---------------------------- Original Message ---------------------------- Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise? From: "Ross Bencina" <rossb-li...@audiomulch.com> Date: Tue, November 3, 2015 11:51 pm To: music-dsp@music.columbia.edu -------------------------------------------------------------------------- > 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 spectrum. I had assumed > (wrongly?) that in the limit (over an infinite length series), that the > fourier integral would converge. And modeling in that way would be > (slightly) more familiar to me. However, If autocorrelation or psd is > the better way to characterize the spectra of random signals then I > should learn about that. � 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. � so if you're gonna look at the spectrum in dB, it's 10*log10() not 20*log10(). � but it ain't gonna be easy. �however, i *think* you gotta 'nuf information. �this is basically a Markov process. � setting aside a complex random signal, autocorrelation is first expressed as a time-average of the product of your random signal times itself with a given lag. �it's an even function, so the PSD will be real. � with the assumption of ergodicity, the time average can be replaced with a probabilistic average for the same quantity. �i think there is enough information in your description to calculate the probabilistic average of the product of your random signal times itself displaced by a given lag. � i have a sneaky suspicion that this Markov process is gonna be something like pink noise. �maybe with different slopes (of dB vs. log frequency) depending on parameter P. �probabilistically holding on to a previous sample will have an LPF effect. -- � r b-j � � � � � � � � � r...@audioimagination.com � "Imagination is more important than knowledge."
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