112
Digital Communication Course file
Digital Communication
Course file
Contents 1. Syllabus
2. Lesson plan
3. Complete notes for 6 Units
4. Assignment Questions
5. Students‟ mid Marks, Assignment Marks
6. CO-PO Mapping
7. References, Journals, websites and E-links
1. Syllabus Copy III B.Tech, II-Sem (ECE) T C
3+1* 3
(A0419126) DIGITAL COMMUNICATIONS
UNIT I
DIGITIZATION TECHNIQUES FOR ANALOG MESSAGES-I: Introduction - Importance
of Digitization Techniques, Elements of Pulse Code Modulation (PCM) - Generation and
Reconstruction, Quantization and coding, Quantization error, PCM with Noise, Companding in
PCM,
UNIT II
DIGITIZATION TECHNIQUES FOR ANALOG MESSAGES-II: Delta modulation,
Adaptive Delta Modulation, Differential PCM systems (DPCM), Adaptive differential PCM
systems.
UNIT III
BASE BAND DIGITAL TRANSMISSION: Digital Signals and Systems – Digital PAM
Signals, Transmission Limitations, Power Spectra of Digital PAM, Noise and Errors – Binary
Error Probabilities, Matched Filtering, Optimum filtering.
UNIT IV
BAND PASS DIGITAL TRANSMISSSION: Digital modulation formats, Coherent binary
modulation techniques, Coherent quadrature modulation techniques, Non coherent binary
modulation techniques, Comparison of binary and quaternary modulation techniques, M-ary
modulation techniques.
UNIT V
INFORMATION THEORY: Uncertainty, information and entropy, source coding theorem,
Huffman coding, discrete memory less channels, mutual information, channel capacity, channel
coding theorem, differential entropy and mutual information for continuous ensembles, channel
capacity theorem.
UNIT VI
CHANNEL CODING: Linear block codes, Cyclic codes: CRC, Golay codes, BCH codes, RS
codes. Convolution codes.
TEXT BOOKS:
1. A. Bruce Carlson, & Paul B. Crilly, “Communication Systems – An Introduction to
Signals & Noise in Electrical Communication”, McGraw-Hill International Edition, 5th
Edition, 2010 .
2. Digital communications - Simon Haykin, John Wiley, 2005.
REFERENCES:
1. Herbert Taub & Donald L Schilling, “Principles of Communication Systems”, Tata
McGraw-Hill, 3rd
Edition, 2009.
2. Digital Communications – John Proakis, TMH, 1983. Communication Systems Analog
& Digital – Singh & Sapre, TMH, 2004.
3. Digital Communications by Bernard Sklar, Tata McGraw Hill.
2. LESSON PLAN
Academic Year: 2016-17
Year & Semester: B-Tech., III Year II-Sem (ECE)
Subject: Digital Communications Total Hours: 70
1. S.No
2. Unit
TOPICS Estimated Lectures
1 I
DIGITIZATION TECHNIQUES FOR ANALOG MESSAGES-I:
Introduction - Importance of Digitization Techniques
- Sampling- its Types and Reconstruction
- Quantization
Elements of Pulse Code Modulation (PCM) –
Generation and Reconstruction,
Quantization and Coding
Quantization Error
PCM with Noise
- Transmission Noise
- Quantization Noise
- Decoding Noise
Companding in PCM.
08
2 II
DIGITIZATION TECHNIQUES FOR ANALOG MESSAGES-II:
Delta Modulation
- Generation & Detection
- Advantages & Disadvantages
- Applications
Adaptive Delta Modulation
Differential PCM systems (DPCM)
-Generation & Detection
-Advantages
-Disadvantages
- Applications
Adaptive Differential PCM [ADPCM] Systems.
- Adaptive Quantization
- Adaptive Prediction
15
3
III
BASE BAND DIGITAL TRANSMISSION:
Digital Signals and Systems –
- Necessity
- Digital PAM Signals
- Transmission Limitations- Inter Symbol Interference
Power Spectra of Digital PAM
Noise and Errors – Binary Error Probabilities
Optimum filtering
15
Matched Filtering
4 IV
BAND PASS DIGITAL TRANSMISSSION:
Digital modulation Formats:
- Binary ASK, FSK, PSK, DPSK
- M-ary ASK,FSK,PSK
Coherent binary modulation techniques,
Coherent Quadrature modulation techniques
Non coherent binary modulation techniques
Comparison of binary and quaternary modulation techniques
M-ary modulation techniques.
10
5 V
INFORMATION THEORY:
Uncertainty, Information and Entropy
Source Coding Theorem
- Prefix Coding
- Huffman coding
Discrete Memory Less Channels
- Binary Symmetric Channel
Mutual Information - Properties
Channel Capacity
Channel Coding Theorem
Differential Entropy and Mutual Information For Continuous
Ensembles
Channel Capacity Theorem.
12
6 VI
CHANNEL CODING:
Linear block codes
Cyclic codes:
- CRC
- Golay codes
- BCH codes
- RS codes
Convolution codes.
10
3.DETAILED NOTES
UNIT I: DIGITIZATION TECHNIQUES FOR ANALOG MESSAGES-I
Elements Of Digital Communication Systems: Model of digital communication system
Digital representation of analog signal
Certain issues of digital transmission
Advantages of digital communication systems
Bandwidth- S/N trade off,
Hartley Shannon Law
Sampling theorem
The term communication (or telecommunication) means the transfer of some form of
information from one place (known as the source of information) to another place (known as
the destination of information) using some system to do this function (known as a
communication system).
Old Methods of Communication
Pigeons
Horseback
Smoke
Fire
Post Office
Drums
Problems with Old Communication Methods
Slow
Difficult and relatively expensive
Limited amount of information can be sent
Some methods can be used at specific times of the day
Information is not secure.
Examples of Today’s Communication Methods
All of the following are electric (or electromagnetic) communication systems
Satellite (Telephone, TV, Radio, Internet, … )
Microwave (Telephone, TV, Data, …)
Optical Fibers (TV, Internet, Telephone, … )
Copper Cables (telephone lines, coaxial cables, twisted pairs, … etc)
Advantages of Today’s Communication Systems
Fast
Easy to use and very cheap
Huge amounts of information can be transmitted
Secure transmission of information can easily be achieved
Can be used 24 hours a day.
Basic Construction of Electrical Communication System
Sound, picture, ...
Electric signal (like
audio and video
outputs of a video
camera
Electric Signal
(transmitted signal)
Electric Signal
(received signal)
Electric Signal (like
the outputs of a
satellite receiver)
Sound, picture, ...
Added Noise
Input
Input
Transducer Transmitter
Channel (distorts
transmitted
signal)
Receiver
Output
Transducer
Output
Converts the input
signal from its
original form (sound,
picture, … etc) to an
electric signal
Adapts the electric
signal to the channel
(changes the signal
to a form that is
suitable for
transmission)
Medium though
which the
information is
transmitted
Extracts the original
electric signal from
the received signal
Converts the electric
signal to its original
form (sount, picture,
… etc)
A communication system may transmit information in one direction such as TV
and radio (simplex), two directions but at different times such as the CB (half-duplex), or
two directions simultaneously such as the telephone (full-duplex).
Basic Terminology Used in this Communications Course
A Signal is a function that specifies how a specific variable changes versus an
independent variable such as time, location, height (examples: the age of people versus their
coordinates on Earth, the amount of money in your bank account versus time).
A System operates on an input signal in a predefined way to generate an output signal.
Analog Signals are signals with amplitudes that may take any real value out of an infinite
number of values in a specific range (examples: the height of mercury in a 10cm–long
thermometer over a period of time is a function of time that may take any value between 0
and 10cm, the weight of people setting in a class room is a function of space (x and y
coordinates) that may take any real value between 30 kg to 200 kg (typically)).
Digital Signals are signals with amplitudes that may take only a specific number of
values (number of possible values is less than infinite) (examples: the number of days in a
year versus the year is a function that takes one of two values of 365 or 366 days, number
of people sitting on a one-person chair at any instant of time is either 0 or 1, the
number of students registered in different classes at KFUPM is an integer number between 1
and 100).
Noise is an undesired signal that gets added to (or sometimes multiplied with) a desired
transmitted signal at the receiver. The source of noise may be external to the communication
system (noise resulting from electric machines, other communication systems, and noise from
outer space) or internal to the communication system (noise resulting from the collision of
electrons with atoms in wires and ICs).
Signal to Noise Ratio (SNR) is the ratio of the power of the desired signal to the power of
the noise signal.
Bandwidth (BW) is the width of the frequency range that the signal occupies. For example
the bandwidth of a radio channel in the AM is around 10 kHz and the
bandwidth of a radio channel in the FM band is 150 kHz.
Rate of Communication is the speed at which DIGITAL information is transmitted. The
maximum rate at which most of today‟s modems receive digital
information is around 56 k bits/second and transmit digital information is
around 33 k bits/second. A Local Area Network (LAN) can theoretically
receive/transmit information at a rate of 100 M bits/s. Gigabit networks
would be able to receive/transmit information at least 10 times that rate.
Modulation is changing one or more of the characteristics of a signal (known as the carrier
signal) based on the value of another signal (known as the information or
modulating signal) to produce a modulated signal.
Analog and Digital Communications
Since the introduction of digital communication few decades ago, it has been gaining a steady
increase in use. Today, you can find a digital form of almost all types of analog
communication systems. For example, TV channels are now broadcasted in digital form
(most if not all Ku–band satellite TV transmission is digital). Also, radio now is being
broadcasted in digital form (see sirus.com and xm.com). Home phone systems are starting to
go digital (a digital phone system is available at KFUPM). Almost all cellular phones are now
digital, and so on. So, what makes digital communication more attractive compared to analog
communication?
Advantages of Digital Communication over Analog Communication
Immunity to Noise (possibility of regenerating the original digital signal if signal
power to noise power ratio (SNR) is relatively high by using of devices called
repeaters along the path of transmission).
Efficient use of communication bandwidth (through use of techniques like
compression).
Digital communication provides higher security (data encryption).
The ability to detect errors and correct them if necessary.
Design and manufacturing of electronics for digital communication systems is
much easier and much cheaper than the design and manufacturing of electronics
for analog communication systems.
