class_02-elementary+signals.pdf

Chengbin Ma UM-SJTU Joint Institute

Class#2

- Elementary Signals (1.6)

- Properties of Systems (1.8)

Slide 1


Review of Previous Lecture Systems: components + relationships

Structure, Behavior, Interconnectivity and Functions

Signals: a carrier of information or power

Applications: Information and computer science, Engineering

and physics, Social and cognitive sciences and management

research, Strategic thinking

Signals: continuous-time/discrete, even/odd (decomposition),

periodical/non-periodical, deterministic/random, energy/power

(instantaneous power, total energy, average power, RMS)

Basic Operations:

Dependent variable: amplitude scaling, addition, multiplication,

differentiation, integration

Independent variable: time scaling, reflection, time shifting,

combination of time shifting and time scaling

Slide 2

MichaelRectangle


This Lecture

Elementary signals: building blocks and

modeling of many physical signals

exponential, sinusoidal, exponentially damped

sinusoidal, step, impulse, ramp

NOTE: Behavior: inputs/processing/outputs of

material, energy, information, etc.

Properties of Systems:

stability, memory, causality, invertibility, time

invariance, linearity

Slide 3


Class#2



Slide 4


Exponential Signals

The 1st order dynamics

Decaying exponential: a0

#Example: RC circuit (RC: time constant)

Slide 5

atBetx )(

0)()( tvtvdt

dRC

RCteVtv /0)(

class2_exp.m

NOTE: It is the time required to charge the capacitor, through the resistor, by 63.2 % of the

difference between the initial value and final value or discharge the capacitor to 36.8 %.

If a = 0, the signals x(t) reduces to a DC signal equal to the constant B.

time constant. the larger the resistor R, the less lossy the capacitor, the slower will be the rate of decay of v(t) with time.

MichaelHighlight

MichaelHighlight

MichaelHighlight

MichaelHighlight

MichaelHighlight

MichaelHighlight

MichaelRectangle

MichaelOval

MichaelLine


Exponential Signals

Discrete-time exponential signals:

Complex exponential signals:

Slide 6

aerBrnx n where,][

njtj ee and

atBetx )(

Michael

Michael

Michael


Sinusoidal signals

The 2nd-order dynamics

Amplitude

Frequency (rad/sec)

Phase (rad)

#Example: LC circuit

Natural frequency:

Slide 7

)cos()( tAtx

0,)1

cos()(

0)()(

0

2

2

ttLC

Vtv

tvtvdt

dLC

LC

1

LCs

LC1

1

:function transfer Laplace

2

Bode plot:

L=0.001H,

C=0.001F

T = 2*pi/w

MichaelRectangle

MichaelHighlight


Sinusoidal Signals

Slide 8

Discrete sinusoidal signals:

Periodical:

Complex exponential signals:

)cos(][ nAnx

2 ,cos

cos][][

NNnA

NnANnxnx

)sin()cos(

sincos

njne

je

nj

j

ne

ne

nj

nj

4sin}Im{

4cos}Re{

/4:Example#

4

4

class2_sin.m

Michael


Exponentially Damped Sinusoidal Signals

Damping: 1/a (the period of transient response)

Natural frequency: the period of transient

response

Slide 9

)4

11cos()(

0)(1

)(1

)(

circuit RLC Parallel :Example#

22

)2/(

0 tRCLC

eVtv

dvL

tvR

tvdt

dc

CRt

t

0,)sin()( aa tAetx t

class2_sinDam

ped.m

- Time constant ?

- Natural frequency ?

- Order of system


Step Function

Discontinuity at t=0 (ex. ON/OFF of a switch)

Can be used to construct other discontinuous

waveforms.

Slide 10

0,0

0,1][

0,0

0,1)(

n

nnu

t

ttu

class2_step.m

not defined at t = 0


Impulse Function

A physical signal of extremely short duration

and high amplitude.

