Power spectra obtained from exponentially increasing spacings of sampling positions and frequencies

An estimate of the spectrum is based on the Laplace transform which is approximated at exponentially spaced samples and analysis frequencies. In this approximation the ratio of the intervals between pairs of adjacent sampling positions is a constant greater than one. The choice of this constant is influenced by the desired analysis bandwidth and by sampling effects. If analysis frequencies are spaced the same as sampling positions, this approximation becomes a discrete correlation. which can be computed by a fast Fourier transform (FFT) or a number theoretic transform. Except at low-analysis frequencies, the analysis bandwidth is "constant-Q," i.e., it is proportional to the analysis frequency. With a white noise input the noise in the computed spectrum is roughly constant at each analysis frequency. The numbers of samples and computations required for exponential spacing of samples and frequencies can be less than those required for equidistant spacing. Better performance at some (but not all) analysis frequencies is provided by a two-sided sampling arrangement consisting of a juxtaposition of the basic one-sided sampling arrangement and its mirror image.