1-bit which was sent using the AMI encoding rule. This

prevents long runs of 0's in the data stream which may otherwise prevent a receiver

from tracking the centre of each bit. By introducing violations, extra "edges" are

introduced, enabling a Digital PLL to reliably reconstruct the clock signal at the

receiver. The HDB3 is transparent to the sequence of bits being transmitted (i.e.

whatever data is sent, the Digital PLL can reconstruct the data and extract the bits at

the receiver).

To prevent a DC being introduced by excessive runs of zeros any run of more than

four zeros encodes as B00V. The value of B is assigned + or - alternately throughout

the bit stream.

Example 1 1 1 1 1 1 1 1 = + - + - + - + -

B BBBBBBB

1 0 1 0 1 0 1 0 = + 0 - 0 + 0 - 0

B 0 B 0 B 0 B 0

1 0 0 0 0 0 0 1 + 0 0 0 + 0 0 -

= B 0 0 0 V 0 0 B

1 0 0 0 0 1 1 0 = + 0 0 0 + - + 0

= B 0 0 0 V B B 0

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PCM Timing and Synchronisation

The PCM receiver must be able to identify the start and finish of each full sampling

sequence and to identify each bit position. The sampling clock needs to be either sent

to, or regenerated at, the receiving side to determine when each full sequence of

sampling begins and ends. The data clock is also needed to determine exactly when

to read each bit of information.

A PCM channel is sampled at 8,000 Hz or

once every 125 s. If there is one channel or

30 TDM channels the sampling period is fixed

at 125 s and this period is known as a frame.

Therefore the frame clock must have a period

of 125 s. The rising edge of the frame clock

informs the receiver that the next bit will be

Bit 1 of a new sample. The falling edge of the

data clock informs the receiver that it must read the data bit.

When the bit stream is transmitted along a line the pulses become distorted and the

rise and fall times become significant. Ideally, a 1 will be “high” for 15.625 s. In

practice the pulse may only be above the “high” threshold for a few s so it is very

important that the bit is read within a certain time limit of the clock pulse.

The simplest way to synchronise a PCM sender to a PCM receiver is to send the

clock signals on different circuits to the data This would be done in a self-contained

system such as private branch exchange (PBX). Telephony is full duplex so that there

is a coder and a decoder at each port, but each would use the same clock.

To minimise the number of circuits it is possible to use a line-coding scheme which

allows the receiver to extract the clocks from the PCM signal. In this case the

receiver will have free running clocks that lock (using a PLL) to the phase and

frequency of the transitions in the data stream. The line-coding scheme ensures that

there is a transition for every data bit.

Differential pulse coding schemes

PCM transmits the absolute value of the signal for each frame. Instead we can

transmit information about the difference between each sample. The two main

differential coding schemes are:

Delta Modulation

Differential PCM and Adaptive Differential PCM (ADPCM)

B 1

1

B 2

0

B 3

1

B 4

0

B 5

0

B 6

1

B 7

1

B 8

1

B 1

?

fra m e c lo c k 1 2 5 s

d a ta c lo c k 6 4 k b it /s1 5 .6 2 5 s

Fig. 2.9

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Delta Modulation

Delta modulation converts an analogue signal, normally voice, into a digital signal.

The analogue signal is sampled as in the

PCM process. Then the sample is

compared with the previous sample. The

result of the comparison is quantified

using a one bit coder. If the sample is

greater than the previous sample a 1 is

generated. Otherwise a 0 is generated.

The advantage of delta modulation over

PCM is its simplicity and lower cost. But the noise performance is not as Fig.

2.10

good as PCM.

To reconstruct the original from the quantization, if a 1 is received the signal is

increased by a step of size q, if a 0 is received the output is reduced by the same size

step. Slope overload occurs when the encoded waveform is more than a step size

away from the input signal. This condition happens when the rate of change of the

input exceeds the maximum change that can be generated by the output. Overload

will occur if:

dx(t)/dt q /T = q * fs

where: x(t) = input signal, q = step size, T = period between samples, fs = sampling

frequency

Assume that the input signal has maximum amplitude A and maximum frequency F.

The most rapidly changing input is provided by x(t) = A * sin (2 * * F * t).

For this dx(t)/dt = 2 * * F * A * sin (2 * * F * t).

This slope has a maximum value of 2 * * F * A

Overload occurs if 2 * * F * A > q * fs

To prevent overload we require q * fs> 2 * * F * A

Example A = 2 V, F = 3.4 kHz, and the signal is sampled 1,000,000 times per

second, requires q > 2 * 3.14 * 3,400 * 2 /1,000,000 V > 42.7 mV

0

1

0

0 0

0

1

0

0

0

1

1

1

1 1

0

1

g ra n u la r n o ise

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Granular noise occurs if the slope changes more slowly than the step size. The

reconstructed signal oscillates by 1 step size in every sample. It can be reduced by

decreasing the step size. This requires that the sample rate be increased. Delta

Modulation requires a sampling rate much higher than twice the bandwidth. It

requires oversampling in order to obtain an accurate prediction of the next input,

since each encoded sample contains a relatively small amount of information. Delta

Modulation requires higher sampling rates than PCM.

Differential PCM (DPCM) and ADPCM

DPCM is also designed to take advantage of the redundancies in a typical speech

waveform. In DPCM the differences between samples are quantized with fewer bits

that would be used for quantizing an individual amplitude sample. The sampling rate

is often the same as for a comparable PCM system,

unlike Delta Modulation.

Adaptive Differential Pulse Code Modulation ADPCM is standardised by ITU-T

recommendations G.721 and G.726. The method uses 32,000 bits/s per voice

channel, as compared to standard PCM’s 64,000 bits/s. Four bits are used to describe

each sample, which represents the difference between two adjacent samples.

Sampling is 8,000 times a second. It makes it possible to reduce the bit flow by half

while maintaining an acceptable quality. While the use of ADPCM (rather than

PCM) is imperceptible to humans, it can significantly reduce the throughput of high

speed modems and fax transmissions.

The principle of ADPCM is to use our knowledge of the signal in the past time to

predict the signal one sample period later, in the future. The predicted signal is then

compared with the actual signal. The difference between these is the signal which is

sent to line - it is the error in the prediction. However this is not done by making

Fig. 2.11

A c c u m u la to r

D A C

Q u a n tis e r

E n o d e r

A D C

B a n d L im itin g

F ilte r+

-

D iffe re n tia to r

A n a lo g u e

In p u t

E n c o d e d

D iffe re n c e

S a m p le s

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comparisons on the incoming audio signal - the comparisons are done after PCM

coding.

To implement ADPCM the original (audio) signal is sampled as for PCM to produce

a code word. This code word is manipulated to produce the predicted code word for

the next sample. The new predicted code word is compared with the code word of the

second sample. The result of this comparison is sent to line. Therefore we need to

perform PCM before ADPCM.

The ADPCM word represents the prediction error of the signal, and has no

significance itself. Instead the decoder must be able to predict the voltage of the

recovered signal from the previous samples received, and then determine the actual

value of the recovered signal from this prediction and the error signal, and then to

reconstruct the original waveform.

ADPCM is sometimes used by telecom operators to fit two speech channels onto a

single 64 kbit/s link. This was very common for transatlantic phone calls via satellite

up until a few years ago. Now, nearly all calls use fibre optic channels at 64 kbit/s.

6.9 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

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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.

Fig. 2.12: Waveforms illustrative of LDM operation

Deltamodulation principleofoperation

Deltamodulationwasintroducedinthe1940sasasimplifiedformofpulsecodemodulatio

n(PCM),whichrequiredadifficult-to-implementanalog-to-digital(A/D)converter.

Theoutputofadeltamodulatorisabitstreamofsamples,atarelatively

highrate(eg,100kbit/sor

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moreforaspeechbandwidthof4 kHz)thevalueofeachbitbeing determinedaccordingas

towhethertheinputmessagesampleamplitudehasincreasedordecreasedrelativetothepr

evioussample.Itisan exampleofdifferentialpulsecodemodulation(DPCM).

Blockdiagram

Theoperationofadeltamodulatoristoperiodicallysampletheinputmessage,tomakeac

omparisonofthecurrentsamplewiththatprecedingit,andtooutputasinglebit which

indicatesthesignofthe differencebetweenthe twosamples.Thisinprinciple would

requirea sample-and-hold type circuit.

DeJager(1952)hitonanideafordispensingwiththeneedforasampleandholdcircuit.Her

easonedthatifthesystemwasproducingthedesiredoutputthenthisoutputcouldbesentb

acktotheinputandthetwoanalogsignalscomparedinacomparator.Theoutput isa

delayedversionoftheinput,andsothecomparison

isineffectthatofthecurrentbitwiththepreviousbit,asrequiredbythedeltamodulationpri

nciple.