Modulation
Famous Types
Amplitude Modulation (AM): changing the amplitude of the carrier based
on the information signal as done for radio channels that are transmitted in
the AM radio band. Phase Modulation (PM): changing phase of the carrier based on the
information signal. Frequency Modulation (FM): changing the frequency of the carrier based on
the information signal as done for channels transmitted in the FM radio band.
Purpose of Modulation
For a signal (like the electric signals coming out of a microphone) to
be transmitted by an antenna, signal wavelength has to be comparable to the
length of the antenna (signal wavelength is equal to 0.1 of the antenna length
or more).If the wavelength is extremely long, modulation must be used
to reduce the wavelength of the signal to make the length of the required
antenna practical. To receive transmitted signals from multiple sources without interference
between them, they must be transmitted at different frequencies (frequency multiplexing) by modulating carriers that have different frequencies with the different information signals.
Exercise 1–1: Specify if the following communication systems are Analog or Digital:
a) TV in the 1970s:
b) TV in the 2030s:
c) Fax machines
d) Local area networks (LANs):
e) First–generation cellular phones
f) Second–generation cellular phones
g) Third–generation cellular phones
These are the basic elements of any digital communication system and It gives a basic
understanding of communication systems.
1. Information Source and Input Transducer:
The source of information can be analog or digital, e.g. analog: aurdio or video signal,
digital: like teletype signal. In digital communication the signal produced by this
source is converted into digital signal consists of 1′s and 0′s. For this we need source
encoder.
1.
2. Source Encoder
In digital communication we convert the signal from source into digital signal as
mentioned above. The point to remember is we should like to use as few binary digits as
possible to represent the signal. In such a way this efficient representation of the source
output results in little or no redundancy. This sequence of binary digits is
called information sequence.
Source Encoding or Data Compression: the process of efficiently converting the output
of wither analog or digital source into a sequence of binary digits is known as source
encoding.
3. Channel Encoder:
The information sequence is passed through the channel encoder. The purpose of the
channel encoder is to introduced, in controlled manner, some redundancy in the binary
information sequence that can be used at the receiver to overcome the effects of noise and
interference encountered in the transmission on the signal through the channel.
e.g. take k bits of the information sequence and map that k bits to unique n bit sequence
called code word. The amount of redundancy introduced is measured by the ratio n/k and
the reciprocal of this ratio (k/n) is known as rate of code or code rate.
4. Digital Modulator:
The binary sequence is passed to digital modulator which in turns convert the sequence
into electric signals so that we can transmit them on channel (we will see channel later).
The digital modulator maps the binary sequences into signal wave forms , for example if
we represent 1 by sin x and 0 by cos x then we will transmit sin x for 1 and cos x for 0. ( a
case similar to BPSK)
5. Channel:
The communication channel is the physical medium that is used for transmitting signals
from transmitter to receiver. In wireless system, this channel consists of atmosphere , for
traditional telephony, this channel is wired , there are optical channels, under water
acoustic channels etc.
we further discriminate this channels on the basis of their property and characteristics,
like AWGN channel etc.
6. Digital Demodulator:
The digital demodulator processes the channel corrupted transmitted waveform and
reduces the waveform to the sequence of numbers that represents estimates of the
transmitted data symbols.
7. Channel Decoder:
This sequence of numbers then passed through the channel decoder which attempts to
reconstruct the original information sequence from the knowledge of the code used by the
channel encoder and the redundancy contained in the received data
The average probability of a bit error at the output of the decoder is a measure of the
performance of the demodulator – decoder combination. THIS IS THE MOST
IMPORTANT POINT, We will discuss a lot about this BER (Bit Error Rate) stuff in
coming posts.
8. Source Decoder
At the end, if an analog signal is desired then source decoder tries to decode the sequence
from the knowledge of the encoding algorithm. And which results in the approximate
replica of the input at the transmitter end
9. Output Transducer:
Finally we get the desired signal in desired format analog or digital.
The points worth noting are :
1. The source coding algorithm plays important role in higher code rate
2. The channel encoder introduced redundancy in data
3. The modulation scheme plays important role in deciding the data rate and immunity
of signal towards the errors introduced by the channel
4. Channel introduced many types of errors like multi path, errors due to thermal noise etc.
5. The demodulator and decoder should provide high BER.
Advantages of digital communication:
1. It is fast and easier.
2. No paper is wasted.
3. The messages can be stored in the device for longer times, without being damaged,
unlike paper files that easily get damages or attacked by insects.
4. Digital communication can be done over large distances through internet and other
things.
5. It is comparatively cheaper and the work which requires a lot of people can be done
simply by one person as folders and other such facilities can be maintained.
6. It removes semantic barriers because the written data can be easily channel to
different languages using software.
7. It provides facilities like video conferencing which save a lot of time, money and effort.
Disadvantages:
1. It is unreliable as the messages cannot be recognised by signatures. Though software can
be developed for this, yet the softwares can be easily hacked.
2. Sometimes, the quickness of digital communication is harmful as messages can be sent
with the click of a mouse. The person does not think and sends the message at an
impulse.
3. Digital Communication has completely ignored the human touch. A personal touch cannot
be established because all the computers will have the same font!
4. The establishment of Digital Communication causes degradation of the environment in
some cases. "Electronic waste" is an example. The vibes given out by the telephone and
cell phone towers are so strong that they can kill small birds. In fact the common sparrow
has vanished due to so many towers coming up as the vibrations hit them on the head.
5. Digital Communication has made the whole world to be an "office." The people carry their
6. Work to places where they are supposed to relax. The whole world has been made into an
office. Even in the office, digital communication causes problems because personal
messages can come on your cell phone, internet, etc.
7. Many people misuse the efficiency of Digital Communication. The sending of hoax
messages, the usage by people to harm the society, etc cause harm to the society on
the whole.
Definition of Digital – A method of storing, processing and transmitting information through the
use of distinct electronic or optical pulses that represent the binary digits 0 and 1.
Advantages of Digital
Less expensive
More reliable
Easy to manipulate
Flexible
Compatibility with other digital
systems
Only digitized information can be transported through a noisy channel without
degradation
Integrated
networks
Disadvantages of Digital
Sampling Error
Digital communications require greater bandwidth than analogue to transmit the
same information.
The detection of digital signals requires the communications system to be
synchronized, whereas generally speaking this is not the case with analogue systems.
Some more explanation of advantages and disadvantages of analog vs digital
communication.
1. The first advantage of digital communication against analog is it‟s noise immunity. In any
transmission path some unwanted voltage or noise is always present which cannot be
eliminated fully. When signal is transmitted this noise gets added to the original
signal causing the distortion of the signal. However in a digital communication at the
receiving end this additive noise can be eliminated to great extent easily resulting in
better recovery of actual signal. In case of analog communication it‟s difficult to remove
the noise once added to the signal.
2. Security is another priority of messaging services in modern days. Digital communication
provides better security to messages than the analog communication. It can be achieved
through various coding techniques available in digital communication.
3. In a digital communication the signal is digitized to a stream of 0s and 1s. So at the
receiver side a simple decision has to me made whether received signal is a 0 or
a 1.
4. Accordingly the receiver circuit becomes simpler as compared to the analog
receiver circuit.
5. Signal when travelling through its transmission path gets faded gradually. So on it‟s path it
needs to be reconstructed to its actual form and re-transmitted many times. For that reason
AMPLIFIERS are used for analog communication and REPEATERS are used in digital
communication. Amplifiers are needed every 2 to 3 Kms apart whereas repeaters are needed
every 5 to 6 Kms apart. So definitely digital communication is cheaper. Amplifiers also often
add non-linearity that distorts the actual signal.5. Bandwidth is another scarce
resource. Various Digital communication techniques are available that use the
available bandwidth much efficiently than analog communication techniques.
6. When audio and video signals are transmitted digitally an AD (Analog to Digital)
converter is needed at transmitting side and a DA (Digital to Analog) converter is again
needed at receiver side. While transmitted in analog communication these devices are not
needed.
7. Digital signals are often an approximation of the analog data (like voice
or video) that is obtained through a process called quantization. The digital representation is
never the exact signal but its most closely approximated digital form. So it‟s accuracy
depends on the degree of approximation taken in quantization process.
Sampling Theorem:
There are 3 cases of sampling:
Ideal impulse sampling
Consider an arbitrary lowpass signal x (t ) shown in Fig. 6.2(a). Let
Pulse Code Modulation
o P C M generation and reconstruction
o Q u a n t i z a t i o n noise
o D i f f e r e n t i a l PCM systems (DPCM)
o D e l t a modulation, adaptive delta modulation
o N o i s e in PCM and DM systems
Digital Transmission of Analog Signals:
o PCM
o DPCM
o DM
6.1 Introduction
Quite a few of the information bearing signals, such as speech, music, video, etc., are analog
in nature; that is, these are the functions of the continuous variable t and for any t = t1, their value
can lie anywhere in the interval, say − A to A. Also, these signals are of the baseband variety. If
there is a channel that can support baseband transmission, we can easily set up a baseband
communication system. In such a system, the transmitter could be as simple as just a power
amplifier so that the signal that is transmitted could be received at the destination with some
minimum power level, even after being subject to attenuation during propagation on the channel. In
such a situation, even the receiver could have a very simple structure; an appropriate filter (to
eliminate the out of band spectral components) followed by an amplifier. If a baseband channel is
not available but have access to a passband channel, (such as ionospheric channel, satellite channel
etc.) an appropriate CW modulation scheme discussed earlier could be used to shift the baseband
spectrum to the passband of the given channel. Interesting enough, it is possible to transmit the
analog information in a digital format.
