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SIGNALS AND STOCHASTIC PROCESS
Subject Code: (EC304ES) Regulations : R16 JNTUH
Class :II Year B.Tech ECE I Semester
Department of Electronics and communication Engineering
BHARAT INSTITUTE OF ENGINEERING AND TECHNOLOGY
Ibrahimpatnam -501 510, Hyderabad
Page 21
SIGNALS AND STOCHASTIC PROCESS (EC304ES) COURSE PLANNER
I. COURSE OVERVIEW
The course introduces the basic concepts of signals and systems. which is the basic of all subjects of
signal processing. It then introduces the concept of Stochastic Processes. A discussion is made about
the temporal and Spectral Characteristic of Random processes viz The concept of Stationary, Auto
and Cross correlation, Concept of Power Spectrum density. The course also deals the response of
Linear Systems for a Random process input. Finally it covers the concept of Noise and its modeling
II. PREREQUISITE:
1. Mathematics – I
2. Mathematical Methods
3. Mathematics – III
4. Electrical Circuits
III. COURSE OBJECTIVE
1.
This gives the basics of Signals and Systems required for all Electrical Engineering
related courses.
2.
This gives concepts of Signals and Systems and its analysis using different transform techniques.
3.
This gives basic understanding of random process which is essential for random Signals and systems encountered in Communications and Signal Processing areas.
IV. COURSE OUTCOME:
S.No Description Bloom‘s Taxonomy Level
1. Understand the principles of vector spaces, including
how to relate the concepts of basis, dimension, inner product, and norm to signals. Know how to analyze,
design, approximate, and manipulate signals using vector-space concepts.
Understand (Level 2)
2. Understand and classify signals (e.g. periodic, even) and systems (e.g. causal, linear) and an understanding
of the difference between discrete and continuous time signals and systems, understand the principles of impulse
functions, step function and signum function.
Understand(Level 2)
3. Analyze the implications of linearity, time-invariance, causality, memory, and bounded-input, bounded-out (BIBO) stability.
Analyze (Level 4)
4. Determine the response of linear systems to any input signal by convolution in the time domain, and by transformation to the frequency domain, filter
characteristics of a system and its bandwidth, the concepts of auto correlation and cross correlation and power density
spectrum.
Understand(Level 2)
5. Understand the definitions and basic properties (e.g.
time-shift, modulation, Parseval's Theorem) of Fourier series, Fourier transforms, Laplace transforms, Z transforms, and an ability to compute the transforms and
inverse transforms of basic examples using methods such as partial fraction expansions, ROC of Z Transform/
Laplace Transform.
Understand(Level 2)
6. Understand the concepts of Random Process and its
characteristics, the response of linear time Invariant system
for a Random Processes.
Understand(Level 2)
V. HOW PROGRAM OUTCOMES ARE ASSESSED
Program Outcomes (PO)
Level Proficiency
assessed by
PO1 Engineering knowledge: An ability to apply knowledge of basic sciences, mathematical skills, engineering and
technology to solve complex electronics and communication engineering
problems (Fundamental Engineering Analysis Skills).
3
Assignments and tutorials
PO2 Problem analysis: An ability to
identify, formulate and analyze engineering problems using knowledge of Basic Mathematics and Engineering
Sciences (Engineering Problem
3
Assignments
PO3 Design/development of solutions: An ability to provide solution and to design
Electronics and Communication Systems as per social needs (Social Awareness).
3
Seminars
PO4 Conduct investigations of complex problems: An ability to investigate the
problems in Electronics and Communication field and develop suitable
solutions (Creative Skills).
3
Projects
PO5 Modern tool usage An ability to use latest hardware and software tools to solve complex engineering problems (Software
and Hardware Interface).
3
Projects
PO6 The engineer and society: An ability to apply knowledge of contemporary
issues like health, Safety and legal which influences engineering design
(Social Awareness).
2
Oral Discussions
PO7 Environment and sustainability: Understand the impact of the Electronics and Communication Engineering
2
Oral Discussions
Page 23
professional engineering solutions in social and environmental contexts, and demonstrate the knowledge of and need for
sustainable development.
PO8 Ethics: Apply ethical principles and commit to professional ethics and
responsibilities and norms of the engineering practice.
-
--
PO9 Individual and team work: Function
effectively as an individual, and as a member or leader in diverse teams, and in multidisciplinary settings.
3
Development of prototype
models
PO10 Communication: Communicate effectively
on complex engineering activities with the engineering community and with society at
large, such as, being able to comprehend and write effective reports and design documentation, make effective
presentations, and give and receive clear instructions.
