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82-443: A DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers . The DCTs are generally related to Fourier series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since

164-402: A   b   c   d   e {\displaystyle a\ b\ c\ d\ e} is exactly equivalent to a DFT of eight real numbers a   b   c   d   e   d   c   b {\displaystyle a\ b\ c\ d\ e\ d\ c\ b} (even symmetry), divided by two. (In contrast, DCT types II-IV involve

246-409: A byte or word , is referred to, it is usually specified by a number from 0 upwards corresponding to its position within the byte or word. However, 0 can refer to either the most or least significant bit depending on the context. Similar to torque and energy in physics; information-theoretic information and data storage size have the same dimensionality of units of measurement , but there

328-603: A DCT algorithm, an image (or frame in an image sequence) is divided into square blocks which are processed independently from each other, then the DCT blocks is taken within each block and the resulting DCT coefficients are quantized . This process can cause blocking artifacts, primarily at high data compression ratios . This can also cause the mosquito noise effect, commonly found in digital video . DCT blocks are often used in glitch art . The artist Rosa Menkman makes use of DCT-based compression artifacts in her glitch art, particularly

410-588: A conducting path at a certain point of a circuit. In optical discs , a bit is encoded as the presence or absence of a microscopic pit on a reflective surface. In one-dimensional bar codes , bits are encoded as the thickness of alternating black and white lines. The bit is not defined in the International System of Units (SI). However, the International Electrotechnical Commission issued standard IEC 60027 , which specifies that

492-401: A continuous extension at the boundaries (although the slope is generally discontinuous). This is why DCTs, and in particular DCTs of types I, II, V, and VI (the types that have two even boundaries) generally perform better for signal compression than DFTs and DSTs. In practice, a type-II DCT is usually preferred for such applications, in part for reasons of computational convenience. Formally,

574-406: A data point or the point halfway between two data points (2 choices per boundary), for a total of 2 × 2 × 2 × 2 = 16 possibilities. Half of these possibilities, those where the left boundary is even, correspond to the 8 types of DCT; the other half are the 8 types of DST. These different boundary conditions strongly affect the applications of the transform and lead to uniquely useful properties for

656-482: A function over a finite domain , such as the DFT or DCT or a Fourier series , can be thought of as implicitly defining an extension of that function outside the domain. That is, once you write a function f ( x ) {\displaystyle f(x)} as a sum of sinusoids, you can evaluate that sum at any x {\displaystyle x} , even for x {\displaystyle x} where

738-693: A half-sample shift in the equivalent DFT.) Note, however, that the DCT-I is not defined for N {\displaystyle N} less than 2, while all other DCT types are defined for any positive N . {\displaystyle N.} Thus, the DCT-I corresponds to the boundary conditions: x n {\displaystyle x_{n}} is even around n = 0 {\displaystyle n=0} and even around n = N − 1 {\displaystyle n=N-1} ; similarly for X k . {\displaystyle X_{k}.} The DCT-II

820-454: A length-one DFT (odd length) of a single number a  , corresponds to a DCT-V of length N = 1. {\displaystyle N=1.} ) Using the normalization conventions above, the inverse of DCT-I is DCT-I multiplied by 2/( N  − 1). The inverse of DCT-IV is DCT-IV multiplied by 2/ N . The inverse of DCT-II is DCT-III multiplied by 2/ N and vice versa. Like for the DFT ,

902-431: A number of bytes which is a low power of two. A string of four bits is usually a nibble . In information theory , one bit is the information entropy of a random binary variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known. As a unit of information , the bit is also known as a shannon , named after Claude E. Shannon . The symbol for

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984-498: A practical DCT algorithm with his PhD students T. Raj Natarajan, Wills Dietrich, and Jeremy Fries, and his friend Dr. K. R. Rao at the University of Texas at Arlington in 1973. They presented their results in a January 1974 paper, titled Discrete Cosine Transform . It described what is now called the type-II DCT (DCT-II), as well as the type-III inverse DCT (IDCT). Since its introduction in 1974, there has been significant research on

1066-458: A practical video compression algorithm, called motion-compensated DCT or adaptive scene coding, in 1981. Motion-compensated DCT later became the standard coding technique for video compression from the late 1980s onwards. A DCT variant, the modified discrete cosine transform (MDCT), was developed by John P. Princen, A.W. Johnson and Alan B. Bradley at the University of Surrey in 1987, following earlier work by Princen and Bradley in 1986. The MDCT