Though there are many ways of doing it, in this chapter, we shall explore three such
techniques, which have found widespread acceptance. These are: Pulse Code Modulation (PCM),
Differential Pulse Code Modulation (DPCM)
and Delta Modulation (DM). Before we get into the details of these techniques, let us summarize
the benefits of digital transmission. For simplicity, we shall assume that information is being
transmitted by a sequence of binary pulses. i) During the course of propagation on the channel, a
transmitted pulse becomes gradually distorted due to the non- ideal transmission characteristic of
the channel. Also, various unwanted signals (usually termed interference and noise) will cause
further deterioration of the information bearing pulse. However, as there are only two types of
signals that are being transmitted, it is possible for us to identify (with a very high probability) a
given transmitted pulse at some appropriate intermediate point on the channel and regenerate a
clean pulse. In this way, be completely eliminating the effect of distortion and noise till the point
of regeneration. (In long-haul PCM telephony, regeneration is done every few Kilometers, with the
help of regenerative repeaters.) Clearly, such an operation is not possible if the transmitted signal
was analog because there is nothing like a reference waveform that can be regenerated.
ii) Storing the messages in digital form and forwarding or redirecting them at a later point in time
is quite simple.
iii) Coding the message sequence to take care of the channel noise, encrypting for secure
communication can easily be accomplished in the digital domain.
iv) Mixing the signals is easy. All signals look alike after conversion to digital form
independent of the source (or language!). Hence they can easily be multiplexed (and
demultiplexed)
6.2 The PCM System
Two basic operations in the conversion of analog signal into the digital is time discretization
and amplitude discretization. In the context of PCM, the former is accomplished with the sampling
operation and the latter by means of quantization. In addition, PCM involves another step, namely,
conversion of quantized amplitudes into a sequence of simpler pulse patterns (usually binary),
generally called as code words. (The word code in pulse code modulation refers to the fact that
every quantized sample is converted to an R -bit code word.)
Fig. 6.1 illustrates a PCM system.
Here, m(t ) is the information bearing message signal that is to be transmitted
digitally. m(t ) is first sampled and then quantized. The output of the sampler is
Ts is the sampling period and n is the appropriate integer.
is called the sampling rate or sampling frequency.
The quantizer converts each sample to one of the values that is closest to it from among a pre-
selected set of discrete amplitudes. The encoder represents each one of these quantized samples by
an R -bit code word. This bit stream travels on the channel and reaches the receiving end. With fs as
the sampling rate and R -bits per code word, the bit rate of the PCM System is
The decoder converts the R -bit code words into the corresponding (discrete) amplitudes.
Finally, the reconstruction filter, acting on these discrete amplitudes, produces the analog
signal, denoted by m‟(t ) . If there are no channel errors, then m‟(t ) approx= m(t).
The most common technique for sampling voice in PCM systems is to a
sample-and- hold circuit.
The instantaneous amplitude of the analog (voice) signal is held as a
constant charge on a capacitor for the duration of the sampling period Ts.
This technique is useful for holding the sample constant while other
processing is taking place, but it alters the frequency spectrum and
introduces an error, called aperture error, resulting in an inability to recover
exactly the original analog signal.
The amount of error depends on how much the analog changes during the
holding time, called aperture time.
To estimate the maximum voltage error possible, determine the maximum slope of the
analog signal and multiply it by the aperture time DT
Recovering the original message signal m(t) from PAM signal :
Where the filter bandwidth is W
The filter
output is
fs M ( f )H ( f ) . Note that the
Fourier transform of h(t) is given by
H ( f ) T sinc( f T ) exp( j f T )
(3.19)
amplitude
distortion delay T 2
aparture effect
Let the equalizer
responseis
1
1 f
(3.20)H ( f ) T sinc( f T ) sin( f T )
Ideally the original signal m(t ) can be recovered completely.
Other Forms of Pulse Modulation:
In pulse width modulation (PWM), the width of each pulse is made directly proportional
to the amplitude of the information signal.
In pulse position modulation, constant-width pulses are used, and the position or time of
occurrence of each pulse from some reference time is made directly proportional to the
amplitude of the information signal.
Pulse Code Modulation (PCM) :
Pulse code modulation (PCM) is produced by analog-to-digital conversion process.
As in the case of other pulse modulation techniques, the rate at which samples are
taken and encoded must conform to the Nyquist sampling rate.
The sampling rate must be greater than, twice the highest frequency in the analog
signal,
fs > 2fA(max)
Quantization Process:
Define partition cell
Jk : mk m mk 1 , k 1,2,, L
(3.21)
Where mk is the decision level or the decision threshold.
Amplitude quantizati on : The process of transforming the
Figure 3.10 Two types of quantization: (a) midtread and (b) midrise.
Quantization Noise:
m 2
2
Figure 3.11 Illustration of the quantization process
Let the quantizati on error be denoted by the random
variable Q of sample value q
q m
(3.23)
Q M V , ( E[M ] 0) (3.24)
Assuming a uniform quantizer
of the midrise
type
the step - size is
2m m a x
L
(3.25)
m m a x m m m a x , L : total number of levels
1 ,
fQ
(q)
q
2 2
(3.26)0, otherwise
2 E[Q 2 ] 2 q 2 f Q
Q
2
(q)dq 1
2 q 2 dq
2
(3.28) 12
When the quatized sample is expressed in binary form,
L 2 R
(3.29)
where R is the number of bits per sample
R log 2 L (3.30)
2m m a x
2 R
(3.31)
2 1
m 2
2 2 R
(3.32)Q
3 max
Let P denote the average power of m(t )
P (SNR)
o 2
Q
( 3P
)2 2 R
max
(3.33)
(SNR)o increases exponentia lly with increasing R (bandwidth).
2
..,_ EXAMPLE 3.1 Sinusoidal Modulating Signal
Consider the special case of a full-load sinusoidal modulating signal of amplitude Am, which
utilizes all the representation levels provided. The average signal power is (assuming a load of 1 ohm)
p = A~
2
The total range of the quantizer input is 2Am, because the modulating signal swings between
-Am and Am. We may therefore set mmax = Am, in which case the use of Equation (3.32) yields the average power (variance) of the quantization noise as
a1 = fA~.2-2R
Thus the output signal-to-noise ratio of a uniform quantizer, for a full-load test tone, is
- A~/2 - 3 2R (SNR)o - A2 2-2R;3 - (2 )
rn
Expressing the signal-to-noise ratio in decibels, we get
10 log10(SNR)0 = 1.8 + 6R
(3.34) (3.35)
Pulse Code Modulation (PCM):
Source of continuous•
time message signal
Low-pass
filter Sampler Quantizer
(a) Transmitter
Encoder
PCM signal applied to
channel input
Distorted PCM signal produced
at channel output
Regenerative
repeater
Regenerative
repeater
Regenerated PCM signal
applied to the receiver
(b) Transmission path
Final channel output
Regeneration circuit
Decoder
(c) Receiver
Reconstruction filter
Destination
Figure 3.13 The basic elements of a PCM system
Quantization (nonuniform quantizer):
Compression laws. (a) m -law. (b) A-law.
- law
A - law
d m
d
log(1 m )
log(1 )
log(1 )
(1 m )
(3.48)
(3.49)
A(m) 1 log A
0 m 1 A 1 A m
(3.50)
log( ) 1 log A
1
A
m 1
1 log A d m 0 m
1
A A (3.51)
d (1 A) m
1
A
m 1
Figure 3.15 Line codes for the electrical representations of binary data.
(a) Unipolar NRZ signaling. (b) Polar NRZ signaling.
(c) Unipolar RZ signaling. (d) Bipolar RZ signaling.
(e) Split-phase or Manchester code.
Noise consideration in PCM systems:
(Channel noise, quantization noise)
Time-Division Multiplexing(TDM):
Digital Multiplexers :
Virtues, Limitations and Modifications of PCM:
Advantages of PCM
1. Robustness to noise and interference
2. Efficient regeneration
3. Efficient SNR and bandwidth trade-off
4. Uniform format
5. Ease add and drop
6. Secure
UNIT II: DIGITIZATION TECHNIQUES FOR ANALOG MESSAGES-II
Delta Modulation (DM) :
Let mn m(nTs )
, n 0,1,2,
where Ts is the sampling period and m(nTs ) is a sample of m(t ).
The error signal is
en mn mq n 1
eq n sgn(en)
mq n mq n 1 eq n
(3.52)
(3.53)
(3.54)
where mq nis
the quantizer output , eq nis
the quantized version of en, and is
the step size
The modulator consists of a comparator, a quantizer, and an accumulator
The output of the accumulator is
n
mq n sgn(ei) i 1
n
eq i (3.55) i 1
Two types of quantization errors: Slope Overload Distortion and Granular Noise:
Denote the quantizati on error by qn,mq
n mn qn
Recall (3.52) , we have
en mn mn 1 qn 1
(3.56)
(3.57)
Except for qn 1, the quantizer
input is
a first
backward difference of
the input
signal
To avoid slope - overload distortion , we require
(slope)
max
Ts
dm(t )
dt
(3.58)
Beneficial effects of using integrator:
1. Pre-emphasize the low-frequency content
2. Increase correlation between adjacent samples
(reduce the variance of the error signal at the quantizer input)
k
3. Simplify receiver design
Because the transmitter has an integrator , the receiver
consists simply of a low-pass filter.
(The differentiator in the conventional DM receiver is cancelled by the integrator )
Linear Prediction (to reduce the sampling rate):
Consider a finite-duration impulse response (FIR)
discrete-time filter which consists of three blocks :
1. Set of p ( p: prediction order) unit-delay elements (z-1)
2. Set of multipliers with coefficients w1,w2,…wp
3. Set of adders ( )
The filter output (The linear
p
predition of the input ) is
x̂n wk
k 1
x(n k ) (3.59)
The prediction error is
en xn x̂n
(3.60)
Let the index
of performance be
J Ee 2 n
(mean square error)
(3.61)
Find w1 , w
2 ,, w
p to minimize J
From (3.59) (3.60) and (3.61)
p
we have
J Ex 2 n 2 w Exnxn k
k 1
Assume X (tp) ispstationary process with zero mean ( E[ x[n]] 0)
Figure 3.27
Block diagram illustrating the linear adaptive prediction process
Differential Pulse-Code Modulation (DPCM):
Usually PCM has the sampling rate higher than the Nyquist rate .The encode signal contains
redundant information. DPCM can efficiently remove this redundancy.
Figure 3.28 DPCM system. (a) Transmitter. (b) Receiver.
Input signal to the quantizer is defined by:
M Q
E
E
G
en mn m̂ n
(3.74)
m̂ nis
a prediction
value.
The quantizer output is
eq n
where
en
qnis
qn
quantizati on
(3.75)
error.
