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
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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
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.
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
Page 35
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
Page 37
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