Li Ti I i t (LTI)Linear Time Invariant (LTI)...

Li Ti I i t (LTI)Linear Time Invariant (LTI) SystemsSystems

Rui Wang, Assistant professorDept. of Information and Communication

T ji U i itTongji University

Email: ruiwang@tongji.edu.cn

OutlineOutline Discrete-time LTI system: The convolution y

Sum Continuous time LTI systems: The Continuous-time LTI systems: The

convolution integral Property of Linear Time-Invariant Systems Causal LTI Systems Described by Causal LTI Systems Described by

Differential and Difference Equations Singularity Functions

2.1 Discrete-time LTI system: SThe convolution Sum

Using delta function to represent discrete Using delta function to represent discrete-time signal

An example:

The response of a linear system to x[n] will The response of a linear system to x[n] will be the superposition of the scaled

f th t t h f thresponses of the system to each of these shifted delta functions.

The property of time invariance tells us that the responses to the time-shifted deltathat the responses to the time-shifted delta functions are simply time-shifted version of

thone another.

Let denote the response of the linear Let denote the response of the linear system to With

The output of the system The output of the system

An example: An example:

If linear system is time invariant we have If linear system is time invariant, we have

This simplify as This simplify as

This result is referred to as the convolution sum

Example 1: Consider an LTI system with Example 1: Consider an LTI system with impulse response h[n] and input x[n] as f ll D t i th t tfollows. Determine the output

Solution: Solution:

Example 2: Consider an LTI system with Example 2: Consider an LTI system with

Determine the output

Solution: Solution:

Example 3: Consider an LTI system with Example 3: Consider an LTI system with

Determine the output

Solution: Solution: Interval 1: n<0, y[n] =0 Interval 2: 0≤n ≤4

Solution: Solution: Interval 3: for n>4, but n-6 ≤0, i.e., 4≤n ≤6

Solution: Solution: Interval 4: for n>6, but n-6 ≤4, i.e., 6≤n ≤10

Solution: Solution: Interval 4: for n-6>4, i.e., n>10i.e., 6≤n ≤10

Solution: Solution:

2.2 Continuous-time LTI system: The convolution integral 2 2 1 Representing the continuous time 2.2.1 Representing the continuous-time

signals in terms of impulses

2.2 Continuous-time LTI system: The convolution integral

2.2 Continuous-time LTI system: The convolution integral If we define a pulse function as If we define a pulse function as

Since has unit amplitude we have Since has unit amplitude, we have

In above, for any value of t, only one term in , y , ythe summation on the right-hand side is non-zero.

2.2 Continuous-time LTI system: The convolution integral As △0 the summation approaches an As △0, the summation approaches an

integral.

Check the unit step function:

2.2 Continuous-time LTI system: The convolution integral 2 2 2 the unit impulse response and the 2.2.2 the unit impulse response and the

convolution integral As x(t) is approximated as the sum of shifted

version of the basic versions of the pulse signal

The response of the linear system will be

convolution integral As , we have

convolution integral When the linear system is time invariant, we

Define we have Define , we have

Referred to the above as the convolution integral.

2.2 Continuous-time LTI system: The convolution integral Example 1: Let x(t) be the input to an LTI

system with unit impulse response h(t), y p p ( ),where

determine the output y(t).

2.2 Continuous-time LTI system: The convolution integral Solution:

2.2 Continuous-time LTI system: The convolution integralWhen t<0, y(t) = 0

2.2 Continuous-time LTI system: The convolution integralWhen t>0

1 0 t T 0 2t t T 1 0( )

x totherwise

0 2( )

0t t T

h totherwise

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

21( )t

t d t 2

2y t d t

21( )t

y t d Tt T ( )2t T

12 2 21( ) 2 ( )2

t Ty t d T t T

( ) 0y t

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

2.2 Continuous-time LTI system: The convolution integralWhen t-3<=0, i.e., t<=3

When t-3>0, i.e., t>3

2 3 Properties of LTI Systems2.3 Properties of LTI Systems

Use convolution sum and convolution integral to obtain the output of discrete-g ptime and continuous-time systems, based on the unit impulse responsebased on the unit impulse response

An LIT system is completely An LIT system is completely characterized by its impulse response ( l f LTI t )(only for LTI system)

2 3 Properties of LTI Systems

交换律

2.3 Properties of LTI Systems

2.3.1 The commutative property (交换律)

Proof: Proof:

