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<title>Sampling (signal processing)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Sampling (signal processing)</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Sampling_(disambiguation)" class="mw-redirect mw-disambig" title="Sampling (disambiguation)">Sampling (disambiguation)</a>.</div>

<p>In <a href="Signal_processing" title="Signal processing">signal processing</a>, <b>sampling</b> is the reduction of a <a href="Continuous-time_signal" class="mw-redirect" title="Continuous-time signal">continuous-time signal</a> to a <a href="Discrete-time_signal" class="mw-redirect" title="Discrete-time signal">discrete-time signal</a>. A common example is the conversion of a <a href="Sound_wave" class="mw-redirect" title="Sound wave">sound wave</a> to a sequence of "samples".

A <b>sample</b> is a value of the <a href="Signal" title="Signal">signal</a> at a point in time and/or space; this definition differs from <a href="Sampling_(statistics)" title="Sampling (statistics)">the term's usage in statistics</a>, which refers to a set of such values.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>A<span class="cite-bracket">]</span></a></sup>

</p><p>A <b>sampler</b> is a subsystem or operation that extracts samples from a <a href="Continuous_signal" class="mw-redirect" title="Continuous signal">continuous signal</a>. A theoretical <b>ideal sampler</b> produces samples equivalent to the instantaneous value of the continuous signal at the desired points.

</p><p>The original signal can be reconstructed from a sequence of samples, up to the <a href="Nyquist_limit" class="mw-redirect" title="Nyquist limit">Nyquist limit</a>, by passing the sequence of samples through a <a href="Reconstruction_filter" title="Reconstruction filter">reconstruction filter</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a></div>

<p>Functions of space, time, or any other dimension can be sampled, and similarly in two or more dimensions.

</p><p>For functions that vary with time, let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}">

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<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>

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</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span> be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> seconds, which is called the <b>sampling interval</b> or <b>sampling period</b>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Moir_3-0" class="reference"><a href="#cite_note-Moir-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Then the sampled function is given by the sequence:

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<annotation encoding="application/x-tex">{\displaystyle s(nT)}</annotation>

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</math></span><img src="./88618d85a88d84d826a8b0d09b81010beb5559ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.931ex; height:2.843ex;" alt="{\displaystyle s(nT)}" loading="lazy"></span>, for integer values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>.</dd></dl>

<p>The <b>sampling frequency</b> or <b>sampling rate</b>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{s}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{s}}</annotation>

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</math></span><img src="./34f933ad7a8dc310b3fa8e9f7b0b2558cba136db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.143ex; height:2.509ex;" alt="{\displaystyle f_{s}}" loading="lazy"></span>, is the average number of samples obtained in one second, thus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{s}=1/T}">

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{s}=1/T}</annotation>

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</math></span><img src="./004344797e3e0aabccd69becff32ef5c31063cc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.202ex; height:2.843ex;" alt="{\displaystyle f_{s}=1/T}" loading="lazy"></span>, with the unit <i>samples per second</i>, sometimes referred to as <a href="Hertz" title="Hertz">hertz</a>, for example 48 kHz is 48,000 <i>samples per second</i>.

</p><p>Reconstructing a continuous function from samples is done by interpolation algorithms. The <a href="Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">Whittaker–Shannon interpolation formula</a> is mathematically equivalent to an ideal <a href="Low-pass_filter" title="Low-pass filter">low-pass filter</a> whose input is a sequence of <a href="Dirac_delta_functions" class="mw-redirect" title="Dirac delta functions">Dirac delta functions</a> that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (T)}</annotation>

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</math></span><img src="./c5f9a82bdcb95d511ed6965b0534a3eda98ceb62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.446ex; height:2.843ex;" alt="{\displaystyle (T)}" loading="lazy"></span>, the sequence of delta functions is called a <a href="Dirac_comb" title="Dirac comb">Dirac comb</a>. Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>

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</p><p>Most sampled signals are not simply stored and reconstructed. The fidelity of a theoretical reconstruction is a common measure of the effectiveness of sampling. That fidelity is reduced when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>

</semantics>

</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span> contains frequency components whose cycle length (period) is less than 2 sample intervals (see <i><a href="Aliasing#Sampling_sinusoidal_functions" title="Aliasing">Aliasing</a></i>). The corresponding frequency limit, in <i>cycles per second</i> (<a href="Hertz" title="Hertz">hertz</a>), is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.5}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 0.5}</annotation>

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</math></span><img src="./c867fe7d5d53ce2c0790852289b794c6ed185f36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 0.5}" loading="lazy"></span> cycle/sample × <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{s}}</annotation>

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</math></span><img src="./34f933ad7a8dc310b3fa8e9f7b0b2558cba136db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.143ex; height:2.509ex;" alt="{\displaystyle f_{s}}" loading="lazy"></span> samples/second = <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{s}/2}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f_{s}/2}</annotation>

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</math></span><img src="./9ec3aa7a5391f6df43f4893c2a0565d910b1fbb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.467ex; height:2.843ex;" alt="{\displaystyle f_{s}/2}" loading="lazy"></span>, known as the <a href="Nyquist_frequency" title="Nyquist frequency">Nyquist frequency</a> of the sampler. Therefore, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>

</semantics>

</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span> is usually the output of a <a href="Low-pass_filter" title="Low-pass filter">low-pass filter</a>, functionally known as an <i>anti-aliasing filter</i>. 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.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Practical_considerations">Practical considerations</h2></div>

<p>In practice, the continuous signal is sampled using an <a href="Analog-to-digital_converter" title="Analog-to-digital converter">analog-to-digital converter</a> (ADC), a device with various physical limitations. This results in deviations from the theoretically perfect reconstruction, collectively referred to as <a href="Distortion" title="Distortion">distortion</a>.

