Nyquist-Shannon sampling theorem

src: i.ytimg.com

In the field of digital signal processing, the sampling theorem is the fundamental bridge between continuous time signals (often called "analog signals") and discrete time signals (often called "digital signals"). This sets a sufficient condition for the sample rate that allows the discrete sequence sample to capture all information from the time-continuous signal of the limited bandwidth.

Strictly speaking, the theorem applies only to classes of mathematical functions that have a fourier transform that is zero outside the finite frequency region. Intuitively we expect that when a person reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the loyalty of the result depends on the density (or sample rate) of the original sample. The sampling theorem introduces the concept of sufficient sample level for perfect loyalty to a class of functions that is limited to the given bandwidth, so that no actual information is lost in the sampling process. It states a sufficient sample rate in terms of bandwidth for the function class. This theorem also leads to a formula for reconstructing the original time-continuous function of the sample.

Perfect reconstruction is possible when sample rate criteria are not met, as long as other constraints on the signal are known. (See § Non-baseband signal samples below, and compressed sensing.) In some cases (when sample rate criteria are not met), using additional constraints allows for approximate reconstruction. The allegiance of this reconstruction can be verified and quantified by using Bochner's theorem.

The name Nyquist-Shannon sampling theorems respects Harry Nyquist and Claude Shannon. This theorem is also found separately by E. T. Whittaker, by Vladimir Kotelnikov, and by others. It is also known by the names of Nyquist-Shannon-Kotelnikov, Whittaker-Shannon-Kotelnikov, Whittaker-Nyquist-Kotelnikov-Shannon , and the cardinal theorem of interpolation .

Video Nyquist-Shannon sampling theorem

Introduction

Sampling is the process of converting signals (eg, continuous time and/or space functions) into numerical sequences (discrete time and/or space functions). Shannon's version of the theorem states:

If the function x (t) does not contain a higher frequency than B Ã, hertz, it is actually determined by giving its ordination at a set of point spaces 1/(2 B ) a few seconds apart.

Therefore, the sample rate is considerably greater than 2 B samples/second. In other words, for a given sample rate f _s , perfect reconstruction is guaranteed to be possible for bandlimit B & lt; f _s /2 .

When bandlimit is too high (or no bandlimit), reconstruction shows imperfections known as aliasing. Modern theorem statements are sometimes cautious to express explicitly that x ( t ) should not contain sinusoidal components at the exact frequency of B , or B should be really less than ½ the sample rate. Both thresholds, 2 B and f _s /2 are each called Nyquist level and Nyquist frequency . And respectively, they are the attributes of x ( t ) and the sampling equipment. The conditions described by this inequality are called Nyquist criteria , or sometimes the condition Raabe . This theorem also applies to other domain functions, such as space, in the case of digital images. The only change, in the case of other domains, is the unit of measure applied to t , f _s , and B .

The T = 1/ f _s symbol is usually used to represent the interval between samples and is called sample period or sampling interval . And the example function x ( t ) is usually denoted by x n ] = x ( nT ) (alternatively " x _n " in the old signal processing literature), for all integer values n . The ideal mathematical way to interpolate sequences involves the use of sinc functions. Each sample in 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 [ n ]. Furthermore, the sinc functions are added to a continuous function. The mathematical equivalent method is to wrap a sinc function with a series of Dirac delta pulses, weighed by the sample values. The second method is not practical. Instead, some types of approximate sinc functions, limited in length, are used. The imperfections caused by this approach are known as interpolation errors .

A digital-to-analog practical converter does not produce scalable and delayed sinc functionality, as well as the ideal Dirac pulse. Instead they generate a piecewise-constant sequence of scale and delayed rectangular pulses (zero-sequence order), usually followed by "anti-imaging filters" to clean up high frequency fake content.

