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A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images (such as JPEG and HEIF, where small high-frequency components can be discarded), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC), digital television (such as SDTV, HDTV and VOD), digital radio (such as AAC+ and DAB+), and speech coding (such as AAC-LD, Siren and Opus). DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.

The use of cosine rather than sine functions is critical for compression, since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions. In particular, 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 the Fourier transform of a real and even function is real and even), whereas in some variants the input and/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, 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 MD signals. There are several algorithms to compute MD DCT. A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT. One of these is the integer DCT[1] (IntDCT), an integer approximation of the standard DCT,[2] used in several ISO/IEC and ITU-T international standards.[1][2]

DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks.[3] DCT blocks can have a number of sizes, including 8x8 pixels for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels.[1][4] The DCT has a strong "energy compaction" property,[5][6] capable of achieving high quality at high data compression ratios.[7][8] However, blocky compression artifacts can appear when heavy DCT compression is applied.

## History

Nasir Ahmed, the inventor of the discrete cosine transform (DCT), which he first proposed in 1972.

The discrete cosine transform (DCT) was first conceived by Nasir Ahmed, while working at Kansas State University, and he proposed the concept to the National Science Foundation in 1972. He originally intended DCT for image compression.[9][1] Ahmed developed a practical DCT algorithm with his PhD student T. Raj Natarajan and friend K. R. Rao at the University of Texas at Arlington in 1973, and they found that it was the most efficient algorithm for image compression.[9] They presented their results in a January 1974 paper, titled Discrete Cosine Transform.[5][6][10] It described what is now called the type-II DCT (DCT-II),[2]: 51 as well as the type-III inverse DCT (IDCT).[5] It was a benchmark publication,[11][12] and has been cited as a fundamental development in thousands of works since its publication.[13] The basic research work and events that led to the development of the DCT were summarized in a later publication by Ahmed, "How I Came Up with the Discrete Cosine Transform".[9]

Since its introduction in 1974, there has been significant research on the DCT.[10] In 1977, Wen-Hsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm.[14][10]Further developments include a 1978 paper by M.J. Narasimha and A.M. Peterson, and a 1984 paper by B.G. Lee.[10] 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.[10][15]

In 1975, John A. Roese and Guner S. Robinson adapted the DCT for inter-frame motion-compensated video coding. They experimented with 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.[16][17] In 1979, Anil K. Jain and Jaswant R. Jain further developed motion-compensated DCT video compression,[18][19] also called block motion compensation.[19] This led to Chen developing a practical video compression algorithm, called motion-compensated DCT or adaptive scene coding, in 1981.[19] Motion-compensated DCT later became the standard coding technique for video compression from the late 1980s onwards.[20][21]

The integer DCT is used in Advanced Video Coding (AVC),[22][1] introduced in 2003, and High Efficiency Video Coding (HEVC),[4][1] 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.[4]

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,[23] following earlier work by Princen and Bradley in 1986.[24] The MDCT is used in most modern audio compression formats, such as Dolby Digital (AC-3),[25][26] MP3 (which uses a hybrid DCT-FFT algorithm),[27] Advanced Audio Coding (AAC),[28] and Vorbis (Ogg).[29]

The discrete sine transform (DST) was derived from the DCT, by replacing the Neumann condition at x=0 with a Dirichlet condition.[2] The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao.[5] 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.[30]

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 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.[31] Lossless DCT is also known as LDCT.[32]

## Applications

The DCT is the most widely used transformation technique in signal processing,[33] and by far the most widely used linear transform in data compression.[34] Uncompressed digital media as well as lossless compression had impractically high memory and bandwidth requirements, which was significantly reduced by the highly efficient DCT lossy compression technique,[7][8] capable of achieving data compression ratios from 8:1 to 14:1 for near-studio-quality,[7] up to 100:1 for acceptable-quality content.[8] DCT compression standards are used in digital media technologies, such as digital images, digital photos,[35][36] digital video,[20][37] streaming media,[38] digital television, streaming television, video-on-demand (VOD),[8] digital cinema,[25] high-definition video (HD video), and high-definition television (HDTV).[7][39]

The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy compression, because it has a strong "energy compaction" property:[5][6] 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 is optimal in the decorrelation sense). As explained below, this stems from the boundary conditions implicit in the cosine functions.

