explain what aliasing is, with respect to sampling an analog

Sampling Theorem

Converting analog to digital signals and vice versa

Edmund Lai PhD, BEng , in Practical Digital Point Processing, 2003

ii.2.1 Sampling theorem

The sampling theorem specifies the minimum-sampling charge per unit at which a continuous-time signal needs to be uniformly sampled and so that the original bespeak tin be completely recovered or reconstructed past these samples alone. This is unremarkably referred to as Shannon's sampling theorem in the literature.

Sampling theorem:

If a continuous time betoken contains no frequency components college than W hz, and then information technology tin can be completely determined by compatible samples taken at a rate f _{due south} samples per second where

or, in term of the sampling period

A signal with no frequency component above a certain maximum frequency is known equally a bandlimited signal. Figure 2.4 shows two typical bandlimited point spectra: one low-laissez passer and one ring-pass.

Figure 2.four. Two bandlimited spectra

The minimum sampling rate allowed past the sampling theorem (f _s = twoW) is chosen the Nyquist rate.

Information technology is interesting to note that even though this theorem is usually chosen Shannon's sampling theorem, it was originated past both E.T. and J.M. Whittaker and Ferrar, all British mathematicians. In Russian literature, this theorem was introduced to communications theory by Kotel'nikov and took its name from him. C.E. Shannon used it to report what is now known as information theory in the 1940s. Therefore in mathematics and engineering literature sometimes it is too called WKS sampling theorem after Whittaker, Kotel'nikov and Shannon.

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Acquisition of Medical Image Information

Bernhard Preim , Charl Botha , in Visual Calculating for Medicine (Second Edition), 2014

two.iii.1 Sampling Theorem

The main basis in signal theory is the sampling theorem that is credited to Nyquist [1924] —who first formulated the theorem in 1928.

The sampling theorem substantially says that a signal has to be sampled at least with twice the frequency of the original signal. Since signals and their corresponding speed can be easier expressed past frequencies, most explanations of artifacts are based on their representation in the frequency domain. The sampling frequency required past the sampling theorem is chosen the Nyquist frequency.

The transformation of signals into the frequency domain (Fig. 2.five) is performed by the Fourier transformation, which essentially reformulates the bespeak into a cosine office space. If, for case, we transform the sine function into the frequency domain, this results in a tiptop at the frequency of the sine function (Fig. two.five, correct). Due to the symmetry of the Fourier transform for real values, in that location are two peaks on both sides of the ordinate (y) axis. More details on the frequency-domain-based interpretation tin can be found in Glassner [1995].

Figure 2.5. Sine -function is sampled (arrows) at aforementioned speed equally sine periodicity of $T=2\pi /\mathrm{westward}$ . Left: spatial domain representation, correct: frequency domain representation, where the frequency is represented by two symmetric peaks at −w and w.

(Courtesy of Dirk Bartz, University of Leipzig)

Equally all epitome information tin exist interpreted as a spatial or Fourier-transformed bespeak, we base of operations our discussion on a simple example of an 1D point, the sine function. This function has a straightforward representation in the spatial and the frequency domain. Effigy 2.five (left) shows a section of the sine part effectually the origin. If we transform this continuous representation into a discrete representation, nosotros need to have samples of the continuous sine function to measure its characteristics. Figure ii.five (left) demonstrates what happens if we take the samples at the same frequency as our original function. Since the sine function has the periodicity $T=2\pi /\text{w}$ (or the frequency of $\text{w}/2\pi$ ), this sampling frequency would exist also T. Equally Figure 2.5 (left) shows, sampling the sine function at the same speed would recover always the aforementioned sine value in dissimilar periods, thus pretending that information technology is a constant function.

If we increase the sampling speed to half of the periodicity of the continuous function, the minimum need of the sampling theorem, nosotros can recover the correct characteristics of the sine office, every bit it can exist seen in Figure 2.vi (left). Yet, depending on what exact position in the period T of the original function nosotros take the sample, we recover different amplitudes of the original point. In an unfortunate case, we always sample the nada crossing of the sine function, as shown in Effigy 2.6 (correct). In this case, the characteristics could exist correctly recovered, but the aamplitude of the signal was recovered in an unfortunate style, so nosotros are back with a abiding signal. Overall, sampling at a rate satisfying the sampling theorem does non guarantee that the full signal strength is reconstructed, although higher sampling rates usually approach the original strength.

Figure two.6. Sine-function is sampled (arrows) at Nyquist charge per unit. Left: Sampling at optimal position (detects peaks and valleys), right: Sampling phase shift—only a constant (zero) amplitude is sampled.

(Courtesy of Dirk Bartz, Academy of Leipzig)

In the frequency domain, sampling of the original point is described as the convolution of the original signal with a comb part (with peaks repeating at the sampling frequency). Due to the periodicity of the comb office (nosotros take samples at regular positions), the convolved signal likewise exposes a replicating design. Since exactly i of these signal patterns is used, we demand to select one re-create of it with a low-laissez passer filter, a technique that is explained in the next section. If the sampling rate is not high enough, it will upshot in an overlap of the replicating patterns, and hence in the inability to select one re-create of the blueprint.

Annotation that medical image information does consist of a full spectrum of frequencies, which are revealed once the Fourier transform of the data is calculated. Therefore, the respective limiting cut-off frequency of this spectrum should exist taken into account, when estimating the right sampling frequency.