3
Presentations
PO11 Project management and finance:
Demonstrate knowledge and understanding of the engineering and management principles and apply these to one‘s own
work, as a member and leader in a team, to manage
projects and in multidisciplinary environments.
3
Seminars.
Discussions
PO12 Life-long learning: Recognize the need
for, and have the preparation and ability to engage in independent and life-long learning in the broadest context of
technological change.
--
`
1: Slight (Low) 2: Moderate (Medium) 3: Substantial (High) - : None
VI. HOW PROGRAM SPECIFIC OUTCOMES ARE ASSESSED
Program Specific Outcomes (PSO) Level Proficiency
assessed by
PSO1
Professional Skills: An ability to understand the basic concepts in Electronics &
Communication Engineering and to apply them to various areas, like Electronics,
Communications, Signal processing, VLSI, Embedded systems etc., in the design and implementation of complex systems.
3 Lectures, Assignments
PSO2
Problem-solving skills: An ability to solve
complex Electronics and communication Engineering problems,using latest hardware
and software tools, along with analytical skills to arrive cost effective and appropriate solutions.software aspects of computer systems.
2 Tutorials
PSO3
Successful career and Entrepreneurship: An
understanding of social-awareness & environmental-wisdom along with ethical
responsibility to have a successful career and to sustain passion and zeal for real-world applications using optimal resources
1 Seminars and
Projects
1: Slight (Low) 2: Moderate (Medium) 3: Substantial (High) -: None
VII. SYLLABUS
UNIT I
Signal Analysis: Analogy between Vectors and Signals, Orthogonal Signal Space, Signal
approximation using Orthogonal functions, Mean Square Error, Closed or complete set of Orthogonal functions, Orthogonality in Complex functions, Exponential and Sinusoidal signals, Concepts of Impulse function, Unit Step function, Signum function.
Signal Transmission through Linear Systems: Linear System, Impulse response, Response of a Linear System, Linear Time Invariant (LTI) System, Linear Time Variant (LTV) System,
Transfer function of a LTI system, Filter characteristics of Linear Systems, Distortion less transmission through a system, Signal bandwidth, System bandwidth, Ideal LPF, HPF and BPF characteristics, Causality and Paley-Wiener criterion for physical realization,
Relationship between Bandwidth and Rise time. Concept of convolution in Time domain and Frequency domain, Graphical representation of Convolution, Convolution property of Fourier
Transforms.
UNIT II
Fourier series, Transforms, and Sampling: Fourier series: Representation of Fourier series, Continuous time periodic signals, Properties of Fourier Series, Dirichlet‘s conditions,
Trigonometric Fourier Series and Exponential Fourier Series, Complex Fourier spectrum. Fourier Transforms: Deriving Fourier Transform from Fourier series, Fourier Transform of
arbitrary signal, Fourier Transform of standard signals, Fourier Transform of Periodic Signals, Properties of Fourier Transform, Fourier Transforms involving Impulse function and Signum function.
Sampling: Sampling theorem – Graphical and analytical proof for Band Limited Signals,
Page 25
Reconstruction of signal from its samples, Effect of under sampling – Aliasing.
UNIT III
Laplace Transforms and Z–Transforms: Laplace Transforms: Review of Laplace
Transforms (L.T), Partial fraction expansion, Inverse Laplace Transform, Concept of Region of Convergence (ROC) for Laplace Transforms, Constraints on ROC for various classes of
signals, Properties of L.T, Relation between L.T and F.T of a signal, Laplace Transform of certain signals using waveform synthesis. Z–Transforms: Fundamental difference between Continuous and Discrete time signals,
Discrete time signal representation using Complex exponential and Sinusoidal components, Periodicity of Discrete time signal using complex exponential signal, Concept of ZTransform
of a Discrete Sequence, Distinction between Laplace, Fourier and Z Transforms, Region of Convergence in Z-Transform, Constraints on ROC for various classes of signals, Inverse Z-transform, Properties of Z-transforms.
UNIT IV
Random Processes – Temporal Characteristics: The Random Process Concept, Classification of Processes, Deterministic and Nondeterministic Processes, Distribution and Density Functions, concept of Stationarity and Statistical Independence. First-Order
Stationary Processes, Second- Order and Wide-Sense Stationarity, (N-Order) and Strict- Sense Stationarity, Time Averages and Ergodicity, Autocorrelation Function and Its
Properties, Cross-Correlation Function and Its Properties, Covariance Functions, Gaussian Random Processes, Poisson Random Process. Random Signal, Mean and Mean-squared Value of System Response, autocorrelation Function of Response, Cross-Correlation
Functions of Input and Output.
