D.1 Chi-Square Distribution

Consider the random variable

where the { A _i} are independent Gaussian random variables with means { m _i} and common variance ? ². The random variable Z is said to have a noncentral chi-square ( ? ²) distribution with N degrees of freedom and a noncentral parameter

To derive the probability density function of Z we first note that each A _i has the density function

From elementary probability, the density of Y _i = A ² _i is

where u(x) = l, x ? 0, and u(a) = 0, x < 0. Substituting (D-3) into (D-4), expanding the exponentials, and simplifying, we obtain the density

The characteristic function of a random variable X is defined as

where , and x(x) is the density of X. Since C _X(j ?) is the conjugate Fourier transform of _X(x),

From Laplace or Fourier transform tables, it is found that the characteristic function of _Yi(x) is

The characteristic function of a sum of independent random variables is equal to the product of the individual characteristic functions. Because Z is the sum of the Y _i, the characteristic function of Z is

where we have used (D-2). From (D-9), (D-7), and Laplace or Fourier transform tables, we obtain the probability density function of noncentral ? ² random variable