Discrete-time:

Continuous-time:

Relationship with step function (energy and

power signals):

Slide 11

0,0

0,1][

n

nn

1)(

00)(

dtt

tfort

dtutudt

dt

t

)()( ),()(

Michael


Impulse Function

Properties

Sampling (sample the value of t=0)

Shifting

Time-scaling

Slide 12

)()()()(

)()0()()(

000 tttxtttx

txttx

)()()( 00 txdttttx

)(1

)( ta

at

1)(

00)(

dtt

tfort

The impulse function serves a mathematical purpose by providing an approximation to a physical signal of extremely short duration and high amplitude.

Michael

Michael

Michael

Michael

Michael


Impulse Function

Doublet: the first order derivative of the unit impulse

Slide 13

)2/()2/(lim)(lim)(

0

)1(

0

)1( tttxt

0)()1(

dtt

0)11

(lim

)2/()2/(lim)(lim

)(lim)(]PROOF[

0

0

)1(

0

)1(

0

)1(

dttt

dttx

dttxdtt

50) Pagein 1.72 (Equ. )()()(00

)1(

tttfdt

ddttttf


Ramp Function

can be used to test tracking performance

Relationship between step and ramp functions

Slide 14

)()( ttutr

][]1[][

)()(

nrnrnu

trdt

dtu


Class#2



Slide 15


Stability

A system is said to be bounded-input,

bounded-output (BIBO) stable if and only if

every bounded input results in a bounded

output.

Slide 16

tMty

tMtx

y

x

allfor )(

allfor )(

0,)sin()(

not?or Stable :Example#

aa tAetx t

class2_stability.m

A = 1,

1/a = 0.1, -0.1,

t = 0.1sec


Memory

Memory: a system is said to possess memory

if its output signal depends on past or future

values of the input signal.

Memoryless: a system is said to be

memoryless if its output signal depends only

on the present value of the input signal.

Slide 17

t

dttiC

tv

tRitv

0)(

1)( :r2)capacito

)()( :ce1)resistan

ipsrelationsh V-I :Problem#

memoryless

has memory

MichaelRectangle


Causality

Causal: a system is said to be causal if the present

value of the output signal depends only on the present

or past values of the input signal.

Noncausal: a system is said to be noncausal if its

output signal depends on one or more future values of

the input signal.

Slide 18

])2[]1[]1[(3

1][)2

])2[]1[][(3

1][1)

:Problem#

nxnxnxny

nxnxnxny causal

non-causal

future value of the input

MichaelHighlight

MichaelHighlight

MichaelHighlight

MichaelLine


Invertibility

Invertibility: a system is said to be invertible if

the input of the system can be recovered from

the output.

Slide 19

IHH

txHH

txHH

tyHtx

txHty

inv

inv

inv

inv

)}({

)}}({{

)}({)(

)}({)(

)()(2)

)(2)()1

:Problem#

2 txty

txty

I is the identity operator

one-to-one mapping / bijective mapping / distinct inputs -> distinct outputs


Linearity

Linearity: a system is said to be linear in terms

of the system input and the system output if it

satisfies the following two properties of

superposition and homogeneity:

Slide 20

)()(2) ),(2)()1 :Problem# 2 txtytxty

consider input signal as weight sum


Time Invariance

Time Invariance: a system is said to be time

invariant if a time delay or time advance of the

input signal leads to an identical time shift in

the output signal. Otherwise, the system is said

to be time varying.

Slide 21

The notion of time invariance. (a) Time-shift operator St0 preceding operator H. (b)

Time-shift operator St0 following operator H. These two situations are equivalent,

provided that H is time invariant.

)(

)()( 2) ,

)()()1 :Quiz#

tR

tvti

R

tvti

HSt0 = St0H

MichaelHighlight


Homework Problem 1.44

Problem 1.45

Problem 1.46

Problem 1.52 (c)(f)

Problem 1.55

Problem 1.60

Problem 1.62

Problem 1.66

Problem 1.68

Problem 1.70

Problem 1.71

- Due: before 2:00PM, next Thursday

Slide 22

class_02-elementary+signals.pdf

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