In digital signal processing Digital signal processing is concerned with the representation of signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing. DSP includes subfields like: audio and speech signal processing, sonar and radar signal processing, sensor, quantization is the process of approximating ("mapping") a continuous range of values (or a very large set of possible discrete values) by a relatively small ("finite") set of ("values which can still take on continuous range") discrete symbols or integer values. For example, rounding Rounding is often done on purpose to obtain a value that is easier to write and handle than the original. It may be done also to indicate the accuracy of a computed number; for example, a quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is better stated as about 123,500 a real number In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an in the interval [0,100] to an integer The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6
In other words, quantization can be described as a mapping that represents a finite continuous interval I = [a,b] of the range of a continuous valued signal, with a single number c, which is also on that interval. For example, rounding to the nearest integer (rounding ½ up) replaces the interval [c − .5,c + .5) with the number c, for integer c. After that quantization we produce a finite set of values which can be encoded by binary techniques for example.
In signal processing, quantization refers to approximating the output by one of a discrete and finite set of values, while replacing the input by a discrete set is called discretization, and is done by sampling In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples (a discrete-time signal): the resulting sampled signal is called a discrete signal A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities. In other words, it is a time series that is a function over a domain of discrete integers. Each value in the sequence is called a sample (discrete time), and need not be quantized (it can have continuous values). To produce a digital signal The term digital signal is used, to refer to more than one concept. It can refer to discrete-time signals that have a discrete number of levels, for example a sampled and quantified analog signal, or to the continuous-time waveform signals in a digital system, representing a bit-stream. In the first case, a signal that is generated by means of a (discrete time and discrete values), one both samples (discrete time) and quantizes the resulting sample values (discrete values).
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In electronics, adaptive quantization is a quantization process that varies the step size based on the changes of the input signal, as a means of efficient compression. Two approaches commonly used are forward adaptive quantization and backward adaptive quantization.
In signal processing the quantization process is the necessary and natural follower of the sampling operation. It is necessary because in practice the digital computer with its general purpose CPU is used to implement DSP algorithms. And since computers can only process finite word length (finite resolution/precision) quantities, any infinite precision continuous valued signal should be quantized to fit a finite resolution, so that it can be represented (stored) in CPU registers and memory.
We shall be aware of the fact that, it is not the continuous values of the analog function that inhibits its binary encoding, rather it is the existence of infinitely many such values due to the definition of continuity,(which therefore requires infinitely many bits to represent). For example we can design a quantizer such that it represents a signal with a single bit (just two levels) such that, one level is "pi=3,14..." (say encoded with a 1) and the other level is "e=2.7183..." ( say encoded with a 0), as we can see, the quantized values of the signal take on infinite precision, irrational numbers. But there are only two levels. And we can represent the output of the quantizer with a binary symbol. Concluding from this we can see that it is not the discreteness of the quantized values that enable them to be encoded but the finiteness enabling the encoding with finite number of bits.
In theory there is no relation between quantization values and binary code words used to encode them (rather than a table that shows the corresponding mapping, just as examplified above). However due to practical reasons we may tend to use code words such that their binary mathematical values has a relation with the quantization levels that is encoded. And this last option merges the first two paragrahs in such a way that, if we wish to process the output of a quantizer within a DSP/CPU system (which is always the case) then we can not allow the representation levels of the quantizers to take on arbitrary values, but only a restricted range such that they can fit in computer registers.
A quantizer is identified with its number of levels M, the decision boundaries {di} and the corresponding representation values {ri}.
The output of a quantizer has two important properties: 1) a Distortion resulting from the approximation and 2) a Bit-Rate resulting from binary encoding of its levels. Therefore the Quantizer design problem is a Rate-Distortion optimization type.
If we are only allowed to use fixed length code for the output level encoding (the practical case) then the problem reduces into a distortion minimization one.
The design of a quantizer usually means the process to find the sets {di} and {ri} such that a measure of optimality is satisfied (such as MMSEQ (Minimum Mean Squared Quantization Error))
Given the number of levels M, the optimal quantizer which minimizes the MSQE with regards to the given signal statistics is called the Max-Lloyd quantizer, which is a non-uniform type in general.
The most common quantizer type is the uniform one. It is simple to design and implement and for most cases it suffices to get satisfactory results. Indeed by the very inherent nature of the design process, a given quantizer will only produce optimal results for the assumed signal statistics. Since it is very difficult to correctly predict that in advance, any static design will never produce actual optimal performance whenever the input statistics deviates from that of the design assumption. The only solution is to use an adaptive quantizer.
