NPTEL COURSE MATERIAL
Course:
Digital Communication
Course Contents (Video)
Prof.Bikash Kumar Dey
Department of Electrical Engineering
IIT Bombay, Powai
Mumbai 400 076, India
Subject Expert:
P. Nagaraju
Associate Professor
R V College of Engineering
Bangalore.
Page 1 of 103
Video No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Topics
Introduction to Digital Communication
Sampling
Quantization , PCM and Delta Modulation
Probability and Random Process
Probability and Random Processes (Part - 2)
Channels and their Models
Channels and their Models (Part -2)
Information Theory (Part - 1)
Information Theory (Part - 2)
Bandpass Signal Representation (Part 1)
Bandpass Signal Representation (Part - 2)
Digital Modulation Techniques (Part - 1)
Digital Modulation Techniques (Part - 2)
Digital Modulation Techniques (Part - 3)
Digital Modulation Techniques (Part - 4 )
Digital Modulation Techniques (Part – 5)
Digital Modulation Techniques (Part - 6)
Digital Modulation Techniques (Part - 7)
Digital Modulation Techniques (part - 8)
Digital Modulation Techniques (Part - 9)
Digital Modulation Techniques (Part - 10)
Probability of Error Calculation
Calculation of Probability of Error
Calculation of Probability of Error
Equalizers
Source Coding (Part - 1)
Source Coding (Part - 2)
Source Coding (Part – 3)
Source Coding (Part - 4)
Channel Coding
Fundamentals of OFDM
Conclusion
Page 2 of 103
Module-1
Page 3 of 103
1.1) NPTEL Video Link
Sl.
Module
No.
No.
1
Mod 01
Lecture
No.
Lec-01
2
Mod 01
Lec-02
3
Mod 01
4
Module-1: Video Lecture number 1 to 7
Topics covered
Video Link
Introduction to Digital
Communication Systems
(54:29 mins)
Sampling, Sampling Theorem
Spectrum, Recovery of signal.
(53:16 mins)
http://nptel.ac.in/courses/117101051/1
Lec-03
Quantization , PCM and Delta
Modulation (51:43 mins)
http://nptel.ac.in/courses/117101051/3
Mod 01
Lec-04
Probability and Random Process
(55:20 mins)
http://nptel.ac.in/courses/117101051/4
5
Mod 01
Lec-05
Probability and Random Processes
(Part - 2)
(59:25 mins)
http://nptel.ac.in/courses/117101051/5
6
Mod 01
Lec-06
Channels and their Models
(54:38 mins)
http://nptel.ac.in/courses/117101051/6
7
Mod 01
Lec-07
Channels and their Models (Part -2)
(53:14 mins)
http://nptel.ac.in/courses/117101051/7
http://nptel.ac.in/courses/117101051/2
NPTEL Web Link:
http://nptel.ac.in/courses/Webcourse-contents/IIT%20Kharagpur/Digi%20Comm/New_index1.html
Page 4 of 103
1.2) Questions:
Questions from Video Lectures of NPTEL
Sl no
1
Questions
Give some examples for the communication systems which use ‘space’ as the
Video
Time in
Number
Minutes
1
2
1
4
1
2-5
channel.
2
Give some examples for the communication systems which use wire-line
channel.
3
Mention the source and destination for each of the following communication
systems:
(i) Radio
(ii) Television
(iii) Telephone system
(iv) Cellular Mobile system
4
What are Storage Channels? Give some examples.
1
5
5
Name the two types of sources used in digital communication.
1
6
6
Mention the two basic steps involved in the conversion of audio signal to
1
9
1
11
digital/binary form.
7
Explain the characteristics of different channels used in digital communication
systems.
8
Describe the different types of noise encountered in communication systems.
1
14
9
What is meant AWGN channel model?
1
15
10
Why frequent characters were represented by short sequences in Morse
1
19
code?
11
Explain the International Morse code with some examples.
1
23
12
What are the fundamental functional blocks of a digital communication
1
26-42
1
43
system? Explain the functions of each block.
13
What are the advantages of Digital Communication over analog
communication?
14
What are the Resources and constraints in digital communication system?
1
44-50
15
Differentiate continuous time signal and discrete time signal.
2
3
16
Define Sampling period and sampling frequency.
2
4
Page 5 of 103
17
How to recover the analog signal from its discrete time signal.
2
5
18
What is a band limited signal? Define the spectrum of a band limited signal.
2
7
19
With a neat spectrum of an analog signal and spectrum of the sampled signal,
2
13
explain the process of sampling analog signal for different sampling rates.
20
What is meant by Aliasing? How it is overcome?
2
18
21
Show that the continuous time signal cannot be recovered from its samples if
2
19
2
20
2
31
2
34
2
39
2
42
the sampling rate is less than the Nyquist rate.
22
Explain the concept of sampling and recovery of the signal from its samples
using sinusoidal signal.
23
How to reconstruct x(t) from x[n] using Interpolation principle? What type of
filter is used?
24
Consider the impulse response h(t) of the reconstruction filter used in the
recovery of the signal x(t) from its samples x[n] as given below: Explain the
process of recovery of the signal x(t) using this reconstruction filter.
25
Describe the recovery of x(t) using filter with impulse response
h(t) = Sinc[ fs( t – nTs) ].
26
Consider the impulse response h(t) of the reconstruction filter used in the
recovery of the signal x(t) from its samples x[n] is h(t) = Sinc(fs t). Derive the
expression for the perfect reconstruction for sampling rate of more than
twice the Nyquist rate.
27
What is meant by Stochastic Process? Define with an example.
2
47
28
What is uniform quantization? How the number of bits/samples, the number
3
3-6
of quantization levels and step size are related?
29
What is quantization noise?
3
4
30
Define the principle of non-uniform quantization.
3
8
31
Draw the input-output characteristics of compressor and expander in a Non-
3
16
uniform Quantizer and explain.
32
tŝƚŚŵĂƚŚĞŵĂƚŝĐĂůĞdžƉƌĞƐƐŝŽŶ͕ĞdžƉůĂŝŶƚŚĞʅ-law companding scheme.
3
19
33
What are the different functional blocks in a PCM system? Explain their
3
20
Page 6 of 103
functions.
34
What is DPCM? What is the basic principle of DPCM scheme?
3
21
35
Why Delta modulation is called one-bit DPCM? Justify your answer.
3
23
36
What are the two types of distortions occur in delta modulation schemes?
3
31
37
What is basic principle of Adaptive delta modulation scheme? Explain with
3
35-40
the block diagrams of modulator and demodulator.
