1. Signals and Systems
EE235
Leo Lam
Leo Lam ©

2. Today’s menu From yesterday (Signals x and y relationships)
More: Describing Common Signals
Periodicity
Leo Lam ©

3. Your turn! Find the period of: No LCM exists! Why?
No LCM because the period of the second term is NOT rational. It is NOT periodic.
No LCM exists! Why?
Leo Lam ©

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

5. Summary
Description of common signals
Periodicity
Leo Lam ©

6. 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 ©

7. Adding signals 1 2 3 t +
= ??
x(t)
y(t)
x(t)+y(t)
Leo Lam ©

8. Delay signals 1 t 1 3 4 unit pulse signal
P(t) t 1
What does y(t)=p(t-3) look like? 3 4
Leo Lam ©

9. Multiply signals
Leo Lam ©

10. Scaling time
Speed-up: y(t)=x(2t) is x(t) sped up by a factor of 2
y(t)=x(t)
y(t)=x(2t) 1 1 t t 1 .5
How could you slow x(t) down by a factor of 2?
Leo Lam ©

11. Scaling time
y(t)=x(t/2) is x(t) slowed down by a factor of 2 t 1 -1
y(t)=x(t/2) 2 -2
y(t)=x(t)
Leo Lam ©

12. Playing with signals 8 t 3 5 1 t 1
What is y(t) in terms of the unit pulse p(t)?
Need:
Wider (x-axis) factor of 2
Taller (y-axis) factor of 8
Delayed (x-axis) 3 seconds
Not to scale! But the need should be clear, the pulse needs to be wider (factor of 2) and taller (factor of 8), and then “shifted/delayed” to the right by 3 seconds. 1 t 1
Leo Lam ©

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

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

15. Playing with signals 8 t 3 5 Double-check: pulse starts: pulse ends:
Leo Lam ©

16. Do it in reverse t
Sketch 1
Leo Lam ©

17. Do it in reverse 3 4 t Let then Double-check pulse starts: 3t-3 = 0
that is, y(t) is a delayed pulse p(t-3) sped up by 3. 1
1 4/3
Double-check
pulse starts:
3t-3 = 0
pulse ends:
3t-3=1
Leo Lam ©

18. 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 ©

19. 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 ©

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

21. 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 ©

22. 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 ©

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