s4stooo2 lecture - montefiore institute€¦ · s4stooo2 lecture introduction to frequency domain...

S4STooo2 LEcTURE

INTRODUCTION TO FREQUENCY DOMAIN

IN Plot output APPROACH LTI SESTENAS

µ yinput DYNAMocs output

NOT Explicitly DESCRIBED

Superposition PRINCIPLE

if p II dipi then y II digifi simple SIGNALS

Yi RESPONSE of 544TEh ToSimplerSIGNALS

noise su ffgtsii fr.FI ZMH ftp.lal.sit olds

SH O ISOLATES THE VALUE of MCHATTIEINTEGRATE FOR ALL 8 Cites µ b

THEN yltkf.tn frl27 hlt dd3h HPYkspwse

Cm hllH Convolution

PROPERTIESOFCONVOLUTION

commutative If g 1H Cg f tAssociative f lg HKh llf gHhHHDistributive lftlgthillh lfx

gllhtlfthll4E.ge

jyIhsyM

hIhE ya

rE F

g ifFhH y

PROPERTIES OF LTI 54STEMS BASED ON h h

CAUSALITY n h 14 0 tf tho

IMPULSE RESPONSE

µlH 814CAUSAL

BECAUSE

A gal is AFTERTHEM

RESPONSE BEFORE µ HNON CAUSAL

CONVOLUTION

yin ft prod htt 2 doV 2 t effect of Future

inputs

Hz t htt 01 0

f t Z co htt 01 0

Cv Zz t Z 41321 0 tf Zac O

h h fin 1114 with NH unit STEPrich

i

um III II r

MEMORY WINDOW of IMPULSE RESPONSE

if hut SCH a

yell Lt Malhar olds

Stoical SH Eto do

µ t.toSTATIC RESPONSE

MEMORY CESI

LTI SYSTEMS CAUSALITY

hurt Ged'ttaeditt NHT

INSTANTANEOUS CHANGE

OF initial ambitions

INFINITE IMPULSE RESPONSE

RESPONSE TIME INSTEAD

TIMECONJTANTI

Thto hindt

HH mathSEEhaldt

max

max.ch Th ft hCb1dt

AREA of AREA UNDERh14

RECTANGLE

FOR ID RESPONSES Real d

hurt Ged 1114d

Th If c edt dt f 3

0AM

hurt aedtrich a e act

Iso Blows UP UNSTABLE 56 STEM

OTHER INPUT OUTPUT APPROACH

i e USE ANOTHER Soup CE SIGNAC µCHt SN

is _Ax xlH a edit

g ectitjwitt

facies't 5 9 tjwi

Si COMPLEX FREQUENCY

LET'S USE put est so Totjwo

YIH http In

L htd.yct ddzfffhczl.es.lt do

If htol.esesmdz

esot.IIhcolesoZdz

µCH AlsoSAME FREQUENCY

HIST Complex NUMBERIm Also

f htreHoo L ARGAlso Also 1 e

YIH est HadLSot Also ecHoo

Also I esottLHboSo FREQUENCE

NOT MODIFIEDCHANGE f

CHANGE in PHASEIN AMPLITUDE

ALL is CHARACTERIZED BYHCSo

AT FREQUEG So

FOR ALL FREQUENCIES5 CHANGE IN

AMPLITUDE AND IN PHASE is

CHARACTERIZED BY A SINCE function

HCS THAT DEPENDS ON FREQUENCES

if µlH CAN BE DECOMPOSED As A Sum

of COMPLEX EXPONENTIALS

C µlH a es't ta es't ta est

yltl ai.Hlsil.esittauHLS4es t

013 HCS est15 LG Lt

1 FOURIER SERVES FOURIER TRANSFORM

4 LAPLACE TRANSFORM TRANSFER FUNCTION

18 Lg Io

17 CAN WE DECIMPOSE ANY SIGNAL AS

A SUM OF COMPLEX EXPONENTIAL

µCH est s Ttjw

14MH eeiwtlr.co FL

SIMPLE SIGNAL µLH ejwtjsincut

toi an

fywscwH PERIS au

FIRST WE START TO RECONSTRUCT

PERIODIC SIGNAL

PERIODIC WITH PERIOD T ifXlttt KH ft

T IgT PERIOD Csec

f c FREQUENCY HzW ANGULAR FREQUENCY

croaks

µcH eiwt eitt

Let's CHOOSE A FREQ Wo

µCH ejwot ej7EtIT HAS A SET OF HARMONICALLY

RELATED COMPLEX EXPONENTIALS

4h14 etkwot ejh7It jk 0,1412THEY ARE ALL PERIODIC WITH ProhioDT

AramT

too jkwotµ ble E ah e

k no

to

E Oke jht

KEN

is Also PERIODIC WITH PERIOD T

o Tamme I

imk 0 CONSTANT IDC COMPONENT

OFFSET Ao

Kitt FUNDAMENTAL coupon ENT

LONGEST PERIOD T

k 12 SECOND HARM evics components

eras

DECOUPOSONG A PERIOD0C SIGNAL

into A Wm ok e JE wot

DETERMINE PERIOD T v

DETERMINE SET OF SIMPLE SIGNALS

0h14 ejk7t j k 0,11112COMPUTE THE SUM

to

Mltk E ah e'thatk 0

tooE ah.ejhYI.hrke N

WHERE ate DETERMINES THE

CONTRIBUTION OF THE KTH HARMONICS

EIr I I a its

Q 2 0

Ocos wt aorta ewttai htt

L ejwttf.et.at

edutte dah2

4 s a

Q E Q c O

HOW DO WE COMPUTE SOLH COEFFICIENTS

FOR ANY ARBITRAM PERIODIC SIGNAL

X t ah k 0,11 t.ly

to jhwotµlH E ah e

k D

jmwot jnwotl e e

Eun.intfotIEIan.edhwt.etinwotdf

tfE anfoTejlh Huddy

di ftp.h nlwotdrC

fotanllkmlwohdttjffsimlchnlwd.dk

he m t Mi Mai

O

k ng e

si owirdf JI dt T

InanimateEE.an Ka m.T

am touch e tweet dt

tf MH ei Ent dtWo YI

Any periodic SIGNAL OF PERIOD T

CAN BE DECOMPOSED INTO A Sum

of complex EAPONENTIALS

to jhwot FOURIERµlH E ak.ee SERIES

ke a

M t EighEEEicients

do tf MH.dkAVERAGE VALVE

of plyOVER A PERIOD Dc component

OFFSET

ANY SIGNALCONVERGENCE OF E Mlbah fff dt Conover CES

Con VERGE NCE of Fou ki Ek SERIES

FINITE ENERGY OVER A PERIOD

DIRICHLET conditions

TE akethwt µ lbha o

f t WHERE perch is continuous

IN PRACTICEN jhwoot

µnlH E ah e

k N

N 10 N't

enter