EE3054
Signals and Systems
Continuous Time
Convolution
Yao Wang
Polytechnic University
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McClellan and Schafer
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3/14/2008
© 2003, JH McClellan & RW Schafer
2
LECTURE OBJECTIVES
Review of C-T LTI systems
Evaluating convolutions
Examples
Impulses
LTI Systems
Stability and causality
Cascade and parallel connections
3/14/2008
© 2003, JH McClellan & RW Schafer
3
Linear and Time-Invariant
(LTI) Systems
If a continuous-time system is both linear and
time-invariant, then the output y(t) is related to
the input x(t) by a convolution integral
y (t ) =
∞
∫ x(τ )h(t − τ )dτ = x(t ) ∗ h(t )
−∞
where h(t) is the impulse response of the system.
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© 2003, JH McClellan & RW Schafer
4
Evaluating a Convolution
x (t ) = u(t − 1)
y (t ) =
h (t ) = e − t u (t )
∞
h
(
τ
)
x
(
t
−
τ
)
d
τ
=
h
(
t
)
∗
x
(
t
)
∫
−∞
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© 2003, JH McClellan & RW Schafer
5
“Flipping and Shifting”
x (τ )
“flipping”
g(τ ) = x(−τ ) = u(− τ −1)
“flipping and shifting”
g(τ − t) = x(−(τ − t)) = x(t − τ )
3/14/2008
t −1
© 2003, JH McClellan & RW Schafer
t
6
Evaluating the Integral
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 0
t −1
y (t ) =  −τ
e dτ
∫

0
© 2003, JH McClellan & RW Schafer
t −1 < 0
t −1 > 0
7
Solution
y (t ) =
=
y (t ) = 0
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t −1
∫e
0
−τ
1− e
dτ = − e
−τ t −1
−( t −1)
0
t ≥1
t<1
© 2003, JH McClellan & RW Schafer
8
Convolution GUI
3/14/2008
© 2003, JH McClellan & RW Schafer
9
Another Example
x (t ) = e − at u(t )
y (t ) =
h(t ) = e − bt u (t ), b ≠ a
∞
∫ x(τ )h(t − τ )dτ = x(t ) ∗ h(t )
−∞
t

∞
−bt
− aτ bτ
e
e
e dτ

−aτ
−b ( t −τ )
∫
u ( t − τ ) dτ = 
= ∫ e u(τ )e
0

−∞
0

 e −at − e −bt
e −at − e −bt
t>0

u (t )
=
=  −a+b
b−a

0
t<0

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© 2003, JH McClellan & RW Schafer
t>0
t<0
10
Special Case: u(t)
x (t ) = e − at u(t ), a ≠ 0
y (t ) =
h (t ) = u (t )
∞
∫ x(τ )h(t − τ )dτ = x(t ) ∗ h(t )
−∞
=
1
− at
(1 − e )u(t )
a
if a = 2
1
−2 t
y (t ) = (1 − e )u(t )
2
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© 2003, JH McClellan & RW Schafer
11
Convolve Unit Steps
x (t ) = u (t )
y (t ) =
h (t ) = u (t )
∞
∫ x(τ )h(t − τ )dτ = x(t ) ∗ h(t )
−∞
t

∞
 ∫ 1 dτ t > 0
= ∫ u (τ )u(t − τ )dτ = 
0
 0
−∞
t<0

 t t > 0
=
= t u (t )
Unit Ramp
0 t < 0
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© 2003, JH McClellan & RW Schafer
12
“Flipping and Shifting”
x(τ )
“flipping”
“flipping and shifting”
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g(τ ) = x(−τ ) = u(− τ −1)
g(τ − t) = x( −(τ − t)) = x(t − τ )
t −1
© 2003, JH McClellan & RW Schafer
t
13
More examples
Rectangular pulses
Another Convolution Example
−t
h(t) = e u(t)
∞
y(t) = ∫ x(τ )h(t − τ )dτ = x(t) ∗ h(t)
−∞
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© 2003, JH McClellan & RW Schafer
15
Evaluating the Integral
y(t) =
t
= ∫e
1
=
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t <1
0
−(t−τ )
2
∫e
1
dτ
−(t−τ )
dτ
1≤ t ≤ 2
2≤t
© 2003, JH McClellan & RW Schafer
16
Solution
t
y(t) =
∫e
2
−(t−τ )
1
= ∫e
−(t−τ )
1
y(t) = 0
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dτ
dτ
−(t−τ ) t
=e
1
−(t−τ ) 2
=e
1
=1- e
−(t−1)
1≤ t ≤ 2
= e−(t−2) - e−(t−1) 2 ≤ t
t <1
© 2003, JH McClellan & RW Schafer
17
Convolution GUI
3/14/2008
© 2003, JH McClellan & RW Schafer
18
Convolution with Impulses, etc.
Convolution with impulses
x(t ) * δ (t − t1 ) = x(t − t1 )
Convolution with step function = integrator
3/14/2008
© 2003, JH McClellan & RW Schafer
19
Convolution is Commutative
h(t ) ∗ x(t ) =
∞
h
x
t
d
(
)
(
)
τ
−
τ
τ
∫
−∞
let σ = t − τ and dσ = −dτ
−∞
h(t ) ∗ x(t ) = − ∫ h(t − σ ) x(σ )dσ
=
∞
∞
∫ h(t − σ ) x(σ )dσ = x(t ) ∗ h(t )
−∞
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© 2003, JH McClellan & RW Schafer
20
Stability
A system is stable if every bounded input
produces a bounded output.
A continuous-time LTI system is stable if
and only if
∞
∫ h(t) dt < ∞
−∞
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© 2003, JH McClellan & RW Schafer
21
Integrator is unstable
Causal Systems
A system is causal if and only if y(t0)
depends only on x(τ) for τ< t0 .
An LTI system is causal if and only if
h(t ) = 0 for t < 0
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© 2003, JH McClellan & RW Schafer
23
Convolution is Linear
Substitute x(t)=ax1(t)+bx2(t)
∞
y (t ) = ∫ [ax1 (τ ) + bx2 (τ )]h(t − τ )dτ
−∞
∞
∞
−∞
−∞
= a ∫ x1 (τ )h(t − τ )dτ + b ∫ x2 (τ )h(t − τ )dτ
= ay1 (t ) + by2 (t )
Therefore, convolution is linear.
3/14/2008
© 2003, JH McClellan & RW Schafer
24
Convolution is Time-Invariant
Substitute x(t-t0)
∞
w(t) =
∫ h(τ )x((t − τ ) − t )dτ
o
−∞
∞
=
∫ h(τ )x((t − t
o
) − τ )dτ
−∞
= y(t − to )
3/14/2008
© 2003, JH McClellan & RW Schafer
25
Cascade of LTI Systems
δ (t)
h1 (t)
h1 (t) ∗ h2 (t)
h(t) = h1 (t) ∗ h2 (t) = h2 (t) ∗ h1(t)
δ (t)
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h2 (t)
© 2003, JH McClellan & RW Schafer
h2 (t) ∗ h1(t)
26
Parallel LTI Systems
h1 (t)
δ (t)
h1 (t) + h2 (t)
h2 (t)
h(t) = h1 (t) + h2 (t)(t
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© 2003, JH McClellan & RW Schafer
27
Example: More complicated
combinations
READING ASSIGNMENTS
This Lecture:
Chapter 9, Sects. 9-6, 9-7, and 9-8
Other Reading:
Ch. 9, all
Next Lecture: Start reading Chapter 10