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Lecture 4: Linear Time Invariant (LTI) systems

2. Linear systems, Convolution (3 lectures): Impulse

response, input signals as continuum of impulses.

Convolution, discrete-time and continuous-time. LTI

systems and convolution

Specific objectives for today:

We’re looking at discrete time signals and systems

• Understand a system’s impulse response properties

• Show how any input signal can be decomposed into

a continuum of impulses

• DT Convolution for time varying and time invariant

systems

EE-2027 SaS, L4: 2/17

LTI systems

• Two important basic Properties of systems:

• Linearity.

• Time-invariance (superposition property).

• Plays an important role in signals and systems

analysis.

• Many of physical processes possess these properties.

• They are modeled as LTI systems

• Any system possess these two properties is called

linear time- invariant (LTI) system.

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LTI systems Properties

• Develop a complete characterization of LTI system in terms

to its impulse response using convolution sum for DTS and

convolution integral for CTS. 4/17

Representation of DTS in Terms of Impulses

• A DTS x[n] can be viewed as sequence of individual

impulses or as a linear combination of time-shifted

impulses:

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Representation of DTS in Terms of Impulses

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Representation of DTS in Terms of Impulses

EE-2027 SaS, L4: 7/17

Discrete Impulses & Time Shifts

Basic idea: use a (infinite) set of of discrete time impulses to represent any signal.

Consider any discrete input signal x[n]. This can be written as the linear sum of a set of unit impulse signals:

Therefore, the signal can be expressed as:

In general, any discrete signal can be represented as:

k

knkxnx ][][][

101]1[

]1[]1[

000]0[

][]0[

101]1[

]1[]1[

nnx

nx

nnx

nx

nnx

nx

]1[]1[ nx

actual value Impulse, time

shifted signal

The sifting property

]1[]1[][]0[]1[]1[]2[]2[][ nxnxnxnxnx

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Representation of DTS in Terms of Impulses

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Example

The discrete signal x[n]

Is decomposed into the

following additive

components

x[-4][n+4] +

x[-3][n+3] + x[-2][n+2] + x[-1][n+1] + …

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DT LTI systems: the convolution sum

• Is the mathematical relationship that links the input and

output signals in any LTI discrete-time system.

• Given an:

• LTI system,

• Input signal x[n].

• Impulse Response H[n]: the response to one of the

basic signals such as impulse signal;

• The convolution sum will allow us to compute the

corresponding output signal y[n] of the system.

• Compute y[n] ??????????

EE-2027 SaS, L4: 11/17

Introduction to Convolution

Definition Convolution is an operator that takes an input signal and

returns an output signal, based on knowledge about the system’s unit

impulse response h[n].

The basic idea behind convolution is to use the system’s response to a

simple input signal to calculate the response to more complex signals

This is possible for LTI systems because they possess the

superposition property (lecture 3):

k kk nxanxanxanxanx ][][][][][ 332211

k kk nyanyanyanyany ][][][][][ 332211

System y[n] = h[n]x[n] = [n]

System: h[n] y[n]x[n]

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Response of LTI as a linear combination of

impulse response

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Response of LTI as a linear combination of

impulse response

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Response of LTI as a linear combination of impulse response

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Response of LTI as a linear combination of

impulse response

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Convolution Sum

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Convolution Sum

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Response of LTI as a combination of H[n]

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Convolution sum

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Convolution sum

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Example : LTI Convolution

A LTI system with the following

unit impulse response:

h[n] = [0 0 1 1 1 0 0]

For the input sequence:

x[n] = [0 0 0.5 2 0 0 0]

The result is:

y[n] = … + x[0]h[n] + x[1]h[n-1] +

…

= 0 +

0.5*[0 0 1 1 1 0 0] +

2.0*[0 0 0 1 1 1 0] +

0

= [0 0 0.5 2.5 2.5 2 0]

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Example 2: LTI Convolution

Consider the problem

described for example 1

Sketch x[k] and h[n-k] for any

particular value of n, then

multiply the two signals and

sum over all values of k.