</p><p>Various types of distortion can occur, including:

</p>

<ul><li><a href="Aliasing" title="Aliasing">Aliasing</a>. Some amount of aliasing is inevitable because only theoretical, infinitely long, functions can have no frequency content above the Nyquist frequency. Aliasing can be made <a href="Arbitrarily_small" class="mw-redirect" title="Arbitrarily small">arbitrarily small</a> by using a <a href="Sufficiently_large" class="mw-redirect" title="Sufficiently large">sufficiently large</a> order of the anti-aliasing filter.</li>

<li><a href="Analog-to-digital_converter#Jitter" title="Analog-to-digital converter">Aperture error</a> 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.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In a <a href="Capacitor" title="Capacitor">capacitor</a>-based <a href="Sample_and_hold" title="Sample and hold">sample and hold</a> circuit, aperture errors are introduced by multiple mechanisms. For example, the capacitor cannot instantly track the input signal and the capacitor can not instantly be isolated from the input signal.</li>

<li><a href="Jitter" title="Jitter">Jitter</a> or deviation from the precise sample timing intervals.</li>

<li><a href="Noise_(physics)" class="mw-redirect" title="Noise (physics)">Noise</a>, including thermal sensor noise, <a href="Analog_circuit" class="mw-redirect" title="Analog circuit">analog circuit</a> noise, etc..</li>

<li><a href="Slew_rate" title="Slew rate">Slew rate</a> limit error, caused by the inability of the ADC input value to change sufficiently rapidly.</li>

<li><a href="Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a> as a consequence of the finite precision of words that represent the converted values.</li>

<li>Error due to other <a href="Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> effects of the mapping of input voltage to converted output value (in addition to the effects of quantization).</li></ul>

<p>Although the use of <a href="Oversampling" title="Oversampling">oversampling</a> can completely eliminate aperture error and aliasing by shifting them out of the passband, this technique cannot be practically used above a few GHz, and may be 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.

</p><p>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 <a href="Low-pass_filter" title="Low-pass filter">low-pass filtering</a>. The non-linearities of either ADC or DAC are analyzed by replacing the ideal <a href="Linear_function" title="Linear function">linear function</a> mapping with a proposed <a href="Nonlinear_system" title="Nonlinear system">nonlinear function</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Audio_sampling">Audio sampling</h3></div>

<p><a href="Digital_audio" title="Digital audio">Digital audio</a> uses <a href="Pulse-code_modulation" title="Pulse-code modulation">pulse-code modulation</a> (PCM) 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.

</p><p>When it is necessary to capture audio covering the entire 20–20,000 Hz range of <a href="Auditory_system" title="Auditory system">human hearing</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz (<a href="Compact_Disc_Digital_Audio" title="Compact Disc Digital Audio">CD</a>), 48 kHz, 88.2 kHz, or 96 kHz.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The approximately double-rate requirement is a consequence of the <a href="Nyquist_theorem" class="mw-redirect" title="Nyquist theorem">Nyquist theorem</a>. Sampling rates higher than about 50 kHz to 60 kHz cannot supply more usable information for human listeners. Early <a href="Professional_audio" title="Professional audio">professional audio</a> equipment manufacturers chose sampling rates in the region of 40 to 50 kHz for this reason.

</p><p>There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Even though <a href="Ultrasound" title="Ultrasound">ultrasonic</a> frequencies are inaudible to humans, recording and mixing at higher sampling rates is effective in eliminating the distortion that can be caused by <a href="Aliasing#Folding" title="Aliasing">foldback aliasing</a>. Conversely, ultrasonic sounds may interact with and modulate the audible part of the frequency spectrum (<a href="Intermodulation_distortion" class="mw-redirect" title="Intermodulation distortion">intermodulation distortion</a>), <i>degrading</i> the fidelity.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for <a href="Analog-to-digital_converter" title="Analog-to-digital converter">ADCs</a> and <a href="Digital-to-analog_converter" title="Digital-to-analog converter">DACs</a>, but with modern oversampling <a href="Delta-sigma_modulation" title="Delta-sigma modulation">delta-sigma-converters</a> this advantage is less important.

</p><p>The <a href="Audio_Engineering_Society" title="Audio Engineering Society">Audio Engineering Society</a> recommends 48 kHz sampling rate for most applications but gives recognition to 44.1 kHz for CD and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed <a href="Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filtering</a>.<sup id="cite_ref-AES5_11-0" class="reference"><a href="#cite_note-AES5-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Both Lavry Engineering and J. Robert Stuart state that the ideal sampling rate would be about 60 kHz, but since this is not a standard frequency, recommend 88.2 or 96 kHz for recording purposes.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

</p><p>A more complete list of common audio sample rates is:

</p>

<th>Sampling rate

</th>

<th>Use

</th></tr>

<tr>

<td>5,512.5 Hz

</td>

<td>Supported in <a href="Adobe_Flash" title="Adobe Flash">Flash</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</td></tr>

<tr>

<td>8,000 Hz

</td>

<td><a href="Telephone" title="Telephone">Telephone</a> and encrypted <a href="Walkie-talkie" title="Walkie-talkie">walkie-talkie</a>, <a href="Wireless_intercom" title="Wireless intercom">wireless intercom</a> and <a href="Wireless_microphone" title="Wireless microphone">wireless microphone</a> transmission; adequate for human speech but without <a href="Sibilance" class="mw-redirect" title="Sibilance">sibilance</a> (<i>ess</i> sounds like <i>eff</i> (<span class="IPA nowrap" lang="und-Latn-fonipa" title="Representation in the International Phonetic Alphabet (IPA)">/<a href="Voiceless_alveolar_fricative" title="Voiceless alveolar fricative">s</a>/</span>, <span class="IPA nowrap" lang="und-Latn-fonipa" title="Representation in the International Phonetic Alphabet (IPA)">/<a href="Voiceless_labiodental_fricative" title="Voiceless labiodental fricative">f</a>/</span>)).

</td></tr>

<tr>

<td>11,025 Hz

</td>

<td>One quarter the sampling rate of audio CDs; used for lower-quality PCM, MPEG audio and for audio analysis of subwoofer bandpasses.