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Data Manual Media

John S. Sobolewski , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

VI.E Pulse Code Moudulation

In pulse amplitude modulation, the aamplitude of the pulse can assume whatever value betwixt zippo and some maximum value. Pulse code modulation (PCM) is derived from PAM simply is distinguished from the latter by two additional signal-processing steps, chosen quantizing and encoding, that take place before the signal is transmitted. Quantizing replaces the exact amplitude of the samples with the nearest value from a limited set of specific amplitudes. The sample amplitude is then encoded, and the codes are transmitted typically as binary codes. This ways that, unlike other modulation techniques described then far, in PCM both the sampling time and the amplitude are in discrete grade.

Representing an exact sample amplitude by one of two ⁿ predetermined and discrete amplitudes introduces an error chosen quantization noise that can exist made negligible by using a sufficiently large number of quantizing levels. Studies have shown that using viii $.25 per sample to stand for one of 256 quantizing levels provides a satisfactory signal-to-noise ratio for voice communication signals (meet Rey, 1983 ). The sampling rate is usually determined from the sampling theorem, which states that a baseband (information) signal of finite energy with no frequency components higher than Westward Hz is completely specified by the amplitudes of its samples taken at a rate of 2 W/sec. The corollary of the sampling theorem states that a baseband analog channel can be used to transmit a train of independent pulses at a maximum rate that is twice the channel bandwidth West. These results are important in determining appropriate sample rates and bandwidths for conversion between analog and digital signals.

Applying the sampling theorem to speech signals that are limited to 4000   Hz, we observe that they demand to be sampled 8000   times/sec to be completely specified. Using PCM with 8 bits to represent one of 256 discrete amplitude samples, 8   ×   8000 or 64,000   bits/sec are required to transmit the 4000-Hz voice signal. If we at present use the corollary to the sampling theorem, we find that a channel with a bandwidth of 32,000   Hz is required to transmit the 64,000   bits/sec needed to specify the 4000-Hz vox signal. Although it is true that PCM requires more than bandwidth than the baseband analog signal (32,000   Hz bandwidth for the 4000-Hz phonation signal in the above example), this is more than get-go by the following:

i.

PCM has very loftier amnesty to noise.

ii.

PCM repeater pattern is relatively simple.

3.

The PCM point can be completely reconstructed at each repeater location by a process chosen regeneration.

4.

PCM provides a uniform modulation technique suitable for other signals on many dissimilar types of media including wire, coaxial cable, free infinite, and optical fibers.

5.

PCM is uniform with time partitioning multiplexing.

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Prototype Acquisition

Eastward.R. DAVIES , in Machine Vision (Third Edition), 2005

27.4 The Sampling Theorem

The Nyquist sampling theorem underlies all situations where continuous signals are sampled and is especially important where patterns are to exist digitized and analyzed by computers. This makes it highly relevant both with visual patterns and with acoustic waveforms. Hence, it is described briefly in this department.

Consider the sampling theorem kickoff in respect of a one-D time-varying waveform. The theorem states that a sequence of samples (Fig. 27.9) of such a waveform contains all the original information and can be used to regenerate the original waveform exactly, just only if (one) the bandwidth W of the original waveform is restricted and (2) the rate of sampling f is at least twice the bandwidth of the original waveform—that is, f ≥ 2W. Assuming that samples are taken every T seconds, this ways that 1/T ≥ 2W.

Figure 27.9. The process of sampling a time-varying indicate. A continuous time-varying 1-D signal is sampled by narrow sampling pulses at a regular rate f _r = i/T, which must exist at least twice the bandwidth of the signal.

At start, it may be somewhat surprising that the original waveform tin be reconstructed exactly from a set of detached samples. Withal, the 2 conditions for achieving perfect reconstruction are very stringent. What they are demanding in effect is that the signal must non be permitted to alter unpredictably (i.east., at too fast a rate) or else accurate interpolation betwixt the samples will non prove possible (the errors that ascend from this source are called "aliasing" errors).

Unfortunately, the starting time status is virtually unrealizable, since information technology is nearly incommunicable to devise a depression-pass filter with a perfect cutoff. Retrieve from Chapter 3 that a low-pass filter with a perfect cutoff volition have infinite extent in the time domain, then any attempt at achieving the same issue by time domain operations must be doomed to failure. However, adequate approximations tin exist accomplished past allowing a "baby-sit-band" between the desired and bodily cutoff frequencies. This means that the sampling rate must therefore be higher than the Nyquist rate. (In telecommunication, satisfactory performance can generally be accomplished at sampling rates effectually 20% above the Nyquist rate—see Brown and Glazier, 1974.)

One way to recover the original waveform is to apply a low-pass filter. This approach is intuitively correct because it acts in such a way equally to augment the narrow discrete samples until they coalesce and sum to give a continuous waveform. This method acts in such a style as to eliminate the "repeated" spectra in the transform of the original sampled waveform (Fig. 27.ten). This in itself shows why the original waveform has to be narrow-banded before sampling—so that the repeated and basic spectra of the waveform do not cross over each other and become impossible to separate with a low-pass filter. The idea may be taken further because the Fourier transform of a square cutoff filter is the sinc (sin u/u) part (Fig. 27.eleven). Hence, the original waveform may be recovered by convolving the samples with the sinc function (which in this case means replacing them by sinc functions of corresponding amplitudes). This broadens out the samples equally required, until the original waveform is recovered.

