spfirst l02 ev

Copyright Monash University 2009

Signal Processing

First

Lecture2Phase&

TimeShiftComplex

Exponentials

1


READING ASSIGNMENTS

ThisLecture: Chapter2,Sects.23to25

AppendixA:ComplexNumbers AppendixB:MATLAB NextLecture:finishChap.2,

Section26toend

2


LECTURE OBJECTIVES

DefineSinusoidFormulafromaplot RelateTIMESHIFTtoPHASE

3

tjXetz )(

Introduce an ABSTRACTION:Complex Numbers represent SinusoidsComplex Exponential Signal


SINUSOIDAL SIGNAL

FREQUENCY Radians/sec or,Hertz(cycles/sec)

PERIOD (insec)

AMPLITUDE Magnitude

PHASE

4

)cos( tA

f)2(

21

f

T

A


PLOTTING COSINE SIGNAL from the FORMULA

Determineperiod:

Determineapeak locationbysolving

Peakatt=4

5

)2.13.0cos(5 t

0)( t

3/203.0/2/2 T


ANSWER for the PLOT

UseT=20/3andthepeaklocationatt=4

6

)2.13.0cos(5 t

320


TIME-SHIFT

Inamathematicalformulawecanreplacetwithttm

Thenthet=0pointmovestot=tm

Peakvalueofcos((ttm))isnowatt=tm

7

))(cos()( mm ttAttx


TIME-SHIFTED SINUSOID

8

))4((3.0cos(5))4(3.0cos(5)4( tttx


PHASE TIME-SHIFT

Equatetheformulas:

andweobtain:

or,

9

)cos())(cos( tAttA m mt

mt


SINUSOID from a PLOT

Measure theperiod,T Betweenpeaksorzerocrossings

Compute frequency: =2/T

Measure timeofapeak:tm Compute phase: =tm

Measure heightofpositivepeak:A

10

3 steps


(A, , ) from a PLOT

11

25.0))(200( mm tt

20001.022 T1001period1 sec01.0 T

sec00125.0mt


SINE DRILL (MATLAB GUI)

12


PHASE is AMBIGUOUS

Thecosinesignalisperiodic Periodis2

Thusaddinganymultipleof2 leavesx(t)unchanged

13

Ttt

t

mm

m

2)2(2

then, if

)cos()2cos( tAtA


COMPLEX NUMBERS

Tosolve:z2 =1 z=j MathandPhysicsusez=i

Complexnumber:z=x+jy

14

x

y zCartesiancoordinatesystem


PLOT COMPLEX NUMBERS

15


COMPLEX ADDITION = VECTOR Addition

1626

)53()24()52()34(

213

jj

jjzzz


*** POLAR FORM ***

VectorForm Length =1 Angle =

CommonValues j hasangleof0.5 1hasangleof j hasangleof1.5 also,angleofjcould be0.51.52 becausethePHASEisAMBIGUOUS

17


POLAR RECTANGULAR

Relate(x,y)to(r,)

18

r

x

y

Need a notation for POLAR FORM

xyyxr

1

222

Tan

sincos

ryrx

Most calculators doPolar-Rectangular


Eulers FORMULA

ComplexExponential Realpartiscosine Imaginarypartissine Magnitudeisone

19

)sin()cos( jrrre j

)sin()cos( je j


COMPLEX EXPONENTIAL

Interpretthisasa RotatingVector t Anglechangesvs.time ex:rad/s Rotates in0.01secs

20

)sin()cos( tjte tj

)sin()cos( je j


cos = REAL PART

21

Real Part of Eulers}{)cos( tjeet

General Sinusoid )cos()( tAtx

So,

}{}{)cos( )(

tjj

tj

eAeeAeetA


REAL PART EXAMPLE

22

Answer:

Evaluate:

tjj eAeetA )cos( tjjeetx 3)(

)5.0cos(33

)3()(5.0

teee

ejetxtjj

tj


COMPLEX AMPLITUDE

23

Then, any Sinusoid = REAL PART of Xejt

tjjtj eAeeXeetx )(

General Sinusoid

tjj eAeetAtx )cos()(Complex AMPLITUDE = X

jtj AeXXetz )(

spfirst l02 ev

Documents

ftacopyright monash

complex addition

complex amplitude23then

j mathandphysicsusez

complex numbers tosolve

tm compute phase

commonvalues j hasangleof0

t measure timeofapeak