For n<0, we see that x[k]h[n-k]

= 0 for all k, since the non-

zero values of the two

signals do not overlap.

y[0] = Skx[k]h[0-k] = 0.5

y[1] = Skx[k]h[1-k] = 0.5+2

y[2] = Skx[k]h[2-k] = 0.5+2

y[3] = Skx[k]h[3-k] = 2

As found in Example 1

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Convolution sumExample 5: Compute y[0] for the input signal and impulse response of

an LTI system shown in the following Figure.

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Convolution sumExample 5: Compute y[1] for the input signal and impulse response of

an LTI system shown in the following Figure.

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Convolution sum

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Convolution sum

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Example 3: LTI Convolution

Consider a LTI system that has a step

response h[n] = u[n] to the unit

impulse input signal

What is the response when an input

signal of the form

x[n] = anu[n]

where 0<a<1, is applied?

For n0:

Therefore,

a

a

a

1

1

][

1

0

n

n

k

kny

][1

1][

1

nunyn

a

a

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Convolution sum

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Convolution sum

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Convolution sum

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Convolution sum

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Discrete, Unit Impulse System Response

A very important way to analyse a system is to study the

output signal when a unit impulse signal is used as

an input

Loosely speaking, this corresponds to giving the system

a kick at n=0, and then seeing what happens

This is so common, a specific notation, h[n], is used to

denote the output signal, rather than the more

general y[n].

The output signal can be used to infer properties about

the system’s structure and its parameters q.

System: q h[n][n]

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Types of Unit Impulse Response

Looking at unit impulse

responses, allows you to

determine certain system

properties

Causal, stable, finite impulse response

y[n] = x[n] + 0.5x[n-1] + 0.25x[n-2]

Causal, stable, infinite impulse response

y[n] = x[n] + 0.7y[n-1]

Causal, unstable, infinite impulse response

y[n] = x[n] + 1.3y[n-1] EE-2027 SaS, L4: 34/17

Linear, Time Varying Systems

If the system is time varying, let hk[n] denote the response

to the impulse signal [n-k] (because it is time varying,

the impulse responses at different times will change).

Then from the superposition property (Lecture 3) of linear

systems, the system’s response to a more general input

signal x[n] can be written as:

Input signal

System output signal is given by the convolution sum

i.e. it is the scaled sum of impulse responses

k

k nhkxny ][][][

k

knkxnx ][][][

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Example: Time Varying Convolution

x[n] = [0 0 –1 1.5 0 0 0]

h-1[n] = [0 0 –1.5 –0.7 .4 0 0]

h0[n] = [0 0 0 0.5 0.8 1.7 0]

y[n] = [0 0 1.4 1.4 0.7 2.6 0]

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Linear Time Invariant Systems

When system is linear, time invariant, the unit impulse

responses are all time-shifted versions of each other:

It is usual to drop the 0 subscript and simply define the

unit impulse response h[n] as:

In this case, the convolution sum for LTI systems is:

It is called the convolution sum (or superposition sum)

because it involves the convolution of two signals x[n]

and h[n], and is sometimes written as:

knhnhk 0][

nhnh 0][

k

knhkxny ][][][

][*][][ nhnxny

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System Identification and Prediction

Note that the system’s response to an arbitrary input signal is

completely determined by its response to the unit impulse.

Therefore, if we need to identify a particular LTI system, we

can apply a unit impulse signal and measure the system’s

response.

