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Nyquist-Shannon sampling theorem

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Nyquist-Shannon sampling theorem

The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, in the literature more commonly referred to as the Nyquist sampling theorem or simply as the sampling theorem, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space). Shannon's version of the theorem states:

If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.

In other words, a bandlimited function can be perfectly reconstructed from a countable sequence of samples if the bandlimit, B, is no greater than half the sampling rate (samples per second). The theorem also leads to a formula for reconstruction of the original function from its samples. When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing. The Poisson summation formula provides a graphic understanding of aliasing and an alternative derivation of the theorem, using the perspective of the function's Fourier transform.

Of course, in practice, infinite sequences, perfect sampling, and perfect interpolation are all replaced by approximations, deviating from the ideal mathematical reconstruction.

Modern statements of the theorem are sometimes careful, explicitly stating that x(t) must contain no sinusoidal component at exactly frequency B, or that B must be strictly less than ½ the sample rate. (See Critical frequency below.) Also, the theorem does not preclude the possibility of perfect reconstruction, under special circumstances, that do not satisfy the sample-rate criterion. It is a sufficient, but not necessary, condition. (See Sampling of non-baseband signals below, and Compressed sensing.)

Introduction

To formalize the statements above, let X(f) be the Fourier transform of bandlimited function x(t):

$X\left(f\right)\ \stackrel\left\{\mathrm\left\{def\right\}\right\}\left\{=\right\}\ \int_\left\{-\infty\right\}^\left\{\infty\right\} x\left(t\right) \ e^\left\{- i 2 \pi f t\right\} \ \left\{\rm d\right\}t,$     and $X\left(f\right) = 0 \quad$  for all  $|f| > B.\,$

When x(t) is sampled uniformly at intervals of T seconds, the resultant sequence is denoted by x(nT), for all integer values of n.  And the sample-rate (samples per second) is:

$f_s\ \stackrel\left\{\mathrm\left\{def\right\}\right\}\left\{=\right\}\ 1/T.\,$

A sufficient condition to reconstruct x(t) from its samples is $f_s > 2B\!$  and equivalently $B < f_s/2.$  The two thresholds, $2B$ and $f_s/2,$ are respectively called the Nyquist rate and Nyquist frequency. And respectively, they are attributes of x(t) and of the sampling equipment. The condition described by these inequalities is called the Nyquist criterion, or sometimes the Raabe condition. The theorem is also applicable to functions of other domains, such as space, in the case of a digitized image. The only change, in the case of other domains, is the units of measure applied to t, T, f, fs, and B.

One way to ensure the Nyquist criterion is met (as an approximation) is to understand the physics of the origination of x(t), and selecting an appropriate rate. For example, sounds that are made by the human voice normally contain relatively insignificant frequency components at or above 10 kHz. Sampling such an audio signal at 20k sam/sec, or more, provides an excellent approximation to meeting the criterion. But often the sample-rate is pre-determined by other considerations, such as an industry standard format (e.g. 8k sam/sec). In this situation, the human voice should be filtered, to remove frequency components above 4 kHz, before being sampled. The type of filter required is a lowpass filter, and in this type of application it is referred to as an anti-aliasing filter.

Methods that reconstruct a continuous function, from the x(nT) sequence, are called interpolation. As demonstrated below, the mathematically ideal way to reconstruct x(t) involves the use of sinc functions, like those shown in Fig 2. Each sample in the sequence is replaced by a sinc function, centered on the time axis at the original location of the sample (nT), with the amplitude of the sinc function scaled to the sample value, x(nT). In the following step, all the sinc functions are summed into a continuous function. A mathematically equivalent method is to convolve one sinc function with a series of Dirac delta pulses, weighted by the sample values. Neither method is numerically practical. Instead, some type of approximation of the sinc functions, finite in length, should be utilized. The imperfections attributable to the approximation are known as interpolation error.

Practical digital-to-analog converters produce neither scaled and delayed sinc functions, nor ideal Dirac pulses. Instead they produce a piecewise-constant sequence of scaled and delayed rectangular pulses, usually followed by a "shaping filter" to clean up spurious high-frequency content.

Aliasing

Main article: Aliasing

The Poisson summation formula shows that the samples, x(nT), of function x(t) are sufficient to create a periodic summation of function X(f). The result is (T=1/fs):

$X_s\left(f\right)\ \stackrel\left\{\mathrm\left\{def$

(})

{=} \sum_{k=-\infty}^{\infty} X\left(f - k f_s\right) = \sum_{n=-\infty}^{\infty} \underbrace{T\cdot x(nT)}_{x[n]}\ e^{-i 2\pi n T f},|Eq.1}}

which is a periodic function and its equivalent representation as a Fourier series, whose coefficients are x[n]. This function is also known as the discrete-time Fourier transform (DTFT). As depicted in Figures 4 and 5, copies of X(f) are shifted by multiples of fs and combined by addition.

If the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous X(f). Any frequency component above fs/2 is indistinguishable from a lower-frequency component, called an alias, associated with one of the copies. In such cases, the reconstruction technique described below produces the alias, rather than the original component.

Derivation as a special case of Poisson summation

From Figure 5, it is apparent that when there is no overlap of the copies (aka "images") of X(f), the k = 0 term of Xs(f) can be recovered by the product:

$X\left(f\right) = H\left(f\right) \cdot X_s\left(f\right),\,$      where:
$H\left(f\right)\ \stackrel\left\{\mathrm\left\{def\right\}\right\}\left\{=\right\}\ \begin\left\{cases\right\}1 & |f| < B \\ 0 & |f| > f_s - B. \end\left\{cases\right\}$

At this point, the sampling theorem is proved, since X(f) uniquely determines x(t).