Figure2.13illustratesthebasicsysteminblockdiagramform,andthiswillbethemodulat

oryouwill bemodelling.

Thesystemisintheformofafeedbackloop.Thismeansthatitsoperationisnotn ecessaril

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yobvious,anditsanalysisnon-

trivial.Butyoucanbuildit,andconfirmthatitdoesbehaveinthe manner adelta

modulatorshould.

Thesystemisacontinuoustimetodiscretetimeconverter.Infact,itisaformofanalogtodi

gitalconverter,andis thestarting pointfrom which more

sophisticateddeltamodulatorscanbe developed.

Thesamplerblockisclocked.Theoutputfromthesamplerisabipolarsignal,intheblockd

iagrambeingeither Vvolts.Thisisthedeltamodulatedsignal,thewaveformof

whichisshowninFigure 2.Itisfedback,inafeedbackloop,viaanintegrator,toasummer.

Theintegratoroutputisasawtooth-likewaveform,alsoillustratedinFigure 2.15.Itis

shownoverlaid uponthemessage,ofwhich itisanapproximation.

Figure 2.15:integrator output superimposed on the messagewith the delta

modulated signal below

Thesawtoothwaveformissubtractedfromthemessage,alsoconnectedtothesummer,

andthe difference-anerror signal-isthe signalappearingatthe summeroutput.

Anamplifierisshowninthefeedbackloop..Thiscontrolstheloopgain.Inpracticeitmaybe

aseparateamplifier,partoftheintegrator,orwithinthesummer.Itisusedtocontrolthesize

ofthe‘teeth’ofthesawtoothwaveform,inconjunctionwiththeintegratortimeconstant.

WhenanalysingtheblockdiagramofFigure

2.13itisconvenienttothinkofthesummerhavingunitygainbetweenbothinputsandtheou

tput.Themessagecomes in at

afixedamplitude.Thesignalfromtheintegrator,whichisasawtoothapproximationtothe

message,isadjustedwiththeamplifiertomatchitasclosely aspossible.

t ime

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stepsizecalculation

InthedeltamodulatorofFigure2.13theoutputoftheintegratorisasawtooth-

likeapproximationtotheinputmessage.Theteethofthesaw

mustbeabletorise(orfall)fastenoughtofollowthemessage.Thustheintegratortimecons

tantisanimportantparameter.

Foragivensampling(clock)ratethestepslope(volt/s)determinesthesize(volts)oftheste

p withinthesamplinginterval.

Supposetheamplitudeof therectangularwavefromthesampleris±V

volt.Forachangeofinputsampleto theintegratorfrom(say)negativeto

positive,thechangeofintegrator output will be,

afteraclockperiodT:

wherekisthegainoftheamplifier precedingthe integrator(asinFigure2.13).

AnswerTutorialQuestions1and

2beforeattemptingtheexperiment.Youcanlatercheckyouranswerbymeasurement.

slopeoverloadandgranularity

ThebinarywaveformillustratedinFigure2.15isthesignaltransmitted.Thisisthedeltamo

dulatedsignal.

Theintegralofthebinarywaveformisthesawtoothapproximationto themessage.

IntheexperimententitledDeltademodulation(inthisVolume)youwillseethatthissawto

othwave istheprimaryoutputfromthedemodulatoratthereceiver.

Lowpassfilteringofthesawtooth(fromthedemodulator)givesabetterapproximationtot

hemessage.Buttherewillbeaccompanyingnoiseanddistortion,productsoftheapproxi

mationprocessatthemodulator.

Theunwantedproductsofthemodulationprocess,observedatthereceiver,areoftwo

kinds.Thesearedue to‘slopeoverload’, and ‘granularity’.

slopeoverload

Thisoccurswhenthesawtoothapproximationcannotkeepupwiththerate-of-

changeoftheinput signalinthe regionsofgreatestslope.

Thestepsizeisreasonableforthosesectionsofthesampledwaveformofsmallslope,butt

heapproximationispoorelsewhere.Thisis‘slopeoverload’,duetotoosmalla step.

Slopeoverloadisillustrated inFigure2.16.

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slo p e o v e rlo a d

Figure2.16:slopeoverload

Toreducethepossibilityofslopeoverloadthestepsizecanbeincreased(forthesamesamp

ling rate).This isillustratedin Figure

2.17.Thesawtoothisbetterabletomatchthemessageinthe regionsofsteep slope.

Figure2.17:increasedstepsize to reduce slope overload

An alternativemethodofslopeoverloadreductionis

toincreasethesamplingrate.ThisisillustratedinFigure

2.18,wheretheratehasbeenincreasedbyafactorof2.4times, but thestep isthe same size

asinFigure2.15.

tim e

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Figure2.18:increasedsampling rate to reduce slope overload

1.4 Granularnoise

ReferbacktoFigure

2.16.Thesawtoothfollowsthemessagebeingsampledquitewellintheregionsofsmallslo

pe.Toreducetheslopeoverloadthestepsizeisincreased,andnow(Figure

2.17)thematchovertheregionsofsmallslopehasbeendegraded.

Thedegradationshowsup,atthedemodulator,asincreasedquantizingnoise,or‘granulari

ty’.

1.5 noiseanddistortionminimization

Thereisaconflictbetweentherequirementsforminimizationofslopeoverloadandthegra

nularnoise.Theonerequiresanincreasedstepsize,theotherareducedstepsize.You

shouldrefertoyourtextbook formorediscussion

ofwaysandmeansofreachingacompromise.Youwillmeetanexampleintheexperimente

ntitledAdaptivedeltamodulation(inthisVolume).Anoptimumstepcanbedeterminedby

minimizingthequantizingerroratthesummer output, or thedistortionatthe

demodulatoroutput.

Adaptive Delata Modulation

tim e

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The Operation Theory of ADM Modulation

From previous chapter, we know that the disadvantage of delta modulation is when

the input audio signal frequency exceeded the limitation of delta modulator, i.e.

Then this situation will produce the occurrence of slope overload and cause signal

distortion. However, the adaptive delta modulation (ADM) is the modification of

delta modulation to improve the disadvantage of the occurrence of slope overload.

Figure 2.20 is the block diagram of ADM modulator. In figure 2.20, we can see that

the delta modulator is comprised by comparator, sampler and integrator, then the

slope controller and the level detect algorithm comprise a quantization level adjuster,

which can control the gain of the integrator in the delta modulator. ADM modulator

is the modification of delta modulator, therefore, due to the delta modulator has the

problem of slope overload at low and high frequencies. The reason is the magnitude

of the Δ(t) of delta modulator is fixed, i.e. the increment of Δ or -Δ is unable to

follow the variation of the slope of the input signal. When the variation of the slope

of the input signal is large, the magnitude of Δ(t) still can increase by following the

variation, then this situation will not occur the problem of slope overload. On the

other hand, there is another technique, which is known as continuous variable slope

delta (CVSD) modulation. This technique is commonly used in Bluetooth

application. CVSD modulator is also the modification of delta modulator, use to

improve the occurrence of slope overload. The different between the CVSD and

ADM modulators are the quantization level adjuster A. ADM modulator is discrete

values and the quantization level adjuster of CVSD modulator is continuous. Simply,

the quantization value of ADM modulator is the variation of digital, such as the

quantization values of +1, +2, +3, -2, -3 and so on. As for CVSD modulator, the

quantization value is the variation of analog, such as the quantization values of +1,

+1.1, +1.2, -1.5, -0.3, -0.9 and so on.

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Fig. 2.20 The Operation Theory of ADM Modulation

UNIT - II

Digital modulation techniques

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

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

modulating wave consists of binary data or an M-ary encoded version of it and the

carrier is sinusoidal wave. Different Shift keying methods that are used in digital modulation techniques are

Amplitude shift keying [ASK]

Frequency shift keying [FSK]

Phase shift keying [PSK]

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Fig 3.1 Different modulations

1. ASK[Amplitude Shift Keying]:

In a binary ASK system symbol ‘1’ and ‘0’ are transmitted as

S (t) 2Eb Cos2f t for symbol 1

1 1

Tb

S 2 (t) 0 for symbol 0

2. FSK[Frequency Shift Keying]:

In a binary FSK system symbol ‘1’ and ‘0’ are transmitted as

S (t) 2Eb Cos2f t for symbol 1

1 1

Tb

S 2(t) 2Eb

Cos2f 2t for symbol 0

Tb

3. PSK[Phase Shift Keying]:

In a binary PSK system the pair of signals S1(t) and S2(t) are used to

represent binary symbol ‘1’ and ‘0’ respectively.