38
How the granular noise is reduced by adaptive delta modulation?
3
44
39
How the Slope overload distortion can be reduced by adaptive delta
3
45
4
2
4
3-6
modulation.
40
Define the following:
(i) Random experiment and (ii) Sample space.
41
For each of the following experiments write the sample space:
(i) Tossing a fair coin
(ii) Rolling of a dice
Also write the probabilities of all the outcomes.
42
Write the sample space for Noise voltage observed at the ends of a resistor?
4
3
43
Define the following with respect to a random experiment:
4
8
4
11
4
14-19
(i) Event (ii) Complement of an event (iii) Union of events.
44
Consider two experiments, experiment-1 and experiment-2 with sample
spaces S1= { x 1 , x 2 , x 3 , …… x M } and S2 = { y 1 , y 2 , y 3 , …… y N } . Find the
following probabilities – p(x i ) and p(y j ).
45
Consider two events; event-A and event-B. Define the following:
(i) Conditional Probability
(ii) Marginal Probability
(iii) Chain rule
(iv) Statistically Independent.
46
Differentiate discrete random variable and continuous random variable?
4
23
47
Define the cumulative distribution function (cdf) of a random variable. For the
4
29-32
4
33
4
34-38
experiment tossing of a coin, sketch the cdf.
48
Consider the random experiment: rolling of a dice. Compute the cdf assuming
that the probability of each outcome is same.
49
Define probability density function, p X (x).
Page 7 of 103
50
The probability density function of a random variable X is,
4
39
4
41
4
43-51
4
51
p X (x) = 0.5 for a < x < b and
zero otherwise.
Compute and sketch the cumulative distribution function, F X (x).
51
When a random variable is called a Gaussian random variable? Define the
probability density function and cumulative distribution function of a
Gaussian random variable. Sketch them.
52
Consider two random variables X1 and X2. Define the following:
(i) Joint distribution function
(ii) Cumulative density function
(iii) Marginal density functions.
(iv) Conditional probability distribution functions
(v) Conditional density functions.
53
Consider random variables X 1 , X 2 , X 3 …… X N . Under what conditions these
variables are called independent random variables.
54
Define function of random variables.
5
2
55
Let X and Y be two random variables and Y=g(X) = aX + b, where ‘a’ and ‘b’ are
5
4
constants.
(i) Express cumulative distribution F Y (y) in terms of F X (y).
(ii) Find the relation between the density functions.
56
Develop the Jacobian of the transformation.
5
10
57
How the density function of a new variable by using density function of the
5
11
other variable?
58
Define the mean value nth order moments and variance of a random variable.
5
14-17
59
What is meant by joint moments and covariance of the random variables?
5
18
60
What is Random Process? How are they classified?
5
29
61
What is meant by stationarity? What are WSS and SSS?
5
34
62
What is Autocorrelation of a random Process? If X(t) is stationary, then what
5
38
is the auto correlation of X(t)?
63
For a complex valued random process define the autocorrelation.
5
42
64
What is meant by independent random process and uncorrelated random
5
43
5
45
process?
65
Explain the relation between autocorrelation function and power density
spectrum of a random process.
Page 8 of 103
66
What are the different types of physical channels? Classify them with
6
2-7
examples.
67
Describe an additive noise channel.
6
14
68
What are the basic channel models? Explain their features.
6
18
69
Compare the features of binary symmetric channel and a binary asymmetric
6
20, 22
6
27
6
28
6
32
channel.
70
Consider three different binary symmetric channels, BSC-1, BSC-2 and BSC-3
with probability of error p=0.01, p=0.3 and p=0.9 . Compare the performance
characteristics of these channels.
71
A binary symmetric channel has probability of error, p=0.9. How its
performance is improved with the use of inverter?
72
The input voltage levels of a binary erasure channel are +5 and -5. The output
symbols are 1, 0 and e. Suggest suitable voltage levels to represent these
three output possibilities.
73
Write the channel diagram for a binary error and erasure channel.
6
37
74
What is a Discrete memory less channel (DMC)? Explain with M-input
6
38-43
alphabets and N-output alphabets. Also define conditional probability.
75
Can we transmit more information through DMC with feedback?
6
47
76
Compare the features of continuous-time and discrete-time channels.
7
2-6
77
With a simple model explain the discrete form of Additive White Gaussian
7
6
Noise channel.
78
Describe the Continuous time AWGN channel model with a diagram.
7
10
79
Explain the Gaussian Noise channel with memory.
7
13
80
What is meant by linear filter channel? Explain how the signals are related in
7
17
7
25
this model of channel.
81
For a linear filter channel based communication, how the signal is recovered?
Explain using the convolution.
82
Describe the features of colored Gaussian noise.
7
28
83
Explain the role of whitening filter in the receiver to remove colored Gaussian
7
29
noise.
84
What are the different ways in which the signal travels in wireless channels?
7
31
85
Mention the different features of fading channel/multipath channel.
7
33
Page 9 of 103
86
Explain the following phenomenon:
7
35
(i) Reflection
(ii) Defraction
(iii) Scattering
87
Explain the different ways the signal travels in a multi path channel.
7
33-39
88
Describe how the gain of the signal changes in the fading environment.
7
41
Page 10 of 103
1.3) Quiz Questions
Questions: Fill in the blanks.
Q.
Question
Answer
No.
1
The channel for Radio communication system is ________.
Space
2
The channel used for Television signal transmission is ________.
Space
3
The channel used for the plain old telephone system is _________
Wire-line
4
The channel for cellular mobile communication system is ________.
Space
5
The noise which corrupts the signal in the channel is _________
Random
nature.
6
The most pre-dominant type of noise in communication system is
Thermal noise/
_____.
Gaussian noise.
7
The statistical parameters of noise are ________.
Mean, PSD.
8
The two major resource constraints digital communication system are
Bandwidth
_________ and _____________.
Transmitted Power
9
The most crucial channel parameter is ____________.
Noise Variance
10
Non-uniform quantizer device consists of ________ and _________.
Nonlinear device and
uniform quantization.
11
The minimum sampling rate required to sample the following signal, 8530 samples/sec
x(t) = 10 ^ŝŶ;ϯϱϴϬʋƚͿнϱŽƐ;ϴϱϯϬʋƚͿ is ___________.
12
The minimum sampling rate required to sample the following signal, 12110 samples/sec
dž;ƚͿсϭϬ^ŝŶ;ϯϱϴϬʋƚͿ͘ŽƐ;ϴϱϯϬʋƚͿ is _____________.