Figure 27.10. Effect of low-pass filtering to eliminate repeated spectra in the frequency domain f _r, sampling rate; L, depression-pass filter characteristic). This diagram shows the repeated spectra of the frequency transform F(f) of the original sampled waveform. It as well demonstrates how a depression-pass filter can be expected to eliminate the repeated spectra to recover the original waveform.

Figure 27.xi. The sinc (sin u/u) function shown in (b) is the Fourier transform of a foursquare pulse (a) corresponding to an ideal depression-pass filter. In this case, u = 2πf_ct, f _c being the cutoff frequency.

Then far we have considered the situation only for 1-D time-varying signals. However, recalling that an exact mathematical correspondence exists between fourth dimension and frequency domain signals on the ane paw and spatial and spatial frequency signals on the other, the in a higher place ideas may all be applied immediately to each dimension of an image (although the status for accurate sampling now becomes one/X ≥ 2W_X, where X is the spatial sampling period and W_X is the spatial bandwidth). Here we accept this correspondence without further give-and-take and go along to apply the sampling theorem to image acquisition.

Consider next how the signal from a TV camera may be sampled rigorously co-ordinate to the sampling theorem. First, it is obviously that the analog voltage comprising the time-varying line signals must be narrow-banded, for example, by a conventional electronic low-pass filter. All the same, how are the images to exist narrow-banded in the vertical direction? The same question clearly applies for both directions with a solid-state area camera. Initially, the most obvious solution to this problem is to perform the procedure optically, possibly past defocussing the lens. Still, the optical transform part for this case is frequently (i.e., for farthermost cases of defocusing) very odd, going negative for some spatial frequencies and causing contrast reversals; hence, this solution is far from ideal (Pratt, 2001). Alternatively, nosotros could utilise a diffraction-express optical system or perhaps pass the focused beam through some sort of patterned or frosted glass to reduce the spatial bandwidth artificially. None of these techniques will be particularly easy to use nor will accurate solutions be probable to event. However, this trouble is not as serious as might be imagined. If the sensing region of the photographic camera (per pixel) is reasonably large, and close to the size of a pixel, then the averaging inherent in obtaining the pixel intensities will in fact perform the necessary narrow-banding (Fig. 27.12). To analyze the situation in more than detail, annotation that a pixel is essentially square with a sharp cutoff at its borders. Thus, its spatial frequency design is a 2-D sinc office, which (taking the cardinal positive elevation) approximates to a low-pass spatial frequency filter. This approximation improves somewhat as the border between pixels becomes fuzzier.

Figure 27.12. Depression-pass filtering carried out past averaging over the pixel region. An paradigm with local high-frequency banding is to be averaged over the whole pixel region by the action of the sensing device.

The indicate here is that the worst example from the point of view of the sampling theorem is that of extremely narrow detached samples, but this worst case is unlikely to occur with nigh cameras. Nevertheless, this does not mean that sampling is automatically ideal—and indeed it is not, since the spatial frequency pattern for a sharply defined pixel shape has (in principle) infinite extent in the spatial frequency domain. The review by Pratt (2001) clarifies the situation and shows that there is a tradeoff between aliasing and resolution error. Overall, it is underlined hither that quality of sampling will exist one of the limiting factors if ane aims for the greatest precision in image measurement. If the bandwidth of the presampling filter is too low, resolution will be lost; if it is too loftier, aliasing distortions will creep in; and if its spatial frequency response curve is not suitably shine, a guard ring volition accept to be included and performance will again suffer.

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Digital Communication Organization Concepts

Vijay Grand. Garg , Yih-Chen Wang , in The Electrical Engineering Handbook, 2005

2.three Sampling Process

Analog information must be transformed into a digital format. The process starts with sampling the waveform to produce a discrete pulse-amplitude-modulated waveform (meet Figure ii.3). The sampling process is commonly described in a fourth dimension domain. This is an functioning that is bones to digital signal processing and digital communication. Using the sampling process, nosotros convert the analog signal in a respective sequence of samples that are usually spaced uniformly in fourth dimension. The sampling process can be implemented in several ways, the most popular being the sample-and-hold operation. In this operation, a switch and storage mechanism (such equally a transistor and a capacitor, or shutter and a film strip) form a sequence of samples of the continuous input waveform. The output of the sampling procedure is called pulse amplitude modulation (PAM) because the successive output intervals tin be described as a sequence of pulses with amplitudes derived from the input waveform samples. The analog waveform can exist approximately retrieved from a PAM waveform by elementary depression-laissez passer filtering, provided nosotros cull the sampling charge per unit properly. The ideal course of sampling is called instantaneous sampling.

FIGURE two.3. Sampling Process

We sample the betoken k(t) instantaneously at a compatible rate of f _{due south} one time every T_s sec. Thus, we tin can write:

(2.ane) $g_{\delta }(t)=\underset{n=-\infty }{\overset{\infty }{\Sigma }}\mathrm{chiliad}(\mathrm{due\; north}{T}_s)\delta (t-n{T}_s),$

where g_δ (t) is the ideal sampled signal and where δ(t − nT_s ) is the delta role positioned at time t = nT_s .