.UNIT V
Random Processes – Spectral Characteristics: The Power Spectrum: Properties,
Relationship between Power Spectrum and Autocorrelation Function, The Cross-Power Density Spectrum, Properties, Relationship between Cross-Power Spectrum and Cross- Correlation Function. Spectral Characteristics of System Response: Power Density Spectrum
of Response, Cross-Power Density Spectrums of Input and Output.
TEXT BOOKS:
1. Signals, Systems & Communications - B.P. Lathi , 2013, BSP.
2. Signal and systems principles and applications, shaila dinakar Apten, Cambridez university press, 2016.
3. Probability, Random Variables & Random Signal Principles - Peyton Z. Peebles, MC GRAW HILL EDUCATION, 4th Edition, 2001
REFERENCE BOOKS:
1. Signals and Systems - A.V. Oppenheim, A.S. Willsky and S.H. Nawab, 2 Ed.,
2. Signals and Signals – Iyer and K. Satya Prasad, Cengage Learning
NPTEL Web Course:
1. http://nptel.ac.in/courses/117104074
NPTEL Video Course:
1. http://nptel.ac.in/courses/117104074
GATE Syllabus
Definitions and properties of Laplace transform, continuous-time and discrete-time Fourier
series, continuous-time and discrete-time Fourier Transform, DFT and FFT, z-transform.
Sampling theorem. Linear Time-Invariant (LTI) Systems: definitions and properties;
causality, stability, impulse response, convolution, poles and zeros, parallel and cascade
structure, frequency response, group delay, phase delay. Signal transmission through LTI
systems.
IES Syllabus
Classification of signals and systems: System modeling in terms of differential and difference
equations; State variable representation; Fourier series; Fourier transforms and their
application to system analysis; Laplace transforms and their application to system analysis;
Convolution and superposition integrals and their applications; Z-transforms and their
applications to the analysis and characterization of discrete time systems; Random signals
and probability, Correlation functions; Spectral density; Response of linear system to random
inputs.
VIII. COURSE PLAN (WEEK- WISE):
Sess
ion
Week
Un
it
To
pic
s
Co
urse
Lea
rn
ing
Ou
tco
mes
Refe
ren
ce
1
1
1 Signal Analysis: Analogy between Vectors and
Signals
Discuss the analogy between
vectors and signals.
T1, R1
2 Signal approximation using Orthogonal functions Describe the signal
approximation using
orthogonal functions
T1, R1
3 Closed or complete set of Orthogonal functions, Describe the signal
approximation using
orthogonal functions
T1, R1
4 Orthogonality in Complex functions, Exponential
and Sinusoidal signals,
Discuss aboutExponential and
sinusoidal signals
T1, R1
5 Concepts of Impulse function, Unit Step function,
Signum function
Concepts of Impulse
function, Unit step function,
Signum function.
T1, R1
6
2
Signal Transmission through Linear Systems Demonstration of Linear
system
T1, R1
7 Linear System, Impulse response Demonstration of Linear
system
T1, R1
8 Response of a Linear System, Linear Time
Invariant (LTI) System
Demonstration of LTI
System
T1, R1
9 Linear Time Variant (LTV) System, Transfer
function of a LTI system,
Compute Transfer function
of a LTI system.
T1, R1
10 1 Filter characteristics of Linear Systems, Discuss Filter characteristics
of linear systems
T1, R1
11
3
Distortion less transmission through a system,
Signal bandwidth,
Discuss Filter characteristics
distortion less transmission
through a system, Signal
bandwidth
T1, R1
12 System bandwidth, Ideal LPF, HPF and
BPF characteristics, Causality and Paley-Wiener
Discuss characteristics of
system bandwidth, Ideal LPF,
T1, R1
Page 27
criterion for physical realization, HPF and BPF
13 Relationship between Bandwidth and Rise time. Analyze Relationship
between bandwidth and rise
time
T1, R1
14 Concept of convolution in Time domain and
Frequency domain
Express concept of
convolution in time domain
and frequency domain
T1, R1
15 Graphical representation of Convolution,
Convolution property of Fourier Trans forms
Express concept of
convolution in time domain
and frequency domain,
Graphical representation of
convolution
T1, R1
16
4
2
Fourier Series, Transforms and Sampling Illustrate Fourier
series, Transforms and
Sampling
T1, R1
17 Representation of Fourier series, Continuous time
periodic signals, Properties of Fourier Series,
Dirichlet‘s conditions,
Illustrate Fourier series,
Continuous time periodic
signals,properties of Fourier
series, IllustrateDirichlet‘s
conditions,
T1, R1
18 Trigonometric Fourier Series Illustrate,Trigonometric
Fourier series
T1, R1
19 Exponential Fourier Series, Complex Fourier
spectrum.