S1 (t) 2Eb Cos2fc t --------- for Symbol ‘1’

Tb

S2 (t) 2Eb

Cos(2fc t ) 2Eb

Cos2fc t ------- for Symbol ‘0’

Tb Tb

Hierarchy of digital modulation technique

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Digital Modulation Technique

Coherent Non - Coherent

Binary M - ary Hybrid Binary M - ary (m) = 2 (m) = 2

* ASK M-ary ASK M-ary APK * ASK M-ary ASK * FSK M-ary FSK M-ary QAM * FSK M-ary FSK * PSK M-ary PSK * DPSK M-ary DPSK (QPSK)

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Coherent Binary PSK:

Binary

Data Sequence

Non Return to Product

Zero Level

Modulator

Encoder

Binary PSK Signal

(t) 2 Cos2f t

1 c

Tb

Fig. 3.2 Block diagram of BPSK transmitter

x(t)

T b x1

Choose 1 if x1>0

Decision

dt

Device

0

Choose 0 if x1<0

Correlator

1 (t) Threshold λ = 0

Fig 3.3 Coherent binary PSK receiver

In a Coherent binary PSK system the pair of signals S1(t) and S2(t) are used

to represent binary symbol ‘1’ and ‘0’ respectively.

S1 (t) 2Eb

Cos2fc t ---------

for Symbol ‘1’

Tb

S2 (t) 2Eb

Cos(2fc t ) 2Eb

Cos2fc t ------- for Symbol ‘0’

Tb

Tb

Where Eb= Average energy transmitted per bit Eb

Eb0

Eb1

2

In the case of PSK, there is only one basic function of Unit energy which is given

by

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(t) 2 Cos2f t 0 t T

c

1 Tb

b

Therefore the transmitted signals are given by

(t)

(t)

0 t T

Symbol

1

S 1

E for

b 1 B

(t)

(t)

0 t T

Symbol 0

S 2

E for

b 1 B

A Coherent BPSK is characterized by having a signal space that is

one dimensional (N=1) with two message points (M=2)

Tb

S11 S1 (t)1 (t) dt Eb 0

Tb

S21 S2 (t)1 (t) dt Eb 0

The message point corresponding to S1(t) is located at S11 Eb and S2(t) is

located at S21 Eb . To generate a binary PSK signal we have to represent the input binary sequence in

polar form with symbol ‘1’ and ‘0’ represented by constant amplitude levels of

Eb & Eb

level encoder.

respectively. This signal transmission encoding is performed by a NRZ

The resulting binary wave [in polar form] and a sinusoidal carrier 1 (t)

[whose frequency fc nc ] are applied to a product modulator. The desired BPSK wave

Tb is obtained at the modulator output.

To detect the original binary sequence of 1’s and 0’s we apply the noisy PSK signal x(t) to a Correlator, which is also supplied with a locally generated coherent

reference signal 1 (t) as shown in fig (b). The correlator output x1 is compared with a

threshold of zero volt.

If x1 > 0, the receiver decides in favour of symbol 1.

If x1 < 0, the receiver decides in favour of symbol 0.

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Coherent Binary FSK

In a binary FSK system symbol ‘1’ and ‘0’ are transmitted as

S (t) 2Eb Cos2f t for symbol 1

1 Tb

1

S 2(t) 2Eb

Cos2f 2t for symbol 0

Tb

The basic functions are given by

(t) 2 Cos2f t And

1 Tb

1

(t) 2 Cos2f t for 0 t T And Zero Otherwise

2

Tb

2 b

Therefore FSK is characterized by two dimensional signal space with two

message points i.e. N=2 and m=2. The two message points are defined by the signal vector

Generation and Detection:-

(a)

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(b)

Fig. 3.4: (a) FSK transmitter ( b) FSK Receiver

A binary FSK Transmitter is as shown in fig. (a). The incoming binary data

sequence is applied to on-off level encoder. The output of encoder is Eb volts for symbol

1 and 0 volts for symbol ‘0’. When we have symbol 1 the upper channel is

switched on with oscillator frequency f1, for symbol ‘0’, because of inverter the lower

channel is switched on with oscillator frequency f2. These two frequencies are combined

using an adder circuit and then transmitted. The transmitted signal is nothing but

required BFSK signal.

The detector consists of two correlators. The incoming noisy BFSK signal x(t) is

common to both correlator. The Coherent reference signal 1 (t) and 2 (t) are supplied

to upper and lower correlators respectively.

The correlator outputs are then subtracted one from the other and resulting a

random vector ‘l’ (l=x1 - x2). The output ‘l’ is compared with threshold of zero volts.

If l > 0, the receiver decides in favour of symbol 1.

l < 0, the receiver decides in favour of symbol0

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BINARY ASK SYSTEM:-

Binary ON-OFF Product Binary ASK

Data Sequence Level Modulator

Signal

Encoder

(t) 2 Cos2f t

1 e

Tb

Fig. 3.5 BASK transmitter

Tb

x(t) X dt Decision If x > λ choose symbol 1

Device

0

If x < λ choose symbol 0

1 (t) Threshold λ

Fig. 3.6 Coherent binary ASK demodulator

In Coherent binary ASK system the basic function is given by

(t) 2 Cos2f t 0 t T

1 e b

Tb

The transmitted signals S1(t) and S2(t) are given by

S1 (t) Eb 1 (t) for Symbol 1

S2 (t) 0 for Symbol 0 The BASK system has one dimensional signal space with two messages (N=1, M=2)

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Region E2 Region E1

Message Point 2

Eb

1 (t)

0 E

b Message

2

Point 1

Fig. 3.7 Signal Space representation of BASK signal

In transmitter the binary data sequence is given to an on-off encoder. Which gives an

output Eb volts for symbol 1 and 0 volt for symbol 0. The resulting binary wave [in unipolar

form] and sinusoidal carrier 1 (t) are applied to a product modulator. The desired BASK wave is

obtained at the modulator output.

In demodulator, the received noisy BASK signal x(t) is apply to correlator with coherent

reference signal 1 (t) as shown in fig. (b). The correlator output x is compared with threshold λ.

If x > λ the receiver decides in favour of symbol 1.

If x < λ the receiver decides in favour of symbol 0.

Incoherent detection:

Fig. 3.8 : Envelope detector for OOK BASK

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Incoherent detection as used in analog communication does not require carrier for

reconstruction. The simplest form of incoherent detector is the envelope detector as shown in

Fig. 3.8. The output of envelope detector is the baseband signal. Once the baseband signal is

recovered, its samples are taken at regular intervals and compared with threshold. If Z(t) is greater than threshold ( ) a decision will be made in favour of symbol ‘1’ If Z(t) the sampled value is less than threshold ( ) a decision will be made in favour of symbol ‘0’.

Non- Coherenent FSK Demodulation:-

Fig. 3.9 : Incoherent detection of FSK

Fig. 3.9 shows the block diagram of incoherent type FSK demodulator. The detector

consists of two band pass filters one tuned to each of the two frequencies used to

communicate ‘0’s and ‘1’s., The output of filter is envelope detected and then baseband detected

using an integrate and dump operation. The detector is simply evaluating which of two possible

sinusoids is stronger at the receiver. If we take the difference of the outputs of the two envelope

detectors the result is bipolar baseband.

The resulting envelope detector outputs are sampled at t = kTb and their values are

compared with the threshold and a decision will be made infavour of symbol 1 or 0.

Differential Phase Shift Keying:- [DPSK]

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(a) DPSK Transmitter

(b) DPSK Receiver

Fig. 3.10 DPSK

A DPSK system may be viewed as the non coherent version of the PSK. It eliminates the

need for coherent reference signal at the receiver by combining two basic operations at the

transmitter (1) Differential encoding of the input binary wave and (2) Phase shift keying

Hence the name differential phase shift keying [DPSK]. To send symbol ‘0’ we

phase advance the current signal waveform by 1800 and to send symbol 1 we leave the

phase of the current signal waveform unchanged.

The differential encoding process at the transmitter input starts with an arbitrary

first but, securing as reference and thereafter the differentially encoded sequence{dk} is

generated by using the logical equation.

d k d k 1 bk

d k 1 bk

Where bk is the input binary digit at time kTb and dk-1 is the previous value of the

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differentially encoded digit. Table illustrate the logical operation involved in the

generation of DPSK signal.

Input Binary Sequence {bK} 1 0 0 1 0 0 1 1

Differentially Encoded 1 1 0 1 1 0 1 1 1

sequence {dK}

Transmitted Phase 0 0 Π 0 0 Π 0 0 0

Received Sequence 1 0 0 1 0 0 1 1

(Demodulated Sequence)

A DPSK demodulator is as shown in fig(b). The received signal is first passed

through a BPF centered at carrier frequency fc to limit noise power. The filter output and

its delay version are applied to correlator the resulting output of correlator is proportional

to the cosine of the difference between the carrier phase angles in the two correlator

inputs. The correlator output is finally compared with threshold of ‘0’ volts .