1148-473: A scaling can be chosen that allows the DCT to be computed with fewer multiplications. The DCT-II implies the boundary conditions: x n {\displaystyle x_{n}} is even around n = − 1 / 2 {\displaystyle n=-1/2} and even around n = N − 1 / 2 ; {\displaystyle n=N-1/2\,;} X k {\displaystyle X_{k}}

1230-531: A strong energy compaction property, capable of achieving high quality at high data compression ratios . However, blocky compression artifacts can appear when heavy DCT compression is applied. The DCT was first conceived by Nasir Ahmed , T. Natarajan and K. R. Rao while working at Kansas State University . The concept was proposed to the National Science Foundation in 1972. The DCT was originally intended for image compression . Ahmed developed

1312-468: Is a portmanteau of binary digit . The bit represents a logical state with one of two possible values . These values are most commonly represented as either " 1 " or " 0 " , but other representations such as true / false , yes / no , on / off , or + / − are also widely used. The relation between these values and the physical states of the underlying storage or device is a matter of convention, and different assignments may be used even within

1394-499: Is a modification of the original DCT algorithm, and incorporates elements of inverse DCT and delta modulation . It is a more effective lossless compression algorithm than entropy coding . Lossless DCT is also known as LDCT. The DCT is the most widely used transformation technique in signal processing , and by far the most widely used linear transform in data compression . Uncompressed digital media as well as lossless compression have high memory and bandwidth requirements, which

1476-478: Is also possible using 2 N signal followed by a multiplication by half shift. This is demonstrated by Makhoul . Some authors further multiply the X 0 {\displaystyle X_{0}} term by 1 / N {\displaystyle 1/{\sqrt {N\,}}\,} and multiply the rest of the matrix by an overall scale factor of 2 / N {\textstyle {\sqrt {{2}/{N}}}} (see below for

1558-503: Is by far the most commonly used format for the recording, compression and distribution of video content, used by 91% of video developers, followed by HEVC which is used by 43% of developers. Multidimensional DCTs (MD DCTs) have several applications, mainly 3-D DCTs such as the 3-D DCT-II, which has several new applications like Hyperspectral Imaging coding systems, variable temporal length 3-D DCT coding, video coding algorithms, adaptive video coding and 3-D Compression. Due to enhancement in

1640-503: Is correspondingly often called simply the inverse DCT or the IDCT . Two related transforms are the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions , and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT to multidimensional signals. A variety of fast algorithms have been developed to reduce

1722-614: Is even around k = 0 {\displaystyle k=0} and odd around k = N . {\displaystyle k=N.} Because it is the inverse of DCT-II up to a scale factor (see below), this form is sometimes simply referred to as "the inverse DCT" ("IDCT"). Some authors divide the x 0 {\displaystyle x_{0}} term by 2 {\displaystyle {\sqrt {2}}} instead of by 2 (resulting in an overall x 0 / 2 {\displaystyle x_{0}/{\sqrt {2}}} term) and multiply

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1804-677: Is even around n = 0 {\displaystyle n=0} and odd around n = N ; {\displaystyle n=N;} X k {\displaystyle X_{k}} is even around k = − 1 / 2 {\displaystyle k=-1/2} and even around k = N − 1 / 2. {\displaystyle k=N-1/2.} The DCT-IV matrix becomes orthogonal (and thus, being clearly symmetric, its own inverse) if one further multiplies by an overall scale factor of 2 / N . {\textstyle {\sqrt {2/N}}.} A variant of

1886-542: Is even or odd), since the corresponding DFT is of length 2 ( N − 1 ) {\displaystyle 2(N-1)} (for DCT-I) or 4 N {\displaystyle 4N} (for DCT-II & III) or 8 N {\displaystyle 8N} (for DCT-IV). The four additional types of discrete cosine transform correspond essentially to real-even DFTs of logically odd order, which have factors of N ± 1 / 2 {\displaystyle N\pm {1}/{2}} in