Illustrate and Exponential
Fourier series, Complex
Fourier spectrum
T1, R1
20 Deriving Fourier Transform from Fourier series,
Fourier Transform of arbitrary signal,
Compute Fourier transform
from Fourier series, Fourier
transform of arbitrary signal,
T1, R1
21
5
Fourier Transform of standard signals, Fourier
Transform of Periodic Signals
Compute Fourier transform of
standard signals, Fourier
transform of periodic signals
T1, R1
22 Properties of Fourier Transform, Fourier
Transforms involving Impulse function and
Signum function.
Illustrate the Properties of
Fourier transforms, Fourier
transforms involving impulse
function andSignum function
T1, R1
23 Sampling theorem – Graphical and analytical proof
for Band Limited Signals
Illustrate Sampling theorem
and , Types of sampling
T1, R1
24 Reconstruction of signal from its samples, Illustrate Reconstruction of
signal from its samples
T1, R1
25 Effect of under sampling – Aliasing. Illustrate Effect of under
sampling – Aliasing.
T1, R1
26
6
3
Laplace Transforms and Z–Transforms:
Laplace Transforms
Describe Laplace transforms,
and Z–Transforms
T1, R1
27 Review of Laplace Transforms (L.T), Partial
fraction expansion
Describe Laplace transforms,
Partial fraction expansion,
T1, R1
28 Inverse Laplace Transform, Concept of Region
of Convergence (ROC) for Laplace Transforms
Describe Inverse Laplace
transform Concept of region
of convergence
(ROC) for Laplace
transforms.
T1, R1
29 Constraints on ROC for various classes of signals,
Properties of L.T, Relation between L.T and F.T of
a signal
Examine theconstraints on
ROC for various classes of
signals DescribeProperties of
L.T‘s relation between
L.T‘s, and F.T. of a signal.
T1, R1
30 Laplace Transform of certain signals using
waveform synthesis.
Describe Laplace transform
of certain signals using
waveform synthesis.
T1, R1
31
7
Fundamental difference between Continuous and
Discrete time signals
Examine thefundamental
difference between
continuous and discrete time
signals
T1, R1
32
3
Discrete time signal representation using Complex
exponential and Sinusoidal components
Analyzediscrete time signal
representation using complex
exponential and
sinusoidal components
T1, R1
33 Periodicity of Discrete time signal using complex
exponential signal, Concept of ZTransform
of a Discrete Sequence
Analyze Periodicity of
discrete time using complex
exponential signal
T1, R1
34 Distinction between Laplace, Fourier and Z
Transforms,
Region of Convergence in Z-Transform
Describe Distinction between
Laplace, Fourier and Z
transforms, Region of
convergence
T1, R1
35 Constraints on ROC for various classes of signals Describe constraints on
ROC for various classes of
signals
T1, R1
36
8
Inverse Z-transform, Properties of Z-transforms Describe conceptof Inverse Z-
transform, Properties of Z-
transforms
T1, R1
37
4
Random Processes - temporal Charecteristics Understand the concept of The Random Process
T1, R1
38 The Random Process concept, Classification of
processes
Understand the concept of
The Random Process
T3
39 Deterministic and Non – Deterministic Processes Classification of Processes, Deterministic and
Nondeterministic Processes
T3
40 Distribution and Density functions, concept of
stationarity and statistical independence
Understand the concept of
Distribution and Density
Functions, concept of stationarity and statistical
independence
T3
41
9
First-Order Stationary Processes, Second- Order
and Wide-Sense Stationarity, (N-Order) and Strict-
Sense Stationarity, Time Averages and Ergodicity,
Autocorrelation Function and Its Properties,
Understand the concept of
First-Order Stationary Processes, Second- Order and
Wide-Sense Stationary, (N-Order) and Strict-Sense
Stationary, Autocorrelation
Function and Its Properties,
T3
42 Cross-Correlation Function and Its Properties,
Covariance Functions, Gaussian Random
Processes,
Understand the concept of Cross- Correlation Function
and Its Properties, Covariance
Functions, Gaussian Random Processes,
T3
43 Poisson Random Process. Random Signal, Understand the concept of Gaussian Random Processes,
Poisson Random Process, Random signal
T3
44 Mean and Mean-squared Value of System
Response, autocorrelation Function of Response,
Understand the concept of
Mean and Mean-squared
Value of System Response,
autocorrelation Function of
Response,
T3
45 Cross-Correlation Functions of Input and Output. Understand the concept of
Cross-Correlation Functions of
Input and Output.
