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Leo Lam © 2010-2013 Signals and Systems EE235 Leo Lam
Leo Lam © 2010-2013 Today’s menu From Wednesday: Manipulating signals How was Lab 1? To Do: Really memorize u(t), r(t), p(t) Today: More of that! Even and odd signals Dirac Delta function
People types There are 10 types of people in the world: Those who know binary and those who don’t. Leo Lam © 2010-2013
Playing with time Leo Lam © 2010-2013 t 1 What does look like? 2 1 -2 Time reverse of speech: Also a form of time scaling, only with a negative number
Playing with time Leo Lam © 2010-2013 t 1 2 Describe z(t) in terms of w(t) 1 -213 t
Playing with time Leo Lam © 2010-2013 time reverse it: x(t) = w(-t) delay it by 3: z(t) = x(t-3) so z(t) = w(-(t-3)) = w(-t + 3) t 1 2 1 -21 3 x(t) you replaced the t in x(t) by t-3. so replace the t in w(t) by t-3: x(t-3) = w(-(t-3))
Playing with time Leo Lam © 2010-2013 z(t) = w(-t + 3) t 1 2 1 -21 3 x(t) Doublecheck: w(t) starts at 0 so -t+3 = 0 gives t= 3, this is the start (tip) of the triangle z(t). w(t) ends at 2 So -t+3=2 gives t=1, z(t) ends there
Summary: Arithmetic: Add, subtract, multiple Time: delay, scaling, shift, mirror/reverse And combination of those Leo Lam © 2010-2013
Even and odd signals Leo Lam © 2010-2013 An even signal is such that: t Symmetrical across the t=0 axis t Asymmetrical across the t=0 axis An odd signal is such that:
Even and odd signals Leo Lam © 2010-2013 Every signal sum of an odd and even signal. Even signal is such that: The even and odd parts of a signal Odd signal is such that:
Even and odd signals Leo Lam © 2010-2013 Euler’s relation: What are the even and odd parts of Even part Odd part
Summary: Even and odd signals Breakdown of any signals to the even and odd components Leo Lam © 2010-2013
Delta function δ(t) Leo Lam © 2010-2013 “a spike of signal at time 0” 0 The Dirac delta is: The unit impulse or impulse Very useful Not a function, but a “generalized function”)
Delta function δ(t) Leo Lam © 2010-2013 Each rectangle has area 1, shrinking width, growing height ---limit is (t)
Dirac Delta function δ(t) Leo Lam © 2010-2013 “a spike of signal at time 0” 0 It has height = , width = 0, and area = 1 δ(t) Rules 1.δ(t)=0 for t≠0 2.Area: 3. If x(t) is continuous at t 0, otherwise undefined 0 t0t0 Shifted to time instant t 0 :
Dirac Delta example Evaluate Leo Lam © 2010-2013 = 0. Because δ(t)=0 for all t≠0
Dirac Delta – Your turn Evaluate Leo Lam © 2010-2013 = 1. Why? Change of variable: 1
Dirac Delta – Another one Evaluate Leo Lam © 2010-2013
Is this function periodic? If so, what is the period? (Sketch to prove your answer) Slightly harder Leo Lam © 2010-2013 Not periodic – delta function spreads with k 2 for t>0 And x(t) = 0 for t<0
Signals and Systems EE235 Leo Lam ©
Leo Lam © Signals and Systems EE235. Leo Lam © Pet Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's.
Leo Lam © Signals and Systems EE235. Leo Lam © Happy Tuesday! Q: What is Quayle-o-phobia? A: The fear of the exponential (e).
Leo Lam © Signals and Systems EE235. Courtesy of Phillip Leo Lam ©
Leo Lam © Signals and Systems EE235. People types There are 10 types of people in the world: Those who know binary and those who don’t. Leo.
Chapter 1 Lesson 1 Variables and Expressions
Leo Lam © Signals and Systems EE235. Leo Lam © Breeding What do you get when you cross an elephant and a zebra? Elephant zebra sin.
Leo Lam © Signals and Systems EE235 Oh beer… An infinite amount of mathematicians walk into a bar. The first one orders a beer. The second.
“Teach A Level Maths” Vol. 1: AS Core Modules
In our lesson today we will learn how to find the area of a building.
Leo Lam © Signals and Systems EE235 October 14 th Friday Online version.
Leo Lam © Signals and Systems EE235. Fourier Transform: Leo Lam © Fourier Formulas: Inverse Fourier Transform: Fourier Transform:
Race Car Integers October 18, Race Car Rules / Positive numbers mean “forward” / Negative numbers mean “reverse” / Add means “keep going” / Subtract.
Continuous-Time Convolution EE 313 Linear Systems and Signals Fall 2005 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian.
Definite Integration and Areas 01 It can be used to find an area bounded, in part, by a curve e.g. gives the area shaded on the graph The limits of integration...
Leo Lam © Signals and Systems EE235. Leo Lam © Today’s menu Homework 2 due now Convolution!
Leo Lam © Signals and Systems EE235. Leo Lam © Convergence Two mathematicians are studying a convergent series. The first one says:
Leo Lam © Signals and Systems EE235. So stable Leo Lam ©
Leo Lam © Signals and Systems EE235. Leo Lam © Arthur’s knights Who was the largest knight at King Arthur’s round table? Sir Cumfrence,
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