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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

a)RealExponentialSequence

b)Compl

exExpone

ntialSequ

ence

c)SineorCosineS

equence

d)IncreasingSequenceandD

ecreasing

Sequence

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EXPONE

NTIALANDSINUSO

IDAL

SEQUENCES

a)RealExponentialSequence

Thissequencecan

occurinm

anydiverse

situation

suchaspopulationgrowth,a

chemical

half-life,o

rinradioa

ctivedatin

g

use

o

eec

orgeresoamouspan

ngs

andalsooccurastermsintheresponseof

digitalfilt

ers.

x(n)

=an,-

n

Eq(1

)

aisrealconstant

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

a)RealExponentialSequence

Theexp

onentialsequencec

ouldresult

from

samplingac

ontinuous

-time

exponentialgiving

x(nT)=et

t=nT

=enT

=(eT)n

whereet=atorelateEq.(1).

Itiscomm

on

practice

toconsiderthisexpon

entialsequence

forn0.

Thenwecanwrite

x(n)=anu(n)

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

a)RealE

xponentia

lSequenc

es

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

b)Comp

lexExpon

entialSequence

Thissequ

ence,whic

hprovides

thefounda

tion

fordiscre

te-time(dig

ital)freque

ncyanalys

is,

is

describedby

p

n

=

e

n

whereNisapositiverealconstantthatde

fines

theperio

dofthese

quence.Th

evalueso

fthis

complexexponentia

latsample

numbersn=

0,1,2,...a

re

p(0)=ej0,p(1)=ej(2/N)1,p(2)=ej(2/N)2,

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

b)Comp

lexExpon

entialSequence

andatsamplenumbersn=N,

N+1,...,we

have

j(2/N)N

j(2/N)(N+1)

Noticethat p(N)=ej(2/N)N=ej2

=ej0=p(0

)

andthat

p(N+1)=

ej(2/N)(N+1)=ej(2/N)N

ej(2/N)1=

ej(2/N)

=p(1)

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

b)Comp

lexExpon

entialSequence

Ingenera

l,ej(2/N)nisacomplexnumbero

f

magnitud

e1androtatingina

-

-j(2/N)

n

rotatinginaclockwisedirection

.

Figure(a)belowsho

wnthatforN=8,

the

resultp(0

)=p(8)=p

(l6)=p(24

)andsofo

rth,

andthatp(1)=p(9)

=p(17)=p

(25)inthe

same

pattern.

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

b)Comp

lexExpon

entialSequence

Figure(b)

belowsho

wnthatthe

rotatinglin

e

startsatanangleof

0forn=0

and

progresses2

/Nra

diansasthesample

numberg

oesfrom0

to1.

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

b)Comp

lexExpon

entialSequence

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

b)Comp

lexExpon

entialSequence

Answer:

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

c)Sineo

rCosineS

equence

TheEuleridentity

et=c

os

+jsin

sequenceinaverystraightforw

ardway.

Startingw

iththeper

iodiccomp

lexexpone

ntial

sequence

p(n)=e

j(2/N)n=co

s(2/N)n

+jsin(2/N)n

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

c)Sineo

rCosineS

equence

wenotice

thatthere

alpartofp

(n)isacosine

sequence,or

2/Nn

=cos(2/N)n

and,

ina

similarway

,theimaginarypartof

p(n)isasinesequencenamely

x(n)=Im[p(n)]=Im

[ej(2/N)n]

=s

in(2/N)n

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

c)Sineo

rCosineS

equence

Shownin

Figure(a)

isaplotof

thecosine

sequenceforN=8wherex(n)

=cos(2n

/8)

whilesine,x(n)=sin(2n/8)a

ppearsinF

igure

(b).

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

c)Sineo

rCosineS

equence

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

c)Sineo

rCosineS

equence

Forexam

ple,thesinusoidals

equencein

Figure(c

)canbedescribedc

orrectlyas

either

xn=cos2n/8-

/4

or

x(n)=sin

(2n/8-

/4)

Californiansgenerallyprefer

tousecos

ines

Sowewilldescribe

allsinuso

idalsequences

asx(n)=Ac

os(2n/N

+)..=phase(rad

)

whereAisamplitudeofthes

equence

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

c)Sineo

rCosineS

equence

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EXPONENTIALAND

SINUSOI

DAL

SEQUENCES

d)Increa

singSequ

enceandDecreasing

Seque

nce

Thissequ

enceissim

plyacombinationofthe

realandc

omplexexponentials

equencesand

isdescrib

edb

x(n)=A

anej{2/N}n

ej

whereaa

ndAarepositiverealn

umbersa

nd

isarealn

umber.Consequently,

x(n)

0asn

for0