NPTEL COURSE MATERIAL Digital Communicationnptel.vtu.ac.in/VTU-NMEICT/digicomm/Module1.pdf · NPTEL...

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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

Transcript of NPTEL COURSE MATERIAL Digital Communicationnptel.vtu.ac.in/VTU-NMEICT/digicomm/Module1.pdf · NPTEL...

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. Topics

1 Introduction to Digital Communication 2 Sampling 3 Quantization , PCM and Delta Modulation 4 Probability and Random Process 5 Probability and Random Processes (Part - 2) 6 Channels and their Models 7 Channels and their Models (Part -2) 8 Information Theory (Part - 1) 9 Information Theory (Part - 2)

10 Bandpass Signal Representation (Part 1) 11 Bandpass Signal Representation (Part - 2) 12 Digital Modulation Techniques (Part - 1) 13 Digital Modulation Techniques (Part - 2) 14 Digital Modulation Techniques (Part - 3) 15 Digital Modulation Techniques (Part - 4 ) 16 Digital Modulation Techniques (Part – 5) 17 Digital Modulation Techniques (Part - 6) 18 Digital Modulation Techniques (Part - 7) 19 Digital Modulation Techniques (part - 8) 20 Digital Modulation Techniques (Part - 9) 21 Digital Modulation Techniques (Part - 10) 22 Probability of Error Calculation 23 Calculation of Probability of Error 24 Calculation of Probability of Error 25 Equalizers 26 Source Coding (Part - 1) 27 Source Coding (Part - 2) 28 Source Coding (Part – 3) 29 Source Coding (Part - 4) 30 Channel Coding 31 Fundamentals of OFDM 32 Conclusion

Page 2 of 103

Module-1

Page 3 of 103

1.1) NPTEL Video Link Module-1: Video Lecture number 1 to 7

Sl. No.

Module No.

Lecture No. Topics covered Video Link

1 Mod 01 Lec-01 Introduction to Digital Communication Systems (54:29 mins)

http://nptel.ac.in/courses/117101051/1

2 Mod 01 Lec-02 Sampling, Sampling Theorem Spectrum, Recovery of signal. (53:16 mins)

http://nptel.ac.in/courses/117101051/2

3 Mod 01 Lec-03 Quantization , PCM and Delta Modulation (51:43 mins)

http://nptel.ac.in/courses/117101051/3

4 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

NPTEL Web Link: http://nptel.ac.in/courses/Webcourse-contents/IIT%20Kharagpur/Digi%20Comm/New_index1.html

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1.2) Questions:

Questions from Video Lectures of NPTEL

Sl no Questions Video

Number

Time in

Minutes

1 Give some examples for the communication systems which use ‘space’ as the

channel.

1 2

2 Give some examples for the communication systems which use wire-line

channel.

1 4

3 Mention the source and destination for each of the following communication

systems:

(i) Radio

(ii) Television

(iii) Telephone system

(iv) Cellular Mobile system

1 2-5

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

digital/binary form.

1 9

7 Explain the characteristics of different channels used in digital communication

systems.

1 11

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

code?

1 19

11 Explain the International Morse code with some examples. 1 23

12 What are the fundamental functional blocks of a digital communication

system? Explain the functions of each block.

1 26-42

13 What are the advantages of Digital Communication over analog

communication?

1 43

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

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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,

explain the process of sampling analog signal for different sampling rates.

2 13

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

the sampling rate is less than the Nyquist rate.

2 19

22 Explain the concept of sampling and recovery of the signal from its samples

using sinusoidal signal.

2 20

23 How to reconstruct x(t) from x[n] using Interpolation principle? What type of

filter is used?

2 31

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.

2 34

25 Describe the recovery of x(t) using filter with impulse response

h(t) = Sinc[ fs( t – nTs) ].

2 39

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.

2 42

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

of quantization levels and step size are related?

3 3-6

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-

uniform Quantizer and explain.

3 16

32 -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

the block diagrams of modulator and demodulator.

3 35-40

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

modulation.

3 45

40 Define the following:

(i) Random experiment and (ii) Sample space.

4 2

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.

4 3-6

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:

(i) Event (ii) Complement of an event (iii) Union of events.