13
An analog low pass signal x(t) is bandlimited to 2500Hz. The maximum 0.2 milli secs
sampling period that can be used is _________, is the x(t) is to be
recovered without distortion, form its samples
14
A lowpass signal x(t) is bandlimited to W Hz and having a spectrum
Sampling rate
X(f). The ideally sampled version, x ɷ (t), of x(t), will have a spectrum
X ɷ (f) which is a periodic repetition of X(f) with a period of ______.
15
A lowpass bandlimited signal, sampled at a frequency higher than the Low pass filter
Nyquist rate, may be recovered from its samples by passing them
through a ____________.
16
The minimum sampling rate f s to be used for sampling a continuous-
Page 11 of 103
Nyquist Rate
17
time lowpass signal x(t), for proper recovery of x(t) from its samples, is
and
called _____ and it is equal to ___________.
fs= 2fm.
The phenomenon where by the higher frequency components of
Aliasing
signal x(t) reappear as low frequency components in the spectrum of
the sampled version of x(t) is called ___________.
18
Anti-aliasing filter is lowpass filter used before the sampling
Aliasing
phenomenon to overcome __________.
19
The three steps in the digitization of an analog message signal are Sampling
_______, ________ and _____
Quantization
Encoding
20
The distortion in the received signal caused by flat-top sampling is Aperture effect
called _______.
21
A lowpass signal bandlimited to 1200 Hz was sampled and it was 1400 sps
found that the 1000 Hz frequency component was re-appearing in the
recovered signal, because of aliasing, as 400 Hz component. The
sampling frequency used is __________.
22
Consider a signal x(t) = 3 + 5cos2;ϰϱϬϬʋƚͿŝƐƐĂŵƉůĞĚĂƚƌĞŐƵůĂƌ
0.222 milli secs
intervals of ‘T’ second. The maximum value of T for which x(t) may be
recovered from the sampled version without any distortion, is equal
to _________.
23
ĐŽƐŝŶƵƐŽŝĚĂůƐŝŐŶĂůdž;ƚͿсϯƐŝŶ;ϮϰϬʋƚͿǁĂƐƐĂŵpled at a frequency f s . 150 Hz
The signal recovered from the samples was, however, found to be
ϯ^ŝŶ;ϭϴϬʋƚͿ͘dŚĞƐĂŵƉůŝŶŐĨƌĞƋƵĞŶĐLJĨ s is equal to ___________.
24
A binary PCM system with 256 quantizing levels has a sampling 56 k bps
frequency of 7 kHz. The bit rate of the system is _________.
25
The number of quantization levels used in a uniform quantizer is ‘L’
levels and the number of bits used to represent a sample is ‘R’ bits. L= 2R or L=2R-1.
Write the relation between L and R.
26
ȴ2/12
The mean square error of the quantization noise in a uniform
quantizer is equal to _____.
27
The two types of uniform quantizers are ________ and _________.
Mid rise and Midtread
28
In a mid-rise type quantizer the number of quantization levels is ___.
EVEN
29
In a mid-tread type quantizer the number of quantization levels is ___. ODD
30
In the mid-tread type of quantizer, any input value lying between -0.5
Page 12 of 103
Zero
to +0.5 is mapped into an output value of ______.
31
In a 8-bit binary PCM system operating very much above the 6 dB
threshold, if the number of bits/code word is increased by one, the
(SNR) q increases by ______ dB.
32
For an 8-bit binary PCM, if the sampling frequency is 8000 sps, the bit
64 K bps
rate of the PCM signal is ________ bps.
33
If the message signal bandwidth is W Hz, the minimum transmission ‘nW’ Hz
bandwidth required for an n-bit PCM is _______ Hz
The two compression laws in practice are _______ and _______.
34
dŚĞ ŽƵƚƉƵƚ ƐŝŐŶĂů ŽĨ ƚŚĞ ʅ-law compander is related to the input Logarithmic
signal in a _______ fashion.
35
A-ůĂǁ͕ʅ-law
Or Non-linear
When companding is used, expansion is done at the _____ and Receiver
compression is done at the ______ .
and transmitter.
36
One bit Differential pulse code modulation (DPCM) is called ______.
Delta modulation.
37
Granular noise can be minimized by reducing the __________.
Step size
38
Slope over-load distortion occurs in delta modulation due to _____.
Mismatch in the step
size of the signal and
modulator.
39
Slope overload distortion can be reduced by _____________.
Increasing the step size.
40
The adaptive delta modulation scheme uses _______ step size.
Variable
41
The Set of outcomes in a random experiment is called ______.
42
Consider an experiment: tossing of a coin, then what is probability of 1/2
Sample Space
the event, getting a head?
43
44
Given two events: Event-A and Event-B. They are called Statistically
P(A|B) = P(A)
independent if _____________.
P(B|A) = P(B)
or
A noisy transmission channel has a per-digit error probability p = 0.01, 0.0042
find the probability of more than one error in ten received digits.
45
The Fourier transform of the autocorrelation of WSS process is called
Power density
_________.
spectrum.
46
The area under the power spectrum density is ___________.
Auto correlation at t=0.
47
If the random process, X(t) is real then its autocorrelation function is
Real and Even.
____________.
48
If the random process, X(t) is Complex then its power spectrum
density function ੮(f) is equal to ______
Page 13 of 103
੮*(f).
49
50
The power spectral density of white noise is _______ for all
Constant
frequency and its autocorrelation is ______.
Delta function.
If X(t) is the input to a LTI system then its output Fourier transform is
Product of input FT and
___________.
Transfer function of
system.
51
The different types of channels are _____________.
Wired
Wireless
Recording channels
52
Some of the examples for the recording channels are ___________.
Magnetic discs
Tapes, CD
Hard disk
53
The transition probability of a BSC is ________ irrespective of the
same
symbol is transmitted.
54
The three output possibilities in a binary erasure channel are ______.
0, e and 1.
55
In a communication channel if the puncturing code is used then such
Binary erasure channel
channel can be modeled as __________.
56
A discrete memory less channel has M-input symbols and N-output
MxN
symbols. The order of the transition matrix for this channel is ____.
57
The mean value of additive white Gaussian noise is ___________.
Zero
58
In a continuous time AWGN channel the auto correlation function of
Delta function
the noise is ____________.
59
In line-of-sight propagation the signal travels ____________ from the Directly
transmitting antenna to receiving antenna.
60
The gain of the signal depends on the _______ when the signal environment
propagation is direct.
61
The receiving antenna receives signals from multiple paths, then the
signal received will have frequency spread, this is called as _______.
Page 14 of 103
Doppler Spread.
1.4) True or False:
Q: State whether the following statements are
True or False?
