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| {{merge from|Sampling rate|discuss=Talk:Sampling rate#Merge to Sampling (signal processing)|date=September 2013}}
| | Many individuals are seeking an effective hemorrhoid treatment because they like to do away with their hemorrhoids as soon because possible. They do not desire to see any blood on their bathroom papers anymore, plus they never like to have any pain during irregularity anymore either. This really is the reason why they are trying so difficult inside purchase to eliminate hemorrhoid in a short period of time.<br><br>Now that the hemorrhoids aren't getting worse, it's time to employ [http://hemorrhoidtreatmentfix.com/internal-hemorrhoids-treatment how to treat internal hemorrhoids] which will relieve itching plus pain. Soaking in a a sitz bathtub for regarding fifteen minutes at a time is 1 method to discover relief. You are able to do this numerous times a day. We can furthermore heat a wet towel and apply it to the hemorrhoid area. Ice for ten minutes, followed by a warm compress for an equal or slightly longer length of time is another technique to relieve the pain and itching.<br><br>And then there was the casual bleeding. Once whenever I removed my light colored trousers at the finish of the day, I observed a small blood stain spot showing on the outside. How embarrassing! I question how several persons saw that and were too polite to mention anything!<br><br>Since there are actually only 3 or four key causes for hemorrhoids, the answer lies inside going to the source and basically doing elements differently, with a small aid from nature.<br><br>Right today, there are a great deal of hemorrhoid treatments. And yes, there are the painless hemorrhoid treatments additionally accessible. Examples of such as use of petroleum jelly, the use of ointment phenylephrine or Preparation H, and even the easy use of soft cotton underwear. These are typically painless for with them we don't should go beneath the knife.<br><br>Step 5 - Consume Sufficient Fiber. The mother may possibly have told we this whenever a kid and today you'll understand it's correct. Not consuming an adequate amount of fiber can truth be told cause irregularity and harder stools. A significant fiber diet may assist to avert flare ups and keep a stools soft. Fresh fruits, veggies, beans and peanut butter are all splendid sources of fiber to help keep we normal.<br><br>When using a sitz bath tub, you are able to use specialized soaps and lotions that is produced to be use with all the bathtub. This will aid treat different symptoms of the hemorrhoids, as well. You can pick up any sitz tub at a regional wellness store or you are able to purchase one online. |
| [[Image:Signal Sampling.png|thumb|300px|Signal sampling representation. The continuous signal is represented with a green colored line while the discrete samples are indicated by the blue vertical lines.]]
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| In [[signal processing]], '''sampling''' is the reduction of a [[continuous signal]] to a [[discrete signal]]. A common example is the conversion of a [[sound wave]] (a continuous signal) to a sequence of samples (a discrete-time signal).
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| A '''sample''' refers to a value or set of values at a point in time and/or space.
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| A '''sampler''' is a subsystem or operation that extracts samples from a [[continuous signal]].
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| A theoretical '''ideal sampler''' produces samples equivalent to the instantaneous value of the continuous signal at the desired points.
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| == Theory ==
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| :''See also: [[Nyquist–Shannon sampling theorem]]''
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| Sampling can be done for functions varying in space, time, or any other dimension, and similar results are obtained in two or more dimensions.
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| For functions that vary with time, let ''s''(''t'') be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every ''T'' seconds, which is called the [[sampling interval]]. Thus, the sampled function is given by the sequence''':'''
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| :''s''(''nT''), for integer values of ''n''.
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| The [[sampling frequency]] or sampling rate '''f<sub>s</sub>''' is defined as the number of samples obtained in one second (samples per second), thus '''f<sub>s</sub> = 1/T'''.
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| Reconstructing a continuous function from samples is done by interpolation algorithms. The [[Whittaker–Shannon interpolation formula]] is mathematically equivalent to an ideal [[lowpass filter]] whose input is a sequence of [[Dirac delta functions]] that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant (''T''), the sequence of delta functions is called a [[Dirac comb]]. Mathematically, the modulated [[Dirac comb]] is equivalent to the product of the comb function with ''s''(''t''). That purely mathematical function is often loosely referred to as the sampled signal.
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| Most sampled signals are not simply stored and reconstructed. But the fidelity of a theoretical reconstruction is a customary measure of the effectiveness of sampling. That fidelity is reduced when ''s''(''t'') contains frequency components higher than '''f<sub>s</sub>'''/2 [[Hz]], which is known as the [[Nyquist frequency]] of the sampler. Therefore ''s''(''t'') is usually the output of a [[lowpass filter]], functionally known as an "anti-aliasing" filter. Without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process. For details, see [[Aliasing]].