Maps Nyquist-Shannon sampling theorem

Aliasing

Ketika x ( t ) adalah fungsi dengan transformasi Fourier, X ( f ) :

${\ displaystyle X (f) \ {\ stackrel {\ mathrm {def}} {=}} \ \ int _ {- \ infty} ^ {\ infty} x ( t) \ e ^ {- i2 \ pi ft} \ {\ rm {d}} t,}  ÂÂ$

the Poisson summation formula shows that the sample, x ( nT ), from x ( t ) is sufficient to make a periodical sum X ( f ). The result is :

which is a periodic function and an equal representation as the Fourier series, whose coefficients are T o x ( nT ). This function is also known as discrete-time fourier transformation (DTFT) of the sequence T o x ( nT ), for integers n.

As illustrated, copies of X are shifted by a multiple of f _s and combined with additions. For limited function bands Ã, ( X ) f ) = 0 for all | f |> = B ), and large enough f _s , it is possible for the copy to remain different from each other. But if the Nyquist criterion is not met, adjacent copies overlap, and it is not generally possible to distinguish the unambiguous X ( f ). Any frequency component above f _s /2 can not be distinguished from a low frequency component, called alias , associated with one copy. In such cases, the usual interpolation techniques produce aliases, not the original components. When the sample rate is determined earlier by other considerations (such as industry standards), x ( t ) is typically filtered to reduce its high frequency to acceptable levels before it is sampled. The type of filter required is a lowpass filter, and in this application is called an anti-aliasing filter.

src: www.mathworks.com

Derived as special case of Poisson formulation

Ketika tidak ada tumpang tindih dari salinan (alias "gambar") dari X ( f ), k Ã, = Ã, 0 istilah X _s ( f ) dapat dipulihkan oleh produk :

${\ displaystyle X (f) = H (f) \ cdot X_ {s} (f),} Â Â$ di mana :

${\ displaystyle H (f) \ {\ stackrel {\ mathrm {def}} {=}} \ {\ begin {cases} 1 & amp; | f | & lt; B \ \ 0 & amp; | f | & gt; f_s} -B. \ End {cases}}} Â Â$

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

What's left is to lower the formula for reconstruction. H ( f ) does not need to be properly defined in the [ B , f _s - B ] because X _s ( f ) is zero in that case region. However, the worst case is when B Ã, = Ã, f _s /2, the Nyquist frequency. Sufficient functions for that and all less severe cases are :

${\ displaystyle H (f) = \ mathrm {rect} \ left {{frac {f} {f_ {s}}} \ right) = { \ begin {cases} 1 & amp; | f | & lt; {\ frac {f_ {s}} {2}} \\ 0 & amp; | f | & gt; {\ frac {f_ {s}} {2}}, \ end {cases}}}$

Transformasi terbalik dari kedua sisi menghasilkan rumus interpolasi Whittaker-Shannon :

${\ displaystyle x (t) = \ jumlah _ {n = - \ infty} ^ {\ infty} x (nT) \ cdot \ mathrm {sinc} \ kiri ({\ frac {t-nT} {T}} \ right),} Â Â$

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

• A value greater than required f _s (a value smaller than T ), is called oversampling , has no effect on the outcome of the reconstruction and has the benefit of leaving room for band transition where H ( f ) freely takes the intermediate value. Undersampling, which causes aliasing, is not generally an invertible operation.
• Theoretically, the interpolation formula can be implemented as a low pass filter, whose impulse response is sinc ( t / T ) and whose input is < math xmlns = "http://www.w3.org/1998/Math/MathML" alttext = "{\ displaystyle \ textstyle \ sum _ {n = - \ infty} ^ {\ infty} x (nT) \ cdot \ delta (t-nT),} ">                                    ?                  Â ·              =      Â              ?                                     ?                                x          (           n           T          )           ?           ?          (           t          -           n           T          )           ,                      {\ displaystyle \ textstyle \ sum _ {n = - \ infty} ^ {\ infty} x (nT) \ cdot \ delta (t-nT) }   which is a Dirac intake function modulated by signal samples. Precise digital-to-analog converter (DAC) implements approaches such as zero-delayed sequences. In this case, oversampling can reduce approximation errors.

src: i.ytimg.com

Shannon's original proof,

Shannon proofs the complete theorem at that point, but he goes on to discuss the reconstruction through the sinc function, which we now call the Whittaker-Shannon interpolation formula as discussed above. He did not derive or prove the properties of the sinc function, but this was familiar to the engineers who read his work at the time, because the relationship of the Fourier pair between the rect (rectangular function) and the sinc was already known.

