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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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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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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

][][][

][][][][][