# Talk:Nyquist rate

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## Nyquist rate for bandpass and base band signals

Section 2 of this article should clarify what happens for a bandpass signal. I suppose that the formula is not valid in that case, since the total bandwidth definition is different. It is only valid for the complex valued equivalent baseband signal, provided that the bandwidth is defined as half the total bandwidth.

Is it correct to say that in the bandpass signal case, the maximum symbol rate is equal to the bandwidth?

To my understanding, pulse shaping might change the bandwidth requirement for a certain symbol rate.

Section 1 may also discuss intentional undersampling of bandpass signals, where the sampling rate must be at least equal to the bandwidth rather than twice the upper cutoff frequency.

Mange01 (talk) 17:05, 11 April 2008 (UTC)

In either the passband or baseband case, the number of independent symbols per second is 2B. There is no difference. Your attempt to show a difference relies on your statement that a passband signal of width B can be downconverted to an equivalent baseband signal of top edge B/2. This is not correct. Consider single-sideband modulated channels for an obvious concrete example. Dicklyon (talk) 22:07, 27 April 2008 (UTC)
Dicklyon, thanks for taking your time helping me on these issues. You mention single-sideband, but I have never seen that used in the case of digital modulation. It would introduce severe phase distorsion, causing an inter-symbol interference, resulting in that the baud rate must be reduced.
I must admit that I am kind of frustrated over that you once again deleted ALL my changes to an article - although I understand why you deleted the paragraph about the above issue. I am not used to that people delete what I write. At Wikipedia we typically improve each others texts rather than deleting them. At the same time you should have credit for guarding "your" articles so carefully. The deleted text can be found in the article history.
Is it okay if I put the text back, if I include references supporting all my claims?
My points, here supported by references, are:
• The Nyquist rate is not twice the bandwidth in case of a passband signal, it is twice the baseband bandwidth, i.e. twice the upper cutoff frequency. This is a common misunderstanding. Reference: [1]
• In this context, the bandwidth is not defined by the 3dB cut-off frequency, but by the stopband corner frequency or limit frequency.
• The Nyqvist sample rate is not applicable to intentional undersampling (also known as bandpass sampling or super-Nyquist sampling), used for example in software defined radio and other DSP based radio receivers. This is possible if a passband anti-aliasing filter, and a passband interpolation filter, are used instead of low-pass filters. In that case the sampling rate must be higher than the bandwidth of the passband filters (not twice the bandwidth). If you draw a graph showing the periodic spectrum, you will realize this. (There are also some other requirements.) An example of a source explaining this can be found here: [2]. It is also described in the IF/RF (bandpass) sampling section of the Sampling (signal processing) article.
• You have taught me to not confuse Nyquist rate with Hartley's law, as many modern books in computer networking do. (I have given you references to these books in another talk page.) I think that this article should clarify the difference, or at least link to the well-written article section on Hartley's law, which I believe you are the father of. (I am happy that you did not delete ALL my changes to that section...)
• The Nyquist rate is applicable to the pulse rate of a line code, but is it applicable to symbol rate (baud rate) of a digital modulation scheme? All digital modulation schemes require at least "double side bands", i.e. twice the required baseband bandwidth of equivalent baseband signal. This means that the required passband bandwidth is equal to the symbol rate, in case of an ideal band-pass filter. Okay, a pulse shaping filter might reduce the 3dB bandwidth requirement to much less, but do we really talk about the 3dB bandwidth here? I have still not found a good source answering these questions, but to my understanding, the answer is no. (Perhaps I should explain the term "equivalent baseband signal". In case of ASK modulation, it is an unmodulated pulse train, with the same pulse rate as the baud rate of the modulated signal. The waveform might be changed by a pulse shaping filter. In the case of QAM and PSK modulation it is the unmodulated in-phase and quadrature phase components, denoted I and Q, combined into a complex valued signal, I+jQ, where j is the imaginary unit. This means that the equivalent baseband signal is a pulse train of complex valued voltages. Most other digital modulation methods can be generated and detected using the same principle as QAM. And ALL digitally modulated signals can be converted to a complex-valued equivalent baseband signal, such that the carrier frequency is moved down to 0 Hz, the lower side band to negative frequencies and the upper sideband to positive frequencies.)
