Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor.
The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal as analyzed in terms of its frequency content, is called its spectrum.
In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy. Due to the measurement error of the recorded signal or image, it can be demonstrated that the worse the SNR, the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem.
In mathematics, a time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.
In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods. It is the most common tool for examining the amplitude vs frequency characteristics of FIR filters and window functions. FFT spectrum analyzers are also implemented as a time-sequence of periodograms.
Welch's method, named after Peter D. Welch, is an approach for spectral density estimation. It is used in physics, engineering, and applied mathematics for estimating the power of a signal at different frequencies. The method is based on the concept of using periodogram spectrum estimates, which are the result of converting a signal from the time domain to the frequency domain. Welch's method is an improvement on the standard periodogram spectrum estimating method and on Bartlett's method, in that it reduces noise in the estimated power spectra in exchange for reducing the frequency resolution. Due to the noise caused by imperfect and finite data, the noise reduction from Welch's method is often desired.
In 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.
A pitch detection algorithm (PDA) is an algorithm designed to estimate the pitch or fundamental frequency of a quasiperiodic or oscillating signal, usually a digital recording of speech or a musical note or tone. This can be done in the time domain, the frequency domain, or both.
The Goertzel algorithm is a technique in digital signal processing (DSP) for efficient evaluation of the individual terms of the discrete Fourier transform (DFT). It is useful in certain practical applications, such as recognition of dual-tone multi-frequency signaling (DTMF) tones produced by the push buttons of the keypad of a traditional analog telephone. The algorithm was first described by Gerald Goertzel in 1958.
MUSIC is an algorithm used for frequency estimation and radio direction finding.
Geophysical survey is the systematic collection of geophysical data for spatial studies. Detection and analysis of the geophysical signals forms the core of Geophysical signal processing. The magnetic and gravitational fields emanating from the Earth's interior hold essential information concerning seismic activities and the internal structure. Hence, detection and analysis of the electric and Magnetic fields is very crucial. As the Electromagnetic and gravitational waves are multi-dimensional signals, all the 1-D transformation techniques can be extended for the analysis of these signals as well. Hence this article also discusses multi-dimensional signal processing techniques.
In signal processing, multitaper is a spectral density estimation technique developed by David J. Thomson. It can estimate the power spectrum SX of a stationary ergodic finite-variance random process X, given a finite contiguous realization of X as data.
Maximum entropy spectral estimation is a method of spectral density estimation. The goal is to improve the spectral quality based on the principle of maximum entropy. The method is based on choosing the spectrum which corresponds to the most random or the most unpredictable time series whose autocorrelation function agrees with the known values. This assumption, which corresponds to the concept of maximum entropy as used in both statistical mechanics and information theory, is maximally non-committal with regard to the unknown values of the autocorrelation function of the time series. It is simply the application of maximum entropy modeling to any type of spectrum and is used in all fields where data is presented in spectral form. The usefulness of the technique varies based on the source of the spectral data since it is dependent on the amount of assumed knowledge about the spectrum that can be applied to the model.
In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the 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.
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum based on a least-squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in the long and gapped records; LSSA mitigates such problems. Unlike in Fourier analysis, data need not be equally spaced to use LSSA.
Phase dispersion minimization (PDM) is a data analysis technique that searches for periodic components of a time series data set. It is useful for data sets with gaps, non-sinusoidal variations, poor time coverage or other problems that would make Fourier techniques unusable. It was first developed by Stellingwerf in 1978 and has been widely used for astronomical and other types of periodic data analyses. Source code is available for PDM analysis. The current version of this application is available for download.
Within statistics, the Kolmogorov–Zurbenko (KZ) filter was first proposed by A. N. Kolmogorov and formally defined by Zurbenko. It is a series of iterations of a moving average filter of length m, where m is a positive, odd integer. The KZ filter belongs to the class of low-pass filters. The KZ filter has two parameters, the length m of the moving average window and the number of iterations k of the moving average itself. It also can be considered as a special window function designed to eliminate spectral leakage.
The Blackman–Tukey transformation is a digital signal processing method to transform data from the time domain to the frequency domain. It was originally programmed around 1953 by James Cooley for John Tukey at John von Neumann's Institute for Advanced Study as a way to get "good smoothed statistical estimates of power spectra without requiring large Fourier transforms." It was published by Ralph Beebe Blackman and John Tukey in 1958.
Multidimension spectral estimation is a generalization of spectral estimation, normally formulated for one-dimensional signals, to multidimensional signals or multivariate data, such as wave vectors.