1968-486: Is in general no meaning to adding, subtracting or otherwise combining the units mathematically, although one may act as a bound on the other. Units of information used in information theory include the shannon (Sh), the natural unit of information (nat) and the hartley (Hart). One shannon is the maximum amount of information needed to specify the state of one bit of storage. These are related by 1 Sh ≈ 0.693 nat ≈ 0.301 Hart. Some authors also define

2050-554: Is more compressed—the same bucket can hold more. For example, it is estimated that the combined technological capacity of the world to store information provides 1,300 exabytes of hardware digits. However, when this storage space is filled and the corresponding content is optimally compressed, this only represents 295 exabytes of information. When optimally compressed, the resulting carrying capacity approaches Shannon information or information entropy . Certain bitwise computer processor instructions (such as bit set ) operate at

2132-710: Is optimal in the decorrelation sense). As explained below, this stems from the boundary conditions implicit in the cosine functions. DCTs are widely employed in solving partial differential equations by spectral methods , where the different variants of the DCT correspond to slightly different even and odd boundary conditions at the two ends of the array. DCTs are closely related to Chebyshev polynomials , and fast DCT algorithms (below) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in Clenshaw–Curtis quadrature . The DCT

2214-1097: Is probably the most commonly used form, and is often simply referred to as "the DCT". This transform is exactly equivalent (up to an overall scale factor of 2) to a DFT of 4 N {\displaystyle 4N} real inputs of even symmetry where the even-indexed elements are zero. That is, it is half of the DFT of the 4 N {\displaystyle 4N} inputs y n , {\displaystyle y_{n},} where y 2 n = 0 , {\displaystyle y_{2n}=0,} y 2 n + 1 = x n {\displaystyle y_{2n+1}=x_{n}} for 0 ≤ n < N , {\displaystyle 0\leq n<N,} y 2 N = 0 , {\displaystyle y_{2N}=0,} and y 4 N − n = y n {\displaystyle y_{4N-n}=y_{n}} for 0 < n < 2 N . {\displaystyle 0<n<2N.} DCT-II transformation

2296-540: Is significantly reduced by the DCT lossy compression technique, capable of achieving data compression ratios from 8:1 to 14:1 for near-studio-quality, up to 100:1 for acceptable-quality content. DCT compression standards are used in digital media technologies, such as digital images , digital photos , digital video , streaming media , digital television , streaming television , video on demand (VOD), digital cinema , high-definition video (HD video), and high-definition television (HDTV). The DCT, and in particular

2378-490: Is the unit byte , coined by Werner Buchholz in June 1956, which historically was used to represent the group of bits used to encode a single character of text (until UTF-8 multibyte encoding took over) in a computer and for this reason it was used as the basic addressable element in many computer architectures . The trend in hardware design converged on the most common implementation of using eight bits per byte, as it

2460-405: Is used in most modern audio compression formats, such as Dolby Digital (AC-3), MP3 (which uses a hybrid DCT- FFT algorithm), Advanced Audio Coding (AAC), and Vorbis ( Ogg ). Nasir Ahmed also developed a lossless DCT algorithm with Giridhar Mandyam and Neeraj Magotra at the University of New Mexico in 1995. This allows the DCT technique to be used for lossless compression of images. It

2542-541: Is widely implemented in digital signal processors (DSP), as well as digital signal processing software. Many companies have developed DSPs based on DCT technology. DCTs are widely used for applications such as encoding , decoding, video, audio, multiplexing , control signals, signaling , and analog-to-digital conversion . DCTs are also commonly used for high-definition television (HDTV) encoder/decoder chips . A common issue with DCT compression in digital media are blocky compression artifacts , caused by DCT blocks. In

Discrete cosine transform - Misplaced Pages Continue

2624-412: Is widely used in many applications, which include the following. The DCT-II is an important image compression technique. It is used in image compression standards such as JPEG , and video compression standards such as H.26x , MJPEG , MPEG , DV , Theora and Daala . There, the two-dimensional DCT-II of N × N {\displaystyle N\times N} blocks are computed and

2706-452: Is widely used today. However, because of the ambiguity of relying on the underlying hardware design, the unit octet was defined to explicitly denote a sequence of eight bits. Computers usually manipulate bits in groups of a fixed size, conventionally named " words ". Like the byte, the number of bits in a word also varies with the hardware design, and is typically between 8 and 80 bits, or even more in some specialized computers. In

