Re: [Discuss-gnuradio] White Noise detection and elimination

[Robert James]

Got it.  I was assuming that since sinc looks like white noise, white
noise would look like sinc.  That's not true - white noise is plain
ol' random.

Thank you everyone for the great discussion.

On 11/18/13, West, Nathan <address@hidden> wrote:
> On Mon, Nov 18, 2013 at 11:25 AM, Robert James <address@hidden>
> wrote:
>> 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.
>>
>
> This is not going to filter any noise. Remember that the
> auto-correlation of white noise (should be) an impulse. If you're
> taking the minimum value and subtracting that value from other FFT
> bins that's not really meaningful. One of the DFT properties:
>
> a*x1[n] + b*x1[n] -> a*X1[k] + b*X2[k].
>
> Let's make the actual signal x1[n], and a=1. You're making X2[k] a
> step since you subtract the same value from the original signal's FFT.
> The value of b you're using is the smallest random number you find,
> the specific value isn't important. In the time domain you're just
> subtracting a sort of sinc with amplitude that changes every FFT
> length. Definitely not reducing any noise.
>
> The noise in each sample should be independent if you're assuming white
> noise.
>