Charles Henry escribió:
When it comes to the general class of functions with flat spectra, the
only difference is in phase, right?
But the error is the same in time domain as in frequency domain thanks
to the isometric property of the Fourier transform. Our interpolation
is the same as a convolution, so we're still just multiplying our
spectra and the phase comes out differently in each frequency.
I'm not sure I understand what you're saying here about the phase, buy I
think the misleading part of youre reasoning is that you take a concept
that makes sense in the context of stochastic processes, namely assuming
a "flat spectrum", and acritically apply it in the context of
deterministic signals where it has a completely different meaning.
You're trying to restrict the analysis to a convenient (but reasonable)
class of signals, and to assume that the signal to be interpolated, x,
belongs to that class. Right?
It doesn't make any sense, as far as I can see, to assume that the
signal being interpolated belongs to the class of function whose
spectrum has a flat modulus (and any phase).
Why not assuming then, for example, that x(t) is a constant?
(please don't take my tone as sarchastic)
What does make some sense (it is a strong hypothesis but discussing its
plausibility would bring the discussion to a much higher level) is to
treat the signal x as a stochastic process with a given power spectrum -
such as flat, or pink.
But that means that the quantity you're minimizing is no longer an
integral of the signal minus some other signal all squared: it is the
expectation of something.
The power spectrum of a stochastic process x(t) is not the fourier
transform of x(t), it is the fourier transform of tha autocorrelation
function of x (or something like that).
--
Matteo Sisti Sette
matteosistise...@gmail.com
http://www.matteosistisette.com
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