What Frequencies Does A K1 Photographic Filter Pass

Uniform Sampling of Signals and Automatic Proceeds Control

Tony J. Rouphael , in RF and Digital Indicate Processing for Software-Defined Radio, 2009

vii.ii.5 Nyquist Zones

Nyquist zones subdivide the spectrum into regions spaced uniformly at intervals of F_s /2. Each Nyquist zone contains a re-create of the spectrum of the desired signal or a mirror image of it. The odd Nyquist zones contain verbal replicas of the signal's spectrum—that is, if the original signal is centered at F_c (F_c =0 is the lowpass signal case)—the exact spectral replica will appear at F_c +kF_{due south} for k=0, one, 2, 3 … . The zone respective to k=0 is known as the first Nyquist zone, whereas the third and 5th Nyquist zones correspond to thousand=ane and k=2, respectively.

Similarly, mirrored replicas of the signal's spectrum occur in fifty-fifty numbered Nyquist zones, that is the spectra are centered at kF_southward – F_c for k=1, 2, iii … . The 2nd Nyquist zone corresponds to k=1 whereas the fourth and 6th Nyquist zones correspond to k=two and k=3, respectively. An example depicting odd and even Nyquist zones is given in Figure seven.seven.

Figure 7.vii. Spectral images of desired indicate depicted in various Nyquist zones

A human relationship between the indicate'due south center frequency F_c at RF or IF can be expressed every bit:

(7.42) $F=\{\begin{array}{cc}\hfill \mathit{rem}(F_c,F_{\mathrm{due\; south}})\hfill & \hfill \lfloor \frac{F_c}{F_s/2}\rfloor \mathrm{is}\mathrm{even}\mathrm{implying}\mathrm{that}\mathrm{prototype}\mathrm{is}\mathrm{exact}\mathrm{replica}\hfill \\ \hfill F_{\mathrm{due\; south}}-\mathit{rem}(F_c,F_s)\hfill & \hfill \lfloor \frac{F_c}{F_{\mathrm{due\; south}}/2}\rfloor \mathrm{is}\mathrm{odd}\mathrm{implying}\mathrm{that}\mathrm{image}\mathrm{is}\mathrm{mirrored}\mathrm{replica}\hfill \end{array}$

where rem is the residue subsequently partition, $\lfloor .\rfloor$ denotes the floor function, which rounds a number towards goose egg, and F is the center frequency of the epitome in the first Nyquist zone.

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Application of FBMC in Optical Communications

Jessica Fickers , ... François Horlin , in Orthogonal Waveforms and Filter Banks for Futurity Communication Systems, 2017

four.3.1 Nyquist Wavelength Segmentation Multiplexing (Nyquist-WDM)

North-WDM systems operate with shaping pulses having nearly rectangular frequency spectrum of bandwidth equal to the symbol charge per unit [fourteen]. The class of root-raised-cosine (RRC) functions is of particular interest because they satisfy the Nyquist benchmark of zero inter-symbol interference (ISI) whatever the coil-off factor when applied at transmitter and receiver. During the last years, RRC filtering with a close to 0 roll-off cistron has attracted much research involvement in gild to maximize SE [sixteen]. However, N-WDM suffers from hardware implementation limitations such equally the finite length of the pulse shaping filters, the timing jitter of the data sampling, and the finite resolution of the analog/digital converters (ADC, DAC). These constraints translate into ISI and ICI and therefore affect significantly the performance. Allowing for nonzero curl-off factors relaxes the constraints on the filter length and the tolerable jitter at expense of increasing ICI. Most recent works on Due north-WDM presume very small roll-off factors [17].