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| Data compression In computer science and information theory, data compression or source coding is the process of encoding information using fewer bits than an unencoded representation would use, through use of specific encoding schemes methods |
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| Lossless Lossless data compression is a class of data compression algorithms that allows the exact original data to be reconstructed from the compressed data. The term lossless is in contrast to lossy data compression, which only allows an approximation of the original data to be reconstructed in exchange for better compression rates |
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Theory Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Since its inception it
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Entropy In information theory, entropy is a measure of the uncertainty associated with a random variable. The term by itself in this context usually refers to the Shannon entropy, which quantifies, in the sense of an expected value, the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the · Complexity In algorithmic information theory , the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov-Chaitin complexity, stochastic complexity, algorithmic entropy, or program-size complexity) of an object such as a piece of text is a measure of the computational resources needed to specify the object. For example, consider the · Redundancy Redundancy in information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message. Informally, it is the amount of wasted "space" used to transmit certain data. Data compression is a way to reduce or eliminate unwanted redundancy, while checksums are a way of adding desired · Lossy A lossy compression method is one where compressing data and then decompressing it retrieves data that is different from the original, but is close enough to be useful in some way. Lossy compression is most commonly used to compress multimedia data , especially in applications such as streaming media and internet telephony. By contrast, lossless
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Entropy encoding In information theory an entropy encoding is a lossless data compression scheme that is independent of the specific characteristics of the medium
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Shannon–Fano · Shannon–Fano–Elias · Huffman In computer science and information theory, Huffman coding is an entropy encoding algorithm used for lossless data compression. The term refers to the use of a variable-length code table for encoding a source symbol where the variable-length code table has been derived in a particular way based on the estimated probability of occurrence for each · Adaptive Huffman · Arithmetic Arithmetic coding is a form of variable-length entropy encoding used in lossless data compression. Normally, a string of characters such as the words "hello there" is represented using a fixed number of bits per character, as in the ASCII code. When a string is converted to arithmetic encoding, frequently-used characters will be stored · Range · Golomb Golomb coding is a lossless data compression method using a family of data compression codes invented by Solomon W. Golomb in the 1960s. Alphabets following a geometric distribution will have a Golomb code as an optimal prefix code, making Golomb coding highly suitable for situations in which the occurrence of small values in the input stream is · Universal (Gamma · Exp-Golomb An exponential-Golomb code of order k is a type of universal code, parameterized by a whole number k. To encode a nonnegative integer in an order-k exp-Golomb code, one can use the following method: · Fibonacci · Levenshtein)
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Dictionary A dictionary coder, also sometimes known as a substitution coder, is a class of lossless data compression algorithms which operate by searching for matches between the text to be compressed and a set of strings contained in a data structure maintained by the encoder. When the encoder finds such a match, it substitutes a reference to the string's
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RLE Run-length encoding is a very simple form of data compression in which runs of data (that is, sequences in which the same data value occurs in many consecutive data elements) are stored as a single data value and count, rather than as the original run. This is most useful on data that contains many such runs: for example, relatively simple graphic · Byte pair encoding · DEFLATE Deflate is a lossless data compression algorithm that uses a combination of the LZ77 algorithm and Huffman coding. It was originally defined by Phil Katz for version 2 of his PKZIP archiving tool, and was later specified in RFC 1951 · Lempel–Ziv (LZ77/78 LZ77 and LZ78 are the names for the two lossless data compression algorithms published in papers by Abraham Lempel and Jacob Ziv in 1977 and 1978. They are also known as LZ1 and LZ2 respectively. These two algorithms form the basis for most of the LZ variations including LZW, LZSS, LZMA and others · LZSS · LZW · LZWL In the initialization step the dictionary is filled up with all characters from the alphabet. In each next step it is searched for the maximal string S, which is from the dictionary and matches the prefix of the still non-coded part of the input. The number of phrase S is sent to the output. A new phrase is added to the dictionary. This phrase is · LZO · LZMA · LZX · LZRW Lempel–Ziv Ross Williams, refers to variants of the LZ77 lossless data compression algorithms with an emphasis on improving compression speed through the use of hash tables and other techniques. This family was explored by Ross Williams, who published a series of algorithms beginning with LZRW1 in 1991 · LZJB LZJB is a lossless data compression algorithm invented by Jeff Bonwick to compress crash dumps and data in ZFS. It includes a number of improvements to the LZRW1 algorithm, a member of the Lempel-Ziv family of compression algorithms · LZT · ROLZ)
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CTW The context tree weighting method is a lossless compression and prediction algorithm by Willems, Shtarkov, and Tjalkens (1995) . The CTW algorithm is among the very few such algorithms that offer both theoretical guarantees and good practical performance (see, e.g., Begleiter, El-Yaniv, and Yona (2004) ). The CTW algorithm is an “ensemble method, · BWT · PPM · DMC · Delta Delta encoding is a way of storing or transmitting data in the form of differences between sequential data rather than complete files; more generally this is known as data differencing. Delta encoding is sometimes called delta compression, particularly where archival histories of changes are required