That data can then be used to predict the system’s response

to any input signal

Note that describing an LTI system using h[n], is equivalent to

a description using a difference equation. There is a direct

mapping between h[n] and the parameters/order of a

difference equation such as:y[n] = x[n] + 0.5x[n-1] + 0.25x[n-2]

System: h[n]

y[n]x[n]

Discrete LTI Convolution in Matlab

In Matlab to find out about a command, you can search the help files or type:

>> lookfor convolution

at the Matlab command line. This returns all Matlab functions that contain the term “convolution” in the basic description

These include:

conv()

To see how this works and other functions that may be appropriate, type:

>> help conv

at the Matlab command line

Example:

>> h = [0 0 1 1 1 0 0];

>> x = [0 0 0.5 2 0 0 0];

>> y = conv(x, h)

>> y = [0 0 0 0 0.5 2.5 2.5 2 0 0 0 0 0]

Consider the DT SISO system:

If the input signal is and the system has no energy at

, the output is called the impulse response

of the system

DT Unit-Impulse Response

[ ]y n[ ]x n

[ ]h n[ ]n

[ ] [ ]x n n[ ] [ ]y n h n

System

System

0n

General Response

Impulse Response

Consider the DT system described by

Its impulse response can be found to be

Example

[ ] [ 1] [ ]y n ay n bx n

( ) , 0,1,2,[ ]

0, 1, 2, 3,

na b nh n

n

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Let x[n] be an arbitrary input signal to a DT LTI system

Suppose that for

This signal can be represented as

Representing Signals in Terms ofShifted and Scaled Impulses

0

[ ] [0] [ ] [1] [ 1] [2] [ 2]

[ ] [ ], 0,1,2,i

x n x n x n x n

x i n i n

1, 2,n [ ] 0x n

Exploiting Time-Invariance and Linearity

0

[ ] [ ] [ ], 0i

y n x i h n i n

This particular summation is called the convolution sum

Equation is called the convolution representation of the system

Remark: a DT LTI system is completely described by its impulse

response h[n]

0

[ ] [ ] [ ]i

y n x i h n i

The Convolution Sum

[ ] [ ]x n h n

[ ] [ ] [ ]y n x n h n

Since the impulse response h[n] provides the complete

description of a DT LTI system, we write

Block Diagram Representation of DT LTI Systems

[ ]y n[ ]x n [ ]h n

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Example:

Suppose that both x[n] and v[n] are equal

Plot of [ ] [ ]x n v n

Associativity

Commutativity

Distributivity w.r.t. addition

Properties of the Convolution Sum

[ ] ( [ ] [ ]) ( [ ] [ ]) [ ]x n v n w n x n v n w n

[ ] [ ] [ ] [ ]x n v n v n x n

[ ] ( [ ] [ ]) [ ] [ ] [ ] [ ]x n v n w n x n v n x n w n

Shift property: define

Convolution with the unit impulse

Convolution with the shifted unit impulse

Properties of the Convolution Sum -Cont’d

[ ] [ ] [ ]w n x n v n

[ ] [ ] [ ] [ ] [ ]q qw n q x n v n x n v n

[ ] [ ]qx n x n q

[ ] [ ]qv n v n q

then

[ ] [ ] [ ]x n n x n

[ ] [ ] [ ]qx n n x n q

Example: Computing Convolution with Matlab

Consider the DT LTI system

impulse response:

input signal:

[ ]y n[ ]x n [ ]h n

[ ] sin(0.5 ), 0h n n n [ ] sin(0.2 ), 0x n n n

n=0:40;

x=sin(0.2*n);

h=sin(0.5*n);

y=conv(x,h);

stem(n,y(1:length(n)))

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Convolution sum

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Convolution sum

Consider the CT SISO system:

If the input signal is and the system has no

energy at , the output

is called the impulse response of the system

CT Unit-Impulse Response

( )h t( )t

( ) ( )x t t

( ) ( )y t h t

( )y t( )x t System

System

0t

Let x[n] be an arbitrary input signal with

for

Using the sifting property of , we may write

Exploiting time-invariance, it is

Exploiting Time-Invariance

( ) 0, 0x t t

( )t

0

( ) ( ) ( ) , 0x t x t d t

( )h t ( )t System

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Exploiting Time-Invariance

Exploiting linearity, it is

If the integrand does not contain an impulse

located at , the lower limit of the integral can be

taken to be 0,i.e.,

Exploiting Linearity

0

( ) ( ) ( ) , 0y t x h t d t

( ) ( )x h t

0

0

( ) ( ) ( ) , 0y t x h t d t

This particular integration is called the convolution integral

Equation is called the convolution representation of the system

Remark: a CT LTI system is completely described by its

impulse response h(t)

The Convolution Integral

( ) ( )x t h t

( ) ( ) ( )y t x t h t

0

( ) ( ) ( ) , 0y t x h t d t

Since the impulse response h(t) provides the complete

description of a CT LTI system, we write

Block Diagram Representation of CT LTI Systems

( )y t( )x t ( )h t

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Suppose that where p(t) is the rectangular

pulse depicted in figure

Example: Analytical Computation of the Convolution Integral

( ) ( ) ( ),x t h t p t

( )p t

tT0

• In order to compute the convolution integral

we have to consider four cases:

0

( ) ( ) ( ) , 0y t x h t d t

Example –Cont’d

Case 1: 0t ( )x

T0

( )h t

t T t

( ) 0y t

0 t T • Case 2:

( )x

T0

( )h t

t T t

0

( )

t

y t d t

• Case 3:0 2t T T T t T

( )x

T0

( )h t

t T t

( ) ( ) 2

T

t T

y t d T t T T t

• Case 4: 2T t T T t ( )x

T0

( )h t

t T t( ) 0y t

( ) ( ) ( )y t x t h t

T0 t2T

Associativity

Commutativity

Distributivity w.r.t. addition

Properties of the Convolution Integral

( ) ( ( ) ( )) ( ( ) ( )) ( )x t v t w t x t v t w t

( ) ( ) ( ) ( )x t v t v t x t

( ) ( ( ) ( )) ( ) ( ) ( ) ( )x t v t w t x t v t x t w t

Shift property: define

Convolution with the unit impulse

Convolution with the shifted unit impulse

Properties of the Convolution Integral - Cont’d

( ) ( ) ( )w t x t v t

( ) ( ) ( ) ( ) ( )q qw t q x t v t x t v t

( ) ( )qx t x t q

( ) ( )qv t v t q

then

( ) ( ) ( )x t t x t

( ) ( ) ( )qx t t x t q

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Convolution Integral - Properties

)](*)([)](*)([)]()([*)(

)](*)([*)()(*)](*)([

)(*)()(*)(

2121

2121

thtxthtxththtx

ththtxththtx

txththtx

• Commutative

• Associative

• Distributive

Example 1

Consider a CT-LTI system. Assume the impulse response of

the system is h(t)=e^(-at) for all a>0 and t>0 and input

x(t)=u(t). Find the output.

h(t)=e^-atu(t) y(t)

)()1(1

)1(1

)()(

)()()(

)()()()()(

0

tuea

ea

de

dtuue

dtuhty

tuthtxthty

at

at

t

a

a

Draw x(), h(), h(t-),etc. next slide

Because t>0

The fact that a>0 is not an issue!

Example 1 – Cont.

y(t)

t>0

t<0

Remember we are plotting it over

and t is the variable

U(-(-t))

U(-(-t))

)()1(1

)1(1

)()(

)()()(

)()()()()(

0

tuea

ea

de

dtuue

dtuhty

tuthtxthty

at

at

t

a

a

t

y(t); for a=3

t

64

t

Example 2.5: Convolution Integral.

Given a RC circuit below (RC=1s). Use convolution to determine the voltage across the capacitor y(t). Input voltage x(t)=u(t)-u(t-2).

Solution:

y(t)=x(t)*h(t)

- capacitor start charging

at t=0 and discharging

at t=2.

a

b

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65

Cont’d…

66 .

Cont’d…