If correlator output is +ve -- A decision is made in favour of symbol ‘1’

If correlator output is -ve --- A decision is made in favour of symbol ‘0’

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COHERENT QUADRIPHASE – SHIFT KEYING Fig. 3.11(a) QPSK Transmitter

Fig. 3.11(b) QPSK Receiver In case of QPSK the carrier is given by

s (t) 2E Cos[2f t (2i 1) / 4] 0 t T i 1 to 4

i T c

s (t) 2E Cos[(2i 1) / 4]cos( 2f t) 2E sin[(2i 1) / 4]sin( 2f t) 0 t T i 1 to 4

i T c T c

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Fig. 3.11(c) QPSK Waveform

In QPSK system the information carried by the transmitted signal is contained in

the phase. The transmitted signals are given by

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Where the carrier frequency f C n

C for some fixed integer nc

7

E = the transmitted signal energy per symbol. T = Symbol duration.

The basic functions 1 (t) and 2 (t) are given by

cos2 f c t

1 (t)

2

0 t T

T b

sin2 f c t

2 (t)

2

0 t T

T b

There are four message points and the associated signal vectors are defined by

2i 1

E cos

Si 4 i 1,2,3,4

2i 1

E sin

4

The table shows the elements of signal vectors, namely Si1 & Si2

Table:-

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Input dibit Phase of Coordinates of message

QPSK points

signal(radians)

Si1 Si2

10

E E

4 2 2

00

3

E E

4 2 2

01

5

E E

4 2 2

11

7

E E

4 2 2

Therefore a QPSK signal is characterized by having a two dimensional signal

constellation(i.e.N=2)and four message points(i.e. M=4) as illustrated in fig(d) .Fig 3.11(d) Signal-space diagram of coherent QPSK system.

Unit III

Base Band Transmission and Optimal Reception of Digital Signal

BASEBAND:

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Pulse Shaping for Optimum Transmissions

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Base Band Reception Techniques

Receiving Filter:

Correlative receiver

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For an AWGN channel and for the case when the transmitted signals are

equally likely, the optimum receiver consists of two subsystems

1) Receiver consists of a bank of M product-integrator or

correlators Φ1(t) ,Φ2(t) …….ΦM(t) orthonormal function

2) The bank of correlator operate on the received signal x(t) to produce

observation vector x

Implemented in the form of maximum likelihood detector that operates

on observationvector x to produce an estimate of the transmitted symbol mii = 1

to M, in a way that would minimize the average probability of symbol error.

The N elements of the observation vector x are first multiplied by the

corresponding N elements of each of the M signal vectors s1, s2… sM , and

the resulting products are successively summed in accumulator to form the

corresponding set of

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Inner products {(x, sk)} k= 1, 2 ..M. The inner products are corrected for the

fact that the transmitted signal energies may be unequal. Finally, the largest in

the resulting set of numbers is selected and a corresponding decision on the

transmitted message made.

The optimum receiver is commonly referred as a correlation receiver

MATCHED FILTER

Science each of the orthonormal basic functions are Φ1(t) ,Φ2(t) …….ΦM(t) is

assumed to be zero outside the interval 0<t<T. we can design a linear filter with

impulse response hj(t), with the received signal x(t) the fitter output is given by

the convolution integral

yj(t) = xj

Where xj is the j th correlator output produced by the received signal x(t).

A filter whose impulse response is time-reversed and delayed version of the

input signal is said to be matched to xj (t)correspondingly, the optimum

receiver based on this isreferred as the matched filter receiver.

For a matched filter operating in real time to be physically realizable, it must be

causal.

MATCHED FILTER

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Φ(t) = input signal

h(t) = impulse response

W(t) =white noise

The impulse response of the matched filter is time-reversed and delayed

version of the input signal

MATCHED FILTER PROPERTIES

PROPERTY 1

The spectrum of the output signal of a matched filter with the matched signal as

input is, except for a time delay factor, proportional to the energy spectral

density of the input signal.

PROPERTY 2

The output signal of a Matched Filter is proportional to a shifted version of the

autocorrelation function of the input signal to which the filter is matched.

PROPERTY 3

The output Signal to Noise Ratio of a Matched filter depends only on the ratio

of the signal energy to the power spectral density of the white noise at the filter

input.

PROPERTY 4

The Matched Filtering operation may be separated into two matching

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conditions; namely spectral phase matching that produces the desired output

peak at time T, and the spectral amplitude matching that gives this peak value

its optimum signal to noise density ratio.

EYE PATTERN

The quality of digital transmission systems are evaluated using the bit

error rate. Degradation of quality occurs in each process modulation,

transmission, and detection. The eye pattern is experimental method that

contains all the information concerning the degradation of quality. Therefore,

careful analysis of the eye pattern is important in analyzing the degradation

mechanism.

• Eye patterns can be observed using an oscilloscope. The received wave is

applied to the vertical deflection plates of an oscilloscope and the

sawtooth wave at a rate equal to transmitted symbol rate is applied to the

horizontal deflection plates, resulting display is eye pattern as it

resembles human eye.

• The interior region of eye pattern is called eye opening

We get superposition of successive symbol intervals to produce eye pattern as

shown below.

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• The width of the eye opening defines the time interval over which the

received wave can be sampled without error from ISI

• The optimum sampling time corresponds to the maximum eye opening

• The height of the eye opening at a specified sampling time is a measure

of the margin over channel noise.

The sensitivity of the system to timing error is determined by the rate of closure

of the eye as the sampling time is varied.

Any non linear transmission distortion would reveal itself in an asymmetric or

squinted eye. When the effected of ISI is excessive, traces from the upper

portion of the eye pattern cross traces from lower portion with the result that the

eye is completely closed.

INFORMATION THEORY

Information and Entropy

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Although it is in principle a very old concept, entropy is generally credited to Shannon

because it is the fundamental measure in information theory. Entropy is often defined as an

expectation:

where 0 log(0) = 0. The base of the logarithm is generally 2. When this is the case, the units

of entropy are bits.

Entropy captures the amount of randomness or uncertainty in a variable. This, in turn,is

aeasure of the average length of a message that would have to be sent to describe a sample.

Recall our fair coin from § 1. It’s entropy is:–0.5log0.5 + 0.5log0.5= 1; that is, thereis one

bit of information in the random variable.This means that on average we need to sendone bit

per trial to describe a sample. This should fit your intuitions: if I flip a coin 100 times,I’ll

need 100 numbers to describe those flips, if order matters. By contrast, our two-headedcoin

has entropy . Even if I flip this coin 100 times, it doesn’t matterbecause the outcome is

always heads. I don’t need to send any message to describe a sample. This should fit your

intuitions: if I flip a coin 100 times, I’ll need 100 numbers to describe those flips, if order matters.

By contrast, our two-headed coin has entropy –1log1+ 0log0= 0.

Even if I flip this coin 100 times, it doesn’t matter because the outcome is always heads. I don’t

need to send any message to describe a sample.

There are other possibilities besides being completely random and completely deter-mined.

Imagine a weighted coin, such that heads occurr 75% of the time. The entropy would be: –

0.75log0.75 + 0.25log0.25= 0.8113. After 100 trials, I’d only need a message of about 82 bits

on average to describe the sample. Shannon showed that there exists a coder thatcan construct

messages of length H(X)+1, nearly matching this ideal rate.

Just as with probabilities, we can compute joint and conditional entropies. Joint

entropy is the randomness contained in two variables, while conditional entropy is a measure

of the randomness of one variable given knowledge of another. Joint entropy is defined as:

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while the conditional entropy is:

There are several interesting facts that follow from these definitions. For example, two

random variables, X and Y, are considered independent if and only if HY| X= HY

or HXY= HX+HYIt is also the case that HY|X≤HY(knowing more information

can never increase our uncertainty). Similarly, HXY≤HX+HYIt is alsothe case that

HXY=HY|X+HX=HX Y+HYThese relations hold in thegeneral case of more than

two variables.

There are several facts about discrete entropy, H(), that do not hold for continuous

ordifferential entropy, h(). The most important is that while HX≥0 h() can actually be

negative. Worse, even a distribution with an entropy of –∞can still have uncertainty.

Luckily, for us, even though differential entropy cannot provide us with an absolute measure

of randomness, it is still that case that if hX≥hY then X has more randomness than Y.

