Sampling (signal processing): Difference between revisions

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{{Short description|Measurement of a signal at discrete time intervals}}

{{Other uses|Sampling (disambiguation)}}

[[Image:Signal Sampling.svg|thumb|300px|Signal sampling representation. The continuous signal ''S''(''t'') is represented with a green colored line while the discrete samples are indicated by the blue vertical lines.]]

In [[signal processing]], '''sampling''' is the reduction of a [[continuous-time signal]] to a [[discrete-time signal]]. A common example is the conversion of a [[sound wave]] to a sequence of "samples".

A '''sample''' is a value of the [[signal]] at a point in time and/or space; this definition differs from [[Sampling (statistics)|the term's usage in statistics]], which refers to a set of such values.{{efn-ua|For example, "number of samples" in signal processing is roughly equivalent to "[[sample size]]" in statistics.}}

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

For functions that vary with time, let ''S''<math>s(''t'')</math> be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every ''<math>T''</math> seconds, which is called the '''sampling interval''' or '''sampling period'''.<ref>{{cite book | title = Communications Standard Dictionary | author = Martin H. Weik | publisher = Springer | year = 1996 | isbn = 0412083914 | url = https://books.google.com/books?id=jxXDQgAACAAJ&q=Communications+Standard+Dictionary}}</ref><ref name=Moir>{{cite book | title = Rudiments of Signal Processing and Systems | author = Tom J. Moir | publisher = Springer International Publishing AG | year = 2022|pages=459 | isbn = 9783030769475 | url = https://public.ebookcentral.proquest.com/choice/publicfullrecord.aspx?p=6809637|doi=10.1007/978-3-030-76947-5 }}</ref>&nbsp; Then the sampled function is given by the sequence:

: ''S''<math>s(''nT'')</math>, &nbsp; for integer values of ''<math>n''</math>.

{{anchor|Sampling rate}}The '''sampling frequency''' or '''sampling rate''', ''f''<submath>sf_s</submath>, is the average number of samples divided by the interval length overobtained in whichone occursecond, thus {{nowrap|1=''f''<submath>s</sub> f_s= 1/''T''}}</math>, with the unit ''samplesamples per second'', sometimes referred to as [[hertz]], for example e.g. 48&nbsp;kHz is 48,000 ''samples per second''.

Reconstructing a continuous function from samples is done by interpolation algorithms. The [[Whittaker–Shannon interpolation formula]] is mathematically equivalent to an ideal [[low-pass 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 <math>(''T'')</math>, 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 ''<math>s''(''t'')</math>. That mathematical abstraction is sometimes referred to as ''impulse sampling''.<ref>{{cite book |title=Signals and Systems |author=Rao, R. |isbn=9788120338593 |url=https://books.google.com/books?id=4z3BrI717sMC |publisher=Prentice-Hall Of India Pvt. Limited|year=2008 }}</ref>

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 ''<math>s''(''t'')</math> contains frequency components whose cycle length (period) is less than 2 sample intervals (see ''[[Aliasing#Sampling sinusoidal functions|Aliasing]]''). The corresponding frequency limit, in ''cycles per second'' ([[hertz]]), is <math>0.5&nbsp;</math> cycle/sample&nbsp;×&nbsp;''f'' <submath>sf_s</submath>&nbsp; samples/second = ''f''<submath>sf_s/2</submath>/2, known as the [[Nyquist frequency]] of the sampler. Therefore, ''<math>s''(''t'')</math> is usually the output of a [[low-pass 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.<ref>[[Claude E. Shannon|C. E. Shannon]], "Communication in the presence of noise", [[Proc. Institute of Radio Engineers]], vol. 37, no.1, pp. 10–21, Jan. 1949. [http://www.stanford.edu/class/ee104/shannonpaper.pdf Reprint as classic paper in: ''Proc. IEEE'', Vol. 86, No. 2, (Feb 1998)] {{webarchive|url=https://web.archive.org/web/20100208112344/http://www.stanford.edu/class/ee104/shannonpaper.pdf |date=2010-02-08 }}</ref>

== Practical considerations==

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Various types of distortion can occur, including:

* [[Aliasing]]. 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 [[arbitrarily small]] by using a [[sufficiently large]] order of the anti-aliasing filter.

* [[Analog-to-digital converter#Jitter|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.<ref>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.</ref> In a [[capacitor]]-based [[sample and hold]] 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.

* [[Jitter]] or deviation from the precise sample timing intervals.

* [[Noise (physics)|Noise]], including thermal sensor noise, [[analog circuit]] noise, etc..