2788-496: The frequency spectrum . (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component. Applied to functions of continuous arguments, Fourier-related transforms include: For usage on computers , number theory and algebra, discrete arguments (e.g. functions of a series of discrete samples) are often more appropriate, and are handled by

2870-410: The yottabit (Ybit). When the information capacity of a storage system or a communication channel is presented in bits or bits per second , this often refers to binary digits, which is a computer hardware capacity to store binary data ( 0 or 1 , up or down, current or not, etc.). Information capacity of a storage system is only an upper bound to the quantity of information stored therein. If

2952-407: The 1950s and 1960s, these methods were largely supplanted by magnetic storage devices such as magnetic-core memory , magnetic tapes , drums , and disks , where a bit was represented by the polarity of magnetization of a certain area of a ferromagnetic film, or by a change in polarity from one direction to the other. The same principle was later used in the magnetic bubble memory developed in

3034-418: The 1980s, and is still found in various magnetic strip items such as metro tickets and some credit cards . In modern semiconductor memory , such as dynamic random-access memory , the two values of a bit may be represented by two levels of electric charge stored in a capacitor . In certain types of programmable logic arrays and read-only memory , a bit may be represented by the presence or absence of

3116-518: The DCT and the fast Fourier transform (FFT), developing inter-frame hybrid coders for both, and found that the DCT is the most efficient due to its reduced complexity, capable of compressing image data down to 0.25- bit per pixel for a videotelephone scene with image quality comparable to an intra-frame coder requiring 2-bit per pixel. In 1979, Anil K. Jain and Jaswant R. Jain further developed motion-compensated DCT video compression, also called block motion compensation. This led to Chen developing

3198-400: The DCT blocks found in most digital media formats such as JPEG digital images and MP3 audio. Another example is Jpegs by German photographer Thomas Ruff , which uses intentional JPEG artifacts as the basis of the picture's style. Like any Fourier-related transform, DCTs express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes . Like

3280-450: The DCT with slightly modified definitions. The N real numbers   x 0 ,   …   x N − 1   {\displaystyle ~x_{0},\ \ldots \ x_{N-1}~} are transformed into the N real numbers X 0 , … , X N − 1 {\displaystyle X_{0},\,\ldots ,\,X_{N-1}} according to one of

3362-608: The DCT, is used in Advanced Video Coding (AVC), introduced in 2003, and High Efficiency Video Coding (HEVC), introduced in 2013. The integer DCT is also used in the High Efficiency Image Format (HEIF), which uses a subset of the HEVC video coding format for coding still images. AVC uses 4 x 4 and 8 x 8 blocks. HEVC and HEIF use varied block sizes between 4 x 4 and 32 x 32 pixels . As of 2019, AVC

Discrete cosine transform - Misplaced Pages Continue

3444-388: The DCT-I matrix orthogonal but breaks the direct correspondence with a real-even DFT . The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a DFT of 2 ( N − 1 ) {\displaystyle 2(N-1)} real numbers with even symmetry. For example, a DCT-I of N = 5 {\displaystyle N=5} real numbers

3526-509: The DCT-II, is often used in signal and image processing, especially for lossy compression, because it has a strong energy compaction property. In typical applications, most of the signal information tends to be concentrated in a few low-frequency components of the DCT. For strongly correlated Markov processes , the DCT can approach the compaction efficiency of the Karhunen-Loève transform (which

3608-650: The DCT-IV, where data from different transforms are overlapped , is called the modified discrete cosine transform (MDCT). The DCT-IV implies the boundary conditions: x n {\displaystyle x_{n}} is even around n = − 1 / 2 {\displaystyle n=-1/2} and odd around n = N − 1 / 2 ; {\displaystyle n=N-1/2;} similarly for X k . {\displaystyle X_{k}.} DCTs of types I–IV treat both boundaries consistently regarding

3690-565: The DCT. In 1977, Wen-Hsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm. Further developments include a 1978 paper by M. J. Narasimha and A. M. Peterson, and a 1984 paper by B. G. Lee. These research papers, along with the original 1974 Ahmed paper and the 1977 Chen paper, were cited by the Joint Photographic Experts Group as the basis for JPEG 's lossy image compression algorithm in 1992. The discrete sine transform (DST)

