Discrete uniform distributionIn probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6.
Nyquist–Shannon sampling theoremThe Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample rate must be at least twice the bandwidth of the signal to avoid aliasing distortion. In practice, it is used to select band-limiting filters to keep aliasing distortion below an acceptable amount when an analog signal is sampled or when sample rates are changed within a digital signal processing function.
Spectral density estimationIn statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.
Continuous uniform distributionIn probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, and which are the minimum and maximum values. The interval can either be closed (i.e. ) or open (i.e. ). Therefore, the distribution is often abbreviated where stands for uniform distribution.
Sampling (signal processing)In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or space; this definition differs from the term's usage in statistics, which refers to a set of such values. A sampler is a subsystem or operation that extracts samples from a continuous signal. A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points.
Discrete-time Fourier transformIn mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
Inverse transform samplingInverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the golden rule) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number between 0 and 1, interpreted as a probability, and then returns the smallest number such that for the cumulative distribution function of a random variable.
Discrete Fourier transformIn mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies.
Rejection samplingIn numerical analysis and computational statistics, rejection sampling is a basic technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in with a density. Rejection sampling is based on the observation that to sample a random variable in one dimension, one can perform a uniformly random sampling of the two-dimensional Cartesian graph, and keep the samples in the region under the graph of its density function.
Fourier analysisIn mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.
