Lecturelibrations

.

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 )