15/03/1439

1

EE-2027 SaS, L6 1/18

Lecture 6: Differential and Difference 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

EE-2027 SaS, L6 2/18

Lecture 6: Differential and Difference LTI

systems

Specific objectives for today:

• Properties of an LTI system

• Differential and difference systems

.

LTI Systems and Impulse Response

)(*)()()()(

][*][][][][

thtxdthxty

nhnxknhkxnyk

However, the following systemsalso have the same impulseresponse

Example: The discrete-time impulse response

Using convolution sum we can find the system model

LTI Systems and Impulse Response

The characteristics of LTI system are completelydescribed by its impulse response. (This onlyholds for LTI systems)

]1[][][ nxnxny

]1[],[max][

]1[][][2

nxnxny

nxnxny

otherwise0

1,01][

nnh

If the system is non-linear, it is not completely characterised by the impulse response

15/03/1439

2

1. Commutative Property (Series Systems)Convolution is a commutative operator (in both DT and CT time)

dtxhtxththtx

knxkhnxnhnhnxk

)()()(*)()(*)(

][][][*][][*][

][*][][][][][][*][ nxnhrhrnxknhkxnhnxrk

Set n-k = r

The output of DT LTI System (or CT LTI) withimpulse response h[n] (or h(t)) and input x[n] (orx(t)) is the same as The output of LTI systemwith impulse response x[n] (or x(t)) and inputh[n] (or h(t)).

Distributive Property (Parallel Systems)

The two systems are equivalent.

)()()(*)()(*)())()((*)(

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

212121

212121

tytythtxthtxththtx

nynynhnxnhnxnhnhnx

A set of parallel of LTI systems can be replaced by a

single LTI system with impulse response equal to

sum of the impulse responses of individual systems.

Distributive Property (Parallel Systems)

The two systems are equivalent.

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

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

2121

2121

txthtxthtxtxth

nxnhnxnhnxnxnh

The response of LTI system to the sum of tow

inputs must equal the sum of the response to these

signals individually.

توضيح بالرسم

Distributive Property (Parallel Systems)

Let Y[n] denote the convolution of the following two

sequences

15/03/1439

3

Distributive Property (Parallel Systems)

Let Y[n] denote the convolution of the following two

sequences

Example: Distributive Property

Let y[n] denote the convolution of the following two sequences:

x[n] is non-zero for all n. We will use the distributive property to express

y[n] as the sum of two simpler convolution problems. Let

x1[n] = 0.5nu[n], x2[n] = 2nu[-n], it follows that

and y[n] = y1[n]+y2[n], where y1[n] = x1[n]*h[n], y1[n] = x1[n]*h[n].

][][

][2][5.0][

nunh

nununx nn

][*])[][(][ 21 nhnxnxny

][5.01

5.01][

1

1 nunyn

12

02][

1

2n

nny

n

Associative Property (Serial Systems)

)(*))(*)(())(*)((*)(

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

2121

2121

ththtxththtx

nhnhnxnhnhnx

][*][*][][ 21 nhnhnxny

This is not true for non-linear systems (y1[n] = 2x[n], y2[n] = x2[n])

Or

Multiply by 2 and then square the signal not equal the reverse

The impulse response

of the cascade of two

LTI system is the

convolution of their

individual impulse

responses regardless

of the order of them.

LTI System Memory

)()(

][][

tkxty

nkxny

An LTI system is memory-less if

its output depends only on the

input value at the same time

Continuous Time Discrete Time

IF C=1 Identity system

The output = Input

Y[n]=x[n]*[n]=x[n]

15/03/1439

4

System Inevitability

Widely used concept for:

• control of physical systems, where the aim is to calculate a

control signal such that the system behaves as specified

• filtering out noise from communication systems, where the

aim is to recover the original signal x(t)

)()()(

][][][

tthth

nnhnh

inv

inv

Does there exist a system with impulse response hinv (t) such that y(t)=x(t)?