T3
46
Random Processes – Spectral Characteristics:
The Power Spectrum: Properties,
Understand the concept of
The Power Spectrum
T3
47 Relationship between Power Spectrum and Understand Relationship T3
Page 29
10
5
Autocorrelation Function, between Power Spectrum and
Autocorrelation Function,
48 The Cross-Power Density Spectrum Understand the concept of
The Cross-Power Density
Spectrum
T3
49 Density Spectrum Properties, Understand the concept of
Density Spectrum Properties,
T3
50 Relationship between Cross-Power Spectrum and
Cross-Correlation Function.
Understand Relationship
between Cross-Power
Spectrum and Cross-
Correlation Function.
T3
51
11
Spectral Characteristics of System Response: Understand the concept of
Spectral Characteristics of
System Response:
T3
52 Power Density Spectrum of Response, Understand the concept of
Power Density Spectrum of
Response,
T3
53 Cross-Power Density Spectrums of Input and
Output.
Understand the concept of
Cross-Power Density
Spectrums of Input and
Output.
T3
IX. MAPPING COURSE OUTCOMES LEADING TO THE ACHIEVEMENT OF
PROGRAM OUTCOMES AND PROGRAM SPECIFIC OUTCOMES:
Course Outco
mes
Program Outcomes Program Specific Outcomes
PO1 PO
2
PO
3
PO
4
PO
5
PO
6
PO
7
PO
8
PO
9
PO10 PO1
1
PO
12
PSO
1
PS
O2
PSO
3
CO1 3 3 2 2 2 1 2 - 2 3 2 - 2 2 3
CO2 2 2 2 2 2 2 1 - 2 2 3 - 3 1 2
CO3 CO4
3 3 3 3 3 2 2 - 3 2 2 - 2 2 3
2 3 2 2 2 2 1 - 2 2 2 - 2 2 3
CO5 3 2 3 3 3 1 2 - 3 3 3 - 2 1 2
CO6 3 2 3 3 3 1 1 - 3 3 3 - 2 1 2
AVG 2.66 2.5 2.5 2.5 2.5 1.5 1.5 0 2.5 2.5 2.5 0 2.16 1.5 2.5
X. QUESTION BANK (JNTUH)
UNIT 1
Long Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 A rectangular function is defined as
Approximate the above function by a single sinusoid sint
between the intervals (0,2π) , Apply the mean square error in
this approximation.
Remember
1
2 Show that f(t) is orthogonal to signals cost, cos2t, cos3t, … cosnt for all integer values of n, n≠0, over the interval (0,2π) if
Apply
1
3 Sketch the following signals
ii) f(t)=3u(t)+tu(t)-(t-1)u(t-1)-5u(t-2)
Understand 1
4 Apply the following integrals
ii)
Apply 1
5 Determine whether each of the following sequences are periodic or not, if periodic determine the fundamental period. x(n)= sin(6πn/7) ii) y(n)= sin(n/8)
Remember 2
6 Determine whether the following input-output equations are linear or non linear. y(t)=x
2(t) b) y(t)=x(t
2) c) y(t)=t
2x(t-1) d) y(t)=x(t)
cos 50πt
Understand 3
7 Find whether the following system are static or dynamic
y(t)= x(t2) b) y(t)=e
x(t) c)
Apply 3
8 Find whether the following systems are causal or non-causal y(t)=x(-t) b) y(t)=x(t+10)+x(t) c) y(t)=x(sin(t)) d) y(t)=x(t) sin(t+1)
Apply 3
9 Find the impulse response of a system characterized by the differential equations
a)
b)
Where x(t) is the input and y(t) is the output
Apply 3
10 Test whether the system described in the figure is BIBO stable or not
Understand 4
Short Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 Define Signal. Remember 1
2 Define system. Understand 1
3 What are the major classifications of the signal? Understand 1
4 Define discrete time signals and classify them Remember 1
5 Define continuous time signals and classify them. Understand 1
6 What are the Conditions for a System to be LTI System? Remember 3
7 Define time invariant and time varying systems. Understand 3
8 Is the system describe by the equation y(t) = x(2t) Time invariant or not? Why?
Understand 3
9 What is the period T of the signal x(t) = 2cos (n/4)? Remember 3
10 Is the system y(t) = y(t-1) + 2t y(t-2) time invariant ? Understand 3
Page 31
UNIT II
Long Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 Find the fourier series expansion of the periodic triangular wave shown below for the interval (0,T) with amplitude of ‗A‘
Apply 5
2 Find the exponential fourier series for the fullwave rectified sinewave as shown below for the interval (0,2π) with an amplitude of ‗A‘
Remember 5
3 Obtain the trigonometric fourier series for the periodic rectangular waveform as shown below for the interval (-T/4,T/4)
Apply 5
4 Distinguish between the exponential form of the fourier
series and fourier transform. What is the nature of the
‗transform pair‘ in the above two cases
Remember 5
5 Find the fourier transform of the following a) real exponential, x(t)= e
-at u(t), a>0
b) rectangular pulse,
x(t)= eat u(-t), a>0
Apply 5
6 Find the fourier transforms of cos wt u(t) b) sin wt u(t) c) cos (wt+Ø) d) e
jwt
Remember 5
7 Find the fourier transforms of the trapezoidal pulse as shown below
Apply 5
8 There are several possible ways of estimating an essential bandwidth of non-band limited signal. For a low pass signal, for example, the essential bandwidth may be chosen as a frequency where the amplitude spectrum of the signal decays to k% of its peak value. The choice of k depends on the nature of application. Choosing k=5, determine the essential bandwidth of g(t)= e
-at u(t).