Mutual Information

Although conditional entropy can tell us when two variables are completely independent, it is

not an adequarte measure of dependence. A small value for HY| Xmay imply that X tells

us a great deal about Y or that H(Y) is small to begin with. Thus, we measure dependence

using mutual information:

IXY= HY–HY|X

Mutual information is a measure of the reduction of randomness of a variable given

knowledge of another variable. Using properties of logarithms, we can derive several equiva-

lent definitions:

IXY= HY–HY | X

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= HX–HX | Y

=HX+HY–HXY

= IYX

In addition to the definitions above, it is useful to realize that mutual information is a

particular case of the Kullback-Leibler divergence. The KL divergence is defined as:

KL divergence measures the difference between two distributions. It is sometimes called the

relative entropy. It is always non-negative and zero only when p=q; however, it is not a

distance because it is not symmetic.

In terms of KL divergence, mutual information is:

In other words, mutual information is a measure of the difference between the joint

probability and product of the individual probabilities. These two distributions are equivalent

only when X and Y are independent, and diverge as X and Y become more dependent.

Shannon-Fano Code

Shannon–Fano coding, named after Claude Elwood Shannon and Robert Fano, is a technique

for constructing a prefix code based on a set of symbols and their probabilities. It is

suboptimal in the sense that it does not achieve the lowest possible expected codeword length

like Huffman coding; however unlike Huffman coding, it does guarantee that all codeword

lengths are within one bit of their theoretical ideal I(x) =−log P(x).

In Shannon–Fano coding, the symbols are arranged in order from most probable to least

probable, and then divided into two sets whose total probabilities are as close as possible to

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being equal. All symbols then have the first digits of their codes assigned; symbols in the first

set receive "0" and symbols in the second set receive "1". As long as any sets with more than

one member remain, the same process is repeated on those sets, to determine successive

digits of their codes. When a set has been reduced to one symbol, of course, this means the

symbol's code is complete and will not form the prefix of any other symbol's code.

The algorithm works, and it produces fairly efficient variable-length encodings; when the two

smaller sets produced by a partitioning are in fact of equal probability, the one bit of

information used to distinguish them is used most efficiently. Unfortunately, Shannon–Fano

does not always produce optimal prefix codes.

For this reason, Shannon–Fano is almost never used; Huffman coding is almost as

computationally simple and produces prefix codes that always achieve the lowest expected

code word length. Shannon–Fano coding is used in the IMPLODE compression method,

which is part of the ZIP file format, where it is desired to apply a simple algorithm with high

performance and minimum requirements for programming.

Shannon-Fano Algorithm:

A Shannon–Fano tree is built according to a specification designed to define an effective

code table. The actual algorithm is simple:

For a given list of symbols, develop a corresponding list of probabilities or frequency

counts so that each symbol’s relative frequency of occurrence is known.

Sort the lists of symbols according to frequency, with the most frequently occurring

symbols at the left and the least common at the right.

Divide the list into two parts, with the total frequency counts of the left part being as

close to the total of the right as possible.

The left part of the list is assigned the binary digit 0, and the right part is assigned the

digit 1. This means that the codes for the symbols in the first part will all start with 0,

and the codes in the second part will all start with 1.

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Recursively apply the steps 3 and 4 to each of the two halves, subdividing groups and

adding bits to the codes until each symbol has become a corresponding code leaf on the tree.

Example:

The source of information A generates the symbols {A0, A1, A2, A3 and A4} with the

corresponding probabilities {0.4, 0.3, 0.15, 0.1 and 0.05}. Encoding the source symbols using

binary encoder and Shannon-Fano encoder gives:

Source Symbol Pi Binary Code Shannon-Fano

A0 0.4 000 0

A1 0.3 001 10

A2 0.15 010 110

A3 0.1 011 1110

A4 0.05 100 1111

Lavg H = 2.0087 3 2.05

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Shannon-Fano code is a top-down approach. Constructing the code tree, we get

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Source Coding

All source models in information theory may be viewed as random process or

random sequence models. Let us consider the example of a discrete memory less source

(DMS), which is a simple random sequence model.

A DMS is a source whose output is a sequence of letters such that each letter is

independently selected from a fixed alphabet consisting of letters; say a1, a2 , ……….ak.

The letters in the source output sequence are assumed to be random and statistically

independent of each other. A fixed probability assignment for the occurrence of each

letter is also assumed. Let us, consider a small example to appreciate the importance of

probability assignment of the source letters.

Let us consider a source with four letters a1, a2, a3 and a4 with P(a1)=0.5,

P(a2)=0.25, P(a3)= 0.13, P(a4)=0.12. Let us decide to go for binary coding of these four

source letters. While this can be done in multiple ways, two encoded representations are

shown below:

Code Representation#1: a1: 00, a2:01, a3:10, a4:11

Code Representation#2: a1: 0, a2:10, a3:001, a4:110

It is easy to see that in method #1 the probability assignment of a source letter has

not been considered and all letters have been represented by two bits each. However in

the second method only a1 has been encoded in one bit, a2 in two bits and the remaining

two in three bits. It is easy to see that the average number of bits to be used per source

letter for the two methods are not the same. ( a for method #1=2 bits per letter and a for

method #2 < 2 bits per letter). So, if we consider the issue of encoding a long sequence of

letters we have to transmit less number of bits following the second method. This is an

important aspect of source coding operation in general. At this point, let us note the

following:

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a) We observe that assignment of small number of bits to more probable

letters and assignment of larger number of bits to less probable letters (or symbols) may

lead to efficient source encoding scheme.

b) However, one has to take additional care while transmitting the encoded

letters. A careful inspection of the binary representation of the symbols in method #2

reveals that it may lead to confusion (at the decoder end) in deciding the end of binary

representation of a letter and beginning of the subsequent letter.

So a source-encoding scheme should ensure that

1) The average number of coded bits (or letters in general) required per source letter is as

small as possible and

2) The source letters can be fully retrieved from a received encodedsequence.

In the following we discuss a popular variable-length source-coding scheme

satisfying the above two requirements.

Variable length Coding

Let us assume that a DMS U has a K- letter alphabet {a1, a 2, ……….aK} with

probabilities P(a 1), P(a2),…………. P(aK). Each source letter is to be encoded into a

codeword made of elements (or letters) drawn from a code alphabet containing D

symbols. Often for ease of implementation a binary code alphabet (D = 2) is chosen. As

we observed earlier in an example, different codeword may not have same number of

code symbols. If nk denotes the number of code symbols corresponding to the source

letter ak , the average number of code letters per source letter ( n ) is:

K

n=∑P (ak)nk 5.1

k =1

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Intuitively, if we encode a very long sequence of letters from a DMS, the number

of code letters per source letter will be close to n .

Now, a code is said to be uniquely decodable if for each source sequence of finite

length, the sequence of code letters corresponding to that source sequence is different

from the sequence of code letters corresponding to any other possible source sequence.

We will briefly discuss about a subclass of uniquely decodable codes, known as

prefix condition code. Let the code word in a code be represented as

xk=(xk,1,xk,2,......,xk,nk), wherexk,1,xk,2,......,xk,nkdenote the individualcode letters

(when D=2, these are 1 or 0). Any sequence made up of an initial part of xk

that is xk,1,xk,2,......,xk,i for i ≤ nk is called a prefix of xk .

A prefix condition code is a code in which no code word is the prefix of

any other codeword.

Example: consider the following table and find out which code out of the four shown is

/are prefix condition code. Also determine n for each code.

Source letters:- a1, a2, a3 and a4

P(ak) :- P(a1)=0.5, P(a2)=0.25, P(a3)= 0.125, P(a4)=0.125

Code Representation#1: a1: 00, a2:01, a3:10, a4:11

Code Representation#2: a1: 0, a2:1, a3:00, a4:11

Code Representation#3: a1: 0, a2:10, a3:110, a4:111

Code Representation#4: a1: 0, a2:01, a3:011, a4:0111

A prefix condition code can be decoded easily and uniquely. Start at the beginning of a

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sequence and decode one word at a time. Finding the end of a code word is not a problem

as the present code word is not a prefix to any other codeword.

Example: Consider a coded sequence 0111100 as per Code Representation #3 of

theprevious example. See that the corresponding source letter sequence is a1 ,a4 , a2 , a1 .

Now, we state one important theorem known as Kraft Inequality theorem

withoutproof.

Binary Huffman Coding (an optimum variable-length source coding scheme)

In Binary Huffman Coding each source letter is converted into a binary code

word. It is a prefix condition code ensuring minimum average length per source letter in

bits.

Let the source letters a1, a 2, ……….aK have probabilities P(a1), P(a2),………….

P(aK) and let us assume that P(a1) ≥ P(a2) ≥ P(a 3)≥…. ≥ P(aK).

We now consider a simple example to illustrate the steps for Huffman coding.

Steps to calculate Huffman Coding

Example Let us consider a discrete memoryless source with six letters having

P(a1)=0.3,P(a2)=0.2, P(a 3)=0.15, P(a 4)=0.15, P(a5)=0.12 and P(a6)=0.08.

Arrange the letters in descending order of their probability (here they are

arranged).

