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en>SporkBot: Repair duplicate template args

2014-12-22T17:21:33Z

144.82.165.186 at 15:18, 11 February 2014

2014-02-11T15:18:59Z

← Older revision		Revision as of 17:18, 11 February 2014
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	~~[[File:bandlimited.svg\|thumb\|Spectrum~~ of a ~~'''bandlimited [[baseband]] signal''' as~~ a ~~function of frequency\|right\|300px]]~~
	~~'''Bandlimiting''' is the limiting~~ of a ~~[[deterministic]] or [[stochastic]] signal~~'s ~~[[Fourier transform]] or [[power spectral density]]~~ to ~~zero above~~ a ~~certain finite [[frequency]]. In other words, if the [[Fourier transform]]~~ or ~~[[power spectral density]] has finite [[support (mathematics)\|support]] then the signal is said to be bandlimited.~~

	~~==Sampling bandlimited signals==~~

	~~A bandlimited signal can be fully reconstructed from its samples, provided that the sampling rate exceeds twice the maximum frequency~~ in ~~the bandlimited signal. This minimum sampling frequency is called the [[Nyquist rate]]. This result~~, ~~usually attributed to [[Harry Nyquist\|Nyquist]]~~ and ~~[[Claude E~~. ~~Shannon\|Shannon]], is known as~~ the [~~[Nyquist–Shannon sampling theorem]]~~.

	~~An example of a simple deterministic bandlimited signal is a [[Sine wave\|sinusoid]] of the form <math>x(t) = \sin(2 \pi ft + \theta) \ </math>~~. ~~If this signal is sampled at a rate <math>f_s =\frac{1}{T} > 2f </math> so that we have the samples <math>x(nT) \ <~~/~~math>, for all integers <math>n<~~/~~math>, we can recover <math>x(t) \ <~~/~~math> completely from these samples~~. ~~Similarly, sums of sinusoids with different frequencies and phases are also bandlimited to the highest of their frequencies.~~

	~~The signal whose Fourier transform~~ is ~~shown~~ in the ~~figure~~ is ~~also bandlimited~~. ~~Suppose~~ <~~math~~>~~x(t)\~~ <~~/math~~> ~~is a signal whose Fourier transform is <math>X(f)\ </math>,~~ the ~~magnitude~~ of ~~which is shown in the figure. The highest frequency component in <math>x~~(~~t)\ <~~/~~math> is <math>B \ <~~/~~math>~~. ~~As a result, the Nyquist rate is~~

	:~~<math> R_N = 2B \, <~~/~~math>~~

	~~or twice the highest frequency component in the signal, as shown in the figure~~. ~~According to the sampling theorem, it is possible to reconstruct <math>x(t)\ <~~/~~math> completely and exactly using the samples~~

	~~:<math> x[n~~] ~~\ \stackrel{\mathrm{def}}{=}\ x(nT) = x \left( { n \over f_s } \right~~) ~~</math> for all integers <math>n \~~, ~~</math> and <math>T \ \stackrel{\mathrm{def}}{=}\ { 1 \over f_s } </math>~~

	~~as long as~~

	~~:<math>f_s > R_N \, </math>~~

	~~The reconstruction of a signal~~ from ~~its samples can be accomplished using the [[Whittaker–Shannon interpolation formula]]~~.

	~~==Bandlimited versus timelimited==~~
	~~{{redirect\|Timelimit\|narrow field of time by which an objective or task must be accomplished\|Time limit}}~~
	~~{{Elucidate\|date=June 2013}}~~
	~~A bandlimited signal cannot be also timelimited. More precisely,~~ a ~~function and its Fourier transform cannot both have finite [[support (mathematics)\|support]]~~. This ~~fact can be proved by using complex analysis and properties of Fourier transform.~~

	'~~''Proof:''' Assume that a signal f(t) which has finite support~~ in ~~both domains exists. Let~~'s ~~sample it faster than the Nyquist frequency, and compute respective [[Fourier transform]] <math> FT(f) = F_1(w) </math> and [[discrete-time fourier transform]] <math> DTFT(f) = F_2(w)</math>~~. ~~According to properties of DTFT, <math> F_2(w) = \sum_{n=~~-~~\infty}^{+\infty} F_1(w+n f_x~~) ~~</math>, where <math>f_x</math> is the frequency used for discretization~~. ~~If f~~ is ~~bandlimited, <math> F_1 </math> is zero outside~~ of ~~a certain interval~~, ~~so with large enough <math> f_x </math>, <math> F_2 </math>~~ will ~~be zero in some intervals~~ too~~, since individual [[Support_(mathematics)\|supports]]~~ of ~~<math> F_1 </math> in sum~~ of ~~<math> F_2 </math> won't overlap~~. ~~According to DTFT definition,~~ <~~math~~> ~~F_2~~ <~~/math~~> is a ~~sum of trigonometric functions,~~ and ~~since f(t) is time-limited~~, ~~this sum~~ will be ~~finite~~, ~~so <math> F_2 </math> will be actually a [[trigonometric polynomial]]~~. ~~All trigonometric polynomials~~ are ~~[[Entire function\|holomorphic on a whole complex plane]]~~, ~~and there~~ is a ~~simple theorem in complex analysis that says that [[Zero_(complex_analysis)\|all zeros of non-constant holomorphic function are isolated]]~~. ~~But this contradicts our earlier finding that~~ <~~math~~> ~~F_2~~ <~~/math~~> has ~~intervals full of zeros~~, ~~because points~~ in ~~such intervals are not isolated~~. ~~Thus the only time- and bandwidth-limited signal~~ is a ~~constant zero~~.