</td></tr>

<tr>

<td>16,000 Hz

</td>

<td><a href="Wideband" title="Wideband">Wideband</a> frequency extension over standard <a href="Telephone" title="Telephone">telephone</a> <a href="Narrowband" title="Narrowband">narrowband</a> 8,000 Hz. Used in most modern <a href="VoIP" class="mw-redirect" title="VoIP">VoIP</a> and VVoIP communication products.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

</td></tr>

<tr>

<td>22,050 Hz

</td>

<td>One half the sampling rate of audio CDs; used for lower-quality PCM and MPEG audio and for audio analysis of low frequency energy. Suitable for digitizing early 20th century audio formats such as <a href="78_rpm" class="mw-redirect" title="78 rpm">78s</a> and <a href="AM_Broadcasting" class="mw-redirect" title="AM Broadcasting">AM Radio</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</td></tr>

<tr>

<td>32,000 Hz

</td>

<td><a href="MiniDV" class="mw-redirect" title="MiniDV">miniDV</a> digital video <a href="Camcorder" title="Camcorder">camcorder</a>, video tapes with extra channels of audio (e.g. <a href="DVCAM" class="mw-redirect" title="DVCAM">DVCAM</a> with four channels of audio), <a href="Digital_Audio_Tape" title="Digital Audio Tape">DAT</a> (LP mode), Germany's <a href="https://de.wikipedia.org/wiki/Digitales_Satellitenradio" class="extiw external" title="de:Digitales Satellitenradio">Digitales Satellitenradio</a>, <a href="NICAM" title="NICAM">NICAM</a> digital audio, used alongside analogue television sound in some countries. High-quality digital <a href="Wireless_microphone" title="Wireless microphone">wireless microphones</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Suitable for digitizing <a href="FM_broadcasting" title="FM broadcasting">FM radio</a>.

</td></tr>

<tr>

<td>37,800 Hz

</td>

<td><a href="CD-ROM#CD-ROM_XA_extension" title="CD-ROM">CD-XA audio</a>

</td></tr>

<tr>

<td>44,055.9 Hz

</td>

<td>Used by digital audio locked to <a href="NTSC" title="NTSC">NTSC</a> <i>color</i> video signals (3 samples per line, 245 lines per field, 59.94 fields per second = 29.97 <a href="Frames_per_second" class="mw-redirect" title="Frames per second">frames per second</a>).

</td></tr>

<tr>

</td>

<td><a href="Audio_CD" class="mw-redirect" title="Audio CD">Audio CD</a>, also most commonly used with <a href="MPEG-1" title="MPEG-1">MPEG-1</a> audio (<a href="VCD" class="mw-redirect" title="VCD">VCD</a>, <a href="SVCD" class="mw-redirect" title="SVCD">SVCD</a>, <a href="MP3" title="MP3">MP3</a>). Originally chosen by <a href="Sony" title="Sony">Sony</a> because it could be recorded on modified video equipment running at either 25 frames per second (PAL) or 30 frame/s (using an NTSC <i>monochrome</i> video recorder) and cover the 20 kHz bandwidth thought necessary to match professional analog recording equipment of the time. A <a href="PCM_adaptor" title="PCM adaptor">PCM adaptor</a> would fit digital audio samples into the analog video channel of, for example, <a href="PAL" title="PAL">PAL</a> video tapes using 3 samples per line, 588 lines per frame, 25 frames per second.

</td></tr>

<tr>

<td>47,250 Hz

</td>

<td>world's first commercial <a href="Pulse-code_modulation" title="Pulse-code modulation">PCM</a> sound recorder by <a href="Nippon_Columbia" title="Nippon Columbia">Nippon Columbia</a> (Denon)

</td></tr>

<tr>

</td>

<td>The standard audio sampling rate used by professional digital video equipment such as tape recorders, video servers, vision mixers and so on. This rate was chosen because it could reconstruct frequencies up to 22 kHz and work with 29.97 frames per second NTSC video – as well as 25 frame/s, 30 frame/s and 24 frame/s systems. With 29.97 frame/s systems it is necessary to handle 1601.6 audio samples per frame delivering an integer number of audio samples only every fifth video frame.<sup id="cite_ref-AES5_11-1" class="reference"><a href="#cite_note-AES5-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Also used for sound with consumer video formats like DV, <a href="Digital_TV" class="mw-redirect" title="Digital TV">digital TV</a>, <a href="DVD" title="DVD">DVD</a>, and films. The professional <a href="Serial_digital_interface" title="Serial digital interface">serial digital interface</a> (SDI) and High-definition Serial Digital Interface (HD-SDI) used to connect broadcast television equipment together uses this audio sampling frequency. Most professional audio gear uses 48 kHz sampling, including <a href="Mixing_console" title="Mixing console">mixing consoles</a>, and <a href="Digital_recording" title="Digital recording">digital recording</a> devices.

</td></tr>

<tr>

<td>50,000 Hz

</td>

<td>First commercial digital audio recorders from the late 70s from <a href="3M" title="3M">3M</a> and <a href="Soundstream" title="Soundstream">Soundstream</a>.

</td></tr>

<tr>

<td>50,400 Hz

</td>

<td>Sampling rate used by the <a href="X-80" class="mw-redirect" title="X-80">Mitsubishi X-80</a> digital audio recorder.

</td></tr>

<tr>

<td>64,000 Hz

</td>

<td>Uncommonly used, but supported by some hardware<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and software.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</td></tr>

<tr>

<td>88,200 Hz

</td>

<td>Sampling rate used by some professional recording equipment when the destination is CD (multiples of 44,100 Hz). Some pro audio gear uses (or is able to select) 88.2 kHz sampling, including mixers, EQs, compressors, reverb, crossovers, and recording devices.

