Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?

[Theo Verelst]

In the course of these discussions, let's not forget the difference between a convolution 
with 1/(Pi*t) (a Hilbert transform kernel) and the inversion of the transfer function of a 
linear system.


T.
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Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?

[robert bristow-johnson]

 Original Message 

Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?

From: "Ethan Duni" 

Date: Tue, November 10, 2015 8:58 pm

To: "A discussion list for music-related DSP" 

--



>>(Semi-)stationarity, I'd say. Ergodicity is a weaker condition, true,

>>but it doesn't then really capture how your usual L^2 correlative

>>measures truly work.

>

> I think we need both conditions, no?
all ergodic processes are stationary. �(not necessarily the other way around.)
�
the reason (besides forgetting stuff i learned 4 decades ago) i left out 
"stationary" was that i was sorta conflating the two. �i just
wanted to be able to turn the time-averages in the whatever norm (and L^2 is as 
good as any) with probabilistic averages, which is the root meaning of the 
property "ergodic". �but probably "stationary" is a better (stronger) 
assumption to make.
�

>

>>Something like that, yes, except that you have to factor in aliasing.

>

> What aliasing? Isn't this process generated directly in the discrete time

> domain?
i'm thinking the same thing. �it's a discrete-time Markov process. �just model 
it and analyze it as such. assuming stationarity, we should be able to derive 
an autocorrelation function (and i think you guys did) and from that (and the 
DTFT) you have the (periodic) power
spectrum.
worry about frequency aliasing when you decide to output this to a DAC.



--
�


r b-j � � � � � � � � � r...@audioimagination.com
�


"Imagination is more important than knowledge."
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Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?

[Ethan Duni]

>(Semi-)stationarity, I'd say. Ergodicity is a weaker condition, true,
>but it doesn't then really capture how your usual L^2 correlative
>measures truly work.

I think we need both conditions, no?

>Something like that, yes, except that you have to factor in aliasing.

What aliasing? Isn't this process generated directly in the discrete time
domain?

E

On Tue, Nov 10, 2015 at 5:43 PM, Sampo Syreeni  wrote:

> On 2015-11-04, robert bristow-johnson wrote:
>
> it is the correct way to characterize the spectra of random signals. the
>> spectra (PSD) is the Fourier Transform of autocorrelation and is scaled as
>> magnitude-squared.
>>
>
> The normal way to derive the spectrum of S/H-noise goes a bit around these
> kinds of considerations. It takes as given that we have a certain sampling
> frequency, which is the same as the S/H frequency. Under that assumption,
> sample-and-hold takes any value, and holds it constant for a sampling
> period. You can model that by a convolution with a rectangular function
> which takes the value one for one sampling period, and which is zero
> everywhere else. Then the rest of the modelling has to do with normal
> aliasing analysis.
>
> That's at least how they did it before the era of delta-sigma converters.
>
> with the assumption of ergodicity, [...]
>>
>
> (Semi-)stationarity, I'd say. Ergodicity is a weaker condition, true, but
> it doesn't then really capture how your usual L^2 correlative measures
> truly work.
>
> i have a sneaky suspicion that this Markov process is gonna be something
>> like pink noise.
>>
>
> Something like that, yes, except that you have to factor in aliasing.
>
>
> r[n] = uniform_random(0, 1)
> if (r[n] <= P)
>x[n] = uniform_random(-1, 1);
> else
>x[n] = x[n-1];
>
>
> If P==1, that give uniform white noise. If P==0, it yields a constant. If
> P==.5, half of the time it holds the previous value.
>
> In a continuous time Markov process you'd get something like pink noise,
> yes. But in a discrete time process you have to factor in aliasing. It goes
> pretty bad, pretty fast.
>
> --
> Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front
> +358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
>
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Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?

[Sampo Syreeni]

On 2015-11-04, robert bristow-johnson wrote:

it is the correct way to characterize the spectra of random signals. 
the spectra (PSD) is the Fourier Transform of autocorrelation and is 
scaled as magnitude-squared.


The normal way to derive the spectrum of S/H-noise goes a bit around 
these kinds of considerations. It takes as given that we have a certain 
sampling frequency, which is the same as the S/H frequency. Under that 
assumption, sample-and-hold takes any value, and holds it constant for a 
sampling period. You can model that by a convolution with a rectangular 
function which takes the value one for one sampling period, and which is 
zero everywhere else. Then the rest of the modelling has to do with 
normal aliasing analysis.


That's at least how they did it before the era of delta-sigma 
converters.



with the assumption of ergodicity, [...]


(Semi-)stationarity, I'd say. Ergodicity is a weaker condition, true, 
but it doesn't then really capture how your usual L^2 correlative 
measures truly work.


i have a sneaky suspicion that this Markov process is gonna be 
something like pink noise.


Something like that, yes, except that you have to factor in aliasing.


r[n] = uniform_random(0, 1)
if (r[n] <= P)
   x[n] = uniform_random(-1, 1);
else
   x[n] = x[n-1];


If P==1, that give uniform white noise. If P==0, it yields a constant. 
If P==.5, half of the time it holds the previous value.


In a continuous time Markov process you'd get something like pink noise, 
yes. But in a discrete time process you have to factor in aliasing. It 
goes pretty bad, pretty fast.

--
Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front
+358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2___
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