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Definition A Fourier series on 0,253 is an

expression f thefirm fat

zCuduct which is conga

Eleni

where factint

conges

I

iii I

to get coff en ca Et It 4 Hdtik t

e

today's thirty getty caffas the

Coard of a weakrepresentationof fat via

arth projector

Ck fat 4,9

iHermitron aka sequitur innerproduct

121 A Hermitian inner product ona complex neckspare

Visi a mapUav IC

v w Lu w

want an Hull real

suchthatwaxntlmntysyw.twLu.w 7tLywD

and Lu Xu D Gw

Xtcand Gu twin Lu w threw

Givin It cu w

a u am EzE

ywT w I STEbi

and Su w

Def view su w7 o

Basis en semisorthogonormal if Lei ej i

ijif i j

is orthogonal if 0 else

Lei ej7 0 unless i j

Det L is nandgemek if su v 0 onlywhew o

4,2 is pasime if suis 20 all u.to

Deke a Hermit mm product onfunctors on o za

anon pride fratesal

prod ut

via f g GITAgutdt

note HfIR Cf f Ifu lfdt

note Ldn.fm JTpnuT4mCHdtint at

n e go e inteimtdt

these are orthogonal to zip

eachotherigo eilmNtdf

if fct Ecuolnut Yoif n th

21T if men

ofmud FED 4dm Scuola

E ColmCafu

g cholm old

cmcdm.fm

cm Lcfmuffled25cm

ImttT fut dt

SEint field

Back to discretetransfer d samply

Input signal i fCt assume periodicsampleat

N pants daytheperiod

to tu tu tu equally speed in priced

Caneucence rescale sothat to O ti l t 2

Cn EN I

soyndpriad in Co N

what frequencies are a holy fr

From prion discussionconsider gn.ae

N nit

fezitifuynJt

oh Yt wut usefulnumbfrsamphynten

main propertyof co is

wN 1 w eatin

if we are only samphyat t o 1,2 N I

then In 44mW look sane

OhNlt od 2 IwktMt.wnttNt.wutnyty.wt

ofnut

Morali if you aretryto sample a high foamy

signal your samplerateneeds to be big as

well

alias z

Nygoisterierion sample ratemust be more than tie

thehighest frequency youare lookingfan

IT21T

O

samphyexactly Triceas fast not

sufficient

But more than Trice isalways enough

Sore 01N sane sampledas 00

Olney

when we are tryingtoreconduct asampled

function fG

from fats itmakes senseto only

consider

o too I ION i

Ideai approximate en Cola FED

tfo Heldt

as f m.IO ya fam Cm standsin Art

IN Coffin fsample frstandard

inunpaidutDoesthsw.sk

Sampledfunctun i f lo fld ACND

u ethinkof as acolumn vector

IIIteeninpotato fnkh wntnq.ie

j.nfeeNskedadmmpnadut en f

N I

II 3 Emp Evinn O

H Santo has'fffationsEn Y I

A Z fo f

fiago.t

Erikt o

f

l D1

e1

IfA 4 Go 04,02,03

tT44Ytn oin.s

I

t 91,43

Eiht ethi j

int 4

n o t l ltil I l f

I HFadi L En Eml N Sn m l

So candetermine en's by inn product

Suppose have asampled signal fee fu of f

suppose can wite it F E en En

then i en IN Lem I