</td></tr>

<tr>

<td>96,000 Hz

</td>

<td><a href="DVD-Audio" title="DVD-Audio">DVD-Audio</a>, some <a href="LPCM" class="mw-redirect" title="LPCM">LPCM</a> DVD tracks, <a href="BD-ROM" class="mw-redirect" title="BD-ROM">BD-ROM</a> (Blu-ray Disc) audio tracks, <a href="HD_DVD" title="HD DVD">HD DVD</a> (High-Definition DVD) audio tracks. Some professional recording and production equipment is able to select 96 kHz sampling. This sampling frequency is twice the 48 kHz standard commonly used with audio on professional equipment.

</td></tr>

<tr>

<td>176,400 Hz

</td>

<td>Sampling rate used by <a href="HDCD" class="mw-redirect" title="HDCD">HDCD</a> recorders and other professional applications for CD production. Four times the frequency of 44.1 kHz.

</td></tr>

<tr>

<td>192,000 Hz

</td>

<td><a href="DVD-Audio" title="DVD-Audio">DVD-Audio</a>, some <a href="LPCM" class="mw-redirect" title="LPCM">LPCM</a> DVD tracks, <a href="BD-ROM" class="mw-redirect" title="BD-ROM">BD-ROM</a> (Blu-ray Disc) audio tracks, and <a href="HD_DVD" title="HD DVD">HD DVD</a> (High-Definition DVD) audio tracks, High-Definition audio recording devices and audio editing software. This sampling frequency is four times the 48 kHz standard commonly used with audio on professional video equipment.

</td></tr>

<tr>

<td>352,800 Hz

</td>

<td><a href="Digital_eXtreme_Definition" title="Digital eXtreme Definition">Digital eXtreme Definition</a>, used for recording and editing <a href="Super_Audio_CD" title="Super Audio CD">Super Audio CDs</a>, as 1-bit <a href="Direct_Stream_Digital" title="Direct Stream Digital">Direct Stream Digital (DSD)</a> is not suited for editing. 8 times the frequency of 44.1 kHz.

</td></tr>

<tr>

<td>384,000 Hz

</td>

<td>Maximum sample rate available in common software.

</td></tr>

<tr>

<td>2,822,400 Hz

</td>

<td><a href="Super_Audio_CD" title="Super Audio CD">SACD</a>, 1-bit <a href="Delta-sigma_modulation" title="Delta-sigma modulation">delta-sigma modulation</a> process known as <a href="Direct_Stream_Digital" title="Direct Stream Digital">Direct Stream Digital</a>, co-developed by <a href="Sony" title="Sony">Sony</a> and <a href="Philips" title="Philips">Philips</a>.

</td></tr>

<tr>

<td>5,644,800 Hz

</td>

<td>Double-Rate DSD, 1-bit <a href="Direct_Stream_Digital" title="Direct Stream Digital">Direct Stream Digital</a> at 2× the rate of the SACD. Used in some professional DSD recorders.

</td></tr>

<tr>

<td>11,289,600 Hz

</td>

<td>Quad-Rate DSD, 1-bit <a href="Direct_Stream_Digital" title="Direct Stream Digital">Direct Stream Digital</a> at 4× the rate of the SACD. Used in some uncommon professional DSD recorders.

</td></tr>

<tr>

<td>22,579,200 Hz

</td>

<td>Octuple-Rate DSD, 1-bit <a href="Direct_Stream_Digital" title="Direct Stream Digital">Direct Stream Digital</a> at 8× the rate of the SACD. Used in rare experimental DSD recorders. Also known as DSD512.

</td></tr>

<tr>

<td>45,158,400 Hz

</td>

<td>Sexdecuple-Rate DSD, 1-bit <a href="Direct_Stream_Digital" title="Direct Stream Digital">Direct Stream Digital</a> at 16× the rate of the SACD. Used in rare experimental DSD recorders. Also known as DSD1024.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>B<span class="cite-bracket">]</span></a></sup>

</td></tr></tbody></table>

<div class="mw-heading mw-heading4"><h4 id="Bit_depth">Bit depth</h4></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Audio_bit_depth" title="Audio bit depth">Audio bit depth</a></div>

<p>Audio is typically recorded at 8-, 16-, and 24-bit depth; which yield a theoretical maximum <a href="Signal-to-quantization-noise_ratio" title="Signal-to-quantization-noise ratio">signal-to-quantization-noise ratio</a> (SQNR) for a pure <a href="Sine_wave" title="Sine wave">sine wave</a> of, approximately; 49.93 <a href="Decibel" title="Decibel">dB</a>, 98.09 dB, and 122.17 dB.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> CD quality audio uses 16-bit samples. <a href="Thermal_noise" class="mw-redirect" title="Thermal noise">Thermal noise</a> limits the true number of bits that can be used in quantization. Few analog systems have <a href="Signal-to-noise_ratio" title="Signal-to-noise ratio">signal to noise ratios</a> (SNR) exceeding 120 dB. However, <a href="Digital_signal_processing" title="Digital signal processing">digital signal processing</a> 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.

</p>

<div class="mw-heading mw-heading4"><h4 id="Speech_sampling">Speech sampling</h4></div>

<p>Speech signals, i.e., signals intended to carry only human <a href="Speech_communication" class="mw-redirect" title="Speech communication">speech</a>, can usually be sampled at a much lower rate. For most <a href="Phoneme" title="Phoneme">phonemes</a>, almost all of the energy is contained in the 100 Hz – 4 kHz range, allowing a sampling rate of 8 kHz. This is the sampling rate used by nearly all <a href="Telephony" title="Telephony">telephony</a> systems, which use the <a href="G.711" title="G.711">G.711</a> sampling and quantization specifications.

</p>

<div class="mw-heading mw-heading3"><h3 id="Video_sampling">Video sampling</h3></div>

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<p><a href="Standard-definition_television" title="Standard-definition television">Standard-definition television</a> (SDTV) uses either 720 by 480 <a href="Pixels" class="mw-redirect" title="Pixels">pixels</a> (US <a href="NTSC" title="NTSC">NTSC</a> 525-line) or 720 by 576 pixels (UK <a href="PAL" title="PAL">PAL</a> 625-line) for the visible picture area.