The prediction filter input is
mq n m̂ nen qn
(3.77)
From (3.74)
mn
mq n mn qn (3.78)
Processing Gain:
The (SNR)o of the DPCM system is
2(SNR)
o M
2
Q
(3.79)
where 2 and 2
2
are variances of mn(E[m[n]] 0) and qn
2(SNR) ( M )( E )o 2 2
E Q
Gp
(SNR )Q
(3.80)
where 2 is
the variance of the predictions error
and the signal - to - quantization on noise ratio is
2(SNR )
Q E
2
Q
(3.81)
2Processing Gain, M
p 2
E
(3.82)
Design a prediction filter to
maximize
G p (minimize 2 )
Adaptive Differential Pulse-Code Modulation (ADPCM):
Need for coding speech at low bit rates , we have two aims in mind:
1. Remove redundancies from the speech signal as far as possible.
2. Assign the available bits in a perceptually efficient manner.
Figure 3.29 Adaptive quantization with backward estimation (AQB).
Figure 3.30 Adaptive prediction with backward estimation (APB).
UNIT III: BASE BAND TRANSMISSION
Digital PAM Signals
Different pulses and power spectrum densities,
Probability of error, optimum receiver,
Optimum of coherent reception,
Signal space representation and probability of error,
Eye diagram,
Cross talk.
1. Line Coding
1.1 Requirements
Digital data can be transmitted by various pulse waveforms, also called line codes. The following
properties are desirable for a line code:
It is important that the pulses stream to be transmitted does not have a DC component. It can case
baseline wander or Galvanic Corrosion.
It should be relatively easy to recover the data clock.
The line coding scheme should be bandwidth efficient.
The line code should be robust in the presence of noise.
It should be possible to recognise a line coding error, sometimes called a line violation. (In some
signalling protocols, a line violation is deliberately generated to mark the start of a frame)
1.2 Analogue Telephone Line Considerations
To review the telephone line. At the local exchange a voltage
is applied, via inductors and resistors to the copper pair. This
allows the transmitting equipment to sink current. The
variations in current correspond to change in the voltage
signal on the line. The receiver terminal reads this voltage. In
most cases the receiving terminal is allowed to take a DC
feed from the line itself.
The bandwidth of an analogue telephone line connection is
300 Hz to 3.4 kHz. A square wave or any pulse train with very fast rise times will be distorted if it is sent
along a telephone line. Therefore an analogue telephone line is not suitable for sending digital pulses as
all frequency components outside the 300 - 3 kHz range will be removed. This bandwidth limitation is
not caused totally by the copper pair but by the filters in the local exchange which are part of the
analogue to digital process. In the past some analogue telephone lines also had loading coils (inductors)
on the line which were intended to give a flat frequency response.
R
+
L
RXTX
R
+
L
Telephone
Line
Terminal
C
Local Exchange
On a digital telephone line all analogue filters are removed so the usable bandwidth of the copper pair
itself is much greater and can extend to a few Megahertz. These lines are suitable for pulse transmission
e.g. ISDN.
1.3 Digital Signalling Formats
1.3.1 Unipolar Non Return to Zero (NRZ)
Symbol 1 is represented by transmitting a pulse of constant amplitude for the entire duration of the bit
interval, and symbol 0 is represented by no pulse. NRZ indicates that the assigned amplitude level is
maintained throughout the entire bit period. This allows for long series without change, which makes
synchronization difficult (difficult to recover the clock). Unipolar also contains a strong DC component.
.
From www.wikipedia.com
In telecommunication, a non-return-to-zero (NRZ) line code is a binary code in which "1's" are
represented by one significant condition and "0's" are represented by the other significant condition, with
no other neutral or rest condition. The pulses have more energy than a RZ code, but it does not have a rest
state, which means a synchronization signal must also be sent alongside the code.
For a given data signaling rate, i.e., bit rate, the NRZ code requires only half the bandwidth required by
the Manchester code.
When used to represent data in an asynchronous communication scheme, the absence of a neutral state
requires other mechanisms for data recovery, to replace methods used for error detection when using
synchronization information when a separate clock signal is available.
1.3.2 Bipolar NRZ
Pulses of equal positive and negative amplitudes represent symbols 1 and 0. (e.g. ± 5 volts, ± 12 volts) In
either case, the assigned pulse amplitude level is maintained throughout the bit interval. Because of the
positive and negative levels the average voltage will tend towards zero and hence little DC component.
Again synchronisation will be difficult.
1.3.3 Unipolar Return to Zero (RZ)
Symbol 1 is represented by a positive pulse that returns to zero before the end of the bit interval and
symbol 0 is represented by the absence of pulse.
1.3.4 Bipolar RZ
Positive and negative pulses of equal amplitude are used for symbol 1 and symbol 0. In either case the
pulse returns to 0 before the end of the bit interval.
From www.wikipedia.com
Return-to-zero (RZ) describes a line code used in telecommunications signals in which the signal drops
(returns) to zero between each pulse. This takes place even if a number of consecutive zeros or ones
occur in the signal. The signal is self-clocking. This means that a separate clock does not need to be sent
alongside the signal, but suffers from using twice the bandwidth to achieve the same data-rate as
compared to non-return-to-zero format.
The "zero" between each bit is a neutral or rest condition, such as a zero amplitude in pulse amplitude
modulation (PAM).
1.3.5 Alternate Mark Inversion (AMI) RZ Signalling
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. An advantage of AMI is
that it is easy to recognise a line violation.
From www.wikipedia.com
A binary 0 is encoded as zero volts as in unipolar encoding. A binary 1 is encoded alternately as a
positive voltage and a negative voltage. This prevents a significant build-up of DC, as the positive and
negative pulses average to zero volts. Little or no DC-component is considered an advantage because the
cable may then be used for longer distances and to carry power for intermediate equipment such as line
repeaters. The DC-component can be easily and cheaply removed before the signal reaches the decoding
circuitry.
Bipolar encoding is preferable to non-return-to-zero where signal transitions are required to maintain
synchronization between the transmitter and receiver. Other systems must synchronize using some form
of out-of-band communication, or add frame synchronization sequences that don't carry data to the signal.
These alternative approaches require either an additional transmission medium for the clock signal or a
loss of performance due to overhead, respectively. A bipolar encoding is an often good compromise: runs
of ones will not cause a lack of transitions, however long sequences of zeroes are still an issue. Long
sequences of zero bits result in no transitions and a loss of synchronization. Where frequent transitions
are a requirement, a self-clocking encoding such as return-to-zero or some other more complicated line
code may be more appropriate, though they introduce significant overhead.
1.3.6 Manchester Coding
Symbol 1 is represented by a positive pulse followed by a negative pulse - with each pulse being of equal
amplitude and duration of half a pulse. For symbol 0 the polarities of these pulses are reversed. An
advantage of this coding is that it is easy to recover the original data clock.
From www.wikipedia.com
Manchester coding provides a simple way to encode arbitrary binary sequences without ever having long
periods without level transitions, thus preventing the loss of clock synchronization, or bit errors from
low-frequency drift on poorly-equalized analog links (see ones-density).
If transmitted as a bipolar signal (i.e. where the two signaling levels are of opposite polarity), the DC
component of the encoded signal is zero, again preventing baseline drift of the repeated signal, making it
easy to regenerate and preventing waste of energy.
Time is divided into periods, and one bit is transmitted per period
A "0" is expressed by a low-to-high transition, a "1" by high-to-low transition (according to G.E.
Thomas' convention--in the IEEE 802.3 convention, the reverse is true)
The transitions signifying 0 or 1 occur at the midpoint of a period
Manchester codes always have a transition at the middle of each bit period, and depending on the state of
the signal, may have a transition at the beginning of the period as well. The direction of the mid-bit
transition is what carries the data, with a low-to-high transition indicating one binary value, and a high-
to-low transition indicating the other.
1.3.7 Coding comparison
1 0 1 1 0 0 1 0
Unipolar NRZ
Bipolar NRZ
Unipolar RZ
Biplolar RZ
AMI
Manchester
2. M-Ary Line Coding
The utilisation of bandwidth can be made more efficient by adopting an M-Ary format for the
representation of the input binary data . A Binary code consists of two symbols- '1' and '0'. A quaternary
(i.e. 4 level) code would consist of 4 symbols. The 4 symbols could be assigned to 00, 01, 10 and 11 for
example. This would allow us to half the symbol rate on a transmission line compared to one bit per
symbol.
Note that binary data rate is measured in bits/second whereas the symbol rate is measured in Baud.
(Symbols per second).
An example of M-Ary Line coding is 2B1Q line code, as above, used on
ISDN basic rate telephone lines between a subscribers premises and the
local telephone exchange. In this case the baud rate will be half the bit rate.
(There are 4 possible symbols, each of which requires 2 bits. If the
probability of each symbol is ¼ then the information in each symbol is log2
(1/¼) = log2 4 = 2 bits, so that the information rate is 2 * baud rate = bit
rate).
For example, the input binary sequence 11100001 is viewed as a new
sequence of dibits (pairs of bits); 11 10 00 01. Each dibit symbol is assigned one of 4 levels. If we
increase the number of levels there will be a trade-off between noise performance and bandwidth.
Example: Explain how a ternary line coding system can code 3 bits per symbol.
Answer: At 3 bits per symbol, 23 = 8, therefore we need at least 8 symbols. A single ternary pulse
would only allow one of 3 symbols to be represented. Two ternary pulses in a particular order,
however would allow for 9 combinations of levels. This code could be abbreviated as 3B2T. It is
not as efficient as 2B1Q but one advantage is that zero volts is one of the levels and the wave form
would resemble a binary bipolar format, Note that 4B3T coding is also used on ISDN lines in some
countries.
11
10
00
01
Time
3. Line Transmission Systems
Problems are encountered when a digital signal is sent through a channel. The diagram shows the basic
stages in a digital signal transmission. A simple non-return-to-zero (NRZ) code is assumed, as already
defined. The transmission medium might be a coaxial cable or a copper twisted pair, as often used in
local area networks or digital telephone systems. Similar principles apply, however, to systems using
other transmission media and/or more complicated codes.
Typical waveforms at the points labelled A to F in the system are shown. After passing down the cable
the original waveform A is attenuated and a noise component is added. Also because of a finite system
response time and propagation delays, the clear transition between voltage levels become indistinct.
