Credible interval - Revision history

en>Terrycojones: /* Choosing a credible interval */ Fixed typo in book title.

2014-12-14T00:16:24Z

Choosing a credible interval: Fixed typo in book title.

← Older revision		Revision as of 02:16, 14 December 2014
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	~~Jayson Berryhill~~ is ~~how I'm~~ known as and ~~my spouse doesn't like it at all~~. ~~Credit authorising~~ is ~~where my primary earnings arrives from~~. ~~What me~~ and ~~my family members love is performing ballet but~~ I~~'ve been using on new issues recently. For a while I've been~~ in Mississippi ~~but now~~ I~~'m considering other choices~~.<br><br>my ~~blog - phone~~ psychic ~~readings~~ ([http://~~www~~.~~onbizin~~.co.kr/xe/~~?document_srl=320614~~ Suggested ~~Online site~~])		The person who wrote the article is known as Jayson Hirano and he completely digs that name. Playing badminton is a thing that he is totally addicted to. My spouse and I live in Mississippi and I love every working day living right here. Invoicing is what I do.<br><br>Stop by my weblog online psychic ([http://Www.octionx.sinfauganda.co.ug/node/22469 Suggested Internet page])

en>Wra2: /* Contrasts with confidence interval */ Added that Antelman's confidence interval is a 95% confidence interval.

2014-03-01T21:18:57Z

Contrasts with confidence interval: Added that Antelman's confidence interval is a 95% confidence interval.

← Older revision		Revision as of 23:18, 1 March 2014
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	~~In [[mathematics]], [[computer science]], and [[electrical engineering]], the '''discrete Fourier transform (DFT)''', occasionally called the [[finite Fourier transform]],~~ is a transform for [[Fourier analysis]] of finite-domain [[discrete-time signal]]s. As with most Fourier analysis, it expresses an input function in terms of a sum of sinusoidal components by determining the amplitude and phase of each component. Unlike the [[Fourier transform]], which operates upon continuous functions assumed to extend to infinity, the DFT operates upon ''discrete'' and '~~'finite'' sets of values: the input to the DFT is a finite sequence of [[real number\|real]] or [[complex number]]s, which makes the DFT ideal for processing information stored in [[computer]]s~~. ~~In particular, the DFT~~ is widely employed in [[Digital signal processing\|signal processing]] and related fields to analyze the frequencies contained in a sampled [[signal (information theory)\|signal]], to solve [[partial differential equations]], and to perform other operations such as [[convolution]]s.		Jayson Berryhill is how I'm known as and my spouse doesn't like it at all. Credit authorising is where my primary earnings arrives from. What me and my family members love is performing ballet but I've been using on new issues recently. For a while I've been in Mississippi but now I'm considering other choices.<br><br>my blog - phone psychic readings ([http://www.onbizin.co.kr/xe/?document_srl=320614 Suggested Online site])

	~~The article [[discrete Fourier transform]] presents the definition of the transform, without derivation, as''':'''~~

	~~{{NumBlk\|:\|<math>X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-i \frac{2 \pi}{N} k n} \quad \quad k = 0, \dots, N-1</math>~~
	~~\|{{EquationRef\|Eq~~.~~1}}}}~~

	~~Here we take the view that the DFT is motivated by a desire to study continuous functions or waveforms~~ and ~~their continuous Fourier transforms using only a finite amount of data.~~
	When the sequence {''x''[''n'']} represents a subset of the samples of a waveform ''x''(''t''), we can model the process that created {''x''[''n'']} as applying a [[window function]] to ''x''(''t''), followed by sampling (or vice versa). It is ~~instructive to envision how those operations affect our ability to observe the Fourier transform, ~~'~~'X''(''&fnof;''). The window function widens every frequency component of ''X''(''&fnof;'') in a way that depends~~ on ~~the type of window used~~. ~~That effect is called [[spectral leakage]]. We can think of it as causing ''X''(''&fnof;'') to blur'''...''' thus~~ a ~~loss of resolution. The sampling operation causes the Fourier transform to become periodic. More precisely, what happens is that {~~''~~x''[''n'']} has no Fourier transform. It is undefined~~. But using the [[Poisson summation formula]] a periodic function of continuous frequency can be constructed from the samples, and it comprises copies of the blurred ''X''(''&fnof;'') repeated at regular multiples of the sampling frequency (''F''<sub>''s''<~~/sub~~> ~~= ''1''/''T'') and summed together where they overlap (called [[periodic summation]])''':'''~~

	~~{{NumBlk\|:\|~~<~~math~~>~~X_{1/T}(f)\ \stackrel{\text{def}}{=}\ \sum_{k=~~-~~\infty}^{\infty} X~~(~~f-k/T)= \sum_{n=-\infty}^{\infty} \underbrace{T\cdot x(nT)}_{x~~[~~n]}\cdot e^{-i 2\pi f nT}</math>    <ref group="note">With this definition'''~~:~~'''  <math>\textstyle{\omega = {2 \pi f \over F_s} = 2 \pi f T,}<~~/~~math>  the right-hand side of {{EquationNote\|Eq.2}} becomes <math>\sum_{n=-\infty}^{\infty} x[n]\cdot e^{-i \omega n},<~~/~~math>  which is a normalized-frequency form of the [[discrete-time Fourier transform]].</ref>~~
	~~\|{{EquationRef\|Eq.2}}}}~~

