Jordan curve theorem: Difference between revisions

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[[File:Sinc function (normalized).svg|thumb|The normalized [[sinc function]], the [[impulse response]] of the sinc filter.]]
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[[File:Rectangular function.svg|thumb|The [[rectangular function]], the [[frequency response]] of the sinc filter.]]
 
In [[signal processing]], a '''sinc filter''' is an idealized [[Filter (signal processing)|filter]] that removes all frequency components above a given [[cutoff frequency]], without affecting lower frequencies, and has [[linear phase]] response. The filter's [[impulse response]] is a [[sinc function]] in the time domain, and its [[frequency response]] is a [[rectangular function]].
 
It is an "ideal" [[low-pass filter]] in the frequency sense, perfectly passing low frequencies, perfectly cutting high frequencies; and thus may be considered to be a brick-wall filter.
 
Real-time filters can only approximate this ideal, since an ideal sinc filter (aka ''rectangular filter'') is [[causal filter|non-causal]] and has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the [[Nyquist–Shannon sampling theorem|sampling theorem]] and the [[Whittaker–Shannon interpolation formula]].
 
In mathematical terms, the desired frequency response is the [[rectangular function]]:
 
:<math>H(f) = \mathrm{rect} \left( \frac{f}{2B} \right)</math>
 
where <math>B\,</math> is an arbitrary cutoff frequency (aka ''bandwidth''). The impulse response of such a filter is given by the [[Continuous Fourier transform#Table of important Fourier transforms|inverse Fourier transform]] of the frequency response:
 
: <math>
\begin{align}
h(t) = \mathcal{F}^{-1} \{ H (f)\} & = 2B \frac{\sin(2\pi Bt)}{2\pi Bt} \\
& = 2B \, \mathrm{sinc}(2 B t)
\end{align}
</math>
 
the normalized [[sinc function]].
 
As the sinc filter has infinite impulse response in both positive and negative time directions, it must be approximated for real-world (non-abstract) applications; a [[window function|windowed]] sinc filter is often used instead. Windowing and truncating a sinc filter [[Convolution kernel|kernel]] in order to use it on any practical real world data set destroys its ideal properties.
 
==Brick-wall filters==
An idealized [[electronic filter]], one that has full transmission in the pass band, and complete attenuation in the stop band, with abrupt transitions, is known colloquially as a "brick-wall filter", in reference to the shape of the [[transfer function]]. The sinc filter is a brick-wall [[low-pass filter]], from which brick-wall [[band-pass filter]]s and [[high-pass filter]]s are easily constructed.
 
The lowpass filter with brick-wall cutoff at frequency ''B''<sub>''L''</sub> has impulse response and transfer function given by:
 
:<math> h_{LPF}(t) = 2B_L \, \mathrm{sinc}\left(2B_L t\right)</math>
 
:<math> H_{LPF}(f) = \mathrm{rect}\left( \frac{f}{2B_L} \right).</math>
 
The band-pass filter with lower band edge ''B''<sub>''L''</sub> and upper band edge ''B''<sub>''H''</sub> is just the difference of two such sinc filters (since the filters are zero phase, their magnitude responses subtract directly):<ref>{{cite book
| title = Practical signal processing
| author = Mark Owen
| publisher = Cambridge University Press
| year = 2007
| isbn = 978-0-521-85478-8
| page = 81
| url = http://books.google.com/?id=lx-tqq-MkK0C&pg=RA1-PA81&dq=sinc-function+high-pass+band-pass+difference&q=sinc-function%20high-pass%20band-pass%20difference
}}</ref>
 
:<math> h_{BPF}(t) = 2B_H \, \mathrm{sinc}\left(2B_H t\right) - 2B_L \, \mathrm{sinc}\left(2B_L t\right)</math>
 
:<math> H_{BPF}(f) = \mathrm{rect}\left( \frac{f}{2B_H} \right) - \mathrm{rect}\left( \frac{f}{2B_L} \right).</math>
 
The high-pass filter with lower band edge ''B''<sub>''H''</sub> is just a transparent filter minus a sinc filter, which makes it clear that the [[Dirac delta function]] is the limit of a narrow-in-time sinc filter:
 
:<math> h_{HPF}(t) = \delta(t) - 2B_H \, \mathrm{sinc}\left(2B_H t\right)</math>
 
:<math> H_{HPF}(f) = 1 - \mathrm{rect}\left( \frac{f}{2B_H} \right).</math>
 
Brick-wall filters that run in realtime are not physically realizable as they have infinite latency (i.e., its [[compact support]] in the [[frequency domain]] forces its time response not to have compact support meaning that it is ever-lasting) and infinite order (i.e., the response cannot be expressed as a [[linear differential equation]] with a finite sum), but approximate implementations are sometimes used and they are frequently called brick-wall filters.{{Citation needed|date=May 2009}}
 
==Frequency-domain sinc==
The name "sinc filter" is applied also to the filter shape that is rectangular in time and a sinc function in frequency, as opposed to the ideal low-pass sinc filter, which is sinc in time and rectangular in frequency. In case of confusion, one may refer to these as '''sinc-in-frequency''' and '''sinc-in-time,''' according to which domain the filter is sinc in.
 
