Signals and Systems
[Ch – 01]
Introduction to Signals and
Systems
Instructor: Engr. Furqan Haider
DEE, NUST College of E & ME
Signals and Systems
About myself !!
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Where you can find me: 1st floor, DEE
E-mail: furqan_haider31@yahoo.com
Mobile Contact: CR can ask after the class.
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Best way to contact me: Come and talk to me during
discussion hours.
Research Interests: Wireless Communication, Fiber
Optics Communication and Acoustic Systems.
DEE, NUST College of E & ME
Signals and Systems
Course Organization
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Course Folder link:
https://www.dropbox.com/sh/rek9gj0e313dzb6/AACDg2WZ5cEQRT3jv3dcRiWa?dl=0
Visit the folder frequently
For further details please see the course
outline
DEE, NUST College of E & ME
Signals and Systems
Grading Policy

Grading Policy
 Quizzes (4 ~ 6)
 Assignments (4 ~ 6)
 Lab+Project
 Midterm
 Final

DEE, NUST College of E & ME
7.5%
7.5%
25%
22.5%
37.5%
100 %
Signals and Systems
Text & Reference Books
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Text book:
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Signals and Systems, 2nd edition by Alan V.
Oppenheim, Alan S. Willsky with S. Hamid Nawab.
Reference Book(s):
Signal Processing First by James H. McClellan, Mark A. Yoder,
and Ronald W. Schafer.
DEE, NUST College of E & ME
Signals and Systems
Pre-Requisite
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EE-211 Electrical Network Analysis
MATH-121 Linear Algebra and ODEs
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Signals and Systems
Helpful Hints!!
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Read each and every word of the text book--Very Important
Participate actively in the class
Do not miss any lecture
Do not be late in the class
Try to apply the theory in the lab
Do not get behind. You are encouraged to work with other students but
avoid plagiarism.
Work in groups, whenever appropriate. However, Assignments and Quizzes
must be attempted alone. No plagiarism will be tolerated.
Do the end problems of each chapter by yourself.
We will not proceed until everybody says “YES”.
Interrupt me during the lecture if I forget to deal with these 2 questions:
 Why? and
 How?
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Signals and Systems
A big WHY?
Why studying this course ?
DEE, NUST College of E & ME
Signals and Systems
A big HOW?
TOOLS required to perform analysis of
Signals & Systems?
DEE, NUST College of E & ME
Signals and Systems
Grading Rules
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Assignment Submission on time: graded out of 100%
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Assignment Submission (1 day late): graded out of 70%
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Assignment Submission (2 days late): graded out of 50%
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No Assignment / Project will be accepted after 2
days.
Only ONE make-up QUIZ in last week of the
semester. (whether you have missed one or more
quizzes throughout the semester).
DEE, NUST College of E & ME
Signals and Systems
What is expected from you?
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Prepare the agendas of Monday & Wednesday
class on weekly basis (You will do relevant End
Problems by yourself – seek my help if
necessary).
Expect a quiz on each Wednesday.
DEE, NUST College of E & ME
Signals and Systems
1.
COURSE OUTLINE
Introduction, Types of Signals
Motivation, Applications, Signal Classification
CT, DT, Analog, Digital, Deterministic, Random, Periodic, aperiodic; Even & Odd signal decomposition
Signal Transformations/Signal Fundamentals
Signal Transformations
Fundamental signals : Complex Exponentials; Decaying exponentials; sinusoids; Unit Impulse; Unit Step
Signal representation using fundamental signals
System Classification
Continuous/Discrete ; Analog/Digital
Linear/Nonlinear ; Time-invariant/Time varying; Causal/Anti-causal; Stable/Unstable
CH-01
CH-01
CH-01
LTI Systems Theory
Intro to LTI Systems, Impulse response as system characterization
LTI System Properties, Linearity, Convolution (CT and DT)
Difference equations for LTI system
CH-02
Fourier Series
Frequency domain view of LTI systems, Concept of complex frequency
Fourier series representation of CT periodic signals (CTFS), Properties of CTFS
Fourier series representation of DT periodic signals (DTFS), Properties of DTFS
CH-03
Continuous Time Fourier Transform (CTFT)
FT of continuous time aperiodic signals, Properties of CTFT
Fourier Transform of periodic signals
CH-04
Discrete Time Fourier Transform (DTFT)
FT of discrete time aperiodic signals, Properties of DTFT
CH-05
CH-07
Introduction to Sampling
Time Domain and frequency domain description; Nyquist criterion
Aliasing; Under/Over sampling
Laplace transform (LT)
Convergence of CTFT and motivation of Laplace transform, Properties of LT
Pole-zero plots;
Filter design by pole zero placement (time permitting)
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CH-09
Signals and Systems
CHAPTER – 1
INTRODUCTION TO
SIGNALS AND SYSTEMS
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Signals and Systems
What is a Signal?

