Corecursion - Revision history

en>WillNess: /* Factorial */ wikify

2014-11-12T16:36:12Z

Factorial: wikify

← Older revision		Revision as of 18:36, 12 November 2014
Line 1:		Line 1:
	~~{{Refimprove\|date=April 2011}}~~		The writer's name is Christy Brookins. Playing badminton is a factor that he is completely addicted to. She functions as a journey agent but quickly she'll be on her own. Alaska is exactly where I've usually been residing.<br><br>Also visit my web site; authentic psychic readings ([http://Blog.laors.co.kr/category/%EA%B0%80%EB%B0%A9?page=40 laors.co.kr])

	~~In [[signal processing]], '''oversampling''~~' is the process of [[sampling (signal processing)\|sampling]] a signal with a [[sampling frequency]] significantly higher than the [[Nyquist frequency]]. Theoretically a bandwidth-limited signal can be perfectly reconstructed if sampled at or above the Nyquist frequency. ~~Oversampling improves [[Resolution (audio)\|resolution]], reduces [[noise]] and helps avoid [[aliasing]] and [[phase distortion]] by relaxing [[anti-aliasing filter]] performance requirements.~~

	~~A signal~~ is ~~said to be oversampled by~~ a factor ~~of N if it~~ is ~~sampled at N times the Nyquist frequency.~~

	~~==Motivation==~~
	~~There are three main reasons for performing oversampling:~~

	~~===Anti-aliasing===~~
	~~Oversampling can make it easier~~ to ~~realize analog [[anti-aliasing filter]]s~~. Without oversampling, it is very difficult to implement filters with the sharp cutoff necessary to maximize use of the available bandwidth without exceeding the [[Nyquist limit]]. By increasing the bandwidth of the sampled signal, design constraints for the anti-aliasing filter may be relaxed.<ref>{{cite magazine \|url=http://www.audioholics.com/education/audio-formats-technology/upsampling-vs-oversampling-for-digital-audio \|title=Upsampling vs. Oversampling for Digital Audio \|quote=Without increasing the sample rate, we would need to design a ~~very sharp filter that would have to cutoff at just past 20kHz and be 80-100dB down at 22kHz. Such a filter is not only very difficult and expensive to implement,~~ but ~~may sacrifice some of the audible spectrum in its rolloff. \|author=Nauman Uppal \|date=2004-08-30 \|accessdate=2012-10-06}}</ref> Once sampled, the signal can~~ be [[digital filter\|digitally filtered]] and [[downsampling\|downsampled]] to the desired sampling frequency. In modern [[integrated circuit]] technology, digital filters are easier to implement than comparable [[analog filter]]s.

	~~===Resolution===~~
	~~In practice, oversampling~~ is ~~implemented in order to achieve cheaper higher-resolution [[Analog-to-digital converter\|A/D]] and [[Digital-to-analog converter\|D/A]] conversion~~. For instance, to implement a 24-bit converter, it is sufficient to use a 20-bit converter that can run at 256 times the target sampling rate. Combining 256 consecutive 20-bit samples can increase the [[signal-to-noise ratio]] at the voltage level by a factor of 16 (the square root of the number of samples averaged), adding 4 bits to the resolution and producing a single sample with 24-bit resolution.

	~~The number of samples required to get~~ <~~math~~>n<~~/math~~> ~~bits of additional data precision is~~

	~~:<math>\mbox{number of samples} =~~ (~~2^n)^2 = 2^{2n}.</math>~~

	~~To get the mean sample scaled up to an integer with <math>n</math> additional bits, the sum of <math>2^{2n}</math> samples is divided by <math>2^n</math>:~~

	:~~<math>\mbox{scaled mean} = \frac{ \sum\limits^{2^{2n}-1}_{i=0} 2^n data_i}{2^{2n}} = \frac{\sum\limits^{2^{2n}-1}_{i=0} data_i}{2^n}.</math>~~