Mange01 (talk) 02:12, 1 May 2008 (UTC)
As is probably already clear, this is really just a matter of definition of "bandwidth". What I'm about to say is all unsourced "original research" on my part, but here goes:
• In all cases, the Nyquist rate is equivalent to the total support of the two-sided Fourier spectrum of the signal (ignoring pulse-shaping considerations).
• Thererfore, in the baseband case, assuming the spectrum occupies -B to +B, then the Nyquist rate is 2B.
• Similarly, in the passband case, assuming the spectrum occupies (fc-B to fc+B) and (-fc-B to -fc+B), again the total support of the the spectrum.
• Therefore, using the traditional definition of passband spectrum, i.e. the spectrum occupies (fc-B/2 to fc+B/2) and (-fc-B/2 to -fc+B/2), then the Nyquist rate is 2B.
• Of course, in a practical system, our passband signal is restricted to real-only, so the spectrum centred on fc is the mirror image of that around -fc. Therefore, we lose half our available dimensions.
• Digital single-sideband schemes do exist (see the single-sideband modulation article). In contrast to what Dick said above, an SSB scheme of bandwidth B (in the sense of fc-B/2 to fc+B/2) can of course be downconverted to a passband signal with "top-edge" B/2 (in the sense of -B/2 to +B/2). It's just that it's asymmetrical (not even Hermitian symmetrical), and not readily demodulated in this form.
What conclusions we can draw for the article content, I'm not sure, but I hope that we can at least start by agreeing on the above statements (or correcting me if I've made an error!). Oli Filth(talk) 19:25, 2 June 2008 (UTC)
Without a source, none, I'd say. As to your SSB example, if you downconvert the SSB band of width B to the range plus and minus B/2, that's not symmetric about zero, as you note, so that signal has to be complex. In that case, the real and imaginary parts each need B samples per second; the Nyquist rate is 2B total samples per second. It's always twice the bandwidth, by the usual baseband and passband definitions on real signals. Dicklyon (talk) 19:34, 2 June 2008 (UTC)
I think I must have missed Mange01's long reply above. I would welcome him to make some careful edits that we can discuss, but only where sourced. If the definitions of bandwidth or Nyquist rate are at issues, let's see some sources that do it his way (with quotes, please, if not accessible online). But he also continues to assert untrue stuff like All digital modulation schemes require at least "double side bands", i.e. twice the required baseband bandwidth of equivalent baseband signal. This is not so. Both SSB and quadrature techniques are routinely used in digital radio to put twice as many symbols per second through a bandpass channel as a double-sideband technique would achieve. Dicklyon (talk) 19:43, 2 June 2008 (UTC)
I agree. However, even without sources, I feel that some of the wording of the article body could be tweaked to be less ambiguous. For instance, the sentence
"the Nyquist rate is equal to the two-sided bandwidth of the signal (the upper and lower sidebands)"
has a further qualifier ("two-sided") not present in the lead. Furthermore, I believe that in the passband case, what is meant by "two-sided" is not "the upper and lower sidebands"; rather, it's the inclusion of negative frequencies (which, of course, is also the meaning in the baseband case). Oli Filth(talk) 19:45, 2 June 2008 (UTC)
Yikes! I tried to make that more better. Take a look. This is still awkward, since sampling at the Nyquist rate is not generally sufficient to avoid aliasing in the bandpass case. Dicklyon (talk) 20:06, 2 June 2008 (UTC)
I have some reservations over the sentence "For complex-valued signals ... another doubling is necessary to get the Nyquist rate." In general, a baseband signal will be complex, but there is no doubling necessary (although the total dimensionality will be doubled). Oli Filth(talk) 22:59, 2 June 2008 (UTC)
I had some reservations, too, about introducing complex signals. If you want to get by with Nyquist rate samples of complex baseband signals, you need complex samples. That's twice as many degrees of freedom as real samples. I think maybe we should just stick to everything real and avoid the confusion. But then it doesn't really make sense to talk about adding the widths of the negative and positive frequency bands, either, since they're always the same. Sticking to normal bandwidth and the normal difinition of Nyquist rate as twice the bandwidth is the easiest way to keep it consistent and correct. Dicklyon (talk) 23:04, 2 June 2008 (UTC)