DCTs are also widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array.

DCTs are also 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 is the coding standard for multimedia telecommunication devices. It is widely used for bit rate reduction, and reducing network bandwidth usage.[1] DCT compression significantly reduces the amount of memory and bandwidth required for digital signals.[8]

### General applications

The DCT is widely used in many applications, which include the following.

### DCT visual media standards

 Further information: JPEG § Discrete cosine transform

The DCT-II, also known as simply the DCT, is the most important image compression technique.[citation needed] 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 ${\displaystyle N\times N}$ blocks are computed and the results are quantized and entropy coded. In this case, ${\displaystyle N}$ 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 ${\displaystyle (0,0)}$ 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.

Advanced Video Coding (AVC) uses the integer DCT[22][1] (IntDCT), an integer approximation of the DCT.[2][1] It uses 4x4 and 8x8 integer DCT blocks. High Efficiency Video Coding (HEVC) and the High Efficiency Image Format (HEIF) use varied integer DCT block sizes between 4x4 and 32x32 pixels.[4][1] As of 2019, AVC 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.[47]

#### Image formats

Image compression standard Year Common applications
JPEG[1] 1992 The most widely used image compression standard[56][57] and digital image format,[50]
JPEG XR 2009 Open XML Paper Specification
WebP 2010 A graphic format that supports the lossy compression of digital images. Developed by Google.
High Efficiency Image Format (HEIF) 2013 Image file format based on HEVC compression. It improves compression over JPEG,[58] and supports animation with much more efficient compression than the animated GIF format.[59]
BPG 2014 Based on HEVC compression
JPEG XL[60] 2020 A royalty-free raster-graphics file format that supports both lossy and lossless compression.

#### Video formats

Video coding standard Year Common applications
H.261[61][62] 1988 First of a family of video coding standards. Used primarily in older video conferencing and video telephone products.
Motion JPEG (MJPEG)[63] 1992 QuickTime, video editing, non-linear editing, digital cameras
MPEG-1 Video[64] 1993 Digital video distribution on CD or Internet video
MPEG-2 Video (H.262)[64] 1995 Storage and handling of digital images in broadcast applications, digital television, HDTV, cable, satellite, high-speed Internet, DVD video distribution
DV 1995 Camcorders, digital cassettes
H.263 (MPEG-4 Part 2)[61] 1996 Video telephony over public switched telephone network (PSTN), H.320, Integrated Services Digital Network (ISDN)[65][66]
Advanced Video Coding (AVC / H.264 / MPEG-4)[1][22] 2003 Most common HD video recording/compression/distribution format, Internet video, YouTube, Blu-ray Discs, HDTV broadcasts, web browsers, streaming television, mobile devices, consumer devices, Netflix,[46] video telephony, Facetime[45]
Theora 2004 Internet video, web browsers
VC-1 2006 Windows media, Blu-ray Discs
Apple ProRes 2007 Professional video production.[54]
WebM Video 2010 A multimedia open source format developed by Google intended to be used with HTML5.
High Efficiency Video Coding (HEVC / H.265)[1][4] 2013 The emerging successor to the H.264/MPEG-4 AVC standard, having substantially improved compression capability.
Daala 2013 Research video format by Xiph.org.
AV1[67] 2018 An open source format based on VP10 (VP9's internal successor), Daala and Thor; used by content providers such as YouTube[68][69] and Netflix.[70][71]

### MDCT audio standards

 Further information: Modified discrete cosine transform

#### General audio

Audio compression standard Year Common applications
Dolby Digital (AC-3)[25][26] 1991 Cinema, digital cinema, DVD, Blu-ray, streaming media, video games
Adaptive Transform Acoustic Coding (ATRAC)[25] 1992 MiniDisc
MPEG Layer III (MP3)[27][1] 1993 Digital audio distribution, MP3 players, portable media players, streaming media
Perceptual Audio Coder (PAC)[25] 1996 Digital audio radio service (DARS)
Advanced Audio Coding (AAC / MP4 Audio)[28][25] 1997 Digital audio distribution, portable media players, streaming media, game consoles, mobile devices, iOS, iTunes, Android, BlackBerry
Cook Codec 1998 RealAudio
Windows Media Audio (WMA)[25] 1999 Windows Media
Vorbis[29][25] 2000 Digital audio distribution, radio stations, streaming media, video games, Spotify, Wikipedia
Dynamic Resolution Adaptation (DRA)[25] 2008 China national audio standard, China Multimedia Mobile Broadcasting, DVB-H
Opus[73] 2012 VoIP,[74] mobile telephony, WhatsApp,[75][76][77] PlayStation 4[78]
Dolby AC-4[79] 2017 ATSC 3.0, ultra-high-definition television (UHD TV)
MPEG-H 3D Audio[80]

