SIGNALS and
SYSTEMS
Lectures
Sr.# Lecture Topic CLO LEVEL
required
1. Continuous-time (CT) and discrete-time (DT) signals 01 1 C2
2. Signal energy and power, Time shift, Reversal, Scaling 02 1 C2
3. Periodic signals, Even and odd signals 01 1 C2
4. CT and DT complex exponential and sinusoidal signals 02 1 C2
5. Periodicity properties 01 1 C2
6. Unit impulse and unit step signals 01 1 C2
7. Memory, Invertibility, Causality, Stability, Time Invariance, 1 C2
02
Linearity
8. DT and CT representation in terms of impulses 01 2 C4
9. DT Unit impulse response 01 2 C4
10. Convolution-Sum representation of LTI Systems 02 2 C4
11. CT Unit impulse response 01 2 C4
12. Convolution-Integral representation of LTI Systems 02 2 C4
13. Fourier series representation of continuous and discrete time 3 C5
01
periodic signals
14. Properties of continuous and discrete-time Fourier series 01 3 C5
15. Continuous and discrete-time Fourier transform 02 3 C5
16. Sampling 01 3 C5
17. Laplace and Z transforms 02 3 C5
18. Region of convergence 02 3 C5
19. BIBO stability 02 3 C5
20. LTV systems 02 3 C5
Total 30
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After successful completion of the course, student will be able to,
CLO Taxonomy Level PLO
1 Comprehend various types of signals and systems C2 1
2 Analyze signals and systems in time and frequency domains C4 2,5
Design continuous and discrete time systems with desired properties.
3 C5 3, 5
RELEVANT PROGRAM LEARNING OUTCOMES (PLOs):
The course is designed so that students will achieve the following PLOs:
1 Engineering Knowledge √ 7 Environment and Sustainability:
2 Problem Analysis: √ 8 Ethics:
3 Design/Development of Solutions: √ 9 Individual and Team Work:
4 Investigation: 10 Communication:
5 Modern Tool Usage: √ 11 Project Management:
6 The Engineering Society: 12 Lifelong Learning:
3
4
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SIGNAL
A signal is a description of how one
parameter varies with another
parameter. For instance, voltage
changing over time in an electronic
circuit, or brightness varying with
distance in an image.
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SIGNAL (EXAMPLES)
Human voice
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SIGNAL (EXAMPLES)
Sine wave
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SYSTEM
System is a described by input and output. A
system is composed of interconnected
elements to perform some action.
System could be:
• a mathematical model (eg. f = m a or a =
1/m f) [y = mx]
• a piece of code/software
• a physical device
• a black box
whose input is a signal and it performs some
processing on that signal, and the output is a
signal. 9
SYSTEM (Example)
A television station is a system. There
are cameras, microphones, audio/video
recording devices, and equipment to
route and switch the signals from those
sources to a transmitter and antenna.
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SYSTEM (Example)
Input analog signal.
Output digital signal.
The system is a conversion system that
converts analog signals to digital signals.
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INTRODUCTION
In our modern world, signals of all kinds emanate
from different types of devices— radios and TVs,
cell phones, global positioning systems (GPS),
radars, and sonars.
to communicate messages
to control processes, and
to sense or measure signals
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EXAMPLES
Compact-Disk (CD) Player
Width of track = micrometer
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Height of bump = nanometer
Computer-control system for an analog plant
(e.g., cruise control for a car)
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CONTINUOUS and DISCRETE SIGNALS
The inputs and outputs of electrical,
mechanical, chemical, and biological
processes are measured as
functions of time with amplitudes
expressed in terms of voltage,
current, torque, pressure, etc. These
functions are called continuous-time
signals.
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CONTINUOUS and DISCRETE SIGNALS
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Continuous-time signals are converted into binary
sequences by means of an analog-to digital converter
(ADC) which converts the continuous-time signal into a
discrete-time signal such that each sample is represented
by a string of ones and zeros giving a binary signal.
Both time and signal amplitude are made discrete in this
process.
Likewise, digital signals can be transformed into
continuous-time signals by means of a digital-to-analog
converter (DAC) that uses the reverse process of the
ADC converter.
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Continuous and Discrete Representations
Significant differences between continuous-time
and discrete-time signals and in their processing.
Discrete-time signal x[n]:
Sequence of measurements typically made at
uniform times
Continuous-time signal x(t) :
Depends continuously on time.
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Discrete-time continuous-time signals
are related by a sampling process:
That is, the signal x[n] is obtained by sampling
x(t) at times t = nTs, where n is an integer and
Ts is the sampling period or the time between
samples.
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This above process is called sampling or
discretization of a continuous-time signal
A small value for Ts makes the continuous-
and the discrete-time signals look similar
(at the expense of memory space)
A large value of Ts improves memory
requirements (at the risk of losing
information contained in the original signal)
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CLASSIFICATION OF TIME-DEPENDENT SIGNALS
1. According to the predictability of their behavior
Signals can be random or deterministic.
