Kluwer Academic Publishers. doi:10.1109/18.532878 ^ Bernard Widrow, "A study of rough amplitude quantization by means of Nyquist sampling theory", IRE Trans. IT-51, No. 5, pp. 1739â€“1755, May 2005. Around the quantum limit, the distinction between analog and digital quantities vanishes.[citation needed] See also[edit] Analog-to-digital converter Beta encoder Data binning Discretization Discretization error Posterization Pulse code modulation Quantile Regression dilution

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[edit] The Lloydâ€“Max quantizer is Quantization noise power can be derived from N = ( δ v ) 2 12 W {\displaystyle \mathrm {N} ={\frac {(\delta \mathrm {v} )^{2}}{12}}\mathrm {W} \,\!} where δ v {\displaystyle \delta With Δ = 1 {\displaystyle \Delta =1} or with Δ {\displaystyle \Delta } equal to any other integer value, this quantizer has real-valued inputs and integer-valued outputs, although this property is Quantization also forms the core of essentially all lossy compression algorithms.

Gray and David L. Please try the request again. For a given supported number of possible output values, reducing the average granular distortion may involve increasing the average overload distortion, and vice versa. The calculations above, however, assume a completely filled input channel.

The input and output sets involved in quantization can be defined in a rather general way. However, it is common to assume that for many sources, the slope of a quantizer SQNR function can be approximated as 6dB/bit when operating at a sufficiently high bit rate. The members of the set of output values may have integer, rational, or real values (or even other possible values as well, in general â€“ such as vector values or complex The additive noise model for quantization error[edit] 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:10.1109/18.720541 ^ a b Allen Gersho, "Quantization", IEEE Communications Society Magazine, pp. 16â€“28, Sept. 1977. Principles of Digital Audio 2nd Edition. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. In an ideal analog-to-digital converter, where the quantization error is uniformly distributed between âˆ’1/2 LSB and +1/2 LSB, and the signal has a uniform distribution covering all quantization levels, the Signal-to-quantization-noise

But both types of approximation errors can, in theory, be made arbitrarily small by good design. This two-stage decomposition applies equally well to vector as well as scalar quantizers. Finding an optimal solution to the above problem results in a quantizer sometimes called a MMSQE (minimum mean-square quantization error) solution, and the resulting pdf-optimized (non-uniform) quantizer is referred to as 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 actuality, the quantization error (for quantizers defined as described here) is deterministically related to the signal rather than being independent of it.[8] Thus, periodic signals can create periodic quantization noise. 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. IT-6, pp. 7â€“12, March 1960. 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 }

The difference between the original signal and the reconstructed signal is the quantization error and, in this simple quantization scheme, is a deterministic function of the input signal. Entropy coding techniques can be applied to communicate the quantization indices from a source encoder that performs the classification stage to a decoder that performs the reconstruction stage. 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 + For a fixed-length code using N {\displaystyle N} bits, M = 2 N {\displaystyle M=2^{N}} , resulting in S Q N R = 20 log 10 2 N = N

Especially for compression applications, the dead-zone may be given a different width than that for the other steps. Audio Buildings Electronics Environment Government regulation Human health Images Radio Rooms Ships Sound masking Transportation Video Class of noise Additive white Gaussian noise (AWGN) Atmospheric noise Background noise Brownian noise Burst For example, for N {\displaystyle N} =8 bits, M {\displaystyle M} =256 levels and SQNR = 8*6 = 48dB; and for N {\displaystyle N} =16 bits, M {\displaystyle M} =65536 and However, for a source that does not have a uniform distribution, the minimum-distortion quantizer may not be a uniform quantizer.

At asymptotically high bit rates, cutting the step size in half increases the bit rate by approximately 1 bit per sample (because 1 bit is needed to indicate whether the value The system returned: (22) Invalid argument The remote host or network may be down. Contents 1 Basic properties of quantization 2 Basic types of quantization 2.1 Analog-to-digital converter (ADC) 2.2 Rateâ€“distortion optimization 3 Rounding example 4 Mid-riser and mid-tread uniform quantizers 5 Dead-zone quantizers 6 The use of this approximation can allow the entropy coding design problem to be separated from the design of the quantizer itself.

Jay Jones, Modern Communication Principles, McGrawâ€“Hill, ISBN 978-0-07-061003-3, 1967 (p. 196). ^ a b c Herbert Gish and John N. The use of sufficiently well-designed entropy coding techniques can result in the use of a bit rate that is close to the true information content of the indices { k } Most commonly, these discrete values are represented as fixed-point words (either proportional to the waveform values or companded) or floating-point words. Quantization error models[edit] In the typical case, the original signal is much larger than one least significant bit (LSB).

Your cache administrator is webmaster. p.60. ^ Okelloto, Tom (2001). Mid-tread quantizers have a zero-valued reconstruction level (corresponding to a tread of a stairway), while mid-riser quantizers have a zero-valued classification threshold (corresponding to a riser of a stairway).[9] The formulas It can be modelled in several different ways.

Lloyd, "Least Squares Quantization in PCM", IEEE Transactions on Information Theory, Vol. The resulting bit rate R {\displaystyle R} , in units of average bits per quantized value, for this quantizer can be derived as follows: R = ∑ k = 1 M It is common for the design of a quantizer to involve determining the proper balance between granular distortion and overload distortion. However using an FLC eliminates the compression improvement that can be obtained by use of better entropy coding.

Your cache administrator is webmaster. IT-28, No. 2, pp. 149â€“157, Mar. 1982. 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 ) = Quantization replaces each real number with an approximation from a finite set of discrete values (levels), which is necessary for storage and processing by numerical methods.

doi:10.1109/TCT.1956.1086334 ^ a b c Bernard Widrow, "Statistical analysis of amplitude quantized sampled data systems", Trans. Quantizing a sequence of numbers produces a sequence of quantization errors which is sometimes modeled as an additive random signal called quantization noise because of its stochastic behavior. It is a rounding error between the analog input voltage to the ADC and the output digitized value. Please try the request again.

When the input signal has a high amplitude and a wide frequency spectrum this is the case.[16] In this case a 16-bit ADC has a maximum signal-to-noise ratio of 98.09dB. 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 In this second setting, the amount of introduced distortion may be managed carefully by sophisticated techniques, and introducing some significant amount of distortion may be unavoidable. Generated Mon, 24 Oct 2016 22:42:02 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

Rateâ€“distortion quantizer design[edit] A scalar quantizer, which performs a quantization operation, can ordinarily be decomposed into two stages: Classification: A process that classifies the input signal range into M {\displaystyle M} The error introduced by this clipping is referred to as overload distortion. AIEE Pt. doi:10.1109/TIT.1984.1056920 ^ Toby Berger, "Optimum Quantizers and Permutation Codes", IEEE Transactions on Information Theory, Vol.

Also see noise shaping.) For complex signals in high-resolution ADCs this is an accurate model. Your cache administrator is webmaster.