Answer
1
In Digital Communication the information is transmitted in digital form.
T
2
Video signal captured in a video camera is digitized using sampling and quantization
T
process.
3
The noise added to the signal in channel is Random in nature.
T
4
The channels behave in deterministic manner.
F
5
The statistical parameters of noise are found experimentally.
T
6
Morse code was used in early telephony systems.
F
7
The source encoder always reduces the number of bits used to represent the
T
information or the analog source.
8
The number of bits at the output of a channel encoder is always less than the number
F
of bits at the input of the channel encoder.
9
The basic purpose of sampling is to discretize the analog signal.
T
10
The critical rate of sampling is also called the Nyquist rate of sampling.
T
11
The basic principle used to reconstruct the signal x(t) from x[n] is Interpolation.
T
12
Band limited signal is a signal whose spectrum is zero outside a certain range of
T
frequency.
13
If a signal is sampled at a value more than the Nyquist rate, it leads to aliasing.
F
14
A band limited lowpass signal is sampled at twice its Nyquist Rate with fs = 3000 sps.
F
The signal is band limited to 2000 Hz.
15
Aperture effect is observed in Flat top sampling.
T
16
In uniform quantization, as the step is decreased the mean-square value of the
F
quantization error will also decrease.
17
dŚĞŵĞĂŶƐƋƵĂƌĞĞƌƌŽƌŽĨƋƵĂŶƚŝnjĂƚŝŽŶŶŽŝƐĞŝŶĂƵŶŝĨŽƌŵƋƵĂŶƚŝnjĞƌŝƐȴ2/12.
Page 15 of 103
T
18
Non-uniform quantization uses linear quantization
F
19
In a PCM system each sample is coded independently of the previous samples.
T
20
A sinusoidal message signal is being transmitted by an 8-bit binary PCM. If the
T
bits/codeword is reduced by a factor of 2, the output signal to quantization noise
ratio will reduce by 24 dB
21
Two bit quantizer is used in delta modulation.
F
22
The sampling rate used in delta modulation is always very much more than the
T
Nyquist rate of sampling rate.
23
The step size is fixed in adaptive delta modulation scheme.
F
24
Delta-sigma modulators find extensive use in digital video.
T
25
Companding is used in PCM in order to keep the quantization noise low for low-
T
amplitude segments of a signal.
26
dŚĞʅ-law based compression scheme is used for voice in India.
F
27
In a linear Delta Modulation system, a large step size will reduce the slope overload
T
noise but will increases the granular noise.
28
An event is called the subset of a sample space.
T
29
Event-A and Event-B are called Statistically independent if P(A|B) = P(B).
F
30
The Cumulative distribution function is always a non-decreasing function
T
31
A random process X(t) is said to be covariance-stationary if the covariance of X(t)
T
depends only on the time difference.
32
The Power density spectrum is the Fourier transform of the autocorrelation function
T
for a random process.
33
If the random process, X(t) is real then its autocorrelation function is complex.
F
34
If the random process, X(t) is real then power density spectrum is real and even.
T
35
If the random process, X(t) is Complex then its autocorrelation function ੮(t) is equal
T
to ੮*(-t).
36
If Random variables X and Y are statistically independent then f XY (x,y) is equal to
F
f X (x) + f Y (y)
37
A process is said to be an ergodic process if its ensemble averages are different from
F
time averages
38
In a binary symmetric channel, the probability of error is independent of the bit
T
transmitted.
39
In a binary erasure channel the bit is correct or it is corrupted and no decision can be
Page 16 of 103
T
made when the received bit is corrupted.
40
A discrete memory less channel does not remember any of the previous and past
T
symbols.
41
We can transmit more information through a DMC if feedback is provided.
F
42
Continuous valued channels carry continuous valued signals.
T
43
The power spectral density of noise in AWGN channel is constant.
T
44
Noise is independent of previous values in AWGN channel with memory
F
45
In a linear filter channel model, the channel is modeled as a linear filter.
T
46
In direct propagation of signal the attenuation is directly proportional to the distance
F
between the transmitting and receiving antennas.
Page 17 of 103
1.5) Frequently Asked Questions [ FAQ ]:
Questions
Video
No
1
Channel encoders provide more security to the information being transmitted
1
through a channel in a digital communication system. Justify this statement.
30
2
What are the advantages of Wireless channels?
1
3
Why Bandwidth and Power are considered as critical parameters in the digital
1
communication system?
4
Why ideal sampling cannot be implemented in practice?
2
5
What are the different sampling methods used in practical systems?
2
6
Why the reconstruction filter with sinc function based impulse response is
2
preferred?
7
Why Quantization is necessary in digitization of an analog signal?
3
8
Nonuniform quantization is used for Speech/voice signal digitization. Why?
3
9
Is it compulsory that the sampling rate should be always 8000 samples
3
per/second for voice signal?
10
Does the performance of Delta modulation is better than PCM?
11
Can the adaptive delta modulation overcome both the granular error and
3
slope-overload errors in delta modulation?
12
Why Gaussian distribution is considered as Normal distribution?
4
13
What is the difference between Random variable and Random Process?
14
Is there any difference between Random Process and Stochastic Process?
5
15
Why variance is considered as representative for power of a random signal?
5
16
What is the difference between wide sense stationary and strict sense
5
4,5
stationary random process?
17
What are the different channel performance measures?
6
18
What is meant by memory-less in a discrete memory less channel?
6
19
Why Binary erasure channel is preferred more than binary channel in practice?
6
20
Why AWGN model is preferred for Communication channels?
7
21
Why fading effect is predominant in wireless channels?
7
22
Do Gaussian Noise channel with memory have additive nature?
7
Page 18 of 103
1.6) Assignment Questions.
Sl.
No.
1
Questions
What are the different signal processing operations involved in the digital communication
systems?
2
Describe the characteristic features of the various channels used in Digital Communication
systems.
3
A discrete signal x(nT) is produced by sampling a continuous time sinusoid of frequency f 0 at a
sampling frequency of f s =1/T. Show that x(nT) is periodic if (f 0 /f s ) is a rational number.
4
Determine the Nyquist rate of sampling for the following signals.
(a) g(t) = 8 sin( 100t)
5
(b) x(t) = 10 cos2 (ϭϬϬʋƚ)
(c) x(t) =sinc(300t)
A certain low pass signal g(t) is sampled and the spectrum of the sampled version has guard
band from 1500 Hz to 1900 Hz. Find the sampling frequency used.
6
For each of the following signals, determine the minimum sampling frequency to be used to
avoid aliasing.