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New Radio Admission Physical Layer Aspects (Part 1)

Sassan Ahmadi , in 5G NR, 2019

three.two.3.five Faster Than Nyquist Bespeak Processing

Faster than Nyquist (FTN) is a non-orthogonal manual scheme which was ane of the approaches initially considered for 5G systems that was expected to improve the spectrum efficiency by increasing the data charge per unit. It was observed a few decades agone that the binary $\text{sinc}$ pulse can be transmitted faster than what the Nyquist theorem states without increasing the fleck mistake rate and despite of ISI. The idea was extended to frequency domain to reduce subcarrier spacing. The transmit bespeak of FTN tin be expressed every bit follows [iv]:

$x(t)=\sum _{k=-\infty }^{\infty }\sum _{n=0}^{N_{\mathrm{sub}}-\mathrm{i}}{s}_{\mathrm{thousand},n}\mathrm{grand}\left(t-k{\mathrm{\Delta }}_{T}T\right)e^{j\mathrm{two}\pi {\mathrm{\Delta }}_F\mathrm{due\; north}t/T}$

where $\mathrm{\Delta }T\le \mathrm{i}$ is the fourth dimension compression gene, which ways that the pulses are transmitted faster a factor of $1/\mathrm{\Delta }T$ and $\mathrm{\Delta }F\le 1$ is the frequency compression cistron, which ways the spectral efficiency is increased by a gene of $\mathrm{ane}/\mathrm{\Delta }F$ . The FTN transmitter construction is depicted in Fig. 3.26. Since the time and frequency spacing varies for different FTN systems, direct implementation cannot provide sufficient implementation flexibility. An FTN mapper based on project scheme has been designed and used in FTN signaling systems to provide flexibility. The FTN mapper is shown in Fig. 3.26. A cyclic extension is needed in the modulation block, with which the arrangement can switch easily betwixt FTN and Nyquist modes. Every bit mentioned before, FTN signaling inevitably introduces ISI in the fourth dimension domain and/or ICI in the frequency domain when its baud rate is over the Nyquist one (see Fig. 3.27). Therefore a very important issue is how to blueprint a receiver with ISI and/or ICI suppression capability to recover the original transmitted information.

Effigy 3.26. Illustration of FTN concept and transmitter and receiver architecture [44].

Figure 3.27. Comparison of PAPR performance of some prominent variants [47].

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Headend Signal Processing

Walter Ciciora , ... Michael Adams , in Mod Cablevision Television Technology (Second Edition), 2004

viii.5.6 Nyquist Slope Filter

The Nyquist slope filter shown every bit FL1 in Figure eight.10 has a shape shown past the solid line in Figure 8.12. At the low-frequency side (aural carrier at IF), the shape is fairly straightforward. The bandpass feature should pass the color carrier and its sidebands while rejecting the sound carrier.

Effigy eight.12. Nyquist slope filter response, with haystack.

The high-frequency side (well-nigh the picture carrier) is rather unusual. The purpose of this shape is to complement the shape of the vestigial sideband (VSB) filter used in the modulator, and shown in Figure viii.12 every bit a dotted line. The Nyquist shape is apartment to 0.75 MHz below the flick carrier; then it begins falling off at a prescribed rate in a higher place that frequency. The rolloff must be linear with frequency on a voltage plot, not a logarithmic plot. The response of the filter must exist 0.v at the motion picture carrier, continuing to driblet until the response is zero at 0.75 MHz above the picture carrier.

Other features of the Nyquist slope filter are emphasized in the next section, in which we draw the differences between synchronous and envelope detection. Ane of those features is non really a office of the Nyquist gradient filter response. Information technology is shown in Figure 8.12 equally a dashed line and was mentioned in conjunction with the chroma rolloff filter (FL2 of Figure 8.10). This is the reduction of response nigh the color carrier practical in the case of an envelope detector. It is called a haystack response after traditional television usage: taken with the Nyquist slope effectually the picture carrier, the filter shape looks like a haystack. The haystack response is used to reduce the furnishings of luminance-to-chrominance crosstalk.

In consumer television receivers using envelope detectors, the haystack response is built into the Nyquist slope filter (FL1 of Figure 8.10). In professional receivers, the chroma rolloff may be realized independently of the Nyquist slope filter so that it is not in the betoken path in the synchronous detector mode. Not that the haystack response would not injure the response of the synchronous detector if it were perfectly compensated at baseband. Rather, it is difficult to compensate perfectly, and for measurement purposes, it is better not to introduce a response fault that must later be compensated.

In mod practice, the Nyquist slope filter is normally realized as a SAW filter in consumer receivers though the shape is compromised to allow the SAW filter to be built with adept toll effectivity. In professional demodulators, the filter may be realized either with a SAW device or as a conventional L-C filter with delay equalization.

