# Signals and Systems ```Signals &amp; Systems
Spring 2009
Instructor: Mariam Shafqat
UET Taxila
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Today's lecture
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The course
Course contents
Recommended books
Course structure
Assessments breakdown
Before we start…
Introduction to signals and systems
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The Course
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Core course
First course in Computer Engineering
A strong foundation for advanced courses
and research
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Mathematical &amp; theoretical
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Analysis and processing of information
System design for required processing
Calculus, Linear Algebra, Differential
Expectations
Extensive and tough
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Labs Session
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Performance criteria:
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Performance within the lab
Lab report
Lab report submission after one week
Lab report submission only in the lab
Viva from each individual student from his/her lab
report
Announcement of marks obtained by each
individual students in the lab at the end of lab
session.
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Course contents
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Introduction to Signals and Systems
Sinusoids
Spectrum Representation
Analysis of Periodic Waveforms
Sampling and Aliasing
Z-Transform
Convolution
Frequency response
Fourier Series and Transforms
Continuous-time &amp; Discrete-time Systems
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Books
Signals &amp; Systems (Second Edition)  Text Book
by
Alan V. Oppenheim, Alan S. Willsky,
S. Hamid Nawab
Signal Processing First  Reference Book
by
James H. McClellan, Ronald W. Schafer,
Mark A. Yoder
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Assessments
Quizzes
10%
Assignments 10%
Mid Term
20%
Labs
16%
FinalExam
40%
Attendence 4%
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Signal
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What is a signal
A description of how one parameter is
related to another parameter.
Examples
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v
The voltage varies with time
t
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Signal
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The Speech Signal
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The ECG Signal
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Signal
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The image
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Signal
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The image
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Signal
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It is the variation pattern that conveys the
information, in a signal
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Signal may exist in many forms like acoustic, image,
video, electrical, heat &amp; light signal
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System
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An entity that responds to a signal
input
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system
output
Examples
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Circuit
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System
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The camera
Image
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The Speech Recognition System
Identified
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System
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The audio CD-player
Block Diagram representation of a system
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Visual representation of a system
Input Signal
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system
Output Signal
Shows inter-relations of many signals involved in
the implementation of a complex system
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Mathematical Representation
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A signal can be represented as a function of one or
more independent variables
Examples
t
vt   sin t 
0  t  2
s t 
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Mathematical Representation
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The image is a function of two spatial variables
sx, y 
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Continuous-Time Signals
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Most signals in the real world are
continuous time, as the scale is
infinitesimally fine.
E.g. voltage, velocity,
Denote by x(t), where the time interval
may be bounded (finite) or infinite
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Continuous-time signals
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A value of signal exists at every instant of time
t
Independent variable
t
Independent variable
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Discrete-Time Signals
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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 varies discretely.
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Discrete-time signals
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The value of signal exists only at equally spaced
discrete points in time
n
Independent variable
n
Independent variable
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Discrete-time signals
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Why to discretize
How to discretize
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How closely spaced are the samples
Distinction between discrete &amp; digital signals
How to denote discrete signals
Is the image a discrete or continuous signal
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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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Analog vs Digital signals
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the difference is with
respect to the value of
the function (y-axis).
Analog corresponds to
a continuous y-axis,
while digital
corresponds to a
discrete y-axis.
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Notation
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A continuous-time signal is represented by
enclosing the independent variable (time) in
xt 
parentheses ()
t
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A discrete-time signal is represented by
enclosing the independent
variable (index) in
xn
square brackets []
n
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Important Parameters
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Signal power
Signal energy
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Continuous time Signal power
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Our usual notion of the energy of a signal is the area
under the curve f(t)2.
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 Some
further classification of
signals
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Periodic vs Aperiodic signals
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Periodic signals repeat with some period T, while
aperiodic, or nonperiodic, signals do not.
We can define a periodic function through the
following mathematical expression, where t can be
any number and T is a positive constant:
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f (t) = f (T + t)
The fundamental period of our function, f (t), is the
smallest value of T that the still allows equation to
be true.
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Periodic vs Aperiodic signals
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Causal vs. Anticausal vs. Noncausal
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Causal signals are signals that are zero for
all negative time,
Anticausal are signals that are zero for
all positive time.
Noncausal signals are signals that have
nonzero values in both positive and negative
time
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Causal vs. Anticausal vs. Noncausal
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Even vs. Odd
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An even signal is any signal f such that f (t) =
f (-t)
Even signals can be easily plotted as they
are vertical about the vertical axis.
An odd signal is a signal such that f(t)=-f(t).
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Even vs. Odd
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Deterministic vs. Random
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Deterministic signal is a signal in which
each value of the signal is fixed and can be
determined by a mathematical expression,
rule, or table. Because of this the future
values of the signal can be calculated from
past values with complete confidence.
Random signal has a lot of uncertainty
about its behavior. The future values of a
random signal cannot be accurately predicted
and can usually only be guessed based on
the averages of sets of signals
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Deterministic vs. Random
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Finite vs. Infinite Length
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f (t) is a finite-length signal if it is nonzero over a
finite interval
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t1 &lt; f (t) &lt; t2
Infinite-length signal, f (t), is defined as nonzero
over all real numbers:
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Signal Operations/Transformations
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Signal operations are operations on the time
variable of the signal.
Two signal operations are considered
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Time shifting/Time reversal
Time scaling
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Time Shifting
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Time shifting is, the shifting of a signal in
time. This is done by adding or subtracting
the amount of the shift to the time variable in
the function. Subtracting a fixed amount from
the time variable will shift the signal to the
right (delay) that amount, while adding to the
time variable will shift the signal to the left
Delay
x(t-2)
x(t+2)
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Time Shifting
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Time Shifting
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Time Scaling
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Sinusoidal signals
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x(t) = A cos(ωt + Φ)
A is the maximum amplitude of the sinusoidal
signal
 is the phase shift
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Unit impulse function
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Unit step
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Unit step
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Unit Step
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Continuous time unit step
Discontinuous at time t=0
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Continuous time unit impulse
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Relation b/w unit step and unit impulse
Running integral for t&lt;0 and t&gt;0
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Scaled unit impulse function
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Some properties of impulse function
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Some properties of impulse function
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Continuous time and discrete time systems
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Interconnection of systems
Series
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Interconnection of systems
Parallel
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Interconnection of systems
Series-parallel
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Interconnection of systems
Feedback
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Examples of Feedback Systems
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Some basic systems properties
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Memory and memory less
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Memory less depends upon the input at the
same time.
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Memory systems
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System must store or remember something
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