1. Number of Degrees of Freedom, Fisher Information, and Frequency-Time Products of a Random Process.
- Author
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Kikkawa, Sho
- Subjects
- *
GAUSSIAN processes , *STOCHASTIC processes , *BANDWIDTHS , *ELECTRONICS , *STATISTICAL correlation , *PROBABILITY theory - Abstract
First, the number of nth order degrees of freedom (n-th order NDF) of stationary random process is defined in terms of the sample moment of order n of a finite length sample. Correlation times and equivalent band- widths are derived from the NDFs. It is shown that the second-order NDF asymptotically approaches the degrees of freedom of a gamma distribution by which the distribution of the ample variance is approximated it is shown also for the case of a Gaussian process that the first-order NDF is the same as the standardized Fisher information about the mean; and, furthermore, the second-order NDF of a Gaussian AR process is the same as the standardized Fisher information about the variance. A problem with the Fisher information is that it cannot always be calculated, which is not convenient from the practical viewpoint. Since the NDF defined in this paper is based on the sample moments, it can easily be calculated from the observed data. Further, from the NDF, useful features can be derived such as the correlation time and the equivalent bandwidth. Therefore, it is of practical use. Finally, for a Gaussian process, the significance of the approximation of the NDF by 2 WT, where W denotes the equivalent bandwidth and T is time duration, is discussed. It is shown, in particular, that the approximation error for the second-order NDF decreases only slowly, depending on the Logarithm of 2 WT. This result is interesting in examining the dimension of the signal space for each moment of a random process since the situation is the same as in the case where the dimension of the signal space for the deterministic signal is approximated by 2 WT. [ABSTRACT FROM AUTHOR]
- Published
- 1994
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