3772-483: The DFT, a DCT operates on a function at a finite number of discrete data points . The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines (in the form of complex exponentials ). However, this visible difference is merely a consequence of a deeper distinction: a DCT implies different boundary conditions from the DFT or other related transforms. The Fourier-related transforms that operate on

3854-413: The Fourier transform of a real and even function is real and even), whereas in some variants the input or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common. The most common variant of discrete cosine transform is the type-II DCT, which is often called simply the DCT . This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT,

3936-424: The average. This principle is the basis of data compression technology. Using an analogy, the hardware binary digits refer to the amount of storage space available (like the number of buckets available to store things), and the information content the filling, which comes in different levels of granularity (fine or coarse, that is, compressed or uncompressed information). When the granularity is finer—when information

4018-672: The binary digit is either "bit", per the IEC 80000-13 :2008 standard, or the lowercase character "b", per the IEEE 1541-2002 standard. Use of the latter may create confusion with the capital "B" which is the international standard symbol for the byte. The encoding of data by discrete bits was used in the punched cards invented by Basile Bouchon and Jean-Baptiste Falcon (1732), developed by Joseph Marie Jacquard (1804), and later adopted by Semyon Korsakov , Charles Babbage , Herman Hollerith , and early computer manufacturers like IBM . A variant of that idea

4100-399: The boundary conditions are responsible for the "energy compactification" properties that make DCTs useful for image and audio compression, because the boundaries affect the rate of convergence of any Fourier-like series. In particular, it is well known that any discontinuities in a function reduce the rate of convergence of the Fourier series, so that more sinusoids are needed to represent

4182-443: The columns (or vice versa). That is, the 2D DCT-II is given by the formula (omitting normalization and other scale factors, as above): List of Fourier-related transforms This is a list of linear transformations of functions related to Fourier analysis . Such transformations map a function to a set of coefficients of basis functions , where the basis functions are sinusoidal and are therefore strongly localized in

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4264-435: The computational complexity of implementing DCT. One of these is the integer DCT (IntDCT), an integer approximation of the standard DCT, used in several ISO/IEC and ITU-T international standards. DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks. DCT blocks sizes including 8x8 pixels for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels. The DCT has

4346-504: The corresponding change in DCT-III). This makes the DCT-II matrix orthogonal , but breaks the direct correspondence with a real-even DFT of half-shifted input. This is the normalization used by Matlab , for example, see. In many applications, such as JPEG , the scaling is arbitrary because scale factors can be combined with a subsequent computational step (e.g. the quantization step in JPEG), and

4428-432: The data are even about the sample a , in which case the even extension is dcbabcd , or the data are even about the point halfway between a and the previous point, in which case the even extension is dcbaabcd ( a is repeated). These choices lead to all the standard variations of DCTs and also discrete sine transforms (DSTs). Each boundary can be either even or odd (2 choices per boundary) and can be symmetric about

4510-462: The denominators of the cosine arguments. However, these variants seem to be rarely used in practice. One reason, perhaps, is that FFT algorithms for odd-length DFTs are generally more complicated than FFT algorithms for even-length DFTs (e.g. the simplest radix-2 algorithms are only for even lengths), and this increased intricacy carries over to the DCTs as described below. (The trivial real-even array,

4592-402: The discrete cosine transform is a linear , invertible function f : R N → R N {\displaystyle f:\mathbb {R} ^{N}\to \mathbb {R} ^{N}} (where R {\displaystyle \mathbb {R} } denotes the set of real numbers ), or equivalently an invertible N × N square matrix . There are several variants of

4674-415: The early 21st century, retail personal or server computers have a word size of 32 or 64 bits. The International System of Units defines a series of decimal prefixes for multiples of standardized units which are commonly also used with the bit and the byte. The prefixes kilo (10 ) through yotta (10 ) increment by multiples of one thousand, and the corresponding units are the kilobit (kbit) through

4756-457: The electrical state of a flip-flop circuit. For devices using positive logic , a digit value of 1 (or a logical value of true) is represented by a more positive voltage relative to the representation of 0 . Different logic families require different voltages, and variations are allowed to account for component aging and noise immunity. For example, in transistor–transistor logic (TTL) and compatible circuits, digit values 0 and 1 at

