Signals and Systems EE235 Today’s Cultural Education: Rachmaninov’s Symphonic Dances III Leo Lam © 2010-2012

People types There are 10 types of people in the world: Those who know binary and those who don’t. Leo Lam © 2010-2012

Today’s menu From Friday: Even and odd signals Dirac Delta Function Manipulation of signals To Do: Really memorize u(t), r(t), p(t) Even and odd signals Dirac Delta Function Leo Lam © 2010-2012

Order matters With time operations, order matters y(t)=x(at+b) can be found by: Shift by b then scale by a (delay signal by b, then speed it up by a) w(t)=x(t+b)  y(t)=w(at)=x(at+b) Scale by a then shift by b/a w(t)=x(at)  y(t)=w(t+b/a)=x(a(t+b/a))=x(at+b) Leo Lam © 2010-2011

Playing with time 1 t 2 look like? What does 1 -2 Time reverse of speech: Also a form of time scaling, only with a negative number Leo Lam © 2010-2012

Playing with time t 1 2 1 -2 3 t Describe z(t) in terms of w(t) Leo Lam © 2010-2012

Playing with time 1 t 2 x(t) 1 -2 3 time reverse it: x(t) = w(-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)) 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) Leo Lam © 2010-2012

Playing with time 1 t 2 x(t) 1 -2 3 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 1 -2 3 x(t) z(t) = w(-t + 3) Leo Lam © 2010-2012

Summary: Arithmetic: Add, subtract, multiple Time: delay, scaling, shift, mirror/reverse And combination of those Leo Lam © 2010-2012

How to find LCM Factorize and group Your turn: 225 and 270’s LCM Answer: 1350 Leo Lam © 2010-2012

Even and odd signals t t An even signal is such that: Symmetrical across the t=0 axis An odd signal is such that: t Asymmetrical across the t=0 axis Leo Lam © 2010-2012

Even and odd signals 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: Leo Lam © 2010-2012

Even and odd signals Euler’s relation: What are the even and odd parts of Euler’s relation: Even part Odd part Leo Lam © 2010-2012

Summary: Even and odd signals Breakdown of any signals to the even and odd components Leo Lam © 2010-2012

Delta function δ(t) “a spike of signal at time 0” The Dirac delta is: The unit impulse or impulse Very useful Not a function, but a “generalized function”) “a spike of signal at time 0” Leo Lam © 2010-2012

Delta function δ(t) Each rectangle has area 1, shrinking width, growing height ---limit is (t) Leo Lam © 2010-2012

Dirac Delta function δ(t) “a spike of signal at time 0” It has height = , width = 0, and area = 1 δ(t) Rules δ(t)=0 for t≠0 Area: If x(t) is continuous at t0, otherwise undefined t0 Shifted to time instant t0: Leo Lam © 2010-2012

Dirac Delta example Evaluate = 0. Because δ(t)=0 for all t≠0 Leo Lam © 2010-2012

Dirac Delta – Your turn = 1. Why? 1 Evaluate Change of variable: Or just realizing that the integral at t=pi/2 produces 1. 1 Leo Lam © 2010-2012