* [[Slew rate]] limit error, caused by the inability of the ADC input value to change sufficiently rapidly.

* [[Quantization (signal processing)|Quantization]] as a consequence of the finite precision of words that represent the converted values.

* Error due to other [[non-linear]] effects of the mapping of input voltage to converted output value (in addition to the effects of quantization).

Although the use of [[oversampling]] 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.

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 system|nonlinear function]].

== Applications ==

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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&nbsp;kHz ([[Compact Disc Digital Audio|CD]]), 48&nbsp;kHz, 88.2&nbsp;kHz, or 96&nbsp;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&nbsp;kHz to 60&nbsp;kHz cannot supply more usable information for human listeners. Early [[professional audio]] equipment manufacturers chose sampling rates in the region of 40 to 50&nbsp;kHz for this reason.

There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96&nbsp;kHz and even 192&nbsp;kHz<ref>{{cite web |url=http://www.digitalprosound.com/Htm/SoapBox/soap2_Apogee.htm |title=Digital Pro Sound |access-date=8 January 2014 |archive-date=20 October 2008 |archive-url=https://web.archive.org/web/20081020231427/http://www.digitalprosound.com/Htm/SoapBox/soap2_Apogee.htm |url-status=dead }}</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&nbsp;kHz sampling rate for most applications but gives recognition to 44.1&nbsp;kHz for CD and other consumer uses, 32&nbsp;kHz for transmission-related applications, and 96&nbsp;kHz for higher bandwidth or relaxed [[anti-aliasing filter]]ing.<ref name=AES5>{{citation |url=http://www.aes.org/publications/standards/search.cfm?docID=14 |title=AES5-2008: AES recommended practice for professional digital audio – Preferred sampling frequencies for applications employing pulse-code modulation |publisher=Audio Engineering Society |year=2008 |access-date=2010-01-18}}</ref> Both Lavry Engineering and J. Robert Stuart state that the ideal sampling rate would be about 60&nbsp;kHz, but since this is not a standard frequency, recommend 88.2 or 96&nbsp;kHz for recording purposes.<ref>{{Cite web|url=http://www.lavryengineering.com/pdfs/lavry-white-paper-the_optimal_sample_rate_for_quality_audio.pdf|title=The Optimal Sample Rate for Quality Audio|last=Lavry|first=Dan|date=May 3, 2012|website=Lavry Engineering Inc.|quote=Although 60 &nbsp;KHz would be closer to the ideal; given the existing standards, 88.2 &nbsp;KHz and 96 &nbsp;KHz are closest to the optimal sample rate.}}</ref><ref>{{Cite web|url=https://www.gearslutz.com/board/showpost.php?p=7883017&postcount=15&s=b05e50b41d1789054724882582d8351b|title=The Optimal Sample Rate for Quality Audio|last=Lavry|first=Dan|website=Gearslutz|language=en|access-date=2018-11-10|quote=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 48KHz48&nbsp;KHz is pretty good compromise, but 88.2 or 96KHz96&nbsp;KHz yields some additional margin.}}</ref><ref>{{Cite web|url=https://www.gearslutz.com/board/showpost.php?p=1234224&postcount=74|title=To mix at 96k or not?|last=Lavry|first=Dan|website=Gearslutz|language=en|access-date=2018-11-10|quote=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.}}</ref><ref>{{Cite book|title=Coding High Quality Digital Audio|last=Stuart|first=J. Robert|date=1998|quote=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 58kHz58&nbsp;kHz. ... there are strong arguments for maintaining integer relationships with existing sampling rates – which suggests that 88.2kHz2&nbsp;kHz or 96kHz96&nbsp;kHz should be adopted.|citeseerx = 10.1.1.501.6731}}</ref>

A more complete list of common audio sample rates is:

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! Sampling rate

! Use

|-

|5,512.5&nbsp;Hz

|Supported in [[Adobe Flash|Flash]].<ref>{{Cite web |date=2013 |title=SWF File Format Specification - Version 19 |url=https://open-flash.github.io/mirrors/swf-spec-19.pdf}}</ref>

|-

| 8,000&nbsp;Hz

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| [[CD-ROM#CD-ROM XA extension|CD-XA audio]]

|-

| 44,056055.9&nbsp;Hz

| Used by digital audio locked to [[NTSC]] ''color'' video signals (3 samples per line, 245 lines per field, 59.94 fields per second = 29.97 [[frames per second]]).