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| == Practical implications ==
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| In practice, the continuous signal is sampled using an [[analog-to-digital converter]] (ADC), a device with various physical limitations. This results in deviations from the theoretically perfect reconstruction, collectively referred to as distortion.
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| Various types of distortion can occur, including:
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| * [[Aliasing]]. A precondition of the sampling theorem is that the signal be bandlimited. However, in practice, no time-limited signal can be bandlimited. Since signals of interest are almost always time-limited (e.g., at most spanning the lifetime of the sampling device in question), it follows that they are not bandlimited. However, by designing a sampler with an appropriate [[guard band]], it is possible to obtain output that is as accurate as necessary.
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| * [[Analog-to-digital_converter#Aperture_error|Aperture error]] results from the fact that the sample is obtained as a time average within a sampling region, rather than just being equal to the signal value at the sampling instant. In a [[capacitor]]-based [[sample and hold]] circuit,aperture error is introduced because the capacitor cannot instantly change voltage thus requiring the sample to have non-zero width.
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| * [[Jitter]] or deviation from the precise sample timing intervals.
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| * [[Noise (physics)|Noise]], including thermal sensor noise, [[analog circuit]] noise, etc.
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| * [[Slew rate]] limit error, caused by the inability of the ADC input value to change sufficiently rapidly.
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| * [[Quantization (signal processing)|Quantization]] as a consequence of the finite precision of words that represent the converted values.
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| * Error due to other [[non-linear]] effects of the mapping of input voltage to converted output value (in addition to the effects of quantization).
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| Although the use of [[oversampling]] can completely eliminate aperture error and aliasing by shifting them out of the pass band, this technique cannot be practically used above a few GHz, and maybe prohibitively expensive at much lower frequencies. Furthermore, while oversampling can reduce quantization error and non-linearity, it cannot eliminate these entirely. Consequently, practical ADCs at audio frequencies typically do not exhibit aliasing, aperture error, and are not limited by quantization error. Instead, analog noise dominates. At RF and microwave frequencies where oversampling is impractical and filters are expensive, aperture error, quantization error and aliasing can be significant limitations.
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| Jitter, noise, and quantization are often analyzed by modeling them as random errors added to the sample values. Integration and zero-order hold effects can be analyzed as a form of [[low-pass filter]]ing. The non-linearities of either ADC or DAC are analyzed by replacing the ideal [[linear function]] mapping with a proposed [[Nonlinear|nonlinear function]].
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| == Applications ==
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| === Audio sampling ===
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| [[Digital audio]] uses [[pulse-code modulation]] and digital signals for sound reproduction. This includes analog-to-digital conversion (ADC), digital-to-analog conversion (DAC), storage, and transmission. In effect, the system commonly referred to as digital is in fact a discrete-time, discrete-level analog of a previous electrical analog. While modern systems can be quite subtle in their methods, the primary usefulness of a digital system is the ability to store, retrieve and transmit signals without any loss of quality.
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| ==== Sampling rate ====
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| When it is necessary to capture audio covering the entire 20–20,000 Hz range of [[auditory system|human hearing]], such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz ([[Compact disc|CD]]), 48 kHz ([[professional audio]]), or 96 kHz. The approximately double-rate requirement is a consequence of the [[Nyquist theorem]].
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| There has been an industry trend towards sampling rates well beyond the basic requirements; 96 kHz and even 192 kHz are available.<ref>{{cite web|url=http://www.digitalprosound.com/Htm/SoapBox/soap2_Apogee.htm |title=Digital Pro Sound |accessdate=8 January 2014}}</ref> This is in contrast with laboratory experiments, which have failed to show that [[Ultrasound|ultrasonic]] frequencies are audible to human observers; however in some cases ultrasonic sounds do interact with and modulate the audible part of the frequency spectrum ([[intermodulation distortion]]). It is noteworthy that intermodulation distortion is not present in the live audio and so it represents an artificial coloration to the live sound.<ref>{{cite web |url=http://world.std.com/~griesngr/intermod.ppt |archiveurl=https://web.archive.org/web/20080501000000/http://world.std.com/~griesngr/intermod.ppt |archivedate=2008-05-01 |title=Perception of mid frequency and high frequency intermodulation distortion in loudspeakers, and its relationship to high-definition audio |author=David Griesinger |format=Powerpoint presentation}}</ref>
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| One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for [[analog-to-digital converter|ADCs]] and [[digital-to-analog converter|DACs]], but with modern oversampling [[sigma-delta converter]]s this advantage is less important.