A delta office is closely approximated by a rectangular pulse of duration Δt and amplitude g(nT_S )/Δt; the smaller we make Δt, the amend will be the approximation:

(2.2) ${\mathrm{1000}}_{\delta }(t)=f_s\underset{m=-\infty }{\overset{\infty }{\Sigma }}G(f-\mathrm{1000}{f}_s),$

where G(f) is the Fourier transform of the original signal grand(t) and f_{due south} is sampling rate.

Equation 2.2 states that the process of uniformly sampling a continuous-time bespeak of finite energy results in a periodic spectrum with a menstruum equal to the sampling rate.

Taking the Fourier transform of both side, of Equation 2.1 and noting that the Fourier transform of the delta function δ(t − nT_s ) is equal to $e^{-j\mathrm{ii}\pi nf{T}_{\mathrm{south}}}$ :

(2.iii) $G_{\delta }(f)\underset{\mathrm{north}=-\infty }{\overset{\infty }{\Sigma }}g(n{T}_s){\mathrm{eastward}}^{-j2\pi nf{T}_S}.$

Equation 2.3 is called the discrete-time Fourier transform. It is the complex Fourier serial representation of the periodic frequency function G_δ (t), with the sequence of samples 1000(nT_s ) defining the coefficients of the expansion.

We consider any continuous-time bespeak g(t) of finite energy and infinite duration. The betoken is strictly band-express with no frequency component college than W Hz. This implies that the Fourier transform G(f) of the signal g(t) has the property that G(f) is nada for |f| ≥ W. If we choose the sampling period T_s = 1/ii Due west, so the corresponding spectrum is given as:

(2.4) $G_{\delta }(f)=\underset{n=\infty }{\overset{\infty }{\Sigma }}g\left(\frac{n}{2\mathrm{West}}\right)e^{-\frac{j\pi nf}{W}}=f_{\mathrm{southward}}G(f)+f_s\underset{m=-\infty ,m\ne 0}{\overset{\infty }{\Sigma }}G(f-m{f}_s)$

Consider the post-obit two conditions:

(1)

G(f) = 0 for |f| ≥ W.

(2)

f_southward = ii W.

Nosotros find from equation 2.four by applying these weather,

(2.v) $\begin{array}{l}G(f)=\frac{\mathrm{one}}{2W}{G}_{\delta }(f)-W<f<\mathrm{Due\; west}.\\ \therefore G(f)=\frac{1}{2\mathrm{Westward}}\underset{n=-\infty }{\overset{\infty }{\Sigma }}g\left(\frac{n}{2W}\right)e^{-(\frac{j\pi nf}{W})}-W<f<W.\end{array}$

Thus, if the sample value thou(n/ii W) of a signal g(t) is specified for all north, and then the Fourier transform G(f) of the signal is uniquely adamant past using the detached-time Fourier transform of equation 2.five. Because g(t) is related to G(f) by the inverse Fourier transform, it follows that the signal m(t) is itself uniquely determined by the sample values one thousand(n/2 W) for −∞ < n < ∞. In other words, the sequence {g(n/ii W)} has all the information independent in g(t).

We state the sampling theorem for band-limited signals of finite energy in two parts that apply to the transmitter and receiver of a pulse modulation arrangement, respectively.

(1)

A band-express signal of finite energy with no frequency components college than Due west Hz is completely described by specifying the values of signals at instants of time separated by 1/ii West sec.

(2)

A band-limited signal of finite free energy with no frequency components higher than W Hz may be completely recovered from a cognition of its samples taken at the rate of 2 W samples/sec.

This is as well known equally the compatible sampling theorem. The sampling rate of 2 W samples per second for a bespeak bandwidth W Hz is called the Nyquist rate and i/ii W sec is called the Nyquist interval.

Nosotros discuss the sampling theorem by assuming that signal grand(t) is strictly band-limited. In practice, all the same, an information-begetting signal is not strictly band-limited, with the upshot that some degree of under sampling is encountered. Consequently, some aliasing is produced by the sampling process. Aliasing refers to the phenomenon of a loftier-frequency component in the spectrum of the bespeak seemingly taking on the identity of a lower frequency in the spectrum of its sampled version.

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Discrete Revolution

Stéphane Mallat , in A Wavelet Tour of Indicate Processing (Tertiary Edition), 2009

Removal of Aliasing

To apply the sampling theorem, f is approximated by the closest signal $\tilde{f}$ , the Fourier transform of which has a back up in [–π/s, π/due south]. The Plancherel formula (2.26) proves that

$\begin{array}{ll}{\Vert f-\tilde{f}\Vert }^2& =\frac{\mathrm{ane}}{\mathrm{two}\pi }{\int }_{-\infty }^{+\infty }|\hat{f}\left(\omega \right)-\hat{\tilde{f}}\left(\omega \right){|}^{2}d\omega \\ & =\frac{1}{\mathrm{two}\pi }{\int }_{\left|\omega \right|>\pi /s}|\hat{f}\left(\omega \right){|}^{2}d\omega +\frac{1}{2\pi }{\int }_{\left|\omega \right|\le \pi /s}|\hat{f}\left(\omega \right)-\hat{\tilde{f}}\left(\omega \right){|}^{2}d\omega \mathrm{.}\end{array}$

This distance is minimum when the second integral is null and therefore

(3.12) $\hat{\tilde{f}}\left(\omega \right)=\hat{f}\left(\omega \right){\mathbf{i}}_{[-\pi /\mathrm{southward},\pi /s]}\left(\omega \right)=\frac{1}{s}{\hat{\phi }}_s\left(\omega \right)\hat{f}\left(\omega \right)\mathrm{.}$

Information technology corresponds to $\tilde{f}=\frac{1}{\mathrm{due\; south}}f\star {\phi }_s$ .