4 8

44 Consider two experiments, experiment-1 and experiment-2 with sample

spaces S1= { x1, x2, x3, …… xM} and S2 = { y1, y2, y3, …… yN} . Find the

following probabilities – p(xi) and p(yj).

4 11

45 Consider two events; event-A and event-B. Define the following:

(i) Conditional Probability

(ii) Marginal Probability

(iii) Chain rule

(iv) Statistically Independent.

4 14-19

46 Differentiate discrete random variable and continuous random variable?

4 23

47 Define the cumulative distribution function (cdf) of a random variable. For the

experiment tossing of a coin, sketch the cdf.

4 29-32

48 Consider the random experiment: rolling of a dice. Compute the cdf assuming

that the probability of each outcome is same.

4 33

49 Define probability density function, pX(x). 4 34-38

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50 The probability density function of a random variable X is,

pX(x) = 0.5 for a < x < b and

zero otherwise.

Compute and sketch the cumulative distribution function, FX(x).

4 39

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.

4 41

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.

4 43-51

53 Consider random variables X1, X2, X3…… XN. Under what conditions these

variables are called independent random variables.

4 51

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

constants.

(i) Express cumulative distribution FY(y) in terms of FX(y).

(ii) Find the relation between the density functions.

5 4

56 Develop the Jacobian of the transformation. 5 10

57 How the density function of a new variable by using density function of the

other variable?

5 11

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

is the auto correlation of X(t)?

5 38

63 For a complex valued random process define the autocorrelation. 5 42

64 What is meant by independent random process and uncorrelated random

process?

5 43

65 Explain the relation between autocorrelation function and power density

spectrum of a random process.

5 45

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66 What are the different types of physical channels? Classify them with

examples.

6 2-7

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

channel.

6 20, 22

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.

6 27

71 A binary symmetric channel has probability of error, p=0.9. How its

performance is improved with the use of inverter?

6 28

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.

6 32

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

alphabets and N-output alphabets. Also define conditional probability.

6 38-43

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

Noise channel.

7 6

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

this model of channel.

7 17

81 For a linear filter channel based communication, how the signal is recovered?

Explain using the convolution.

7 25

82 Describe the features of colored Gaussian noise. 7 28

83 Explain the role of whitening filter in the receiver to remove colored Gaussian

noise.

7 29

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

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86 Explain the following phenomenon:

(i) Reflection

(ii) Defraction

(iii) Scattering

7 35

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.

No.

Question Answer

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 _________

nature.

Random

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

_________ and _____________.

Bandwidth

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,

x(t) = 10 is ___________.

8530 samples/sec

12 The minimum sampling rate required to sample the following signal,

is _____________.

12110 samples/sec

13 An analog low pass signal x(t) is bandlimited to 2500Hz. The maximum

sampling period that can be used is _________, is the x(t) is to be

recovered without distortion, form its samples

0.2 milli secs

14 A lowpass signal x(t) is bandlimited to W Hz and having a spectrum

X(f). The ideally sampled version, x (t), of x(t), will have a spectrum

X (f) which is a periodic repetition of X(f) with a period of ______.

Sampling rate

15 A lowpass bandlimited signal, sampled at a frequency higher than the

Nyquist rate, may be recovered from its samples by passing them

through a ____________.

Low pass filter

16 The minimum sampling rate fs to be used for sampling a continuous- Nyquist Rate

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time lowpass signal x(t), for proper recovery of x(t) from its samples, is

called _____ and it is equal to ___________.

and

fs= 2fm.

17 The phenomenon where by the higher frequency components of

signal x(t) reappear as low frequency components in the spectrum of

the sampled version of x(t) is called ___________.

Aliasing

18 Anti-aliasing filter is lowpass filter used before the sampling

phenomenon to overcome __________.

Aliasing

19 The three steps in the digitization of an analog message signal are

_______, ________ and _____

Sampling

Quantization

Encoding

20 The distortion in the received signal caused by flat-top sampling is

called _______.

Aperture effect

21 A lowpass signal bandlimited to 1200 Hz was sampled and it was

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 __________.

1400 sps

22 Consider a signal x(t) = 3 + 5cos2

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 _________.

0.222 milli secs

23 pled at a frequency fs.