</p><p><a href="High-definition_television" title="High-definition television">High-definition television</a> (HDTV) uses <a href="720p" title="720p">720p</a> (progressive), <a href="1080i" title="1080i">1080i</a> (interlaced), and <a href="1080p" title="1080p">1080p</a> (progressive, also known as Full-HD).

</p><p>In <a href="Digital_video" title="Digital video">digital video</a>, the temporal sampling rate is defined as the <a href="Frame_rate" title="Frame rate">frame rate</a> – or rather the <a href="Field_rate" class="mw-redirect" title="Field rate">field rate</a> – rather than the notional pixel clock. The image sampling frequency is the repetition rate of the sensor integration period. Since the integration period may be significantly shorter than the time between repetitions, the sampling frequency can be different from the inverse of the sample time:

</p>

<ul><li>50 Hz – <a href="PAL" title="PAL">PAL</a> video</li>

<li>60 / 1.001 Hz ~= 59.94 Hz – <a href="NTSC" title="NTSC">NTSC</a> video</li></ul>

<p>Video <a href="Digital-to-analog_converter" title="Digital-to-analog converter">digital-to-analog converters</a> operate in the megahertz range (from ~3 MHz for low quality composite video scalers in early game consoles, to 250 MHz or more for the highest-resolution VGA output).

</p><p>When analog video is converted to <a href="Digital_video" title="Digital video">digital video</a>, a different sampling process occurs, this time at the pixel frequency, corresponding to a spatial sampling rate along <a href="Scan_line" title="Scan line">scan lines</a>. A common pixel sampling rate is:

</p>

<ul><li>13.5 MHz – <a href="CCIR_601" class="mw-redirect" title="CCIR 601">CCIR 601</a>, <a href="D1_video" class="mw-redirect" title="D1 video">D1 video</a></li></ul>

<p>Spatial sampling in the other direction is determined by the spacing of scan lines in the <a href="Raster_graphics" title="Raster graphics">raster</a>. The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height.

</p><p>Spatial <a href="Aliasing" title="Aliasing">aliasing</a> of high-frequency <a href="Luma_(video)" title="Luma (video)">luma</a> or <a href="Chrominance" title="Chrominance">chroma</a> video components shows up as a <a href="Moir%C3%A9_pattern" title="Moiré pattern">moiré pattern</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="3D_sampling">3D sampling</h3></div>

<p>The process of <a href="Volume_rendering" title="Volume rendering">volume rendering</a> samples a 3D grid of <a href="Voxel" title="Voxel">voxels</a> to produce 3D renderings of sliced (tomographic) data. The 3D grid is assumed to represent a continuous region of 3D space. Volume rendering is common in medical imaging, <a href="X-ray_computed_tomography" class="mw-redirect" title="X-ray computed tomography">X-ray computed tomography</a> (CT/CAT), <a href="Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging</a> (MRI), <a href="Positron_emission_tomography" title="Positron emission tomography">positron emission tomography</a> (PET) are some examples. It is also used for <a href="Seismic_tomography" title="Seismic tomography">seismic tomography</a> and other applications.

</p>

<div class="mw-heading mw-heading2"><h2 id="Undersampling">Undersampling</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Undersampling" title="Undersampling">Undersampling</a></div>

<p>When a <a href="Bandpass" class="mw-redirect" title="Bandpass">bandpass</a> signal is sampled slower than its <a href="Nyquist_rate" title="Nyquist rate">Nyquist rate</a>, the samples are indistinguishable from samples of a low-frequency <a href="Aliasing" title="Aliasing">alias</a> of the high-frequency signal. That is often done purposefully in such a way that the lowest-frequency alias satisfies the <a href="Nyquist_rate" title="Nyquist rate">Nyquist criterion</a>, because the bandpass signal is still uniquely represented and recoverable. Such <a href="Undersampling" title="Undersampling">undersampling</a> is also known as <i>bandpass sampling</i>, <i>harmonic sampling</i>, <i>IF sampling</i>, and <i>direct IF to digital conversion.</i><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Oversampling">Oversampling</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Oversampling" title="Oversampling">Oversampling</a></div>

<p>Oversampling is used in most modern analog-to-digital converters to reduce the distortion introduced by practical <a href="Digital-to-analog_converter" title="Digital-to-analog converter">digital-to-analog converters</a>, such as a <a href="Zero-order_hold" title="Zero-order hold">zero-order hold</a> instead of idealizations like the <a href="Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">Whittaker–Shannon interpolation formula</a>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Complex_sampling">Complex sampling </h2></div>

<p><b>Complex sampling</b> (or <b>I/Q sampling</b>) is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>C<span class="cite-bracket">]</span></a></sup> When one waveform, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {s}}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {s}}(t)}</annotation>

</semantics>

</math></span><img src="./68c41a48746327134561c7e5ef1b34f835472348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.941ex; height:2.843ex;" alt="{\displaystyle {\hat {s}}(t)}" loading="lazy"></span>, is the <a href="Hilbert_transform" title="Hilbert transform">Hilbert transform</a> of the other waveform, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>

</semantics>

</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span>, the complex-valued function, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{a}(t)\triangleq s(t)+i\cdot {\hat {s}}(t)}">

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{a}(t)\triangleq s(t)+i\cdot {\hat {s}}(t)}</annotation>

</semantics>

</math></span><img src="./9270669f6978da5ea193138bcd66d8660d26ed93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.942ex; height:3.009ex;" alt="{\displaystyle s_{a}(t)\triangleq s(t)+i\cdot {\hat {s}}(t)}" loading="lazy"></span>, is called an <a href="Analytic_signal" title="Analytic signal">analytic signal</a>, whose Fourier transform is zero for all negative values of frequency. In that case, the <a href="Nyquist_rate" title="Nyquist rate">Nyquist rate</a> for a waveform with no frequencies ≥ <i>B</i> can be reduced to just <i>B</i> (complex samples/sec), instead of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2B}</annotation>