Thank you both for your help with this issue! A few comments: I don't see how complex-valued signals gives a doubling of the symbol rate or sampling rate. Perhaps a doubling of the data signalling rate in bit/s in some sense (in comparison to what?), but that is not the topic of this article. The text you wrote was in the sampling section, where symbol rate should not be discussed, but we should also write something about passbands signals in the signalling section. Any suggestions are welcome.
A complex-valued signaling channel is just two normal channels in parallel, so the aggregate rate is double; you can count N complex values or 2N real values, however you want to call them. In sampling, it takes 2N real numbers to represent N complex samples, but what we usually count as samples are real scalars, and that's the sense in which I meant it. But it's a can of worms, easily misinterpreted. I'm still unclear on why you want to say something about passband signals; is what I added not clear enough yet? Dicklyon (talk) 06:49, 4 June 2008 (UTC)
I have now tried to merge your suggestion with parts of my own deleted suggestion and Olis text above, and added a couple of references. I removed the parts of your text that I did not understand, or that was not related to the sampling section. For example, I was not sure on what you meant by "frequency translation" - but I guessed you had the superhetorodyne principle in mind, i.e. multiplication by a sine wave. The reason for the super-Nyquist discussion is that the article should clarify if and to what extent the term "Nyquist rate" is applicable to passband bandwidth, and why. Perhaps I have added too many details on super-Nuqist sampling - later we can move parts of it to some other article if you want to. We still have not written anything about passband signals in the data transmission section.
A related issue: The OFDM article says "The orthogonality also allows high spectral efficiency, near the Nyquist rate". Is this true? OFDM is using a symbol rate near the passband channel bandwidth. Perhaps it should be "near the limit given by Hartley's law" , or "near the Shannon-Hartley channel capacity"? All these articles should be modified or clarified with the passband case in mind. Mange01 (talk) 06:34, 4 June 2008 (UTC)
It looks right by the "signalling at the Nyquist rate" intepretation. The orthogonality of the subcarriers lets you use a number of symbols per second equal to twice the total bandwidth, whether baseband or passband. It doesn't get into Hartley's stuff about numbers of levels, or bits; just symbols per second. Notice that they use QAM or QPSK, which is how they get the factor of two and why the spectra are not symmetrical about the carriers. Dicklyon (talk) 06:49, 4 June 2008 (UTC)
Thankyou for your answer. However, also if we were using ASK modulation, the maximum symbol rate would be equal to the passband bandwidth. 16ASK and 16QAM gives the same symbol rate and data rate for the same bandwidth, but 16QAM can also be considered as two paralell 4ASK signals. This discussion is about the passband signal, which is the physical signal, which does not have any complex valued samples. The equivalent baseband signal may consist of an I and a Q compenent, or it may be considered as a complex valued signal of I+jQ, but that is not the topic of this article. The equivalent baseband signal can be retreived after coherent detection, which today typically is done digitally, after sampling, and should not be discussed here.
My question is: is it correct to use the term "Nyquist rate" for the passband bandwidth? Mange01 (talk) 07:30, 4 June 2008 (UTC)
I have some serious reservations about your new "Maximum symbol rate of a modulated bandpass signal" section. If we're counting complex symbols as two real symbols, then clearly QAM obtains twice the real-symbol-rate as ASK. SSB can only be used for phase-less signals (i.e. ASK in this comparison), and will take it back up to the same spectral efficiency. However you choose to measure bandwidth and symbol rate, OFDM gets you no closer to the Nyquist rate than the underlying modulation method (i.e. real or complex).
I would strongly suggest this section is removed, either:
1. until we can all come to sort of agreement on what is correct (what is written certainly isn't!), preferably with unequivocal sources,
2. or permanently because the passband case only serves to confuse the issue. Oli Filth(talk) 18:49, 5 June 2008 (UTC)

Thanks for all your helpful comments. The article should clarify these issues. Dicklyon, you have a point in several of your comments, but I don't agree on everything. My last version can be found here: [3]. The version that is published now suffers from several incorrect statements, as discussed above. I will not have time to work on this artile for a while now. Perhpaps you have time?

