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Encyclopedia > Quantization (signal processing)

In digital signal processing, quantization is the process of approximating a continuous range of values (or a very large set of possible discrete values) by a relatively-small set of discrete symbols or integer values. More specifically, a signal can be multi-dimensional and quantization need not be applied to all dimensions. Discrete signals (a common mathematical model) need not be quantized, which can be a point of confusion. See ideal sampler. Image File history File links Quantized. ... Image File history File links Quantized. ... Image File history File links Digital. ... Image File history File links Digital. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... A discrete signal is a signal that has been sampled from a continuous signal. ... In digital signal processing, an ideal sampler is a sampler that samples in an ideal fashion. ...

A common use of quantization is in the conversion of a discrete signal (a sampled continuous signal) into a digital signal by quantizing. Both of these steps (sampling and quantizing) are performed in analog-to-digital converters with the quantization level specified in bits. A specific example would be compact disc (CD) audio which is sampled at 44,100 Hz and quantized with 16 bits (2 bytes) which can be one of 65,536 (i.e. 216) possible values per sample. A discrete signal is a signal that has been sampled from a continuous signal. ... A sample refers to a value or set of values at a point in time and/or space. ... A continuous signal or a continuous time signal is a varying quantity (a signal) that can be, or is expressed, as a continuous function of an independent variable, usually time. ... The term digital signal is used to refer to more than one concept. ... An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic circuit that converts continuous signals to discrete digital numbers. ... BIT is an acronym for: Bangalore Institute of Technology Bilateral Investment Treaty Bhilai Institute of Technology - Durg Birla Institute of Technology - Mesra Battles in Time (Doctor Who magazine) Category: ... The Compact Disc logo was inspired by that of the previous Compact Cassette. ... The hertz (symbol: Hz) is the SI unit of frequency. ... In computer architecture, 16-bit is an adjective used to describe integers, memory addresses or other data units that are at most 16 bits (2 octets) wide, or to describe CPU and ALU architectures based on registers, address buses, or data buses of that size. ... A byte is commonly used as a unit of storage measurement in computers, regardless of the type of data being stored. ...

## Contents

The simplest and best-known form of quantization is referred to as scalar quantization, since it operates on scalar (as opposed to multi-dimensional vector) input data. In general, a scalar quantization operator can be represented as In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... $Q(x) = g(lfloor f(x) rfloor)$

where

• x is a real number,
• $lfloor x rfloor$ is the floor function, yielding the integer $i = lfloor f(x) rfloor$
• f(x) and g(i) are arbitrary real-valued functions.

The integer value i is the representation that is typically stored or transmitted, and then the final interpretation is constructed using g(i) when the data is later interpreted. The integer value i is sometimes referred to as the quantization index. The floor and fractional part functions In mathematics, the floor function of a real number x, denoted or floor(x), is the largest integer less than or equal to x (formally, ). For example, floor(2. ...

In computer audio and most other applications, a method known as uniform quantization is the most common. There are two common variations of uniform quantization, called mid-rise and mid-tread uniform quantizers.

If x is a real-valued number between -1 and 1, a mid-rise uniform quantization operator that uses M bits of precision to represent each quantization index can be expressed as $Q(x) = frac{leftlfloor 2^{M-1}x rightrfloor+0.5}{2^{M-1}}$.

In this case the f(x) and g(i) operators are just multiplying scale factors (one multiplier being the inverse of the other) along with an offset in g(i) function to place the representation value in the middle of the input region for each quantization index. The value 2 − (M − 1) is often referred to as the quantization step size. Using this quantization law and assuming that quantization noise is approximately uniformly distributed over the quantization step size (an assumption typically accurate for rapidly varying x or high M) and further assuming that the input signal x to be quantized is approximately uniformly distributed over the entire interval from -1 to 1, the signal to noise ratio (SNR) of the quantization can be computed as Quantization noise is a noise error introduced by the analogue to digital conversion (ADC) process in telecommunication systems and signal processing. ... In mathematics, the continuous uniform distributions are probability distributions such that all intervals of the same length are equally probable. ... The phrase signal-to-noise ratio, often abbreviated SNR or S/N, is an engineering term for the ratio between the magnitude of a signal (meaningful information) and the magnitude of background noise. ... $frac{S}{N_q} approx 20 log_{10}(2^M) = 6.0206 M operatorname{dB}$.