Let ${\ displaystyle \ scriptstyle x_ {n}}$ to sample n ^th . Then the ${\ displaystyle \ scriptstyle x (t)}$ is represented by:

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂxÂ\; Â\; Â\; Â\; Â\; Â\; Â(Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂtÂ\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\underset{Â\; Â\; Â\; Â\; Â\; ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ\; ·\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ,Â-Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\mathrm{?}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}{\overset{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\mathrm{?}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}{?}}Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          x                 Â ·                                                     Â Â <Â>                           ?             (    Â 2      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, <Â> B              t      Â       Â ·              )                                       ?             (    Â 2      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, <Â> B              t      Â       Â ·              )                                      .               {\ displaystyle x (t) = \ sum _ {n = - \ infty} ^ {\ infty} x_ {n} {\ sin \ pi (2Bt -n) \ more than \ pi (2Bt-n)}.}  Â}

As with other evidence, the existence of the Fourier transform of the original signal is assumed, so the proof does not say whether the sampling theorem extends to stationary random process without limit.

Note

src: images.slideplayer.com

Apps to multivariable signals and images

The sampling theorem is usually formulated for a single variable function. Consequently, the theorem applies directly to time dependent signals and is usually formulated in that context. However, the sampling theorem can be extended in a direct way to the function of many random variables. Grayscale images, for example, are often depicted as two-dimensional array (or matrix) real numbers representing the relative intensity of pixels (image elements) located at the intersection of the row and column sample locations. As a result, the image requires two independent variables, or indexes, to specify each pixel uniquely - one for the row, and one for the column.

The color image usually consists of a combination of three separate grayscale images, one to represent each of the three main colors - red, green, and blue, or RGB for short. Other colorpaces that use 3-vector for colors include HSV, CIELAB, XYZ, etc. Some color spaces like cyan, magenta, yellow, and black (CMYK) can represent colors with four dimensions. All of this is treated as a vector-valued function over a two-dimensional sample domain.

Similar to one-dimensional discrete time signals, images can also experience aliasing if sampling resolution, or pixel density, is inadequate. For example, digital photos of high-frequency striped shirts (in other words, the distance between small lines), can cause aliasing of shirts when sampled by camera image sensors. Aliasing appears as a moirÃÆ'Â © pattern. The "solution" for higher sampling in the spatial domain for this case will move closer to the shirt, using a higher resolution sensor, or optically blurring the image before acquiring it with the sensor.

Another example is shown to the right in a brick pattern. The upper figure shows the effect when the sampling theorem condition is not satisfied. When the image rescales software (the same process that creates the thumbnail is displayed on the lower image) it, basically, runs the image through the first low-pass filter and then downsamples the image to produce a smaller image that does not show the moirÃÆ' Â © pattern. The top image is what happens when the image is downsamped without a low-pass filter: the result of aliasing.

The sampling theorem applies to camera systems, where scenes and lenses are sources of analog spatial signals, and image sensors are spatial sampling devices. Each of these components is characterized by a modulation transfer function (MTF), representing the exact resolution (spatial bandwidth) available in that component. The aliasing or blur effect can occur when MTF lenses and MTF sensors do not match. When the optical image sampled by the sensor device contains a higher spatial frequency than the sensor, the sampling below acts as a low-pass filter to reduce or eliminate aliasing. When the sampling area (pixel sensor size) is not large enough to provide sufficient spatial anti-aliasing, a separate anti-aliasing filter (low-pass optical filter) can be incorporated in the camera system to reduce the MTF of the optical image. Instead of needing an optical filter, the camera graphics processor unit performs digital signal processing to remove aliasing with a digital filter. Digital filters also apply sharpening to strengthen the contrast of the lens at high spatial frequencies, which otherwise fall rapidly on the diffraction limit.

