Leo Lam 2010-2011 Signals and Systems EE235 October 14 th
Friday Online version
Slide 3
Leo Lam 2010-2011 Todays menu Superposition (Quick recap)
System Properties Summary LTI System Impulse response
Slide 4
Superposition Leo Lam 2010-2011 Superposition is Weighted sum
of inputs weighted sum of outputs Divide & conquer
Slide 5
Superposition example Leo Lam 2010-2011 Graphically 4 x 1 (t) T
1 1 y 1 (t) 1 1 2 x 2 (t) T 1 1 y 2 (t) 1 1 32 T 1 ? 2 y 1 (t) 1 -y
2 (t)
Slide 6
Superposition example Leo Lam 2010-2011 Slightly aside (same
system) Is it time-invariant? No idea: not enough information
Single input-output pair cannot test positively 5 x 1 (t) T 1 1 y 1
(t) 1 1 2 x 2 (t) T 1 1 y 2 (t) 1 1 32
Slide 7
Superposition example Leo Lam 2010-2011 Unique case can be used
negatively 6 x 1 (t) T 1 1 y 1 (t) 1 1 2 x 2 (t) T 1 y 2 (t) 1 -2
NOT Time Invariant: Shift by 1 shift by 2 x 1 (t)=u(t) S y 1
(t)=tu(t) NOT Stable: Bounded input gives unbounded output
Slide 8
Summary: System properties Causal: output does not depend on
future input times Invertible: can uniquely find system input for
any output Stable: bounded input gives bounded output
Time-invariant: Time-shifted input gives a time-shifted output
Linear: response to linear combo of inputs is the linear combo of
corresponding outputs Leo Lam 2010-2011
Slide 9
Impulse response (Definition) Any signal can be built out of
impulses Impulse response is the response of any Linear Time
Invariant system when the input is a unit impulse Leo Lam 2010-2011
Impulse Response h(t)
Slide 10
Using superposition Leo Lam 2010-2011 Easiest when: x k (t) are
simple signals (easy to find y k (t)) x k (t) are similar for
different k Two different building blocks: Impulses with different
time shifts Complex exponentials (or sinusoids) of different
frequencies
Slide 11
Briefly: recall Dirac Delta Function Leo Lam 2010-2011 3t t
x(t) t-3) 3 t x t-3) Got a gut feeling here?
Slide 12
Building x(t) with (t) Leo Lam 2010-2011 Using the sifting
properties: Change of variable: t t0 tt0 t From a constant to a
variable =
Slide 13
Building x(t) with (t) Leo Lam 2010-2011 Jumped a few
steps
Slide 14
Building x(t) with (t) Leo Lam 2010-2011 Another way to see
x(t) t (t) t 1/ Compensate for the height of the unit pulse Value
at the tip
Slide 15
So what? Leo Lam 2010-2011 Two things we have learned If the
system is LTI, we can completely characterize the system by how it
responds to an input impulse. Impulse Response h(t)
Slide 16
h(t) Leo Lam 2010-2011 For LTI system T x(t)y(t) T (t) h(t)
Impulse Impulse response T (t-t 0 ) h(t-t 0 ) Shifted Impulse
Shifted Impulse response
Slide 17
Finding Impulse Response (examples) Leo Lam 2010-2011 Let
x(t)=(t) What is h(t)?
Slide 18
Finding Impulse Response Leo Lam 2010-2011 For an LTI system,
if x(t)=(t-1) y(t)=u(t)-u(t-2) What is h(t)? h(t) (t-1) u(t)-u(t-2)
h(t)=u(t+1)-u(t-1) An impulse turns into two unit steps shifted in
time Remember the definition, and that this is time invariant
Slide 19
Finding Impulse Response Leo Lam 2010-2011 Knowing T, and let
x(t)=(t) What is h(t)? 18 This system is not linear impulse
response not useful.
Slide 20
Summary: Impulse response for LTI Systems Leo Lam 2010-2011 19
T (t- )h(t- ) Time Invariant T Linear Weighted sum of impulses in
Weighted sum of impulse responses out First we had
Superposition
Slide 21
Summary: another vantage point Leo Lam 2010-2011 20 LINEARITY
TIME INVARIANCE Output! An LTI system can be completely described
by its impulse response! And with this, you have learned
Convolution!
Slide 22
Convolution Integral Leo Lam 2010-2011 21 Standard Notation The
output of a system is its input convolved with its impulse
response
Slide 23
Leo Lam 2010-2011 Summary LTI System Impulse response Leading
into Convolution!