;ŝͿdž;ƚͿсϱƐŝŶ;ϭϬϬʋƚͿĐŽƐ;ϰϬϬʋƚͿ
(ii) x(t) =10 cos2 ;ϭϬϬʋƚͿ
7
dŚĞƐŝŐŶĂůdž;ƚͿсϭϬĐŽƐ;ϱϬϬʋƚͿнϲĐŽƐ;ϯϱϬʋƚͿŝƐŝĚĞĂůůLJƐĂŵƉůĞĚĂƚƚŚĞƌĂƚĞŽĨϮϬϬƐĂŵƉůĞƐ
per second. The sampled version, x ɷ (t), of x(t), is passed through a unit-gain ideal LPF with a
cutoff frequency of 100 Hz. What frequency components will be present in the output of the
LPF? Write down an expression for its output signal.
8
What is the necessity for Non-uniform quantization? Explain the two forms of compression
laws used.
9
What is meant by Companding? What is the need for it? Define A-ůĂǁĂŶĚʅ-law.
10
A commonly used value A for the A-law compander is A = 87. If the signal peak amplitude is
10V and 256 quantizing levels are employed, what is the smallest and what is the largest
effective separation between levels?
11
A sinusoidal message signal of 15 Volt peak-to-peak is to be transmitted using 8-bit PCM.
Determine the values of quantization levels, L, step-ƐŝnjĞѐĂŶĚŝƚƐƐŝŐŶĂů-to-quantization noise
ratio.
12
In a communication using binary PCM encoder, the output signal-to-quantizing noise ratio is
to be held to a minimum value of 30dB. Determine the number of required quantization
levels and find the corresponding output signal-to-quantizing noise ratio.
Page 19 of 103
13
A PCM system uses a uniform quantizer followed by an 8-bit binary encoder. If the bit rate of
the system is 56 x 106 bits/second, determine,
(i) The maximum message bandwidth for which the system operates satisfactorily
(ii) The output (S/N) q when a full load sinusoidal message signal of 2 MHz frequency is
applied to the quantizer of the system.
14
Consider an audio signal with spectral components limited to the frequency band of 300 Hz to
3400 Hz. A PCM signal is generated with a sampling rate of 8000 samples /sec. The required
output signal-to-quantizing noise ratio is 30 dB.
(i) What is the minimum number of uniform quantizing levels needed, and is the
minimum number of bits per sample needed?
(ii) Calculate the minimum system bandwidth needed?
15
Determine the processing gain of a DPCM system with a first-order predictor, if the message
signal has a normalized auto-correlation function of 0.85 for a lag of one sampling period,
assuming that the predictor is designed to minimize the mean-square value of the prediction
error.
16
A DPCM system has processing gain of 6dB. Show that a code word of this DPCM system
needs one bit less than that required for binary PCM system, all other factors remaining the
same.
17
A PCM-TDM system multiplexes 24 voice channels, each of 0 Hz to 4 kHz bandwidth. If 7-bit
PCM is used and framing bit is added to each frame, what is the minimum line speed in
bits/second and the corresponding minimum bandwidth needed?
18
A number of audio channels, each band limited to 15 kHz, are to be transmitted using 10-bit
binary PCM. Calculate how many of these PCM signals can be accommodated if the available
bandwidth is 500 kHz.
19
Consider binary PCM transmission of a video signal with a bandwidth of 6MHz. Calculate the
signaling rate needed to achieve a minimum of 40 dB SNR.
20
The bandwidth of a TV radio plus audio signal is 5 MHz. If this signal is converted to PCM with
1024 quantizing levels, determine the bit rate of the resulting PCM signal. Assume that the
signal is sampled at a rate 25 percent above the Nyquist rate.
21
Show that in a PCM system, the output signal-to-quantization noise ratio (SNR) is function of
channel bandwidth and message bandwidth.
22
What is slope-overload error in delta modulation scheme? Derive a condition for no slope
overload error for a sinusoidal input signal.
Page 20 of 103
23
A given DM system operates with Ă ƐĂŵƉůŝŶŐ ƌĂƚĞ ĨƐ ĂŶĚ ĨŝdžĞĚ ƐŝnjĞ ѐ͘ /Ĩ ƚŚĞ ŝŶƉƵƚ ƚŽ ƚŚĞ
system is m(t) = at for t > 0, determine the condition for no slope overload error in terms of
‘a’.
24
A DM system can handle message signals of bandwidth up to 5kHz and has a sampling rate of
50 KHz. A sinusoidal signal of 5 volts peak amplitude and frequency 2 kHz is applied to the
system. Determine the step-size required to avoid slope overload. Also find the (S/N) q for
the system for a given sinusoidal signal.
25
Let A and B be events defined in a sample space S.
i)
For P(A) = 0.9 and P(B) = 0.8. Show that ܲ(‫ )ܤ ת ܣ‬൒ 0.7
ii) Show that if both P(A) and P(B) are nonzero then events A and B cannot be both
mutually exclusive and independent
26
Consider a telegraph source generating two symbols: dot and dash. It was found that the dots
were twice as likely to occur as the dashes. Find the probabilities of the dot are occurring and
dash’s occurring.
27
A binary source generates digits 1 and 0 randomly with probabilities 0.7 and 0.3, respectively.
i)
What is the probability that two 1s and 0s will occur in a ten-digit sequence?
ii) What is the probability that at least three 1s will occur in a twenty-digit sequence?
28
The probability density function (pdf) of X is given by
݂௫ (‫ି ݁݇ = )ݔ‬௔௫ ‫)ݔ(ݑ‬
Where ‘a’ is a positive constant. Determine the value of the constant k.
29
The probability density function (pdf) of a random variable X is given by
݇ ܽ൑‫ݔ‬൑ܾ
݂௫ (‫ = )ݔ‬ቄ
0 ‫݁ݏ݅ݓݎ݄݁ݐ݋‬
Where k is a constant.
i)
Determine the value of k.
ii) Let a = 0 and b = 2. Calculate P(|X|< c ) for c = 1/3.
30
Let X and Y be two random variables and Y= 3X + 5. If a random variable X is uniformly
distributed over [-2, 2], find the pdf f Y (y).
31
The joint pdf of X and Y is given by
݂௑௒ (‫ݔ‬, ‫(ି ݁݇ = )ݕ‬௔௫ା௕௬) ‫)ݕ(ݑ)ݔ(ݑ‬
Where a and b are positive constant. Determine the value of constant k.
32
Consider the transformations
Z = aX +bY and W = cX + dY
Find the joint density function f zw (z, w) in terms of f xy (x, y).