	~~One important consequence~~ of ~~this result is that it is impossible to generate~~ a ~~truly bandlimited signal in~~ any ~~real~~-~~world situation~~, ~~because~~ a ~~bandlimited signal would require infinite time~~ to ~~transmit~~. ~~All real~~-~~world signals are~~, ~~by necessity~~, ~~''timelimited''~~, ~~which means that they ''cannot'' be bandlimited~~. ~~Nevertheless~~, the ~~concept of a bandlimited signal is a useful idealization for theoretical and analytical purposes~~. ~~Furthermore, it~~ is ~~possible to approximate a bandlimited signal to any arbitrary level~~ of ~~accuracy desired.~~

	~~A similar relationship between duration in time and bandwidth in frequency also forms~~ the ~~mathematical basis for the [[uncertainty principle]] in [[quantum mechanics]]~~. ~~In that setting~~, the ~~"width"~~ of ~~the time domain and frequency domain functions are evaluated~~ with ~~a [[variance]]-like measure. Quantitatively, the uncertainty principle imposes the following condition on any real waveform:~~

	~~:<math> W_B T_D \ge 1 </math>~~

	where

	~~:<math>W_B</math> is~~ a ~~(suitably chosen) measure of bandwidth (in hertz),~~ and

	~~:<math>T_D</math> is a (suitably chosen) measure of time duration (~~in ~~seconds)~~.

	~~In [[time–frequency analysis]], these limits are known as~~ the ~~''[[Gabor limit]],''~~ and ~~are interpreted~~ as a ~~limit~~ on ~~the ''simultaneous'' time–frequency resolution one may achieve~~.

	~~==References==~~

	~~{{Refimprove\|date=January 2011}}~~
	~~{{No footnotes\|date=January 2011}}~~

	~~{{cite book \| author = William McC. Siebert \| title = Circuits, Signals, and Systems \| year = 1986 \| location = Cambridge, MA \| publisher = MIT Press }}~~

	~~==See also==~~
	*[[Bandwidth (signal processing)\|Bandwidth]]
	*[[Nyquist–Shannon sampling theorem]]
	*[[Nyquist rate]]
	*[[Nyquist frequency]]


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

en>Slawekb: Restored article (see discussion page). Reorganized material on sublevel sets. Removed excessive boldface.

2013-12-05T16:56:23Z

Restored article (see discussion page). Reorganized material on sublevel sets. Removed excessive boldface.

← Older revision		Revision as of 18:56, 5 December 2013
Line 1:		Line 1:
	~~Dalton is~~ [~~http~~:~~//www~~.~~britannica.com/search?query=what%27s+written what~~'~~s written~~] ~~by his birth certificate but he never really liked that name. The most-loved hobby for him plus his kids is farming but he~~'~~s been bringing on new things~~ as of ~~late. Auditing~~ is ~~where his primary income originates from. Massachusetts is where~~ the ~~doctor and his wife survive. He~~'s ~~not godd at design but consider want~~ to ~~check its website: http://prometeu~~.~~net<br><br>~~		{{technical\|date=January 2013}}
			[[File:bandlimited.svg\|thumb\|Spectrum of a '''bandlimited [[baseband]] signal''' as a function of frequency\|right\|300px]]
			'''Bandlimiting''' is the limiting of a [[deterministic]] or [[stochastic]] signal's [[Fourier transform]] or [[power spectral density]] to zero above a certain finite [[frequency]]. In other words, if the [[Fourier transform]] or [[power spectral density]] has finite [[support (mathematics)\|support]] then the signal is said to be bandlimited.

	~~Feel free~~ to ~~surf~~ to ~~my page~~ ... ~~clash~~ of ~~clans cheat~~ ([~~http~~://~~prometeu~~.~~net mouse click~~ the ~~next web site~~])		==Sampling bandlimited signals==

			A bandlimited signal can be fully reconstructed from its samples, provided that the sampling rate exceeds twice the maximum frequency in the bandlimited signal. This minimum sampling frequency is called the [[Nyquist rate]]. This result, usually attributed to [[Harry Nyquist\|Nyquist]] and [[Claude E. Shannon\|Shannon]], is known as the [[Nyquist–Shannon sampling theorem]].

