Transcript ppt slides

Linear Time-Invariant
(“LTI”) Systems
Montek Singh
Thurs., Feb. 7, 2002
3:30-4:45 pm, SN115
1
What we will learn
 How to represent a circuit as an input-output
system (“black box”)
 What are LTI systems?
 How is their behavior described?
2
Why treat circuits as I/O systems?
A system representation …
 is not bound to a particular input
 allows us to distill the essence of an arbitrarily complex circuit into a
concise description
f
output
input
output
input
 e.g., Thevenin and Norton equivalents
 can incorporate other (non-electrical) technologies
 e.g., acoustic, optical, magnetic etc.
3
What are LTI systems?
LTI systems are linear and time-invariant:
 Linearity:
 output for a sum of inputs = sum of individual outputs
 i.e.,
y1 (t )  f ( x1 (t )) and y2 (t )  f ( x2 (t ))
 Ay1 (t )  By 2 (t )  f ( Ax1 (t )  Bx 2 (t ))
 Time-Invariance:
 inherent system properties do not change with time
 delaying the input by time  simply delays the output by 
 i.e.,
y (t )  f ( x(t ))
 y (t   )  f ( x(t   ))
4
Examples
LTI systems:
 Most physical systems when operated at small amplitudes:
 an LCR electrical network
 a mechanical spring, a glass prism, a loudspeaker …
Non-linear systems:
 Most physical systems when “stretched to the limit”:
 a blaring loudspeaker
 Some systems that are intentionally operated in that mode:
 diodes, transistors, logic gates, digital systems …
Time-variant systems:
 Systems whose properties change with time:
 a resistor getting hotter
 the human eye
5
6
An LTI system’s behavior
System’s behavior = mapping from input to output
How to represent?
 Describe the underlying physical phenomena
 goes back to circuit theory
 Enumerate all (interesting) input-output pairs
 unwieldy description
 Describe output for a select set of inputs
 choose some special input
 compute output behavior for that input
 infer behavior for arbitrary inputs
7
Choosing that special input …
Unit impulse function: (t)
F(t)
(t)
1
F(t)
1/
1
1
t

t
t
 (t )  lim F (t )
Unit impulse = a pulse of:
 0
 infinitesimal duration
 infinite amplitude
 unit area
Also known as: Dirac delta function
8
Unit impulse: properties
 (t )  0, for t  0
 (t ) is undefined ( ), for t  0

  (t )dt  1
-
Examples:
(t-a)
2(t)
2
-½(t+1)
1
t
a
-1
t
½
t
9
Unit impulse: used as a sampler
x(t)(t-a)
(t-a)
x(t)
1
t
1
a
a
t
Sampling Theorem:
t

 x(t ) (t  a)dt
-

Multiplying a signal by
(t-a) and integrating
has the effect of
sampling it at t = a.

 x(a) (t  a)dt
-

 x ( a )   (t  a ) dt
-
 x(a)
10
Reconstituting a signal from samples
(1)

 x(t ) (t  a)dt  x(a)
Sampling Theorem:
-
Swap the roles of t and a:

x(t ) 
 x(a) (a  t )da
-

or, x(t ) 
 x(a) (t  a)da
-
x(t) can be regarded as an infinite
sum of infinitesimal samples, i.e.,
sample x(a) summed over all a.
11
Reconstituting a signal from samples
x(t)
(t-a)
a
x(t)
t
da
1/da
a
x(t)
x(a)(t-a)da
a
t
(2)
t

x(t)
 x(a) (t  a)da
-

 x(a) (t  a)da
-
a
t
 x(t )
12
Unit impulse: system’s response
Output of a system when input = (t) is called the
“unit impulse response”
Denoted by h(t):
y (t )  f ( x(t ))
 h(t )  f ( (t ))
Example: human eye
h(t)
latency
peak
persistence
t
13
Generalization: Arbitrary input
Given: unit impulse response h(t), i.e., f ( (t ))  h(t )
Find: system response y(t) to an arbitrary input x(t)
Method:
 express input x(t) as an infinite sum of weighted impulses

x(t ) 
 x(a) (t  a)da
-
 compute response to each individual impulse
f ( (t  a))  h(t  a)
 weight and add up all the individual responses

y (t )  f ( x(t )) 
 x(a) f ( (t  a)) da
-

or,
y (t ) 
 x(a)h(t  a)da
-
14
Convolution
Definition: y(t) is the “convolution of” x(t) and h(t) if:

y (t ) 
 x(a)h(t  a)da
-
Notation:
Properties:
1. commutativity:
2. associativity:
3. distributivity:
y (t )  x(t )  h(t )  ( x  h)(t )
x h  h x
( x  h1 )  h2  x  (h1  h2 )
x  (h1  h2 )  ( x  h1 )  ( x  h2 )
4. scalability:
a( x  h)  (ax)  h  x  (ah)
5. derivatives:
d
dx
dh
( x  h) 
h  x
dt
dt
dt
15
Convolution: example
16
Check it out!
http://www.jhu.edu/~signals/convolve/index.html
17
Homework: Due 2/19
1.
The output of a particular system S is the time derivative of
its input.
Prove that system S is linear time-invariant (LTI).
b) What is the unit impulse response of this system?
a)
2.
Prove Property 5. That is, prove that, for an arbitrary LTI
system, for a given input waveform x(t), the time derivative
of its output is identical to the output of that system when
subjected to the time derivative of its input. In other words,
differentiation on the input and output sides are equivalent.
18