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Nuclear reactor kinetics

A. John Arul , ... Om Pal Singh , in Physics of Nuclear Reactors, 2021

Contour mapping

A contour is a directed smooth closed curve in the complex plane and C exist a profile that does not pass through any poles and zeros of Grand. Thousand(southward) be an analytic function in the complex airplane except at some isolated points. The function 1000(south) maps a closed contour C in the s-plane (x, y aeroplane) to another profile {G(s): south  ∈ C} in the complex plane G. From complex analysis, it is learned that such a mapping retains the angles of the southward-airplane profile. That is, a closed contour in the form of a square is mapped into a closed contour with four right angles. This form of mapping is called a conformal mapping.

Cauchy'southward argument principle: If a contour C in the southward-aeroplane encircles Z zeros and P poles of a function Thou(s) and does not pass through any poles or zeros of 1000(due south) and the direction of travel is the clockwise direction along the contour, the corresponding contour G(C) in the G-plane encircles the origin N times in the clockwise management, where Northward  = Z  − P. This is illustrated in Fig. 7.thirteen. The s-plane contour encloses 7 zeros and v poles. Therefore in the G-plane the contour encircles the origin twice in the clockwise direction.

Fig. 7.13. Illustration of Cauchy's statement principle.

By applying Cauchy's principle of argument to open loop transfer part (OLTF), the stability data about the airtight loop transfer office (CLTF) tin can be obtained. Additionally, it gives information about the relative (degree of) stability of the system in terms of the proceeds margin and phase margin.

The airtight loop transfer function of a linear fourth dimension-invariant (LTI) system is written as,

(7.146) $T\left(s\right)=\frac{G\left(s\right)}{\mathrm{ane}+G\left(s\right)H\left(s\right)}$

The system is (BIBO) stable if and merely if in that location are no poles on the Correct One-half Plane (RHP   = Re(south)   ≥   0) of T(southward). This is equivalent to the requirement that there are no zeros of 1   + G(s)H(southward) in the RHP. To utilize Cauchy's argument principle to this trouble, starting time generate the Nyquist profile (Fig. 7.14), which encompass the whole right half of the complex airplane s, with R  →   ∞. Farther, any poles or zeros on the contour are encircled with infinitesimally pocket-size semicircles as shown in the Fig. 7.14.

Fig. 7.14. Nyquist profile.

Now the requirement that in that location are no zeros of D(s)   =   1   + Yard(south)H(s) in the RHP translates to the condition that the number of encirclements of origin in the D-plane plus the number of poles of D are equal to nix. Nyquist criteria further simplifies the problem by considering only the OLTF Grand(south)H(s). Since the poles of D are the aforementioned as the poles of OLTF and the zeros of OLTF left shifted past zeros of D past −   one, the Nyquist criteria requires that for stability of T(s), in the mapping of the Nyquist contour by the OLTF, the number of encirclements of the point (−   1,0) plus the number of poles of OLTF must exist nada for stability. If the transfer function T(s), has whatever poles on the imaginary axis information technology needs to be treated separately. In summary the Nyquist stability criteria can be formally stated as follows.

Nyquist stability criteria: A feedback organisation is stable if and just if the contour in the Grand(s)H(s) plane does not encircle the betoken (−   one, 0) when at that place are no poles on the right half airplane. When poles are present on the right half plane, the number of counter clockwise encirclements must be equal to the number of poles.

Exercise vii.xv

For OLTF, $\mathrm{One\; thousand}\left(s\right)H\left(\mathrm{south}\right)=\frac{240}{\left(s-4\right)\left(s+9\right)\left(\mathrm{southward}+\mathrm{v}\right)}$ , determine whether the system is stable. For a stable systems Z  =   0, that is, the roots of the characteristic equations are not at RHS. Hence, for stability, system must satisfy N  =   − P. The Nyquist plot of above system is shown in Fig. seven.15. As per diagram, Nyquist plot encircles the point (−   1, 0) once (Due north  =   1). There is one pole in the RHP at +   4 and hence P  =   1. However, since North  ≠   − P, the system is unstable.

Fig. seven.fifteen. Nyquist diagram for a simple unstable organisation in Exercise 15.

