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 one orders half a beer. The third one orders one quarter of a beer. The fourth one starts to order, but the bartender interrupts "Here's two beers; you lot can figure the rest out yourself." Leo Lam ©

Today’s menu To do: –If you still haven’t: join the Facebook Group! –Homework 1 posted, due Tuesday. From Wednesday: Describing Common Signals –Introduced the three “building blocks” –General description for sinusoidal signals Today: Periodicity

Periodic signals Definition: x(t) is periodic if there exists a T (time period) such that: The minimum T is the period Fundamental frequency f 0 =1/T Leo Lam © For all integers n

Periodic signals: examples Sinusoids Complex exponential (non-decaying or increasing) Infinite sum of shifted signals v(t) (more later) Leo Lam © x(t)=A cos(   t+  ) T0T0

Periodicity of the sum of periodic signals Question: If x 1 (t) is periodic with period T 1 and x 2 (t) is periodic with period T 2 –What is the period of x 1 (t)+x 2 (t)? Can we rephrase this using our “language” in math? Leo Lam ©

Rephrasing in math Leo Lam © Goal: find T such that

Rephrasing in math Leo Lam © Goal: find T such that Need:  T=LCM(T 1,T 2 ) Solve it for r=1, true for all r

Periodic sum example If x 1 (t) has T 1 =2 and x 2 (t) has T 2 =3, what is the period of their sum, z(t)? LCM (2,3) is 6 And you can see it, too. Leo Lam © T 1 T 2

Your turn! Find the period of: Leo Lam © No LCM exists! Why? Because LCM exists only if T 1 /T 2 is a rational number

A few more Leo Lam © Not rational, so not periodic Decaying term means pattern does not repeat exactly, so not periodic

Summary Description of common signals Periodicity Leo Lam ©

Playing with signals Operations with signals –add, subtract, multiply, divide signals pointwise –time delay, scaling, reversal Properties of signals (cont.) –even and odd Leo Lam ©

Adding signals Leo Lam © t t + = ?? x(t) y(t) 1 t 123 x(t)+y(t)

Delay signals Leo Lam © unit pulse signal (memorize) t What does y(t)=p(t-3) look like? P(t) 0 34

Multiply signals Leo Lam ©

Scaling time Leo Lam © Speed-up: y(t)=x(2t) is x(t) sped up by a factor of 2 t t y(t)=x(2t) How could you slow x(t) down by a factor of 2? y(t)=x(t)

Scaling time Leo Lam © y(t)=x(t/2) is x(t) slowed down by a factor of 2 t 01 t 01 y(t)=x(t/2) 2 -2 y(t)=x(t)

Playing with signals Leo Lam © What is y(t) in terms of the unit pulse p(t)? t t Need: 1.Wider (x-axis) factor of 2 2.Taller (y-axis) factor of 8 3.Delayed (x-axis) 3 seconds

Playing with signals Leo Lam © t in terms of unit pulse p(t) t 8 2 first step: 3 5 t 8 second step:

Playing with signals Leo Lam © t in terms of unit pulse p(t) t 8 2 first step: 3 5 t 8 second step: replace t by t-3: Is it correct?

Playing with signals Leo Lam © t 8 Double-check: pulse starts: pulse ends:

Do it in reverse Leo Lam © t Sketch 1

Do it in reverse Leo Lam © t Let then that is, y(t) is a delayed pulse p(t-3) sped up by / Double-check pulse starts: 3t-3 = 0 pulse ends: 3t-3=1

Order matters Leo Lam © 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)

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 ©