From this equation, it is often said that the SNR is approximately 6 dB per bit. The decibel (dB) is a measure of the ratio between two quantities, and is used in a wide variety of measurements in acoustics, physics and electronics. ... BIT is an acronym for: Bangalore Institute of Technology Bilateral Investment Treaty Bhilai Institute of Technology - Durg Birla Institute of Technology - Mesra Battles in Time (Doctor Who magazine) Category: ...

For mid-tread uniform quantization, the offset of 0.5 would be added within the floor function instead of outside of it.

Sometimes, mid-rise quantization is used without adding the offset of 0.5. This reduces the signal to noise ratio by approximately 6.02 dB, but may be acceptable for the sake of simplicity when the step size is small.

In digital telephony, two popular quantization schemes are the 'A-law' (dominant in Europe) and 'μ-law' (dominant in North America and Japan). These schemes map discrete analog values to an 8-bit scale that is nearly linear for small values and then increases logarithmically as amplitude grows. Because the human ear's perception of loudness is roughly logarithmic, this provides a higher signal to noise ratio over the range of audible sound intensities for a given number of bits. An old rotary telephone This article is about telephone technology. ... An a-law algorithm is a standard companding algorithm, used in European digital communications systems to optimize, modify, the dynamic range of an analog signal for digitizing. ... World map showing Europe Political map Europe is one of the seven traditional continents of Earth; the term continent here referring to a cultural and political distinction, rather than a physiographic one, thus leading to various perspectives about Europes precise borders. ... In telecommunication, a mu-law algorithm (Î¼-law) is a standard analog signal compression or companding algorithm, used in digital communications systems of the North American and Japanese digital hierarchies, to optimize (in other words, modify) the dynamic range of an audio analog signal prior to digitizing. ... World map showing North America A satellite composite image of North America. ... Loudness is the quality of a sound which is high in volume (amplitude, or sound pressure). ...

## Quantization and data compression

Quantization plays a major part in lossy data compression. In many cases, quantization can be viewed as the fundamental element that distinguishes lossy data compression from lossless data compression, and the use of quantization is nearly always motivated by the need to reduce the amount of data needed to represent a signal. In some compression schemes, like MP3 or Vorbis, compression is also achieved by selectively discarding some data, an action that can be analyzed as a quantization process (e.g., a vector quantization process) or can be considered a different kind of lossy process. A lossy data compression method is one where compressing data and then decompressing it retrieves data that may well be different from the original, but is close enough to be useful in some way. ... A lossy data compression method is one where compressing data and then decompressing it retrieves data that may well be different from the original, but is close enough to be useful in some way. ... Lossless data compression is a class of data compression algorithms that allows the exact original data to be reconstructed from the compressed data. ... This article or section does not cite its references or sources. ... Vorbis is an open and free lossy audio compression codec project headed by the Xiph. ...

One example of a lossy compression scheme that uses quantization is JPEG image compression. During JPEG encoding, the data representing an image (typically 8-bits for each of three color components per pixel) is processed using a discrete cosine transform and is then quantized and entropy coded. By reducing the precision of the transformed values using quantization, the number of bits needed to represent the image can be reduced substantially. For example, images can often be represented with acceptable quality using JPEG at less than 3 bits per pixel (as opposed to the typical 24 bits per pixel needed prior to JPEG compression). Even the original representation using 24 bits per pixel requires quantization for its PCM sampling structure. In computing, JPEG (pronounced JAY-peg) is a commonly used standard method of lossy compression for photographic images. ... 2-D DCT compared to the DFT The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. ... An entropy encoding is a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols. ... Pulse-code modulation (PCM) is a digital representation of an analog signal where the magnitude of the signal is sampled regularly at uniform intervals, then quantized to a series of symbols in a digital (usually binary) code. ...

