Lecture #4:
THE SUPERPOSITION SUM (DT SYSTEM) AND THE SUPERPOSITION INTEGRAL (CT SYSTEM) OF LTI SYSTEMS
Motivation:
- The superposition sum/integral gives an insight into the operation of LTI systems that complements the insight given by the transform methods
- Solutions of LTI systems in the time domain using the superposition sum/integral can give efficient methods of solution
- The unit sample response of an LTI DT system and the unit impulse response of an LTI CT system characterize those systems
Outline:
- The superposition sum for DT systems
- The superposition integral for CT systems
- Relation of time domain and transform domain characterization of LTI systems
- The unit sample/impulse response
- Unit impulse/sample responses of different classes of LTI systems
I. DERIVATION OF SUPERPOSITION SUM/INTEGRAL
We will show that for a DT system
{}
y
[
n
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=
∑
m
=
−
∞
∞
x
[
m
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h
[
n
−
m
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y
[
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=
∑
m
=
−
∞
∞
x
[
m
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h
[
n
−
m
]
size 12{y \[ n \] = Sum cSub { size 8{m= - infinity } } cSup { size 8{ infinity } } {x \[ m \] h \[ n - m \] } } {}
where x[n] is an arbitrary input, h[n] is the unit sample response, y[n] is the output, and the above relation is called the superposition sum.
We will show that for a CT system
y
(
t
)
=
∫
−
∞
∞
x
(
τ
)
h
(
t
−
τ
)
dτ
y
(
t
)
=
∫
−
∞
∞
x
(
τ
)
h
(
t
−
τ
)
dτ
size 12{y \( t \) = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {x \( τ \) h \( t - τ \) dτ} } {}
where x(t) is an arbitrary input, h(t) is the unit impulse response, y(t) is the output, and the above relation is called the superposition integral.
II. THE SUPERPOSITION SUM FOR DT SYSTEMS
1/ Graphic view of superposition sum
2/ Derivation of superposition sum
3/ Convolution sum
The relation
x
[
n
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=
∑
m
x
[
m
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δ
[
n
−
m
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x
[
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=
∑
m
x
[
m
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δ
[
n
−
m
]
size 12{x \[ n \] = Sum cSub { size 8{m} } {x \[ m \] δ \[ n - m \] } } {}
expresses the sifting property of the unit sample. Note that the only non-zero term in this sum occurs when m = n, hence demonstrating the validity of the equation. The major conclusion of the derivation is that for an arbitrary input x[n], the output is
y
[
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=
∑
m
x
[
m
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h
[
n
−
m
]
y
[
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=
∑
m
x
[
m
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h
[
n
−
m
]
size 12{y \[ n \] = Sum cSub { size 8{m} } {x \[ m \] h \[ n - m \] } } {}
which is called the superposition sum. Such a relation is called a convolution sum when it involves arbitrary functions, i.e.,
z
[
n
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=
∑
m
x
1
[
m
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x
2
[
n
−
m
]
z
[
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=
∑
m
x
1
[
m
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x
2
[
n
−
m
]
size 12{z \[ n \] = Sum cSub { size 8{m} } {x rSub { size 8{1} } } \[ m \] x rSub { size 8{2} } \[ n - m \] } {}
Thus, the superposition sum is a special case of the convolution sum.