A description of how one parameter is related to
another parameter
Examples of signal include:
 Electrical signals : Voltages and currents in a circuit
 Acoustic signals: Acoustic pressure (sound) over time
 Mechanical signals: Velocity of a car over time
 Video signals: Intensity level of a pixel (camera, video)
over time
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Signals and Systems
How is a Signal Represented?
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Mathematically, signals are represented as a function of
one or more independent variables.
For instance a black & white video signal intensity is
dependent on x, y coordinates and time t f(x,y,t)
On this course, we shall be exclusively concerned with
signals that are a function of a single variable: time
f(t)
t
Signal
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The Speech Signal
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The ECG Signal
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Signals and Systems
Signal
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The image
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Signals and Systems
Signal
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The image
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Signals and Systems
Signal
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It is the variation pattern that conveys the information, in
a signal
Signal may exist in many forms like acoustic, image,
video, electrical, heat & light signal
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Signals and Systems
Example: Signals in an Electrical
Circuit
R
v (t )  v (t )
i (t ) 
vs
i
C
vc
c
R
dv (t )
i (t )  C c
dt
dvc (t ) 1
1

vc (t ) 
vs (t )
dt
RC
RC
The signals vc and vs are patterns of variation over time
Step (signal) vs at t=1
RC = 1
First order (exponential)
response for vc
vs, vc
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+
-
s
t
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Note, we could also have considered the voltage across the resistor
or the current as signals
Continuous-time signals
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A value of signal exists at every instant of time
Independent variable
t
Independent variable
Discrete-time signals
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The value of signal exists only at equally spaced
discrete points in time
t
Independent variable
t
Independent variable
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Signals and Systems
Discrete-time signals
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Why to discretize ?
How to discretize ?
 How closely spaced are the samples
Distinction between discrete & digital signals
How to denote discrete signals
Is image a discrete or continuous signal
 The image is generally considered to be a
continuous variable
 Sampling can however be used to obtain a discrete,
two dimensional signal (sampled image)
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Signals and Systems
Notation
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A continuous-time signal has independent variable
xt 
(time) in parentheses ()
t
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A discrete-time signal is represented by enclosing
the independent variable in square brackets []
xn
n
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Signals and Systems
Continuous & Discrete-Time Signals
Continuous-Time Signals
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Most signals in the real world are continuous
time, as the scale is infinitesimally fine e.g x(t)
voltage, velocity,
Denote by x(t), where the time interval may be
bounded (finite) or infinite
Discrete-Time Signals
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t
Some real world and many digital signals are
discrete time, as they are sampled e.g. pixels,
daily stock price (anything that a digital
computer processes)
Denote by x[n], where n is an integer value that x[n]
varies discretely
Sampled continuous signal

x[n] =x(nk) , where k is sample time
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n
Signals and Systems
Types of Signals
Particular interest in signals with certain properties:

Periodic signals: a signal that repeats itself after a fixed
period T, i.e. x(t) = x(t+T) for all t. e.g. A sin(t).
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Even and odd signals: even if x(-t) = x(t), and odd if
x(-t) = -x(t). Examples are cos(t) and sin(t) signals.
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Exponential and sinusoidal signals: a signal is (real)
exponential if it can be represented as x(t) = Ceat. The same
example is (complex) exponential C and a are complex.
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Step and Impulse signals: A pulse signal is one which is
nearly completely zero, apart from a short spike, δ(t). A
step signal is zero up to a certain time, and then a constant
value after that time, u(t).
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Signals and Systems
Odd and Even Signals
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An even signal is identical to its time reversed signal, i.e. it can
be reflected in the origin and is equal to the original:
x( t )  x(t )
Examples:
x(t) = cos(t)
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An odd signal is identical to its negated, time reversed signal,
i.e. it is equal to the negative reflected signal
x( t )   x (t )
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Examples:
x(t) = sin(t)
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This is important because any signal can be expressed as the
sum of an odd signal and an even signal.
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Signals and Systems
Exponential and Sinusoidal Signals
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Exponential and sinusoidal signals are characteristic of realworld signals and also from a basis (a building block) for other
signals.
A generic complex exponential signal is of the form:
x(t )  Ce at
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where C and a are, in general, complex numbers. Lets
investigate some special cases of this signal
Real exponential signals
Exponential growth
Exponential decay
a0
a0
C 0
C 0
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Signals and Systems
Periodic Complex Exponential &
Sinusoidal Signals
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Consider when a is purely imaginary:
x(t )  Ce jw0t
By Euler’s relationship, this can be
expressed as:
cos(1)
e jw0t  cosw0t  j sin w0t
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This is a periodic signals because:
e jw0 (t T )  cosw0 (t  T )  j sin w0 (t  T )
 cosw0t  j sin w0t  e jw0t
when T=2p/w0
A closely related signal is the sinusoidal
signal:
x(t )  cosw0t   
w0  2pf 0
We can always use:
A cosw0t     A e j (w0t  )