2.3.2 The distributive property (分配律) Definition:

结合律

2.3.3 The associative property (结合律) Definition:

结合律

2.3.3 The associative property (结合律) Definition:

2.3.4 LTI system with and without memory (有记忆和无记忆系统)y (有无 ) Definition: the output only depends on value

of the input at the same timeof the input at the same time. We have

i e i.e.,

2.3.5 Inevitability of LTI system (可逆性)

Identical systemIdentical system

Example: Delayed or advanced systemy y

W h We have

Can we find such that Can we find such that

Example: accumulator systemy

The inverse system is a first different ti ioperation, i.e.,

2.3.6 Causality for LTI (因果性) LTI system should satisfyy y

2.3.7 Stability for LTI (稳定性) LTI system should satisfy: absolutely y y y

summable and absolutely integrable

How to prove??

Proof: Proof:

Example: time shift

2.3.8 The unit step response of LTI Use unit step response to describe the p p

system behavior

2.4 Causal LTI system described ff & ffby differential & difference equs

An important class practical system can be described by usingy g Linear constant-coefficient differential

equationsequations Linear constant-coefficient difference

equationsequations

Linear constant-coefficient differential equations

( ) ( )k kN Md y t d x t 0 0

( ) ( ),k kk kk k

d y t d x ta bdt dt

are constant, N is the order,k ka b The response to an input x(t) generally consists of

the sum of particular solution (特解) to the diff ti l ti & h l ti (其differential equation & a homogeneous solution (其次解)

a homogeneous solution (nature response: 自然相应) l ti t th diff ti l ti ith i t应): a solution to the differential equation with input set to zero

Different choices of auxiliary conditions leads to Different choices of auxiliary conditions leads to different relationships between input and output

Linear constant-coefficient difference equations

( ) ( )N M

k kk k

a y n k b x n k

are constant, N is the order,k ka b The response to an input x[n] generally consists of

the sum of particular solution (特解) to the diff ti l ti & h l ti (其differential equation & a homogeneous solution (其次解)

a homogeneous solution (nature response: 自然相应) l ti t th diff ti l ti ith i t应): a solution to the differential equation with input set to zero

Re express difference equation as recursive Re-express difference equation as recursive equation

1( ) ( ) ( )M N

( ) ( ) ( )k kk k

y n b x n k a y n ka

We see that

( )x n ( 1), ( 2), , ( )y y y N (0)y

(0)y ( 1), ( 2), , ( 1)y y y N (1)y

0 When N = 0, non-recursive equation

2.4 Causal LTI system described ff & ffNon recursive case:

by differential & difference equs Non-recursive case:

( ) ( )M

kby n x n k The unit impulse response is

0 0k a

Finite impulse response (FIR) system For recursive case: Infinite Impulse Response

(IIR) system

Block diagram representation of the first-order systems described by differential and difference equationsequations

Addition

Multiplication by a coefficient

The following first-order difference equation can be described as

1 M N 0 10

1( ) ( ) ( )k kk k

( )x n ( )w n0b

( ) ( )M

kw n b x n k D

0( ) ( )k

kw n b x n k

TypeⅠ

( )w n ( )y n01/ a

1( ) ( ) ( )N

y n w n a y n ka

a10 ka

1 M N 0 10

1( ) ( ) ( )k kk k

( )x n ( )y n0b01 / a

TypeⅡ

For differential equation

Addition

Multiplication by a coefficient

differentiation

The following first-order differential equation can be described as

( ) ( )k kN N

k kk kk k

d y t d x ta bdt dt

0 0k k

( ) ( )( ) ( )N N

k N k k N ka y t b x t ( ) ( )0 0

( ) ( )k N k k N kk k

11 N N 1

( ) ( )0 0

1( ) ( ) ( )N N

k N k k N kk kN

y t b x t a y ta

( )x t ( )w tNb ( )w t ( )y t1/ Na

TypeⅠ2Nb

TypeⅠ1b

11( ) ( ) ( )N N

t b t t

( ) ( )0 0

( ) ( ) ( )k N k k N kk kN

y t b x t a y ta

( )x t ( )y t1/ a b ( )x t ( )y t1/ Na Nb

TypeⅡ0a 0b

2 5 Singularity functions2.5 Singularity functions

The delta function is the impulse response of the identity systemp y y

Unit doublet function : the derivative of the unit impulsep

We define as the k-th derivative of delta function

Property of unit doublet p y

For t = 0, yields

Unit step function:

The unit ramp function The unit ramp function

Similarly, we have

We have We have

Li Ti I i t (LTI)Linear Time Invariant (LTI)...

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