	~~This averaging is only possible if the [[signal (information theory)\|signal]] contains equally distributed [[noise]] which is enough to be observed by the A~~/~~D converter. If not, in the case of a stationary input signal, all <math>2^n<~~/~~math> samples would have the same value and the resulting average would be identical to this value; so in this case, oversampling would have made no improvement~~. ~~(In similar cases where the A/D converter sees no noise and the input signal is changing over time, oversampling still improves the result, but to an inconsistent/unpredictable extent~~.) This is an interesting counter-intuitive example where adding some [[dither\|dithering]] noise to the input signal can improve (rather than degrade) the final result because the dither noise allows oversampling to work to improve resolution (or dynamic range). In many practical applications, a small increase in noise is well worth a substantial increase in measurement resolution. In practice, the dithering noise can often be placed outside the frequency range of interest to the measurement, so that this noise can be subsequently filtered out in the digital domain--resulting in a final measurement (in the frequency range of interest) with both higher resolution and lower noise.

	~~===Noise===~~
	~~If multiple samples are taken of the same quantity with [[uncorrelated]] noise added to each sample, then averaging ''N'' samples reduces the [[noise power]] by a factor of 1~~/~~''N''.<ref>See [[standard error (statistics)]]<~~/ref> If, for example, we oversample by a factor of 4, the [[signal-to-noise ratio]] in terms of power improves by factor of 4 which corresponds to a factor of 2 improvement in terms of voltage.<ref group=note>A system's signal-to-noise ratio cannot necessarily be increased by simple over-sampling, since noise samples are partially correlated (only some portion of the noise due to sampling and analog-to-digital conversion will be uncorrelated).~~</ref>~~

	Certain kinds of A/D converters known as [[Delta-sigma modulation\|delta-sigma converter]]s produce disproportionately more [[Quantization (signal processing)\|quantization]] noise in the upper portion of their output spectrum. ~~By running these converters at some multiple of the target sampling rate, and [[low-pass filter]~~]~~ing the oversampled signal down to half the target sampling rate, a final result with ''less'' noise (over the entire band of the converter~~) ~~can be obtained. Delta-sigma converters use a technique called [[noise shaping]] to move the quantization noise to the higher frequencies.~~

	~~==Example==~~
	For example, consider a signal with a bandwidth or highest frequency of ''B'' = 100 [[Hertz\|Hz]]. The [[Nyquist-Shannon sampling theorem\|sampling theorem]] states that sampling frequency would have to be greater than 200 Hz. Sampling at four times that rate requires a sampling frequency of 800 Hz. This gives the anti-aliasing filter a [[transition band]] of 300 Hz ((''f''<sub>s</sub>/2) − ''B'' = (800 Hz/2) − 100 Hz = 300 Hz) instead of 0 Hz if the sampling frequency was 200 Hz.

	~~Achieving an anti-aliasing filter with 0 Hz transition band is unrealistic whereas an anti-aliasing filter with a transition band of 300 Hz is not difficult to create.~~

	~~==Oversampling in reconstruction==~~
	The term oversampling is also used to denote a process used in the reconstruction phase of [[digital-to-analog conversion]], in which an intermediate high sampling rate is used between the digital input and the analogue output. Here, samples are interpolated in the digital domain to add additional samples in between, thereby converting the data to a higher sample rate, which is a form of [[upsampling]]. When the resulting higher-rate samples are converted to analog, a less complex/expensive analog low pass filter is required to remove the high-frequency content, which will consist of reflected images of the real signal created by the [[zero-order hold]] of the [[digital-to-analog converter]]. Essentially, this is a way to shift some of the complexity of the filtering into the digital domain and achieves the same benefit as oversampling in analog-to-digital conversion.

	~~== See also ==~~
	*[[Upsampling]]
	*[[Undersampling]]
	*[[Oversampling and undersampling in data analysis]]
	*[[Oversampled binary image sensor]]

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

	~~==References==~~
	~~{{Reflist}}~~

	~~==Further reading==~~
	*{{cite book \|author=John Watkinson \|title=The Art of Digital Audio \|ISBN=0-240-51320-7}}