The filtering of f by φ _s avoids aliasing by removing whatever frequency larger than π/s. Since $\hat{\tilde{f}}$ has a support in [–π/southward, π/s], the sampling theorem proves that ˜(t) can be recovered from the samples ˜(ns). An analog-to-digital converter is therefore equanimous of a filter that limits the frequency band to [–π/s, π/s], followed by a uniform sampling at interval s.

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Stochastic Processes

Yûichirô Kakihara , in Encyclopedia of Physical Scientific discipline and Technology (3rd Edition), 2003

VII.B Sampling Theorem

Shannon's (1949) sampling theorem was obtained for deterministic functions on $\mathbb{R}$ (signal functions). This was extended to be valid for weakly stationary stochastic processes past Balakrishnan (1957).

First we consider bandlimited and finite-energy bespeak functions. A signal function X(t) is said to be of finite energy if X  ∈ L ²( $\mathbb{R}$ ) with the Lebesgue measure out. Then its Fourier transform F X is defined in the mean-square sense past

$(\mathcal{F}\text{X})(\text{u})=\frac{\mathrm{ane}}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }\text{Ten}(\text{t}){\text{due east}}^{\text{<mi mathvariant="italic" is="true">iut</mi>}}\text{<mi mathvariant="italic" is="true">dt</mi>}=\underset{\text{T}\to \infty }{\text{l}.\text{i}.\text{m}.}{\displaystyle {\int }_{-\text{T}}^{\text{T}}\text{Ten}(\text{t}){\text{due east}}^{\text{<mi mathvariant="italic" is="true">iut</mi>}}\text{<mi mathvariant="italic" is="true">dt</mi>}}},$

where 50.i.chiliad. means "limit in the mean" and F: L ⁱⁱ( $\mathbb{R}$ )   → L ²( $\mathbb{R}$ ) turns out to be a unitary operator. A signal function X(t) is said to exist bandlimited if there exists some abiding W  >   0 such that

$(\mathcal{F}\text{X})(\text{u})=0\text{for}\text{almost}\text{every}\text{u}\text{with}|\text{u}|>\text{W}.$

Here, W is chosen a bandwidth and [−W, W] a frequency interval.

Let W  >   0 and define a function Due south _W on $\mathbb{R}$ by

${\text{S}}_{\text{W}}(\text{t})=\{\begin{array}{ll}\frac{\text{Due west}}{\pi }\frac{\text{sin}{\text{Due west}}_{\text{t}}}{{\text{W}}_{\text{t}}},\hfill & \text{t}\ne 0\hfill \\ \frac{\text{W}}{\pi },\hfill & \text{t}=0.\hfill \end{array}$

Then, South _Westward is a typical instance of a bandlimited function and is called a sample part. In fact, one tin verify that

$(\mathcal{F}{\text{S}}_{\text{Due west}})(\text{u})=\frac{1}{\sqrt{2\pi }}{\mathrm{i}}_{\text{Westward}}(\text{u}),\text{u}\in \mathbb{R},$

where one _W   =   1_{[−W, West]}, the indicator function of [−W, W]. Denote by BLW the set of all bandlimited indicate functions with bandwidth Westward  >   0. Then it is easily seen that BLW is a closed subspace of L ^two( $\mathbb{R}$ ). Now the sampling theorem for a function in BLW is stated as follows: Any 10  ∈ BLW has a sampling expansion in L ² - and L ^∞-sense given by

(20) $\text{Ten}(\text{t})={\displaystyle \sum _{\text{north}=-\infty }^{\infty }\text{X}\left(\frac{\text{north}\pi }{\text{W}}\right)\sqrt{\frac{\text{W}}{\pi }}{\varphi }_{\text{n}}(\text{t})},$

where the ϕ _n are divers past

${\varphi }_{\text{n}}(\text{t})=\sqrt{\frac{\pi }{\text{W}}}{\text{South}}_{\text{W}}\left(\text{t}-\frac{\text{due north}\pi }{\text{W}}\right),\text{t}\in \mathbb{R},\text{n}\in \mathbb{Z},$

{ϕ _n }_−∞ ^∞ forms a complete orthonormal organisation in BLW, and information technology is called a system of sampling functions. Nosotros can say that a sampling theorem is a Fourier expansion of an L ²-function with respect to this system of sampling functions.