Consider the last two probabilities. Tie up the last two probabilities. Assign, say,

0 to the last digit of representation for the least probable letter (a6) and 1 to the

last digit of representation for the second least probable letter (a5). That is, assign

‘1’ to the upper arm of the tree and ‘0’ to the lower arm.

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1

P(a5)=0.12

0.2

P(a6)=0.08 0

(3) Now, add the two probabilities and imagine a new letter, say b1, substituting for

a6 and a5. So P(b1) =0.2. Check whether a4 and b1are the least likely letters. If

not, reorder the letters as per Step#1 and add the probabilities of two least likely

letters. For our example, it leads to:

P(a1)=0.3, P(a2)=0.2, P(b1)=0.2, P(a3)=0.15 and P(a4)=0.15

(4) Now go to Step#2 and start with the reduced ensemble consisting of a1 , a2 , a3 ,

a4 and b1. Our example results in:

P(a3)=0.15 1

0.3

0

P(a4)=0.15

Here we imagine another letter b1, with P(b2)=0.3.

t Continue till the first digits of the most reduced ensemble of two letters are

assigned a ‘1’ and a ‘0’.

Again go back to the step (2): P(a1)=0.3, P(b2)=0.3, P(a2)=0.2 and P(b1)=0.2.

Now we consider the last two probabilities:

P(a2)=0.2

0.4

P(b1)=0.2

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So, P(b3)=0.4. Following Step#2 again, we get, P(b3)=0.4, P(a1)=0.3 and

P(b2)=0.3.

Next two probabilities lead

to: 1

P(a1)=0.3

0.6

P(b2)=0.3 0

with P(b4) = 0.6. Finally we get only two probabilities

P(b4)=0.6 1

1.00

P(b3)=0.4

0

6. Now, read the code tree inward, starting from the root, and construct the

codewords. The first digit of a codeword appears first while reading the code tree

inward.

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Hence, the final representation is: a1=11, a2=01, a3=101, a4=100, a5=001, a6=000.

A few observations on the preceding example

1. The event with maximum probability has least number of bits.

2. Prefix condition is satisfied. No representation of one letter is prefix for other.

Prefix condition says that representation of any letter should not be a part of any

other letter.

3. Average length/letter (in bits) after coding is

= ∑P (ai )ni = 2.5 bits/letter.

i

4. Note that the entropy of the source is: H(X)=2.465 bits/symbol. Average length

per source letter after Huffman coding is a little bit more but close to the source

entropy. In fact, the following celebrated theorem due to C. E. Shannon sets the

limiting value of average length of codewords from a DMS.

CONVOLUTIONAL CODES

Convolutional codes are commonly described using two parameters: the code rate

and the constraint length. The code rate, k/n, is expressed as a ratio of the number of bits

into the convolutional encoder (k) to the number of channel symbols output by the

convolutional encoder (n) in a given encoder cycle. The constraint length parameter, K,

denotes the "length" of the convolutional encoder, i.e. how many k-bit stages are

available to feed the combinatorial logic that produces the output symbols. Closely

related to K is the parameter m, which indicates how many encoder cycles an input bit is

retained and used for encoding after it first appears at the input to the convolutional

encoder. The m parameter can be thought of as the memory length of the encoder.

Convolutional codes are widely used as channel codes in practical communication

systems for error correction. The encoded bits depend on the current k input bits and a

few past input bits. The main decoding strategy for convolutional codes is based on the

widely used Viterbi algorithm. As a result of the wide acceptance of convolutional codes,

there have been several approaches to modify and extend this basic coding scheme.

Trellis coded modulation (TCM) and turbo codes are two such examples. In TCM,

redundancy is added by combining coding and modulation into a single operation. This is

achieved without any reduction in data rate or expansion in bandwidth as required by

only error correcting coding schemes.

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A simple convolutional encoder is shown in Fig. 7.1. The information bits are fed

in small groups of k-bits at a time to a shift register. The output encoded bits are obtained

by modulo-2 addition (EXCLUSIVE-OR operation) of the input information bits and the

contents of the shift registers which are a few previous information bits.

If the encoder generates a group of ‘n’ encoded bits per group of ‘k’ information bits,

the code rate R is commonly defined as R = k/n. In Fig. 7.1, k = 1 and n = 2. The

number, K of elements in the shift register which decides for how many codewords

one information bit will affect the encoder output, is known as the constraint length

of the code. For the present example, K = 3.

The shift register of the encoder is initialized to all-zero-state before encoding

operation starts. It is easy to verify that encoded sequence is 00 11 10 00 01 ….for an

input message sequence of 01011….

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Fig. 7.1A convolutional encoder with k=1, n=2 and r=1/2

The operation of a convolutional encoder can be explained in several but

equivalent ways such as, by a) state diagram representation, b) tree diagram

representation and c) trellis diagram representation.

a) State Diagram Representation

A convolutional encoder may be defined as a finite state machine. Contents of the rightmost (K-1) shift register stages define the states of the encoder. So, the encoder in Fig. 7.2has four states. The transition of an encoder from one state to another, ascaused by input bits, is depicted in the state diagram. Fig. 7.2 shows the state diagram of the encoder in Fig. 7.1. A new input bit causes a transition from one state to another. The

path information between the states, denoted as b/c1c2 , represents input information bit

‘b’ and the corresponding output bits (c1c2). Again, it is not difficult to verify from the

state diagram that an input information sequence b = (1011) generates an encoded sequence c = (11, 10, 00, 01).

Fig. 7.2 State diagram representation for the encoder in Fig. 7.1

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b)Tree Diagram Representation

The tree diagram representation shows all possible information and encoded

sequences for the convolutional encoder. Fig. 7.3 shows the tree diagram for the encoder

in Fig. 7.1. The encoded bits are labeled on the branches of the tree. Given an input

sequence, the encoded sequence can be directly read from the tree. As an example, an

input sequence (1011) results in the encoded sequence (11, 10, 00, 01).

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c) Trellis Diagram Representation

The trellis diagram of a convolutional code is obtained from its state diagram. All

state transitions at each time step are explicitly shown in the diagram to retain the time

dimension, as is present in the corresponding tree diagram. Usually, supporting

descriptions on state transitions, corresponding input and output bits etc. are labeled in the

trellis diagram. It is interesting to note that the trellis diagram, which describes the

operation of the encoder, is very convenient for describing the behavior of

thecorresponding decoder, especially when the famous ‘Viterbi Algorithm (VA)’ is

followed. Figure 7.4 shows the trellis diagram for the encoder in Figure 7.1.

Fig. 7.3 A tree diagram for the encoder in Fig. 7.1

Fig. 7.4Trellis diagram, used in the decoder corresponding to the encoder in Fig.7.1

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Hard-Decision and Soft-Decision Decoding

Hard-decision and soft-decision decoding are based on the type of quantization

used on the received bits. Hard-decision decoding uses 1-bit quantization on the received

samples. Soft-decision decoding uses multi-bit quantization (e.g. 3 bits/sample) on the

received sample values.

Hard-Decision Viterbi Algorithm

The Viterbi Algorithm (VA) finds a maximum likelihood (ML) estimate of a

transmitted code sequence c from the corresponding received sequence r by maximizing

the probability p(r|c) that sequence r is received conditioned on the estimated code

sequence c. Sequence c must be a valid coded sequence.

The Viterbi algorithm utilizes the trellis diagram to compute the path metrics. The

channel is assumed to be memory less, i.e. the noise sample affecting a received bit is

independent from the noise sample affecting the other bits. The decoding operation starts

from state ‘00’, i.e. with the assumption that the initial state of the encoder is ‘00’. With

receipt of one noisy codeword, the decoding operation progresses by one step deeper into

the trellis diagram. The branches, associated with a state of the trellis tell us about the

corresponding codewords that the encoder may generate starting from this state. Hence,

upon receipt of a codeword, it is possible to note the ‘branch metric’ of each branch by

determining the Hamming distance of the received codeword from the valid codeword

associated with that branch. Path metric of all branches, associated with all the states are

calculated similarly Now, at each depth of the trellis, each state also carries some

‘accumulated path metric’, which is the addition of metrics of all branches that construct

the ‘most likely path’ to that state. As an example, the trellis diagram of the code shown

in Fig. 7.1, has four states and each state has two incoming and two outgoing branches.

At any depth of the trellis, each state can be reached through two paths from the previous

stage and as per the VA, the path with lower accumulated path metric is chosen. In the

process, the ‘accumulated path metric’ is updated by adding the metric of the incoming

branch with the ‘accumulated path metric’ of the state from where the branch originated.

No decision about a received codeword is taken from such operations and the decoding

decision is deliberately delayed to reduce the possibility of erroneous decision.

