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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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o Tamme I
imk 0 CONSTANT IDC COMPONENT
OFFSET Ao
Kitt FUNDAMENTAL coupon ENT
LONGEST PERIOD T
k 12 SECOND HARM evics components
eras
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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
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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
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to jhwotµlH E ah e
k D
jmwot jnwotl e e
Eun.intfotIEIan.edhwt.etinwotdf
tfE anfoTejlh Huddy
di ftp.h nlwotdrC
fotanllkmlwohdttjffsimlchnlwd.dk
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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
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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
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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