Signals and Systems 1 Lecture 7 Dr. Ali. A. Jalali September 4, 2002

Signals and Systems 1Lecture 7

Dr. Ali. A. JalaliSeptember 4, 2002

Signals and Systems 1

Lecture # 7Continuous Systems

EE 327 fall 2002

Continuous Systems1. Definition of system.2. Examples of systems.3. Classifications of systems.4. Linear time invariant systems.5. Conclusions.

EE 327 fall 2002 Signals and Systems 1

Continuous Systems1. Preview2. A system is transforms input signals into output signals.

3. A continuous-time system receives an input signal x(t) and generates an output signals y(t).

4. y(t)=h(t)x(t) means the system h(t) acts on input signal x(t) to produce output signal y(t).

5. We concentrate on systems with one input and one output signal, i.e., Single-input, single output (SISO) systems.

6. Systems often denoted by block diagram.

7. Lines with arrows denote signals (not wires). Arrows show inputs and outputs


Continuous-timeSystem

h(t)

x(t)Input

y(t)Output

Continuous SystemsExamples of C.S.:A filter to eliminate unwanted signals.

An automobile.

An algorithm. A circuit. (input voltage, output current)

EE 327 (input instructor and student’s effort and output instructor evaluation and student’s grade)

Stock Market (input buy orders and sell orders, output IBM stock price and Intel stock price)

We have MIMO, MISO, SIMO and SISO systems.

Many physical systems have the same mathematical model.


Continuous Systems

Example: Robot car block diagram

This is a subsystem.

Continuous SystemsExample: Human speech production system block diagram

Continuous Systems

Example: Mechanical free-body diagram

Continuous Systems

Example: Electric Network

Continuous Systems The mechanical and electrical systems are dynamically analogous.

Thus, understanding one of these systems gives insights into the other.

Continuous Systems Example: Block-diagram

using integrations, adders, and gains.

This is a subsystem.

Systems

Classifications of systems:1. Linear and nonlinear systems.

2. Time Invariant and time varying systems.

3. Causal, noncausal and anticausal systems.

4. Stable and unstable systems.

5. Memoryless systems and systems with memory.

6. Continuous and Discrete time systems.EE 327 fall 2002 Signals and Systems 1

Continuous SystemsProperties of continuous systems:1- Linearity: One of the most important concepts

in system theory is linearity.

2- Using linear system theory, you have easier and more convenient methods of analysis and design of systems.

3- A system is linear if and only if it satisfies the principle of homogeneity and the principle of additively.


Continuous SystemsLinearity:

principle of homogeneity (c is real constant).

principle of additively

homogeneity and additively (Principle of superposition)


x(t)=C 1 x1(t) y(t)=C 1 y1(t) LSx1(t) y1(t) LS

x1(t) y1(t)

x2(t) y2(t)

x(t)=x1(t)+x2(t) y(t)=y1(t)+y2(t) LS

LS LS

x1(t) y1(t)

x2(t) y2(t)

LS

LS LS

X(t)=C1x1(t)+C2x2(t) y(t)=C1y1(t)+C2y2(t)

21 , cc

Continuous SystemsLinearity example: Let the response of a linear system at rest due to the

system input be given by and let the response of the same system at rest due to another system input be

Then the response of the same system at rest due to input given by

Is simply obtained as:

,0,1)(1 ttf


,0,5.05.0)( 21 teety tt

,0),sin()(2 tttf

;0),cos()10/10(2.05.0)( 22 tteety tt

,0),sin(32)(3)(2 21 tttftf

.0),cos(10

1034.05.01)(3)(2)( 2

21 tteetytyty tt

Continuous Systems


LS

LSTT

t t

t t

x(t) y(t)

x(t-T) y(t-T)

x(t-T) y(t-T)

x(t) y(t)

1

1 1 4

31

Time Invariance:

Shifted inputShifted outputFor All value of t and T.

Continuous SystemsTime invariant example: Let the response of a time-invariant linear system at rest

due to be given by Then, the system response due to the shifted system input defined by

Is

,0),(1 ttf


,0,43)( 21 teety tt

6,0

6),6()( 1

2 t

ttftf

6,0

6,43)(

)6(2)6(

2t

teety

tt

Continuous SystemsLinear and Time Invariant systems: Many man-made and naturally occurring

systems can be modeled as LTI systems.