|-

| [[44,100 &nbsp;Hz]]

| [[Audio CD]], also most commonly used with [[MPEG-1]] audio ([[VCD]], [[SVCD]], [[MP3]]). Originally chosen by [[Sony]] because it could be recorded on modified video equipment running at either 25 frames per second (PAL) or 30 frame/s (using an NTSC ''monochrome'' video recorder) and cover the 20&nbsp;kHz bandwidth thought necessary to match professional analog recording equipment of the time. A [[PCM adaptor]] would fit digital audio samples into the analog video channel of, for example, [[PAL]] video tapes using 3 samples per line, 588 lines per frame, 25 frames per second.

|-

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| world's first commercial [[Pulse-code modulation|PCM]] sound recorder by [[Nippon Columbia]] (Denon)

|-

| [[48,000&nbsp;Hz]]

| 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&nbsp;kHz and work with 29.97 &nbsp;frames per second NTSC video – as well as 25 &nbsp;frame/s, 30 &nbsp;frame/s and 24 &nbsp;frame/s systems. With 29.97 &nbsp;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.<ref name=AES5/>&nbsp; Also used for sound with consumer video formats like DV, [[digital TV]], [[DVD]], and films. The professional Serial Digital Interface [[Serialserial Digitaldigital interface]] Interface|(SDI)]] and High-definition Serial Digital Interface [[Serial Digital Interface|(HD-SDI)]] used to connect broadcast television equipment together uses this audio sampling frequency. Most professional audio gear uses 48&nbsp;kHz sampling, including [[mixing console]]s, and [[digital recording]] devices.

|-

| 50,000&nbsp;Hz

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|-

| 64,000&nbsp;Hz

| Uncommonly used, but supported by some hardware<ref>{{Cite web|url=http://www.rme-audio.de/en/products/hdsp_9632.php|title=RME: Hammerfall DSP 9632|website=www.rme-audio.de|access-date=2018-12-18|quote=Supported sample frequencies: Internally 32, 44.1, 48, 64, 88.2, 96, 176.4, 192 &nbsp;kHz.}}</ref><ref>{{Cite web|url=https://www.pioneer-audiovisual.eu/uk/products/sx-s30dab|title=SX-S30DAB {{!}} Pioneer|website=www.pioneer-audiovisual.eu|access-date=2018-12-18|quote=Supported sampling rates: 44.1 &nbsp;kHz, 48 &nbsp;kHz, 64 &nbsp;kHz, 88.2 &nbsp;kHz, 96 &nbsp;kHz, 176.4 &nbsp;kHz, 192 &nbsp;kHz|archive-date=2018-12-18|archive-url=https://web.archive.org/web/20181218145630/https://www.pioneer-audiovisual.eu/uk/products/sx-s30dab|url-status=dead}}</ref> and software.<ref>{{Cite web|url=https://steinberg.help/wavelab_pro/v9.5/en/wavelab/topics/master_section/master_section_customize_sample_rate_menu_dialog_r.html|title=Customize Sample Rate Menu|last1=Cristina Bachmann|first1=Heiko Bischoff|last2=Schütte|first2=Benjamin|website=Steinberg WaveLab Pro|language=en-US|access-date=2018-12-18|quote=Common Sample Rates: 64 000 Hz}}</ref><ref>{{Cite web|url=https://getsatisfaction.com/m-audio/topics/m-track-2x2m-cubase-pro-9-can-t-change-sample-rate|title=M Track 2x2M Cubase Pro 9 can ́t change Sample Rate|website=M-Audio|language=en-US|access-date=2018-12-18|quote=[Screenshot of Cubase]|archive-date=2018-12-18|archive-url=https://web.archive.org/web/20181218102147/https://getsatisfaction.com/m-audio/topics/m-track-2x2m-cubase-pro-9-can-t-change-sample-rate|url-status=dead}}</ref>

|-

| 88,200&nbsp;Hz

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

|-

| 96,000&nbsp;Hz

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|-

| 352,800&nbsp;Hz

| [[Digital eXtreme Definition]], used for recording and editing [[Super Audio CD]]s, as 1-bit [[Direct Stream Digital|Direct Stream Digital (DSD)]] is not suited for editing. Eight8 times the frequency of 44.1&nbsp;kHz.

|-

|384,000 Hz

|Maximum sample rate available in common software.{{cn|date=January 2025}}

|-

| 2,822,400&nbsp;Hz

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{{See also|Audio bit depth}}

Audio is typically recorded at 8-, 16-, and 24-bit depth,; which yield a theoretical maximum [[signal-to-quantization-noise ratio]] (SQNR) for a pure [[sine wave]] of, approximately,; 49.93&nbsp;[[Decibel|dB]], 98.09&nbsp;dB, and 122.17&nbsp;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 |access-date=2010-01-19 |archive-date=2022-10-09 |archive-url=https://ghostarchive.org/archive/20221009/http://www.analog.com/static/imported-files/tutorials/MT-001.pdf |url-status=dead }}</ref> CD quality audio uses 16-bit samples. [[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&nbsp;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.