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| ==== Bit depth (quantization) ====
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| Audio is typically recorded at 8-, 16-, and 20-bit depth, which yield a theoretical maximum [[Signal-to-quantization-noise ratio]] (SQNR) for a pure [[sine wave]] of, approximately, 49.93 [[Decibel|dB]], 98.09 dB and 122.17 dB.<ref>{{cite web|url=http://www.analog.com/static/imported-files/tutorials/MT-001.pdf |title=MT-001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N + 1.76dB," and Why You Should Care}}</ref> CD quality audio is recorded at 16-bit. [[Thermal noise]] limits the true number of bits that can be used in quantization. Few analog systems have [[Signal-to-noise_ratio|signal to noise ratios (SNR)]] exceeding 120 dB. However, [[digital signal processing]] operations can have very high dynamic range, consequently it is common to perform mixing and mastering operations at 32-bit precision and then convert to 16 or 24 bit for distribution.
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| ==== Speech sampling ====
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| Speech signals, i.e., signals intended to carry only human [[Speech communication|speech]], can usually be sampled at a much lower rate. For most [[phoneme]]s, almost all of the energy is contained in the 5Hz-4 kHz range, allowing a sampling rate of 8 kHz. This is the [[sampling rate]] used by nearly all [[telephony]] systems, which use the [[G.711]] sampling and quantization specifications.
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| === Video sampling ===
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| [[Standard-definition television]] (SDTV) uses either 720 by 480 [[pixels]] (US [[NTSC]] 525-line) or 704 by 576 [[pixels]] (UK [[PAL]] 625-line) for the visible picture area.
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| [[High-definition television]] (HDTV) is currently moving towards three standards referred to as [[720p]] (progressive), [[1080i]] (interlaced) and [[1080p]] (progressive, also known as Full-HD) which all 'HD-Ready' sets will be able to display.
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| [[File:Samplerates.svg|thumb|right|255px|Plot of sample rates (y axis) versus the upper edge frequency (x axis) for a band of width 1; grays areas are combinations that are "allowed" in the sense that no two frequencies in the band alias to same frequency. The darker gray areas correspond to [[undersampling]] with the lowest allowable sample rate.]]
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| == Undersampling ==
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| {{main|Undersampling}}
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| When a [[bandpass]] signal is sampled slower than its [[Nyquist rate]], the samples are indistinguishable from samples of a low-frequency [[aliasing|alias]] of the high-frequency signal. That is often done purposefully in such a way that the lowest-frequency alias satisfies the [[Nyquist criterion]], because the bandpass signal is still uniquely represented and recoverable. Such [[undersampling]] is also known as ''bandpass sampling'', ''harmonic sampling'', ''IF sampling'', and ''direct IF to digital conversion.''<ref>
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| {{cite book
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| | title = Mixed-signal and DSP design techniques
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| | author = Walt Kester
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| | publisher = Newnes
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| | year = 2003
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| | isbn = 978-0-7506-7611-3
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| | page = 20
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| | url = http://books.google.com/books?id=G8XyNItpy8AC&pg=PA20
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| | accessdate = 8 January 2014
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| }}</ref>
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| == Oversampling ==
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| {{main|Oversampling}}
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| Oversampling is used in most modern analog-to-digital converters to reduce the distortion introduced by practical [[digital-to-analog converter]]s, such as a [[zero-order hold]] instead of idealizations like the [[Whittaker–Shannon interpolation formula]].
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| == Complex sampling ==
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| ''Complex sampling'' refers to the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as [[complex numbers]]. Usually one waveform<math>, \hat s(t),</math> is the [[Hilbert transform]] of the other waveform<math>, s(t),\,</math> and the complex-valued function, <math>s_a(t)\ \stackrel{\text{def}}{=}\ s(t) + j\cdot \hat s(t),</math> is called an [[analytic signal]], whose Fourier transform is zero for all negative values of frequency. In that case, the [[Nyquist rate]] for a waveform with no frequencies '''≥ B''' can be reduced to just ''B'' (complex samples/sec), instead of ''2B'' (real samples/sec).<ref group="note">When the complex sample-rate is B, a frequency component at 0.6 B, for instance, will have an alias at -0.4 B, which is unambiguous because of the constraint that the pre-sampled signal was analytic. Also see [[Aliasing#Complex_sinusoids]]</ref> More apparently, the
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| [[Baseband#Equivalent baseband signal|equivalent baseband waveform]], <math>s_a(t)\cdot e^{-j 2\pi \frac{B}{2} t},</math> also has a Nyquist rate of ''B'', because all of its non-zero frequency content is shifted into the interval [-B/2, B/2).