4838-775: The formulas: Some authors further multiply the x 0 {\displaystyle x_{0}} and x N − 1 {\displaystyle x_{N-1}} terms by 2 , {\displaystyle {\sqrt {2\,}}\,,} and correspondingly multiply the X 0 {\displaystyle X_{0}} and X N − 1 {\displaystyle X_{N-1}} terms by 1 / 2 , {\displaystyle 1/{\sqrt {2\,}}\,,} which, if one further multiplies by an overall scale factor of 2 N − 1 , {\displaystyle {\sqrt {{\tfrac {2}{N-1\,}}\,}},} , makes

4920-411: The function is even or odd at both the left and right boundaries of the domain (i.e. the min- n and max- n boundaries in the definitions below, respectively). Second, one has to specify around what point the function is even or odd. In particular, consider a sequence abcd of four equally spaced data points, and say that we specify an even left boundary. There are two sensible possibilities: either

5002-496: The function with a given accuracy. The same principle governs the usefulness of the DFT and other transforms for signal compression; the smoother a function is, the fewer terms in its DFT or DCT are required to represent it accurately, and the more it can be compressed. (Here, we think of the DFT or DCT as approximations for the Fourier series or cosine series of a function, respectively, in order to talk about its "smoothness".) However,

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5084-404: The hardware, software and introduction of several fast algorithms, the necessity of using MD DCTs is rapidly increasing. DCT-IV has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks, lapped orthogonal transform and cosine-modulated wavelet bases. DCT plays an important role in digital signal processing specifically data compression . The DCT

5166-486: The implicit periodicity of the DFT means that discontinuities usually occur at the boundaries: any random segment of a signal is unlikely to have the same value at both the left and right boundaries. (A similar problem arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.) In contrast, a DCT where both boundaries are even always yields

5248-409: The level of manipulating bits rather than manipulating data interpreted as an aggregate of bits. In the 1980s, when bitmapped computer displays became popular, some computers provided specialized bit block transfer instructions to set or copy the bits that corresponded to a given rectangular area on the screen. In most computers and programming languages, when a bit within a group of bits, such as

5330-412: The normalization factor in front of these transform definitions is merely a convention and differs between treatments. For example, some authors multiply the transforms by 2 / N {\textstyle {\sqrt {2/N}}} so that the inverse does not require any additional multiplicative factor. Combined with appropriate factors of √ 2 (see above), this can be used to make

5412-447: The original f ( x ) {\displaystyle f(x)} was not specified. The DFT, like the Fourier series, implies a periodic extension of the original function. A DCT, like a cosine transform , implies an even extension of the original function. However, because DCTs operate on finite , discrete sequences, two issues arise that do not apply for the continuous cosine transform. First, one has to specify whether

5494-473: The output of a device are represented by no higher than 0.4 V and no lower than 2.6 V, respectively; while TTL inputs are specified to recognize 0.8 V or below as 0 and 2.2 V or above as 1 . Bits are transmitted one at a time in serial transmission , and by a multiple number of bits in parallel transmission . A bitwise operation optionally processes bits one at a time. Data transfer rates are usually measured in decimal SI multiples of

5576-452: The point of symmetry: they are even/odd around either a data point for both boundaries or halfway between two data points for both boundaries. By contrast, DCTs of types V-VIII imply boundaries that are even/odd around a data point for one boundary and halfway between two data points for the other boundary. In other words, DCT types I–IV are equivalent to real-even DFTs of even order (regardless of whether N {\displaystyle N}

5658-570: The resulting matrix by an overall scale factor of 2 / N {\textstyle {\sqrt {2/N}}} (see above for the corresponding change in DCT-II), so that the DCT-II and DCT-III are transposes of one another. This makes the DCT-III matrix orthogonal , but breaks the direct correspondence with a real-even DFT of half-shifted output. The DCT-III implies the boundary conditions: x n {\displaystyle x_{n}}

5740-608: The results are quantized and entropy coded . In this case, N {\displaystyle N} is typically 8 and the DCT-II formula is applied to each row and column of the block. The result is an 8 × 8 transform coefficient array in which the ( 0 , 0 ) {\displaystyle (0,0)} element (top-left) is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies. The integer DCT, an integer approximation of