	The copies are [[aliasing\|aliases]] of the original frequency components. In particular, due to the overlap, aliases can significantly distort the region containing the original ''X''(''&fnof;'') (if ''F''<sub>''s''</sub> is not sufficiently large enough to prevent it). ~~ But if the windowing and sampling are done with sufficient care, the Poisson summation still contains a reasonable semblance of ''X''(''&fnof;'')~~. ~~It is therefore a common practice to compute an arbitrary number of samples (N) of one cycle of the periodic function <math>X_{1/T}</math> ''':'''~~

	~~:<math>\underbrace{X_{1/T}\left(\frac{k}{NT}\right)}_{X[k]} = \sum_{n=-\infty}^{\infty} x[n]\cdot e^{-i 2\pi \frac{kn}{N}} \quad \quad k = 0, \dots, N-1</math>~~

	~~Since the kernel, <math>e^{-i 2\pi \frac{kn}{N}}\,</math>  is N-periodic, it can readily be shown that this is equivalent to the following '''[[discrete Fourier transform\|DFT]]:'''~~

	~~{{NumBlk\|:\|<math>X[k] = \sum_{N} x_N[n]\cdot e^{-i 2\pi \frac{kn}{N}},</math>~~
	~~\|{{EquationRef\|Eq~~.~~3}}}}~~

	~~where <math>\textstyle{\sum_{N}}<~~/~~math> is a summation over any interval of length N, and <math>x_N\,<~~/~~math> is another [[periodic summation]]''':'''~~

	~~:<math>x_N[n]\ \stackrel{\text{def}}{~~=~~}\ \sum_{m=-\infty}^{\infty} x[n-mN~~]~~.</math>~~

	~~{{EquationNote\|Eq.1}} (the standard DFT~~) is just a simplification of {{EquationNote\|Eq.3}} when the x[n] sequence is zero outside the interval [0, N-1].<ref group="note">The discussion of longer sequences can be found at [[Discrete-time_Fourier_transform#Sampling_the_DTFT\|Sampling the DTFT]].</ref> But regardless of the duration of the x[n] sequence, the inverse DFT produces the periodic <math>x_N\,</math> sequence. That can be thought of as a consequence of substituting a discrete set of frequencies for the continuous <math>X_{1/T}</math>.

	~~==Notes==~~
	~~{{reflist\|group=note}}~~

	~~[[Category:Fourier analysis]]~~
	~~[[Category:Digital signal processing]]~~

en>ClueBot NG: Reverting possible vandalism by 216.129.226.194 to version by Addbot. False positive? Report it. Thanks, ClueBot NG. (1612614) (Bot)

2013-12-10T16:39:09Z

Reverting possible vandalism by 216.129.226.194 to version by Addbot. False positive? Report it. Thanks, ClueBot NG. (1612614) (Bot)

← Older revision		Revision as of 18:39, 10 December 2013
Line 1:		Line 1:
	~~Alyson Meagher~~ is the ~~name her mothers~~ and ~~fathers gave her but she doesn~~'t ~~like when people use her complete name~~. I'~~ve always cherished residing~~ in ~~Alaska~~. He is ~~an info officer~~. ~~What me and my family members adore~~ is ~~doing ballet but I~~'~~ve been taking on new things lately~~.<br><br>~~Also visit my blog love psychics~~ ([~~http~~://~~ustanford~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~38218~~/~~info~~/ ~~http:~~//~~ustanford~~.~~com~~/~~index~~.~~php?do~~=/~~profile-38218~~/~~info~~/])		In [[mathematics]], [[computer science]], and [[electrical engineering]], the '''discrete Fourier transform (DFT)''', occasionally called the [[finite Fourier transform]], is a transform for [[Fourier analysis]] of finite-domain [[discrete-time signal]]s. As with most Fourier analysis, it expresses an input function in terms of a sum of sinusoidal components by determining the amplitude and phase of each component. Unlike the [[Fourier transform]], which operates upon continuous functions assumed to extend to infinity, the DFT operates upon ''discrete'' and ''finite'' sets of values: the input to the DFT is a finite sequence of [[real number\|real]] or [[complex number]]s, which makes the DFT ideal for processing information stored in [[computer]]s. In particular, the DFT is widely employed in [[Digital signal processing\|signal processing]] and related fields to analyze the frequencies contained in a sampled [[signal (information theory)\|signal]], to solve [[partial differential equations]], and to perform other operations such as [[convolution]]s.

