electrical engineering week 1 slides

7/25/2019 electrical engineering week 1 slides

1/58

EECS 2602

Signals and Systems in Continuous Time

Winter 2016


2/58

Instructor: John Lam

Office: Lassonde Building 1012G

E-mail: johnlam@se!yor"u!a

Office hours: #onday $%$0&m ' (%$0&m) *riday 11am ' 12&m

2

EECE 2602 -- Signals and Systems in Continuous Time

2

Contat in+ormation


3/58

Course website:http://www.eecs.yorku.ca/coursearchi!e/20"#-"6/$/2602/

$$

Grading% "0 marked assignments &"' each( 10, in total)

% * la+s in total &,' each( 12, in total)

% , in-class uies &6' each( 1-, in total)

% idterm test &20,)

% inal e1am &(0,)

% a+s +egin ne1t onday &3an "") and riday &3an "#)



4/58

((

Course ./er/ie% 4ntroduction

% Continuous time &CT) signals and systems% 5roperties o CT signals

% Signal representations% Comple1 num+ers( comple1 signals

% ourier series

% inear Systems 7nalysis% inear time-in!ariant &T4) systems% Con!olution% aplace Transorm% ourier 7nalysis

% 7pplications% CT ilter design and characteristics% 8C circuit analysis% inear eed+ack systems



5/58

Course Shedule% $eek " 9 ,:

% 5roperties o CT signals( comple1 signals( classiication o systems LB1% 3ui4 1 5Wee" $

% $eek * 9 6:% inear time-in!ariant &T4) systems( impulse response( con!olution LB2

% 3ui4 2 5Wee" 6

% $eek 9 ;:% ourier analysis( reuency response o a system LB$% #idterm 5Wee" -

% $eek < 9 "":% aplace transorm( transer unction o linear systems( ilter design LB(% 3ui4 $ 5Wee" 11

% $eek "2:% inear eed+ack and control% 8e!iew



6/58

66


Wee" 1 ' 7ntrodution to signals5Te8t9oo"% Ch! 1


7/58

::


7ntrodution ' Signals% $hat is a =Signal>?

% Con!ey inormation a+out time-!arying physical phenomena.

% E1ample in electrical engineering: current or !oltage signal

% athematically speaking( signalsare modeled as unction o one or more

independent !aria+les. E1amples o independent !aria+les used to represent

signals are time( reuency.


8/58

--


7ntrodution ' Systems% $hat is a =System>?

% athematically speaking( a system is deined as an entity that processes a set

o signals and produces another set o signals

% 7lso called ;Single


9/58

==


7ntrodution ' Signals and Systems

E1ample: image signal

E1ample: electrical signal

Signal System

Signal System


10/58

1010


Continuous


11/58

1111


>isreteT signals% 4 a signal is deined only at discrete !alues o time &i.e. data points o the signal

are not continuously connected)( it is called a discrete-time &@T) signal.

% E1amples:

% @T signals will +e discussed in your ,rdyear signal course


12/58

1212


Ty&es o+ CT signals% 7 signal is usually descri+ed +ased on the ollowing properties:

". 5eriodic or aperiodic

2. E!en or odd

,. Signal operations: time-shited or time-scaled

*. Com+ination o elementary signals

#. Energy and power signal

6. Comple1 signals


13/58

1$1$


Ty&es o+ CT signals% 7 signal is usually descri+ed +ased on the ollowing properties:


2. E!en or odd



#. Energy or power signal

6. Comple1 signals


14/58

1(1(

?eriodi signals% 7 CT signalx&t) is periodic i:

% The smallest positi!e !alue o T0that satisies the periodicity condition is reerred

to as the undamental period ox&t)

( ) ( )0Ttxtx +=



15/58

11

?eriodi signals

% AA amplitude

% f0A undamental reuency BertD

% A phase shit BradiansD% 2f0A undamental angular reuency &0) Bradians/secD

% T0A "/ f0 A undamental period BsecD

% 7 typical e1ample o a periodic signal is a sinusoidal unction represented +y:

( ) ( ) += tfAtx 02sin



16/58

1616

?eriodi signals

% 7 typical e1ample o a periodic signal is a sinusoidal unction represented +y:

( ) ( ) += tfAtx 02sin

0

0

2

=T% The undamental period o a sinusoidal unction is also gi!en +y:

% 0 = 2f0A undamental angular reuency Bradians/secD

% T0A "/ f0 A undamental period BsecD



17/58

1:1:

?eriodi signals

% E1ample: $hat is the undamental period o the ollowing CT signal:

( ) ( )ttx 3cos5.0=

undamental period:

3

2

3

22

00

===

T

% 7ll the sine( cosine( tangent signals are periodic signals.



18/58

1-1-

?eriodi signals


( ) ( )

=otherwise

tttx

0

2/32/3cos

% This is FT a periodic signal.

