Legendre polynomial expressions for the probability density function of magnitude-squared coherence estimates

Carter, Knapp, and Nuttall [1] presented a discrete Fourier transform (DFT) method for estimating the magnitude-squared coherence between two zero-mean wide-sense-stationary random processes. This paper shows that a simple Legendre polynomial expression can replace the hypergeometric function used in the probability density function expression of Carter et al. [1] (see also Fisher [2] and Goodman [3]) for the coherence estimate obtained from nonoverlapped DFT's. A standard Legendre polynomial recursion provides a recursive method for evaluating the density expression.