Oli Filth wrote: "If we're counting complex symbols as two real symbols, then clearly QAM obtains twice the real-symbol-rate as ASK." I don't agree. See the baud rate article. A baud rate = symbol rate = pulse rate = modulation rate of for example 2400 symbols/s means 2400 transitions per second in the physical passband signal. In the equivalent baseband signal I+jQ, this corresponds to 2400 complex valued signals per second. A 16QAM modem offers the same symbol rate and bitrate as a 16ASK modem using the same bandwidth. One 16QAM signal can be considered as two orthogonal 4ASK signals. Do you agree?

Oli wrote "SSB can only be used for phase-less signals (i.e. ASK in this comparison), and will take it back up to the same spectral efficiency. " I don't understand. Conventional M-ASK as well as M-QAM, using M different signal levels, of baud rate $f_{sym}$ requires a passband bandwidth of f_{sym}, and gives a spectral efficiency of $log_2(M)$. "M-ASK over SSB" would require a passband bandwidth of $f_{sym}/2$, and a spectral efficiency of $log_2(M)/2$.

Oli wrote: "However you choose to measure bandwidth and symbol rate, OFDM gets you no closer to the Nyquist rate than the underlying modulation method (i.e. real or complex)." Real or complex does not matter. Howeever, I must reconsider this issue and check in the literature.

Mange01 (talk) 08:50, 6 June 2008 (UTC)

## "Nyquist bit rate" vs "Nyquist symbol rate"

In many modern computer networking course books, the term "Nyquist rate" is a maximum bit rate rather than a baud rate, as opposed to literature on data transmission and wireless communication where this is known as the bit rate given by the Hartley's law:

$R \le 2B \log_2(M),$

where M is the number of symbols or voltage levels, and B is the (equivalent) low-pass signal bandwidth.

Perhaps the article should also state the relationship between the maximum bitrate given my Nyquist/Hartley, and the Shannon-Hartley channel capacity, since there often is confusion about this issue in the literature. The first is a maximum gross bitrate (data signalling rate), inclusive of redundant forward error correction coding (FEC), while the latter a maximum net bitrate (useful bitrate excluding FEC), provided that an ideal FEC is assumed. Many teachers and authors incorrectly claims that Shannon-Hartley includes FEC code. (I wrote something similar in two other articles, but someone thaught my formulations were inaccurate and deleted them instead of improving them. That's why I ask here for permission, and ask you to improve my formulations.)

Mange01 (talk) 17:05, 11 April 2008 (UTC)

The best thing is to find a source that makes the connection you want to clarify. I can help get it right if it's based on a source. Some of that is already done in Shannon–Hartley law, based on a book by John Robinson Pierce. Dicklyon (talk) 22:09, 27 April 2008 (UTC)

## Real Signals vs Complex Signals:

I just adopted this article, and am currently working on assembling all of my relevant sources to complete it. Just as a point from the discussion above:

Sampling a signal faster than the nyquist rate preserves the phase information in the signal. That means that it also preserves all of the complex information in the signal. HatlessAtless (talk) 20:57, 10 June 2008 (UTC)

It depends what you mean by phase. A passband signal can have phase information (when viewed in the frequency domain), but be entirely real. Of course, its baseband equivalent will be complex.
Also, I would have thought "preserving the phase information" would be implicit in the definition "the sample rate for alias-free signal sampling". Well, certainly no less explicit than the property of "preserving the amplitude information". Oli Filth(talk) 21:02, 10 June 2008 (UTC)
Phase information is synonymous with a complex signal. More accurately put, all sinusoids are complex signals, but some may have a zero coefficient in the complex part. The key to my point here is that a band-limited physical signal can be perfectly reconstructed. Because it can be perfectly reconstructed, all complex information is preserved as well as all real part information. HatlessAtless (talk) 14:18, 11 June 2008 (UTC)

