Introduction to signals and systems - NPRUhome.npru.ac.th/sopapun/Lecture3.pdf · Systems 6552111 Signals and Systems Sopapun Suwansawang 2 For the most part, our view of systems

Signals and Systems

6552111 Signals and Systems

Sopapun Suwansawang

1

Lecture #3

Elementary Signals and Systems

Week#3

Systems


Sopapun Suwansawang 2

For the most part, our view of systems will be

from an input-output perspective:

A system responds to applied input signals, and

its response is described in terms of one or

more output signals

Examples of Systems

Sopapun Suwansawang 3


An RLC circuit

Dynamics of an aircraft or space vehicle

An algorithm for analyzing financial and

economic factors to predict bond prices

An edge detection algorithm for medical images

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System Interconnections

An important concept is that of interconnecting

systems

-To build more complex systems by interconnecting

simpler subsystems

-To modify response of a system

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Signal flow (Block) diagram


Cascade Interconnection

The cascade interconnection is a successive application

of two (or more) systems on an input signal:



1212 )( yGxGGy

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Parallel Interconnection

The parallel interconnection is an application of two (or

more) systems to the same input signal, and the output

is taken as the sum of the outputs of the individual

systems.



xGGy

xGxGy

yyy

)( 21

21

21

Feedback Interconnection

The feedback interconnection of two systems is a

feedback of the output of system G1 to its input,

through system G2. In this context, signal e is the error

between a desired output signal and a direct

measurement of the output.

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eGy

yGxe

1

2

Systems may be interconnections of other systems. Example of the discrete-time system :

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][][][],[][

][][],[][

2

31

nznwnsnvGnw

nxGnznxGnv

][][ 4 nsGny



System Properties

Causality - Causal & Noncausal

Linearity - Linear & Nonlinear

Time-invariance - Time-invariant & Time-varying

Memory - Memoryless & Memory

Stability - Stable & Unstable

Invertibility - Invertible & Noninvertible



Causality

A system is causal if its output at time t or n depends

only on past or current values of the input.

Mathematically (in CT): A system x(t) →y(t) is causal if

when

and

then

Causal or Noncausal ?



Causality

]1[2

1][.4

][][.3

)1()(.2

)1()(.1

31

2

nxny

nxny

txty

txty

n



Solve 1.

Solve 2.

Causality



Solve 3.

Solve 4.

Causality



Linearity

Linear systems

Additivity:

Given that Tx1= y1 and Tx2 = y2, then

for any signals x1 and x2.

Homogeneity (or Scaling):

for any signals x and any scalar .

yxT }{

2121 }{ yyxxT (1)

(2)

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Equations (1) and ( 2) can be combined into a

single condition as

where 1 and 2 are arbitrary scalars. Equation(3)

is known as the superposition property.

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Linearity

22112211 }{ yyxxT

Any system that does not satisfy Eq.(1) and/or Eq. (2)

is classified as a nonlinear system.



Linearity

Summary•Superposition

•For linear systems, zero input → zero output

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Linear or nonlinear?

Sopapun Suwansawang

Linearity

][

2

2

2

][.6

)()(.5

][][.4

)()(.3

][][.2

)()(.1

nxeny

txty

nxny

txty

nnxny

ttxty

19

Solve 2.

1 1

2 2

( ) ( )

( ) ( )

y n nx n

y n nx n

3 1 1 2 2

1 1 2 2

1 1 2 2

1 1 2 2

( ) [ ( ) ( )]

[ ( ) ( )]

( ) ( )

( ) ( )

y n T a x n a x n

n a x n a x n

na x n na x n

a y n a y n

][][ nnxny


Sopapun Suwansawang

Linearity

Linear

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2

1 1

2

2 2

( ) ( )

( ) ( )

y n x n

y n x n

3 1 1 2 2

2

1 1 2 2

2 2 2 2

1 1 1 2 1 2 2 2

2 2 2

1 1 2 2 1 1 2 2

( ) [ ( ) ( )]

( ) ( )

( ) 2 ( ) ( ) ( )

( ) ( ) ( ) ( )

y n T a x n a x n

a x n a x n

a x n a a x n x n a x n

a y n a y n a x n a x n

Nonlinear


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Linearity

Solve 4. ][][ 2 nxny

Informally, a system is time-invariant (TI) if its

behavior does not depend on what time it is.

Mathematically (in DT): A system x[n] → y[n] is

TI if for any input x[n] and anytime shift n0,

Similarly for a CT time-invariant system,

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Time-invariance (TI)

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Time-invariant or time-varying?


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)1()()(.4

][][.3

][][.2

))(sin()(.1

txtxty

nxny

nnxny

txty

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Solve 1.Let y1(t) be the output produced by the shifted

input x1(t) = x(t – to). Then

and

Hence, the system is time-invariant.

))(sin()}({)( 001 ttxttxTty

)())(sin()( 100 tyttxtty

))(sin()( txty

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Sopapun Suwansawang


Solve 2.Let y1[ n ] be the response to xl[ n ] = x[n - no].

Then

but

Hence, the system is time-varying.

][][ nnxny

][]}[{][ 001 nnnxnnxTny

][][)(][ 1000 nynnxnnnny

Linear Time-Invariant Systems

If the system is linear and also time-invariant,

then it is called a linear time-invariant (LTI)

system.

A basic fact: If we know the response of an LTI

system to some inputs, we actually know the

response to many inputs.

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Memory

A system is memoryless if its output at time t

or n depends only on the input at that same

time.

Conversely, a system has memory if its output

at time t or n depends on input values at some

other times.

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Sopapun Suwansawang

)(1

)()(,][][ 2

tx

txtynxny

]1[][]1[][ nxnxnxny

Stability

Bounded-Input Bounded-Output Stability

A system S is bounded-input bounded-output

(BIBO) stable if for any bounded input x, the

corresponding output y is also bounded.

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A system is called invertible if we can determine

its input signal x uniquely by observing its output

signal y.

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Invertibility


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)(21 txy )(2

11 tyy

Exercise

Consider the system shown in Fig. E-1. Determine

whether it is

(a) memoryless,

(b) causal,

(c) linear

(d) time-invariant,

(e) stable.

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Fig. E-1

Documents

Introduction to signals and systems - NPRUhome.npru.ac.th/sopapun/Lecture3.pdf · Systems 6552111 Signals and Systems Sopapun Suwansawang 2 For the most part, our view of systems