Exercise seven.xvi

For OLTF, $G\left(s\right)H\left(s\right)=\frac{\mathrm{Yard}}{s{\left(s+1\right)}^2}$ , decide the system stability, when (i) 1000  =   i, (two) 1000  =   ii, and (3) Thousand  =   3.Since there is no pole in the RHP, for stability N  =   0,

(i)

For Grand  =   one, the Nyquist plot is shown in Fig. vii.sixteenA . From the effigy, Due north  =   0, hence the arrangement is stable.

Fig. 7.xvi. (A) Nyquist diagram near (−   one, 0j) for Exercise 16, when K  =   1. (B) Nyquist diagram almost (−   1, 0j), for Exercise xvi, when G  =   2. (C) Nyquist diagram near (−   1, 0j), for Exercise xvi, when K  =   3.

(ii)

For, K  =   2, from Fig. vii.16B, the contour is most to encircle (−   1, 0j). Therefore the organization is merely marginally stable.

(iii)

For K  =   3, from Fig. 7.16C, the contour encircles (−   i, 0j). Therefore the system is clearly unstable.

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Nyquist diagrams

W. Bolton , in Instrumentation and Control Systems, 2004

12.3 Stability

Equally indicated in Department 11.five: the critical point which separates stable from unstable systems is when the open-loop phase shift is −180° and the open-loop magnitude is 1. If a Nyquist diagram of the open-loop frequency response is plotted so for the arrangement to exist stable in that location must not exist any phasor with length greater than 1 and phase −180°. Thus the line traced past the tips of the phasors, the so-termed loci, must not enclose the −1 point. If an open-loop system is stable, then the corresponding unity feedback airtight-loop system is stable.

Nyquist stability benchmark: Airtight-loop systems whose open up-loop frequency response G(jω)H(jω), as ω goes from 0 to ∞, does not encircle the −1 betoken will be stable, those which encircle the −1 indicate are unstable and those which laissez passer through the −1 signal are marginally stable. Encircling the point may be taken every bit passing to the left of the point.

Figure 12.six illustrates the above with examples of stable, marginally stable and unstable systems. The Nyquist plots, non to scale, correspond to the open up-loop frequency response of:

Figure 12.six. Stability and the Nyquist plot

$G\left(j\omega \right)H\left(j\omega \right)=\frac{k}{\left(\mathrm{i}+j\omega \mathrm{0.two}\right)\left(1+j\omega \right)\left(1+j\omega 10\right)}$

with K = 10 for the stable plot, Grand = 137 for the marginally stable plot and Chiliad = 500 for the unstable plot.

The vertical axis of the Nyquist plot corresponds to the phase equal to 90° and so is the imaginary function of the open-loop frequency response. The horizontal axis corresponds to the phase equal to 0 ° and and then is the real part of the open-loop frequency response.

Figure 12.eight. M(southward)Thousand(s) = K/southward(s + a), stable for all values of K > 0

Figure 12.9. Thou(s)H(s) = K/(s + a)(s + b), stable for all of K > 0

Figure 12.10. G(southward)H(south) = Chiliad/s(s + a)(s + b); this is values unstable with large K simply can become stable if K is reduced, the point at which the plot crosses the axis beingness -K/(a + b) and so stability is when -K/(a + b) > −1

Figures 12.7 to 12.10 show examples of Nyquist plots for common forms of open-loop transfer functions and their conditions for stability.

Figure 12.7. One thousand(southward)H(s) = 1000/(s + a), stable for all values of Yard > 0

Example

Plot the Nyquist diagram for a system with the open-loop transfer function 1000/(s + 1)(s + two)(due south + iii) and consider the value of K needed for stability.

The open-loop frequency response is:

$\frac{k}{j\omega \left(j\omega +\mathrm{ane}\right)\left(j\omega +\mathrm{ii}\right)\left(j\omega +3\right)}$

The magnitude and phase are:

$\text{magnitude=}\frac{M}{\sqrt{\left({\omega }^{\mathrm{two}}+\mathrm{one}\right)\left({\omega }^2+\mathrm{iv}\right)\left({\omega }^2+9\right)}}$

$\text{phase}={tan}^{-1}(\frac{\omega }{1})+{tan}^{-\mathrm{ane}}(\frac{\omega }{2})+{tan}^{-1}(\frac{\omega }{3})$

When ω = 0 and so the magnitude is M/6 and the phase is 0°. When ω = ∞ then the magnitude is 0 and the phase is 270°. Nosotros tin use these, and other points to plot the polar graph.

