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7/31/2019 Causal Lt i Systems 03
1/17
1
Properties of Linear-Time Invariant SystemCommutative Property
Convolution is a commutative operation.
k
x n h n h n x n h k x n k
x t h t h t x t h x t d
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Properties of Linear-Time Invariant System
In the preceding two sections, we developed the
extremely important representations of CT and DT LTI.
The LTI systems are represented in terms of their unit
impulse responses.
[ ] [ ] [ ] [ ] [ ]
( )
k
y n x k h n k x n h n
y t x h t d x t h t
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Properties of Linear-Time Invariant SystemDistributive Property
Convolution is distributive over addition:
As a consequence of distributive property,
we have the following interpretation of
parallel interconnected LTI systems
1 2 1 2
1 2 1 2
x n h n h n x n h n x n h n
x t h t h t x t h t x t h t
)()()(
)()()()(
)()()(
ththtx
thtxthtx
tytyty
21
21
21
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Properties of Linear-Time Invariant SystemDistributive Property
Lety[n] denote the convolution of the following two
sequences:
Direct evaluation of such convolution is tedious. Instead
we may use the distributive property as follows:
1
[ ] 2
2[ ] [ ]
n
nx n u n u n
h n u n
1 2
1 2
1 2
[ ] [ ] [ ]
[ ] [ ] [ ] [ ]
y n x n x n h n
x n h n x n h n
= y n y n
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Properties of Linear-Time Invariant SystemDistributive Property
y1[n] can be obtained from
example 2.3 (with = ).
y2[n] can be obtained fromexample 2.5.
Their sum is shown in the figure
below.
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Properties of Linear-Time Invariant SystemAssociative Property
Convolution is associative:
Impulse response of cascaded
systems is the convolution of their
individual impulse responses.
1 2 1 2
1 2 1 2
x n h n h n x n h n h n
x t h t h t x t h t h t
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Properties of Linear-Time Invariant SystemSystems with and without Memory
System is memoryless if its output at any time depends
only on the value of input at the same time.
For DT-LTI systems and from the convolution operation,
this can be true if
and the convolution sum reduces to
[ ] [ ]y n bx n
][][][][ nbxnbnxny
[ ] 0 0 [ ] [ ]h n for n h n b n
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Properties of Linear-Time Invariant SystemSystems with and without Memory
In analogy to discrete-time systems, CT-LTI system is
memoryless if
this can be true if
and the convolution integral reduces to
y t bx t
( ) ( ) ( ) ( )y t x t b t bx t
( ) 0 0 ( ) ( )h t for t h t b t
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Properties of Linear-Time Invariant SystemInvertibility of LTI Systems
The LTI system is invertible if
an inverse system exists.
For the CT system in (a) the
overall impulse response is
Similarly for the DT in (b) the
overall impulse response is
1( ) ( ) ( )h t h t t
1[ ] [ ] [ ]h n h n n
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Properties of Linear-Time Invariant SystemInvertibility of LTI Systems
Consider an LTI with impulse response h[n] = u[n], find the output of the
system.
The system is accumulator computes the running sum of the input values.
This system is invertible and its inverse has impulse response
n
k
k
kx
knukxny
][
][][][
1[ ] [ ] [ 1]h n n n
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Properties of Linear-Time Invariant SystemInvertibility of LTI Systems
The impulse response of inverse system is
We can verify this result by direct calculation:
1[ ] [ ] [ 1]h n n n
1[ ] [ ] [ ] [ ] [ 1
[ ]* [ ]- [ ]* [ -1]
[ ] - [ -1]
[ ]
h n h n u n n n
u n n u n n
u n u n
n
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Prepared by Dr Nidal Kamel 12
Properties of Linear-Time Invariant SystemCausality of LTI Systems
The current output of a causal system depends only on the
present and past values of the input.
LTI system is causal if
For causal LTI system, the convolution sum and integralbecome
( ) 0
[ ] 0 0
h t for t < 0
h n for n .
t
n
k
thxty
knhkxny
)()()(
][][][
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Properties of Linear-Time Invariant SystemCausality of LTI Systems
Causality for linear system is equivalent to the condition of
initial rest.
Initial rest: if the input to causal system is zero up to
some point in time, then the output must also bezero up to the same time.
Both the accumulator and its inverse are causal.
][][ nunh
[ ] [ ] [ 1]h n n n
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Prepared by Dr Nidal Kamel 14
Properties of Linear-Time Invariant SystemStability of LTI Systems
A system is stable if every bounded input produces bounded
output (BIBO).
Consider input to LTI that is bounded in magnitude:
The magnitude of the output is:
Since the magnitude of the sum of set of numbers is nolarger than the sum of their magnitudes, it follows that
[ ]x n B for all n.
[ ]k
y n h k x n k
[ ]
[ ]
k
k
y n h k x n k
B h k for all n
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Properties of Linear-Time Invariant SystemStability of LTI Systems
Thus, we may conclude that LTI system is stable if the
impulse response is absolutely summable, that is, if
In CT-LTI, the system is stable if the impulse response is
absolutely integrable, i.e., if
[ ]k
h k
( )h d
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Properties of Linear-Time Invariant SystemStability of LTI Systems
Consider the pure time shift system in either DT or CT.
thus the system is stable.
Consider the accumulator system, h[n] = u[n]. This system
is unstable, because
0
0
[ ] 1
( ) ( 1
n n
h n n n
h d t d
0
0
[ ] [ ]
( )
n n
u n u n
u d d
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Properties of Linear-Time Invariant SystemUnit step Response of an LTI
With DT systems, step response is obtained when u[n] is
applied at the input of the system
Thus h[n] can be recovered from s[n] using the relation
[ ] [ ] [ 1] [ ] [ ] [ 1]n u n u n h n s n s n
k
n
k
khknukh
nunhnhnuns
][][][
][][][][][