The basic operations which are carried out as per the hard-decision Viterbi Algorithm

after receiving one codeword are summarized below: All the branch metrics of all the states are determined;

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.

Accumulated metrics of all the paths (two in our example code) leading to a state

are calculated taking into consideration the ‘accumulated path metrics’ of the

states from where the most recent branches emerged;

Only one of the paths, entering into a state, which has minimum ‘accumulated

path metric’ is chosen as the ‘survivor path’ for the state (or, equivalently ‘node’);

So, at the end of this process, each state has one ‘survivor path’. The ‘history’ of a

survivor path is also maintained by the node appropriately ( e.g. by storing the

codewords or the information bits which are associated with the branches making

the path);

(5) Steps a) to d) are repeated and decoding decision is delayed till sufficient number

of codewords has been received. Typically, the delay in decision making = Lx k codewords where L is an integer, e.g. 5 or 6. For the code in Fig. 7.1, the decision delay of 5x3 = 15 codewords may be sufficient for most occasions. This means,

we decide about the first received codeword after receiving the 16th

codeword.

The decision strategy is simple. Upon receiving the 16th

codeword and carrying

out steps a) to d), we compare the ‘accumulated path metrics’ of all the states ( four in our example) and chose the state with minimum overall ‘accumulated path metric’ as the ‘winning node’ for the first codeword. Then we trace back the history of the path associated with this winning node to identify the codeword tagged to the first branch of the path and declare this codeword as the most likely transmitted first codeword.

The above procedure is repeated for each received codeword hereafter. Thus, the

decision for a codeword is delayed but once the decision process starts, we decide once

for every received codeword. For most practical applications, including delay-sensitive

digital speech coding and transmission, a decision delay of Lx k codewords is acceptable.

Soft-Decision Viterbi Algorithm

In soft-decision decoding, the demodulator does not assign a ‘0’ or a ‘1’ to

each received bit but uses multi-bit quantized values. The soft-decision Viterbi algorithm

is very similar to its hard-decision algorithm except that squared Euclidean distance is

used in the branch metrics instead of simpler Hamming distance. However, the

performance of a soft-decision VA is much more impressive compared to its HDD (Hard

Decision Decoding) counterpart [Fig. 7.6 (a) and (b)]. The computational requirement of

a Viterbi decoder grows exponentially as a function of the constraint length and hence it

is usually limited in practice to constraint lengths of K = 9.

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Fig. 7.6 (a) Decoded BER vs input BER for the rate – half convolutional codes

withViterbi Algorithm ; 1) k = 3 (HDD),2) k = 5 (HDD),3) k = 3 (SDD), and 4)

k= 5 (SDD). HDD: Hard Decision Decoding; SDD: Soft Decision Decoding.

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Fig. 7.6 (b) Decoded BER vs Eb/No(in dB) for the rate – half convolutional codes withViterbi

Algorithm ; 1) Uncoded system; 2) with k = 3 (HDD) and 3) k = 3 (SDD). HDD: Hard

Decision Decoding; SDD: Soft Decision Decoding.

Spread – Spectrum Modulation

Introduction:

Initially developed for military applications during II world war, that was less sensitive to

intentional interference or jamming by third parties.

Spread spectrum technology has blossomed into one of the fundamental building blocks in

current and next-generation wireless systems

Problem of radio transmission

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Narrow band can be wiped out due to interference

To disrupt the communication, the adversary needs to do two things,

to detect that a transmission is taking place and

to transmit a jamming signal which is designed to confuse the receiver.

Solution

A spread spectrum system is therefore designed to make these tasks as difficult as possible.

Firstly, the transmitted signal should be difficult to detect by an adversary/jammer, i.e.,the

signal should have a low probability of intercept (LPI).

Secondly, the signal should be difficult to disturb with a jamming signal, i.e.,

thetransmitted signal should possess an anti-jamming (AJ) property

Remedy

Spread the narrow band signal into a broad band to protect against interference

In a digital communication system the primary resources are Bandwidth andPower. The

study of digital communication system deals with efficient utilization ofthese two resources,

but there are situations where it is necessary to sacrifice their efficient utilization in order to

meet certain other design objectives.

For example to provide a form of secure communication (i.e. the transmitted signal is

not easily detected or recognized by unwanted listeners) the bandwidth of thetransmitted

signal is increased in excess of the minimum bandwidth necessary to transmit it. This

requirement is catered by a technique known as “Spread SpectrumModulation”.

The primary advantage of a Spread – Spectrum communication system is its

ability to reject ‘Interference’ whether it be the unintentional or the

intentionalinterference.

The definition of Spread – Spectrum modulation may be stated in two parts.

Spread Spectrum is a mean of transmission in which the data sequence occupies a

BW (Bandwidth) in excess of the minimum BW necessary to transmit it.

The Spectrum Spreading is accomplished before transmission through the use of a

code that is independent of the data sequence. The Same code is used in the receiver

to despread the received signal so that the original data sequence may be recovered.

s(t) wide band r(t) wide band b(t) + Noise

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b(t) . . . . . . . . . Narrow Wide Band Band

c(t) n(t) c(t)

Wide band (noise) Wide band

---- Transmitter---- ---- Channel------ --- Receiver--------

Fig: 8.1 spread spectrum technique.

b(t) = Data Sequence to be transmitted (Narrow Band) c(t)

= Wide Band code

s(t) = c(t) * b(t) – (wide Band)

Fig: 8.2Spectrum of signal before & after spreading

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Code division multiple access

CDMA is achannel access methodutilized by various radio communication

technologies.

One of the basic concepts in data communication is the idea of allowing several

transmitters to send information simultaneously over a single communication channel. This

allows several users to share a bandwidth of frequencies. This concept is called multiplexing.

CDMA employs spread-spectrum technology and a special coding scheme (where each

transmitter is assigned a code) to allow multiple users to be multiplexed over the same

physical channel. By contrast, time division multiple access (TDMA) divides access by time,

while frequency-division multiple access (FDMA) divides it by frequency. CDMA is a form

of "spread-spectrum" signaling, since the modulated coded signal has a much higher data

bandwidth than the data being communicated.

An analogy to the problem of multiple access is a room (channel) in which people

wish to communicate with each other. To avoid confusion, people could take turns speaking

(time division), speak at different pitches (frequency division), or speak in different

languages (code division). CDMA is analogous to the last example where people speaking

the same language can understand each other, but not other people. Similarly, in radio

CDMA, each group of users is given a shared code. Many codes occupy the same channel,

but only users associated with a particular code can understand each other.

Technical details

CDMA is a spread spectrum multiple access technique. In CDMA a locally generated

code runs at a much higher rate than the data to be transmitted. Data for transmission is

simply logically XOR (exclusive OR) added with the faster code. The figure shows how

spread spectrum signal is generated. The data signal with pulse duration of Tb is XOR added

with the code signal with pulse duration of Tc. (Note: bandwidth is proportional to 1 / T

where T = bit time) Therefore, the bandwidth of the data signal is 1 / Tb and the bandwidth of

the spread spectrum signal is 1 / Tc. Since Tc is much smaller than Tb, the bandwidth of the

spread spectrum signal is much larger than the bandwidth of the original signal.

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Fig. 8.3

CDMA uses Direct Sequence spreading, where spreading process isdone by directly

combining the baseband information to high chip rate binary code. The Spreading Factor is

the ratio of the chips (UMTS = 3.84Mchips/s) to baseband information rate. Spreading

factors vary from 4 to 512 in FDD UMTS. Spreading process gain can in expressed in dBs

(Spreading factor 128 = 21dB gain).

Fig. 8.4

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Each user in a CDMA system uses a different code to modulate their signal. Choosing

the codes used to modulate the signal is very important in the performance of CDMA

systems. The best performance will occur when there is good separation between the signal

of a desired user and the signals of other users. The separation of the signals is made by

correlating the received signal with the locally generated code of the desired user. If the

signal matches the desired user's code then the correlation function will be high and the

system can extract that signal. If the desired user's code has nothing in common with the

signal the correlation should be as close to zero as possible (thus eliminating the signal); this

is referred to as cross correlation. If the code is correlated with the signal at any time offset

other than zero, the correlation should be as close to zero as possible. This is referred to as

auto-correlation and is used to reject multi-path interference.

Fig. 8.5

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PSUEDO-NOISE SEQUENCE:

Generation of PN sequence:

Clock

Output

Shift Shift Shift

S0

Register1 Register2 S3

Register3

Logic Circuit

Fig 8.6: Maximum-length sequence generator for n=3

A feedback shift register is said to be Linear when the feed back logic consists of

entirely mod-2-address ( Ex-or gates). In such a case, the zero state is not permitted. The

period of a PN sequence produced by a linear feedback shift register with ‘n’ flip flops cannot

exceed 2n-1. When the period is exactly 2

n-1, the PN sequence is called a

‘maximum length sequence’ or ‘m-sequence’.