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| Audio Audio compression is a form of data compression designed to reduce the transmission bandwidth requirement of digital audio streams and the storage size of audio files. Audio compression algorithms are implemented in computer software as audio codecs. Generic data compression algorithms perform poorly with audio data, seldom reducing data size much |
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Theory Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics technology may be called an acoustical or audio engineer
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Companding In telecommunication, signal processing, and thermodynamics, companding is a method of mitigating the detrimental effects of a channel with limited dynamic range. The name is a portmanteau of compressing and expanding · Convolution In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation. It has applications that include statistics, computer vision, image and · Dynamic range Dynamic range is the ratio between the smallest and largest possible values of a changeable quantity, such as in sound and light. It is measured as a ratio, or as a base-10 or base-2 (doublings, bits or stops) logarithmic value · Latency · Sampling In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples (a discrete-time signal) · Nyquist–Shannon theorem · Sound quality Sound quality can be defined as the degree of accuracy with which a device records or emits the original sound waves. For digital recording/digital playback, this accuracy depends on the range of sound which is sampled, the rate at which it is sampled, and the various conversions that occur in any sound reproduction system. With lossy codecs such
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Audio codec In software, a codec is a computer program that compresses/decompresses digital audio data according to a given audio file format or streaming audio format. The object of a codec algorithm is to represent the high-fidelity audio signal with minimum number of bits while retaining the quality. This can effectively reduce the storage space and the parts
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LPC Linear predictive coding is a tool used mostly in audio signal processing and speech processing for representing the spectral envelope of a digital signal of speech in compressed form, using the information of a linear predictive model. It is one of the most powerful speech analysis techniques, and one of the most useful methods for encoding good (LAR Log Area Ratios can be used to represent Reflection Coefficients (another form for Linear Prediction Coefficients) for transmission over a channel. While not as efficient as Line Spectral Pairs (LSPs), Log Area Ratios are much simpler to compute. Let rk be the kth reflection coefficient of a filter, the kth LAR is: · LSP) · WLPC · CELP Code excited linear prediction is a speech coding algorithm originally proposed by M.R. Schroeder and B.S. Atal in 1985. At the time, it provided significantly better quality than existing low bit-rate algorithms, such as RELP and LPC vocoders (e.g., FS-1015). Along with its variants, such as ACELP, RCELP, LD-CELP and VSELP, it is currently the · ACELP · A-law An A-law algorithm is a standard companding algorithm, used in European digital communications systems to optimize, i.e., modify, the dynamic range of an analog signal for digitizing · μ-law The µ-law algorithm is a companding algorithm, primarily used in the digital telecommunication systems of North America and Japan. Companding algorithms reduce the dynamic range of an audio signal. In analog systems, this can increase the signal-to-noise ratio (SNR) achieved during transmission, and in the digital domain, it can reduce the · ADPCM Adaptive DPCM is a variant of DPCM (differential pulse-code modulation) that varies the size of the quantization step, to allow further reduction of the required bandwidth for a given signal-to-noise ratio · DPCM DPCM or differential pulse-code modulation is a signal encoder that uses the baseline of PCM but adds some functionalities based on the prediction of the samples of the signal. The input can be an analog signal or a digital signal · MDCT The modified discrete cosine transform is a Fourier-related transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where subsequent blocks are overlapped so that the last half of one block coincides with the first · Fourier transform In mathematics, the Fourier transform is an operation that transforms one complex-valued function of a real variable into another. In such applications as signal processing, the domain of the original function is typically time and is accordingly called the time domain. The domain of the new function is typically called the frequency domain, and · Psychoacoustic model Psychoacoustics is the study of subjective human perception of sounds. Alternatively it can be described as the study of the psychological correlates of the physical parameters of acoustics
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Bit rate In telecommunications and computing, bitrate is the number of bits that are conveyed or processed per unit of time (CBR Constant bitrate is a term used in telecommunications, relating to the quality of service. Compare with variable bitrate · ABR Average bitrate refers to the average amount of data transferred per unit of time, usually measured per second. This is commonly referred to for digital music or video. An MP3 file, for example, that has an average bit rate of 128 kbit/s transfers, on average, 128,000 bits every second. It can have higher bitrate and lower bitrate parts, and the · VBR Variable bitrate is a term used in telecommunications and computing that relates to the bitrate used in sound or video encoding. As opposed to constant bitrate (CBR), VBR files vary the amount of output data per time segment. VBR allows a higher bitrate (and therefore more storage space) to be allocated to the more complex segments of media files) · Speech compression · Sub-band coding
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Timeline of information theory, data compression, and error-correcting codes
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| See for formats and for codecs |
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Categories: Signal processing