Alternatively nosotros can consider the frequency response in terms of real and imaginary parts. Nosotros tin write the open-loop frequency office as:

$\frac{\mathrm{half-dozen}\mathrm{One\; thousand}\left(\mathrm{i}-{\omega }^2\right)}{\left({\omega }^2+\mathrm{i}\right)\left({\omega }^{\mathrm{ii}}+\mathrm{iv}\right)\left({\omega }^2+9\right)}+j\frac{\omega K\left({\omega }^2-1\right)}{\left({\omega }^{\mathrm{ii}}+\mathrm{i}\right)\left({\omega }^{\mathrm{ii}}+\mathrm{four}\right)\left({\omega }^2+9\right)}$

When ω = 0 then the imaginary part is zero and the existent part is Thou/6. When ω = ∞ and so the imaginary part is goose egg and the existent role is 0. The imaginary part volition be zero when ω = √xi. This is a existent role, and hence magnitude, of -K/60 and is the point at which the plot crosses the real axis. Thus for a stable system we must have -K/60 less than −i, i.e. K must be less than 60. Figure 12.11 shows the complete Nyquist plot (non to scale).

Figure 12.eleven. Case

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Activity and passivity of magnesium (Mg) and its alloys

E. Ghali , in Corrosion of Magnesium Alloys, 2011

EIS measurements

The Nyquist representation of the impedance shows 2 clear semicircles that tin exist associated to two time constants (Fig. 2.17), therefore to ii capacitors. This can be described by a simple electric equivalent excursion made of: one resistor, associated to the solution resistance (R _s), in series with 2 parallel resistor–capacitor (R–C) circuits, both connected in series. One circuit representing the charge transfer process taking identify on the surface of the alloy R _ct–C _dl and a second circuit associated to the film of corrosion products covering the surface of the electrode R _pic–C _film.

Fig. 2.17. EIS spectra, Nyquist representation, at half dozen   h immersion of the sacrificial alloy M1; with the corresponding equivalent excursion (ASTM solution) (CPE   =   constant phase element) (Guadarrama-Muñoz et al., 2006).

It was found that, when an Mg alloy, intended to work as a sacrificial anode, is polarized at a constant anodic potential, near the OCP (E _oc), the dissolution process tin can exist described by an electrical equivalent excursion similar to the i described above (Guadarrama-Mu–oz et al., 2006).

When an Mg blend does not fulfill the chemical composition specified for a sacrificial magnesium anode, features every bit inductive loops at lower frequencies announced in the Nyquist representation of the measured impedance. Every bit the magnesium alloy is polarized further abroad from its E _oc, in the anodic direction, the Nyquist representation of the impedance exhibits anterior loop behavior (Fig. 2.18). This fact leads to the consideration of an inductor component in the corresponding electric equivalent circuit. This inductive loop tin be associated with the adsorption and desorption phenomena occurring on the surface of the sample and leading to the procedure of formation of the corrosion product layer on the surface of the electrode (Guadarrama-Mu–oz et al., 2006).

Fig. ii.18. EIS spectra, Nyquist representation, at six   h immersion; with the corresponding equivalent excursion of the sacrificial alloy M3 (ASTM solution) (Guadarrama-Muñoz et al., 2006).

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Nyquist plot

Yazdan Bavafa-Toosi , in Introduction to Linear Control Systems, 2019

6.2.ii Nyquist stability criterion

What Nyquist did was to cull C _s , the semi-circle shown in Fig. 6.four, with infinite radius so as to embrace the whole ORHP. On the other hand, he noted that the closed-loop poles are the zeros of $1+L=0$ . The application of the principle of argument thus resulted in his stability criterion equally follows: Let P be the number of open-loop poles in ORHP. And then the closed-loop system is stable iff the Nyquist plot of L encircles of the point −1 in the CCW manner P times.

