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EECS 2602
Signals and Systems in Continuous Time
Winter 2016
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Instructor: John Lam
Office: Lassonde Building 1012G
E-mail: johnlam@se!yor"u!a
Office hours: #onday $%$0&m ' (%$0&m) *riday 11am ' 12&m
2
EECE 2602 -- Signals and Systems in Continuous Time
2
Contat in+ormation
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Course website:http://www.eecs.yorku.ca/coursearchi!e/20"#-"6/$/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 "#)
EECE 2602 -- Signals and Systems in Continuous Time
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((
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
EECE 2602 -- Signals and Systems in Continuous Time
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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
EECE 2602 -- Signals and Systems in Continuous Time
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66
EECE 2602 -- Signals and Systems in Continuous Time
Wee" 1 ' 7ntrodution to signals5Te8t9oo"% Ch! 1
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::
EECE 2602 -- Signals and Systems in Continuous Time
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.
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--
EECE 2602 -- Signals and Systems in Continuous Time
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
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==
EECE 2602 -- Signals and Systems in Continuous Time
7ntrodution ' Signals and Systems
E1ample: image signal
E1ample: electrical signal
Signal System
Signal System
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1010
EECE 2602 -- Signals and Systems in Continuous Time
Continuous
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1111
EECE 2602 -- Signals and Systems in Continuous Time
>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
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1212
EECE 2602 -- Signals and Systems in Continuous Time
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
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1$1$
EECE 2602 -- Signals and Systems in Continuous Time
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 or power signal
6. Comple1 signals
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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 +=
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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
EECE 2602 -- Signals and Systems in Continuous Time
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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
EECE 2602 -- Signals and Systems in Continuous Time
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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.
EECE 2602 -- Signals and Systems in Continuous Time
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1-1-
?eriodi signals
% E1ample: $hat is the undamental period o the ollowing CT signal:
( ) ( )
=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
EECE 2602 -- Signals and Systems in Continuous Time
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1=1=
?eriodi signals
% E1ample: $hat is the undamental period o the ollowing CT signal:
( ) ( ) ( )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
EECE 2602 -- Signals and Systems in Continuous Time
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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
EECE 2602 -- Signals and Systems in Continuous Time
C S S C
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2121
% E1ample: $hat is the undamental period o the ollowing CT signal:
( ) ( ) ( )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
Linear om9ination o+ to signals
EECE 2602 -- Signals and Systems in Continuous Time
EECE 2602 Si l d S t i C ti Ti
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2222
EECE 2602 -- Signals and Systems in Continuous Time
% 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 +=
Linear om9ination o+ to signals
EECE 2602 Si l d S t i C ti Ti
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2$2$
EECE 2602 -- Signals and Systems in Continuous Time
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 or power signal
6. Comple1 signals
EECE 2602 Signals and S stems in Contin o s Time
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2(2(
EECE 2602 -- Signals and Systems in Continuous Time
.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
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22
EECE 2602 -- Signals and Systems in Continuous Time
.dd and e/en signals
% 7 CT signalxodd&t) is said to +e an odd signal i :
( ) ( )txtx oddodd = Symmetrical a+out thex-a1is
EECE 2602 Signals and Systems in Continuous Time
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2626
?ro&erties o+ odd and e/en signals
EECE 2602 -- Signals and Systems in Continuous Time
EECE 2602 Signals and Systems in Continuous Time
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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
EECE 2602 -- Signals and Systems in Continuous Time
EECE 2602 Signals and Systems in Continuous Time
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2-2-
.dd and e/en signals
% 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
EECE 2602 -- Signals and Systems in Continuous Time
EECE 2602 Signals and Systems in Continuous Time
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2=2=
?eriodi signals
% E1ample: 4s the pre!ious e1ample CT signal odd or e!en( or neither o them?
( ) ( ) ( )tttx 4sin2sin +=
EECE 2602 -- Signals and Systems in Continuous Time
% E1ample: 4s the ollowing CT signal odd or e!en( or neither o them?
( ) ( ) ( )[ ]14cos2sin = ttty
EECE 2602 -- Signals and Systems in Continuous Time
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$0$0
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 or power signal
6. Comple1 signals
EECE 2602 -- Signals and Systems in Continuous Time
EECE 2602 -- Signals and Systems in Continuous Time
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$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
EECE 2602 -- Signals and Systems in Continuous Time
EECE 2602 -- Signals and Systems in Continuous Time
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$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.
EECE 2602 Signals and Systems in Continuous Time
EECE 2602 -- Signals and Systems in Continuous Time
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$$$$
Signal o&erations
% E1ample: Consider the ollowing CT signalx&t)
a. Sketchx&t)
+. $hat is the undamental period?
( )
+=
4sin1
ttx
EECE 2602 Signals and Systems in Continuous Time
EECE 2602 -- Signals and Systems in Continuous Time
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$($(
Signal o&erations
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5-1
-0.5
0
0.5
1
1.5
2
time
amplitude
% Graphical representation in 7T7H:
( )
+=4
sin1 ttx
5eriod A 2
EECE 2602 Signals and Systems in Continuous Time
EECE 2602 -- Signals and Systems in Continuous Time
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$$
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 or power signal
6. Comple1 signals
EECE 2602 Signals and Systems in Continuous Time
EECE 2602 -- Signals and Systems in Continuous Time
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$6$6
Elementary signals
% Elementary signals are used reuently to represent more complicated signals.
% E1amples o these signals are:
") Knit-step unction
2) 8ectangular unction
,) 8amp unction
*) Signum &or sign) unction
#) Sinc unction
C 60 S g a s a d Syste s Co t uous e
EECE 2602 -- Signals and Systems in Continuous Time
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$:$:
Elementary signals
% Knit step unction u&t):
u&t) is piecewise continuous with a discontinuity at tA 0
( )
=
2/0
2/1
t
ttrect
g y
EECE 2602 -- Signals and Systems in Continuous Time
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$= $=
Elementary signals
% Signum or sign unction sgn&t):
( )
=01
00
01
sgn
t
t
t
t
g y
EECE 2602 -- Signals and Systems in Continuous Time
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(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):
( )
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