Leo Lam 2010-2012 Signals and Systems EE235 Lecture 14
Slide 3
Convolution Properties Leo Lam 2010-2011 2
Slide 4
Commutative Leo Lam 2010-2011 3 Commutative: Doesnt matter
which signal to flip, its the same Pick the easier one!
Slide 5
Associative Leo Lam 2010-2011 4 Associative: Order doesnt
matter h 1 (t)h 2 (t) x(t)y(t) The overall response of two LTI
systems in series is given by
Slide 6
Distributive Leo Lam 2010-2011 5 Distributive: Two types h 1
(t) h 2 (t) x(t) y(t) + The overall response of two systems in
parallel is h(t) x 1 (t) y(t) + x 2 (t) Divide and conquer for
input signals
Slide 7
More Convolution Properties Leo Lam 2010-2011 6 Convolution of
any signal with an impulse, gives the same signal Convolution of
any signal with a shifted impulse, shifts the signal
Slide 8
More Convolution Properties Leo Lam 2010-2012 7 Convolution of
the impulse response of an LTI system with a unit step, gives its
step response s(t).
Slide 9
Another implication for LTI Leo Lam 2010-2012 8 Recall: d/dt
u(t) h(t) s(t) h(t) d/dt u(t) (t) taking the derivative is an LTI
system, and using associative properties: We can find the impulse
response of a system from its step response s(t)
Slide 10
More Convolution Properties Leo Lam 2010-2012 9 Convolution
with a time-shifted signal, gives a time shifted output: If
then
Slide 11
Summary: Leo Lam 2010-2012 10 Convolution properties
Commutative Associative Distributive Convolve with impulse Convolve
with shifted impulse Convolve h(t) with u(t) gives s(t)
Slide 12
Echo Properties Leo Lam 2010-2012 11 Echo properties of impulse
* 3 x(t) (t-3) t t t 3 = What does this system do?
Slide 13
Echo Properties Leo Lam 2010-2012 12 Multiple echoes (your
turn) * 3 x(t) (t) +0.5(t-3)+0.25(t-6) t t 6 3 t 6 = (1) (0.5)
(0.25)
Slide 14
Echo Properties Leo Lam 2010-2012 13 Another example * 2 x(t)
h(t)= (t) +0.5 (t-2) t t (1) (0.5) 1 2 t (0.5) (1.5) (1) 13 Solve
and plot? Hint: Distribute
Slide 15
Echo Properties Leo Lam 2010-2012 14 More With multiple time
shifts, add them all up.
Slide 16
Finding Impulse Response Leo Lam 2010-2012 15 Example: find
h(t) when 1) Plug in (t) for x(t)
Slide 17
System properties testing given h(t) Leo Lam 2010-2012 16
Impulse response h(t) fully specifies an LTI system Gives
additional tools to test system properties for LTI systems
Additional ways to manipulate/simplify problems, too
Slide 18
Causality for LTI Leo Lam 2010-2012 17 A system is causal if
the output does not depend on future times of the input An LTI
system is causal if h(t)=0 for t
