Hi Everyone,
Suppose that I generate a time series x[n] as follows:
>>>
P is a constant value between 0 and 1
At each time step n (n is an integer):
r[n] = uniform_random(0, 1)
x[n] = (r[n] <= P) ? uniform_random(-1, 1) : x[n-1]
Where "(a) ? b : c" is the C ternary operator that takes on the
Has this been answered yet? If not, I'll try a back-of-the-envelope
derivation.
Consider two consecutive samples. By definition, the probability for
them to be equal is (1-P), else they will be different and perfectly
uncorrelated. Hence the expected correlation between two consecutive
sample
Do you mean the literal Fourier spectrum of some realization of this
process, or the power spectral density? I don't think you're going to get a
closed-form expression for the former (it has a random component). For the
latter what you need to do is work out an expression for the
autocorrelation fu
Wait, just realized I wrote that last part backwards. It should be:
So in broad strokes, what you should see is a lowpass spectrum
parameterized by P - for P very small, you approach a DC spectrum, and for
P close to 1 you approach a spectrum that's flat.
On Tue, Nov 3, 2015 at 10:26 AM, Ethan Du
dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ethan Duni"
Date: Tue, November 3, 2015 1:29 pm
To: "A discussion list for music-related DSP"
--
> Wait, just realized I wro
On 4/11/2015 5:26 AM, Ethan Duni wrote:
Do you mean the literal Fourier spectrum of some realization of this
process, or the power spectral density? I don't think you're going to
get a closed-form expression for the former (it has a random component).
I am interested in the long-term magnitude
Original Message
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ross Bencina"
Date: Tue, November 3, 2015 11:51 pm
To: music-dsp@music.co
On 4/11/2015 9:39 AM, robert bristow-johnson wrote:
i have to confess that this is hard and i don't have a concrete solution
for you.
Knowing that this isn't well known helps. I have an idea (see below). It
might be wrong.
it seems to me that, by this description:
r[n] = uniform_random(0,
Original Message
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ross Bencina"
Date: Wed, November 4, 2015 12:22 am
To: r...@audioimagination.com
music-dsp@music.co
Original Message
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ross Bencina"
Date: Wed, November 4, 2015 12:22 am
To: r...@audioimagination.com
music-dsp@music.co
Original Message
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ross Bencina"
Date: Wed, November 4, 2015 12:22 am
To: r...@audioimagination.com
music-dsp@music.co
Original Message
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ross Bencina"
Date: Wed, November 4, 2015 12:22 am
To: r...@audioimagination.com
music-dsp@music.co
riginal Message
> Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold
> noise?
> From: "Ross Bencina"
> Date: Wed, November 4, 2015 12:22 am
> To: r...@audioimagin
It's pretty straightforward to derive the autocorrelation and psd for this
one. Let me restate it with some convenient notation. Let's say there are a
parameter P in (0,1) and 3 random processes:
r[n] i.i.d. ~U(0,1)
y[n] i.i.d. ~(some distribution with at least first and second moments
finite)
x[n]
gt;
> Does that make sense?
>
> -Ethan
>
>
>
>
>
> On Wed, Nov 4, 2015 at 8:21 AM, robert bristow-johnson <
> r...@audioimagination.com> wrote:
>
>>
>>
>> ------------ Original Message
>> Subje
Ross Bencina wrote:
Hi Everyone,
Suppose that I generate a time series x[n] as follows:
>>>
P is a constant value between 0 and 1
At each time step n (n is an integer):
r[n] = uniform_random(0, 1)
x[n] = (r[n] <= P) ? uniform_random(-1, 1) : x[n-1]
Where "(a) ? b : c" is the C ternary opera
Thanks Ethan(s),
I was able to follow your derivation. A few questions:
On 4/11/2015 7:07 PM, Ethan Duni wrote:
It's pretty straightforward to derive the autocorrelation and psd for
this one. Let me restate it with some convenient notation. Let's say
there are a parameter P in (0,1) and 3 rando
>
> What is the method that you used to go from ac[k] to psd[w]? Robert
> mentioned that psd was the Fourier transform of ac. Is this particular case
> a standard transform that you knew off the top of your head?
Yes, this is the Fourier transform of P^|k| (following Ethan D's notation).
To deriv
Thanks Ethan,
I think that I have it working. It would be great is someone could check
the scaling though. I'm not sure whether the FFT values should be
fringing above the psd line or not:
https://www.dropbox.com/s/txc0txhxqr1t274/SH1_2.png?dl=0
I removed the hamming window, which was causin
>
> Let's see if I got this right: each bin contains the power for a frequency
> interval of 2pi/N radians. If I multiply each bin's power by N/2pi I should
> get power values in units of power/radian.
>
Sounds reasonable to me, but I'm not sure I've got it right so who knows!
I think I was sligh
>So for y[n] ~U(-1,1) I should multiply psd[w] by what exactly?
The variance of y[n]. For U(-1,1) this is 1/3. From your subsequent post it
sounds like you got this ironed out?
>What is the method that you used to go from ac[k] to psd[w]?
>Robert mentioned that psd was the Fourier transform of ac
>since the whole signal has infinite power, the units really
>need to be power per unit frequency per unit time, which
>(confusingly) is the same thing as power.
I think you mean to say "infinite energy" and then "energy per unit
frequency per unit time," no?
E
On Thu, Nov 5, 2015 at 8:21 AM, Et
Yes, thank you! I guess most of the places I typed the word power I really
meant energy... units are hard...
-Ethan
On Thu, Nov 5, 2015 at 7:33 PM, Ethan Duni wrote:
> >since the whole signal has infinite power, the units really
> >need to be power per unit frequency per unit time, which
> >(c
> I think I was slightly off when I said that the units of psd are power per
> unit frequency -- since the whole signal has infinite power,
�
no, i don't think so.
�
> the units�really need to be power per unit frequency per unit time, which
> (confusingly) is the same thing as power.
�
the
On 2015-11-05, robert bristow-johnson wrote:
I think I was slightly off when I said that the units of psd are
power per unit frequency -- since the whole signal has infinite
power,
no, i don't think so.
Me neither. Power is already by definition energy per unit time. Even if
an infinitely
Apologies if this question has already been answered, I didn't read the
entire thread, just wanted to share the following idea off the top of my
head FWIW.
Consider a random PCM signal whose period is equal to the SnH period.
This is the same as discrete-time noise and, if I'm not mistaken, gi
On 06-Nov-15 11:03, Vadim Zavalishin wrote:
Apologies if this question has already been answered, I didn't read the
entire thread, just wanted to share the following idea off the top of my
head FWIW.
Oops, nevermind, I didn't realize that the SnH period is also random in
the original question.
Okay, an updated idea. Represent the signal as a sum of time-shifted box
functions of random amplitudes and durations. We assume that the sum is
finite and then we can take the limit (if the values approach the
infinity as the result, we can normalize them according to the length of
the signal)
Hi Ross,
Just spotted this. I don't have an answer for you, but a possible
helpful literature connection...?
The system you describe is a simple Markov model. It's ergodic and
time-homogeneous and reversible, and has no hidden state, so I'd guess
that there must be results from the Markov model l
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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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
>(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 pr
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 mu
mption come from?
E
On Tue, Nov 10, 2015 at 6:33 PM, robert bristow-johnson <
r...@audioimagination.com> wrote:
>
>
> -------------------- Original Message ------------
> Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold
> noise?
Original Message
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ethan Duni"
Date: Wed, November 11, 2015 5:57 pm
To: "robert bristow-johnson"
"A discussion l
or making plots to verify that the answer is correct,
> > but not needed just to derive the autocorrelation/spectrum themselves.
>
>
>
> i totally disagree. i consider this to be fundamental (and it's how i
> remember doing statistical communication theory back in grad sc
Original Message
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Ethan Duni"
Date: Wed, November 11, 2015 7:36 pm
To: "robert bristow-johnson"
"A discussion l
he OP - where we were given
an explicit description of a random signal - is in probabilistic terms
using definitions of acf and psd in terms of expected value.
E
On Wed, Nov 11, 2015 at 5:02 PM, robert bristow-johnson <
r...@audioimagination.com> wrote:
>
>
>
> ---------------
Everywhere in the exact sciences there's the dualism between statistical analysis and
deterministic engineering tools, since the major break through in quantum physics at the
beginning of the 20th century. Whether that's some sort of diabolical duality or, as it
actually is at the higher levels
Am 16.11.2015 20:00, schrieb Martin Vicanek:
[..] the autocorrelation is
= (1/3)*(1-P)^|k|
(I checked that with a little MC code before posting.) So the power
spectrum is (1/3)/(1 + (1-P)z^-1), i.e flat at DC and pink at higher
frequencies. For reasonably small P the corner frequency is
w_
> Am 16.11.2015 20:00, schrieb Martin Vicanek:
>> [..] the autocorrelation is
>>
>> = (1/3)*(1-P)^|k|
>>
>> (I checked that with a little MC code before posting.) So the power
>> spectrum is (1/3)/(1 + (1-P)z^-1), i.e flat at DC and pink at higher
>> frequencies. For reasonably small P
> well, pink is -3 dB/octave and red (a.k.a. brown) is -6 dB/octave. a
roll-off of -12
> dB/octave would be very brown. -- r b-j
Those values are for amplitudes - for a power spectrum the slopes double.
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Original Message
Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise?
From: "Martin Vicanek"
Date: Mon, November 16, 2015 3:50 pm
To: music-dsp@music.co
>> [..] the autocorrelation is
>>
>> = (1/3)*(1-P)^|k|
>>
>> (I checked that with a little MC code before posting.) So the power
>> spectrum is (1/3)/(1 + (1-P)z^-1)
The FT of (1/3)*(1-P)^|k| is (1/3)*(1-Q^2)/(1-2Qcos(w) + Q^2), where Q =
(1-P).
Looks like you were thinking of the expression for
>/ > well, pink is -3 dB/octave and red (a.k.a. brown) is -6 dB/octave. a
/
>/ > roll-off of -12// dB/octave would be very brown. -- r b-j
/
>/ Those values are for amplitudes - for a power spectrum the slopes double.
/
no sir. not with dB. this is why we use
dB = 20 * log10(ampli
I don't recall the various chapters of Information Theory and practical digtial
electronics or something about these kinds of power estimates, so I'm not putting this
thread in some direction, except two small considerations.
When the power of a signal is concerned, which in the continuous case
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