A sampling theorem holds for some stochastic processes. Let {X(t)} exist an L ² ₀(Ω)-valued weakly harmonizable procedure with the representing measure out ξ, i.due east.,

$\text{Ten}(\text{t})={\displaystyle {\int }_{-\infty }^{\infty }{\text{e}}^{\text{<mi mathvariant="italic" is="true">itu</mi>}}\xi (\text{<mi mathvariant="italic" is="true">du</mi>}),\text{t}\in \mathbb{R}}.$

We say that {Ten(t)} is bandlimited if in that location exists a W  >   0 such that the back up of ξ is contained in [−W, Westward], i.e., ξ(A)   =   0 if A ∩ [−W, W]   =   ∅. If this is the instance, the sampling theorem holds:

$\begin{array}{r}\hfill \text{Ten}(\text{t},\omega )={\displaystyle \sum _{\text{n}=-\infty }^{\infty }\text{X}\left(\frac{\text{due north}\pi }{\text{W}},\omega \right)\frac{\text{sin}\left(\text{Westward}(\text{t}-\text{n}\pi /\text{W})\right)}{\text{W}(\text{t}-\text{n}\pi /\text{W})}},\\ \hfill \text{t}\in \mathbb{R},\end{array}$

where the convergence is in ∥ · ∥₂ for each t  ∈ $\mathbb{R}$ .

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Common Diffraction Integral Adding Based on a Fast Fourier Transform Algorithm

Junchang Li , ... Yan Li , in Advances in Imaging and Electron Physics, 2010

Abstract

From the basic theory of sampling theorem and detached Fourier transform, the scalar diffraction theory is studied, which involves the Fresnel diffraction integral, the Kirchhoff formula, the Rayleigh–Sommerfeld formula, the athwart spectrum transmission formula for diffraction, and the Collins formula. The sampling weather that each formula must encounter to ensure correct calculations are deduced. Based on the results, an application case of the binary optical design by the Fresnel diffraction integral and its inverse operation is given, and another example regarding wavefront reconstruction with the Collins inverse formula is also presented.

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Data Conquering

Wim van Drongelen , in Indicate Processing for Neuroscientists, 2007

2.three SAMPLING AND NYQUIST FREQUENCY IN THE FREQUENCY DOMAIN

This department considers the Nyquist sampling theorem in the frequency domain. Unfortunately, this explanation in its simplest form requires a groundwork in the Fourier transform and convolution, both topics that will be discussed subsequently (meet Chapters five through 8 Chapter 5 Affiliate 6 Chapter 7 Chapter 8 ). Readers who are not all the same familiar with these topics are advised to skip this section and return to it later. In this section, we approach sampling in the frequency domain somewhat intuitively and focus on the general principles depicted in Figure ii.half dozen. A more formal handling of the sampling problem can exist constitute in Appendix 2.1.

Figure 2.6. Fourier transform of a sampled role. Sampling a function f(t) (A) in the time domain can be represented by a multiplication (*) of f(t) with a train of δ functions with an interval T_s, as depicted in (B), resulting in a series of samples (C). The Fourier transform of the sampled version is a periodic function, as shown in (D). The Fourier transform of the sampled function can be obtained from the convolution (⊗) of the Fourier transform F(f) of f(t), shown in (Eastward), and the Fourier transform of the railroad train of unit of measurement impulses with an interval F_south = 1/T _{due south} , as shown in (F). From this diagram, information technology can exist appreciated that the width of F(f) should fall within period F_{due south} (i.e., the maximum value of the spectrum of the sampled signal must be less than F_s /ii) to avoid overlap in the spectra (shown in Fig. 2.7). Further details can be establish in Appendix 2.ane.

When sampling a part f(t), using the sifting property of the δ function, every bit in Equation (ii.eight), we multiply the continuous fourth dimension role with a Dirac comb, a series of unit impulses with regular interval T_s :

(2.xi) $\text{Sampled}\text{function}:f\left(t\right)\sum _{\mathrm{northward}=-\infty }^{\infty }\delta \left(t-n{T}_{\mathrm{due\; south}}\right)$

As we will discuss in Chapter 8, multiplication in the time domain is equivalent to a convolution (⊗) in the frequency domain:

(2.12) $F\left(f\right)\otimes \Delta \left(f\right)withF\left(f\right)\iff f\left(t\right)and\Delta \left(f\right)\iff \sum _{\mathrm{northward}=-\infty }^{\infty }\delta \left(t-n{T}_{\mathrm{south}}\right)$

The double pointer ⇔ in Equation (two.12) separates a Fourier transform pair: here the frequency domain is left of the arrow and the fourth dimension domain equivalent is the expression on the right of ⇔. We can use the sifting property to evaluate the Fourier transform integral (Equation (6.4), in Affiliate six): of a single delta office:

(2.13) $\delta \left(t\right)\iff \underset{-\infty }{\overset{\infty }{\int }}\delta \left(t\right)e^{-\mathrm{two}\pi ft}dt={\mathrm{due\; east}}^0=1$

For the series of impulses (the Dirac comb), the transform Δ(f) is a more complex expression, according to the definition of the Fourier transform

(ii.xiv) $\Delta \left(f\right)=\underset{-\infty }{\overset{\infty }{\int }}\sum _{\mathrm{due\; north}=-\infty }^{\infty }\delta \left(t-\mathrm{north}{T}_s\right)e^{-2^{\pi ft}}dt$

Assuming that nosotros tin can interchange the summation and integral operations, and using the sifting holding again, this expression evaluates to

(2.15) $\sum _{\mathrm{north}=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}\delta \left(t-n{T}_s\right)e^{-\mathrm{two}\pi ft}dt=\sum _{n=-\infty }^{\infty }{e}^{-2\pi \mathrm{due\; north}{T}_{\mathrm{southward}}}$

