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Slide 2 Leo Lam 2010-2012 Signals and Systems EE235 Slide 3 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 2010-2012 Slide 4 Todays menu To do: If you still havent: join the Facebook Group! From Wednesday: Describing Common Signals Introduced the three building blocks General description for sinusoidal signals Started Periodicity Today: Finish Periodicity Manipulating signals! Slide 5 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 2010-2012 For all integers n Slide 6 Your turn! Find the period of: Leo Lam 2010-2012 No LCM exists! Why? Because LCM exists only if T 1 /T 2 is a rational number Slide 7 A few more Leo Lam 2010-2012 Not rational, so not periodic Decaying term means pattern does not repeat exactly, so not periodic Slide 8 Summary Periodicity Leo Lam 2010-2012 Slide 9 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 2010-2011 Slide 10 Adding signals Leo Lam 2010-2011 1 1 123 t t + = ?? x(t) y(t) 1 t 123 x(t)+y(t) Slide 11 Delay signals Leo Lam 2010-2011 unit pulse signal (memorize) t 0 1 1 What does y(t)=p(t-3) look like? P(t) 0 34 Slide 12 Multiply signals Leo Lam 2010-2011 Slide 13 Scaling time Leo Lam 2010-2011 Speed-up: y(t)=x(2t) is x(t) sped up by a factor of 2 t 0 1 1 t 0 1.5 y(t)=x(2t) How could you slow x(t) down by a factor of 2? y(t)=x(t) Slide 14 Scaling time Leo Lam 2010-2011 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) Slide 15 Playing with signals Leo Lam 2010-2011 What is y(t) in terms of the unit pulse p(t)? t 8 3 5 t 0 1 1 Need: 1.Wider (x-axis) factor of 2 2.Taller (y-axis) factor of 8 3.Delayed (x-axis) 3 seconds Slide 16 Playing with signals Leo Lam 2010-2011 t 8 3 5 in terms of unit pulse p(t) t 8 2 first step: 3 5 t 8 second step: Slide 17 Playing with signals Leo Lam 2010-2011 t 8 3 5 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? Slide 18 Playing with signals Leo Lam 2010-2011 3 5 t 8 Double-check: pulse starts: pulse ends: Slide 19 Do it in reverse Leo Lam 2010-2011 t Sketch 1 Slide 20 Do it in reverse Leo Lam 2010-2011 t Let then that is, y(t) is a delayed pulse p(t-3) sped up by 3. 1 1 4/3 1 3 4 Double-check pulse starts: 3t-3 = 0 pulse ends: 3t-3=1 Slide 21 Order matters Leo Lam 2010-2011 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) Slide 22 Playing with time Leo Lam 2010-2011 t 1 What does look like? 2 1 -2 Time reverse of speech: Also a form of time scaling, only with a negative number Slide 23 Playing with time Leo Lam 2010-2011 t 1 2 Describe z(t) in terms of w(t) 1 -213 t Slide 24 Playing with time Leo Lam 2010-2011 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)) Slide 25 Playing with time Leo Lam 2010-2011 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 Slide 26 Summary: Arithmetic: Add, subtract, multiple Time: delay, scaling, shift, mirror/reverse And combination of those Leo Lam 2010-2011 Slide 27 How to find LCM Factorize and group Your turn: 225 and 270s LCM Answer: 1350 Leo Lam 2010-2012 Slide 28 Even and odd signals Leo Lam 2010-2012 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: Slide 29 Even and odd signals Leo Lam 2010-2012 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: