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It is the simplest, nontrivial, innite dimension Markov chain which possesses all the nice properties - stationarity, irreducibility, positive recurrency, aperiodicity, : : :, etc. Source Taiwanese J. Zentralblatt MATH identifier Subjects Primary: 60J Markov chains discrete-time Markov processes on discrete state spaces 60F Central limit and other weak theorems. Keywords Markov chain ergodic stationary limiting distribution sum of random variables multivariate compound Poisson distribution.
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Abstract Article info and citation First page Abstract We introduce an innite dimension completely stationary ergodic Markov chain; investigate its properties; establish a sucient condition that the Markov property of a Markov chain is preserved by its induced multi-state sequence and nally derive the limiting distribution of the sum of the Markov chain. It is used for as simple cases as visualizing sound, extracting general sound features or providing basis for accurate sound parameterization techniques. Its popularity lies in the fact, that Fourier transform provides very human readable and sonically meaningful parameterization.
Is a stationary object in space really stationary?
With discovery of Fast Fourier Transform in and ideas like phase vocoding eventually led to development of Spectral Modeling Synthesis. This synthesis method essentially synthesizes sound from a spectrum, a word widely used to denote Fourier transform.
Constructing a reasonably good spectrum from scratch can be a painstakingly long lasting task, so some kind of templates should be used. Such templates can be extracted from sounds of instruments. Such procedure can result in very naturally sounding synthetic sounds.
Of course, a high quality analysis is a must, as the most interesting instruments tend to contain numerous details in form of subtle frequency and amplitude modulations, giving it its distinct character, recognized by producers and music lovers alike.
- Wang , Tang : A COMPLETELY STATIONARY MARKOV CHAIN WITH INFINITE STATE SPACE!
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