A sinw t     Ae
0
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j (w0t  )


T0 = 2p/w0
T0 is the fundamental
time period
w0 is the fundamental
frequency
Signals and Systems
General Complex Exponential
Signals
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So far, considered the real and periodic complex exponential
Now consider when C can be complex. Let us express C is polar form
and a in rectangular form:
C  C e j
a  r  jw0
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So
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Using Euler’s relation

These are damped sinusoids
Ce at  C e j e( r  jw0 )t  C e rt e j (w0  )t
Ce at  C e j e( r  jw0 )t  C e rt cos((w0   )t )  j C e rt sin((w0   )t )
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30/25
Signals and Systems
Discrete Unit Impulse and Step Signals
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The discrete unit impulse signal is
defined:
0 n  0
x[n]   [n]  
1 n  0
Useful as a basis for analyzing other
signals
The discrete unit step signal is defined:
0 n  0
x[n]  u[n]  
1 n  0
Note that the unit impulse is the first
difference (derivative) of the step signal
 [n]  u[n]  u[n  1]
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Similarly, the unit step is the running sum
(integral) of the unit impulse.
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Signals and Systems
Continuous Unit Impulse and Step
Signals
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The continuous unit impulse signal is
defined:
0 t  0
x(t )   (t )  
 t  0
Note that it is discontinuous at t=0
The arrow is used to denote area, rather
than actual value
Again, useful for an infinite basis
The continuous unit step signal is
defined:
t
x(t )  u (t )    ( )d

0 t  0
x(t )  u (t )  
1 t  0
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Signals and Systems
Sinusoidal signal : x(t) = 10cos(2π(440)t - 0.4π)
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Signals and Systems
Recording of a Tuning fork
signal: Fig 2-3
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Signals and Systems
MATLAB Demo of Tuning Fork
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% TuningFork
t = 0:.0001:.01;
y = 10*cos(2*pi*1000*t-0.4*pi);
plot(t,y)
grid
pause;
t = 0:.0001:1;
y = 10*cos(2*pi*1000*t-0.4*pi);
sound (y)
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Signals and Systems
x(t) = 20cos(2π(40)t - 0.4π)
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36
Signals and Systems
x(t) = 5cos(2πfot) for different values of fo
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Signals and Systems
What is a System?
Systems process input signals to produce output signals
Examples:
 A circuit involving a capacitor can be viewed as a
system that transforms the source voltage (signal) to
the voltage (signal) across the capacitor
 A CD player takes the signal on the CD and transforms
it into a signal sent to the loud speaker
 A communication system is generally composed of
three sub-systems, the transmitter, the channel and the
receiver. The channel typically attenuates and adds
noise to the transmitted signal which must be processed
by the receiver

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Signals and Systems
System

An entity that responds to a signal
input

system
output
Examples
 Circuit
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Signals and Systems
System
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The camera
Image
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The Speech Recognition System
Identified
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Signals and Systems
System
The audio CD-player
Block Diagram representation of a system
 Visual representation of a system
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Input Signal
system
Output Signal
Shows inter-relations of many signals involved in
the implementation of a complex system
Look at everything around and try to identify the
signals and systems !!
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Signals and Systems
How is a System Represented?

A system takes a signal as an input and transforms it
into another signal
Input signal
x(t)

System
Output signal
y(t)
In a very broad sense, a system can be represented as
the ratio of the output signal over the input signal

That way, when we “multiply” the system by the
input signal, we get the output signal

This concept will be firmed up in the coming weeks
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Signals and Systems
Example: An Electrical Circuit System
R
vs
i
vc
C
Simulink representation of the electrical circuit
vs(t)
vc(t)
first order
system
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vs, vc