Time shift system is Inevitable system

Accumulator System

EE-2027 SaS, L6 16/18

Example: Accumulator System

Consider a DT LTI system with an impulse response

h[n] = u[n]

Using convolution, the response to an arbitrary input x[n]:

As u[n-k] = 0 for n-k<0 and 1 for n-k0, this becomes

i.e. it acts as a running sum or accumulator. Therefore an

inverse system can be expressed as:

A first difference (differential) operator, which has an impulse

response

k

knhkxny ][][][

n

k

kxny ][][

]1[][][ nxnxny

]1[][][1 nnnh

15/03/1439

5

17/18

Causality for LTI Systems (Initial Rest)

dthxthtx

knhkxnhnx

t

n

k

)()()(*)(

][][][*][

A causal system only depends on present and past values of the input signal. We do not use knowledge about future information.

h[n] = 0 for n<0

- Any signal is causal if it is zero for n<0 or t<0.

- A LTI system is causal if its impulse response signal is causal.

18/18

LTI System Stability

dh )(

Discrete-time system

Remember: A system is stable if every bounded

input produces a bounded output

consider a bounded input signal |x[n]| < B for all n

Continuous-time system

k

kh ][

A LTI system is stable if and only if its

impulse response is absolutely summable or

Integerable

LTI System Stability

Integrator

Time Shift Differential and Difference Systems

important class of systems in engineering

• Circuit analysis

• Filter design

• Controller design

• Process modeling

• Many other applications.

Differential Systems form a subset of

Continuous LTI systems

15/03/1439

6

Differential and Difference EquationsTwo extremely important classes of causal LTI systems:

1) CT systems whose input-output response is described by linear, constant-coefficient, ordinary differential equations with a forced response

2) DT systems whose input-output response is described by linear, constant-coefficient, difference equations

- To “solve” these equations for y(t) or y[n], we need to know the initial conditions

Examine such systems and relate them to the system properties just described

)()()(

tbxtaydt

tdy RC circuit with: y(t) = vc(t),

x(t) = vs(t), a = b = 1/RC.

][]1[][ nbxnayny

Simple bank account with:

a = -1.01, b = 1.

EE-2027 SaS, L6 22/18

Continuous-Time Differential Equations

A general Nth-order LTI differential equation is

If the equation involves derivative operators on y(t) (N>0) or

x(t), it has memory.

The system stability depends on the coefficients ak. For

example, a 1st order LTI differential equation with a0=1:

If a1>0, the system is unstable as its impulse response

represents a growing exponential function of time

If a1<0 the system is stable as its impulse response

corresponds to a decaying exponential function of time

M

kk

k

k

N

kk

k

kdt

txdb

dt

tyda

00

)()(

0)()(

1 tyadt

tdy taAety 1)(

2318

Continuous-Time Differential Equations

2418

Example of CT Differential System

15/03/1439

7

2518

Example of CT Differential System Example of CT Differential System

Calculate the Impulse response Calculate the Impulse response

15/03/1439

8

Discrete-Time Difference EquationsA general Nth-order LTI difference equation is

If the equation involves difference operators on y[n] (N>0) or x[n],

it has memory.

The system stability depends on the coefficients ak. For example,

a 1st order LTI difference equation with a0=1:

If a1>1, the system is unstable as its impulse response represents

a growing power function of time

If a1 <1 the system is stable as its impulse response corresponds

to a decaying power function of time

M

k

k

N

k

k knxbknya00

][][

0]1[][ 1 nyanynAany 1][

Example of DT Difference System

Example of DT Difference System Example of DT Difference System

15/03/1439

9

Find Impulse Response: Find Impulse Response:

Find Impulse Response:

36

Is an interconnection of elementary operations

that act on the input signal.

A more detailed representation of the system than

the impulse response or differential (difference)

equation description since it describes how the

system’s internal computations or operations are

ordered.

Block Diagram Representation.

15/03/1439

10

37

Block diagram representations consists of an

interconnection of three elementary operations on

signals;

(1) Scalar Multiplication.

(2) Addition.

(3) Integration for continuous-time LTI

system.

Block Diagram Representation.

15/03/1439

11

15/03/1439

12

S1 followed by S2 S2 followed by S1

S1 followed by S2 S2 followed by S1

S1 followed by S2 S2 followed by S1

15/03/1439

13

S1 followed by S2 S2 followed by S1

Second Order Difference Equation.

15/03/1439

14

EE-2027 SaS, L6 53/18

Lecture 6: Summary

The standard notions of commutative, associative and distributive properties are valid for convolution operators. These can be used to simplify evaluating convolution, by decomposing the input system/signal into simpler parts, and then solving the transformed problem.

Standard system notations of

• Memory

• Causality

• Invertibility

• Stability

Can be given specific definitions in terms of representing a system via its convolution response or in terms of the derivative/differential equation