4/ Notation
We shall write the convolution sum of two DT signals as
z
[
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=
x
1
[
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∗
x
2
[
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=
∑
m
x
1
[
m
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x
2
[
n
−
m
]
z
[
n
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=
x
1
[
n
]
∗
x
2
[
n
]
=
∑
m
x
1
[
m
]
x
2
[
n
−
m
]
size 12{z \[ n \] =x rSub { size 8{1} } \[ n \] * x rSub { size 8{2} } \[ n \] = Sum cSub { size 8{m} } {x rSub { size 8{1} } \[ m \] x rSub { size 8{2} } \[ n - m \] } } {}
The symbol for convolution in various textbooks includes
x
1
[
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∗
x
2
[
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,
x
1
[
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∗
x
2
[
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and
x
1
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⊗
x
2
[
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x
1
[
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∗
x
2
[
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,
x
1
[
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∗
x
2
[
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and
x
1
[
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⊗
x
2
[
n
]
size 12{x rSub { size 8{1} } \[ n \] * x rSub { size 8{2} } \[ n \] ,x rSub { size 8{1} } \[ n \] * x rSub { size 8{2} } \[ n \] matrix {
{} # {}
} ital "and" matrix {
{} # {}
} x rSub { size 8{1} } \[ n \] ⊗x rSub { size 8{2} } \[ n \] } {}
5/ Mechanics
To compute the superposition sum
y
[
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=
x
[
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]
∗
h
[
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=
∑
m
x
[
m
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h
[
n
−
m
]
y
[
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=
x
[
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∗
h
[
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=
∑
m
x
[
m
]
h
[
n
−
m
]
size 12{y \[ n \] =x \[ n \] * h \[ n \] = Sum cSub { size 8{m} } {x \[ m \] h \[ n - m \] } } {}
Step 1 Plot x and h vs m since the convolution sum is on m.
Step 2 Flip h[m] around the vertical axis to obtain h[−m].
Step 3 Shift h[−m] by n to obtain h[n − m].
{}Step 4 Multiply to obtain x[m]h[n − m].
Step 5 Sum on m to compute.
∑mx[m]h[n−m]∑mx[m]h[n−m] size 12{ Sum cSub { size 8{m} } {x \[ m \] h \[ n - m \] } } {}
Step 6 Index n and repeat Steps 3-6.
Demo of DT convolution
6/ DT convolution properties — commutative property
x[n]*h[n]=h[n]*x[n]
Proof:
y
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=
x
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∗
h
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=
∑
m
x
[
m
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h
[
n
−
m
]
y
[
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=
x
[
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∗
h
[
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=
∑
m
x
[
m
]
h
[
n
−
m
]
size 12{y \[ n \] =x \[ n \] * h \[ n \] = Sum cSub { size 8{m} } {x \[ m \] h \[ n - m \] } } {}
Let n − m = l then
y
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=
x
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∗
h
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=
∑
1
x
[
n
−
1
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h
[
1
]
=
∑
1
h
[
1
]
x
[
n
−
1
]
=
h
[
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∗
x
[
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y
[
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=
x
[
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∗
h
[
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=
∑
1
x
[
n
−
1
]
h
[
1
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=
∑
1
h
[
1
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x
[
n
−
1
]
=
h
[
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∗
x
[
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size 12{y \[ n \] =x \[ n \] * h \[ n \] = Sum cSub { size 8{1} } {x \[ n - 1 \] h \[ 1 \] = Sum cSub { size 8{1} } {h \[ 1 \] x \[ n - 1 \] } =h \[ n \] * x \[ n \] } } {}
7/ DT convolution properties — associative property
{}
y
[
n
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=
(
x
[
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∗
h
1
[
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)
∗
h
2
[
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=