To counteract the distortion illustrated, the
system includes an equaliser which sharpens
the received waveform, so that the
relationship of the equaliser output C to the
original binary symbols is much clearer. The
equaliser would normally consist of an
amplifier stage combined with a filter to
reduce unwanted frequency components. For
example, it is quite common for copper
cables to pick up a 50 Hz noise component
from the mains. It is essential that the
equaliser removes this component. Also, in certain configurations copper cables will pick up
electromagnetic interference, which must be filtered out. Note also that the input to the equaliser must be
protected from over-voltages such as induced lightning and other transients.
Passing the equalised waveform through a threshold detector (e.g. a Schmitt trigger) generates a binary
signal very similar to the transmitted one. If the threshold settings are too small then noise will trigger the
detector. If the settings are too large then the data may not trigger the detector. It is important that the
slew rate of the comparator used in the detector is fast enough for the data rate.
Provided that the noise levels are sufficiently low, and the equaliser and threshold detector are properly
designed, then the only difference between binary waveforms A and D is that the transitions of the latter
Retiming
Extracot
Retiming
Sector
Threshold
DetectorEqualiser
Channel
e.g. Co-ax
Copper pair
Transmitter
NoiseDigital Information
clocked at fc
- may be source
and/or channel
coded
Received
Digital
Information
A B C D F
E
1 0 1 1 0 1 0
1 0 1 1 0 1 0 0 Binary information
A NRZ Line code from Transmitter
B Received signal from TX channel
C Output from equaliser
D Output from threshold detector
E Output from timing extractor
F Output from re-timing circuit
are not perfectly in step with those of the former. The transitions of D will correspond to the threshold-
crossings of waveform C which will not precisely mirror those of the original binary waveform.
The final stage is the re-timing of the received waveform. If this were not carried out, then the
irregularities (jitter) in the waveforms would soon build up to cause error over a long link.
A regular timing reference signal F - the data clock - is derived from the received waveform itself by a
special circuit, (the timing extraction circuit which is based on a Phase Locked Loop). The clock signal
and the output from the threshold detector are then processed to give a final regenerated digital signal F
whose transitions now coincide with the instants at which the clock signal goes from low to high.
A comparison of waveforms C and F shows that the combined effect of threshold detection and re-timing
is equivalent to sampling waveform C near its peaks and troughs to determine the appropriate binary
states. So even in the presence of noise, regenerated signal F can be an almost perfect (delayed) replica of
the transmitted signal, provided only that the noise is not sufficient to cause an incorrect decision to be
made at the threshold detector.
3.1 Equaliser
From www.wikipedia.com
An equalization (EQ) filter is a filter, usually adjustable, chiefly meant to compensate for the unequal
frequency response of some other signal processing circuit or system.
An EQ filter typically allows the user to adjust one or more parameters that determine the overall shape
of the filter's transfer function. It is generally used to improve the fidelity of sound, to emphasize certain
instruments, to remove undesired noises, or to create completely new and different sounds.
4. Transmission Line Impairments
Until now we have assumed that the transmitter sends a rectangular pulse. A transmission line acts like a
filter so the output response of the transmission line to a rectangular pulse can be quite distorted. The
distortion can mean that pulses can become overlapped thus causing receiver errors. We therefore need to
model the effects of transmitting pulses through a transmission line.
a) Rectangular pulse response of a first order lowpass filter,
where the duration of the pulse is approximately equal to the
filter time constant.
b) Response of the same filter to a binary waveform.
The figure shows the response to a single pulse, and the
superimposed pulse responses corresponding to an input pulse
train (binary waveform). Note that because the response to a
single pulse takes longer to decay that the duration of a symbol
period, the output waveform gradually accumulates a DC
offset. In the absence of further processing this would clearly cause problems for threshold detection.
Even in the positions corresponding to a binary 0 there can be a considerable output voltage.
This figure, on the other hand, shows a much more desirable
overall pulse response for a telecommunications channel. It
shows the possible response of a telecommunications channel
to (a) a rectangular pulse and (b) a binary waveform. Here the
system response to a bit stream could be decoded without
difficulty, owing to the clear distinction in the combined
response between binary 1 and 0.
An alternative approach to modelling a linear channel or
component is based on the second definition of linearity. Any
practical message signal can be described in terms of its frequency content - or, to be more precise,
modelled as a frequency spectrum. Similarly, any linear system can be completely specified by its
frequency response function, which is a description of amplitude and phase shifts introduced by the
system for all frequencies.
4.1 Amplitude distortion and phase distortion
An ideal transmission channel would pass all frequency components of a signal with their amplitude and
phase relationships unchanged The simplest frequency
domain model of such behaviour would be a constant
amplitude ratio and zero phase shift for all frequencies
of interest. For example a square wave would be
unaffected by the transmission channel. In practice, the
higher order harmonics will be greatly attenuated by the
transmission channel. Also the phase shift will be
different for each harmonic. For example, the fundamental harmonic may have a phase shift of 45
V
Time
input pulse
pulse response
1 0 1 1 0 1 0 0
(a)
(b)
V
Time
input pulse
pulse response
1 0 1 1 0 1 0 0
(a)
(b)
TimeTransmitted Signal
Phase and
Amplitude Distortion
Amplitude Distortion
Only
Pulse has spread
due to phase delay
for harmonics
degrees whereas the fifth harmonic could have a phase shift of 80 degrees. This will cause components of
the pulse to be delayed or stretched. Specifications for digital receiver systems usually include limits for
phase delay.
4.2 Inter Symbol Interference
Due to the fact that the transmission channels are bandlimited, the transmitted pulses
tend to spread during transmission. This pulse spreading or dispersion causes overlap
of pulses into adjacent pulse time slots. This signal overlap may result in an error at
the point where the receiver makes a decision as to which pulse has been transmitted,
especially when other impairments are present (such as noise, interference).
This effect of pulse overlap and the resultant difficulty of discriminating between
symbols at the receiver are termed inter symbol interference (ISI).
From www.wikipedia.com
In telecommunication, intersymbol interference (ISI) means a form of distortion of a signal that causes
the previously transmitted symbols to have an effect on the currently received symbol. This is usually an
unwanted phenomenon as the previous symbols have similar effect as noise, thus making the
communication less reliable. ISI is usually caused by echoes or non-linear frequency response of the
channel. Ways to fight against intersymbol interference include adaptive equalization or error correcting
codes.
In a digital transmission system, distortion of the received signal, which is manifested in the temporal
spreading and consequent overlap of individual pulses to the degree that the receiver cannot reliably
distinguish between changes of state, i.e., between individual signal elements. At a certain threshold,
intersymbol interference will compromise the integrity of the received data.
t
Intersymbol
Interferencce
Eye Pattern:
Experimental tool for such an evaluation in an insightful manner
– Synchronized superposition of all the signal of interest viewed within a
particular signaling interval
Eye opening : interior region of the eye pattern
In the case of an M-ary system, the eye pattern contains (M-1) eye opening, where M
is the number of discreteamplitude levels
Interpretation of Eye Diagram:
UNIT IV: BAND PASS DIGITAL TRANSMISSSION Introduction, ASK, ASK Modulator, Coherent ASK detector, non-Coherent ASK
detector,
Band width frequency spectrum of FSK,
Non-Coherent FSK detector,
Coherent FSK detector,
FSK Detection using PLL,
BPSK, Coherent PSK detection, QPSK, Differential PSK
ASK, OOK, MASK:
• The amplitude (or height) of the sine wave varies to transmit the ones and zeros
• One amplitude encodes a 0 while another amplitude encodes a 1 (a form of amplitude
modulation)
Binary amplitude shift keying, Bandwidth:
• d ≥ 0-related to the condition of the line
B = (1+d) x S = (1+d) x N x 1/r implementation of binary ASK:
Frequency Shift Keying:
• One frequency encodes a 0 while another frequency encodes a 1 (a form of frequency
modulation)
st
FSK Bandwidth:
Acos2f2t
Acos2f2t
binary 1
binary 0
• Limiting factor: Physical capabilities of the carrier
• Not susceptible to noise as much as ASK
• Applications
– On voice-grade lines, used up to 1200bps
– Used for high-frequency (3 to 30 MHz) radio transmission
– used at higher frequencies on LANs that use coaxial cable
DBPSK:
• Differential BPSK
– 0 = same phase as last signal element
– 1 = 180º shift from last signal element
c
c
A cos 2f c t 11
4
st
A cos 2f t
A cos 2f t
3 01
4
3 00
4
A cos 2f t 10
c
4
Concept of a constellation :
M-ary PSK:
Using multiple phase angles with each angle having more than one amplitude, multiple
signals elements can be achieved
D R
L
R log 2 M
– D = modulation rate, baud
– R = data rate, bps
– M = number of different signal elements = 2L
– L = number of bits per signal element
QAM:
– As an example of QAM, 12 different phases are combined with two different
amplitudes
– Since only 4 phase angles have 2 different amplitudes, there are a total of 16
combinations
– With 16 signal combinations, each baud equals 4 bits of information (2 ^ 4 =
16) – Combine ASK and PSK such that each signal corresponds to multiple bits – More phases than amplitudes
– Minimum bandwidth requirement same as ASK or PSK
QAM and QPR:
• QAM is a combination of ASK and PSK
– Two different signals sent simultaneously on the same carrier frequency
– M=4, 16, 32, 64, 128, 256
• Quadrature Partial Response (QPR)
– 3 levels (+1, 0, -1), so 9QPR, 49QPR
Offset quadrature phase-shift keying (OQPSK):
• QPSK can have 180 degree jump, amplitude fluctuation
• By offsetting the timing of the odd and even bits by one bit-period, or half a symbol-
period, the in-phase and quadrature components will never change at the same time.
Generation and Detection of Coherent BPSK:
Figure 6.26 Block diagrams for (a) binary FSK transmitter and (b) coherent binary FSK
receiver.
Fig. 6.28
FFiigguurree 66..3300 ((aa)) IInnppuutt bbiinnaarryy sseeqquueennccee.. ((bb)) WWaavveeffoorrmm ooff ssccaalleedd ttiimmee
ffuunnccttiioonn ss11ff11((tt)).. ((cc)) WWaavveeffoorrmm ooff ssccaalleedd ttiimmee ffuunnccttiioonn ss22ff22((tt)).. ((dd))
Figure 6.29 Signal-space diagram for MSK system. Generation and Detection of MSK Signals:
Figure 6.31 Block diagrams for (a) MSK transmitter and (b) coherent MSK receiver.