33
Let Z = X + Y. Find the pdf of Z if X and Y are independent random variables.
Page 21 of 103
34
Consider the transformation
ܴ = ξܺ ଶ + ܻ ଶ
ܽ݊݀ ߠ = ‫ି݊ܽݐ‬ଵ
௒
௑
Find f Zɽ ;ƌ͕ɽͿŝŶƚĞƌŵƐŽĨĨ XY (x, y). Assume r > 1 and 0 < ɽ< Ϯʋ͘
35
What are the different Statistical Averages for a random variable? Define them.
36
The binary random variable X takes the values 0 and 1 with probabilities p and q respectively.
Find the mean and the variance of X.
37
Binary data are transmitted over a noisy communication channel in a block of 24 binary digits.
The probability that a received digit is in error due to channel noise is 0.02. Assume that the
errors occurring in various digit positions within a block are independent.
38
i) Find the mean and variance of the number of errors per block.
ii) Find the probability that the number of errors per block is greater than or equal to 5.
Compare the features of Poisson Distribution and Gaussian Distribution.
39
Find the covariance of X and Y if (a) they are independent and (b) Y is related to X by Y = aX+b.
40
>ĞƚyĂŶĚzďĞĚĞĨŝŶĞĚĂƐyсĐŽƐɽĂŶĚzсƐŝŶɽ͕
ǁŚĞƌĞɽŝƐƌĂŶĚŽŵǀĂƌŝĂďůĞƵŶŝĨŽƌŵůLJĚŝƐƚƌŝďƵƚĞĚŽǀĞƌ΀Ϭ͕Ϯʋ΁͘
41
i) Find E[X], E[Y], E[XY], E[X2], E[Y2] and E[X2 Y2].
ii) Show that X and Y are uncorrelated.
iii) Show that X and Y are not independent.
Let X be a random variable with non-zero mean and variance. Find the linear transformation
Y=aX +b such that Y has zero mean and unit variance.
42
Consider a random process X(t) given by
y;ƚͿсĐŽƐ;ʘƚнɽͿ
tŚĞƌĞĂŶĚʘĂƌĞĐŽŶƐƚĂŶƚĂŶĚɽŝƐĂƵŶŝĨŽƌŵƌĂŶĚŽŵǀĂƌŝĂďůĞŽǀĞƌ΀-ʋ͕ʋ΁͘^ŚŽǁƚŚĂƚy;ƚͿŝƐ
Wide sense stationary (WSS).
43
^ŚŽǁ ƚŚĂƚ ƉƌŽĐĞƐƐ y;ƚͿ ĚĞĨŝŶĞĚ ŝŶ y;ƚͿ с ĐŽƐ ;ʘƚн ɽͿ ŝƐ ĞƌŐŽĚŝĐ ŝŶ ďŽƚŚ ƚŚĞ ŵĞĂŶ ĂŶĚ ƚŚĞ
autocorrelation.
44
Two random processes X(t) and Y(t) are given by
y;ƚͿсĐŽƐ;ʘƚнɽ) and z;ƚͿсĐŽƐ;ʘƚнɽ)
tŚĞƌĞĂŶĚʘĂƌĞĐŽŶƐƚĂŶƚĂŶĚɽŝƐĂƵŶŝĨŽƌŵƌĂŶĚŽŵǀĂƌŝĂďůĞŽǀĞƌ΀Ϭ͕Ϯʋ΁. Find the crosscorrelation of X(t) and Y(t).
45
Suppose that a WSS random process X(t) with power spectrum S XX ;ʘͿ is the input to the filter
Y(t) = X(t) – X(t-T). Find the power spectrum of the output process Y(t).
46
Consider a discrete memory less channel which has M-input symbols and N-output symbols.
Write the generalized transition matrix for this channel? If M=3 and N=5, write the transition
matrix and channel diagram for such a channel.
Page 22 of 103
1.7) Test your skill:
Questions
Sl.
No.
1
2
Describe the three basic signal processing operations in digital communication systems.
Explain the advantages and disadvantages of digital communication over analog
communication.
3
State and Prove the Sampling theorem for low pass signals with necessary mathematical
expressions and spectrum.
4
Explain the Quadrature sampling process of band pass signal using necessary
mathematical expressions and block diagram.
5
A signal x(t) consists of two frequency components f 1 = 4100 Hz and f 2 = 3900 Hz in such a
relationship that they cancel each other out when the signal x(t) is sampled at the instants
ƚсϬ͕d͕Ϯd͕͙͕͘͘͘ǁŚĞƌĞdсϭϮϱʅƐĞĐ͘dŚĞƐŝŐŶĂůdž;ƚͿŝƐĚĞĨŝŶĞĚĂƐ͗
dž;ƚͿсϮ^ŝŶ;ϮʋĨ 1 ƚͿн͘ĐŽƐ;ϮʋĨ 2 t + ੮)
Find the values of amplitude A and phase ੮ of the second component.
6
The spectrum of a signal g(t) is shown in fig.Q6. This signal is sampled with a periodic
impulse train. Compute and plot the spectrum of the sampled signal for the following
values of sampling frequency:
(i) Fs = 40 samples/sec
(ii) Fs = 15 samples/sec
Fig. Q6
7
The spectrum of a signal g(t) is shown in fig.Q6. This signal is sampled with a periodic train
of rectangular pulses of duration 20 millisecs. Plot the spectrum of the sampled signal for
frequencies up to 60 Hz. Use the sampling frequency as 25 samples/sec.
8
9
Describe Flat-Top sampling? What is aperture effect and how it is eliminated?
Define Quantization noise. Derive an expression for the signal-to-quantization noise ratio
assuming sinusoidal input signal for
(i) a mid-rise type quantizer
(ii) a mid-tread type quantizer.
Page 23 of 103
10
What is Signal to Quantization Noise Ratio in an uniform quantizer?. If a Sound Blaster
card is an 8-bit card, what is the best SQNR (Signal to Quantization Noise Ratio) it can
achieve?
11
24 voice signals are sampled uniformly and then time division multiplexed. The sampling
operation uses the flat-top samples with 1microsec duration. The multiplexing operation
includes provision for Synchronization by adding an extra pulse of sufficient amplitude and
also 1micro second. Assuming a sampling rate of 8KHz, calculate the spacing between
successive pulses of the multiplexed signal.
12
Ten audio signals, each having a bandwidth of 15KHz, are to be time-division multiplexed
and transmitted over a channel via PAM. A guard band of 5 KHz is required for signal
reconstruction from the PAM samples of each signal. What is the sampling rate for each
signal? Calculate the overall sampling rate and transmission bandwidth of the multiplexed
signal.