5822-415: The same device or program . It may be physically implemented with a two-state device. A contiguous group of binary digits is commonly called a bit string , a bit vector, or a single-dimensional (or multi-dimensional) bit array . A group of eight bits is called one  byte , but historically the size of the byte is not strictly defined. Frequently, half, full, double and quadruple words consist of

5904-415: The states of electrical relays which could be either "open" or "closed". When relays were replaced by vacuum tubes , starting in the 1940s, computer builders experimented with a variety of storage methods, such as pressure pulses traveling down a mercury delay line , charges stored on the inside surface of a cathode-ray tube , or opaque spots printed on glass discs by photolithographic techniques. In

5986-554: The symbol for binary digit should be 'bit', and this should be used in all multiples, such as 'kbit', for kilobit. However, the lower-case letter 'b' is widely used as well and was recommended by the IEEE 1541 Standard (2002) . In contrast, the upper case letter 'B' is the standard and customary symbol for byte. Multiple bits may be expressed and represented in several ways. For convenience of representing commonly reoccurring groups of bits in information technology, several units of information have traditionally been used. The most common

6068-400: The transform matrix orthogonal . Multidimensional variants of the various DCT types follow straightforwardly from the one-dimensional definitions: they are simply a separable product (equivalently, a composition) of DCTs along each dimension. For example, a two-dimensional DCT-II of an image or a matrix is simply the one-dimensional DCT-II, from above, performed along the rows and then along

6150-438: The transforms (analogous to the continuous cases above): The use of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a fast Fourier transform (FFT). The Nyquist–Shannon sampling theorem is critical for understanding the output of such discrete transforms. Bit The bit is the most basic unit of information in computing and digital communication . The name

6232-556: The two possible values of one bit of storage are not equally likely, that bit of storage contains less than one bit of information. If the value is completely predictable, then the reading of that value provides no information at all (zero entropic bits, because no resolution of uncertainty occurs and therefore no information is available). If a computer file that uses n  bits of storage contains only m  <  n  bits of information, then that information can in principle be encoded in about m  bits, at least on

6314-462: The two stable states of a flip-flop , two positions of an electrical switch , two distinct voltage or current levels allowed by a circuit , two distinct levels of light intensity , two directions of magnetization or polarization , the orientation of reversible double stranded DNA , etc. Bits can be implemented in several forms. In most modern computing devices, a bit is usually represented by an electrical voltage or current pulse, or by

6396-507: The unit bit per second (bit/s), such as kbit/s. In the earliest non-electronic information processing devices, such as Jacquard's loom or Babbage's Analytical Engine , a bit was often stored as the position of a mechanical lever or gear, or the presence or absence of a hole at a specific point of a paper card or tape . The first electrical devices for discrete logic (such as elevator and traffic light control circuits , telephone switches , and Konrad Zuse's computer) represented bits as

6478-477: The use of a logarithmic measure of information in 1928. Claude E. Shannon first used the word "bit" in his seminal 1948 paper " A Mathematical Theory of Communication ". He attributed its origin to John W. Tukey , who had written a Bell Labs memo on 9 January 1947 in which he contracted "binary information digit" to simply "bit". A bit can be stored by a digital device or other physical system that exists in either of two possible distinct states . These may be

6560-510: The various DCT types. Most directly, when using Fourier-related transforms to solve partial differential equations by spectral methods , the boundary conditions are directly specified as a part of the problem being solved. Or, for the MDCT (based on the type-IV DCT), the boundary conditions are intimately involved in the MDCT's critical property of time-domain aliasing cancellation. In a more subtle fashion,

6642-518: Was derived from the DCT, by replacing the Neumann condition at x=0 with a Dirichlet condition . The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao. A type-I DST (DST-I) was later described by Anil K. Jain in 1976, and a type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978. In 1975, John A. Roese and Guner S. Robinson adapted the DCT for inter-frame motion-compensated video coding . They experimented with

6724-465: Was the perforated paper tape . In all those systems, the medium (card or tape) conceptually carried an array of hole positions; each position could be either punched through or not, thus carrying one bit of information. The encoding of text by bits was also used in Morse code (1844) and early digital communications machines such as teletypes and stock ticker machines (1870). Ralph Hartley suggested

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