The signal recovered from the samples was, however, found to be

s is equal to ___________.

150 Hz

24 A binary PCM system with 256 quantizing levels has a sampling

frequency of 7 kHz. The bit rate of the system is _________.

56 k bps

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.

Write the relation between L and R.

L= 2R or L=2R-1.

26 The mean square error of the quantization noise in a uniform

quantizer is equal to _____.

2/12

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 Zero

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to +0.5 is mapped into an output value of ______.

31 In a 8-bit binary PCM system operating very much above the

threshold, if the number of bits/code word is increased by one, the

(SNR)q increases by ______ dB.

6 dB

32 For an 8-bit binary PCM, if the sampling frequency is 8000 sps, the bit

rate of the PCM signal is ________ bps.

64 K bps

33 If the message signal bandwidth is W Hz, the minimum transmission

bandwidth required for an n-bit PCM is _______ Hz

‘nW’ Hz

The two compression laws in practice are _______ and _______. A- -law

34 -law compander is related to the input

signal in a _______ fashion.

Logarithmic

Or Non-linear

35 When companding is used, expansion is done at the _____ and

compression is done at the ______ .

Receiver

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 ______. Sample Space

42 Consider an experiment: tossing of a coin, then what is probability of

the event, getting a head?

1/2

43 Given two events: Event-A and Event-B. They are called Statistically

independent if _____________.

P(A|B) = P(A) or

P(B|A) = P(B)

44 A noisy transmission channel has a per-digit error probability p = 0.01,

find the probability of more than one error in ten received digits.

0.0042

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 (f) is equal to ______

*(f).

Page 13 of 103

49 The power spectral density of white noise is _______ for all

frequency and its autocorrelation is ______.

Constant

Delta function.

50 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

symbol is transmitted.

same

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

channel can be modeled as __________.

Binary erasure channel

56 A discrete memory less channel has M-input symbols and N-output

symbols. The order of the transition matrix for this channel is ____.

MxN

57 The mean value of additive white Gaussian noise is ___________. Zero

58 In a continuous time AWGN channel the auto correlation function of

the noise is ____________.

Delta function

59 In line-of-sight propagation the signal travels ____________ from the

transmitting antenna to receiving antenna.

Directly

60 The gain of the signal depends on the _______ when the signal

propagation is direct.

environment

61 The receiving antenna receives signals from multiple paths, then the

signal received will have frequency spread, this is called as _______.

Doppler Spread.

Page 14 of 103

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

process.

T

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

information or the analog source.

T

8 The number of bits at the output of a channel encoder is always less than the number

of bits at the input of the channel encoder.

F

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

frequency.

T

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.

The signal is band limited to 2000 Hz.

F

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

quantization error will also decrease.

F

17

2/12. T

Page 15 of 103

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

bits/codeword is reduced by a factor of 2, the output signal to quantization noise

ratio will reduce by 24 dB

T

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

Nyquist rate of sampling rate.

T

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-

amplitude segments of a signal.

T

26 -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

noise but will increases the granular noise.

T

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)

depends only on the time difference.

T

32 The Power density spectrum is the Fourier transform of the autocorrelation function

for a random process.

T

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 (t) is equal

to *(-t).

T

36 If Random variables X and Y are statistically independent then fXY (x,y) is equal to

fX (x) + fY (y)

F

37 A process is said to be an ergodic process if its ensemble averages are different from

time averages

F

38 In a binary symmetric channel, the probability of error is independent of the bit

transmitted.

T

39 In a binary erasure channel the bit is correct or it is corrupted and no decision can be T

Page 16 of 103

made when the received bit is corrupted.

40 A discrete memory less channel does not remember any of the previous and past

symbols.

T

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

between the transmitting and receiving antennas.

F

Page 17 of 103

1.5) Frequently Asked Questions [ FAQ ]:

Questions Video

No

1 Channel encoders provide more security to the information being transmitted

through a channel in a digital communication system. Justify this statement.

1

30

2 What are the advantages of Wireless channels? 1

3 Why Bandwidth and Power are considered as critical parameters in the digital

communication system?

1

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

preferred?

2

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

per/second for voice signal?