Example1: Consider the linear feed back shift register as shown in fig 2involve

three flip-flops. The input so is equal to the mod-2 sum of S1 and S3. If the initial state of the

shift register is 100. Then the succession of states will be as follows.

100,110,011,011,101,010,001,100 . . . . . .

The output sequence (output S3) is therefore. 00111010 . . . . .

Which repeats itself with period 23–1 = 7 (n=3)

Maximal length codes are commonly used PN codes

In binary shift register, the maximum length sequence is

N = 2m

-1

chips, where m is the number of stages of flip-flops in the shift register.

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Fig.8.7

At each clock pulse

(6) Contents of register shifts one bit right.

(7) Contents of required stages are modulo 2 added and fed back to input.

Fig. 8.8 Initial stages of Shift registers1000

Let initial status of shift register be 1000

1 0 0 0

0 1 0 0

0 0 1 0

1 0 0 1

1 1 0 0

0 1 1 0

1 0 1 1

0 1 0 1

1 0 1 0

1 1 0 1

1 1 1 0

1 1 1 1

0 1 1 1

0 0 1 1

0 0 0 1

1 0 0 0

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•We can see for shift Register of length m=4.

.At each clock the change in state of flip-flop is shown.

•Feed back function is modulo two of X3andX4.

•After 15 clock pulses the sequence repeats.

Output sequence is

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

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Properties of PN Sequence Randomness of PN sequence is tested by following properties

u Balance property

v Run length property

w Autocorrelation property

1. Balance property

In each Period of the sequence , number of binary ones differ from binary zeros by

at most one digit .

Consider output of shift register 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 Seven

zeros and eight ones -meets balance condition.

2. Run length property

Among the runs of ones and zeros in each period, it is desirable that about one half the runs

of each type are of length 1, one- fourth are of length 2 and one-eighth are of length 3 and so-

on.

Consider output of shift register

Number of runs =8

0 0 0 1 0 0 1 1 0 1 01 1 1 1

3 1 2 2 1 1 1 4

{ No. of agreements – No. of disagreements in comparison of one full period } Consider

output of shift register for l=1

Yields PN autocorrelation as

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Range of PN Sequence Lengths

Length 0f Shift Register, m PN Sequence Length,

7 127

8 255

9 511

10 1023

11 2047

12 4095

13 8191

17 131071

19 524287

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A Notion of Spread Spectrum:

An important attribute of Spread Spectrum modulation is that it can provide

protection against externally generated interfacing signals with finite power. Protection

against jamming (interfacing) waveforms is provided by purposely making the information

– bearing signal occupy a BW far in excess of the minimum BW necessary to transmit it.

This has the effect of making the transmitted signal a noise like appearance so as to blend

into the background. Therefore Spread Spectrum is a method of ‘camouflaging’ the

information – bearing signal.

V

b(t) m(t). . r(t) z(t) Tb Decisi

dt

on

Device

0

c(t) n(t) c(t) Threshold=0

<----Transmitter-----> --Channel - - - - - Receiver -------------

Let { bK} denotes a binary data sequence.

{ cK } denotes a PN sequence.

b(t) and c(t) denotes their NRZ polar representation respectively.

The desired modulation is achieved by applying the data signal b(t) and PN signal c(t) to a

product modulator or multiplier. If the message signal b(t) is narrowband and the PN

sequence signal c(t) is wide band, the product signal m(t) is also wide band. The PN

sequence performs the role of a ‘Spreading Code”.

For base band transmission, the product signal m(t) represents the transmitted

signal. Therefore m(t) = c(t).b(t)

The received signal r(t) consists of the transmitted signal m(t) plus an additive

interference noise n(t), Hence

r(t) = m(t) + n(t)

= c(t).b(t) + n(t)

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+1

0

-1

a) Data Signal b(t)

+1

0 -1

b)Spreading Code c(t)

+1

0 -1

c)Product signal or base band transmitted signal m(t)

To recover the original message signal b(t), the received signal r(t) is applied to a

demodulator that consists of a multiplier followed by an integrator and a decision device. The

multiplier is supplied with a locally generated PN sequence that is exact replica of that used

in the transmitter. The multiplier output is given by

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Z(t) = r(t).c(t)

4. [b(t) * c(t) + n(t)] c(t)

5. c2(t).b(t) + c(t).n(t)

The data signal b(t) is multiplied twice by the PN signal c(t), where as unwanted signal

n(t) is multiplied only once. But c2(t) = 1, hence the above equation reduces to

Z(t) = b(t) + c(t).n(t)

Now the data component b(t) is narrowband, where as the spurious component c(t)n(t)

is wide band. Hence by applying the multiplier output to a base band (low pass) filter most of

the power in the spurious component c(t)n(t) is filtered out. Thus the effect of the interference

n(t) is thus significantly reduced at the receiver output.

The integration is carried out for the bit interval 0 ≤ t ≤ Tb to provide the sample

value V. Finally, a decision is made by the receiver.

If V > Threshold Value ‘0’, say binary symbol ‘1’

If V < Threshold Value ‘0’, say binary symbol ‘0’

Frequency – Hop Spread Spectrum:

In a frequency – hop Spread – Spectrum technique, the spectrum of data

modulated carrier is widened by changing the carrier frequency in a pseudo – random

manner. The type of spread – spectrum in which the carrier hops randomly form one

frequency to another is called Frequency–Hop (FH) Spread Spectrum.

Since frequency hopping does not covers the entire spread spectrum

instantaneously. We are led to consider the rate at which the hop occurs. Depending

upon this we have two types of frequency hop.

1. Slow frequency hopping:- In which the symbol rate Rs of the MFSK signal is an

integer multiple of the hop rate Rh. That is several symbols are transmitted on each

frequency hop.

2. Fast – Frequency hopping:- In which the hop rate Rh is an integral multiple of the

MFSK symbol rate Rs. That is the carrier frequency will hoop several times during

the transmission of one symbol.

A common modulation format for frequency hopping system is that of M-

ary frequency – shift – keying (MFSK).

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Slow frequency hopping:-

Fig. 8.12 a) Shows the block diagram of an FH / MFSK transmitter, which involves

frequency modulation followed by mixing.

The incoming binary data are applied to an M-ary FSK modulator. The resulting

modulated wave and the output from a digital frequency synthesizer are then applied to a

mixer that consists of a multiplier followed by a band – pass filter. The filter is designed

to select the sum frequency component resulting from the multiplication process as the

transmitted signal. An ‘k’ bit segments of a PN sequence drive the frequencysynthesizer,

which enables the carrier frequency to hop over 2n distinct values. Since frequency

synthesizers are unable to maintain phase coherence over successive hops, most frequency

hops spread spectrum communication system use non coherent M-ary modulation system.

Fig 8.12:- Frequency hop spread M-ary Frequency – shift – keying

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In the receiver the frequency hoping is first removed by mixing the received signal

with the output of a local frequency synthesizer that is synchronized with the transmitter. The

resulting output is then band pass filtered and subsequently processed by a non coherent M-

ary FSK demodulator. To implement this M-ary detector, a bank of M non coherent matched

filters, each of which is matched to one of the MFSK tones is used. By selecting the largest

filtered output, the original transmitted signal is estimated.

An individual FH / MFSK tone of shortest duration is referred as a chip. The chip rate

Rc for an FH / MFSK system is defined by

Rc = Max(Rh,Rs)

Where Rh is the hop rate and Rs is Symbol Rate

In a slow rate frequency hopping multiple symbols are transmitted per hop. Hence

each symbol of a slow FH / MFSK signal is a chip. The bit rate Rb of theincoming binary

data. The symbol rate Rs of the MFSK signal, the chip rate Rc and the hop rate Rn are

related by

Rc = Rs = Rb /k ≥ Rh

where k= log2M

Fast frequency hopping:-

A fast FH / MFSK system differs from a slow FH / MFSK system in that there

are multiple hops per m-ary symbol. Hence in a fast FH / MFSK system each hop is a chip.

Fig. illustrates the variation of the frequency of a slow FH/MFSK signal with time for one

complete period of the PN sequence. The period of the PN sequence is 24-1 = 15. The

FH/MFSK signal has the following parameters:

Number of bits per MFSK symbol K = 2.

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Number of MFSK tones M = 2K

= 4

Length of PN segment per hop k = 3

Total number of frequency hops 2k = 8

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Fig. illustrates the variation of the transmitted frequency of a fast FH/MFSK signal with time.

The signal has the following parameters:

Number of bits per MFSK symbol K = 2.

Number of MFSK tones M = 2K

= 4

Length of PN segment per hop k = 3

Total number of frequency hops 2k = 8