			The article [[discrete Fourier transform]] presents the definition of the transform, without derivation, as''':'''

			{{NumBlk\|:\|<math>X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-i \frac{2 \pi}{N} k n} \quad \quad k = 0, \dots, N-1</math>
			\|{{EquationRef\|Eq.1}}}}

			Here we take the view that the DFT is motivated by a desire to study continuous functions or waveforms and their continuous Fourier transforms using only a finite amount of data.
			When the sequence {''x''[''n'']} represents a subset of the samples of a waveform ''x''(''t''), we can model the process that created {''x''[''n'']} as applying a [[window function]] to ''x''(''t''), followed by sampling (or vice versa). It is instructive to envision how those operations affect our ability to observe the Fourier transform, ''X''(''&fnof;''). The window function widens every frequency component of ''X''(''&fnof;'') in a way that depends on the type of window used. That effect is called [[spectral leakage]]. We can think of it as causing ''X''(''&fnof;'') to blur'''...''' thus a loss of resolution. The sampling operation causes the Fourier transform to become periodic. More precisely, what happens is that {''x''[''n'']} has no Fourier transform. It is undefined. But using the [[Poisson summation formula]] a periodic function of continuous frequency can be constructed from the samples, and it comprises copies of the blurred ''X''(''&fnof;'') repeated at regular multiples of the sampling frequency (''F''<sub>''s''</sub> = ''1''/''T'') and summed together where they overlap (called [[periodic summation]])''':'''

			{{NumBlk\|:\|<math>X_{1/T}(f)\ \stackrel{\text{def}}{=}\ \sum_{k=-\infty}^{\infty} X(f-k/T)= \sum_{n=-\infty}^{\infty} \underbrace{T\cdot x(nT)}_{x[n]}\cdot e^{-i 2\pi f nT}</math>    <ref group="note">With this definition''':'''  <math>\textstyle{\omega = {2 \pi f \over F_s} = 2 \pi f T,}</math>  the right-hand side of {{EquationNote\|Eq.2}} becomes <math>\sum_{n=-\infty}^{\infty} x[n]\cdot e^{-i \omega n},</math>  which is a normalized-frequency form of the [[discrete-time Fourier transform]].</ref>
			\|{{EquationRef\|Eq.2}}}}

			The copies are [[aliasing\|aliases]] of the original frequency components. In particular, due to the overlap, aliases can significantly distort the region containing the original ''X''(''&fnof;'') (if ''F''<sub>''s''</sub> is not sufficiently large enough to prevent it).  But if the windowing and sampling are done with sufficient care, the Poisson summation still contains a reasonable semblance of ''X''(''&fnof;''). It is therefore a common practice to compute an arbitrary number of samples (N) of one cycle of the periodic function <math>X_{1/T}</math> ''':'''

			:<math>\underbrace{X_{1/T}\left(\frac{k}{NT}\right)}_{X[k]} = \sum_{n=-\infty}^{\infty} x[n]\cdot e^{-i 2\pi \frac{kn}{N}} \quad \quad k = 0, \dots, N-1</math>

			Since the kernel, <math>e^{-i 2\pi \frac{kn}{N}}\,</math>  is N-periodic, it can readily be shown that this is equivalent to the following '''[[discrete Fourier transform\|DFT]]:'''

			{{NumBlk\|:\|<math>X[k] = \sum_{N} x_N[n]\cdot e^{-i 2\pi \frac{kn}{N}},</math>
			\|{{EquationRef\|Eq.3}}}}

			where <math>\textstyle{\sum_{N}}</math> is a summation over any interval of length N, and <math>x_N\,</math> is another [[periodic summation]]''':'''

			:<math>x_N[n]\ \stackrel{\text{def}}{=}\ \sum_{m=-\infty}^{\infty} x[n-mN].</math>

			{{EquationNote\|Eq.1}} (the standard DFT) is just a simplification of {{EquationNote\|Eq.3}} when the x[n] sequence is zero outside the interval [0, N-1].<ref group="note">The discussion of longer sequences can be found at [[Discrete-time_Fourier_transform#Sampling_the_DTFT\|Sampling the DTFT]].</ref> But regardless of the duration of the x[n] sequence, the inverse DFT produces the periodic <math>x_N\,</math> sequence. That can be thought of as a consequence of substituting a discrete set of frequencies for the continuous <math>X_{1/T}</math>.

			==Notes==
			{{reflist\|group=note}}

			[[Category:Fourier analysis]]
			[[Category:Digital signal processing]]

en>Cgranade: Changed name of journal for ref [1]; original citation was to wrong journal.

2012-08-21T16:06:41Z

Changed name of journal for ref [1]; original citation was to wrong journal.

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Alyson Meagher is the name her mothers and fathers gave her but she doesn't like when people use her complete name. I've always cherished residing in Alaska. He is an info officer. What me and my family members adore is doing ballet but I've been taking on new things lately.<br><br>Also visit my blog love psychics ([http://ustanford.com/index.php?do=/profile-38218/info/ http://ustanford.com/index.php?do=/profile-38218/info/])