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CTFT Development 2

Module by: Richard Baraniuk. E-mail the author

This is a "voodoo" development

Fiω=fte(iωt)d t =ft,eiωt F ω t f t ω t f t ω t
(1)
in Hilbert Space L2R L 2 , which is like a generalized CTFS coefficient with "basis" eiωt ω t , given that ω ω is real.

Key:

All ei ω 1 t ω 1 t and ei ω 2 t ω 2 t are orthogonal ω 1 ω 2 ω 1 ω 2 , in other words: ei ω 1 t,ei ω 2 t=0 ω 1 t ω 2 t 0 unless ω 1 = ω 2 ω 1 ω 2 .

Problem:

eiωt ω t has infinite energy eiωt,eiωt= ω t ω t .

In fact:

ei ω 1 t,ei ω 2 t=2πδ( ω 1 ω 2 ) ω 1 t ω 2 t 2 δ ω 1 ω 2 .

If we carry on undeterred then we should be able to resynthesize ft f t from

ft=ωRFiωeiωt f t ω ω R F ω ω t
(2)
where the sum becomes an integral:
ft=12πFiωeiωtd ω f t 1 2 ω F ω ω t
(3)

Interesting:

ft=12π(Fiω,e(iωt)) f t 1 2 F ω ω t where e(iωt) ω t is the inner product over ω ω (why we need the 12π 1 2 ). Fiω=ft,eiωt F ω f t ω t

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