- by Paolo Prandoni and Martin Vetterli
- Fourier analysis on finite groups, applications in signal processing and system design
- Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials

In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note.

One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations. The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis.

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Each transform used for analysis see list of Fourier-related transforms has a corresponding inverse transform that can be used for synthesis. Fourier analysis has many scientific applications — in physics , partial differential equations , number theory , combinatorics , signal processing , digital image processing , probability theory , statistics , forensics , option pricing , cryptography , numerical analysis , acoustics , oceanography , sonar , optics , diffraction , geometry , protein structure analysis, and other areas.

In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum.

The FT method is used to decode the measured signals and record the wavelength data. And by using a computer, these Fourier calculations are rapidly carried out, so that in a matter of seconds, a computer-operated FT-IR instrument can produce an infrared absorption pattern comparable to that of a prism instrument. Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses a variant of the Fourier transformation discrete cosine transform of small square pieces of a digital image.

The Fourier components of each square are rounded to lower arithmetic precision , and weak components are eliminated entirely, so that the remaining components can be stored very compactly. In image reconstruction, each image square is reassembled from the preserved approximate Fourier-transformed components, which are then inverse-transformed to produce an approximation of the original image.

When processing signals, such as audio , radio waves , light waves, seismic waves , and even images, Fourier analysis can isolate narrowband components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation. Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, and it produces a continuous function of frequency, known as a frequency distribution.

One function is transformed into another, and the operation is reversible. When the domain of the input initial function is time t , and the domain of the output final function is ordinary frequency , the transform of function s t at frequency f is given by the complex number:. Evaluating this quantity for all values of f produces the frequency-domain function. Then s t can be represented as a recombination of complex exponentials of all possible frequencies:.

See Fourier transform for much more information, including:. The Fourier transform of a periodic function, s P t , with period P , becomes a Dirac comb function, modulated by a sequence of complex coefficients :. The inverse transform, known as Fourier series , is a representation of s P t in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients:.

When s P t , is expressed as a periodic summation of another function, s t :. See Fourier series for more information, including the historical development. Thus, a convergent periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:.

Parameter T corresponds to the sampling interval, and this Fourier series can now be recognized as a form of the Poisson summation formula. That is a cornerstone in the foundation of digital signal processing. Applications of the DTFT are not limited to sampled functions.

See Discrete-time Fourier transform for more information on this and other topics, including:. It is also N -periodic, so it is never necessary to compute more than N coefficients. The inverse transform is given by:. When s N [ n ] is expressed as a periodic summation of another function:.

### by Paolo Prandoni and Martin Vetterli

In most cases, N is chosen equal to the length of non-zero portion of s [ n ]. Decreasing N , causes overlap adding in the time-domain analogous to aliasing , which corresponds to decimation in the frequency domain. The DFT can be computed using a fast Fourier transform FFT algorithm, which makes it a practical and important transformation on computers. See Discrete Fourier transform for much more information, including:.

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components Fourier series , and the transforms diverge at those frequencies. One common practice not discussed above is to handle that divergence via Dirac delta and Dirac comb functions. But the same spectral information can be discerned from just one cycle of the periodic function, since all the other cycles are identical. Similarly, finite-duration functions can be represented as a Fourier series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact.

But these formulas do not require that condition.

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## Fourier analysis on finite groups, applications in signal processing and system design

When the real and imaginary parts of a complex function are decomposed into their even and odd parts , there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform : [4]. The Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact Abelian topological groups , which are studied in harmonic analysis ; there, the Fourier transform takes functions on a group to functions on the dual group.

This treatment also allows a general formulation of the convolution theorem , which relates Fourier transforms and convolutions. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform.

More specific, Fourier analysis can be done on cosets, [5] even discrete cosets. In signal processing terms, a function of time is a representation of a signal with perfect time resolution , but no frequency information, while the Fourier transform has perfect frequency resolution , but no time information.

As alternatives to the Fourier transform, in time—frequency analysis , one uses time—frequency transforms to represent signals in a form that has some time information and some frequency information — by the uncertainty principle , there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform , the Gabor transform or fractional Fourier transform FRFT , or can use different functions to represent signals, as in wavelet transforms and chirplet transforms , with the wavelet analog of the continuous Fourier transform being the continuous wavelet transform.

A primitive form of harmonic series dates back to ancient Babylonian mathematics , where they were used to compute ephemerides tables of astronomical positions. The classical Greek concepts of deferent and epicycle in the Ptolemaic system of astronomy were related to Fourier series see Deferent and epicycle: Mathematical formalism. In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in to compute an orbit, [10] which has been described as the first formula for the DFT, [11] and in by Joseph Louis Lagrange , in computing the coefficients of a trigonometric series for a vibrating string.

Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: Daniel Bernoulli and Leonhard Euler had introduced trigonometric representations of functions, [11] and Lagrange had given the Fourier series solution to the wave equation, [11] so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.

The subsequent development of the field is known as harmonic analysis , and is also an early instance of representation theory. In signal processing , the Fourier transform often takes a time series or a function of continuous time , and maps it into a frequency spectrum. Inbunden Engelska, Spara som favorit. Skickas inom vardagar.

## Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials

Laddas ned direkt. This book lets you discover applications of Fourier analysis on finite non Abelian groups. The majority of publications in spectral techniques consider Fourier transform on Abelian groups. However, non Abelian groups provide notable advantages in efficient implementations of spectral methods.

Switching functions are included as an example of discrete functions in engineering practice.