## HatlessAtlas changes

User:HatlessAtlas, I've reverted the rest of your changes, the part that OliFilth didn't already revert. Please take another whack at if after consider my objections. First of all, don't put excess capital letters in your headings, or excess line spaces before them, for others to clean up. Second, if you want to extend the scope of an article, it is mandatory to say what sources you're using (e.g. for the event detection section). Third, don't make up stuff like "the Nyquist rate is a term used to describe the minimum sampling rate fir a continuous time signal such that all information of interest in the signal can be reconstructed by an ideal reconstructor." I doubt that you'll find a reputable source to support that, and if you do it's still not correct, since that's no formal relation between "all information of interest" and "an ideal reconstructor." The concept is only defined and sensible with respect to a bandwidth-limited class of signal, with respect to perfect reconstruction. Fourth, since we've already got articles on sampling theorem, sampling, aliasing, etc., there's not a good reason to reconstruct their contents here, bringing up all the same problems that we've already solved there, about how to state results correctly. So take it slow; one edit at a time is a good strategy; the multidimensional stuff, which you added with a ref, might be a good place to start. Dicklyon (talk) 23:00, 10 June 2008 (UTC)

I First, if you don't like minor issues such as capitalization, you will find that you generate much less frustration with others if you fix their changes rather than simply deleting them. Secondly, I don't appreciate your eliminating changes to a work in progress. Sourcing and such takes time. I will add the "under construction" template next time if you feel its necessary. Finally, as to your one section at a time comment, I respectfully refer you to: WP: Be bold I will add my section on the multidimensional component, however I would appreciate cooperative help in improving it incrementally rather than reflexively deleting should you see something you don't like. HatlessAtless (talk) 13:53, 11 June 2008 (UTC)
I understand the possibilities. Sometimes it's best to get an editor's attention to the problem right away by reverting flaky edits. When you're ready to work on it, put it back. But have your sources ready first; what sense does it make to write first and find sources later? I will cooperate on improvements if you go slow, so that I don't have to do too much cleanup. Dicklyon (talk) 14:39, 11 June 2008 (UTC)

I reverted again. The new definition is unsourced, and leaves out application to the signaling case, where the bandwidth is a channel bandwidth, not a signal bandwidth. The assertion of confusion is unsourced. The introduction of baseband and real into the bandpass sampling section is unclear, seemingly wrong. Let's discuss exactly what meaning you intend there, and see if we can find a good way to say it. Dicklyon (talk) 15:40, 11 June 2008 (UTC)

I disagree with your reverts. First:
"In signal processing, the Nyquist rate is defined to be twice the bandwidth of a band - limited signal. This term is used in two different ways under two different circumstances: most commonly, as a lower bound for the sample rate for alias-free signal sampling and as an upper bound for the signaling rate across a bandwidth-limited channel such as a telegraph line."
First, the term 'bandwidth' is a property, and so ending a sentence with the term 'bandwidth' is incorrect gramatically. I attempted to correct this. I preserved your sourcing for the definitions of the two different components of 'Nyquist Rate' but unless you have failed to source a quotation for the opening sentence, your statement is also unsourced. I agree that I failed to incorporate the signalling rate, but the correction to my introduction is minor:
"In signal processing, the Nyquist rate is defined to be twice the bandwidth of a band - limited signal or a band - limited channel. This term is used to mean two different things under two different circumstances: most commonly, as a lower bound for the sample rate for alias-free signal sampling and as an upper bound for the signaling rate across a bandwidth-limited channel such as a telegraph line."
Second: "In the case of bandpass sampling, the situation does not change, but ambiguity in the word 'bandwidth' can cause confusion. For sampling of real signals, all signals must be considered lowpass signals; bandpass signals must be sampled at at least twice the frequency of the highest frequency component of the bandpass signal in order to avoid aliasing. For a bandpass signal of bandwidth B, sampling at 2B will not avoid aliasing if the center frequency of the bandpass signal is greater than B. For this reason, many bandpass signals are mixed down to baseband prior to sampling, in situations where analog mixing hardware is less expensive than high-speed sampling hardware."
An assertion of possible confusion does not need a source. (If there is a Wikipedia policy I am missing, please refer me to it). However, from the talk page above, there fact that the bandpass bandwidth is not the same as the highest frequency component of the signal (also called bandwidth) can cause self-evident confusion. On another note, I used real to indicate physical signals. I can see how, given the discussion of real vs complex signals, you might have misunderstood. I have made both corrections to the article.
"For sampling of physical signals, all signals must be considered lowpass signals; bandpass signals must be sampled at at least twice the frequency of the highest frequency component of the bandpass signal in order to avoid aliasing. For a bandpass signal of bandwidth B, sampling at 2B will not avoid aliasing if the center frequency of the bandpass signal is greater than B. For this reason, many bandpass signals are mixed down to baseband prior to sampling, in situations where analog mixing hardware is less expensive than high-speed sampling hardware." HatlessAtless (talk) 16:05, 11 June 2008 (UTC)
DickLyon; Thanks for your copyedits to my edits. I'll be taking another crack at the multidimensional case probably tomorrow; I look forward to your input on it. HatlessAtless (talk) 19:41, 11 June 2008 (UTC)
Well, it wasn't exactly a copyedit when I tried to put back the meat of the bandpass sampling bit that you had taken out, which is that bandpass signals can be sampled at much less than twice the highest frequency. Dicklyon (talk) 20:30, 11 June 2008 (UTC)

