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Time–frequency Analysis
In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various time–frequency representations. Rather than viewing a 1-dimensional signal (a function, real or complex-valued, whose domain is the real line) and some transform (another function whose domain is the real line, obtained from the original via some transform), time–frequency analysis studies a two-dimensional signal – a function whose domain is the two-dimensional real plane, obtained from the signal via a time–frequency transform. The mathematical motivation for this study is that functions and their transform representation are tightly connected, and they can be understood better by studying them jointly, as a two-dimensional object, rather than separately. A simple example is that the 4-fold periodicity of the Fourier transform – and the fact that two-fold Fourier transform reverses direction – can be interp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomography, seismic signals, Altimeter, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, Data storage, digital storage efficiency, correcting distorted signals, improve subjective video quality, and to detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was publis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner Distribution Function
The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis. The WDF was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, and it is of importance in quantum mechanics in phase space (see, by way of comparison: '' Wigner quasi-probability distribution'', also called the ''Wigner function'' or the ''Wigner–Ville distribution''). Given the shared algebraic structure between position-momentum and time-frequency conjugate pairs, it also usefully serves in signal processing, as a transform in time-frequency analysis, the subject of this article. Compared to a short-time Fourier transform, such as the Gabor transform, the Wigner distribution function provides the highest possible temporal vs frequency resolution which is mathematically possible within the limitations of the uncertainty principle. The downside is the introduction of large cross terms between eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instantaneous Frequency
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''complex-valued'' function ''s''(''t''), is the real-valued function: :\varphi(t) = \arg\, where arg is the complex argument function. The instantaneous frequency is the temporal rate of change of the instantaneous phase. And for a ''real-valued'' function ''s''(''t''), it is determined from the function's analytic representation, ''s''a(''t''): :\begin \varphi(t) &= \arg\ \\ pt &= \arg\, \end where \hat(t) represents the Hilbert transform of ''s''(''t''). When ''φ''(''t'') is constrained to its principal value, either the interval or , it is called ''wrapped phase''. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Fourier Transform
In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' need not be an integer — thus, it can transform a function to any ''intermediate'' domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Canonical Transform
In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(C) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space. The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group. The basic properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filter Design For White Noise
Filtration is a physical process that separates solid matter and fluid from a mixture. Filter, filtering, filters or filtration may also refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that performs some operation on a multimedia stream * Information filtering system ** Email filtering, the processing of email to organize it according to specified criteria * Content-control software also known as an Internet filter * Wordfilter, a script typically used on Internet forums or chat rooms * Berkeley Packet Filter, filter expression used in the qualification of network data * DSL filter, a low-pass filter installed between analog devices and a telephone line * Helicon Filter, a raster graphics editor * Filter (large eddy simulation), a mathematical operation intended to remove a range of small scales from the solution to the Nav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Random Process's WDF And AF
Stationary may refer to: * "Stationary", a song from ''Copacetic'' (Knuckle Puck album) * a common misspelling of "stationery Stationery refers to writing materials, including cut paper, envelopes, continuous form paper, and other office supplies. Stationery usually specifies materials to be written on by hand (e.g., letter paper) or by equipment such as computer p ...", meaning office supplies * stationary process, in mathematics and statistics, a process whose probability distribution does not change over time. See also * * {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Process
In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. More formally, the joint probability distribution of the process remains the same when shifted in time. This implies that the process is statistically consistent across different time periods. Because many statistical procedures in time series analysis assume stationarity, non-stationary data are frequently transformed to achieve stationarity before analysis. A common cause of non-stationarity is a trend in the mean, which can be due to either a unit root or a deterministic trend. In the case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. With a deterministic trend, the process is called trend-stationary, and shocks have only transitory effects, with the variable tending towards a determin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ambiguity Function
In pulsed radar and sonar signal processing, an ambiguity function is a two-dimensional function of propagation delay \tau and Doppler frequency f, \chi(\tau,f). It represents the distortion of a returned pulse due to the receiver matched filter (commonly, but not exclusively, used in pulse compression radar) of the return from a moving target. The ambiguity function is defined by the properties of the pulse and of the filter, and not any particular target scenario. Many definitions of the ambiguity function exist; some are restricted to narrowband signals and others are suitable to describe the delay and Doppler relationship of wideband signals. Often the definition of the ambiguity function is given as the magnitude squared of other definitions (WeissWeiss, Lora G. "Wavelets and Wideband Correlation Processing". ''IEEE Signal Processing Magazine'', pp. 13–32, Jan 1994). For a given complex baseband pulse s(t), the narrowband ambiguity function is given by :\chi(\tau,f)=\int_^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone-shape Distribution Function
The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution,Leon Cohen, Time Frequency Analysis: Theory and Applications, Prentice Hall, (1994) (acronymized as the ZAM distribution or ZAMD), is one of the members of Cohen's class distribution function. It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks II in 1990. The distribution's name stems from the twin cone shape of the distribution's kernel function on the t, \tau plane. The advantage of the cone kernel function is that it can completely remove the cross-term between two components having the same center frequency. Cross-term results from components with the same time center, however, cannot be completely removed by the cone-shaped kernel. Mathematical definition The definition of the cone-shape distribution function is: :C_x(t, f)=\int_^\int_^A_x(\eta,\tau)\Phi(\eta,\tau)\exp (j2\pi(\eta t-\tau f))\, d\eta\, d\tau, where :A_x(\eta,\tau)=\int_^x(t+\tau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time–frequency Representation
A time–frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. Time–frequency analysis means analysis into the time–frequency domain provided by a TFR. This is achieved by using a formulation often called "Time–Frequency Distribution", abbreviated as TFD. TFRs are often complex-valued fields over time and frequency, where the modulus of the field represents either amplitude or "energy density" (the concentration of the root mean square over time and frequency), and the argument of the field represents phase. Background and motivation A signal, as a function of time, may be considered as a representation with perfect ''time resolution''. In contrast, the magnitude of the Fourier transform (FT) of the signal may be considered as a representation with perfect ''spectral resolution'' but with no time information because the magnitude of the FT conveys frequency content but it fails to convey when, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
