Discrete frequency domain

About: Discrete frequency domain is a research topic. Over the lifetime, 1932 publications have been published within this topic receiving 28622 citations.

Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case

Abstract: A discrete time series has associated with it an amplitude spectrum which is a periodic function of frequency. This paper investigates the extent to which a time series can be concentrated on a finite index set and also have its spectrum concentrated on a subinterval of the fundamental period of the spectrum. Key to the analysis are certain sequences, called discrete prolate spheroidal sequences, and certain functions of frequency called discrete prolate spheroidal functions. Their mathematical properties are investigated in great detail, and many applications to signal analysis are pointed out.

Calculation of a constant Q spectral transform

Abstract: The frequencies that have been chosen to make up the scale of Western music are geometrically spaced. Thus the discrete Fourier transform (DFT), although extremely efficient in the fast Fourier transform implementation, yields components which do not map efficiently to musical frequencies. This is because the frequency components calculated with the DFT are separated by a constant frequency difference and with a constant resolution. A calculation similar to a discrete Fourier transform but with a constant ratio of center frequency to resolution has been made; this is a constant Q transform and is equivalent to a 1/24‐oct filter bank. Thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range. This transform against log (frequency) to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted. This is compared to the conventio...

Scale-space for discrete signals

Abstract: A basic and extensive treatment of discrete aspects of the scale-space theory is presented. A genuinely discrete scale-space theory is developed and its connection to the continuous scale-space theory is explained. Special attention is given to discretization effects, which occur when results from the continuous scale-space theory are to be implemented computationally. The 1D problem is solved completely in an axiomatic manner. For the 2D problem, the author discusses how the 2D discrete scale space should be constructed. The main results are as follows: the proper way to apply the scale-space theory to discrete signals and discrete images is by discretization of the diffusion equation, not the convolution integral; the discrete scale space obtained in this way can be described by convolution with the kernel, which is the discrete analog of the Gaussian kernel, a scale-space implementation based on the sampled Gaussian kernel might lead to undesirable effects and computational problems, especially at fine levels of scale; the 1D discrete smoothing transformations can be characterized exactly and a complete catalogue is given; all finite support 1D discrete smoothing transformations arise from repeated averaging over two adjacent elements (the limit case of such an averaging process is described); and the symmetric 1D discrete smoothing kernels are nonnegative and unimodal, in both the spatial and the frequency domain. >

Fast algorithms for the discrete W transform and for the discrete Fourier transform

Abstract: A systematic method of sparse matrix factorization is developed for all four versions of the discrete W transform, the discrete cosine transform, and the discrete sine transform, as well as for the discrete Fourier transform. The factorization leads to fast algorithms in which only real arithmetic is involved. A scheme for reducing multiplications and a convenient index system are introduced. This makes new algorithms more efficient than conventional algorithms for the discrete Fourier transform, the discrete cosine transform, and the discrete sine transform.