#### Speech coding

Speech coding standard Year Common applications
AAC-LD (LD-MDCT)[81] 1999 Mobile telephony, voice-over-IP (VoIP), iOS, FaceTime[45]
Siren[44] 1999 VoIP, wideband audio, G.722.1
G.722.1[82] 1999 VoIP, wideband audio, G.722
G.729.1[83] 2006 G.729, VoIP, wideband audio,[83] mobile telephony
EVRC-WB[42]: 31, 478]  2007 Wideband audio
G.718[84] 2008 VoIP, wideband audio, mobile telephony
G.719[42] 2008 Teleconferencing, videoconferencing, voice mail
CELT[85] 2011 VoIP,[86][87] mobile telephony
Enhanced Voice Services (EVS)[88] 2014 Mobile telephony, VoIP, wideband audio

### MD DCT

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,[89] variable temporal length 3-D DCT coding,[90] video coding algorithms,[91] adaptive video coding [92] and 3-D Compression.[93] Due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using M-D DCTs is rapidly increasing. DCT-IV has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks,[94] lapped orthogonal transform[95][96] and cosine-modulated wavelet bases.[97]

### Digital signal processing

DCT plays a very important role in digital signal processing. By using the DCT, the signals can be compressed. DCT can be used in electrocardiography for the compression of ECG signals. DCT2 provides a better compression ratio than DCT.

The DCT 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.[1]

### Compression artifacts

 Further information: Compression artifact

A common issue with DCT compression in digital media are blocky compression artifacts,[98] caused by DCT blocks.[3] The DCT algorithm can cause block-based artifacts when heavy compression is applied. Due to the DCT being used in the majority of digital image and video coding standards (such as the JPEG, H.26x and MPEG formats), DCT-based blocky compression artifacts are widespread in digital media. In 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 of these blocks is taken, and the resulting DCT coefficients are quantized. This process can cause blocking artifacts, primarily at high data compression ratios.[98] This can also cause the "mosquito noise" effect, commonly found in digital video (such as the MPEG formats).[99]

DCT blocks are often used in glitch art.[3] The artist Rosa Menkman makes use of DCT-based compression artifacts in her glitch art,[100] particularly the DCT blocks found in most digital media formats such as JPEG digital images and MP3 digital audio.[3] Another example is Jpegs by German photographer Thomas Ruff, which uses intentional JPEG artifacts as the basis of the picture's style.[101][102]

## Informal overview

Like any Fourier-related transform, discrete cosine transforms (DCTs) express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes. Like the discrete Fourier transform (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 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 ${\displaystyle f(x)}$ as a sum of sinusoids, you can evaluate that sum at any ${\displaystyle x}$, even for ${\displaystyle x}$ where the original ${\displaystyle f(x)}$ 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.

Illustration of the implicit even/odd extensions of DCT input data, for N=11 data points (red dots), for the four most common types of DCT (types I-IV).

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 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 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 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 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, 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 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, 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 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.

