Arbitrary forces - Solution can be written as convolution
NH ) = § Fit ) get - T ) de where get) is impulse response fun
GAE Fe'±qiwattso
and is ;
o tso
convcfiop = f*g
By laplace transform,
we also found the solution
. XCS )=G( s ) { Eis ) + [ ( no tms not mvo ] where G-is ) = ÷2+ Cst K
Finally ,we can use Fourier transform as well .
Indication :bar ( i ) for Laplace
tilde ( ~ ) for Fourier
Defi Ecw ) = )[axitieiwtdtL
transform variableis w ( Frequency )
Related to a laplace transform by A 90°
rotation ni the Complan plane (Compton variable )
We often detmer the Fourier transform as transform pair .
awi=[axttiejwtdt and. xH=a÷u[
.
do )eiwtdw
4kernel
Kernel - what you one multiplying your function by .
we can think of the Fourier transform mi terms of the limit of the Fourier
Series as our period goes to infinity ( T → a ) .
nuts =n£=→cnei→¥ with q=÷[±gn#iint¥dt
By letting To &, me may think Cn - o , but the member of terms n becomes
mifinite , So we need to be Careful .
let w=*÷ , n= TwxT
IWTNH = [ Cncwle now multiply by one in a special way .
= [ cncw ) eiwt ( Ital ) . bw = 25
NHI = at,w€|TC ( w ) ] eiwtdw
The other mitegral Tc ( w ) = [g[ nctseiwtdt
as T→oo we see that TCLW ) → Tew )
The Fourier transform is the Smooth representation of Fourier Series for mi finite period.
F. T of delta function :
flash , iiwtdt =\
Inverse [* elwtdt= 2a Siw )
look at FH ) of a derivative dat
by parts :{Faze "wtdt =Yteiw+P→. [aeiwsxu , eiwtdt
we assume that NH ) vanishes at too ( a restriction requiredfor the enhance at FT )
= iwkew )
Similarly ,for the Second derivative :
¥hfny = - w2I ( w )FT of a derivative gives us Sth proportional to
FIT of the function .
useful for differential equations :
miitcsitkn= Fit ) Take FT of equation :
- mw2 Ecw ) + iwcpicw ) + KKW ) = ¥ ( w )
we now have an algebraic equation mi w - domain . we can some for K ( W )
¥ ( w )Kew ) =
-or = ¥w ) ~Gcw )
- mwhtiwctk
where ~Gw ) =L ⇒ Eriw ) is FT of get =yEcw )=#'I- mwtiwctk K
Now , we have three representations for the response .
Convolution : Ntt )=↳t Fight - T)dT=F*g
Laplace transform : Jecs) = Elsy GTCS) it No=o , no .
. o ( quiescent Ic . )
Fourier transform : Ecw )=I( w ) G~( w )
Ex : Fourier transform : Nt ) = Got
Kiwi =L :(asztsiiutdt .
. Eat"t¥5 etwtdta
=L; tseilsw "dt+{asset"w⇒ 't
at *
Jeiwtataasiw
) *a
⇒ Ecw )={ ( at ) [ 813 - w ) +8 ( → . w ) ]tkl
LGH:3 W~↳sonar park : net , =/'
- That
0 otherwise
NTWI =L; nttieiwtdt tT T
2T
= ftyeiwtdt = Iwsmiwt An w
*=I
.
T
we Can see an important time - frequency relation : pulse that is narrow in time
is wide ni frequency and vice versa . - The reason why we use impulsehammers .
This idea leads to concepts to digital Signal processing ( Dsp ) .
when we measure a Signal , we usually Sample it → take values at discrete
mitorrds of time.
;¥ik✓nHa tDT
if we Sample at BT miterral for a length of time T . we comet N pointsThen we have a digital representation 1 with N points .
There is going to be rigorous limitation of vet , st . I represents Val well .
Terms : Sampling rate , digitization rate,
ldf = HZ i Samples Hec Signal length Tz Ndt
27N → number of points Sampled KF. - |
• •
look at a sine wane : ~\#µ•
4 points per cycle → not very good
w pts per que is better
#hT✓#.,
If we Sample less than Once per cycle → our digital signed is a how
frequency one
.
Aliasing ! Frequency grater than 1- one aliasedas lower frequencies2dt
.,
fµyg= Nyquist Frequency = fat
freqs.
from 0 to fµyg are represented well ( enapt my dose to fnyg ).
Frequencies from Fnygt 2fµy,
are mapped back between o and fnyg .
,¥¥ "
Ii .
.
w ~ 't0
T defines the frequency resolution
( lowest frequency measurable )
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