==== Speech sampling ====

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 100&nbsp;Hz – 4&nbsp;kHz range, allowing a sampling rate of 8&nbsp;kHz. This is the [[sampling rate]] used by nearly all [[telephony]] systems, which use the [[G.711]] sampling and quantization specifications.{{Citation needed|reason=References are needed for frequency range of human voice, and use of G.711|date=May 2018}}

=== Video sampling ===

{{More citations needed section|date=June 2007}}

[[Standard-definition television]] (SDTV) uses either 720 by 480 [[pixels]] (US [[NTSC]] 525-line) or 720 by 576 [[pixels]] (UK [[PAL]] 625-line) for the visible picture area.

[[High-definition television]] (HDTV) uses [[720p]] (progressive), [[1080i]] (interlaced), and [[1080p]] (progressive, also known as Full-HD).

In [[digital video]], the temporal sampling rate is defined as the [[frame rate]]{{snd}} or rather the [[field rate]]{{snd}} 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:

* 50&nbsp;Hz &nbsp;– [[PAL]] video

* 60 / 1.001&nbsp;Hz ~= 59.94&nbsp;Hz &nbsp;– [[NTSC]] video

Video [[digital-to-analog converter]]s operate in the megahertz range (from ~3&nbsp;MHz for low quality composite video scalers in early gamesgame consoles, to 250&nbsp;MHz or more for the highest-resolution VGA output).

When analog video is converted to [[digital video]], a different sampling process occurs, this time at the pixel frequency, corresponding to a spatial sampling rate along [[scan line]]s. A common [[pixel]] sampling rate is:

* 13.5&nbsp;MHz &nbsp;– [[CCIR 601]], [[D1 video]]

Spatial sampling in the other direction is determined by the spacing of scan lines in the [[raster graphics|raster]]. The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height.

Spatial [[aliasing]] of high-frequency [[luma (video)|luma]] or [[chrominance|chroma]] video components shows up as a [[moiré pattern]].

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The process of [[volume rendering]] samples a 3D grid of [[voxel]]s 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, [[X-ray computed tomography]] (CT/CAT), [[magnetic resonance imaging]] (MRI), [[positron emission tomography]] (PET) are some examples. It is also used for [[seismic tomography]] and other applications.

[[File:Bandpass sampling depiction.svg|thumb|right|255px|The top two graphs depict Fourier transforms of two different functions that produce the same results when sampled at a particular rate. The baseband function is sampled faster than its Nyquist rate, and the bandpass function is undersampled, effectively converting it to baseband. The lower graphs indicate how identical spectral results are created by the aliases of the sampling process.]]

== Undersampling ==

{{Main|Undersampling}}

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 rate|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>

{{cite book

| title = Mixed-signal and DSP design techniques

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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]].

}}&nbsp; When one waveform, <math>, \hat s(t),</math>&nbsp;, is the [[Hilbert transform]] of the other waveform, <math>, s(t),\,</math>&nbsp;, the complex-valued function, &nbsp;<math>s_a(t) \triangleq s(t) + i\cdot \hat s(t),</math>&nbsp;, is called an [[analytic signal]],&nbsp; whose Fourier transform is zero for all negative values of frequency. In that case, the [[Nyquist rate]] for a waveform with no frequencies ≥&nbsp;''B'' can be reduced to just ''B'' (complex samples/sec), instead of 2''B''<math>2B</math> (real samples/sec).{{efn-ua|

When the complex sample-rate is ''B'', a frequency component at 0.6&nbsp;''B'', for instance, will have an alias at −0.4&nbsp;''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]], &nbsp;<math>s_a(t)\cdot e^{-i 2i2\pi \frac{B}{2} t},</math>&nbsp;, 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>&nbsp;, by processing the product sequence, <math>, \left [s(nT)\cdot e^{-i 2 i2\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>

}} &nbsp;through a digital low-pass filter whose cutoff frequency is ''<math>B''/2</math>.{{efn-ua|

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.

}} 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 <math>s(t)</math> waveform can be recovered, if necessary.

== See also ==