## Formal definition

Formally, the discrete cosine transform is a linear, invertible function ${\displaystyle f:\mathbb {R} ^{N}\to \mathbb {R} ^{N))$ (where ${\displaystyle \mathbb {R} }$ denotes the set of real numbers), or equivalently an invertible N × N square matrix. There are several variants of the DCT with slightly modified definitions. The N real numbers ${\displaystyle ~x_{0},\ \ldots \ x_{N-1}~}$ are transformed into the N real numbers ${\displaystyle X_{0},\,\ldots ,\,X_{N-1))$ according to one of the formulas:

### DCT-I

${\displaystyle X_{k}={\frac {1}{2))(x_{0}+(-1)^{k}x_{N-1})+\sum _{n=1}^{N-2}x_{n}\cos \left[\,{\frac {\pi }{\,N-1\,))\,n\,k\,\right]\qquad {\text{ for ))~k=0,\ \ldots \ N-1~.}$

Some authors further multiply the ${\displaystyle x_{0))$ and ${\displaystyle x_{N-1))$ terms by ${\displaystyle {\sqrt {2\,))\,,}$ and correspondingly multiply the ${\displaystyle X_{0))$ and ${\displaystyle X_{N-1))$ terms by ${\displaystyle 1/{\sqrt {2\,))\,,}$ which makes the DCT-I matrix orthogonal. If one further multiplies by an overall scale factor of ${\displaystyle {\sqrt ((\tfrac {2}{N-1\,))\,)),}$ 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 ${\displaystyle 2(N-1)}$ real numbers with even symmetry. For example, a DCT-I of ${\displaystyle N=5}$ real numbers ${\displaystyle a\ b\ c\ d\ e}$ is exactly equivalent to a DFT of eight real numbers ${\displaystyle a\ b\ c\ d\ e\ d\ c\ b}$ (even symmetry), divided by two. (In contrast, DCT types II-IV involve a half-sample shift in the equivalent DFT.)

Note, however, that the DCT-I is not defined for ${\displaystyle N}$ less than 2. (All other DCT types are defined for any positive ${\displaystyle N.}$

Thus, the DCT-I corresponds to the boundary conditions: ${\displaystyle x_{n))$ is even around ${\displaystyle n=0}$ and even around ${\displaystyle n=N-1}$; similarly for ${\displaystyle X_{k}.}$

### DCT-II

${\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\cos \left[\,{\tfrac {\,\pi \,}{N))\left(n+{\frac {1}{2))\right)k\,\right]\qquad {\text{ for ))~k=0,\ \dots \ N-1~.}$

The DCT-II is probably the most commonly used form, and is often simply referred to as "the DCT".[5][6]

This transform is exactly equivalent (up to an overall scale factor of 2) to a DFT of ${\displaystyle 4N}$ real inputs of even symmetry where the even-indexed elements are zero. That is, it is half of the DFT of the ${\displaystyle 4N}$ inputs ${\displaystyle y_{n},}$ where ${\displaystyle y_{2n}=0,}$ ${\displaystyle y_{2n+1}=x_{n))$ for ${\displaystyle 0\leq n ${\displaystyle y_{2N}=0,}$ and ${\displaystyle y_{4N-n}=y_{n))$ for ${\displaystyle 0 DCT-II transformation is also possible using 2N signal followed by a multiplication by half shift. This is demonstrated by Makhoul.

Some authors further multiply the ${\displaystyle X_{0))$ term by ${\displaystyle 1/{\sqrt {2\,))\,.}$ and multiply the resulting matrix by an overall scale factor of ${\textstyle {\sqrt ((2}/{N))))$ (see below for 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.[103] 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[104]), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications.[105][106]

The DCT-II implies the boundary conditions: ${\displaystyle x_{n))$ is even around ${\displaystyle n=-1/2}$ and even around ${\displaystyle n=N-1/2\,;}$ ${\displaystyle X_{k))$ is even around ${\displaystyle k=0}$ and odd around ${\displaystyle k=N.}$

### DCT-III

${\displaystyle X_{k}={1}/{2}x_{0}+\sum _{n=1}^{N-1}x_{n}\cos \left[\,{\tfrac {\,\pi \,}{N))\left(k+{\frac {1}{2))\right)n\,\right]\qquad {\text{ for ))~k=0,\ \dots \ N-1~.}$

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").[6]

Some authors divide the ${\displaystyle ~x_{0}~}$ term by ${\displaystyle {\sqrt {2\,))}$ instead of by 2 (resulting in an overall ${\displaystyle x_{0}/{\sqrt {2\,))}$ term) and multiply the resulting matrix by an overall scale factor of ${\textstyle {\sqrt ((2}/{N}\,))}$ (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: ${\displaystyle x_{n))$ is even around ${\displaystyle n=0}$ and odd around ${\displaystyle n=N;}$ ${\displaystyle X_{k))$ is even around ${\displaystyle k=-{1}/{2))$ and even around ${\displaystyle k=N-{1}/{2}.}$