3

10 Does the performance of Delta modulation is better than PCM?

11 Can the adaptive delta modulation overcome both the granular error and

slope-overload errors in delta modulation?

3

12 Why Gaussian distribution is considered as Normal distribution? 4

13 What is the difference between Random variable and Random Process? 4,5

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

stationary random process?

5

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. Questions

1 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 f0 at a

sampling frequency of fs =1/T. Show that x(nT) is periodic if (f0/fs) is a rational number.

4 Determine the Nyquist rate of sampling for the following signals.

(a) g(t) = 8 sin( 100t) (b) x(t) = 10 cos2 ( ) (c) x(t) =sinc(300t)

5 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.

(ii) x(t) =10 cos2

7

per second. The sampled version, x (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- -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- -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.

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23 A given DM system operates with

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 ( ) 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 ( ) = ( )

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 ( ) = 0

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 fY(y).

31 The joint pdf of X and Y is given by ( , ) = ( ) ( ) ( )

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 fzw(z, w) in terms of fxy(x, y).

33 Let Z = X + Y. Find the pdf of Z if X and Y are independent random variables.

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34 Consider the transformation = + =

Find f XY (x, y). Assume r > 1 and 0 < <

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.

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.

38 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

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.

41 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

-

Wide sense stationary (WSS).

43

autocorrelation.

44 Two random processes X(t) and Y(t) are given by

) and )

. Find the cross-

correlation of X(t) and Y(t).

45 Suppose that a WSS random process X(t) with power spectrum SXX 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.

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1.7) Test your skill:

Sl. No.

Questions

1 Describe the three basic signal processing operations in digital communication systems.

2 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 f1 = 4100 Hz and f2 = 3900 Hz in such a

relationship that they cancel each other out when the signal x(t) is sampled at the instants

1 2t + )

Find the values of amplitude A and phase 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 Describe Flat-Top sampling? What is aperture effect and how it is eliminated?

9 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.

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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 -

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?

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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 [g1(X). g2(Y)] = E [g1(x)] . E[g2(Y)]

23 Let Z = X+Y. If X and Y are independent, then show that

2Z 2

X + 2

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

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 ide 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.

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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

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 2

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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-digital-communications-i-fall-2006/lecture-notes/book_1.pdf

Nos Topic Videos Web Link 1 Telephone System 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/orari-diklat/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-and-computer-science/6-450-principles-of-digital-communications-i-fall-2006/lecture-notes/book_1.pdf

4 Channel encoder https://www.youtube.com/watch?v=1Z3l14k8X1M https://www.youtube.com/watch?v=Q73CBI9VNcE

http://www.britannica.com/EBchecked/topic/105743/channel-encoding http://wiki.answers.com/Q/What_is_difference_between_source_encoder_and_channel_encoder?#slide=1 http://www.indigovision.com/documents/public/datasheets/10-Channel-Encoder-Decoder_Datasheet-A4.pdf

5 Cellular Systems http://www.eng.iastate.edu/ee423/EE421/Lecture/cellcommtutorial.pdf http://www.iitg.ernet.in/engfac/krs/public_html/lectures/ee635/A3.pdf

6 Digital Modulation http://www.youtube.com/watch?v=pZbczyghP8Y

https://www.google.co.in/search?q=Digital+modulation&tbm=isch&tbo=u&source=univ&sa=X&ei=ASQvU-_oKIaNrQfnqYGgBA&ved=0CDIQsAQ&biw=1366&bih=596

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http://www.youtube.com/watch?v=2yXQ6VrbpKk

http://cp.literature.agilent.com/litweb/pdf/5965-7160E.pdf http://web.ee.ccu.edu.tw/~wl/wenclass/95/IDCclass/Ch5_Digital.pdf

7 Channels for digital Communication

http://www.youtube.com/watch?v=5AVsObMRRko http://www.youtube.com/watch?v=Qd3cp1VXrsM

http://www.ee.iastate.edu/~russell/cpre537xf00/Projects/Zeng.pdf

8 Band-limited signal http://www.youtube.com/watch?v=tbLDvajIKdk http://www.youtube.com/watch?v=HO-_DQOiYIs

https://ccrma.stanford.edu/~jos/wgj/Bandlimited_Signals_Interpretation.html http://www.systems.caltech.edu/EE/Groups/dsp/students/bojan/conf/SampTaplenary.pdf