Sinc-in-frequency [[Cascaded integrator-comb|CIC]] filters, among many other applications, are almost universally used for [[Decimation (signal processing)|decimating]] [[Delta-sigma modulation|delta-sigma]] [[Analog-to-digital converter|ADCs]], as they are easy to implement and nearly optimal for this use.<ref>{{cite journal | title=Time domain analysis of sigma delta modulation | author=Chou, W.; Meng, T.H.; Gray, R.M. | journal=Acoustics, Speech, and Signal Processing | year=1990 |pages=1751–1754 | volume=3 | doi=10.1109/ICASSP.1990.115820 }}</ref>
 
==Stability==
 
The sinc filter is not [[BIBO stability|bounded-input–bounded-output (BIBO) stable]]. That is, a bounded input can produce an unbounded output, because the integral of the absolute value of the sinc function is infinite. A bounded input that produces an unbounded output is sgn(sinc(''t'')).  Another is sin(2{{pi}}''Bt'')u(''t''), a sine wave starting at time 0, at the cutoff frequency.
 
==See also==
* [[Lanczos resampling]]
* [[Aliasing]]
* [[Anti-aliasing filter]]
 
==References==
{{reflist}}
 
==External links==
* [http://www.audioholics.com/education/audio-formats-technology/brick-wall-digital-filters-and-phase-deviations Brick Wall Digital Filters and Phase Deviations]
* [http://www.sweetwater.com/expert-center/glossary/t--BrickwallFilter Brick-wall filters]
 
{{DEFAULTSORT:Sinc Filter}}
[[Category:Signal processing]]
[[Category:Digital signal processing]]
[[Category:Filter theory]]
[[Category:Filter frequency response]]

Latest revision as of 13:59, 15 December 2014

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Over the last 100 years the average person has gotten taller and tends to carry more muscle mass. The result is that BMI calculations tend to be a little bit off, many folks can read high than they actually are. Nevertheless for many individuals the results are still fairly accurate. If you are truly tall or you may be carrying a lot of muscle be prepared for the charts to tell we which you are obese.

Not consuming enough calories and nutritients is a major wellness risk which will result in not only disorder, but death even. A healthy diet ought to be balanced and consist of 1,500 to 2,000 calories a day depending on bodily acitivity. Taking multivitamins daily may further confirm that the body's needs for nutrients are met. Avoid fasts for fat reduction or fad diets advocating the consumpion of just one form of food for a certain period of time because these could also cause malnutrition.

The Body mass index (BMI) utilizes height plus weight to estimate the amount of body fat. You are able to purchase a bmi calculator men or figure it out manually. To calculate the BMI, divide fat inside kg by height in meters squared; for pounds and inches, divide fat by height squared and multiply the outcome by the conversion factor 703. Normal weight is between the 5th and lower than the 85th percentile, thus if a child is overweight their BMI is between your 85th-95th percentiles plus overweight is above the 95th percentile. The BMI is not a ideal measurement plus is misleading with a more muscular child. Puberty that is a time of fast growth could have a misleading BMI.

There are lots of rapid and convenient online tools to utilize a own individual BMI, yet basically it happens to be a calculation of your current fat against your height. A 'normal' range of BMI is estimated to be between 18.5 and 24.9. A survey performed in 1994 indicated which over 59% of American men and 49% of American women were over 25%. A 2007 study now states that 63% of people are now over 25%. That is a startling statistic and certainly 1 which indicates that losing weight is more critical than ever as which trend demands to be reduced.

The 3 Day Cardiac Diet has set menu plans for breakfast, lunch plus dinner for three days. Every day is limited to about 800 calories. While certain claim that there is a specialized "chemical response," much of the fat reduction benefit is probably obtained by the restricted calorie consumption. Additionally, there are surprisingly few carbs on the diet which leads to water loss.

Women should constantly rely on a Body mass index calculator before they commence a fat loss system. This can moreover assist females recognize when they need healthcare help for alarming BMI fluctuations or perhaps a quite high / low BMI. Although it is actually possible to calculate the BMI manually with all the help of certain formulae plus charts, it is actually furthermore better in the event you rely on calculator to understand what the ideal BMI is for ladies of the height.

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