### DCT-IV

${\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\cos \left[\,{\tfrac {\,\pi \,}{N))\,\left(n+{\frac {1}{2))\right)\left(k+{\frac {1}{2))\right)\,\right]\qquad {\text{ for ))~k=0,\ \ldots \ N-1~.}$

The DCT-IV matrix becomes orthogonal (and thus, being clearly symmetric, its own inverse) if one further multiplies by an overall scale factor of ${\textstyle {\sqrt {2/N\,)).}$

A variant of the DCT-IV, where data from different transforms are overlapped, is called the modified discrete cosine transform (MDCT).[107]

The DCT-IV implies the boundary conditions: ${\displaystyle x_{n))$ is even around ${\displaystyle n=-{1}/{2))$ and odd around ${\displaystyle n=N-{1}/{2};}$ similarly for ${\displaystyle X_{k}.}$

### DCT V-VIII

DCTs of types I–IV treat both boundaries consistently regarding 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 ${\displaystyle N}$ is even or odd), since the corresponding DFT is of length ${\displaystyle 2(N-1)}$ (for DCT-I) or ${\displaystyle 4N}$ (for DCT-II & III) or ${\displaystyle 8N}$ (for DCT-IV). The four additional types of discrete cosine transform[108] correspond essentially to real-even DFTs of logically odd order, which have factors of ${\displaystyle N\pm {1}/{2))$ in 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, a length-one DFT (odd length) of a single number a , corresponds to a DCT-V of length ${\displaystyle N=1.}$)

## Inverse transforms

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.[6]

Like for the DFT, 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 ${\textstyle {\sqrt {2/N))}$ 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 the transform matrix orthogonal.

## Multidimensional DCTs

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.

### M-D DCT-II

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 the columns (or vice versa). That is, the 2D DCT-II is given by the formula (omitting normalization and other scale factors, as above):

{\displaystyle {\begin{aligned}X_{k_{1},k_{2))&=\sum _{n_{1}=0}^{N_{1}-1}\left(\sum _{n_{2}=0}^{N_{2}-1}x_{n_{1},n_{2))\cos \left[{\frac {\pi }{N_{2))}\left(n_{2}+{\frac {1}{2))\right)k_{2}\right]\right)\cos \left[{\frac {\pi }{N_{1))}\left(n_{1}+{\frac {1}{2))\right)k_{1}\right]\\&=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}x_{n_{1},n_{2))\cos \left[{\frac {\pi }{N_{1))}\left(n_{1}+{\frac {1}{2))\right)k_{1}\right]\cos \left[{\frac {\pi }{N_{2))}\left(n_{2}+{\frac {1}{2))\right)k_{2}\right].\end{aligned))}
The inverse of a multi-dimensional DCT is just a separable product of the inverses of the corresponding one-dimensional DCTs (see above), e.g. the one-dimensional inverses applied along one dimension at a time in a row-column algorithm.

The 3-D DCT-II is only the extension of 2-D DCT-II in three dimensional space and mathematically can be calculated by the formula

${\displaystyle X_{k_{1},k_{2},k_{3))=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}\sum _{n_{3}=0}^{N_{3}-1}x_{n_{1},n_{2},n_{3))\cos \left[{\frac {\pi }{N_{1))}\left(n_{1}+{\frac {1}{2))\right)k_{1}\right]\cos \left[{\frac {\pi }{N_{2))}\left(n_{2}+{\frac {1}{2))\right)k_{2}\right]\cos \left[{\frac {\pi }{N_{3))}\left(n_{3}+{\frac {1}{2))\right)k_{3}\right],\quad {\text{for ))k_{i}=0,1,2,\dots ,N_{i}-1.}$

The inverse of 3-D DCT-II is 3-D DCT-III and can be computed from the formula given by