In modern compression technology, the entropy of the output of a quantizer matters more than the number of possible values of its output (the number of values being 2M in the above example). Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function. ...

In order to determine how many bits are necessary to effect a given precision, logarithms are used. Suppose, for example, that it is necessary to record six significant digits, that is to say, millionths. The number of values that can be expressed by N bits is equal to two to the Nth power. To express six decimal digits, the required number of bits is determined by rounding (6 / log 2)—where log refers to the base ten, or common, logarithm—up to the nearest integer. Since the logarithm of 2, base ten, is approximately 0.30102, the required number of bits is then given by (6 / 0.30102), or 19.932, rounded up to the nearest integer, viz., 20 bits.

This type of quantization—where a set of binary digits, e.g., an arithmetic register in a CPU, are used to represent a quantity—is called Vernier quantization. It is also possible, although rather less efficient, to rely upon equally spaced quantization levels. This is only practical when a small range of values is expected to be captured: for example, a set of eight possible values requires eight equally spaced quantization levels—which is not unreasonable, although obviously less efficient than a mere trio of binary digits (bits)—but a set of, say, sixty-four possible values, requiring sixty-four equally spaced quantization levels, can be expressed using only six bits, which is obviously far more efficient.

## Relation to quantization in nature

At the most fundamental level, all physical quantities are quantized. This is a result of quantum mechanics (see Quantization (physics)). Signals may be treated as continuous for mathematical simplicity by considering the small quantizations as negligible. A physical quantity is either a quantity within physics that can be measured (e. ... Fig. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ...

In any practical application, this inherent quantization is irrelevant. First of all, it is overshadowed by signal noise, the intrusion of extraneous phenomena present in the system upon the signal of interest. The second, which appears only in measurement applications, is the inaccuracy of instruments. In science, and especially in physics and telecommunication, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. ...

An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic circuit that converts continuous signals to discrete digital numbers. ... In electronics, a digital-to-analog converter (DAC or D-to-A) is a device for converting a digital (usually binary) code to an analog signal (current, voltage or charges). ... It has been suggested that this article or section be merged with quantization noise. ... Quantization noise is a noise error introduced by the analogue to digital conversion (ADC) process in telecommunication systems and signal processing. ... Look up discrete in Wiktionary, the free dictionary. ... A digital system is one that uses discrete numbers, especially binary numbers, or non-numeric symbols such as letters or icons, for input, processing, transmission, storage, or display, rather than a continuous spectrum of values (an analog system). ... Dither is a form of noise, or erroneous signal or data which is added to sample data for the purpose of minimizing quantization error. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Rate distortion theory is the branch of information theory addressing the problem of determining the minimal amount of entropy (or information) R that should be communicated over a channel such that the source (input signal) can be reconstructed at the receiver (output signal) with given distortion D. As such, rate... In data compression, vector quantization is a quantization technique often used in lossy data compression in which the basic idea is to code or replace with a key, values from a multidimensional vector space into values from a discrete subspace of lower dimension. ... Results from FactBites:

 Quantization (signal processing) (1020 words) In digital signal processing, quantization is the process of approximating a continuous range of values (or a very large set of possible discrete values) by a relatively-small set of discrete symbols or integer values. A common use of quantization is in the conversion of a discrete signal (a sampled continuous signal) into a digital signal by quantizing. In many cases, quantization can be viewed as the fundamental element that distinguishes lossy data compression from lossless data compression, and the use of quantization is nearly always motivated by the need to reduce the amount of data needed to represent a signal.
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