13
An audio signal of bandwidth 15 kHz is to be transmitted using a PCM system. The
available channel bandwidth is 100 kHz. Design a suitable PCM transmitter system
indicating the specifications of all the blocks.
14
A PCM system uses a uniform quantizer followed by a 7-bit binary encoder. The bit rate
of the system is 56Mega bits/sec. What is the maximum message bandwidth for which the
system operates satisfactorily? Find the output signal-to-quantization noise ratio when a
sinusoidal wave of 2MHz frequency is applied to the input.
15
A Delta modulator (DM) system is designed to operate at four times the Nyquist rate for a
signal of bandwidth 2KHz. Determine the maximum amplitude for a 2KHz input sinusoid
for which the Delta modulator does not have a slope overload. The quantizing step size
used is 250 millivolts. Determine the SNR for the DM system.
16
A Delta modulator (DM) system is designed to operate at ten times the Nyquist rate. The
amplitude for a 10 KHz input sinusoid is 3 Volts (peak-peak). Determine the step size
required to prevent slope overload.
17
A ʅ-ůĂǁĐŽŵƉĂŶĚĞƌŝƐĞŵƉůŽLJĞĚŝŶĂĚŝŐŝƚŝnjĂƚŝŽŶƐLJƐƚĞŵǁŝƚŚʅсϮϱϱ͘dŚĞƐLJƐƚĞŵŝƐƵƐĞĚ
to digitize the audio signal of bandwidth 4000 Hz with a sampling rate of 8000 sps. If the
required output signal-to-quantizing noise ratio is 30 dB. What is the
(i) the minimum number of uniform quantizing levels needed, and is the minimum
number of bits per sample needed?
(ii) the minimum system bandwidth needed?
Page 24 of 103
18
A μ-law compressor is used to compare a message signal having a dynamic range of -40 V
to +40 V, employing 256 quantization levels. Assuming μ =255, determine.
(i) The interval between two consecutive levels if no compression is used
(ii) The minimum interval and the maximum interval between consecutive levels, if
compression is used.
19
A noisy transmission channel has a per-digit error probability p = 0.03, find the probability
of more than five errors in 100 received digits.
20
The input to a noisy communication channel is a binary variable X with P(X=0) = P(X=1) =
1/2. The output of channel Z is given by X +Y, where Y is the additive noise introduced by
the channel. Assuming that X and Y are independent and Y = N(0;1), find the density
function of Z.
21
Suppose that X and Y are independent normalized normal random variables. Find the pdf
of Z = X + Y.
22
If X and Y are independent, then show that
E[XY] = E[X]E[Y]
and E [g 1 (X). g 2 (Y)] = E [g 1 (x)] . E[g 2 (Y)]
23
Let Z = X+Y. If X and Y are independent, then show that
ʍ2 Z сʍ2 X + ʍ2 Y
24
Consider two random variables X and Y with finite second moments. State and prove the
Cauchy-Schwarz inequality for X and Y.
25
Let random variables U and V be defined as U = X + aY and V = X – aY, where ‘a’ is a real
value. Determine ‘a’ such that U and V are orthogonal.
26
Consider a random process X(t) given by
y;ƚͿсĐŽƐ;ʘƚнɽͿ
tŚĞƌĞʘĂŶĚɽĂƌĞĐŽŶƐƚĂŶƚĂŶĚŝƐĂƌĂŶĚŽŵǀĂƌŝĂďůĞƵŶŝĨŽƌŵůLJĚŝƐƚƌŝďƵƚĞĚŽǀĞƌ΀Ϭ͕ϭ΁͘
i)
Determine whether X(t) is WSS.
ii) Find the auto-correlation of X(t)
iii) Find the auto-covariance of X(t).
27
The output of a filter is Y(t) = X( t + a) - X( t – a), where X(t) is a WSS process with power
spectrum Sx(w) and ‘a’ is a constant. Find the power spectrum of Y(t).
28
Consider an ideĂůďĂŶĚƉĂƐƐĨŝůƚĞƌŚĂǀŝŶŐĂďĂŶĚǁŝĚƚŚ͕͚͛ĂŶĚĐĞŶƚĞƌĨƌĞƋƵĞŶĐLJ͚ʘ c ‘ .
i)
Draw the frequency response curve.
ii)
If the input to the filter X(t) is a white noise process, find the total noise power at
the output of the filter.
Page 25 of 103
29
Consider a low pass RC-filter having a bandwidth, W Hz whose input to the filter X(t) is a
white noise process. Find the power spectrum and autocorrelation of the output process.
30
Consider a random process X(t) given by
y;ƚͿсĐŽƐʘƚнƐŝŶʘƚ
tŚĞƌĞʘŝƐĐŽŶƐƚĂŶƚĂŶĚĂŶĚĂƌĞƌĂŶĚŽŵǀĂƌŝĂďůĞƐ͘
a) Show that the condition E[A] = E[B] =0 is necessary for X(t) to be stationary.