</semantics>

</math></span><img src="./5385e37d2c4c6828636b510f2b4f993ec1286536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.926ex; height:2.176ex;" alt="{\displaystyle 2B}" loading="lazy"></span> (real samples/sec).<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>D<span class="cite-bracket">]</span></a></sup> More apparently, the <a href="Baseband#Equivalent_baseband_signal" title="Baseband">equivalent baseband waveform</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{a}(t)\cdot e^{-i2\pi {\frac {B}{2}}t}}">

<msub>

</mrow>

</msub>

<msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{a}(t)\cdot e^{-i2\pi {\frac {B}{2}}t}}</annotation>

</semantics>

</math></span><img src="./62d900fd07b7941b9c547aeaeb256f3dbccd4d7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.888ex; height:4.009ex;" alt="{\displaystyle s_{a}(t)\cdot e^{-i2\pi {\frac {B}{2}}t}}" loading="lazy"></span>, also has a Nyquist rate of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>, because all of its non-zero frequency content is shifted into the interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-B/2,B/2]}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [-B/2,B/2]}</annotation>

</semantics>

</math></span><img src="./3eaba70af89480aab78d7c35025882696835bcb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.314ex; height:2.843ex;" alt="{\displaystyle [-B/2,B/2]}" loading="lazy"></span>.

</p><p>Although complex-valued samples can be obtained as described above, they are also created by manipulating samples of a real-valued waveform. For instance, the equivalent baseband waveform can be created without explicitly computing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {s}}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {s}}(t)}</annotation>

</semantics>

</math></span><img src="./68c41a48746327134561c7e5ef1b34f835472348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.941ex; height:2.843ex;" alt="{\displaystyle {\hat {s}}(t)}" loading="lazy"></span>, by processing the product sequence, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[s(nT)\cdot e^{-i2\pi {\frac {B}{2}}Tn}\right]}">

<mrow>

<mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[s(nT)\cdot e^{-i2\pi {\frac {B}{2}}Tn}\right]}</annotation>

</semantics>

</math></span><img src="./1ae973e6561275c7668442cbdd54557d6a377cec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.982ex; height:6.176ex;" alt="{\displaystyle \left[s(nT)\cdot e^{-i2\pi {\frac {B}{2}}Tn}\right]}" loading="lazy"></span>,<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>E<span class="cite-bracket">]</span></a></sup> through a digital low-pass filter whose cutoff frequency is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B/2}</annotation>

</semantics>

</math></span><img src="./a39b912a2d972cec1195e0f9f1f791dd7cb0f1a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle B/2}" loading="lazy"></span>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>F<span class="cite-bracket">]</span></a></sup> 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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>

</semantics>

</p>

<ul><li><a href="Crystal_oscillator_frequencies" title="Crystal oscillator frequencies">Crystal oscillator frequencies</a></li>

<li><a href="Downsampling" class="mw-redirect" title="Downsampling">Downsampling</a></li>

<li><a href="Upsampling" title="Upsampling">Upsampling</a></li>

<li><a href="Multidimensional_sampling" title="Multidimensional sampling">Multidimensional sampling</a></li>

<li><a href="In-phase_and_quadrature_components" title="In-phase and quadrature components">In-phase and quadrature components</a> and <a href="In-phase_and_quadrature_components#I/Q_data" title="In-phase and quadrature components">I/Q data</a></li>

<li><a href="Sample_rate_conversion" class="mw-redirect" title="Sample rate conversion">Sample rate conversion</a></li>

<li><a href="Digitizing" class="mw-redirect" title="Digitizing">Digitizing</a></li>

<li><a href="Sample_and_hold" title="Sample and hold">Sample and hold</a></li>

<li><a href="Beta_encoder" title="Beta encoder">Beta encoder</a></li>

<li><a href="Kell_factor" title="Kell factor">Kell factor</a></li>

<li><a href="Normalized_frequency_(digital_signal_processing)" class="mw-redirect" title="Normalized frequency (digital signal processing)">Normalized frequency</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-upper-alpha">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">For example, "number of samples" in signal processing is roughly equivalent to "<a href="Sample_size" class="mw-redirect" title="Sample size">sample size</a>" in statistics.</span>

</li>

<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Even higher DSD sampling rates exist, but the benefits of those are likely imperceptible, and the size of those files would be humongous.</span>

</li>

<li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Sample-pairs are also sometimes viewed as points on a <a href="Constellation_diagram" title="Constellation diagram">constellation diagram</a>.</span>

</li>

<li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">When the complex sample-rate is <i>B</i>, a frequency component at 0.6 <i>B</i>, for instance, will have an alias at −0.4 <i>B</i>, which is unambiguous because of the constraint that the pre-sampled signal was analytic. Also see <a href="Aliasing#Complex_sinusoids" title="Aliasing">Aliasing § Complex sinusoids</a>.</span>

</li>

<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">When <i>s</i>(<i>t</i>) is sampled at the Nyquist frequency (1/<i>T</i> = 2<i>B</i>), the product sequence simplifies to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[s(nT)\cdot (-i)^{n}\right].}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[s(nT)\cdot (-i)^{n}\right].}</annotation>

</semantics>

</math></span><img src="./7e0394ce8648cfa875d6274242b21368a4ded905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.576ex; height:2.843ex;" alt="{\displaystyle \left[s(nT)\cdot (-i)^{n}\right].}" loading="lazy"></span></span>

</li>

<li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">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.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">H.O. Johansson and C. Svensson, "Time resolution of NMOS sampling switches", IEEE J. Solid-State Circuits Volume: 33, Issue: 2, pp. 237–245, Feb 1998.</span>