-4 -3 -2 -1 0 1 2 3 4-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1



19/58

1=1=

?eriodi signals


( ) ( ) ( )tttx 4sin2sin +=

% 4 we let : ( ) ( )( ) ( )ttx

ttx

4sin2sin

2

1

==

% undamental angular reuency ox"&t) is: 2rad/s

% undamental angular reuency ox2&t) is: *rad/s

% The angular reuency ox2&t) is 2 times the angular reuency ox"&t). $e

calledx2&t) the 2ndharmonic ox"&t)

7n general) a signalxm5t ith an angular +reueny that is mtimes the

angular +reueny o+ o+x5t! We alledxm5t themthharmoni o+x5t



20/58

2020

Linear om9ination o+ to signals

% 7 signal g&t) that is a linear com+ination o two periodic signals(x"&t) withundamental period T"andx2&t) with undamental period T2as ollows:

is periodic i and only i :

( ) ( ) ( )tbxtaxtg 21 +=

% The undamental period o g&t) is gi!en +y nT"

A mT2

pro!ided that the !alues

o mand nare chosen such that the greatest common di!isor +etween mand

nis ".

numberrationaln

m

T

T==

2

1


C S S C


21/58

2121


( ) ( ) ( )tttx 4sin2sin +=

% 4 we let : ( ) ( )( ) ( )ttx

ttx

4sin2sin

2

1

==

2142122

2

1

====

TT

So: ( which is a rational num+er

The undamental period is nT"A "&") A " or mT2A 2&"/2) A "

2

2

1 =T

T



EECE 2602 Si l d S t i C ti Ti


22/58

2222


% Graphical representation ox&t) in 7T7H:

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

25eriod A "

( ) ( ) ( )tttx 4sin2sin +=


EECE 2602 Si l d S t i C ti Ti


23/58

2$2$


Ty&es o+ CT signals

% 7 signal is usually descri+ed +ased on the ollowing properties:


2. E!en or odd




6. Comple1 signals

EECE 2602 Signals and S stems in Contin o s Time


24/58

2(2(


.dd and e/en signals

% 7 CT signalxeven&t) is said to +e an e!en signal i :

( ) ( )txtx eveneven = Symmetrical a+out the y-a1is

EECE 2602 Signals and Systems in Continuous Time


25/58

22



% 7 CT signalxodd&t) is said to +e an odd signal i :

( ) ( )txtx oddodd = Symmetrical a+out thex-a1is



26/58

2626

?ro&erties o+ odd and e/en signals




27/58

2:2:

?ro&erties o+ odd and e/en signals

*. 7 CT odd signal is always ero at tA 0

#. 4ntegration o a CT odd signal within the limits:

6. 4ntegration o a CT e!en signal within the limits:

( ) 0=T

Todd dttx

( ) ( ) =

T

even

T

T

even dttxdttx

0

2




28/58

2-2-


% 4 a signalx&t) is classiied in the =neither odd nor e!en> category( the signal can+e e1pressed as a sum o e!en and odd signals:

where

and

( ) ( ) ( )txtxtx oddeven +=

( ) ( ) ( )[ ]txtxtxeven += 21

( ) ( ) ( )[ ]txtxtxodd =2

1




29/58

2=2=

?eriodi signals

% E1ample: 4s the pre!ious e1ample CT signal odd or e!en( or neither o them?

( ) ( ) ( )tttx 4sin2sin +=


% E1ample: 4s the ollowing CT signal odd or e!en( or neither o them?

( ) ( ) ( )[ ]14cos2sin = ttty



30/58

$0$0

Ty&es o+ CT signals



2. E!en or odd




6. Comple1 signals




31/58

$1$1

Signal o&erations

% Time shiting:

Consider a CT signalx2&t) o+tained +y shiting another signalx&t) +y Ttime units(

the time-shited signalx2&t) is e1pressed as:( ) ( )Ttxtx +=2




32/58

$2$2

Signal o&erations

% Time scaling:

7 CT signalx&t) scaled +y a actor cin the time domain is denoted +yx&ct).

4 cI"( the signal is compressed +y a actor o c.

4 0 J c J "( the signal is e1panded.




33/58

$$$$

Signal o&erations

% E1ample: Consider the ollowing CT signalx&t)

a. Sketchx&t)

+. $hat is the undamental period?

( )

+=

4sin1

ttx




34/58

$($(

Signal o&erations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5-1

-0.5

0

0.5

1

1.5

2

time

amplitude

% Graphical representation in 7T7H:

( )

+=4

sin1 ttx

5eriod A 2




35/58

$$

Ty&es o+ CT signals



2. E!en or odd




6. Comple1 signals




36/58

$6$6

Elementary signals

% Elementary signals are used reuently to represent more complicated signals.

% E1amples o these signals are:

") Knit-step unction

2) 8ectangular unction

,) 8amp unction

*) Signum &or sign) unction

#) Sinc unction

C 60 S g a s a d Syste s Co t uous e



37/58

$:$:

Elementary signals

% Knit step unction u&t):

u&t) is piecewise continuous with a discontinuity at tA 0

( )

=

2/0

2/1

t

ttrect

g y



39/58

$= $=

Elementary signals

% Signum or sign unction sgn&t):

( )

=01

00

01

sgn

t

t

t

t

g y



40/58

(0 (0

Elementary signals

% 7 ramp unction r&t):

7 ramp unction can +e e1pressed as a unction o the unit step unction u&t):

( )

electrical engineering week 1 slides

Documents