Proof

Let Z exist the number of zeros of 1+L in the ORHP. Thus, Z is the number of unstable closed-loop poles of the organisation. For stability there should hold Z=0, and hence $N=-P$ . We note that the positive and negative values of N, respectively, mean CW and CCW encirclements of the bespeak 0 by 1+L. On the other hand, encirclement of the point 0 by one+L is equivalent to encirclement of the point −1 past L. ⁵ The theorem thus is proven.

Δ.

Effigy half dozen.iv. Nyquist contour, covering the whole ORHP.

In other words, in terms of N, if $\mathrm{North}>0$ (CW encirclements of the point −ane) the airtight-loop system is certainly unstable. If $N<0$ (CCW encirclements of the point −one) the airtight-loop system is stable iff information technology has N unstable open up-loop poles. Too note that the theorem in particular means that if the open-loop arrangement is stable, the Nyquist plot of the system must not encircle the point −1.

How should we count the number of encirclements of the point −1? This question has been treated in several papers through the decades and each has contributed some new insight to the problem. The basic approach, which seems to be the most neutral and comprehensible one, is equally follows (see besides Further Readings). Consummate the plot for both positive and negative frequencies, by cartoon the negative-frequency part every bit the mirror prototype of the positive-frequency part with respect to the real axis, then simply count the number of encirclements. However, with some exercise this tin be simplified, see item 7 of Further Readings and Exercises 6.7, 6.eight.

Remark 6.i

In the original statement of the principle of argument information technology is assumed that the function L(due south) does not have any zeros/poles on the C _south purlieus. Therefore in the Nyquist criterion the organisation L(southward) does not have any (finite) zeros/poles on the j-axis. The example of such zeros/poles will exist addressed later in Section 6.2.seven.

In the next Department we present the basic structure or drawing of the Nyquist plot.

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Image Conquering

E.R. Davies , in Computer and Machine Vision (Fourth Edition), 2012

25.four The Sampling Theorem

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

Consider the sampling theorem first in respect of a one-D time-varying waveform. The theorem states that a sequence of samples (Fig. 25.9) of such a waveform contains all the original information and tin can be used to regenerate the original waveform exactly, simply merely if (a) the bandwidth W of the original waveform is restricted and (b) the rate of sampling f is at least twice the bandwidth of the original waveform—i.e., f≥2W. Assuming that samples are taken every T seconds, this means that 1/T≥2Westward.

Effigy 25.9. The process of sampling a fourth dimension-varying point: a continuous time-varying 1-D signal is sampled by narrow sampling pulses at a regular rate f _r=1/T, which must be at least twice the bandwidth of the indicate.

At first it may be somewhat surprising that the original waveform can be reconstructed exactly from a ready of detached samples. However, the two weather for achieving this are very stringent. What they are enervating in result is that the signal must non be permitted to change unpredictably (i.east., at too fast a rate), else authentic interpolation between the samples will not prove possible (the errors that ascend from this source are called "aliasing" errors).

Unfortunately, the first condition is almost unrealizable, since it is shut to impossible to devise a depression-pass filter with a perfect cut-off. Recall from Affiliate iii that a low-pass filter with a perfect cut-off will have infinite extent in the fourth dimension domain, so any try at achieving the same issue by time domain operations must be doomed to failure. However, acceptable approximations can exist achieved by allowing a "guard ring" between the desired and actual cutting-off frequencies. This ways that the sampling charge per unit must be higher than the Nyquist rate (in telecommunications, satisfactory functioning tin can generally be achieved at sampling rates effectually 20% above the Nyquist rate—see Brown and Glazier, 1974).

1 way of recovering the original waveform is past applying a low-pass filter. This arroyo is intuitively correct, since it acts in such a manner as to broaden the narrow discrete samples until they coalesce and sum to give a continuous waveform. Indeed, this method acts in such a way as to eliminate the "repeated" spectra in the transform of the original sampled waveform (Fig. 25.10). 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 thought may be taken further because the Fourier transform of a square cutting-off filter is the sinc (sin u/u) function (Fig. 25.11). Hence, the original waveform may be recovered past convolving the samples with the sinc function (which in this case ways replacing them by sinc functions of corresponding amplitudes). This has the effect of broadening out the samples as required, until the original waveform is recovered.

