ECE
8443
– PatternContinuous
Recognition
EE
3512
– Signals:
and Discrete
LECTURE 15: CONVOLUTION FOR CT SYSTEMS
• Objectives:
Convolution Definition
Graphical Convolution
Examples
Properties
• Resources:
Wiki: Convolution
MIT 6.003: Lecture 4
JHU: Convolution Tutorial
ISIP: Java Applet
URL:
Representation of CT Signals
• Recall from calculus how we approximated a function by a sum of timeshifted, scaled pulse functions:
• We approximate the signal’s amplitude value as a constant over the interval
k  t  (k  1) : xˆ (t )  x(k) for k  t  (k  1)
• The signal changes discontinuously at the next step.
• What happens as   0 ? Recall our
representation of a CT impulse function:
EE 3512: Lecture 15, Slide 1
Representation of CT Signals Using Impulse Functions
• We approximate a CT signal
as a weighted pulse function.
• The signal can be written as a
sum of these pulses:
xˆ (t ) 

 x(k)
k  

(t  k)
• In the limit, as   0 :

x(t ) 
 x( )

(t   )d

• Mathematical definition of an impulse
function (the equivalent of the unit pulse
for DT signals and systems):
0

 (t )  
for t  0
for t  0
0
  (t )dt  1
0
• Unit pulses can be constructed from many functional shapes (e.g.,
triangular or Gaussian) as long as they have a vanishingly small width. The
rectangular pulse is popular because it is easy to integrate 
EE 3512: Lecture 15, Slide 2
Response of a CT LTI System
x(t )
CT LTI
h(t )
y(t )  x(t ) * h(t )
• Denote the system impulse response, h(t), as the output produced when the
input is a unit impulse function, (t).
• From time-invariance:  (t   )  h(t   )
• From linearity:

x(t ) 
 x( ) (t   )d


 y(t ) 
 x( )h(t   )d  x(t ) * h(t )

• This is referred to as the convolution integral for CT signals and systems.
• Its computation is completely analogous to the DT version:
EE 3512: Lecture 15, Slide 3
Example: Unit Pulse Functions
• t < 0: y(t) = 0
• t > 2: y(t) = 0
• 0  t  1: y(t) = t
• 1  t  2: y(t) = 2-t
EE 3512: Lecture 15, Slide 4
Example: Negative Unit Pulse
• t < 0.5: y(t) = 0
• t > 2.5: y(t) = 0
• 0.5  t  1.5: y(t) = 0.5-t
• 1  t  2: y(t) = -2.5+t
EE 3512: Lecture 15, Slide 5
Example: Combination Pulse
• p(t) = 1 0  t  1
• x(t) = p(t) - p(t-1)
• y(t) = ???
EE 3512: Lecture 15, Slide 6
Example: Unit Ramp
• p(t) = 1 0  t  1
• x(t) = r(t) p(t)
• y(t) = ???
EE 3512: Lecture 15, Slide 7
Properties of Convolution
• Sifting Property:
x(t ) *  (t  t 0 )  x(t  t 0 )
Proof:
x(t ) *  (t  t 0 ) 
  
x( ) (t  t


0
  )d 
t t0
 x( ) (t  t
0
  )d  x(t  t 0 )
 t t0
 
• Integration: t
 x( )d
x(t ) * u(t ) 
Proof:


t


x(t ) * u(t ) 
 x( )u(t   )d   x( )d
because u(t   )  0 for   t
• Step Response (follows from the integration property):
t
u(t ) * h(t )  h(t ) * u(t )   h( )d
Comments:

 Requires proof of the commutative property.
 In practice, measuring the step response of a system is much easier than
measuring the impulse response directly. How can we obtain the impulse
response from the step response?
EE 3512: Lecture 15, Slide 8
Properties of Convolution (Cont.)
• Implications (from DT lecture):
• Commutative Property:
x(t ) * h(t )  h(t ) * x(t )
Proof:
x(t ) * h(t ) 
  
x( )h(t   )d


 
let   t   , or   t   , and d  d
x(t ) * h(t ) 
  
  
 
 
x(t   )h( )(d )   h( ) x(t   )d  h(t ) * x(t )



• Distributive Property:
x(t ) * [h1 (t )  h2 (t )]  x(t ) * h1 (t )  x(t ) * h2 (t )
Proof:
  
x(t ) * [h1 (t )  h2 (t )] 
x( )[h (t   )  h (t   )]d


1
2
 
  



 
x( )h1 (t   )d 
  
x( )h (t   )d


2
 
 x(t ) * h1 (t )  x(t ) * h2 (t )
EE 3512: Lecture 15, Slide 9
Properties of Convolution (Cont.)
• Implications (from DT lecture):
• Associative Property:
x(t ) * h1 (t ) * h2 (t )  ( x(t ) * h1 (t )) * h2 (t )
 ( x(t ) * h2 (t )) * h1 (t )
Proof:
  
[ x(t ) * h1 (t )] 
x( )h (t   )d


1
 
     
[ x(t ) * h1 (t )] * h2 (t ) 
[  x( )h (   )d ]h (t   )d



1
 
  



 
2
 
x( )
  
h (   )h (t   )dd


1
2
 
Let            and d  d
  
  

  x( )   h1 ( )h2 (t  (   ))d  d
  
  

  
  

  x( )   h1 ( )h2 ((t   )   )d  d
  
  

 x(t ) * [h1 (t ) * h2 (t )]
EE 3512: Lecture 15, Slide 10
Useful Properties of CT LTI Systems
• Causality: ht   0
n0
which implies:
This means y(t) only depends on x( < t).

t


 x( )ht   d   x( )ht   d

• Stability:
 h(t )  

Bounded Input ↔ Bounded Output
Sufficient Condition:
for x(t )  xmax  

y (t ) 
 x( )ht   d


 xmax
 ht    d  

Necessary Condition:

if
 ht     

Let x(t )  h * (t ) / h(t ) , then x(t )  1 (bounded)



h * ( )
But y(0)   x( )h0   d  
h0   d   h0   d  
h( )



EE 3512: Lecture 15, Slide 11
Summary
• We introduced CT convolution.
• We worked some analytic examples.
• We also demonstrated graphical convolution.
• We discussed some general properties of convolution.
• We also discussed constraints on the impulse response for bounded input /
bounded output (stability).
EE 3512: Lecture 15, Slide 12