			An example of a simple deterministic bandlimited signal is a [[Sine wave\|sinusoid]] of the form <math>x(t) = \sin(2 \pi ft + \theta) \ </math>. If this signal is sampled at a rate <math>f_s =\frac{1}{T} > 2f </math> so that we have the samples <math>x(nT) \ </math>, for all integers <math>n</math>, we can recover <math>x(t) \ </math> completely from these samples. Similarly, sums of sinusoids with different frequencies and phases are also bandlimited to the highest of their frequencies.

			The signal whose Fourier transform is shown in the figure is also bandlimited. Suppose <math>x(t)\ </math> is a signal whose Fourier transform is <math>X(f)\ </math>, the magnitude of which is shown in the figure. The highest frequency component in <math>x(t)\ </math> is <math>B \ </math>. As a result, the Nyquist rate is

			:<math> R_N = 2B \, </math>

			or twice the highest frequency component in the signal, as shown in the figure. According to the sampling theorem, it is possible to reconstruct <math>x(t)\ </math> completely and exactly using the samples

			:<math> x[n] \ \stackrel{\mathrm{def}}{=}\ x(nT) = x \left( { n \over f_s } \right) </math> for all integers <math>n \, </math> and <math>T \ \stackrel{\mathrm{def}}{=}\ { 1 \over f_s } </math>

			as long as

			:<math>f_s > R_N \, </math>

			The reconstruction of a signal from its samples can be accomplished using the [[Whittaker–Shannon interpolation formula]].

			==Bandlimited versus timelimited==
			{{redirect\|Timelimit\|narrow field of time by which an objective or task must be accomplished\|Time limit}}
			{{Elucidate\|date=June 2013}}
			A bandlimited signal cannot be also timelimited. More precisely, a function and its Fourier transform cannot both have finite [[support (mathematics)\|support]]. This fact can be proved by using complex analysis and properties of Fourier transform.

			'''Proof:''' Assume that a signal f(t) which has finite support in both domains exists. Let's sample it faster than the Nyquist frequency, and compute respective [[Fourier transform]] <math> FT(f) = F_1(w) </math> and [[discrete-time fourier transform]] <math> DTFT(f) = F_2(w)</math>. According to properties of DTFT, <math> F_2(w) = \sum_{n=-\infty}^{+\infty} F_1(w+n f_x) </math>, where <math>f_x</math> is the frequency used for discretization. If f is bandlimited, <math> F_1 </math> is zero outside of a certain interval, so with large enough <math> f_x </math>, <math> F_2 </math> will be zero in some intervals too, since individual [[Support_(mathematics)\|supports]] of <math> F_1 </math> in sum of <math> F_2 </math> won't overlap. According to DTFT definition, <math> F_2 </math> is a sum of trigonometric functions, and since f(t) is time-limited, this sum will be finite, so <math> F_2 </math> will be actually a [[trigonometric polynomial]]. All trigonometric polynomials are [[Entire function\|holomorphic on a whole complex plane]], and there is a simple theorem in complex analysis that says that [[Zero_(complex_analysis)\|all zeros of non-constant holomorphic function are isolated]]. But this contradicts our earlier finding that <math> F_2 </math> has intervals full of zeros, because points in such intervals are not isolated. Thus the only time- and bandwidth-limited signal is a constant zero.

			One important consequence of this result is that it is impossible to generate a truly bandlimited signal in any real-world situation, because a bandlimited signal would require infinite time to transmit. All real-world signals are, by necessity, ''timelimited'', which means that they ''cannot'' be bandlimited. Nevertheless, the concept of a bandlimited signal is a useful idealization for theoretical and analytical purposes. Furthermore, it is possible to approximate a bandlimited signal to any arbitrary level of accuracy desired.

			A similar relationship between duration in time and bandwidth in frequency also forms the mathematical basis for the [[uncertainty principle]] in [[quantum mechanics]]. In that setting, the "width" of the time domain and frequency domain functions are evaluated with a [[variance]]-like measure. Quantitatively, the uncertainty principle imposes the following condition on any real waveform:

			:<math> W_B T_D \ge 1 </math>

			where

			:<math>W_B</math> is a (suitably chosen) measure of bandwidth (in hertz), and

			:<math>T_D</math> is a (suitably chosen) measure of time duration (in seconds).

			In [[time–frequency analysis]], these limits are known as the ''[[Gabor limit]],'' and are interpreted as a limit on the ''simultaneous'' time–frequency resolution one may achieve.

			==References==

			{{Refimprove\|date=January 2011}}
			{{No footnotes\|date=January 2011}}

			{{cite book \| author = William McC. Siebert \| title = Circuits, Signals, and Systems \| year = 1986 \| location = Cambridge, MA \| publisher = MIT Press }}

			==See also==
			*[[Bandwidth (signal processing)\|Bandwidth]]
			*[[Nyquist–Shannon sampling theorem]]
			*[[Nyquist rate]]
			*[[Nyquist frequency]]


			[[Category:Digital signal processing]]

en>Ptbotgourou: r2.6.5) (Robot: Modifying de:Niveaumenge

2012-02-03T00:31:48Z

r2.6.5) (Robot: Modifying de:Niveaumenge

New page

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