On impropriety for a large-sized discrete fourier transform of a real-valued stationary process

Abstract

It is known that a finite-sized discrete Fourier transform (DFT) of a real-valued colored stationary random process results in improperly distributed complex random variables in the frequency domain. It is also well-known that a large-sized DFT of a colored random process approximates the behavior of the Karhunen–Loève (KL) transform, in terms of achieving approximate decorrelation between the frequency domain terms. We show that this decorrelating behavior also manifests itself in terms of the approximate complex circular (proper) nature of the majority of the transform domain terms, except near the band edges, where the nature of the complex distribution may be non-circular. We derive an analytical result relating the DFT size with the degree of impropriety, and also demonstrate the same via simulations. The result is important in the context of wideband xDSL (Digital Subscriber Line) discrete multi-tone (DMT) receivers affected by extraneous spectrally colored noise sources, besides being of general interest.