	~~{{DSP}}~~

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

en>WillNess: /* Factorial */ m

2014-01-29T18:54:09Z

Factorial: m

← Older revision		Revision as of 20:54, 29 January 2014
Line 1:		Line 1:
	~~Hello~~ and ~~welcome~~. My title is ~~Irwin~~ and ~~I totally dig~~ that ~~name~~. ~~What I love doing~~ is to ~~gather badges~~ but ~~I've been using on new things recently~~. ~~Managing individuals~~ is ~~his profession~~. ~~Years~~ in the ~~past he moved~~ to ~~North Dakota~~ and ~~his family members enjoys it~~.<br><br>~~my site~~ ... at ~~home std test~~ ([~~http:~~//~~imama~~.~~com~~.ua/~~groups/what~~-~~are~~-~~lifestyle~~-~~diseases~~-and-~~how~~-to-~~prevent~~-~~them/ mouse click the next web site~~])		{{Refimprove\|date=April 2011}}

			In [[signal processing]], '''oversampling''' is the process of [[sampling (signal processing)\|sampling]] a signal with a [[sampling frequency]] significantly higher than the [[Nyquist frequency]]. Theoretically a bandwidth-limited signal can be perfectly reconstructed if sampled at or above the Nyquist frequency. Oversampling improves [[Resolution (audio)\|resolution]], reduces [[noise]] and helps avoid [[aliasing]] and [[phase distortion]] by relaxing [[anti-aliasing filter]] performance requirements.

			A signal is said to be oversampled by a factor of N if it is sampled at N times the Nyquist frequency.

			==Motivation==
			There are three main reasons for performing oversampling:

			===Anti-aliasing===
			Oversampling can make it easier to realize analog [[anti-aliasing filter]]s. Without oversampling, it is very difficult to implement filters with the sharp cutoff necessary to maximize use of the available bandwidth without exceeding the [[Nyquist limit]]. By increasing the bandwidth of the sampled signal, design constraints for the anti-aliasing filter may be relaxed.<ref>{{cite magazine \|url=http://www.audioholics.com/education/audio-formats-technology/upsampling-vs-oversampling-for-digital-audio \|title=Upsampling vs. Oversampling for Digital Audio \|quote=Without increasing the sample rate, we would need to design a very sharp filter that would have to cutoff at just past 20kHz and be 80-100dB down at 22kHz. Such a filter is not only very difficult and expensive to implement, but may sacrifice some of the audible spectrum in its rolloff. \|author=Nauman Uppal \|date=2004-08-30 \|accessdate=2012-10-06}}</ref> Once sampled, the signal can be [[digital filter\|digitally filtered]] and [[downsampling\|downsampled]] to the desired sampling frequency. In modern [[integrated circuit]] technology, digital filters are easier to implement than comparable [[analog filter]]s.

			===Resolution===
			In practice, oversampling is implemented in order to achieve cheaper higher-resolution [[Analog-to-digital converter\|A/D]] and [[Digital-to-analog converter\|D/A]] conversion. For instance, to implement a 24-bit converter, it is sufficient to use a 20-bit converter that can run at 256 times the target sampling rate. Combining 256 consecutive 20-bit samples can increase the [[signal-to-noise ratio]] at the voltage level by a factor of 16 (the square root of the number of samples averaged), adding 4 bits to the resolution and producing a single sample with 24-bit resolution.

			The number of samples required to get <math>n</math> bits of additional data precision is

			:<math>\mbox{number of samples} = (2^n)^2 = 2^{2n}.</math>

			To get the mean sample scaled up to an integer with <math>n</math> additional bits, the sum of <math>2^{2n}</math> samples is divided by <math>2^n</math>:

			:<math>\mbox{scaled mean} = \frac{ \sum\limits^{2^{2n}-1}_{i=0} 2^n data_i}{2^{2n}} = \frac{\sum\limits^{2^{2n}-1}_{i=0} data_i}{2^n}.</math>

			This averaging is only possible if the [[signal (information theory)\|signal]] contains equally distributed [[noise]] which is enough to be observed by the A/D converter. If not, in the case of a stationary input signal, all <math>2^n</math> samples would have the same value and the resulting average would be identical to this value; so in this case, oversampling would have made no improvement. (In similar cases where the A/D converter sees no noise and the input signal is changing over time, oversampling still improves the result, but to an inconsistent/unpredictable extent.) This is an interesting counter-intuitive example where adding some [[dither\|dithering]] noise to the input signal can improve (rather than degrade) the final result because the dither noise allows oversampling to work to improve resolution (or dynamic range). In many practical applications, a small increase in noise is well worth a substantial increase in measurement resolution. In practice, the dithering noise can often be placed outside the frequency range of interest to the measurement, so that this noise can be subsequently filtered out in the digital domain--resulting in a final measurement (in the frequency range of interest) with both higher resolution and lower noise.