The sampling theorem also applies to post-processing digital images, such as upward or downward sampling. The effect of aliasing, blurring, and sharpening can be adjusted by digital filtering implemented in the software, which must follow theoretical principles.

src: i.ytimg.com

Critical Frequency

Untuk menggambarkan perlunya f _s & gt; 2 B , pertimbangkan keluarga sinusoid yang dihasilkan oleh nilai yang berbeda? go back to the home :

${\ displaystyle x (t) = {\ frac {\ cos (2 \ pi Bt \ theta)} {\ cos (\ theta)}} \ = \ \ cos (2 \ pi Bt) - \ sin (2 \ pi Bt) \ tan (\ theta), \ quad - \ pi/2 & lt; \ theta & lt; \ pi/2.} Â Â$

Denge f _s = 2 B atau secara ekuivalen T = 1/(2 B ), diberikan oleh sampel :

${\ displaystyle x (nT) = \ cos (\ pi n) - \ underbrace {\ sin (\ pi n)} _ {0} \ tan ( \ theta) = (-1) ^ {n}} Â Â$

regardless of its value? . Such ambiguity is the reason for the strictly inequal of the sampling conditions of the theorem.

src: slideplayer.com

Sampling of non-baseband signals

As Shannon talks about:

The same result is true if the band does not start at zero frequency but at some higher value, and can be proved by linear translation (physically fit with single sideband modulation) of zero-frequency cases. In this case the base pulse is obtained from sin ( x )/ x by a single-side-band modulation.

That is, a condition without sufficient loss for a sampling signal that does not have a baseband component involving width of the non-zero frequency interval compared to the highest frequency component. See Sampling for more details and examples.

For example, to sample FM radio signals in the 100-102 MHz frequency range, it is not necessary to sample at 204Ã, MHz (twice the top frequency), but sufficient for samples at 4Ã, MHz (doubled). wide frequency interval).

Kondisi bandpass adalah X ( f ) = 0, untuk semua nonnegatif f di luar gelombang frekuensi terbuka:

${\ displaystyle \ left ({\ frac {N} {2}} f _ {\ mathrm {s}}, {\ frac {N 1} { 2}} f_ {\ mathrm {s}} \ right),} Â Â$

for some non-negative integers N . This formulation includes a normal baseband condition as the case of N = 0.

Fungsi interpolasi yang sesuai adalah respon impuls dari filter bandpass dinding bata yang ideal (dibandingkan dengan filter lowpass dinding bata yang ideal yang digunakan di atas) dengan cutoff di tepi atas dan bawah dari band yang ditentukan, yang merupakan perbedaan antara pasangan tanggapan impuls lowpass:

${\ displaystyle (N 1) \, \ operatorname {sinc} \ kiri ({\ frac {(N 1) t} {T}} \ kanan) -N \, \ operatorname {sinc} \ kiri ({\ frac {Nt} {T}} \ right).}  ÂÂ$

Other generalizations, for example for signals occupying multiple non-contiguous bands, are also possible. Even the most common form of sampling theorem has no proven inverse. That is, one can not conclude that information should be lost only because the conditions of the sampling theorem are not met; from a technical perspective, however, it is generally safe to assume that if the sampling theorem is not satisfied then the information will most likely be lost.

src: i.ytimg.com

Nonuniform sampling

Shannon's sampling theory can be generalized for non-uniform sampling cases, ie, samples are not taken simultaneously in time. Shannon's sampling theory for non-uniform sampling states that a limited band signal can be perfectly reconstructed from the sample if the average sampling rate satisfies the Nyquist condition. Therefore, although a uniformly spaced sample can yield easier reconstruction algorithms, this is not a necessary condition for perfect reconstruction.