9 Sampling in ADC http://www.youtube.com/watch?v=PNR7dx0lk_s http://www.youtube.com/watch?v=9a7b224X4BA

http://www.ni.com/white-paper/3016/en/ http://www.atmel.in/Images/doc8003.pdf

10 Aliasing in Sampling http://www.youtube.com/watch?v=KuaannH5pnM http://www.youtube.com/watch?v=7H4sJdyDztI

http://www.cs.umd.edu/~djacobs/CMSC427/Aliasing.pdf http://redwood.berkeley.edu/bruno/npb261/aliasing.pdf

11 Uniform Quantization

http://www.youtube.com/watch?v=MwjsK2t_G4w http://www.youtube.com/watch?v=MfKTD6WpJ6s

http://www.princeton.edu/~cuff/ele201/kulkarni_text/digitizn.pdf http://care.iitd.ac.in/Academics/Courses/crp_718/exp_sp_2.pdf

12 Non-uniform quantization

http://www.youtube.com/watch?v=qv_rfGtMMGE http://www.youtube.com/watch?v=j0_4rBnQIjM

http://www.ece.unm.edu/faculty/bsanthan/ece539/note6.pdf http://nptel.ac.in/courses/Webcourse-contents/IIT-KANPUR/Digi_Img_Pro/chapter_5/5_5.html

13 Pulse code modulation

http://www.youtube.com/watch?v=YJmUkNTBa8s http://www.youtube.com/watch?v=zW3fWfP8Bg0

http://eng.uokerbala.edu.iq/lectures/electrical_engineering/Fourth_year/Digital%20Communications/Pulse%20Code%20Modulation.pdf

14 Delta Modulation http://www.youtube.com/watch?v=oFEOryECzug

https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Delta_modulation.html

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http://www.youtube.com/watch?v=MmDHhx2lwx0

http://en.wikibooks.org/wiki/Analog_and_Digital_Conversion/Delta_Modulation

15 Wireless channel models

http://www.youtube.com/watch?v=9ujT1upyWVg http://www.youtube.com/watch?v=3YHovkP1JC8

http://people.cs.umass.edu/~arun/653/lectures/channel_models.pdf http://www.eecs.berkeley.edu/~dtse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter2.pdf

16 LTI channel model http://www.youtube.com/watch?v=gF9Q0wNGENc http://www.youtube.com/watch?v=fQcJNoe-q-s

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/readings/MIT6_02F12_chap11.pdf http://www.eit.lth.se/fileadmin/eit/courses/eit140/ofdm_channels.pdf

17 AWGN channel Models

www.youtube.com/watch?v=UyFqESqr4rI www.youtube.com/watch?v=CdWdS0Vv3l0

http://users.crhc.illinois.edu/nhv/09spring.439/slides/2slides.pdf http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-451-principles-of-digital-communication-ii-spring-2005/lecture-notes/chap_2.pdf

18 Probability http://www.ece.utah.edu/~npatwari/ece5520/lectureAll.pdf

http://nptel.ac.in/courses/IIT-MADRAS/Principles_of_Communication1/Pdfs/1_5.pdf

http://www.ece.tamu.edu/~georghiades/courses/ftp455/intro.pdf

19 Joint Probability , conditional Probability

http://www.ele.uri.edu/faculty/kay/New%20web/downloadable%20files/book_total.pdf http://nptel.ac.in/courses/IIT-MADRAS/Principles_of_Communication1/Pdfs/1_5.pdf

20 Random Variables http://www.stanford.edu/~montanar/RESEARCH/BOOK/partA.pdf www.cambridge.org/us/download_file/208509/

21 Distribution Function

http://www.ele.uri.edu/faculty/kay/New%20web/downloadable%20files/book_total.pdf http://nptel.ac.in/courses/IIT-MADRAS/Principles_of_Communication1/Pdfs/1_5.pdf

22 Probability Density Functions

http://www.ele.uri.edu/faculty/kay/New%20web/downloadable%20files/book_total.pdf http://nptel.ac.in/courses/IIT-MADRAS/Principles_of_Communication1/Pdfs/1_5.pdf

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