It appears that I wasn't clear about the intent of your statement then. It makes sense conceptually, however, but I couldn't infer that from the existing text. I think it is much more clear now. HatlessAtless (talk) 21:04, 11 June 2008 (UTC)

## Merge with nyquist frequency?

The nyquist rate and the nyquist frequency are the same fundamental object, separated only by the fourier transform. I think that they could both be properly covered under the same article. Most likely, by turning the nyquist rate article into a nyquist signalling rate article based on Nyquist's paper, and having nyquist rate as it is now discussed as a subsection of the nyquist frequency.HatlessAtless (talk) 13:35, 11 June 2008 (UTC)

I don't understand how you can say they're so closely related, if you've read the article. In the sampling situation, one (Nyquist frequency) is a property of a system, the other (Nyquist rate) of a signal. They are only related in the particular case of the sampling rate being chosen to be twice the signal bandwidth. Dicklyon (talk) 14:36, 11 June 2008 (UTC)
Because they can be related as follows: "For a given band-limited signal to be sampled without aliasing, the sampling system must have a Nyquist frequency equal to or higher than the highest frequency component in the signal to be sampled." As soon as we start talking about sampling a real signal, we are talking about going into a sampled, discrete system. Calling the "nyquist rate" an inherent property of a system is a convention, but is only meaningful and useful in terms of a sampled system. Unless there is some independent identity to the Nyquist sampling rate that requires a separate treatment, I see no reason it can't be handled just as effectively as a subsection of the Nyquist frequency. I'm not arguing that the Nyquist rate exisits, I am asking whether it can be suitably discussed as a subsection of a closely related topic. HatlessAtless (talk) 14:49, 11 June 2008 (UTC)
But there is "some independent identity to the Nyquist sampling rate that requires a separate treatment". You can certainly use Nyquist rate as a lower bound on sampling rate to get alias-free sampling, but you don't have to. Many pairs of signals with systems to NOT meet R_N = 2*fN; they can be off in either direction, depending on whether the signal is over-sampled or under-sampled. And as you note, Nyquist frequency "is only meaningful and useful in terms of a sampled system", which Nyquist rate is only meaningful in terms of a signal (or class of signals). Dicklyon (talk) 15:03, 11 June 2008 (UTC)
Actually, my primary point is that neither the Nyquist rate or the Nyquist frequency are meaningful outside of a sampled system. In general, an over-sampled system is not a problem; it will reconstruct just fine. On the other hand an under sampled signal will not, in the general case, reconstruct without distortion. The key here is that sampling a signal at or above the Nyquist rate ensures that a system processing those samples will have a sufficiently high Nyquist frequency to handle the signal without distortion. Since sampling rate is inherently part of the definition of the Nyquist frequency, I see them as the same entity separated only by the difference between time and frequency domain. The "independent identity" I was looking for (and forgive me if I was misunderstanding your previous post) was one that was unrelated to the issue of aliasing and the Nyquist frequency of a system to process the samples.HatlessAtless (talk) 15:27, 11 June 2008 (UTC)

## So what does Nyquist rate mean?

After reading the intro of the article I still don't know what the Nyquist rate is. Please somebdy enlighten me. — Preceding unsigned comment added by Jangirke (talkcontribs) 01:19, 14 January 2014 (UTC)

It is "twice the bandwidth of a bandlimited function or a bandlimited channel." In different areas it has different interpretations, as the lead explains. Dicklyon (talk) 05:14, 14 January 2014 (UTC)