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Leo Lam © Signals and Systems EE235 Leo Lam
Leo Lam © 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 ©
Playing with time Leo Lam © t 1 What does look like? Time reverse of speech: Also a form of time scaling, only with a negative number
Playing with time Leo Lam © t 1 2 Describe z(t) in terms of w(t) t
Playing with time Leo Lam © 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 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 © z(t) = w(-t + 3) t 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 ©
Even and odd signals Leo Lam © 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 © 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 © 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 ©
Delta function δ(t) Leo Lam © “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 © Each rectangle has area 1, shrinking width, growing height ---limit is (t)
Dirac Delta function δ(t) Leo Lam © “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 © = 0. Because δ(t)=0 for all t≠0
Dirac Delta – Your turn Evaluate Leo Lam © = 1. Why? Change of variable: 1
Dirac Delta – Another one Evaluate Leo Lam ©
Is this function periodic? If so, what is the period? (Sketch to prove your answer) Slightly harder Leo Lam © Not periodic – delta function spreads with k 2 for t>0 And x(t) = 0 for t<0
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.
The. of and a to in is you that it he for.
Graphical Analysis of Motion AP Physics C. Slope – A basic graph model A basic model for understanding graphs in physics is SLOPE. Using the model - Look.
Of. and a to the in is you that it at be.
Question Graph 1 1.What country has the largest column? What are the reasons that you think that this country has the highest amount? -The country that.
Roots & Zeros of Polynomials How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related. 2.5 Zeros of Polynomial Functions.
1.2 The Commutative and Identity Properties Objective (things to learn): How to use the commutative properties for addition and multiplication of whole.
GRAPHING Parts of graphs, and How to set up graphs.
The. of and a to in is you that it he was.
Year 5 Term 3 Unit 8 Day 1. L.O.1 To be able to visualise and name polygons.
Leo Lam © Signals and Systems EE235. Leo Lam © Pet Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's.
Broken Circles Pieces marked with multiple letters are used in different envelopes depending on the number of people in a group.
Objective: Evaluate expressions containing variables and translate verbal phrases into algebraic expressions.
Multiplication Model A Fraction of a Fraction Length X Length = Area.
High Frequency Words List A Group 1. the of and.
Dolch Words the of and to a in that is was.
POLYNOMIAL FUNCTIONS A POLYNOMIAL is a monomial or a sum of monomials. A POLYNOMIAL IN ONE VARIABLE is a polynomial that contains only one variable. Example:
If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.
Page 14 When Can We Plug It in and When We Cant? Are there any cases we cant just plug it in? Yes and No. Yessometimes the simply-plug-it-in method will.
Objective : Calculate the area of regular quadrilaterals. o Area is the space inside a 2D shape. o A quadrilateral is a 4 sided shape such as a square.
Algebraic Expressions – Rules for Exponents Let’s review the rules for exponents you learned in Algebra 1 :
Multiplying Polynomials. How do we find the area of a square?
Introducing: common denominator least common denominator like fractions unlike fractions. HOW TO COMPARE FRACTIONS.
Area and the Definite Integral Tidewater Community College Mr. Joyner, Dr. Julia Arnold and Ms. Shirley Brown using Tans 5th edition Applied Calculus for.
Surds Surds are a special type of number that you need to understand and do calculations with. The are examples of exact values and are often appear in.
The Basics of Physics with Calculus – Part II AP Physics C.
Factors, Prime Numbers & Composite Numbers by Carol Edelstein.
The following slides show one of the 51 presentations that cover the AS Mathematics core modules C1 and C2. Demo Disc Teach A Level Maths Vol. 1: AS Core.
The Rational Zero Theorem. The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives.
2.3 Continuity When you plot function values generated in a laboratory or collected in a field, you can connect the plotted points with an unbroken curve.
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