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<cite id="CITEREFD'AmbroseChoudhary2003" class="citation web cs1">D'Ambrose, Christoper; Choudhary, Rizwan (2003). Elert, Glenn (ed.). <a rel="nofollow" class="external text" href="https://hypertextbook.com/facts/2003/ChrisDAmbrose.shtml">"Frequency range of human hearing"</a>. <i>The Physics Factbook</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-01-22</span></span>.</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFSelf2012" class="citation book cs1">Self, Douglas (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WzYm1hGnCn4C&pg=PT200"><i>Audio Engineering Explained</i></a>. Taylor & Francis US. pp. 200, 446. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0240812731</bdi>.</cite></span>

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081020231427/http://www.digitalprosound.com/Htm/SoapBox/soap2_Apogee.htm">"Digital Pro Sound"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.digitalprosound.com/Htm/SoapBox/soap2_Apogee.htm">the original</a> on 20 October 2008<span class="reference-accessdate">. Retrieved <span class="nowrap">8 January</span> 2014</span>.</cite></span>

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFColletti2013" class="citation journal cs1">Colletti, Justin (February 4, 2013). <a rel="nofollow" class="external text" href="https://sonicscoop.com/2016/02/19/the-science-of-sample-rates-when-higher-is-better-and-when-it-isnt/?singlepage=1">"The Science of Sample Rates (When Higher Is Better—And When It Isn't)"</a>. <i>Trust Me I'm a Scientist</i><span class="reference-accessdate">. Retrieved <span class="nowrap">February 6,</span> 2013</span>. <q>in many cases, we can hear the sound of higher sample rates not because they are more transparent, but because they are less so. They can actually introduce unintended distortion in the audible spectrum</q></cite></span>

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<li id="cite_note-AES5-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-AES5_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-AES5_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.aes.org/publications/standards/search.cfm?docID=14"><i>AES5-2008: AES recommended practice for professional digital audio – Preferred sampling frequencies for applications employing pulse-code modulation</i></a>, Audio Engineering Society, 2008<span class="reference-accessdate">, retrieved <span class="nowrap">2010-01-18</span></span></cite></span>

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<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFLavry2012" class="citation web cs1">Lavry, Dan (May 3, 2012). <a rel="nofollow" class="external text" href="http://www.lavryengineering.com/pdfs/lavry-white-paper-the_optimal_sample_rate_for_quality_audio.pdf">"The Optimal Sample Rate for Quality Audio"</a> <span class="cs1-format">(PDF)</span>. <i>Lavry Engineering Inc</i>. <q>Although 60 KHz would be closer to the ideal; given the existing standards, 88.2 KHz and 96 KHz are closest to the optimal sample rate.</q></cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFLavry" class="citation web cs1">Lavry, Dan. <a rel="nofollow" class="external text" href="https://www.gearslutz.com/board/showpost.php?p=7883017&postcount=15&s=b05e50b41d1789054724882582d8351b">"The Optimal Sample Rate for Quality Audio"</a>. <i>Gearslutz</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-11-10</span></span>. <q>I am trying to accommodate all ears, and there are reports of few people that can actually hear slightly above 20KHz. I do think that 48 KHz is pretty good compromise, but 88.2 or 96 KHz yields some additional margin.</q></cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFLavry" class="citation web cs1">Lavry, Dan. <a rel="nofollow" class="external text" href="https://www.gearslutz.com/board/showpost.php?p=1234224&postcount=74">"To mix at 96k or not?"</a>. <i>Gearslutz</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-11-10</span></span>. <q>Nowdays [sic] there are a number of good designers and ear people that find 60-70KHz sample rate to be the optimal rate for the ear. It is fast enough to include what we can hear, yet slow enough to do it pretty accurately.</q></cite></span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFStuart1998" class="citation book cs1">Stuart, J. Robert (1998). <i>Coding High Quality Digital Audio</i>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.501.6731">10.1.1.501.6731</a></span>. <q>both psychoacoustic analysis and experience tell us that the minimum rectangular channel necessary to ensure transparency uses linear PCM with 18.2-bit samples at 58 kHz. ... there are strong arguments for maintaining integer relationships with existing sampling rates – which suggests that 88.2 kHz or 96 kHz should be adopted.</q></cite></span>

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<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://open-flash.github.io/mirrors/swf-spec-19.pdf">"SWF File Format Specification - Version 19"</a> <span class="cs1-format">(PDF)</span>. 2013.</cite></span>

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<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.voipsupply.com/cisco-hd-voice">"Cisco VoIP Phones, Networking and Accessories - VoIP Supply"</a>.</cite></span>

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<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090914133005/http://www.restoring78s.co.uk/Procedure%20Part%201.htm">"The restoration procedure – part 1"</a>. Restoring78s.co.uk. Archived from <a rel="nofollow" class="external text" href="http://www.restoring78s.co.uk/Procedure%20Part%201.htm">the original</a> on 2009-09-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-01-18</span></span>. <q>For most records a sample rate of 22050 in stereo is adequate. An exception is likely to be recordings made in the second half of the century, which may need a sample rate of 44100.</q></cite></span>

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<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110209050359/http://zaxcom.com/transmitters.htm">"Zaxcom digital wireless transmitters"</a>. Zaxcom.com. Archived from <a rel="nofollow" class="external text" href="http://www.zaxcom.com/transmitters.htm">the original</a> on 2011-02-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-01-18</span></span>.</cite></span>

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<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.rme-audio.de/en/products/hdsp_9632.php">"RME: Hammerfall DSP 9632"</a>. <i>www.rme-audio.de</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-12-18</span></span>. <q>Supported sample frequencies: Internally 32, 44.1, 48, 64, 88.2, 96, 176.4, 192 kHz.</q></cite></span>

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<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20181218145630/https://www.pioneer-audiovisual.eu/uk/products/sx-s30dab">"SX-S30DAB | Pioneer"</a>. <i>www.pioneer-audiovisual.eu</i>. Archived from <a rel="nofollow" class="external text" href="https://www.pioneer-audiovisual.eu/uk/products/sx-s30dab">the original</a> on 2018-12-18<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-12-18</span></span>. <q>Supported sampling rates: 44.1 kHz, 48 kHz, 64 kHz, 88.2 kHz, 96 kHz, 176.4 kHz, 192 kHz</q></cite></span>

