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Electronic instruments called spectrum analyzers are used to observe and measure the power spectra of signals. The spectrum analyzer measures the magnitude of the short-time Fourier transform (STFT) of an input signal. If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral ...
A Fourier series (/ ˈ f ʊr i eɪ,-i ər / [1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are ...
SigSpec (acronym of SIGnificance SPECtrum) is a statistical technique to provide the reliability of periodicities in a measured (noisy and not necessarily equidistant) time series. [1] It relies on the amplitude spectrum obtained by the Discrete Fourier transform (DFT) and assigns a quantity called the spectral significance (frequently ...
Two power spectra (magnitude-squared) (rectangular and Hamming window functions plus background noise), calculated by the periodogram method. For sufficiently small values of parameter T, an arbitrarily-accurate approximation for X ( f ) can be observed in the region − 1 2 T < f < 1 2 T {\displaystyle -{\tfrac {1}{2T}}<f<{\tfrac {1}{2T}}} of ...
Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain. Recall that a Fourier series is only suitable to the analysis of periodic (or finite-domain) functions f(x) with period 2π. It can be expressed as an infinite series ...
An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (2 ed.). Ginn. p. 30. Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and Integrals, Volume 1 (2 ed.). Macmillan and Company. p. 196.
The modulus of the number is the amplitude of that component, and the argument is the relative phase of the wave. For example, using the Fourier transform, a sound wave, such as human speech, can be broken down into its component tones of different frequencies, each represented by a sine wave of a different amplitude and phase. The response of ...
The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum. The Fourier transform of C 3 (t 1,t 2) (third-order cumulant) is called bispectrum or bispectral density. They fall in the category of Higher Order Spectra, or Polyspectra and provide supplementary information to the power ...