UNIT V: INFORMATION THEORY
Information and entropy,
Conditional entropy and redundancy,
Shannon Fano coding
mutual, information,
Information loss due to noise,
Source codings,- Huffman code, variable length coding
Source coding to increase average information per bit,
Lossy source Coding.
INFORMATION THEORY AND CODING
TECHNIQUES Information sources
Definition: The set of source symbols is called the source alphabet, and the elements of
the set are called the symbols or letters.
The number of possible answers „ r ‟ should be linked to
“information.” “Information” should be additive in some sense.
We define the following measure of information:
Where „ r ‟ is the number of all possible outcome so far an do m message U.
Using this definition we can confirm that it has the wanted property of
additivity:
The basis „b‟ of the logarithm b is only a change of units without actually changing
the amount of information it describes.
Classification of information sources
1. Discrete memory less.
2. Memory.
Discrete memory less source (DMS) can be characterized by “the list of the symbols, the
probability assignment to these symbols, and the specification of the rate of generating
these symbols by the source”.
1. Information should be proportion to the uncertainty of an outcome.
2. Information contained in independent outcome should add.
Information content of a symbol:
Let us consider a discrete memory less source (DMS) denoted by X and having the
alphabet
{U1, U2, U3, ……Um}. The information content of the symbol xi, denoted by I(xi) is
defined as
I(U) = logb = - log b
P(U) Where P(U) is the probability of occurrence of symbol
U
Units of I(xi):
For two important and one unimportant special cases of b it has been agreed to use
the following names for these units:
b =2(log2):
bit,
b = e (ln): nat (natural
logarithm), b =10(log10):
Hartley.
The conversation of these units to other units is given as
log2a=
Uncertainty or Entropy (i.e Average information)
Definition:
In order to get the information content of the symbol, the flow information on the symbol
can fluctuate widely because of randomness involved into the section of symbols.
The uncertainty or entropy of a discrete random variable (RV) „U‟ is defined as
H(U)= E[I(u)]=
where PU(·)denotes the probability mass function (PMF)2 of the RV U, and where the
support of P U is defined as
We will usually neglect to mention “support” when we sum over PU(u) · logb PU(u),
i.e., we implicitly assume that we exclude all u With zero probability PU(u)=0.
Entropy for binary source
It may be noted that for a binary souce U which genets independent symbols 0 and 1 with
equal probability, the source entropy H(u) is
H(u) = - log2 - log2 = 1 b/symbol Bounds on H(U)
If U has r possible values, then 0 ≤ H(U) ≤ log r,
0 ≤ H(U) ≤ log r, Where
H(U)=0 if, and only if, PU(u)=1 for some u,
H(U)=log r if, and only if, PU(u)= 1/r ∀ u.
For all u ∈ supp(PU),i.e., PU(u)=1forall u ∈ supp(PU).
Formation theory: We take the deference and try to show that it must be non-positive.
Equality can only be achieved if
1. In the IT Inequality ξ =1,i.e.,if 1r·PU(u)=1=⇒ PU(u)= 1r ,for all u;
Note that if Condition1 is satisfied, Condition 2 is also satisfied.
Conditional Entropy
Similar to probability of random vectors, there is nothing really new about
conditional probabilities given that a particular event Y = y has occurred.
The conditional entropy or conditional uncertainty of the RV X given the event Y = y is
defined as
Note that the definition is identical to before apart from that everything is conditioned on
the event Y = y
Note that the conditional entropy given the event Y = y is a function of y. Since Y is also
a RV, we can now average over all possible events Y = y according to the probabilities of
each event. This will lead to the averaged.
• Forward Error Correction (FEC) – Coding designed so that errors can be corrected at the receiver – Appropriate for delay sensitive and one-way transmission (e.g., broadcast TV)
of data
– Two main types, namely block codes and convolutional codes. We will only
look at block codes
UNIT VI: CHANNEL CODING
Matrix description of linear block codes,
Matrix description of linear block codes,
Error detection and error correction capabilities of linear block codes
Cyclic codes: algebraic structure, encoding, syndrome calculation, decoding :CRC,
Golay codes, BCH codes, RS codes. Convolution codes.
Block Codes:
• We will consider only binary data
• Data is grouped into blocks of length k bits (dataword)
• Each dataword is coded into blocks of length n bits (codeword), where in general n>k
• This is known as an (n,k) block code
• A vector notation is used for the datawords and codewords,
– Dataword d = (d1 d2….dk)
– Codeword c = (c1 c2……..cn)
• The redundancy introduced by the code is quantified by the code rate,
– Code rate = k/n
– i.e., the higher the redundancy, the lower the code rate
Hamming Distance:
• Error control capability is determined by the Hamming distance
• The Hamming distance between two codewords is equal to the number of differences
between them, e.g.,
10011011 11010010 have a Hamming distance = 3
• Alternatively, can compute by adding codewords (mod 2)
=01001001 (now count up the ones)
• The maximum number of detectable errors is
dmin
1
• That is the maximum number of correctable errors is given by,
t d min 1
where dmin is the minimum Hamming distance between 2 codewords and means the
smallest integer
Linear Block Codes:
• As seen from the second Parity Code example, it is possible to use a table to hold all
the codewords for a code and to look-up the appropriate codeword based on the
supplied dataword
• Alternatively, it is possible to create codewords by addition of other codewords. This
has the advantage that there is now no longer the need to held every possible
codeword in the table.
• If there are k data bits, all that is required is to hold k linearly independent codewords,
i.e., a set of k codewords none of which can be produced by linear combinations of 2
or more codewords in the set.
• The easiest way to find k linearly independent codewords is to choose those which
have „1‟ in just one of the first k positions and „0‟ in the other k-1 of the first k
positions.
• For example for a (7,4) code, only four codewords are required, e.g.,
1 0 0 0 1 1 0
0 1 0 0 1 0 1
0 0 1 0 0 1 1
0 0 0 1 1 1 1
• So, to obtain the codeword for dataword 1011, the first, third and fourth codewords in
the list are added together, giving 1011010
• This process will now be described in more detail
• An (n,k) block code has code vectors
d=(d1 d2….dk) and
c=(c1 c2……..cn)
• The block coding process can be written as c=dG
where G is the Generator Matrix
a11 a12
... a1n a1
G a
21 a22 ... a
2 n a
2 . .
... . .
ak1 ak 2 ... akn a k
• Thus,
k
c di a i i 1
• ai must be linearly independent, i.e., Since codewords are given by summations of the ai vectors, then to avoid 2 datawords
having the same codeword the ai vectors must be linearly independent.
• Sum (mod 2) of any 2 codewords is also a codeword, i.e.,
Since for datawords d1 and d2 we have;
d3 d
1 d
2
So,
k k k k
c3 d3i a i (d1i d2i )a i d1i a i d2i a ii 1 i 1 i 1 i 1
c3 c1 c2Error Correcting Power of LBC:
• The Hamming distance of a linear block code (LBC) is simply the minimum
Hamming weight (number of 1‟s or equivalently the distance from the all 0
codeword) of the non-zero codewords
• Note d(c1,c2) = w(c1+ c2) as shown previously
• For an LBC, c1+ c2=c3
• So min (d(c1,c2)) = min (w(c1+ c2)) = min (w(c3))
• Therefore to find min Hamming distance just need to search among the 2k codewords
to find the min Hamming weight – far simpler than doing a pair wise check for all
possible codewords.
•
1 0 .. 0 0 1 .. 0
1
1
Linear Block Codes – example 1:
• For example a (4,2) code, suppose;
1 0 G
0 1
1 1
0
a1 = [1011] a2 = [0101]
• For d = [1 1], then;
1 0 1 1
0 1 0 1 c
1 1 1 0
Linear Block Codes – example 2:
• A (6,5) code wit h
1
0
G 0 0 0
0 0 0
1 0 0
0 1 0
0 0 1
0 0 0
0 1
0
0 1
0 1
1 1• Is an even single parity cod e
Systematic Codes:
• For a systematic block code the dataword appears unaltered in the codeword – usually
at the start
• The generator matrix has the structure,
p11
G p21
p12 ..
p22 ..
p1R
p
2 R I | P..
.. .. .. .. .. .. ..
0 0 .. 1 pk1 pk 2 .. pkR
R = n - k
• is often referred to as parity bits
I is k*k identity matrix. Ensures data word appears as beginning of codeword P is k*R matrix.
Decoding Linear Codes:
• One possibility is a ROM look-up table
• In this case received codeword is used as an address
• Example – Even single parity check code;
Address Data
000000 0
000001 1
000010 1
000011 0
……… .
• Data output is the error flag, i.e., 0 – codeword ok,
• If no error, data word is first k bits of codeword
• For an error correcting code the ROM can also store data words
• Another possibility is algebraic decoding, i.e., the error flag is computed from the
received codeword (as in the case of simple parity codes)
• How can this method be extended to more complex error detection and correction
codes?
Parity Check Matrix:
• A linear block code is a linear subspace S sub of all length n vectors (Space S) • Consider the subset S null of all length n vectors in space S that are orthogonal to all
length n vectors in S sub
• It can be shown that the dimensionality of S null is n-k, where n is the dimensionality of S and k is the dimensionality of S sub
• It can also be shown that S null is a valid subspace of S and consequently S sub is also
the null space of S null
• S null can be represented by its basis vectors. In this case the generator basis vectors (or „generator matrix‟ H) denote the generator matrix for S null - of dimension n-k = R
• This matrix is called the parity check matrix of the code defined by G, where G is obviously the generator matrix for S sub - of dimension k
• Note that the number of vectors in the basis defines the dimension of the subspace • So the dimension of H is n-k (= R) and all vectors in the null space are orthogonal to
all the vectors of the code
• Since the rows of H, namely the vectors bi are members of the null space they are
orthogonal to any code vector
• So a vector y is a codeword only if yHT=0
• Note that a linear block code can be specified by either G or H
Parity Check Matrix:
b11 b12
... b1n b1
H b21 b22 ... b2 n
b2 R = n - k .