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| Although complex-valued samples can be obtained as described above, they are much more commonly created by manipulating samples of a real-valued waveform. For instance, the equivalent baseband waveform can be created without explicitly computing <math>\hat s(t),</math> by processing the product sequence<math>, \left [s(nT)\cdot e^{-j 2 \pi \frac{B}{2}Tn}\right ],</math><ref group="note">When s(t) is sampled at the Nyquist frequency (1/T = 2B), the product sequence simplifies to <math>\left [s(nT)\cdot (-j)^n\right ].</math></ref> through a digital lowpass filter whose cutoff frequency is B/2.<ref group="note">The sequence of complex numbers is convolved with the impulse response of a filter with real-valued coefficients. That is equivalent to separately filtering the sequences of real parts and imaginary parts and reforming complex pairs at the outputs.</ref> Computing only every other sample of the output sequence reduces the sample-rate commensurate with the reduced Nyquist rate. The result is half as many complex-valued samples as the original number of real samples. No information is lost, and the original s(t) waveform can be recovered, if necessary.
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| === Notes ===
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| {{reflist|group=note}}
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| == See also ==
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| * [[Beta encoder]]
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| * [[Digitizing]]
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| * [[Kell factor]]
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| * [[Downsampling]]
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| * [[Upsampling]]
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| * [[Multidimensional sampling]]
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| ==References==
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| * Matt Pharr and Greg Humphreys, ''Physically Based Rendering: From Theory to Implementation'', Morgan Kaufmann, July 2004. ISBN 0-12-553180-X. The chapter on sampling ([http://graphics.stanford.edu/~mmp/chapters/pbrt_chapter7.pdf available online]) is nicely written with diagrams, core theory and code sample.
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| * Shannon, Claude E. (January 1949). Communications in the presence of noise, [[Proc. IRE]], vol. 37, pp. 10–21.
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|
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| {{reflist}}
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| ==External links==
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| * [http://www.vanosta.be/pcrnyq.htm Nyquist sampling in digital microscopy]{{dead link|date=August 2011}}
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| * [http://www.stsip.org Journal devoted to Sampling Theory]
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| * [http://whiteboard.ping.se/SDR/IQ I/Q Data for Dummies] A page trying to answer the question ''Why I/Q Data?''
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| [[Category:Signal processing]]
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Many individuals are seeking an effective hemorrhoid treatment because they like to do away with their hemorrhoids as soon because possible. They do not desire to see any blood on their bathroom papers anymore, plus they never like to have any pain during irregularity anymore either. This really is the reason why they are trying so difficult inside purchase to eliminate hemorrhoid in a short period of time.
Now that the hemorrhoids aren't getting worse, it's time to employ how to treat internal hemorrhoids which will relieve itching plus pain. Soaking in a a sitz bathtub for regarding fifteen minutes at a time is 1 method to discover relief. You are able to do this numerous times a day. We can furthermore heat a wet towel and apply it to the hemorrhoid area. Ice for ten minutes, followed by a warm compress for an equal or slightly longer length of time is another technique to relieve the pain and itching.
And then there was the casual bleeding. Once whenever I removed my light colored trousers at the finish of the day, I observed a small blood stain spot showing on the outside. How embarrassing! I question how several persons saw that and were too polite to mention anything!
Since there are actually only 3 or four key causes for hemorrhoids, the answer lies inside going to the source and basically doing elements differently, with a small aid from nature.
Right today, there are a great deal of hemorrhoid treatments. And yes, there are the painless hemorrhoid treatments additionally accessible. Examples of such as use of petroleum jelly, the use of ointment phenylephrine or Preparation H, and even the easy use of soft cotton underwear. These are typically painless for with them we don't should go beneath the knife.
Step 5 - Consume Sufficient Fiber. The mother may possibly have told we this whenever a kid and today you'll understand it's correct. Not consuming an adequate amount of fiber can truth be told cause irregularity and harder stools. A significant fiber diet may assist to avert flare ups and keep a stools soft. Fresh fruits, veggies, beans and peanut butter are all splendid sources of fiber to help keep we normal.
When using a sitz bath tub, you are able to use specialized soaps and lotions that is produced to be use with all the bathtub. This will aid treat different symptoms of the hemorrhoids, as well. You can pick up any sitz tub at a regional wellness store or you are able to purchase one online.