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6.5: Impulse functions

( \newcommand{\kernel}{\mathrm{null}\,}\)

Unit impulse function = Dirac delta function is a generalized function with the properties:

  1. δ(t)=0,t0
  2. δ(t)dt=1
  3. L(δ(tt0))=est0

Since the first item is just a definition we will prove 2. and 3.

Proof of 2.

We let dk(t)={12kk<t<k0tk0tk

We note that limk0dk(t)=0 if t0 and limk0dk(t)=limk01=1=δ(t)dt.

Proof of 3.

L(δ(tt0))=limk0+L(dk(tt0))=limk00est(dk(tt0))dt=limk012kt0+kt0kestdt=limk012skest|t0+kt0k=limk012skest0(eskesk)=limk0sinh(sk)skest0=limk0scosh(sk)sest0=est0

Note

The following are some useful identities:

  1. sin(t)=eiteit2i
  2. cos(t)=eit+eit2
  3. sinh(t)=etet2
  4. cosh(t)=et+et2
  5. [sinh(t)]=cosh(t)
  6. [cosh(t)]=sinh(t)
  7. sinh(0)=e0e02=0
  8. cosh(0)=e0+e02=1

Intro to Group Theory
Define the product on R2 by
(x1,y1)(x2,y2)=(x1x2y1y2,x1y2y1x2)

Note is
1.) commutative:(x1,y1)(x2,y2)=(x1x2y1y2,x1y2y1x2)=(x2x1y2y1,x2y1y2x1)=(x2,y2)(x1,y1)
2.) associative: (fg)h=f(gh)
3.) distributive w.r.t +:f(g1+g2)=fg1+fg2
4.) (x1,y1)(0,0)=(0,0)

Note: (0,1)(0,1)=(1,0)

 


This page titled 6.5: Impulse functions is shared under a not declared license and was authored, remixed, and/or curated by Isabel K. Darcy.

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