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This is a list of [[linear transformation]]s of [[function (mathematics)|function]]s related to [[Fourier analysis]]. Such transformations [[Map (mathematics)|map]] a function to a set of [[coefficient]]s of [[basis function]]s, where the basis functions are [[trigonometric function|sinusoidal]] and are therefore strongly localized in the [[frequency spectrum]].  (These transforms are generally designed to be invertible.)  In the case of the Fourier transform, each basis function corresponds to a single [[frequency]] component.
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== Continuous transforms ==
Applied to functions of continuous arguments, Fourier-related transforms include:
* [[Two-sided Laplace transform]]
* [[Mellin transform]], another closely related integral transform
* [[Laplace transform]]
* [[Fourier transform]], with special cases''':'''
** [[Fourier series]]
*** When the input function/waveform is periodic, the Fourier transform output is a [[Dirac comb]] function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called '''Fourier series coefficients'''. The term '''Fourier series''' actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients.
*** When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed.  The same discrete set is obtained by treating the duration of the segment as one period of a periodic function and computing the Fourier series coefficients.
** [[Sine and cosine transforms]]''':'''  When the input function has odd or even symmetry around the origin, the Fourier transform reduces to a sine or cosine transform.
* [[Hartley transform]]
* [[Short-time Fourier transform]] (or short-term Fourier transform) (STFT)
* [[Chirplet transform]]
* [[Fractional Fourier transform]] (FRFT)
* [[Hankel transform]]: related to the Fourier Transform of radial functions.
 
== Discrete transforms ==
For usage on [[computer]]s, number theory and algebra, discrete arguments (e.g. functions of a series of discrete samples) are often more appropriate, and are handled by the transforms (analogous to the continuous cases above):
 
* [[Discrete-time Fourier transform]] (DTFT)''':''' Equivalent to the Fourier transform of a "continuous" function that is constructed from the discrete input function by using the sample values to modulate a [[Dirac comb]]. When the sample values are derived by sampling a function on the real line, ƒ(''x''), the DTFT is equivalent to a [[periodic summation]] of the Fourier transform of '''ƒ'''. The DTFT output is always [[Period (physics)|periodic]] (cyclic).  An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded (or ''finite''), the length of one cycle.
** [[discrete Fourier transform]] (DFT)''':'''
*** When the input sequence is periodic, the DTFT output is also a [[Dirac comb]] function, modulated by the coefficients of a Fourier series<ref>The Fourier series represents <math>\scriptstyle \sum_{n=-\infty}^{\infty}f(nT)\cdot \delta(t-nT),\,</math> where T is the interval between samples.</ref> which can be computed as a DFT of one cycle of the input sequence. The number of discrete values in one cycle of the DFT is the same as in one cycle of the input sequence.
*** When the non-zero portion of the input sequence has finite duration, the DTFT is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed. The same discrete set is obtained by treating the duration of the segment as one cycle of a periodic function and computing the DFT'''.
** Discrete [[sine and cosine transforms]]''':'''  When the input sequence has odd or even symmetry around the origin, the DTFT reduces to a [[discrete sine transform]] (DST) or [[discrete cosine transform]] (DCT).
***[[Regressive discrete Fourier series]], in which the period is determined by the data rather than fixed in advance.
**[[Discrete chebyshev transforms]] (on the 'roots' grid and the 'extrema' grid of the chebyshev polynomials of the first kind). This transform is of much importance in the field of spectral methods for solving differential equations because it can be used to swiftly and efficient go from grid point values to chebyshev series coefficients.
* Generalized DFT (GDFT), a generalization of the DFT and constant modulus transforms where phase functions might be of linear with integer and real valued slopes, or even non-linear phase bringing flexibilities for optimal designs of various metrics, e.g. auto- and cross-correlations.
* [[Z-transform]], a generalization of the DTFT.
* [[Modified discrete cosine transform]] (MDCT)
* [[Discrete Hartley transform]] (DHT)
* Also the discretized STFT (see above).
* [[Hadamard transform]] ([[Walsh function]]).
 
The use of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a [[fast Fourier transform]] (FFT). The [[Nyquist&ndash;Shannon sampling theorem]] is critical for understanding the output of such discrete transforms.
 