An essential divergence between this expression and the Fourier transform of a unmarried δ office is the summation for due north from −∞to ∞. Irresolute the sign of the exponent in Equation (ii.15) is equivalent to irresolute the order of the summation from −∞ → ∞ to ∞ → –∞. Therefore we may state

(two.16) $\sum _{\mathrm{northward}=-\infty }^{\infty }{\mathrm{due\; east}}^{-\mathrm{ii}\pi n{T}_s}=\sum _{n=-\infty }^{\infty }{e}^{2\pi n{T}_s}$

From Equation (2.xvi) it can exist established that the sign of the exponent in Equations (2.thirteen) to (2.sixteen) does not matter. Recollect most this a bit: taking into account the similarity between the Fourier transform and the changed transform integrals (Equations (vi.4) and (6.eight) in Chapter 6), the main divergence of the integral being the sign of the exponent, this indicates that the Fourier transform and the inverse Fourier transform of a Dirac comb must evaluate to a similar class. This leads to the conclusion that the (inverse) Fourier transform of a Dirac rummage must exist another Dirac comb. Given that in the fourth dimension domain, we accept $\sum _{n=-\infty }^{\infty }\delta \left(t-\mathrm{northward}{T}_s\right)$ , its Fourier transform in the frequency domain must be proportional to $\sum _{\mathrm{northward}=-\infty }^{\infty }\delta \left(f-n{F}_s\right)$ ).

In these expressions, the sample frequency F_south = 1/T_s. If yous experience that this "proof" is too informal, please consult Appendix 2.1 for a more thorough approach. You will find there that we are indeed ignoring a scaling factor equal to 1/T_s in the preceding expression (come across Equation (A2.1-7), Appendix 2.i).

We will not worry almost this scaling factor here; because for sample rate problems, we are interested in timing and non amplitude. For now, nosotros can constitute the relationship between the Fourier transform F(f) of a part f(t) and the Fourier transform of its sampled version. Using the obtained consequence and Equation (2.12), we find that the sampled version is proportional to

(2.17) $F\left(f\right)\otimes \sum _{n=-\infty }^{\infty }\delta \left(f-\mathrm{due\; north}{F}_s\right)$

This consequence is easiest interpreted past the graphical representation of convolution (Chapter 8 and (Appendix 8.1), which is sliding the Dirac comb (Fig. 2.6F) forth the Fourier transform F(f) (Fig. 2.6E). At whatsoever point in this sliding process, the impulses in the railroad train sift the value in the Fourier transform F(f). When F(f) lies within the gaps between the private δ functions, nosotros obtain a periodic function as shown in Figure 2.6D. This result illustrates the same human relationship between sample frequency and highest frequency component in a bespeak as discussed earlier. For F(f) to fall within the gaps of the δ function train, the highest frequency in betoken f(t) must be < F_s/2, the Nyquist frequency. If, on the contrary, F(f) does not fall inside the gaps of the δ function railroad train, there will be an overlap resulting in distortion due to an aliasing effect (Fig. 2.vii).

Figure 2.vii. Equivalent of Figure ii.6D in the case where the spectra F(f) do non fit inside the impulses in the impulse train. This will cause the sum of the individual contributions (red) to include overlap, resulting in an aliasing effect.

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Advances in Imaging and Electron Physics

Leonid P. Yaroslavsky , in Advances in Imaging and Electron Physics, 2011

2 Discrete Sampling Theorem

Let A _North be a vector of N samples {a_{one thousand} } _k = 0,…,N … one of a discrete point, Φ _N exist an N × Due north orthogonal transform matrix,

(two.1) ${\Phi }_N={\{{\phi }_r(\mathrm{thousand})\}}_{r=0,1,\dots ,\mathrm{North}-\mathrm{i}}$

composed of basis functions φ_r (k), and Γ _Northward exist a vector of signal transform coefficients {γ_r } _{r = 0,…,N − 1} such that

(2.2) $A_{\mathrm{Northward}}={\Phi }_N{\Gamma }_N={\left\{{\displaystyle \sum _{r=0}^{\mathrm{Northward}-1}{\gamma }_r{\phi }_r(k)}\right\}}_{k=0,\mathrm{i},\dots ,N-1}$

Assume now that but Chiliad < North signal samples ${\left\{a_{\widetilde{\mathrm{yard}}}\right\}}_{\tilde{k}\in \widetilde{\mathrm{Yard}}}$ are available, where $\widetilde{\mathrm{One\; thousand}}$ is a Yard-size not-empty subset of indices {0, 1,…,N − 1}. These available One thousand signal samples define a system of Yard equations:

(two.3) ${\left\{a_{\tilde{k}}={\displaystyle \sum _{r=0}^{N-\mathrm{one}}{\gamma }_r{\phi }_r(\tilde{k})}\right\}}_{\widetilde{\mathrm{chiliad}}\in \widetilde{\mathrm{Grand}}}$

for Thousand signal transform coefficients {γ_r } of certain Yard indices r.