All that remains is to derive the formula for reconstruction. H(f) need not be precisely defined in the region [B, fsB] because Xs(f) is zero in that region. However, the worst case is when B = fs/2, the Nyquist frequency. A function that is sufficient for that and all less severe cases is:

$H\left(f\right) = \mathrm\left\{rect\right\} \left\left(\frac\left\{f\right\}\left\{f_s\right\} \right\right) = \begin\left\{cases\right\}1 & |f| < \frac\left\{f_s\right\}\left\{2\right\} \\ 0 & |f| > \frac\left\{f_s\right\}\left\{2\right\}, \end\left\{cases\right\}$

where rect(•) is the rectangular function.  Therefore:

$X\left(f\right) = \mathrm\left\{rect\right\} \left\left(\frac\left\{f\right\}\left\{f_s\right\} \right\right) \cdot X_s\left(f\right)\$
$= \mathrm\left\{rect\right\}\left(Tf\right)\cdot \sum_\left\{n=-\infty\right\}^\left\{\infty\right\} T\cdot x\left(nT\right)\ e^\left\{-i 2\pi n T f\right\}$      (from  Eq.1, above).
$= \sum_\left\{n=-\infty\right\}^\left\{\infty\right\} x\left(nT\right)\cdot \underbrace\left\{T\cdot \mathrm\left\{rect\right\} \left(Tf\right) \cdot e^\left\{-i 2\pi n T f\right\}\right\}_\left\{$

\mathcal{F}\left \{

\mathrm{sinc} \left( \frac{t - nT}{T} \right)


\right \} }.     

The inverse transform of both sides produces the Whittaker–Shannon interpolation formula:

$x\left(t\right) = \sum_\left\{n=-\infty\right\}^\left\{\infty\right\} x\left(nT\right)\cdot \mathrm\left\{sinc\right\} \left\left( \frac\left\{t - nT\right\}\left\{T\right\}\right\right),$

which shows how the samples, x(nT), can be combined to reconstruct x(t).

• From Figure 5, it is clear that larger-than-necessary values of fs (smaller values of T), called oversampling, have no effect on the outcome of the reconstruction and have the benefit of leaving room for a transition band in which H(f) is free to take intermediate values. Undersampling, which causes aliasing, is not in general a reversible operation.
• Theoretically, the interpolation formula can be implemented as a low pass filter, whose impulse response is sinc(t/T) and whose input is $\textstyle\sum_\left\{n=-\infty\right\}^\left\{\infty\right\} x\left(nT\right)\cdot \delta\left(t - nT\right),$ which is a Dirac comb function modulated by the signal samples. Practical digital-to-analog converters (DAC) implement an approximation like the zero-order hold. In that case, oversampling can reduce the approximation error.

Shannon's original proof

Poisson shows that the Fourier series in Eq.1 produces the periodic summation of X(f), regardless of fs and B. Shannon, however, only derives the series coefficients for the case fs = 2B. Quoting Shannon's original paper, which uses f for the function, F for the spectrum, and W instead of B:

Let $F\left(\omega\right)$ be the spectrum of $f\left(t\right).$  Then
 $f\left(t\right)\,$ $= \left\{1 \over 2\pi\right\} \int_\left\{-\infty\right\}^\left\{\infty\right\} F\left(\omega\right) e^\left\{i\omega t\right\}\;\left\{\rm d\right\}\omega \$ $= \left\{1 \over 2\pi\right\} \int_\left\{-2\pi W\right\}^\left\{2\pi W\right\} F\left(\omega\right) e^\left\{i\omega t\right\}\;\left\{\rm d\right\}\omega \$
since $F\left(\omega\right)$ is assumed to be zero outside the band W. If we let
$t = \left\{n \over \left\{2W\right\}\right\}\,$
where n is any positive or negative integer, we obtain
$f \left\left(\left\{n \over \left\{2W\right\}\right\} \right\right) = \left\{1 \over 2\pi\right\} \int_\left\{-2\pi W\right\}^\left\{2\pi W\right\} F\left(\omega\right) e^\left\{i\omega \left\{n \over \left\{2W\right\}\right\}\right\}\;\left\{\rm d\right\}\omega.$
On the left are values of $f\left(t\right)$ at the sampling points. The integral on the right will be recognized as essentially[note 1] the nth coefficient in a Fourier-series expansion of the function $F\left(\omega\right),$ taking the interval –W to W as a fundamental period. This means that the values of the samples $f\left(n/2W\right)$ determine the Fourier coefficients in the series expansion of $F\left(\omega\right).$  Thus they determine $F\left(\omega\right),$ since $F\left(\omega\right)$ is zero for frequencies greater than W, and for lower frequencies $F\left(\omega\right)$ is determined if its Fourier coefficients are determined. But $F\left(\omega\right)$ determines the original function $f\left(t\right)$ completely, since a function is determined if its spectrum is known. Therefore the original samples determine the function $f\left(t\right)$ completely.

Shannon's proof of the theorem is complete at that point, but he goes on to discuss reconstruction via sinc functions, what we now call the Whittaker–Shannon interpolation formula as discussed above. He does not derive or prove the properties of the sinc function, but these would have been familiar to engineers reading his works at the time, since the Fourier pair relationship between rect (the rectangular function) and sinc was well known. Quoting Shannon:

Let $x_n$ be the nth sample. Then the function $f\left(t\right)$ is represented by:
$f\left(t\right) = \sum_\left\{n=-\infty\right\}^\left\{\infty\right\}x_n\left\{\sin \pi\left(2Wt-n\right) \over \pi\left(2Wt-n\right)\right\}.\,$

As in the other proof, the existence of the Fourier transform of the original signal is assumed, so the proof does not say whether the sampling theorem extends to bandlimited stationary random processes.