Apply 4
9 For the analog signal x(t)=3 cos 100πt a) Determine the minimum sampling rate to avoid
aliasing b) Suppose that the signal is sampled at the rate,
fs=200Hz, what is the discrete time signal obtained after sampling
c) Suppose that the signal is sampled at the rate, fs=75Hz, what is the discrete time signal obtained after sampling
What is the frequency 0<f< fs/2 of a sinusoid that yields samples identical to those obtained in (c) above
Understand 6
Short Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 State Time Shifting property in relation to fourier series. Understand 5
2 Obtain Fourier Series Coefficients for 𝑥(𝑛) = 𝑠𝑖𝑛𝑤0𝑛 Remember 5
3 What are the types of Fourier series? Remember 5
4 State properties of fourier transform. Understand 5
5 Define Fourier transform pair. Remember 5
6 Explain time shifting property of fourier transform Apply 6
7 Find the fourier transform of x(t)=cos(wt) Apply 5
8 What is an antialiasing filter? Apply 6
9 What is the condition for avoid the aliasing effect? Apply 6
10 What is the Nyquist‘s Frequency for the signal x(t) =3 cos 100t +10 sin 30t – cos50t ?
Apply 6
UNIT III
Long Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 Determine the function of time x(t) for each of the following Laplace transforms and their associated region of convergence
i) ii)
Understand 5
2 Consider the following signals, find Laplace transform and region of convergence for each signal a) e
-2t u(t) + e
-3t u(t) b) e
-4t u(t) + e
-5t sin 5t u(t)
Apply
5
3 State the properties of Laplace transform Understand 5
4 Determine the function of time x(t) for each of the following Laplace transforms
a) b) c)
Remember
5
5 Determine the Laplace transform and associated region of convergence for each of the following functions of time
i) x(t) = 1; 0 ≤ t ≤ 1 ii) x(t)=
iii) x(t)= cos wt
Apply
5
Page 33
6 Properties of ROC of Laplace transforms Understand 5
7 Find the inverse Z-transform of X(z)= ; |z|>2
using partial fraction
Understand 5
8 Find inverse z-transform of X(z) using long division method
X(z)=
Remember 5
9 Properties of Z-transforms? Apply 5
10 Find the inverse z-transform of X(z)=
Understand 5
Short Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 What is the use of Laplace transform? Understand 5
2 What are the types of laplace transform? Remember 5
3 Define Bilateral and unilateral laplace transform. Understand 5
4 Define inverse laplace transform. Remember 5
5 State the linearity property for laplace transform. Apply 5
6 Define Z transform. Understand 5
7 What are the two types of Z transform? Understand 5
8 Define unilateral Z transform. Apply 5
9 What is the time shifting property of Z transform. Apply 5
10 What is the differentiation property in Z domain Apply 5
UNIT IV
Long Answer Questions
S.No
Question
Bloom’s
Taxonomy Level
Course outcome
1 Given x=6 and Rxx(t,t+ τ)= 36+25 exp(-τ) for a random process
X(t)
.indicate which of the following statements are true based on what
is known
with certainty: X(t)
i. is first order stationary
ii. has total average power of 61W
iii. is ergodic
Understand
6
2 (a) State and prove the properties of Autocorrelation function.
(b) Show that the process X(t)= A Cos (w0t+θ) is wide sense
stationery if it is
assumed that A
and w0 are constants and θ is uniformly distributed random variable
over the
interval (0,2π).
Remember
5
3 a) Write the conditions for a Wide sense stationary random process.
(b) Let two random processes X(t) and Y(t) be defined by X(t) = A
Cos(w0t)+Bsin(wot) and Y(t) = Bcos(wot)-Asin(wot).where A and
B are random
variables and wo is constant. Show that X(t) and Y(t) are jointly
Remember 6
wide sense
stationery , assume A and B are uncorrelated zero-mean random
variables with
same variable
4 a) State and prove the properties of Cross correlation function.
