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In the physical sciences, the spectrum of a physical quantity (such as energy) may be called continuous if it is non-zero over the whole spectrum domain (such as frequency or wavelength) or discrete if it attains non-zero values only in a discrete set over the independent variable, with band gaps between pairs of spectral bands or spectral lines.
The discrete-time Fourier transform, on the other hand, maps functions with discrete time (discrete-time signals) to functions that have a continuous frequency domain. [2] [3] A periodic signal has energy only at a base frequency and its harmonics; thus it can be analyzed using a discrete frequency domain. A discrete-time signal gives rise to a ...
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number is said to be in the spectrum of a bounded linear operator if. or the set-theoretic inverse is either ...
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete 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.
Discrete spectrum (mathematics) (Redirected from Discrete spectrum (Mathematics)) In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite.
The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and ...
Spectral leakage. The Fourier transform of a function of time, s (t), is a complex-valued function of frequency, S (f), often referred to as a frequency spectrum. Any linear time-invariant operation on s (t) produces a new spectrum of the form H (f)•S (f), which changes the relative magnitudes and/or angles (phase) of the non-zero values of S ...
Otherwise it is called unwrapped phase, which is a continuous function of argument t, assuming s a (t) is a continuous function of t. Unless otherwise indicated, the continuous form should be inferred. Instantaneous phase vs time. The function has two true discontinuities of 180° at times 21 and 59, indicative of amplitude zero-crossings.