Powerful techniques have been developed to analyze and to characterize LTI systems.

The analysis of LTI systems is an essential precursor to the analysis of more complex systems.


Continuous SystemsCausality: A causal system is a nonpredictive system in that the output does not precede, or anticipate, the input.

Example for the input signal as an impulse

).()(1 tAtx


-1 -0.5 0 0.5 1 1.5 2 2.5-0.02

0

0.02

0.04

0.06

0.08

0.1 Fig.2.19c Angular velocity

time, seconds

v2(t

)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Fig.2.19c Angular velocity

time, seconds

v2(t

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

10

20

30

40

50

60 Fig.2.19c Angular velocity

time, seconds

v2(t

)

Causal System Anticusal System Noncusal System

Continuous SystemsCausality: For the causal system the output at time depends only on the

input for i.e., the system cannot anticipate the input.


0t,0tt

Continuous SystemsStability: System stability is defined from several points of view.1- In term of input –output behavior. If the system input is bounded and if the system is stable, then the output must also be bounded. (This test is known as BIBO.)2- From the characteristic roots of system we measure the stability. 3- From dynamic matrix A of systems (we learn more later).

Note: a) If we connect some stable systems together there is no guaranty that overall system becomes stable!

b) BIBO test is not an easy job in practical cases.Example:


-1 -0.5 0 0.5 1 1.5 2 2.5-0.02

0

0.02

0.04

0.06

0.08

0.1 Fig.2.19c Angular velocity

time, seconds

v2(t

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

10

20

30

40

50

60 Fig.2.19c Angular velocity

time, seconds

v2(t

)

Stable System

Unstable System

Continuous SystemsStability: Example with the test of BIBO. For the resistor, if i(t) is bounded then so is v(t), but for the capacitor this is not true. Consider i(t)= u(t) then v(t)=tu(t) which is unbounded.


Continuous SystemsMemoryless systems: The output of a memoryless system at some time Depends only on its input at the same time . Example is resistive divider network.

Therefor, depends upon the value of and not on for .

0t

0t

0tt )( 0tvo )( 0tvi

)(tvi

Continuous SystemsSystem with memory:Example: v(t) depends not just on i(t) at one point in time t. Therefor the system that relates v to i exhibits memory.

System with memory and memoryless. If a system is contained with capacitors, inductors and flip-flop the system is known as system with memory. Otherwise if it is contained with pure resistors, is known as memoryless system.

Continuous and Discrete time systems. If input and output of systems are continuous the system is referred as continuous-time systems.

If input and output of systems are discrete the system is referred as discrete-time systems.

Continuous Systems

Continuous SystemsNonlinear example: A continuous system is described by the input-output

Equation where K and A are real constants. This systems is not linear because:

If input is the output is

If input is the output is

For the input the output is

But

And

The system is nonlinear. If A = 0, then the system will be linear.

,)()( AtKxty


),(1 tx

),(2)()( 2313 txatxatx

AtKxty )()( 11

AtKxty )()( 22),(2 tx

,)}()({)()( 221133 AtxatxaKAtKxty },)({})({)()( 22112211 AtKxaAtKxatyatya

).()()( 22113 tyatyaty

Continuous SystemsTime invariant example: A continuous system is described by the input-outputEquation where K and A are real constants.

This systems is time invariant because:An arbitrary input produces an output

and the shifted version on this input produces an output

For the input the output is

But for all

so the system is time invariant (or shift invariant).

,)()( AtKxty


),(1 tx

),(2)()( 2313 txatxatx

AtKxty )()( 11

.)( 01 AttKx ),()( 012 ttxtx

AttKxtty )()( 0101 0t

Continuous SystemsConclution:1. Linear Time Invariant systems arespecial systems for

which powerful mathematical methods of description are available.

2. Classification of systems: linear and nonlinear systems, time-invariant and time-varying systems, causal, noncausal and anticausal systems, stable and unstable systems, systems with memory and memoryless systems, continuous and discrete-time systems.

3. Physical and non physical systems.4. Man-made and natural systems.


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Signals and Systems 1 Lecture 7 Dr. Ali. A. Jalali September 4, 2002