Select a subset $\tilde{R}$ of K transform coefficients indices $\{\tilde{r}\in \tilde{R}\}$ and define a "ThousandofN"-ring-limited approximation ${\widehat{A}}_N^{B\mathrm{Fifty}}$ to the bespeak A_N as

(2.4) ${\widehat{A}}_{\mathrm{North}}^{BL}=\{{\widehat{a}}_k={\displaystyle \sum _{\tilde{r}\in \tilde{R}}{\gamma }_{\tilde{r}}{\phi }_{\tilde{r}}(k)\}}.$

Rewrite this equation in a more general form:

(2.5) ${\widehat{A}}_{\mathrm{North}}^{B\mathrm{Fifty}}=\left\{{\widehat{a}}_k={\displaystyle \sum _{r=0}^{N-1}{\tilde{\gamma }}_r{\phi }_r(k)}\right\},$

assuming that all transform coefficients with indices $r\notin \tilde{R}$ are set to zero:

(ii.half-dozen) ${\tilde{\gamma }}_r=\{\begin{array}{cc}{\gamma }_r,& r\in \tilde{R}\\ 0,& other\mathrm{westward}i\mathrm{southward}e.\end{array}$

And so the vector ${\tilde{A}}_K$ of available signal samples $\left\{a_{\tilde{k}}\right\}$ can be expressed in terms of the basis functions {φ_r (k)} of transform Φ _N as

(2.7) $\begin{array}{l}{\tilde{A}}_K=Kof{\mathrm{Northward}}_{\Phi }\cdot {\tilde{\Gamma }}_{\mathrm{Chiliad}}=\left\{a_{\widetilde{\mathrm{1000}}}={\displaystyle \sum _{\tilde{r}\in \tilde{R}}{\gamma }_{\tilde{r}}{\phi }_{\tilde{r}}(\widetilde{\mathrm{grand}})}\right\},\hfill \end{array}$

where the K × K sub-transform matrix ThouofN_Φ is equanimous of samples ${\phi }_{\tilde{r}}(\tilde{k})$ of the basis functions with indices $\{\tilde{r}\in \tilde{R}\}$ for bespeak sample indices $\tilde{k}\in \tilde{K}$ , and ${\tilde{\Gamma }}_M$ is a vector equanimous of the respective subset $\left\{{\gamma }_{\tilde{r}}\right\}$ of point nonzero transform coefficients {γ_r }. This subset of the coefficients can be found as

(ii.8) ${\tilde{\Gamma }}_K=\left\{{\tilde{\gamma }}_r\right\}=Kof{N}_{\Phi }^{-1}·{\tilde{A}}_{\mathrm{Chiliad}}$

provided the matrix $\text{K}of{N}_{\Phi }^{-\mathrm{ane}}$ inverse to the matrix KofN _Φ exists, which, in full general, is conditioned, for a specific transform, by the positions $\tilde{k}\in \tilde{K}$ of available signal samples and by the selection of the subset $\left\{\tilde{R}\right\}$ of transform basis functions.

Past virtue of the Parseval human relationship for orthonormal transforms, the band-limited bespeak ${\widehat{A}}_N^{B\mathrm{Fifty}}$ approximates the complete indicate A_N with MSE as follows:

(two.9) $\mathrm{One\; thousand}S\mathrm{Due\; east}={\Vert A_{\mathrm{North}}-{\widehat{A}}_{\mathrm{Due\; north}}^{BL}\Vert }^2={\displaystyle \sum _{\mathrm{one\; thousand}=0}^{N-1}|a_k-{\widehat{a}}_{\mathrm{thousand}}{|}^2={\displaystyle \sum _{r\notin R}|{\gamma }_r{|}^2}}.$

This fault tin can be minimized past an appropriate selection of the M basis functions of the sub-transform MofN_Φ. In order to practise so, the energy compaction ordering of ground functions of the transform Φ _N must be known. If, in addition, it is known that, for a form of signals, a sure transform features the best energy compaction in the smallest number of transform coefficients, then selecting this transform tin can secure the best minimum MSE band-limited approximation of the signal {a_k } for the given subset $\left\{{\tilde{a}}_k\right\}$ of its samples.

In this way, we get in at the post-obit discrete sampling theorem that can be formulated in these 2 statements:

Argument i

For any discrete betoken of N samples defined by its K ≤ N sparse and non necessarily regularly arranged samples, its band-limited, in terms of certain transform Φ _N, approximation defined by Eq. (2.5) tin can exist obtained with the MSE defined by Eq. (2.9) provided the positions of the samples secure the being of the matrix $Gof{\mathrm{Due\; north}}_{\Phi }^{-1}$ inverse to the sub-transform matrix KofN _Φ that corresponds to the band limitation. The approximation error tin can be farther minimized by using a transform with the best energy compaction property.

Statement ii

Any bespeak of Due north samples that is known to have only Chiliad ≤ N nonzero transform coefficients for certain transform Φ _N (Φ _N-transform "band-limited" signal) tin can be precisely recovered from exactly K of its samples provided the positions of the samples secure the being of the matrix $Kof{\mathrm{Northward}}_{\Phi }^{-\mathrm{ane}}$ inverse to the sub-transform matrix KofN _Φ that corresponds to the ring limitation.

In this formulation, the detached sampling theorem is applicative to signals of any dimensionality. It too does not require any assumption regarding the firmness of nonzero bespeak spectral coefficients in the transform domain. The signal dimensionality affects merely the formulation of the indicate band limitedness. For 2nd images and transforms such as discrete Fourier, discrete cosine, and Walsh transforms, the most natural is compact "low-laissez passer" band-limitedness by a rectangle or circle sector.

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