. ... . .
bR1 bR 2 ... bRn bR
• So H is used to check if a codeword is valid,
• The rows of H, namely, bi, are chosen to be orthogonal to rows of G, namely ai
• Consequently the dot product of any valid codeword with any bi is zero
This is so since,
k
c di a i i 1
and so,
k
b j .c b
j . di
a i
i 1
k
di (a
i .b
j ) 0
i 1
• This means that a codeword is valid (but not necessarily correct) only if cHT = 0. To
ensure this it is required that the rows of H are independent and are orthogonal to the
rows of G • That is the bi span the remaining R (= n - k) dimensions of the codespace
• For example consider a (3,2) code. In this case G has 2 rows, a1 and a2
• Consequently all valid codewords sit in the subspace (in this case a plane) spanned by
a1 and a2
• In this example the H matrix has only one row, namely b1. This vector is orthogonal to the plane containing the rows of the G matrix, i.e., a1 and a2
• Any received codeword which is not in the plane containing a1 and a2 (i.e., an invalid codeword) will thus have a component in the direction of b1 yielding a non- zero dot product between itself and b1.
Error Syndrome:
• For error correcting codes we need a method to compute the required correction
• To do this we use the Error Syndrome, s of a received codeword, cr
s = crHT
• If cr is corrupted by the addition of an error vector, e, then
cr = c + e
and
s = (c + e) HT = cHT + eHT
s = 0 + eHT
Syndrome depends only on the error
• That is, we can add the same error pattern to different code words and get the same
syndrome.
– There are 2(n - k) syndromes but 2n error patterns
– For example for a (3,2) code there are 2 syndromes and 8 error patterns
– Clearly no error correction possible in this case
– Another example. A (7,4) code has 8 syndromes and 128 error patterns.
– With 8 syndromes we can provide a different value to indicate single errors in
any of the 7 bit positions as well as the zero value to indicate no errors • Now need to determine which error pattern caused the syndrome
• For systematic linear block codes, H is constructed as follows,
G = [ I | P] and so H = [-PT | I]
where I is the k*k identity for G and the R*R identity for H
• Example, (7,4) code, dmin= 3
1 0 0 0 0 1 10 1 0 0 1 0
1
0
r
1
1
G I | P0 0
0 1 0 1
0 0 1 1
1
1 0
1
0
H - PT | I
1
1
1 1 1 1 0
0 1 1 0 1
1 0 1 0 0
0
1
Error Syndrome – Example:
• For a correct received codeword cr = [1101001]
In this case,
s c HT 1 1 0 1 0
0
1 1
0 1 1
1 1
0
1 0
1 1 0 0 0
0 0Standard Array:
0 1 0
• The Standard Array is constructe0d as0foll1ows,
1 0 0 0 0 1 10 1 0 0 1 0
0 0 1 0 1 1 0
0 0 0 1 1 1 1
1 1 1 0 0 0 01 0 0 1 1 0
0 1 0 1 0 1 0
1 1 0 1 0 0 1
0
0 1
c1 (all zero) c2 …… cM s0
e1
e2
e3
… eN
c2+e1
c2+e2
c2+e3
…… c2+eN
…… …… …… …… ……
cM+e1
cM+e2
cM+e3
…… cM+eN
s1
s2
s3
… sN
• The array has 2k columns (i.e., equal to the number of valid codewords) and 2R rows
(i.e., the number of syndromes)
Hamming Codes:
• We will consider a special class of SEC codes (i.e., Hamming distance = 3) where,
– Number of parity bits R = n – k and n = 2R – 1
– Syndrome has R bits
– 0 value implies zero errors
– 2R – 1 other syndrome values, i.e., one for each bit that might need to be
corrected
– This is achieved if each column of H is a different binary word – remember s
= eHT
• Systematic form of (7,4) Hamming code is,
G I | P
1 0
H - PT | I
1
1
1 1 1 1 0
0 1 1 0 1
1 0 1 0 0
0
1
• The original form is non-systematic,
G 0
0
H 1
0 0 1 1
1 1 0 0
0 1 0 1
1 1
1
0 1
• Compared with the systematic code, the column orders of both G and H are swapped
so that the columns of H are a binary count
• The column order is now 7, 6, 1, 5, 2, 3, 4, i.e., col. 1 in the non-systematic H is col. 7
in the systematic H.
Transmission and Storage Transmission and Storage Introduction
◊ A major concern of designing digital data transmission and storage Systems is the control
of errors so that reliable reproduction of data systems is the control of errors so that reliable
reproduction of data can be obtained.
◊ In 1948, Shannon demonstrated that, by proper encoding of the information, errors induced
by a noisy channel or storage medium can be reduced to any desired level without sacrificing
the rate of information transmission or storage, as long as the information rate is less than the
capacity of the channel.
◊ A great deal of effort has been expended on the problem of devising efficient encoding and
decoding methods for error control in a noisy environment
Typical Digital Communications Systems ◊ Block diagram of a typical data transmission or storage system
Types of Codes
◊ There are four types of codes in common use today: ◊ Block codes
◊ Convolutionalcodes
◊ Turbo codes
◊ Low-Density Parity-Check (LDPC) Codes
◊ Block codes
◊ The encoder for a block code divides the information sequence
into message blocks of k information bits each. ◊ A message block is represented by the binary k-tuple ( )lld u=(u1,u2,…,uk) called a
message. ◊ There are a total of 2k different possible messages.
Block Codes
◊ Block codes (cont.) ◊ The encoder transforms each message u into an n-tuple
◊ The encoder transforms each message u into an n-tuple
v=(v1,v2,…,vn) of discrete symbols called a code word.
◊ Corresponding to the 2k different possible messages, there are 2k different possible code
words at the encoder output.
◊ This set of 2k code words of length n is called an (n,k) block code. ◊ The ratio R=k/n is called the code rate.
◊ n-k redundant bits can be added to each message to form a code word
◊ Since the n-symbol output code word depends only on the corresponding k-bit input
message, the encoder is memoryless, and can be implemented with a combinational logic
circuit.
Block Codes
◊ Binary block code with k=4 and n=7 6Finite Field (Galois Field) Finite Field (Galois Field)
◊ Much of the theory of linear block code is highly mathematical in nature and requires an
extensive background in modern algebra nature, and requires an extensive background in
modern algebra.
◊ Finite field was invented by the early 19th century mathematician, ◊ Galois was a young French math whiz who developed a theory of finite fields, now know as
Galois fields, before being killed in a duel at the age of 21. ◊ For well over 100 years, mathematicians looked upon Galois fields as elegant mathematics
but of no practical value.
Convolutional Codes
◊ The encoder for a convolutional code also accepts k-bit blocks of the information sequence
u and produces an encoded sequence (code word) v of n-symbol blocks.
◊ Each encoded block depends not only on the corresponding k-bit message block at the same
time unit, but also on m previous message blocks. Hence the encoder has a memory order of
m message blocks. Hence, the encoder has a memory order of m.
◊ The set of encoded sequences produced by a k-input, n-output encoder of memory order m
is called an (n, k, m) convolutional y ( , , ) code.
◊ The ratio R=k/n is called the code rate. ◊ Since the encoder contains memory, it must be implemented with a sequential logic circuit.
◊ Binary convolutional encoder with k=1, n=2, and m=2
◊ Memorylesschannels are called random-error channels.
Transition probability diagrams for binary symmetric channel (BSC).1.5 Types of Errors 1.5
Types of Errors ◊ On channels with memory, the noise is not independent from Transmission to transmission
◊ Channel with memory are called burst-error channels.
Simplified model of a channel with memory.1.6 Error Control Strategies 1.6 Error Control
Strategies ◊ Error control for a one-way system must be accomplished using
Forward error correction (FEC) that is by employing error- forward error correction (FEC),
that is, by employing error correcting codes that automatically correct errors detected at the
receiver. ◊ Error control for a two-way system can be accomplished using error detection and
retransmission, called automatic repeat request (ARQ).
This is also know as the backward error correction (BEC). ◊ In an ARQ system, when errors are detected at the receiver, a request is sent
For the transmitter to repeat the message and this continues until the message for the
transmitter to repeat the message, and this continues until the message is received correctly.
◊ The major advantage of ARQ over FEC is that error detection requires much simpler
decoding equipment than does error correction.
151.6 Error Control Strategies 1.6 Error Control Strategies
◊ ARQ is adaptive in the sense that information is retransmitted only when errors occur when
errors occur.
◊ When the channel error rate is high, retransmissions must be sent too frequently, and the
system throughput, the rate at which newly generated messages are correctly received, is
lowered by ARQ.
◊ In general, wire-line communications (more reliable) adopts BEC scheme, while wireless
communications (relatively unreliable) adopts FEC scheme.
Error Detecting Codes Error Detecting Codes ◊ Cyclic Redundancy Code (CRC Code) –also know as the polynomial code polynomial
code.
◊ Polynomial codes are based upon treating bit strings as representations of polynomials with
coefficients of 0 and 1 only.
◊ For example, 110001representsasix-termpolynomial:x5+x4+x0
◊ When the polynomial code method is employed, the sender and receiver must agree upon a
generator polynomial, G(x), in advance.
◊ To compute the checksum for some frame with m bits, corresponding to the polynomial
M(x), the frame must be longer than the generator polynomial. Error Detecting Codes
◊ The idea is to append a checksum to the end of the frame in such a way that the polynomial
represented by the check summed frame divisible by G(x).
◊ When the receiver gets the checksummed frame, it tries dividing it by G(x). If there is a
remainder, there has been a transmission error.
◊ The algorithm for computing the checksum is as follows: Calculation of the polynomial code checksum Calculation of the polynomial code checksum
Calculation of the polynomial code checksum Calculation of the polynomial code checksum
Convolution Codes
Encoding,
Decoding using state Tree and trellis diagrams,
Decoding using Viterbi algorithm,
Comparison of error rates in coded and uncoded transmission.
