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Time series. In mathematics, a time series is a series of data points indexed (or listed or graphed) 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 ...
Motivation. In signal processing, time–frequency analysis [3] is a body of techniques and methods used for characterizing and manipulating signals whose statistics vary in time, such as transient signals. It is a generalization and refinement of Fourier analysis, for the case when the signal frequency characteristics are varying with time.
Fourier transforms. 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 ...
The component frequencies, spread across the frequency spectrum, are represented as peaks in the frequency domain. Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers ...
In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time series. [1] Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency ...
A time–frequency representation ( TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. [1] 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.
Decomposition of time series. The decomposition of time series is a statistical task that deconstructs a time series into several components, each representing one of the underlying categories of patterns. [1] There are two principal types of decomposition, which are outlined below.
The (potentially time-dependent) autocorrelation matrix (also called second moment) of a (potentially time-dependent) random vector is an matrix containing as elements the autocorrelations of all pairs of elements of the random vector . The autocorrelation matrix is used in various digital signal processing algorithms.