==Notes==
{{reflist}}
 
==See also==
* [[Integral transform]]
* [[Wavelet transform]]
* [[Fourier transform spectroscopy]]
* [[Harmonic analysis]]
* [[List of transforms]]
* [[List of operators]]
* [[Bispectrum]]
 
== References==
* A. D. Polyanin and A. V. Manzhirov, ''Handbook of Integral Equations'', CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
* A.N. Akansu and H. Agirman-Tosun, [http://web.njit.edu/~akansu/PAPERS/AkansuIEEE-TSP2010.pdf "''Generalized Discrete Fourier Transform With Nonlinear Phase''"], IEEE ''Transactions on Signal Processing'', vol. 58, no. 9, pp. 4547-4556, Sept. 2010.
[[Category:Fourier analysis| ]]
[[Category:Image processing]]
[[Category:Integral transforms|Fourier transforms]]
[[Category:Mathematics-related lists|Fourier transforms]]
 
[[de:Liste der Fourier-Transformationen]]

Revision as of 22:14, 27 February 2014

Nobody enjoys to be unemployed, especially when a great deal of effort is intended to gain employment. The information on this page will help you re-enter today's workforce.

When you can't find work, your searching strategy needs to be reworked. Just because lots of places aren't many jobs doesn't mean you need to give up. You will need to expand your work search area, but be certain it is possible to live in your community should you actually secure a task.

Have questions set up before your interviewer. You will frequently times be asked if you will find any queries.

A lot of places worldwide which are at the top can give their employees saunas, gyms or restaurants on-site.This will likely increase competition and therefore makes other jobs more profitable.This will give the employer the opportunity to recruit from a greater pool of potential candidates from which to choose.

Try to indicate approximately your job early. You never know what could end up hitting traffic and it also will give you an opportunity to talk with the prior shift. This will allow you to be by the due date each time, a trait which is valuable yet surprisingly difficult to get.

Remember that some companies value making income. When you are get yourself ready for an interview, search for approaches to convince them that you just improves their profits together with your skills and talents.

Technology and business are usually changing. Take classes and even seminars if you can.This information could just give you become a more inviting employee over time.

Understand that a lot of companies value making income. When you find yourself ready for an interview and polish your resume, try and figure out how to word your individual sales pitch of methods your talents can boost your prospective employer's revenue.

Set an ordinary schedule with your job. Your employer will appreciate your consistency of a well thought out schedule. They will trust you a lot more when they are conscious of what to expect. Be very specific together with your working hours and lunch time hours. If the needs to be adjusted, you must engage with your boss.

Create a consistent work which you as well as others can trust. Your employer will appreciate your consistency of any well planned out schedule. They may trust you plenty more when they're aware of what to expect. Be specific with the work hours and lunch time hours. If you've have got to adjust, let your manager know.

Consider networking in your desired field. Networking uses techniques that can assist you track of opportunities that happen to be otherwise unavailable. Try becoming immersed within your field by attending webinars, industry networking events, seminars, and industry networking events. Network as much as is possible to find out much more.

Send your resume to your company that you might want to finish up. Right after the initial contact, make sure you send a monthly follow-around question openings. You can even go in person. They'll remember you from the persistence and may even consider you an interview ahead of the job is posted.

Ensure your reference letters under control. So many people are praoclaiming that they're ready with their references, however it is smarter to have the reference letters with you. This will impress the interviewer from having to hunt down your references since they keep these things right looking at their eyes.

Research any business you happen to be applying with before your interview. Have a look at their website, Twitter, or Facebook page. Learn anything you can about them. This data can make you stay ahead of another candidates and show you're sincere with your interest is sincere.

A wise idea for self-employed individuals is to maintain records of your respective purchases.Keep all receipts as some items might be written off on the taxes.Organization could keep your financial situation.

Tend not to interview without learning about the corporation. Have you any idea what their mission statement? It is possible to separate yourself from your pack by demonstrating the initiative to discover whenever you can about them.

Since we stated earlier, the economy is slow and unemployment has risen. It is not necessarily easy to find the job you need and to survive during difficult times. Study the recommendations and tips presented here and placed them into daily use. You can expect to soon be better off with regards to finding and securing your ideal job.

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