Got it: The Fourier coefficients tell you *two* things per freq:
amplitude and phase. In real values (what I'm used to), those are two
distinct numbers. In complex values (welcome to DSP), it's one
complex number, telling you the amplitude of the 0-degree component
(Re) and the amplitude of the -90-degree component (Img).
Going from complex to real is easy: complex r = real amplitude,
complex theta = real phase. But that's besides the point. The point
is that amplitude alone is only half the story, phase is the other
half, and, without phase, you can't reconstruct the original signal.
(The FFT displays might not show the phase, and we might not alwasy
talk about it, but it's half the information, like it or not.)
So while FFT is invertible to signal, PSD isn't.
What about this approach, then: Put the Fourier coefficients of in
polar coordinates. White noise will tend add a fixed complex value,
call it W, to each Fourier coefficient of the signal. Assuming the
original signal has many zero Fourier coefficients (many more than
nonzero), when we add the noise, the most common coefficients will be
close to W.
If there is a complex value, call it W-hat, that is within delta of x%
of Fourier coefficients, we assume W-hat is the value of the additive
white noise. Subtract W-hat from all the Fourier coefficients, IFTT,
and recover the signal with much of the white noise removed.
On 11/18/13, Martin Braun (CEL) <address@hidden> wrote:
On Mon, Nov 18, 2013 at 09:56:03AM -0500, Robert James wrote:
I'm eager to learn what's wrong with my concept, especially from the
experts here.
Robert, there's a lot of basics that need to be covered here. You might
have to go into the textbooks.
First of all, the FFT does *not* give you the power at a frequency. It
gives you a Fourier coefficient. It's amplitude does have something to
do with the power, but it's not the same.
Short tangent: You can estimate a PSD by using an FFT and then
mag-squaring the output. This is called a 'peridogram'.
You can get a better estimate by applying a window, and averaging
several periodograms. This is called 'Welch's method'.
Now here's the difference: You're chucking away the phase, and
squaring the amplitude. So what are you subtracting from what?
This goes on and on. Have look at the concept of 'digital filtering',
and specifically the Fourier method of designing filters. You will find
some similarities to your approach.
Also, be careful when assigning absolute powers (in Watts) to FFT bins!
I guess it's technically correct when you multiply the PSD value here
with the size of the FFT bin, but that assumes a good estimate of the
PSD, and for absolute values, that you have calibrated your system
correctly.
Even if it is *completely* wrong, I'd like to know the format of the
FFT output vector, so I can experiment myself. What is the format?
Complex numbers (representing Fourier coefficients).
MB
--
Karlsruhe Institute of Technology (KIT)
Communications Engineering Lab (CEL)
Dipl.-Ing. Martin Braun
Research Associate
Kaiserstraße 12
Building 05.01
76131 Karlsruhe
Phone: +49 721 608-43790
Fax: +49 721 608-46071
www.cel.kit.edu
KIT -- University of the State of Baden-Württemberg and
National Laboratory of the Helmholtz Association
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