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| editor-first=Glenn
| accessdate=2022-01-22
}}</ref> such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz ([[Compact Disc Digital Audio|CD]]), 48 kHz, 88.2 kHz, or 96 kHz.<ref>{{cite book |url=https://books.google.com/books?id=WzYm1hGnCn4C&pg=PT200 |pages=200, 446 |last=Self |first=Douglas |title=Audio Engineering Explained |publisher=Taylor & Francis US |year=2012 |isbn=978-0240812731}}</ref> The approximately double-rate requirement is a consequence of the [[Nyquist theorem]]. Sampling rates higher than about 50 kHz to 60
There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz<ref>{{cite web|url=http://www.digitalprosound.com/Htm/SoapBox/soap2_Apogee.htm |title=Digital Pro Sound |access-date=8 January 2014}}</ref> Even though [[Ultrasound|ultrasonic]] frequencies are inaudible to humans, recording and mixing at higher sampling rates is effective in eliminating the distortion that can be caused by [[Aliasing#Folding|foldback aliasing]]. Conversely, ultrasonic sounds may interact with and modulate the audible part of the frequency spectrum ([[intermodulation distortion]]), ''degrading'' the fidelity.<ref>{{cite journal|last=Colletti|first=Justin|date=February 4, 2013|title=The Science of Sample Rates (When Higher Is Better—And When It Isn't)|url=https://sonicscoop.com/2016/02/19/the-science-of-sample-rates-when-higher-is-better-and-when-it-isnt/?singlepage=1|journal=Trust Me I'm a Scientist|access-date=February 6, 2013|quote=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}}</ref> 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 [[Delta-sigma modulation|delta-sigma-converters]] this advantage is less important.
The [[Audio Engineering Society]] recommends 48
A more complete list of common audio sample rates is:
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'''Complex sampling''' (or '''I/Q sampling''') is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as [[complex numbers]].{{efn-ua|
Sample-pairs are also sometimes viewed as points on a [[constellation diagram]].
}} When one waveform<math>, \hat s(t),</math> is the [[Hilbert transform]] of the other waveform<math>, s(t),\,</math> the complex-valued function, <math>s_a(t) \triangleq s(t) + i\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 <math>2B</math> (real samples/sec).{{efn-ua|
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 {{slink|Aliasing|Complex sinusoids}}.
}} More apparently, the [[Baseband#Equivalent baseband signal|equivalent baseband waveform]], <math>s_a(t)\cdot e^{-i 2\pi \frac{B}{2} t},</math> also has a Nyquist rate of <math>B,</math> because all of its non-zero frequency content is shifted into the interval <math>[-B/2,B/2].</math>
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 <math>\hat s(t),</math> by processing the product sequence<math>, \left [s(nT)\cdot e^{-i 2 \pi \frac{B}{2}Tn}\right ],</math>{{efn-ua|
When ''s''(''t'') is sampled at the Nyquist frequency (1/''T'' {{=}} 2''B''), the product sequence simplifies to <math>\left [s(nT)\cdot (-i)^n\right ].</math>
}} through a digital low-pass filter whose cutoff frequency is <math>B/2.</math>{{efn-ua|
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