Deterministic signal can be represented by
a formula or a table of values Random
signals can only be approached
probabilistically.
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2. According to the their variation with time
Continuous-time or discrete-time
3. According to whether the signals exhibit
repetitive behavior or not
periodic or aperiodic signals
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4. According to their energy content
Finite energy and finite-power signals or infinite energy
and infinite power.
5. According to their symmetry with
respect to the time origin
even, odd, or neither.
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CONTINUOUS-TIME SIGNALS
• These are functions of time carrying
information
tb = time at which this signal starts
tf = time at which it ends
The function v(t) varies continuously with
time, and its amplitude can take any
possible value.
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Time Amplitude Signal Type
Continuous Continuous Analog
Continuous Discrete Multi-Level
Discrete Continuous Discrete-time
Discrete Discrete Digital
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Time Amplitude Signal Type
Continuous Continuous Analog
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Time Amplitude Signal Type
Continuous Discrete Multi-Level
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Time Amplitude Signal Type
Discrete Continuous Discrete-time
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Time Amplitude Signal Type
Discrete Discrete Digital
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• Analog signal must be converted into a digital
signal because analog signal cannot be processed
with a computer as it would require to store an
innumerable number of signal values.
• Also for an accurate representation of the possible
amplitude values we would need a large number of
bits.
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• Thus, it is necessary to reduce the amount of data
without losing the information provided by the
signal.
• To accomplish that, we sample the signal by taking
signal values at equally spaced times nTs, where n
is an integer and Ts is the sampling period which
is appropriately chosen for this signal
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if we want each sample to be represented by
8 bits we have 28 or 256 possible levels
(quantization)
Assigning a unique binary code to each of the
levels we convert an analog signal into a
digital signal (coding)
The device that converts an analog signal into
a digital signal is called an analog-to-digital
converter or A/D converter
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Example
The signal x(t) is:
• deterministic, as the value of the signal can be
obtained for any possible value of t;
• analog, as there is a continuous variation of the
time variable t from −∞ to ∞
• The amplitude of the signal between − √2 to √2;
and, its frequency is = π/2 (rad/sec), and its
phase is π/4 rad
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What is amplitude, frequency and phase of the
signal plotted in following figure?
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Basic Signal Operations
Signal addition
two signals x(t) and y(t) are added to obtain their sum z(t).
An adder is used.
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Signal addition 40
Signal addition
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Signal addition
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Basic Signal Operations
Multiplication with constant
When a signal x(t) is multiplied by a constant α, a
constant multiplier is used.
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Basic Signal Operations
Multiplication with constant
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Basic Signal Operations
Time shifting
The signal x (t) is delayed seconds to get
X (t − ) and advanced by to get x (t + ).
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Original Signal
Delayed Signal
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Original Signal
Advanced Signal
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Original Signal
Reflected Signal
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EXAMPLE
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EXAMPLE
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Note: when signal is a function of −t (reflected):
• −t + becomes reflection and delay
• −t − becomes reflection and advancing
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Basic Signal Operations
Time scaling
The time variable of a signal x(t) is
scaled by a constant α to give
x(αt). If α = −1, the signal is
reversed in time, i.e., x(−t), or
reflected.
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Basic Signal Operations
Time windowing
• Most digital signals are infinite, or sufficiently large
• So dataset cannot be manipulated as a whole
• Hence, small subsets of the total data are analyzed,
through a process called windowing.
A signal x(t) is multiplied by a window signal w(t)
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Even and Odd Signals
Even: x ( - t ) = x ( t )
Odd: x ( - t ) = - x ( t )
x ( -t ) indicates time reversal
- x ( t ) indicates amplitude reversal
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Example Even Signals
x(-t)=x(t)
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Example Odd Signals
x(-t) = -x(t)
-x(-t) = x(t)
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Even × Even = Even
Odd × Odd = Even
Even × Odd = Odd
Even ± Even = Even
Odd ± Odd = Odd
Even ± Odd = Neither even nor odd
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A signal may be composed of even and odd
parts:
x (t ) = xe (t ) + xo (t ) ………………… Eq. 1
By definition:
x(-t) = x(t) if x (t) is even, and
-x(-t) = x(t) if x (t) is odd
Putting this in Eq. 1;
x(-t) = xe (-t ) + xo (-t )
x(-t) = xe (t ) - xo (t ) …………………….Eq. 2
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x (t ) = xe (t ) + xo (t ) ……………Eq. 1
x (-t) = xe (t ) - xo (t ) …………… Eq. 2
Adding equations 1 and 2
Subtracting equation 2 from 1
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Example
Find the even and odd parts of the signal
x (t)= t + 2
t +t3
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Example
Consider the analog signal
x (t) = cos (2 π t + θ), −∞ < t < ∞
Determine the value of θ for which
x(t) is even and for which it is odd.
