Re: Fourier transform normalization

[brian whatcott <inet@INTELLISYS.NET>]

At 23:16 8/31/01 -0400, John Denker wrote:

> /snip/  Suppose we have two seconds of
> waveform data, sampled at intervals of
>        dt = 1 millisecond
> for a total of
>        N = 2000 samples.
>
> Then the abscissas in the frequency domain will have a spacing of
>        df = 1 / (N dt)         (as a rule)
>           = 0.5 Hz             (in this case)
>
> C) I claim that when doing a forward FFT, the output should be scaled by a
> factor of dt.  In particular, if Vin is a delta function
>        Vin = [1, 0, 0, 0, .... , 0]
> then the transform should be
>        myVf = myFFT(Vin)
>             = [.001, .001, .001, .... , .001]
>
> D) I claim that when doing an inverse FFT, the output should be scaled by a
> factor of df.  In particular, if Vf is a delta function,
>        Vf = [1, 0, 0, 0, .... , 0]
> then the inverse transform should be
>        myVout = myIFFT(Vf)
>               = [.5, .5, .5, .... , .5]
>
> Note that when using the Scilab or Matlab inverse FFT, the software stuffs
> in a factor of 1/N automagically, so you need to undo that (multiply by a
> factor of N) before applying the factor of df.  By the way, note that
> multiplying by (N df) is the same as dividing by (dt).
>
> E) Note that myIFFT is the inverse of myFFT.  Their definitions are
> completely symmetric;  one has a factor of dt and the other has a factor of
> df.  Neither of them needs to have a factor of 1/N.  This all works out
> cleanly because
>        df * dt = 1/N

/snip/

As a physical experiment of some slight applicability, I mention that
in 1947, the German physicist Fritz Zernike set up an arc lamp with an
exit slit and condenser lens to produce a moderately coherent light beam
(like being moderately pregnant, I suppose...)

 He played this light on a second slit, call it slit A, and noted the
pattern on a screen at the back focus of a cylindrical lens interposed
after slit A.  He saw the Fraunhofer diffraction pattern of the
single slit as you would expect.

 This can be described by the function sin(pi q)/(pi q) in terms
of amplitude. (q is a spatial frequency value)
 In terms of observability, we do not detect light
as amplitude but as intensity so the intensity function would be
half the square of this function. We see two intensity peaks per
amplitude cycle then.
 The optical transform of this function is a triangle wave in amplitude,
which evidently does not reconstruct the initial slit.

He next placed a half silvered glass with a narrow clear slit, call it
slit B, immediately in front of the existing slit A.
This new slit B is much narrower than slit A, but allows half the light
to penetrate, while further diffracting light in its central aperture
 into a wide beam . This background illumination is coherent with that
from the main slit, of course.

He noticed something striking: he could now see a pattern at twice the
spacing of the previous pattern. This can be attributed to adding a
constant amplitude coherent field to offset the previous amplitude
pattern.  By lifting a sinusoidal amplitude distribution away from
 the zero of amplitude, the observable intensity now peaks once
per cycle.

This experimental set up is just one step away from wavefront
 reconstitution: shining a laser through an image of this pattern
would reconstitute the tandem slits. This was the mechanism that
Gabor demonstrated a year later.
When Leith and Upatnieks at UMichigan suggested providing a reference
beam at an angle, when the laser became available in 1960,
holography took off.

(This discussion follows the form of P J Holligan,
ST291:9/10 Open University Press 1978)


brian whatcott <inet@intellisys.net>  Altus OK
                Eureka!