b) Show that X(t) is WSS if and only if the random variables A and B are uncorrelated
with equal variance, that is,
E[AB] =0 and E[A2] = E[B2΁сʍ2
Page 26 of 103
1.8) Additional Links:
Module-1: General Links
https://www.youtube.com/watch?v=1Z3l14k8X1M
https://www.youtube.com/watch?v=Q73CBI9VNcE
https://www.youtube.com/watch?v=cVgjgUrz9FM
https://www.youtube.com/watch?v=BHxLsKQcNOQ
https://www.youtube.com/watch?v=JmiM3AzoEsg
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-450-principles-of-digitalcommunications-i-fall-2006/lecture-notes/book_1.pdf
Nos
1
Topic
Telephone System
Videos
Web Link
http://www.sp4comm.org/docs/chapter12.pdf
http://en.wikibooks.org/wiki/Communication_Systems/
Telephone_System
2
Morse code
http://en.wikipedia.org/wiki/Morse_code
http://kambing.ui.ac.id/onnopurbo/oraridiklat/teknik/cw/doc/LEARN%20MORSE%20CODE%20in
%20one%20minute.pdf
3
Source encoder
http://elearning.vtu.ac.in/P6/enotes/EC6/Unit1-NR.pdf
http://ocw.mit.edu/courses/electrical-engineering-andcomputer-science/6-450-principles-of-digitalcommunications-i-fall-2006/lecture-notes/book_1.pdf
4
Channel encoder
https://www.yout
ube.com/watch?v
=1Z3l14k8X1M
https://www.yout
ube.com/watch?v
=Q73CBI9VNcE
5
Cellular Systems
http://www.britannica.com/EBchecked/topic/105743/chann
el-encoding
http://wiki.answers.com/Q/What_is_difference_between_so
urce_encoder_and_channel_encoder?#slide=1
http://www.indigovision.com/documents/public/datasheets/
10-Channel-Encoder-Decoder_Datasheet-A4.pdf
http://www.eng.iastate.edu/ee423/EE421/Lecture/cellc
ommtutorial.pdf
http://www.iitg.ernet.in/engfac/krs/public_html/lectur
es/ee635/A3.pdf
6
Digital Modulation
http://www.youtu
be.com/watch?v=
pZbczyghP8Y
https://www.google.co.in/search?q=Digital+modulation&tb
m=isch&tbo=u&source=univ&sa=X&ei=ASQvU_oKIaNrQfnqYGgBA&ved=0CDIQsAQ&biw=1366&bih=596
Page 27 of 103
http://www.youtu
be.com/watch?v=
2yXQ6VrbpKk
7
8
9
10
11
12
13
14
Channels for digital
Communication
Band-limited signal
Sampling in ADC
Aliasing in Sampling
Uniform
Quantization
Non-uniform
quantization
Pulse code
modulation
Delta Modulation
http://www.youtu
be.com/watch?v=
5AVsObMRRko
http://www.youtu
be.com/watch?v=
Qd3cp1VXrsM
http://www.youtu
be.com/watch?v=t
bLDvajIKdk
http://www.youtu
be.com/watch?v=
HO-_DQOiYIs
http://www.youtu
be.com/watch?v=
PNR7dx0lk_s
http://www.youtu
be.com/watch?v=
9a7b224X4BA
http://www.youtu
be.com/watch?v=
KuaannH5pnM
http://www.youtu
be.com/watch?v=
7H4sJdyDztI
http://www.youtu
be.com/watch?v=
MwjsK2t_G4w
http://www.youtu
be.com/watch?v=
MfKTD6WpJ6s
http://www.youtu
be.com/watch?v=
qv_rfGtMMGE
http://www.youtu
be.com/watch?v=j
0_4rBnQIjM
http://www.youtu
be.com/watch?v=
YJmUkNTBa8s
http://www.youtu
be.com/watch?v=
zW3fWfP8Bg0
http://www.youtu
be.com/watch?v=
oFEOryECzug
http://cp.literature.agilent.com/litweb/pdf/5965-7160E.pdf
http://web.ee.ccu.edu.tw/~wl/wenclass/95/IDCclass/Ch5_Di
gital.pdf
http://www.ee.iastate.edu/~russell/cpre537xf00/Projects/Ze
ng.pdf
https://ccrma.stanford.edu/~jos/wgj/Bandlimited_Signals_In
terpretation.html
http://www.systems.caltech.edu/EE/Groups/dsp/students/b
ojan/conf/SampTaplenary.pdf
http://www.ni.com/white-paper/3016/en/
http://www.atmel.in/Images/doc8003.pdf
http://www.cs.umd.edu/~djacobs/CMSC427/Aliasing.pdf
http://redwood.berkeley.edu/bruno/npb261/aliasing.pdf
http://www.princeton.edu/~cuff/ele201/kulkarni_text/digitiz
n.pdf
http://care.iitd.ac.in/Academics/Courses/crp_718/exp_sp_2.
pdf
http://www.ece.unm.edu/faculty/bsanthan/ece539/note6.p
df
http://nptel.ac.in/courses/Webcourse-contents/IITKANPUR/Digi_Img_Pro/chapter_5/5_5.html
http://eng.uokerbala.edu.iq/lectures/electrical_engineering/
Fourth_year/Digital%20Communications/Pulse%20Code%20
Modulation.pdf
https://www.princeton.edu/~achaney/tmve/wiki100k/docs/
Delta_modulation.html
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http://www.youtu
be.com/watch?v=
MmDHhx2lwx0
15
16
17
Wireless channel
models
LTI channel model
AWGN channel
Models
http://www.youtu
be.com/watch?v=
9ujT1upyWVg
http://www.youtu
be.com/watch?v=
3YHovkP1JC8
http://www.youtu
be.com/watch?v=
gF9Q0wNGENc
http://www.youtu
be.com/watch?v=f
QcJNoe-q-s
www.youtube.co
m/watch?v=UyFqE
Sqr4rI
www.youtube.co
m/watch?v=CdWd
S0Vv3l0
18
Probability
http://en.wikibooks.org/wiki/Analog_and_Digital_Conversion
/Delta_Modulation
http://people.cs.umass.edu/~arun/653/lectures/channel_mo
dels.pdf
http://www.eecs.berkeley.edu/~dtse/Chapters_PDF/Fundam
entals_Wireless_Communication_chapter2.pdf
http://ocw.mit.edu/courses/electrical-engineering-andcomputer-science/6-02-introduction-to-eecs-ii-digitalcommunication-systems-fall2012/readings/MIT6_02F12_chap11.pdf
http://www.eit.lth.se/fileadmin/eit/courses/eit140/ofdm_ch
annels.pdf
http://users.crhc.illinois.edu/nhv/09spring.439/slides/2slides.
pdf
http://ocw.mit.edu/courses/electrical-engineering-andcomputer-science/6-451-principles-of-digital-communicationii-spring-2005/lecture-notes/chap_2.pdf
http://www.ece.utah.edu/~npatwari/ece5520/lectureA
ll.pdf
http://nptel.ac.in/courses/IITMADRAS/Principles_of_Communication1/Pdfs/1_5.pdf
http://www.ece.tamu.edu/~georghiades/courses/ftp45
5/intro.pdf
19
Joint Probability ,
conditional
Probability
http://www.ele.uri.edu/faculty/kay/New%20web/down
loadable%20files/book_total.pdf
http://nptel.ac.in/courses/IITMADRAS/Principles_of_Communication1/Pdfs/1_5.pdf
20
21
22
Random Variables
http://www.stanford.edu/~montanar/RESEARCH/BOOK
/partA.pdf
Distribution
Function
www.cambridge.org/us/download_file/208509/
http://www.ele.uri.edu/faculty/kay/New%20web/down
loadable%20files/book_total.pdf
Probability Density
Functions
http://nptel.ac.in/courses/IITMADRAS/Principles_of_Communication1/Pdfs/1_5.pdf
http://www.ele.uri.edu/faculty/kay/New%20web/down
loadable%20files/book_total.pdf
http://nptel.ac.in/courses/IITMADRAS/Principles_of_Communication1/Pdfs/1_5.pdf
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