If θ = π/4 is x(t) even or odd?
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x(t) = cos (2 πt + θ)
x(-t) = cos (-2 πt + θ)
= cos (-(2 πt – θ))
= cos (2 πt – θ)
x(t) is even if x(t) = x(−t)
Or if
cos(2 πt + θ) = cos(2 πt – θ)
Therefore for θ =0 or π, x(t) is even.
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x(t) is odd if x(-t) = - x(-t)
Or if
cos ( 2 π t - θ) = - cos (2 π t + θ)
Assignment 1 Question 1: due
date 1.3.21 Hand written
1. Find values of θ from above
equation for which function is
odd.
2. Check if the function is odd or
even for θ = π/4 69
Assignment 1 Question 2:
Which one of the following functions
are even or odd or neither?
a) sin(t)
b) cos(t)
c) sin(t)cos(t)
d) tsin(t)
e) t2
f) sin(t)+cos(t)
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(a) Since sin(-t) = - sin (t);
Therefore ODD
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(b) Since cos (-t) = cos (t);
Therefore EVEN
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(c) Since sin(−t)cos(−t) = −sin(t)cos(t)
Therefore ODD
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(d)
(−t) sin(−t) = (−t) (−sin (t) ) = t sin (t)
Therefore EVEN
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(e) Since (−t)2 =t2
Therefore EVEN
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(f) Since sin(−t)+cos(−t)=−sin(t)+cos(t)
Therefore Neither EVEN nor ODD. It has both
parts.
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Determine if following signal is even or odd. If
neither even nor odd, plot its even and odd
components.
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ASSIGNMENT 1 Question 3
The decomposition into even and odd
components depends on the location of the
origin. Shifting the signal changes the
decomposition. Plot the even and odd
components of the previous example, after
shifting x(t) by 1/2 to the right.
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Periodic and Aperiodic Signals
A signal is a periodic signal if it completes a
pattern within a measurable time frame,
called a period and repeats that pattern over
identical subsequent periods.
The completion of a full pattern is called a
cycle. A period is defined as the amount of
time (expressed in seconds) required to
complete one full cycle.
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Periodic and Aperiodic Signals
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Periodic and Aperiodic Signals
Z(t) = x(t) + v(t)
T1 = NT0
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Periodic and Aperiodic Signals
A continuous-time signal x(t) is periodic if:
• it is defined for all possible values of t,
−∞ < t < ∞, and
• there is a positive real value T0, the
fundamental period of x(t), such that
x(t + kT0) = x(t ) for any integer k.
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Periodic and Aperiodic Signals
The square wave signal shown is periodic. The
fundamental period of this square wave is 4,
but 8, 12, and 16 are also periods of the signal.
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Periodic and Aperiodic Signals
The discrete-time signal x[n] = (-1)n is
periodic with fundamental period N = 2.
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Example 1.8 10
Consider the sinusoid
x(t) = A cos(0 t + θ),−∞ < t < ∞.
Determine the fundamental period
of this signal, and indicate for what
frequency 0 the fundamental
period of x(t) is not defined.
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Fundamental period T0 may be found
from frequency period relationship:
0 = 2π/T0
T0 = 2π/ 0 is the fundamental period.
Whenever 0 > 0 these sinusoids are
periodic.
For DC signal, fundamental period can
not be determined; because for DC
signal 0 = 0.
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Example 1.9
Consider a periodic signal x(t) of fundamental
period T , determine whether the following
0
signals are periodic and, if so, find their
corresponding fundamental periods:
1. y(t) = A + x(t)
Adding a constant to a periodic signal does not
change the periodicity. Since x(t) is periodic, so
is y(t) = A + x(t).
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Example 1.9 (Contd.)
2. z(t) = x(t) + v(t) where v(t) is periodic of fundamental
period T1 = NT0, where N is a positive integer, i.e., a
multiple of T0
The fundamental period T1 = NT0 of v(t) is
also a period of x(t) and so z(t) is periodic
of fundamental period T1
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Finite-Energy Signal
Example:
A non-rechargeable battery and
firecracker are energy signals. They hold
a finite amount of energy. It can supply
some power only for a limited time, but if
you compute the average power over your
lifetime you will get a very small number.
Therefore, Energy Signal:
FINITE ENERGY, ZERO AVERAGE
POWER
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Finite-Power Signal
Example:
The sun is an approximation of a power
signal. It radiates at a constant power. If you
add the energy it gives off over your lifetime,
you will get a very large number. The sun
would supply infinite energy when you add
up its energy over all time.
Therefore, Power Signal:
FINITE CONSTANT POWER,
INFINITE ENERGY
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The energy and the power of a continuous-
time signal x(t) are defined as:
Please note:
• Power = Energy/Time.
• Energy is divided by 2T because limits of
integration are from –T to T and not from 0 to T.
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