[Cst-1b] Fourier Transform product

[pm258@cam.ac.uk]

Dear Moritz

Thanks. It maks more sense

Phebe

>Dear Phebe,=20
>
>it's no wonder that you had difficulties with showing that the product=20=

of
>two GWs is another single GW because it isn't, unless you define the
>functional form of GWs differently, ie with an extra real factor in=20
front of
>the formula. See my notes under (2) for more details. This mistake=20
affects
>questions 8.4 and 8.6 of the problem sheet he handed out (corresponding=20=

to
>some Tripos questions).
>
>IMHO there is another mistake in the notes and in problem 6.4 where he
>claims that the GW "unifies" the Fourier and the space domain. Again,=20
this
>can be overcome by adding an extra factor of something like=20
1/Sqrt(2*PI*a)
>(I believe). So I suggest a functional form for the GW of=20
>f(x) =3D 1/Sqrt(2*PI*a) exp(-imx) exp-(x-x0)^2/a^2
>
>See notes under (1).
>
>Moritz
>
>***********************************************************************
>
>(1)
>In the notes and in the answer to problem 6.4 it says:
>   "...in the limit our expansion basis becomes [...] exp(-imx),=20
>   the ordinary Fourier basis".=20
>This is correct since=20
>f(x) =3D integral(F(m) exp(-imx) dm) between -inf ... +inf
>
>But then he claims that if a->0 the limit becomes delta(x-x0) which is=20=

the
>   "expansion basis implementing pure space-domain sampling".=20
>While it's true that the Dirac delta function is the expansion basis=20
for the
>space domain
>since=20
>f(x) =3D integral(f(z) delta(x-z) dz) between -inf ... +inf
>it's wrong that lim(f(x)) =3D Dirac-delta(x-x0) for a->0, m->0. In fact=20=

the
>limit is the *Kronecker* delta function which is surely NOT the=20
expansion
>basis for the time domain.
>
>
>
>(2)
>In problem 8.4 and 8.6 he claims that the product of two Gabor wavelets=20=

(GW)
>is just another Gabor wavelet because "the arguments of the=20
exponentials
>simply add within each exponential". That is wrong because the=20
exponents of
>the Gaussian part of the GW do *not* simply add in general, ie if you=20
don't
>happen to have the same exponent in both arguments of the product. You
>essentially have to prove that the product of the two Gaussian=20
exponentials
>is another Gaussian exponential which is not easy. In fact it only=20
works if
>you put an *extra factor* as parameter in front of the exponential.
>Therefore
>the product of two GW has *not* got the same functional form as stated=20=

in
>the notes and thus the set of all GW is *not* closed under convolution.=20=

It
>would work, however, if the GW was defined as
>f(x) =3D A exp(-imx) exp-(x-x0)^2/a^2
>
>
>
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