Effigy 25.x. Effect of low-pass filtering to eliminate repeated spectra in the frequency domain (f _r, sampling charge per unit; L, low-laissez passer filter feature). This diagram shows the repeated spectra of the frequency transform F(f) of the original sampled waveform. It also demonstrates how a low-pass filter tin can be expected to eliminate the repeated spectra to recover the original waveform.

Effigy 25.eleven. The sinc (sin u/u) office shown in (b) is the Fourier transform of a square pulse (a) corresponding to an ideal depression-pass filter. In this case, u=2πf _c t, f _c being the cut-off frequency.

So far we have considered the state of affairs merely for i-D fourth dimension-varying signals. However, recalling that at that place is an verbal mathematical correspondence between time and frequency domain signals on the ane manus and spatial and spatial frequency signals on the other, the above ideas may all be applied immediately to each dimension of an image (although the condition for authentic sampling now becomes 1/Ten≥iiW ₁₀, where Ten is the spatial sampling period and West ₁₀ is the spatial bandwidth). Hither we take this correspondence without farther discussion and proceed to utilise the sampling theorem to image conquering.

Consider next how the signal from a photographic camera may be sampled rigorously according to the sampling theorem. Starting time, annotation that this has to be achieved both horizontally and vertically. Perhaps the nearly obvious solution to this problem is to perform the process optically, mayhap by defocusing the lens; however, the optical transform role for this case is oft (i.due east., for extreme 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 use a diffraction-limited optical system or peradventure pass the focussed beam through some sort of patterned or frosted drinking glass to reduce the spatial bandwidth artificially. None of these techniques will be particularly easy to apply, nor (apart possibly from the second) will information technology give authentic solutions. However, this problem is non equally serious equally might be imagined. If the sensing region of the camera (per pixel) is reasonably large, and close to the size of a pixel, and then the averaging inherent in obtaining the pixel intensities will in fact perform the necessary narrow-banding (Fig. 25.12). To analyze the situation in more detail, note that a pixel is substantially foursquare with a sharp cut-off at its borders. Thus its spatial frequency pattern is a 2-D sinc part, which (taking the primal positive acme) approximates to a low-pass spatial frequency filter. This approximation improves somewhat as the border between pixels becomes more fuzzy.

Effigy 25.12. Low-pass filtering carried out past averaging over the pixel region: an image with local loftier-frequency banding is to be averaged over the whole pixel region past the action of the sensing device.

The point hither is that the worst case from the indicate of view of the sampling theorem is that of extremely narrow detached samples, but clearly this worst case is most unlikely to occur with most cameras. However, this does not mean that sampling is automatically platonic—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 past Pratt (2001) clarifies the situation and shows that in that location is a tradeoff betwixt aliasing and resolution fault. Overall, quality of sampling will be i of the limiting factors if the greatest precision in image measurement is aimed for: if the bandwidth of the presampling filter is besides low, resolution will exist lost; if it is too loftier, aliasing distortions will creep in; and if its spatial frequency response curve is non suitably smooth, a guard band will have to be included and performance will again suffer.

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Image Acquisition

Eastward.R. DAVIES , in Car 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 past computers. This makes it highly relevant both with visual patterns and with acoustic waveforms. Hence, information technology is described briefly in this section.

Consider the sampling theorem first in respect of a 1-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, but but if (1) the bandwidth W of the original waveform is restricted and (ii) 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 means that ane/T ≥ 2W.

Figure 27.ix. The process of sampling a time-varying point. A continuous fourth dimension-varying ane-D signal is sampled by narrow sampling pulses at a regular rate f _r = 1/T, which must be at least twice the bandwidth of the point.

At first, information technology may be somewhat surprising that the original waveform can be reconstructed exactly from a gear up of discrete samples. However, the two conditions for achieving perfect reconstruction are very stringent. What they are demanding in upshot is that the signal must not exist permitted to alter unpredictably (i.e., at too fast a rate) or else accurate interpolation between the samples will non show possible (the errors that arise from this source are chosen "aliasing" errors).