(b) Find the mean and auto correlation function of a random
process X(t)=A ,
where A is continuous random variable with uniform distribution
over (0,1).
Remember
5
5 State and prove any four properties of cross covariance function. Remember 4
6 Explain classification of random process Understand 6
7 Explain wide sense stationary random process? Understand 6
8 State and prove any four properties of cross correlation function. Remember 5
9 State and prove any four properties of auto correlation function. Remember 5
10 Define second order stationary process? Remember 6
Short Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 Define random process? Remember 6
2 Define ergodicity? Remember 6
3 Define wide sense stationary random process? Remember 6
4 Define auto correlation function of a random process? Remember 6
5 Define cross correlation function of a random process? Remember 6
6 Define mean ergodic process? Remember 6
7 Define correlation ergodic process? Remember 6
8 Define strict sense stationary random process? Remember 6
9 Define auto correlation function of a random process? Remember 6
10 State the condition for a wide sense stationary random process
Remember 6
UNIT V
Long Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 Given x=6 and Rxx(t,t+ τ)= 36+25 exp(-τ) for a random process X(t) .indicate which of the following statements are true based on what is known with certainty: X(t) i. is first order stationary ii. has total average power of 61W
Apply 6
2 Explain the concept of power spectral density and power spectrum
Remember 5
3 State and prove any four properties of auto covariance function.
Remember 5
4 A random processes X(t)= Asin(wt+θ) , where A , w are constants and θ is a uniformly distributed random variable on the interval (-π ,π) .define a new random processes Y(t)= X2(t).
Understand 5
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i. Find the auto correlation function of Y(t) ii. Find the cross correlation function of X(t) and Y(t) iii. Are X(t) and Y(t) wide sense stationary Are X(t) and Y(t) jointly wide sense stationary
5 A wide sense stationary process X(t) has autocorrelation function R X (τ ) =Ae−b|τ where b > 0. Derive the power spectral density function S X( f ) and calculate the average power E[X2(t)].
Analysis 6
Short Answer Questions
S.No
Question
Bloom’s
Taxonomy
Level
Course outcome
1 State any two uses of spectral density. Remember 5
2 Define Spectral analysis? Remember 5
3 Define Spectral density? Remember 5
4 Define cross correlation and its properties. Remember 5
5 State any two properties of cross correlation. Remember 5
6 State any two properties of cross-power density spectrum. Remember 5
7 Define cross –spectral density and its examples. Remember 6
8 State any two properties of an auto correlation function. Remember 5
9 Prove that RXY(t) = RYX(-t) Remember 5
10 Define wiener khinchine relations Remember 6
OBJECTIVE TYPE QUESTIONS
UNIT I
1. The differentiation of unit step signal u(t) is
a) sgn(t) b) r(t) c) б(t) d) none of these
Answer : c
2. Two vectors are said to be orthogonal if their dot product is
a) infinity b) zero c) one d) none of these
Answer : b
3. If we approximate a function by its orthogonal function, the error will be
a) infinity b) large c) zero d) small
Answer : d
4. The relation between unit step function and signum function is
a) sgn(t)=2u(t)-1 b) sgn(t)=2u(t)+1 c) sgn(t)=2u(t) d) none of these
Answer : a
UNIT II
1. Half wave symmetry is also called
a) Rotation symmetry b.Mirror symmetry c.Full symmetry d.Even symmetry
Answer : a
2. The coefficient an is zero for __________ functions
a) Even b.Odd c.Both a and b d.None of these
Answer : b
3. Fourier series could be applied to
a) Power signal
b) Energy signal
c) Aperiodic signal
d) Unit step signal
Answer : a
4. The fourier series of an odd periodic function contains only
a) Odd harmonics
b) Even harmonics
c) Cosine terms
d) Sine terms
Answer : d
5. The fourier transform of real valued signal has
a) Odd symmetry
b) Even symmetry
c) Conjugate symmetry
d) No symmetry
Answer : c
UNIT IV
1. The collection of all the sample functions is referred to as
a) ensemble b) assumble
c) average d) set
Answer : a
2. If the future value of a sample function cannot be predicted based on its past values, the
process is referred to as
a) deterministic process b )non-deterministic process
c) independent process d) statistical process Answer : b
3. For the random process X(t)=Acoswt where wt is a constant and A is uniform random
variable over (0, 1), the mean square value is
a) 1/3
b) 1/3 coswt
c)1/3 cos2wt
d) 1/9
Answer : c