x
[
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∗
(
h
1
[
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∗
h
2
[
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)
y
[
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=
(
x
[
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∗
h
1
[
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)
∗
h
2
[
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=
x
[
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∗
(
h
1
[
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∗
h
2
[
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)
size 12{y \[ n \] = \( x \[ n \] *h rSub { size 8{1} } \[ n \] \) *h rSub { size 8{2} } \[ n \] =x \[ n \] * \( h rSub { size 8{1} } \[ n \] *h rSub { size 8{2} } \[ n \] \) } {}
Proof:
y
[
n
]
=
∑
1
(
∑
m
x
[
m
]
h
1
[
1
−
m
]
)
h
2
[
n
−
1
]
=
∑
m
x
[
m
]
(
∑
1
h
1
[
1
−
m
]
h
2
[
n
−
1
]
)
y
[
n
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=
∑
1
(
∑
m
x
[
m
]
h
1
[
1
−
m
]
)
h
2
[
n
−
1
]
=
∑
m
x
[
m
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(
∑
1
h
1
[
1
−
m
]
h
2
[
n
−
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]
)
size 12{y \[ n \] = Sum cSub { size 8{1} } { \( Sum cSub { size 8{m} } {x \[ m \] h rSub { size 8{1} } \[ 1 - m \] } \) h rSub { size 8{2} } \[ n - 1 \] = Sum cSub { size 8{m} } {x \[ m \] \( Sum cSub { size 8{1} } {h rSub { size 8{1} } } \[ 1 - m \] h rSub { size 8{2} } \[ n - 1 \] \) } } } {}
Let k = l − m to obtain
y
[
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=
∑
m
x
[
m
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(
∑
k
h
1
[
k
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h
2
[
n
−
m
−
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]
)
=
∑
m
x
[
m
]
h
[
n
−
m
]
y
[
n
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=
∑
m
x
[
m
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(
∑
k
h
1
[
k
]
h
2
[
n
−
m
−
k
]
)
=
∑
m
x
[
m
]
h
[
n
−
m
]
size 12{y \[ n \] = Sum cSub { size 8{m} } {x \[ m \] \( Sum cSub { size 8{k} } {h rSub { size 8{1} } } \[ k \] h rSub { size 8{2} } \[ n - m - k \] \) = Sum cSub { size 8{m} } {x \[ m \] h \[ n - m \] } } } {}
where
h
[
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=
h
1
[
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∗
h
2
[
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h
[
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=
h
1
[
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∗
h
2
[
n
]
size 12{h \[ n \] =h rSub { size 8{1} } \[ n \] *h rSub { size 8{2} } \[ n \] } {}
8/ DT convolution properties — distributive property
y
[
n
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=
x
[
n
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∗
(
h
1
[
n
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+
h
2
[
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)
=
x
[
n
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∗
h
1
[
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+
x
[
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∗
h
2
[
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y
[
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=
x
[
n
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∗
(
h
1
[
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+
h
2
[
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)
=
x
[
n
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∗
h
1
[
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]
+
x
[
n
]
∗
h
2
[
n
]
size 12{y \[ n \] =x \[ n \] * \( h rSub { size 8{1} } \[ n \] +h rSub { size 8{2} } \[ n \] \) =x \[ n \] *h rSub { size 8{1} } \[ n \] +x \[ n \] *h rSub { size 8{2} } \[ n \] } {}
Proof:
y
[
n
]
=
∑
m
x
[
m
]
(
h
1
[
n
−
m
]
+
h
2
[
n
−
m
]
)
=
∑
m
x
[
n
]
h
1
[
n
−
m
]
+
∑
m
x
[
m
]
h
2
[
n
−
m
]
y
[
n
]
=
∑
m
x
[
m
]
(
h
1
[
n
−
m
]
+
h
2
[
n
−
m
]
)
=
∑
m
x
[
n
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h
1
[
n
−
m
]
+
∑
m
x
[
m
]
h
2
[
n
−
m
]
alignl { stack {
size 12{y \[ n \] = Sum cSub { size 8{m} } {x \[ m \] \( h rSub { size 8{1} } \[ n - m \] +h rSub { size 8{2} } \[ n - m \] \) } } {} #
matrix {
{} # {}
} = Sum cSub { size 8{m} } {x \[ n \] h rSub { size 8{1} } \[ n - m \] + Sum cSub { size 8{m} } {x \[ m \] h rSub { size 8{2} } \[ n - m \] } } {}
} } {}
9/ DT convolution properties — delay accumulation
If
y[n] = x[n]∗h[n]
then
x[n − j]∗h[n − k] = y[n − k − j]
Proof:
∑
m
x
[
m
−
j
]
h
[
n
−
k
−
m
]
=
∑
1
x
[
1
]
h
[
n
−
k
−
j
−
1
]
=
y
[
n
−
k
−
j
]
