Nyquistin teoreema

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Nyquestin teoreema tai Nyquist-Shannon näytteenottoteoreema on informaatioteorian alan keskeisiä opinkappaleita.

Se tunnetaan myös Whittaker-Nyquist-Kotelnikov-Shannon näytteenottoteoreemana.

Teoreema lausuu, että muunnettaessa analogista signaalia digitaaliseksi (tai muuten näytteytettäessä signaalia diskreetein aikavälein), silloin näytteenottotaajuuden pitää olla ainakin tuplat korkeimmasta ottosignaalin taajuudesta kyetäkseen rekonstruoimaan alkuperäisen täydellisessä muodossaan.

Jos näytteytystaajuus on tämän rajan alla, silloin alkuperäisen signaalin taajuuskomponentit jotka ovat yli puolen näytteytystaajuuden yläpuolella kohtaavat aliasoinnin ja sen seurauksena kuvautuvat alemmille taajuuksille. Jos näytteytystaajuus on täsmälleen kaksinkertainen korkeimpaan syötesignaaliin nähden, näytteenoton ja signaalin keskinäinen vaihe-ero vääristää signaalia.

Aliasoinnin ongelmien takia realisoitujen muuntimien eteen laitetaan yleensä alipäästösuotimet, jotka pitävät nämä aliakset poissa häiritsemästä. Koskapa tällaisen anti-alias suodattimen on oltava varsin laadukas mm. ryhmäkulkuaikojensa suhteen, ne eivät juuri koskaan ole erityisen halpoja

Teoreema toimii myös, kun on tarkoitus vähentää näytteitystaajuutta olemassaolevassa digitaalisessa aineistossa.

KÄÄNNÖS KESKEN

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The theorem was first formulated by Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), but was only formally proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). Mathematically, the theorem is formulated as a statement about the Fourier transformation.

If a function s(x) has a Fourier transform F[s(x)] = S(f) = 0 for |f| ≥ W, then it is completely determined by giving the value of the function at a series of points spaced 1/(2W) apart. The values s_n = s(n/(2W)) are called the samples of s(x).

The minimum sample frequency that allows reconstruction of the original signal, that is 2W samples per unit distance, is known as the Nyquist frequency, (or Nyquist rate). The time inbetween samples is called the Nyquist interval.

If S(f) = 0 for |f| > W, then s(x) can be recovered from its samples by the Nyquist-Shannon interpolation formula.

A well-known consequence of the sampling theorem is that a signal cannot be both bandlimited and time-limited. To see why, assume that such a signal exists, and sample it faster than the Nyquist frequency. These finitely many time-domain coefficients should define the entire signal. Equivalently, the entire spectrum of the bandlimited signal should be expressible in terms of the finitely many time-domain coefficients obtained from sampling the signal. Mathematically this is equivalent to requiring that a (trigonometric) polynomial can have infinitely many zeros since the bandlimited signal must be zero on an interval beyond a critical frequency which has infinitely many points. However, it is well-known that polynomials do not have more zeros than their orders due to the fundamental theorem of algebra. This contradiction shows that our original assumption that a time-limited and bandlimited signal exists is incorrect.

Undersampling

It has to be noted that even if the concept of "twice the highest frequency" is the more commonly used idea, it is not absolute. In fact the theorem stands for "twice the bandwidth", which is totally different. Bandwidth is related with the range between the first frequency and the last frequency that represent the signal. Bandwidth and highest frequency are identical only in baseband signals, that is, those that go very nearly down to DC. This concept led to what is called undersampling, that is very used in software-defined radio.

Imagine that you want to sample all the FM commercial radio stations that broadcast in a given area. They broadcast in channels that span from 88 MHz to 108 MHz, giving a signal with bandwidth of 20 MHz. In the baseband interpretation of the theorem, this would require a sampling frequency more than 216 MHz. In fact, doing undersampling one is only required to sample at more than 40 MHz, as long as the antenna signal is passed by a bandpass filter to keep the signal in the 88-108 MHz range. Sampling at 44 MHz, the frequency 100 MHz will be reflected as a 12 MHz digital frequency.

In certain problems, the frequencies of interest are not an interval of frequencies, but perhaps some more interesting set F of frequencies. Again, the sampling frequency must be proportional to the size of F. For instance, certain domain decomposition methods fail to converge for the 0th frequency (the constant mode) and some medium frequencies. Then the set of interesting frequencies would be something like 10 Hz to 100 Hz, and 110 Hz to 200 Hz. In this case, one would need to sample at 360 Hz, not 400 Hz, to fully capture these signals.

References

• H. Nyquist, "Certain topics in telegraph transmission theory," Trans. AIEE, vol. 47, pp. 617-644, Apr. 1928.
• C. E. Shannon, "Communication in the presence of noise," Proc. Institute of Radio Engineers, vol. 37, no.1, pp. 10-21, Jan. 1949.