4. For an ergodic process
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a) mean is necessarily zero
b) mean square value is infinity
c) all time averages are zero
d) mean square value is independent of time
Answer : d
UNIT V
1. For an ergodic process
a) Mean is necessarily zero
b) Mean square value is infinity
c) All time averages are zero
d) Mean square value is independent of time
Answer: d
2. A stationary random process X(t) is periodic with period 2T. it‘s autocorrelation
function is
a) Non-periodic
b) Periodic with period T
c) Periodic with period 2T
d) Periodic with period T/2
Answer: c
3. The mean square value for the poisson process X(t) with parameter λt
a) λt
b) parameter (λt)2
c) (λt)+ (λt)2
d) (λt)- (λt)2
Answer: c
4. The difference of two independent Poisson process is
a) Poisson process
b) Not a Poisson process
c) Process with mean=0, [λ1t ≠ λ2t]
d) Process with variance=0, [λ1t ≠ λ2t]
Answer: b
5. A white noise process will have
a) A zero mean
b) A constant variance
c) Autocovariances that are constant
d) None Answer: b
GATE Objective Questions
1. Tile trigonometric Fourier series for the waveform f(t) ,shown below contains,
(A) Only cosine terms and zero value for the dc component
(B) Only cosine terms and a positive value for the dc component
(C) Only cosine terms and a negative value for the dc component
(D) Only sine terms and a negative for the dc component
Answer : c
2. Consider the z-transform ; 0 <|z| < ∞ . The inverse z-transform
x[n] is
(A) 5δ[n + 2] + 3δ[n] + 4δ[n – 1]
(B) 5δ[n - 2] + 3δ[n] + 4δ[n + 1]
(C) 5 u[n + 2] + 3 u[n] + 4 u[n – 1]
(D) 5 u[n - 2] + 3 u[n] + 4 u[n + 1]
Answer : b
3. Two discrete time systems with impulse responses h1[n] = δ[n -1] and h2[n] = δ[n– 2] are
connected in cascade. The overall impulse response of the cascaded system is
(A) δ[n - 1] + δ[n - 2]
(B) δ[n - 4]
(C) δ[n - 3]
(D) δ[n - 1] δ[n - 2]
Answer : c
4. For an N-point FFT algorithm with which one of the following statements is
TRUE?
(A) It is not possible to construct a signal flow graph with both input and output in normal
order
(B) The number of butterflies in the stage is N/m
(C) In-place computation requires storage of only 2N node data
(D) Computation of a butterfly requires only one complex multiplication
Answer : d
5. The Fourier series of a real periodic function has only
P. cosine terms if it is even
Q. sine terms if it is even
R. cosine terms if it is odd
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S. sine terms if it is odd
Which of the above statements are correct?
(A) P and S
(B) P and R
(C) Q and S
(D) Q and R
Answer : a
XI. WEBSITES:
1. https://www.edx.org/counse/signals-systems-part-1-iitbombay-ee210-1x-1
2. nptel.ac.in/courses/117104074
3. dsp.rice.edu/courses/elec301
XII. EXPERT DETAILS:
Alan V. Oppenheim received the S.B. and S.M. degrees in 1961 and the Sc.D. degree in
1964, all in electrical engineering, from the Massachusetts Institute of Technology. He is also
the recipient of an honorary doctorate from Tel Aviv University, of Technology. He is also
the recipient of an honorary doctorate from Tel Aviv University, which was conferred upon
him in 1995. In 1964, Dr.Oppenheim joined the faculty at MIT, where he is currently Ford
Professor of Engineering and a MacVicar Faculty Fellow. Since 1967 he has been affiliated
with MIT Lincoln Laboratory and since 1977 with the Woods Hole Oceanographic
Institution. His research interests are in the general area of signal processing and its
applications. He is coauthor of the widely used textbooks Discrete-Time Signal Processing
and Signals and Systems. He is also editor of several advanced books on signal processing
XIII. JOURNALS:
1. IEEE Journal on Selected Areas in Communications (Impact factor 3.121)
2. IEEE Transactions on Signal Processing (Impact factor 2.813)
3. IEEE Transactions on Circuits and Systems (Impact factor 2.24)
4. IEEE Transactions on Audio, Speech, and Language Processing (Impact factor 1.675)
5. The Journal of the Acoustical Society of America (Impact factor 1.55)
6. EURASIP Journal on Advances in Signal Processing (Impact factor 0.81)
7. Journal of Signal Processing Systems (Impact factor 0.551)
XIV. LIST OF TOPICS FOR STUDENT SEMINARS:
1. Signal approximation using orthogonal functions.
2. Fourier series representation of periodic signals.
3. Fourier series properties.
4. Detection of periodic signals in the presence of noise by correlation.
XV. CASE STUDIES/ SMALL PROJECTS
1. Joint Estimation of I/Q Imbalance, CFO and Channel Response for MIMO OFDM
Systems
2. Interference Cancellation and Detection for More than Two Users.