+
-
vs (t )  vc (t )
R
dv (t )
i (t )  C c
dt
dvc (t )
1
1

vc (t ) 
vs (t )
dt
RC
RC
i (t ) 
t
Signals and Systems
Continuous & Discrete-Time Models
Continuous-Time Systems
 Most continuous time systems
represent how continuous
signals are transformed via
differential equations. e.g.
circuit, car velocity
Discrete-Time Systems
 Most discrete time systems
represent how discrete signals
are transformed via difference
equations e.g. bank account,
discrete car velocity system
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dvc (t ) 1
1

vc (t ) 
vs (t )
dt
RC
RC
m
dv(t )
 v(t )  f (t )
dt
First order differential equations
y[n]  1.01y[n  1]  x[n]
First order difference equations
Signals and Systems
Continuous and discrete time
system

Like signals we have continuous and discrete-time
systems as well
xt 
system
y t 
xt   yt 
xn
system
yn
xn  yn
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Signals and Systems
Continuous and discrete
time system
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Examples of continuous and discrete-time systems
Squaring System
xt   xt 
2
2




y t  x t 
Differentiator System
y t  
d
xt 
dt
Accumulator System
yn  
n
 xk 
k  
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Signals and Systems
Transformations

Transformations of the independent variable

Time Shift
xn  3
xn
n
xt 
xt  4
4
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n
t
8
Signals and Systems
t
Transformations

Time reversal
x n
xn
n
n
xt 
x t 
t
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t
Signals and Systems
Transformations
xt 

Time scaling
2
2
t
x2t 
1
1
t
xt / 2
4
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4
t
Signals and Systems
Transformations


1
Example
3
2
1
1
3
2
1
xt
0
1
2
3
4
1
2
3
4
2
3
4
t
x t 
0
t
x2  t / 2
3
2
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1
0
1
t
Signals and Systems
Properties of a System

Memory: Memoryless (Resistor in V-I relationship, identity
system), Memory(V-I relation of a Capacitor, Accumulator)





Invertible: output = input (e.g 2x(t) and 1/2x(t)) (y(t)=
x2(t) is not invertible)
Causal: a system is causal if the output at a time, only
depends on input values up to that time.
Stability: small inputs lead to responses that do not diverge
Linear: a system is linear if the output of the scaled sum of
two input signals is the equivalent scaled sum of outputs
Time-invariance: a system is time invariant if the system’s
output is the same, given the same input signal, regardless
of time.
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Signals and Systems
LINEARITY Check
• Interchanging the operations result in same output, so,
SYSTEM is LINEAR
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Signals and Systems
Time-Invariance Check
• Interchanging the operations does not result in same output,
so, SYSTEM IS NOT TIME-INVARIANT.
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Signals and Systems
How Are Signal & Systems
Related?
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How to design a system to process a signal in particular ways?
Design a system to restore or enhance a particular signal
 Remove high frequency background communication noise
 Enhance noisy images from spacecraft
Assume a signal is represented as
 x(t) = d(t) + n(t)
Design a system to remove the unknown “noise” component n(t),
so that y(t)  d(t)
x(t) = d(t) + n(t)
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System
?
y(t)  d(t)
Signals and Systems
How Are Signal & Systems
Related?



How to design a system to extract specific pieces of
information from signals
 Estimate the heart rate from an electrocardiogram
 Estimate economic indicators (bear, bull) from
stock market values
Assume a signal is represented as: x(t) = g(d(t))
Design a system to “invert” the transformation g(), so
that y(t) = d(t)
x(t) = g(d(t))
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System
?
y(t) = d(t) = g-1(x(t))
Signals and Systems
How Are Signal & Systems
Related?



How to design a (dynamic) system to modify or control the
output of another (dynamic) system
 Control an aircraft’s altitude, velocity, heading by
adjusting throttle, rudder, ailerons
 Control the temperature of a building by adjusting the
heating/cooling energy flow.
Assume a signal is represented as: x(t) = g(d(t))
Design a system to “invert” the transformation g(), so that
y(t) = d(t)
x(t)
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dynamic
system ?
y(t) = d(t)
Signals and Systems
Phase Shift and Time Shift
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Signals and Systems
Phase Shift is Ambiguous
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Signals and Systems
Practice with sinusoid
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Solution
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Signals and Systems
Sinusoid from a Plot
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Signals and Systems
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Signals and Systems
Lecture 1: Summary



Signals and systems are important for:
 Electrical circuits
 Physical models and control systems
 Digital media (music, voice, photos, video)
Study of signals and systems helps in:
 Design systems to remove noise/enhance
measurement from audio and picture/video data
 Investigate stability of physical structures
 Control the performance of mechanical and
electrical devices
This will be the foundation for studying systems and
signals as a generic subject in this course.
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Signals and Systems