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<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFCristina_BachmannSchütte" class="citation web cs1">Cristina Bachmann, Heiko Bischoff; Schütte, Benjamin. <a rel="nofollow" class="external text" href="https://steinberg.help/wavelab_pro/v9.5/en/wavelab/topics/master_section/master_section_customize_sample_rate_menu_dialog_r.html">"Customize Sample Rate Menu"</a>. <i>Steinberg WaveLab Pro</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-12-18</span></span>. <q>Common Sample Rates: 64 000 Hz</q></cite></span>

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<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20181218102147/https://getsatisfaction.com/m-audio/topics/m-track-2x2m-cubase-pro-9-can-t-change-sample-rate">"M Track 2x2M Cubase Pro 9 can ́t change Sample Rate"</a>. <i>M-Audio</i>. Archived from <a rel="nofollow" class="external text" href="https://getsatisfaction.com/m-audio/topics/m-track-2x2m-cubase-pro-9-can-t-change-sample-rate">the original</a> on 2018-12-18<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-12-18</span></span>. <q>[Screenshot of Cubase]</q></cite></span>

</li>

<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.analog.com/static/imported-files/tutorials/MT-001.pdf">"MT-001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N + 1.76dB," and Why You Should Care"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.analog.com/static/imported-files/tutorials/MT-001.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2022-10-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-01-19</span></span>.</cite></span>

</li>

<cite id="CITEREFWalt_Kester2003" class="citation book cs1">Walt Kester (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G8XyNItpy8AC&pg=PA20"><i>Mixed-signal and DSP design techniques</i></a>. Newnes. p. 20. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-7611-3</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">8 January</span> 2014</span>.</cite></span>

</li>

<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilliam_Morris_Hartmann1997" class="citation book cs1">William Morris Hartmann (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3N72rIoTHiEC&pg=PA485"><i>Signals, Sound, and Sensation</i></a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1563962837</bdi>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Matt Pharr, Wenzel Jakob and Greg Humphreys, <i>Physically Based Rendering: From Theory to Implementation, 3rd ed.</i>, Morgan Kaufmann, November 2016. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0128006450</bdi>. The chapter on sampling (<a rel="nofollow" class="external text" href="http://www.pbrt.org/chapters/pbrt_chapter7.pdf">available online</a>) is nicely written with diagrams, core theory and code sample.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.stsip.org">Journal devoted to Sampling Theory</a></li>

<li><a rel="nofollow" class="external text" href="http://whiteboard.ping.se/SDR/IQ">I/Q Data for Dummies</a> – a page trying to answer the question <i>Why I/Q Data?</i></li>

<li><a rel="nofollow" class="external text" href="http://webdemo.inue.uni-stuttgart.de/webdemos/02_lectures/uebertragungstechnik_1/sampling_theorem/">Sampling of analog signals</a> – an interactive presentation in a web-demo at the Institute of Telecommunications, University of Stuttgart</li></ul>

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</style><div id="Digital_signal_processing96" style="font-size:114%;margin:0 4em"><a href="Digital_signal_processing" title="Digital signal processing">Digital signal processing</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Detection_theory" title="Detection theory">Detection theory</a></li>

<li><a href="Discrete_time_and_continuous_time" title="Discrete time and continuous time">Discrete signal</a></li>

<li><a href="Estimation_theory" title="Estimation theory">Estimation theory</a></li>

<li><a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sub-fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Audio_signal_processing" title="Audio signal processing">Audio signal processing</a></li>

<li><a href="Digital_image_processing" title="Digital image processing">Digital image processing</a></li>

<li><a href="Speech_processing" title="Speech processing">Speech processing</a></li>

<li><a href="Statistical_signal_processing" class="mw-redirect" title="Statistical signal processing">Statistical signal processing</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Techniques</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Z-transform" title="Z-transform">Z-transform</a>

<ul><li><a href="Advanced_z-transform" title="Advanced z-transform">Advanced z-transform</a></li>

<li><a href="Matched_Z-transform_method" title="Matched Z-transform method">Matched Z-transform method</a></li></ul></li>

<li><a href="Bilinear_transform" title="Bilinear transform">Bilinear transform</a></li>

<li><a href="Constant-Q_transform" title="Constant-Q transform">Constant-Q transform</a></li>

<li><a href="Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a> (DCT)</li>

<li><a href="Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a> (DFT)</li>

<li><a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT)</li>

<li><a href="Impulse_invariance" title="Impulse invariance">Impulse invariance</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>

<li><a href="Laplace_transform" title="Laplace transform">Laplace transform</a></li>

<li><a href="Post's_inversion_formula" class="mw-redirect" title="Post's inversion formula">Post's inversion formula</a></li>

<li><a href="Starred_transform" title="Starred transform">Starred transform</a></li>

<li><a href="Zak_transform" title="Zak transform">Zak transform</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aliasing" title="Aliasing">Aliasing</a></li>

<li><a href="Anti-aliasing_filter" title="Anti-aliasing filter">Anti-aliasing filter</a></li>

<li><a href="Downsampling_(signal_processing)" title="Downsampling (signal processing)">Downsampling</a></li>

<li><a href="Nyquist_rate" title="Nyquist rate">Nyquist rate</a> / <a href="Nyquist_frequency" title="Nyquist frequency">frequency</a></li>

<li><a href="Oversampling" title="Oversampling">Oversampling</a></li>

<li><a href="Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li>

<li><a href="Sampling_rate" class="mw-redirect" title="Sampling rate">Sampling rate</a></li>

<li><a href="Undersampling" title="Undersampling">Undersampling</a></li>

<li><a href="Upsampling" title="Upsampling">Upsampling</a></li></ul>

</div></td></tr></tbody></table></div>

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