the frequency domain refers to the analysis of mathematicalfunctions or signals withrespect to frequency, rather than time. Put simply, a time-domain graph shows how a signalchanges over time, whereas a frequency-domain graph shows how much of thesignal lies within each given frequency band over a range of frequencies.
Afrequency-domain representation can also include information on the phase shift that must be applied toeach sinusoid in order to be able to recombinethe frequency components to recover the original time signal.A given function or signal can be converted between the time andfrequency domains with a pair of mathematical operatorscalleda transform.An example is the Fourier transform,which converts the time function into a sum of sine waves of differentfrequencies, each of which represents a frequency component. The ‘spectrum’ offrequency components is the frequency domain representation of the signal.The inverse Fouriertransform converts the frequency domain function back to a timefunction. A spectrum analyzer isthe tool commonly used to visualize real-world signals in the frequency domain.
a signal is described by a complexfunction offrequency: the component of the signal at any given frequency is given bya complex number. The magnitude of the number is the amplitude of that component, and the angleis the relative phase of the wave. For example, using the Fourier transforma sound wave, such as human speech, can be broken down into its componenttones of different frequencies, each represented by a sine wave of a differentamplitude and phase. The response of a system, as a function of frequency, canalso be described by a complex function. In many applications, phaseinformation is not important. By discarding the phase information it is possibleto simplify the information in a frequency domain representation to generatea frequency spectrum or spectraldensity.
The Fourier transform is an operation thatassociates one function of a real variable with another function of a realvariable. This new function describes the coefficients (“amplitudes”)in the decomposition of the original function into elementary components -harmonic oscillations with different frequencies (just as a musical chord canbe expressed as the sum of the musical sounds that make it up).The Fourier transform of a function f of a real variable is integral and isgiven by the following formula: In signal processing and related domains, theFourier transform is usually considered as a decomposition of a signal intofrequencies and amplitudes, that is, a reversible transition from a time domaininto a frequency domain. Rich application capabilities are based on severaluseful conversion properties:· Transformationsare linear operators and, with appropriate normalization, unitary (a propertyknown as the Parseval theorem, or, more generally, as a Plancherel theorem, or,more generally, as a Pontryagin dualism).
· Transformationsare reversible, and the inverse transformation has almost the same shape as thedirect transformation.· Sinusoidalbasis functions (rather, complex exponentials) are eigenfunctions ofdifferentiation, which means that this representation transforms lineardifferential equations with constant coefficients into ordinary algebraic ones.(For example, in a linear stationary system, the frequency is a conservativevalue, so the behavior at each frequency can be solved independently).· By theconvolution theorem, the Fourier transform transforms a complex convolutionoperation into simple multiplication, which means that they provide anefficient way of computing convolution-based operations such as polynomialmultiplication and multiplication of large numbers.· Adiscrete version of the Fourier transform can be quickly calculated oncomputers using the Fast Fourier Transform (FFT) algorithm.In the case of this problem, it is necessary touse a discrete Fourier transform.
The discrete Fourier transform is one of theFourier transforms widely used in algorithms of digital signal processing (itsmodifications are applied to the compression of sound in MP3, compression ofimages in JPEG, etc.), as well as in other areas related to the analysis offrequencies in a discrete example, digitized analog) signal. The discreteFourier transform requires a discrete function as an input. Such functions areoften created by sampling (sampling values ??from continuous functions).Discrete Fourier transforms help to solve partial differential equations andperform such operations as convolutions.
Discrete Fourier transforms are alsoactively used in statistics, when analyzing time series. Direct conversion: Inverse transformation: Notation:· N isthe number of signal values ??measured over a period, as well as the number ofdecomposition components;· xn,n = 0, ..
., N-1 are the measured signal values ??(in discrete time points withnumbers n = 0, …
, N-1), which are input data for direct conversion and outputfor reverse;· Xk,k = 0, …, N-1 – N complex amplitudes of the sinusoidal signals composing theoriginal signal; are output data for direct conversion and input for reverse;Since the amplitudes are complex, one can calculate both the amplitude and thephase from them simultaneously;· is the usual (real) amplitude of the kth sinusoidalsignal;· arg(Xk) is the phase of the k-th sinusoidal signal (theargument of a complex number);· k isthe frequency index. The frequency of the k-th signal is k/T, where T is thetime period during which the input data was taken.The latter shows that the transformationdecomposes the signal into sinusoidal components (which are called harmonics)with frequencies from N oscillations for a period up to one oscillation perperiod. Since the sampling rate itself is equal to N samples per period, thehigh-frequency components can not be correctly displayed – a moiré effectoccurs.
This leads to the fact that the second half of N complex amplitudes, infact, is a mirror image of the first and does not carry additional information.