Introduction:
• Convolution codes map information to code bits sequentially by convolving a
sequence of information bits with “generator” sequences
• A convolution encoder encodes K information bits to N>K code bits at one time step
• Convolutional codes can be regarded as block codes for which the encoder has a
certain structure such that we can express the encoding operation as convolution
• Convolutional codes are applied in applications that require good performance with
low implementation cost. They operate on code streams (not in blocks)
• Convolution codes have memory that utilizes previous bits to encode or decode
following bits (block codes are memoryless)
• Convolutional codes achieve good performance by expanding their memory depth
• Convolutional codes are denoted by (n,k,L), where L is code (or encoder) Memory
depth (number of register stages)
• Constraint length C=n(L+1) is defined as the number of encoded bits a message bit
can influence to
• Convolutional encoder, k = 1, n = 2, L=2
– Convolutional encoder is a finite state machine (FSM) processing
information bits in a serial manner
– Thus the generated code is a function of input and the state of the FSM
– In this (n,k,L) = (2,1,2) encoder each message bit influences a span of C=
n(L+1)=6 successive output bits = constraint length C
– Thus, for generation of n-bit output, we require n shift registers in k = 1
convolutional encoders
m m j
m
x ' m m mj j 3 j 2 j
x ''
j j 3
j 1
m j
m
x '''
j j 2
Here each message bit influences a span of C = n(L+1)=3(1+1)=6
successive output bits
Convolution point of view in encoding and generator matrix:
Example: Using generator matrix
g(1)
g ( 2 )
[1 0 1 1]
[1 1 1 1]
m
m
m j
m
j m
Representing convolutional codes: Code tree:
(n,k,L) = (2,1,2) encoder
x ' j j 2
j 1
x ''
j j 2
x x ' x '' x ' x '' x ' x '' ...out 1 1 2 2 3 3
Representing convolutional codes compactly: code trellis and state diagram:
State diagram
Inspecting state diagram: Structural properties of convolutional codes:
• Each new block of k input bits causes a transition into new state
• Hence there are 2k branches leaving each state
• Assuming encoder zero initial state, encoded word for any input of k bits can thus be
obtained. For instance, below for u=(1 1 1 0 1), encoded word v=(1 1, 1 0, 0 1, 0 1, 1
1, 1 0, 1 1, 1 1) is produced:
•
- encoder state diagram for (n,k,L)=(2,1,2) code - note that the number of states is 2L+1 = 8
Distance for some convolutional codes:
THE VITERBI ALGORITHEM:
• Problem of optimum decoding is to find the minimum distance path from the initial
state back to initial state (below from S0 to S0). The minimum distance is the sum of
all path metrics
• that is maximized by the correct path
• Exhaustive maximum likelihood
method must search all the paths
in phase trellis (2k paths emerging/
entering from 2 L+1 states for
an (n,k,L) code)
• The Viterbi algorithm gets its
efficiency via concentrating intosurvivor paths of the trellis•
THE SURVIVOR PATH:
• Assume for simplicity a convolutional code with k=1, and up to 2k = 2 branches can
enter each state in trellis diagram
• Assume optimal path passes S. Metric comparison is done by adding the metric of S
into S1 and S2. At the survivor path the accumulated metric is naturally smaller
(otherwise it could not be the optimum path)
• For this reason the non-survived path can be discarded -> all path alternatives need not
to be considered • Note that in principle whole transmitted
sequence must be received before decision.
However, in practice storing of states for
input length of 5L is quite adequate
The maximum likelihood path:
The decoded ML code sequence is 11 10 10 11 00 00 00 whose Hamming
distance to the received sequence is 4 and the respective decoded
sequence is 1 1 0 0 0 0 0 (why?). Note that this is the minimum distance path.
(Black circles denote the deleted branches, dashed lines: '1' was applied)
How to end-up decoding?
• In the previous example it was assumed that the register was finally filled with zeros
thus finding the minimum distance path
• In practice with long code words zeroing requires feeding of long sequence of zeros to
the end of the message bits: this wastes channel capacity & introduces delay
• To avoid this path memory truncation is applied:
– Trace all the surviving paths to the
depth where they merge
– Figure right shows a common point
at a memory depth J – J is a random variable whose applicable
magnitude shown in the figure (5L) has been experimentally tested for negligible error rate increase
– Note that this also introduces the delay of 5L!
J 5L stages of the trellis
Hamming Code Example:
• H(7,4)
• Generator matrix G: first 4-by-4 identical matrix
• Message information vector p
• Transmission vector x
• Received vector r
and error vector e
• Parity check matrix H
Error Correction:
• If there is no error, syndrome vector z=zeros
• If there is one error at location 2
• New syndrome vector z is
11 00 01 11 01 11 10
Example of CRC:
Example: Using generator matrix:
g(1)
g ( 2 )
[1 0 1 1]
[1 1 1 1]
01
correct:1+1+2+2+2=8;8 (0.11) 0.88
false:1+1+0+0+0=2;2 (2.30) 4.6
total path metric: 5.48
Turbo Codes:
• Backgound
– Turbo codes were proposed by Berrou and Glavieux in the 1993 International
Conference in Communications.
– Performance within 0.5 dB of the channel capacity limit for BPSK was
demonstrated.
• Features of turbo codes
– Parallel concatenated coding
– Recursive convolutional encoders
– Pseudo-random interleaving
– Iterative decoding
Motivation: Performance of Turbo Codes
• Comparison:
– Rate 1/2 Codes.
– K=5 turbo code.
– K=14 convolutional code.
• Plot is from:
– L. Perez, “Turbo Codes”, chapter 8 of Trellis Coding by C. Schlegel. IEEE
Press, 1997
Pseudo-random Interleaving:
• The coding dilemma:
– Shannon showed that large block-length random codes achieve channel
capacity.
– However, codes must have structure that permits decoding with reasonable
complexity. – Codes with structure don‟t perform as well as random codes. – “Almost all codes are good, except those that we can think of.”
• Solution:
– Make the code appear random, while maintaining enough structure to permit
decoding.
– This is the purpose of the pseudo-random interleaver.
– Turbo codes possess random-like properties.
– However, since the interleaving pattern is known, decoding is possible.
Why Interleaving and Recursive Encoding?
• In a coded systems:
– Performance is dominated by low weight code words.
• A “good” code:
– will produce low weight outputs with very low probability.
• An RSC code:
– Produces low weight outputs with fairly low probability. – However, some inputs still cause low weight outputs.
• Because of the interleaver:
– The probability that both encoders have inputs that cause low
weight outputs is very low. – Therefore the parallel concatenation of both encoders will produce
a “good” code.
Iterative Decoding:
• There is one decoder for each elementary encoder.
• Each decoder estimates the a posteriori probability (APP) of each data
bit. • The APP‟s are used as a priori information by the other decoder. • Decoding continues for a set number of iterations.
– Performance generally improves from iteration to iteration, but
follows a law of diminishing returns
4. Assignment Questions Explain the basic elements of a Digital Communication Systems.
Explain the generation and Detection of Pulse-Code modulation System
Enlist the advantages of Delta Modulation over PCM.
Explain the necessity of Adaptive DPCM.
A binary PSK signal is applied to a correlator filter supplied with a phase reference
that differs from the exact carrier phase by pi radians. Determine the effects of phase
error on the average probability of symbol error of the system .
An FSK system transmits binary data at the rate of 2.5*10^6 bites/sec. During the
corse of transmission a white Gaussian of zero mean and power spectral density 10^-
20 W/Hz is added to the signal in the absence of noise the amplitude of sinusoidal
wave for digit 1 or 0 is 1 m/v. detremine the average probibility of symbol error
assuming coherrent detection .
3.a) In a coherrent FSK system ,the signal S1(t) and S2(t) representing symbols 1 and 0 ,respectively are
defined by S1(t),S2(t)= Ac cos[2pi(fc+_del(f)/2)t} 0<_t<_Tb . Assuming that Fc>(del F), show that the
correlation co efficient of the signal S1(T) and S2(t) is approximately given by
b) what is the minimum value of frequency shift del(F) for which the signal S1(t) and S2(t) are orthogonal?
c) what is the value of del (f) that minimises the average probability of symbol error?
d) for the value of del(f) obtained in part(c), determine the increase in Eb/N0 required so that this coherent FSK
system has the noise performance as a coherent binding PSK system.
4.Binary data are transmitted over a microwave link at the rate of 10^6 b/s,and the power spectral density of
noise at the receiver input is 10^-10W/HZ.Find the carrier power required to maintain an average probability
i.e.,Pe>=10^-14 for coherent binary FSK.What is the required channel value?
5.Write the properties of mutual information?
6.State and prove the properties of differential entropy?
7.State and prove information/channel capacity theorem?
8.Differentiate among entropy,conditional entropy and differential entropy?
9.A discrete memoryless source emits each of 5 possible for every signaling intrerval.If these symbols are
encoded in 4 different ways as shown in below table.
Then identify which of these codes are prefix codes and also justify athe answer?
o Code-I Code-II Code-III Code-IV
S0 1 1 1 11
S1 01 10 10 10
S2 001 110 100 01
S3 0001 1101 001 001
S4 0000 1100 000 000
10.A source emits each of 4 possible symbols S0,S1,S2,S3 for each signalling interval.The possibilities of these
symbols are 1/3,1/6,1/4,1/4 repectively.
Find the amount of information gained by observing.The source emitting each of these symbols?Also calculate
the entropy?
11.Write a short note an non-coherent binary modulation techniques?
12.Sketch the waveform of inphase and quadrature phase componentsof QPSK signal produced by the input
binary sequence 010000101000 and also sketch the waveform of QPSK signal?
13.Consider a discrete memoryless source whose alphabet consists of k equiportable symbol.
o What conditions have to be satisfied by k and the code-word length for the coding efficiency to be 100
percent?
14.Write a short note on non-coherent binary modulation technique. Derive an expression for symbol error
probability of non-coherent PSK?
15.What do you understand by the Union Bound?
6. COURSE OUTCOMES
Course Outcomes:
The Engineering Graduates will be able to
1. Understand the basic process of analog to digital conversion by applying the basics of Fourier Transform.
2. Formulate the merits and demerits of various digital modulation systems in order to evaluate their
performance based on output Signal-to-Noise ratio [SNR] and transmission bandwidth.
3. Apply the knowledge of digital electronics and signals & systems to evaluate Power spectral density
[PSD] and Error Probability [Pe] of various digital modulation techniques [binary and m-ary].
4. Design a digital communication system with error control sub-systems by applying various coding
Techniques.
5. Choose and design appropriate modulation and demodulation system for the given specifications of
application.
PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
CO1 3 2
CO2 2 3 3 2
CO3 3 3 2 1
CO4 1 3 2
CO5 3 3