Unfortunately, the offset condition is virtually unrealizable, since information technology is well-nigh impossible to devise a low-pass filter with a perfect cutoff. Think from Chapter 3 that a depression-laissez passer filter with a perfect cutoff will have space extent in the time domain, so any attempt at achieving the same upshot by time domain operations must be doomed to failure. However, adequate approximations tin be achieved by allowing a "guard-ring" betwixt the desired and actual cutoff frequencies. This means that the sampling rate must therefore exist higher than the Nyquist charge per unit. (In telecommunications, satisfactory functioning tin can more often than not be achieved at sampling rates around 20% to a higher place the Nyquist rate—come across Brown and Glazier, 1974.)

One manner to recover the original waveform is to employ a low-pass filter. This approach is intuitively right because it acts in such a way equally to augment the narrow detached samples until they coalesce and sum to requite a continuous waveform. This method acts in such a way as to eliminate the "repeated" spectra in the transform of the original sampled waveform (Fig. 27.x). This in itself shows why the original waveform has to be narrow-banded before sampling—then that the repeated and basic spectra of the waveform do not cross over each other and become incommunicable to separate with a depression-pass filter. The idea may be taken farther because the Fourier transform of a square cutoff filter is the sinc (sin u/u) function (Fig. 27.11). Hence, the original waveform may exist 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 as required, until the original waveform is recovered.

Effigy 27.10. Effect of low-pass filtering to eliminate repeated spectra in the frequency domain f _r, sampling charge per unit; L, depression-laissez passer filter characteristic). This diagram shows the repeated spectra of the frequency transform F(f) of the original sampled waveform. It likewise demonstrates how a depression-pass filter can exist expected to eliminate the repeated spectra to recover the original waveform.

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

And then far we have considered the situation but for 1-D time-varying signals. Nonetheless, recalling that an exact mathematical correspondence exists between time and frequency domain signals on the one hand and spatial and spatial frequency signals on the other, the above ideas may all be practical immediately to each dimension of an prototype (although the condition for accurate sampling now becomes i/X ≥ 2W₁₀, where X is the spatial sampling menses and W_X is the spatial bandwidth). Here we take this correspondence without further discussion and proceed to apply the sampling theorem to image acquisition.

Consider next how the betoken from a TV photographic camera may be sampled rigorously according to the sampling theorem. First, it is plain that the analog voltage comprising the time-varying line signals must exist narrow-banded, for instance, by a conventional electronic depression-pass filter. However, how are the images to exist narrow-banded in the vertical direction? The aforementioned question clearly applies for both directions with a solid-land expanse photographic camera. Initially, the most obvious solution to this problem is to perform the procedure optically, perhaps past defocussing the lens. However, the optical transform role for this case is frequently (i.e., for extreme 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, we could utilise a diffraction-limited optical system or perhaps pass the focused axle through some sort of patterned or frosted glass to reduce the spatial bandwidth artificially. None of these techniques will be particularly easy to apply nor will authentic solutions be likely to result. Notwithstanding, this trouble is not as serious as might be imagined. If the sensing region of the camera (per pixel) is reasonably big, and shut 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 clarify the state of affairs in more particular, notation that a pixel is essentially square with a sharp cutoff at its borders. Thus, its spatial frequency pattern is a 2-D sinc function, which (taking the fundamental positive peak) approximates to a depression-pass spatial frequency filter. This approximation improves somewhat as the border between pixels becomes fuzzier.

Effigy 27.12. Low-laissez passer filtering carried out past averaging over the pixel region. An epitome with local loftier-frequency banding is to exist averaged over the whole pixel region by the action of the sensing device.

The betoken here is that the worst case from the betoken of view of the sampling theorem is that of extremely narrow discrete samples, but this worst example is unlikely to occur with most cameras. Nevertheless, this does not hateful that sampling is automatically ideal—and indeed it is not, since the spatial frequency pattern for a sharply defined pixel shape has (in principle) space extent in the spatial frequency domain. The review past Pratt (2001) clarifies the state of affairs and shows that there is a tradeoff betwixt aliasing and resolution error. Overall, information technology is underlined here that quality of sampling will be 1 of the limiting factors if one aims for the greatest precision in image measurement. If the bandwidth of the presampling filter is besides low, resolution will be lost; if it is too high, aliasing distortions will creep in; and if its spatial frequency response curve is not suitably smooth, a guard ring will have to exist included and performance will over again endure.

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What Frequencies Does A K1 Photographic Filter Pass,

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