Sinusoidal Waves as Random Variables

Abstract

The probability distribution associated with a random variable is a basic notion on which signal analysis of sounds is based. The probability distribution for a sum of a pair of independent random variables underlies the notion of convolution that is basic to signal analysis from the point of view of the linear system theory. The correlation between a pair of random variables is a standard metric to characterize pairs of random variables as well as the independence of random variables. The difference between the uncorrelated (or orthogonality) and statistically independent pairs of random variables renders typical auditory effects on sound image sensations. The perception of sound image experienced during binaural listening to an independent pair of random noise is called subjective diffuseness. An example of the probability density function for a sinusoidal wave is given to help understand orthogonality as being different from the statistical independence. The square correlation is a measure that represents the energy ratio of the uncorrelated pair of variables. Direct sound and its reverberation is a typical example of an uncorrelated pair in reverberant sound fields.