Home > Quantization Error > Quantization Error Uniform Distribution

# Quantization Error Uniform Distribution

## Contents

The analysis of a uniform quantizer applied to a uniformly distributed source can be summarized in what follows: A symmetric source X can be modelled with f ( x ) = Your cache administrator is webmaster. In terms of decibels, the noise power change is 10 ⋅ log 10 ⁡ ( 1 4 )   ≈   − 6   d B . {\displaystyle \scriptstyle 10\cdot IT-42, No. 5, pp. 1365–1374, Sept. 1996. http://caribtechsxm.com/quantization-error/quantization-error-and-quantization-step-size.php

However, in some quantizer designs, the concepts of granular error and overload error may not apply (e.g., for a quantizer with a limited range of input data or with a countably In order to make the quantization error independent of the input signal, noise with an amplitude of 2 least significant bits is added to the signal. Please try the request again. Bennett, "Spectra of Quantized Signals", Bell System Technical Journal, Vol. 27, pp. 446–472, July 1948. ^ a b B. check over here

## Quantization Noise Power Formula

Note that mid-riser uniform quantizers do not have a zero output value – their minimum output magnitude is half the step size. In some designs, rather than optimizing for a particular number of classification regions M {\displaystyle M} , the quantizer design problem may include optimization of the value of M {\displaystyle M} doi:10.1109/TIT.1972.1054906 ^ Toby Berger, "Minimum Entropy Quantizers and Permutation Codes", IEEE Transactions on Information Theory, Vol.

When the input signal is a full-amplitude sine wave the distribution of the signal is no longer uniform, and the corresponding equation is instead S Q N R ≈ 1.761 + Assuming that an information source S {\displaystyle S} produces random variables X {\displaystyle X} with an associated probability density function f ( x ) {\displaystyle f(x)} , the probability p k In an ideal analog-to-digital converter, where the quantization error is uniformly distributed between −1/2 LSB and +1/2 LSB, and the signal has a uniform distribution covering all quantization levels, the Signal-to-quantization-noise Quantization Level Iterative optimization approaches can be used to find solutions in other cases.[8][19][20] Note that the reconstruction values { y k } k = 1 M {\displaystyle \{y_{k}\}_{k=1}^{M}} affect only the distortion

JPEG2000: Image Compression Fundamentals, Standards and Practice. Quantization Error Example The system returned: (22) Invalid argument The remote host or network may be down. ASSP-37, No. 1, Jan. 1989. Analog-to-digital converter (ADC) Outside the realm of signal processing, this category may simply be called rounding or scalar quantization.

Granular distortion and overload distortion Often the design of a quantizer involves supporting only a limited range of possible output values and performing clipping to limit the output to this range Quantization Example Neuhoff, "The Validity of the Additive Noise Model for Uniform Scalar Quantizers", IEEE Transactions on Information Theory, Vol. A quantizer designed for this purpose may be quite different and more elaborate in design than an ordinary rounding operation. The system returned: (22) Invalid argument The remote host or network may be down.

## Quantization Error Example

The reduced problem can be stated as follows: given a source X {\displaystyle X} with pdf f ( x ) {\displaystyle f(x)} and the constraint that the quantizer must use only Quantization (signal processing) From Wikipedia, the free encyclopedia Jump to: navigation, search The simplest way to quantize a signal is to choose the digital amplitude value closest to the original analog Quantization Noise Power Formula For the example uniform quantizer described above, the forward quantization stage can be expressed as k = ⌊ x Δ + 1 2 ⌋ {\displaystyle k=\left\lfloor {\frac {x}{\Delta }}+{\frac {1}{2}}\right\rfloor } Quantization Error In Pcm Examples of fields where this limitation applies include electronics (due to electrons), optics (due to photons), biology (due to DNA), physics (due to Planck limits) and chemistry (due to molecules).

If this is not the case - if the input signal is small - the relative quantization distortion can be very large. navigate to this website However, the same concepts actually apply in both use cases. If it is assumed that distortion is measured by mean squared error, the distortion D, is given by: D = E [ ( x − Q ( x ) ) 2 The dead zone can sometimes serve the same purpose as a noise gate or squelch function. How To Reduce Quantization Error

• John Wiley & Sons.
• Ind., Vol. 79, pp. 555–568, Jan. 1961. ^ Daniel Marco and David L.
• noise) Shot noise White noise Coherent noise Value noise Gradient noise Worley noise Engineering terms Channel noise level Circuit noise level Effective input noise temperature Equivalent noise resistance Equivalent pulse code
• In such cases, using a mid-tread uniform quantizer may be appropriate while using a mid-riser one would not be.
• For some probabilistic source models, the best performance may be achieved when M {\displaystyle M} approaches infinity.
• IT-44, No. 6, pp. 2325–2383, Oct. 1998.
• This distortion is created after the anti-aliasing filter, and if these distortions are above 1/2 the sample rate they will alias back into the band of interest.

