---------------------------- Original Message ----------------------------
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ross Bencina" <rossb-li...@audiomulch.com>
Date: Wed, November 4, 2015 12:22 am
To: r...@audioimagination.com
music-dsp@music.columbia.edu
--------------------------------------------------------------------------
�
with mods....
>�Using ASDF instead of autocorrelation:
>
> let n be an arbitrary time index
> let t be the ASDF lag time of interest
>
> ASDF[t] = (x[n] - x[n-t])^2
>
> there are two cases:
>
> case 1, (holding): x[n-t] == x[n]
this has probability of P^|t|
�
> case 2, (not holding) x[n-t] == uniform_random(-1, 1)
this has probability of 1 - P^|t|
>
> In case 1, ASDF[t] = 0
> In case 2, ASDF[t] = (1/3)^2 �(i think)
�
so maybe it's
�
ASDF[t] = 0 * P^|t| �+ �(1/3)^2 * (1 - P^|t|)
�
now the autocorrelation function (AF) is related to the ASDF as
�
AF[t] = �mean{ x[n] * x[n-t] }
AF[t] = �mean{ (x[n])^2 } �- (1/2)*mean{ (x[n] - x[n-t])^2 }
�
AF[t] = �mean{ (x[n])^2 } �- (1/2)*ASDF[t]
�
AF[t] �= �(1/3) �- �(1/2) *�(1/3)^2 * (1 - P^|t|)
�
this doesn't quite look right to me. �somehow i was expecting �AF[t] to go to
zero as t goes to
infinity.
�
> To get the limit of ASDF[t], weight the values of the two cases by the
> probability of each case case. (Which seems like a textbook waiting-time
> problem, but will require me to return to my textbook).
>
> Then I just need to convert the ASDF to PSD somehow.
�
� ASDF[t] = 2*AF[0] - 2*AF[t]
�
or
�
� AF[t] �= �AF[0] �- (1/2)*ASDF[t]
�
�
PSD = Fourier_Transform{ AF[t] }
> Does that seem like a reasonable approach?
�
it's the approach i am struggling with. � somehow, i don't like the AF i get.
--
�
r b-j � � � � � � � � � r...@audioimagination.com
�
"Imagination is more important than knowledge."
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