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