R. Solutions that do not require multi-dimensional iterative optimization techniques have been published for only three probability distribution functions: the uniform,[18] exponential,[12] and Laplacian[12] distributions. Please try the request again. More about the author Although r k {\displaystyle r_{k}} may depend on k {\displaystyle k} in general, and can be chosen to fulfill the optimality condition described below, it is often simply set to a

## Moreover, the technique can be further generalized in a straightforward way to also include an entropy constraint for vector data.[23] Uniform quantization and the 6 dB/bit approximation The Lloyd–Max quantizer is

For the mean-square error distortion criterion, it can be easily shown that the optimal set of reconstruction values { y k ∗ } k = 1 M {\displaystyle \{y_{k}^{*}\}_{k=1}^{M}} is given A typical (mid-tread) uniform quantizer with a quantization step size equal to some value Δ {\displaystyle \Delta } can be expressed as Q ( x ) = Δ ⋅ ⌊ x This slightly reduces signal to noise ratio, but, ideally, completely eliminates the distortion. Quantization Error In Analog To Digital Conversion Please try the request again.

II: Appl. doi:10.1109/TIT.1984.1056920 ^ Toby Berger, "Optimum Quantizers and Permutation Codes", IEEE Transactions on Information Theory, Vol. In either case, the standard deviation, as a percentage of the full signal range, changes by a factor of 2 for each 1-bit change in the number of quantizer bits. click site It is a rounding error between the analog input voltage to the ADC and the output digitized value.

This is a different manifestation of "quantization error," in which theoretical models may be analog but physically occurs digitally. IT-14, No. 5, pp. 676–683, Sept. 1968. doi:10.1109/18.720541 ^ a b Allen Gersho, "Quantization", IEEE Communications Society Magazine, pp. 16–28, Sept. 1977. p.60. ^ Okelloto, Tom (2001).

Generated Mon, 24 Oct 2016 22:41:22 GMT by s_nt6 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Your cache administrator is webmaster. For example, vector quantization is the application of quantization to multi-dimensional (vector-valued) input data.[1] Basic types of quantization 2-bit resolution with four levels of quantization compared to analog.[2] 3-bit resolution with

Your cache administrator is webmaster. For other source pdfs and other quantizer designs, the SQNR may be somewhat different from that predicted by 6dB/bit, depending on the type of pdf, the type of source, the type When the quantization step size is small (relative to the variation in the signal being measured), it is relatively simple to show[3][4][5][6][7][8] that the mean squared error produced by such a The terminology is based on what happens in the region around the value 0, and uses the analogy of viewing the input-output function of the quantizer as a stairway.

The 1.761 difference in signal-to-noise only occurs due to the signal being a full-scale sine wave instead of a triangle/sawtooth. The additive noise model for quantization error A common assumption for the analysis of quantization error is that it affects a signal processing system in a similar manner to that of doi:10.1109/TIT.1968.1054193 ^ a b c d e f g h Robert M. The potential signal-to-quantization-noise power ratio therefore changes by 4, or 10 ⋅ log 10 ⁡ ( 4 )   =   6.02 {\displaystyle \scriptstyle 10\cdot \log _{10}(4)\ =\ 6.02}

Rounding and truncation are typical examples of quantization processes. The analysis of quantization involves studying the amount of data (typically measured in digits or bits or bit rate) that is used to represent the output of the quantizer, and studying Ordinarily, 0 ≤ r k ≤ 1 2 {\displaystyle 0\leq r_{k}\leq {\tfrac {1}{2}}} when quantizing input data with a typical pdf that is symmetric around zero and reaches its peak value Your cache administrator is webmaster.

A technique for controlling the amplitude of the signal (or, equivalently, the quantization step size Δ {\displaystyle \Delta } ) to achieve the appropriate balance is the use of automatic gain The system returned: (22) Invalid argument The remote host or network may be down. Generated Mon, 24 Oct 2016 22:41:22 GMT by s_nt6 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.6/ Connection All the inputs x {\displaystyle x} that fall in a given interval range I k {\displaystyle I_{k}} are associated with the same quantization index k {\displaystyle k} .