			===Noise===
			If multiple samples are taken of the same quantity with [[uncorrelated]] noise added to each sample, then averaging ''N'' samples reduces the [[noise power]] by a factor of 1/''N''.<ref>See [[standard error (statistics)]]</ref> If, for example, we oversample by a factor of 4, the [[signal-to-noise ratio]] in terms of power improves by factor of 4 which corresponds to a factor of 2 improvement in terms of voltage.<ref group=note>A system's signal-to-noise ratio cannot necessarily be increased by simple over-sampling, since noise samples are partially correlated (only some portion of the noise due to sampling and analog-to-digital conversion will be uncorrelated).</ref>

			Certain kinds of A/D converters known as [[Delta-sigma modulation\|delta-sigma converter]]s produce disproportionately more [[Quantization (signal processing)\|quantization]] noise in the upper portion of their output spectrum. By running these converters at some multiple of the target sampling rate, and [[low-pass filter]]ing the oversampled signal down to half the target sampling rate, a final result with ''less'' noise (over the entire band of the converter) can be obtained. Delta-sigma converters use a technique called [[noise shaping]] to move the quantization noise to the higher frequencies.

			==Example==
			For example, consider a signal with a bandwidth or highest frequency of ''B'' = 100 [[Hertz\|Hz]]. The [[Nyquist-Shannon sampling theorem\|sampling theorem]] states that sampling frequency would have to be greater than 200 Hz. Sampling at four times that rate requires a sampling frequency of 800 Hz. This gives the anti-aliasing filter a [[transition band]] of 300 Hz ((''f''<sub>s</sub>/2) − ''B'' = (800 Hz/2) − 100 Hz = 300 Hz) instead of 0 Hz if the sampling frequency was 200 Hz.

			Achieving an anti-aliasing filter with 0 Hz transition band is unrealistic whereas an anti-aliasing filter with a transition band of 300 Hz is not difficult to create.

			==Oversampling in reconstruction==
			The term oversampling is also used to denote a process used in the reconstruction phase of [[digital-to-analog conversion]], in which an intermediate high sampling rate is used between the digital input and the analogue output. Here, samples are interpolated in the digital domain to add additional samples in between, thereby converting the data to a higher sample rate, which is a form of [[upsampling]]. When the resulting higher-rate samples are converted to analog, a less complex/expensive analog low pass filter is required to remove the high-frequency content, which will consist of reflected images of the real signal created by the [[zero-order hold]] of the [[digital-to-analog converter]]. Essentially, this is a way to shift some of the complexity of the filtering into the digital domain and achieves the same benefit as oversampling in analog-to-digital conversion.

			== See also ==
			*[[Upsampling]]
			*[[Undersampling]]
			*[[Oversampling and undersampling in data analysis]]
			*[[Oversampled binary image sensor]]

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

			==References==
			{{Reflist}}

			==Further reading==
			*{{cite book \|author=John Watkinson \|title=The Art of Digital Audio \|ISBN=0-240-51320-7}}

			{{DSP}}

			[[Category:Digital signal processing]]
			[[Category:Information theory]]

en>R'n'B: Fix links to disambiguation page Infinite

2012-08-23T16:21:06Z

Fix links to disambiguation page Infinite

New page

Hello and welcome. My title is Irwin and I totally dig that name. What I love doing is to gather badges but I've been using on new things recently. Managing individuals is his profession. Years in the past he moved to North Dakota and his family members enjoys it.<br><br>my site ... at home std test ([http://imama.com.ua/groups/what-are-lifestyle-diseases-and-how-to-prevent-them/ mouse click the next web site])