AE06/AC04/AT04 SIGNALS AND SYSTEMS
1. Introduction 3 hours
1.1
Continuous
time and discrete time signals.
1.2
Signal
energy and power.
1.3
Periodic
signals, even and odd signals, exponential and sinusoidal signals.
1.4
The
unit impulse and the unit step function.
1.5
Continuous
time and discrete time systems.
1.6
Interconnection
of systems.
1.7
Classification
of systems – linear and nonlinear, memory and memoryless, causal and
non-causal, stable and unstable, time-varying and time-invariant.
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2.1 Convolution for Discrete Time Systems (DTS).
2.2 Convolution for Continuous Time Systems (CTS).
2.3 Properties of Linear Time-Invariant (LTI) systems – commutative, distributive, associative.
2.4 Characterization of LTI systems – with and without memory, stable and unstable, causal and non-causal.
2.5 Unit step response of LTI systems.
2.6 Causal LTI systems described by differential and difference equations.
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3. Fourier Series Representation of Periodic Signals 6 hours
3.1
Response
of LTI systems to complex exponential.
3.2
Fourier
series representation of continuous time periodic signals.
3.3
Convergence
conditions.
3.4
Properties
of continuous time Fourier series – linearity, time shifting, time reversal,
time scaling, multiplication, Parseval’s relation.
3.5
Application
of Fourier series to analysis of linear systems.
3.6
Discrete
time Fourier series – its finiteness and properties, in brief, leading to the
Discrete Fourier Transform (DFT).
4. Continuous-time Fourier Transform 6 hours
4.1 Aperiodic signals and the Continuous time Fourier Transform (CFT).
4.2 Convergence of CFT.
4.3 CFT of periodic signals.
4.4 Properties of the CFT – linearity, time shifting, conjugation and conjugate symmetry, differentiation and integration, time and frequency scaling, duality, Parseval’s relation.
4.5 CFT and convolution.
4.6 Multiplication property.
4.7 Application of CFT to LTI system analysis.
5. Discrete-time Fourier Transform
6 hours
5.1 Aperiodic signals and the Discrete Time Fourier Transform (DTFT).
5.2 Convergence of DTFT.
5.3 DTFT of periodic signals.
5.4 Properties of the DTFT – periodicity, linearity, time and frequency shifting, conjugation and conjugate symmetry, differencing and accumulation, time reversal, time expansion, differentiation in frequency, Parseval’s relation.
5.5 DTFT and convolution .
5.6 Multiplication property .
5.7 Application of DTFT to LTI system analysis.
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6. Time and Frequency Characterization of Linear Systems 3 hours
6.1
Magnitude
phase representation of the frequency response of LTI systems.
6.2
Importance
of linear phase.
6.3
Group
delay.
6.4
Time
domain properties of ideal frequency selective filters.
6.5
Time
and frequency domain aspects of nonideal filters.
6.6
First
and second order continuous time systems.
6.7
First
and second order discrete time systems.
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7. Sampling 3 hours
7.1 The sampling theorem.
7.2 Zero order hold.
7.3 Reconstruction of signals from its samples.
7.4 Aliasing.
7.5 Discrete time processing of continuous time signals.
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8. Laplace Transform 9 hours
8.1
Definition
of the Laplace Transform (LT).
8.2
Region
of convergence.
8.3
Poles
and zeros.
8.4
Inverse
LT.
8.5
Graphical
evaluation of CFT from LT pole zero plot.
8.6
Properties
of the LT – linearilty, time shifting, shifting in the s-domain, time scaling,
conjugation, convolution, differentiation in the time domain, differentiation
in the frequency domain.
8.7
Initial
and final value theorems .
8.8
Standard
LT pairs.
8.9
Analysis
and characterization of LTI systems using the LT.
8.10
The
unilateral LT – Examples, Properties, Applications.
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9. Z-Transform 9 hours
9.1
Definition
of the Z-Transform (ZT) .
9.2
Region
of convergence.
9.3
Poles
and zeros .
9.4
The
inverse ZT .
9.5
Graphical
evaluation of the DTFT from pole zero plot.
9.6
Properties
of the ZT – linearity, time shifting, scaling in the z-domain, time reversal,
time expansion, conjugation, convolution, differentiation in the z-domain.
9.7
Initial
and final value theorems .
9.8
Standard
ZT pairs.
9.9
Analysis
and characterization of LTI systems using the ZT.
9.10
The
unilateral ZT; Examples; Properties ; Applications.
10. Random Signals and
Systems 12 hours
10.1
Basic
concepts of probability and Random Variables (RV).
10.2
Distribution
and density functions.
10.3
Mean
values and moments.
10.4
The
Gaussian RV.
10.5
Joint
probability.
10.6
Statistical
independence.
10.7
Random
Processes (RP) – continuous and discrete, deterministic and non-deterministic,
stationary and non-stationary, ergodic and non-ergodic.
10.8
Correlation
functions – auto and cross, and their properties.
10.9
Sums of RP’s.
10.10
Spectral
density and its properties.
10.11
Mean
squared value from spectral density.
10.12
Spectral
density and autocorrelation function (ACF).
10.13
White
noise.
10.14
Cross
spectral density.
10.15 Response of Linear systems to random inputs – mean and mean squared values of the output, ACF of system output, cross correlation between input and output spectral density at the output, cross spectral density between input and output.
I.
A V Oppenheim, A S Willsky and S H Nawab, “Signals and Systems”,
Third Indian
Reprint,
Prentice Hall, 2002.
II. G R Cooper and C D
McGillam, “Methods of Signal and System Analysis”, Holt,
Rinehart and Winston, 1990.
III.
S.Hary
Kin, “Communication Systems”, 3rd Edition, John Willey, 1994.
IV.
R
E Ziemen and W H Tranter, “Principles of Communication”, 3rd
Edition, Houghton Mifflin, 1990.
V. B P Lathi, “ Modern Digital and Analog Communication System”, Indian Edition by Prism Books, 1993.