The general theory for non-baseband and nonuniform samples was developed in 1967 by Henry Landau. He proves that the average sampling rate (uniform or other) should be twice the bandwidth occupied of the signal, assuming it is a priori which is known which part of the occupied spectrum. In the late 1990s, this work was partially expanded to include signals when the amount of occupied bandwidth is known, but the portion of the occupied spectrum is actually unknown. In 2000, a complete theory was developed (see the Beyond Nyquist section below) using compressed sensing. In particular, the theory, using the signal processing language, is described in this 2009 paper. They show, inter alia, that if the location of frequencies is unknown, it is necessary to sample at least two Nyquist criteria; in other words, you must pay at least a factor of 2 for not knowing the location of the spectrum. Note that the minimum sampling requirements do not always guarantee stability.

src: images.slideplayer.com

Sampling below the Nyquist level below the additional limit

The Nyquist-Shannon sampling theorem provides sufficient conditions for sampling and reconstruction of limited-band signals. When the reconstruction is done through the Whittaker-Shannon interpolation formula, the Nyquist criterion is also a necessary condition for avoiding aliasing, in that if samples are taken at a slower rate than twice the bandwidth limit, then some signals will not be reconstructed correctly. However, if further restrictions are imposed on the signal, then the Nyquist criterion may no longer be a necessary condition.

A non-trivial example of utilizing additional assumptions about the signal is given by a new field of compressed sensing, which allows for a full reconstruction with a sub-Nyquist sampling rate. Specifically, this applies to signals that are rare (or compressible) across multiple domains. For example, a compressed sensing transaction with a signal may have low over-all bandwidth (eg, effective bandwidth EB ), but the location of the frequency is unknown, rather than all together in one band, so the passband technique does not apply. In other words, the frequency spectrum is rare. Traditionally, the required sampling rate is 2 B . Using a compressed sensing technique, the signal can be perfectly reconstructed if the sample is slightly lower than 2 EB . The disadvantage of this approach is that reconstruction is no longer given by the formula, but by a solution for a convex optimization program that requires a well-studied but non-linear method.

Another example where optimal sub-Nyquist sampling appears under additional constraints is that the sample is quantized optimally, as in a combined sampling system and optimal lossy compression. This setting is relevant in cases where the combined effects of sampling and quantization should be considered, and may provide a lower bound for the minimal reconstruction error that can be achieved in sampling and quantization of random signals. For stationary Gaussian random signals, this lower bound is usually achieved at the sub-Nyquist sampling rate, indicating that sub-Nyquist sampling is optimal for this signal model under optimum quantization.

src: i.ytimg.com

​​â € <â €

The sampling theorem is implied by the work of Harry Nyquist in 1928, in which he indicated that up to 2 B independent pulse samples can be transmitted via the bandwidth system B ; but he did not explicitly consider the issue of sampling and reconstruction of sustainable signals. At about the same time, Karl KÃÆ'¼pfmÃÆ'¼ller showed similar results and discussed the sincon-impulse response of the band-limiting filter, through its integral, integral sin-step response; these bandlimiting and reconstruction filters of great importance to the sampling theorem are sometimes referred to as the KÃÆ'¼pfmÃÆ'¼ller (but rarely so in English) filter.

The sampling theorem is basically two Nyquist results, evidenced by Claude E. Shannon. V. A. Kotelnikov published similar results in 1933, as did the mathematicians E. T. Whittaker in 1915, J. M. Whittaker in 1935, and Gabor in 1946 ("Theory of communication"). In 1999, the Eduard Rhein Foundation gave Kotelnikov their Basic Research Award "for the first exact theorem formulation of the sampling theorem".

By 1948, 1949, Claude E. Shannon, menerbitkan, was a second-party revolutionary journalist in the journal of theoretical theory. Dalam Shannon 1948 theorem sampling dirumuskan sebagai "Theorem 13": Biarkan f ( t ) tidak mengandung frekuensi lebih dari W. Kemudian

${\ displaystyle f (t) = \ jumlah _ {n = - \ infty} ^ {\ infty} X_ {n} {\ frac {\ sin \ pi (2Wt- n)} {\ pi (2Wt-n)}},} Â Â$

Source of the article : Wikipedia