${\displaystyle x_{n_{1},n_{2},n_{3))=\sum _{k_{1}=0}^{N_{1}-1}\sum _{k_{2}=0}^{N_{2}-1}\sum _{k_{3}=0}^{N_{3}-1}X_{k_{1},k_{2},k_{3))\cos \left[{\frac {\pi }{N_{1))}\left(n_{1}+{\frac {1}{2))\right)k_{1}\right]\cos \left[{\frac {\pi }{N_{2))}\left(n_{2}+{\frac {1}{2))\right)k_{2}\right]\cos \left[{\frac {\pi }{N_{3))}\left(n_{3}+{\frac {1}{2))\right)k_{3}\right],\quad {\text{for ))n_{i}=0,1,2,\dots ,N_{i}-1.}$

Technically, computing a two-, three- (or -multi) dimensional DCT by sequences of one-dimensional DCTs along each dimension is known as a row-column algorithm. As with multidimensional FFT algorithms, however, there exist other methods to compute the same thing while performing the computations in a different order (i.e. interleaving/combining the algorithms for the different dimensions). Owing to the rapid growth in the applications based on the 3-D DCT, several fast algorithms are developed for the computation of 3-D DCT-II. Vector-Radix algorithms are applied for computing M-D DCT to reduce the computational complexity and to increase the computational speed. To compute 3-D DCT-II efficiently, a fast algorithm, Vector-Radix Decimation in Frequency (VR DIF) algorithm was developed.

#### 3-D DCT-II VR DIF

In order to apply the VR DIF algorithm the input data is to be formulated and rearranged as follows.[109][110] The transform size N × N × N is assumed to be 2.

The four basic stages of computing 3-D DCT-II using VR DIF Algorithm.
${\displaystyle {\begin{array}{lcl}{\tilde {x))(n_{1},n_{2},n_{3})=x(2n_{1},2n_{2},2n_{3})\\{\tilde {x))(n_{1},n_{2},N-n_{3}-1)=x(2n_{1},2n_{2},2n_{3}+1)\\{\tilde {x))(n_{1},N-n_{2}-1,n_{3})=x(2n_{1},2n_{2}+1,2n_{3})\\{\tilde {x))(n_{1},N-n_{2}-1,N-n_{3}-1)=x(2n_{1},2n_{2}+1,2n_{3}+1)\\{\tilde {x))(N-n_{1}-1,n_{2},n_{3})=x(2n_{1}+1,2n_{2},2n_{3})\\{\tilde {x))(N-n_{1}-1,n_{2},N-n_{3}-1)=x(2n_{1}+1,2n_{2},2n_{3}+1)\\{\tilde {x))(N-n_{1}-1,N-n_{2}-1,n_{3})=x(2n_{1}+1,2n_{2}+1,2n_{3})\\{\tilde {x))(N-n_{1}-1,N-n_{2}-1,N-n_{3}-1)=x(2n_{1}+1,2n_{2}+1,2n_{3}+1)\\\end{array))}$
where ${\displaystyle 0\leq n_{1},n_{2},n_{3}\leq {\frac {N}{2))-1}$

The figure to the adjacent shows the four stages that are involved in calculating 3-D DCT-II using VR DIF algorithm. The first stage is the 3-D reordering using the index mapping illustrated by the above equations. The second stage is the butterfly calculation. Each butterfly calculates eight points together as shown in the figure just below, where ${\displaystyle c(\varphi _{i})=\cos(\varphi _{i})}$.

The original 3-D DCT-II now can be written as

${\displaystyle X(k_{1},k_{2},k_{3})=\sum _{n_{1}=1}^{N-1}\sum _{n_{2}=1}^{N-1}\sum _{n_{3}=1}^{N-1}{\tilde {x))(n_{1},n_{2},n_{3})\cos(\varphi k_{1})\cos(\varphi k_{2})\cos(\varphi k_{3})}$

where ${\displaystyle \varphi _{i}={\frac {\pi }{2N))(4N_{i}+1),{\text{ and ))i=1,2,3.}$

If the even and the odd parts of ${\displaystyle k_{1},k_{2))$ and ${\displaystyle k_{3))$ and are considered, the general formula for the calculation of the 3-D DCT-II can be expressed as

The single butterfly stage of VR DIF algorithm.
${\displaystyle X(k_{1},k_{2},k_{3})=\sum _{n_{1}=1}^((\tfrac {N}{2))-1}\sum _{n_{2}=1}^((\tfrac {N}{2))-1}\sum _{n_{1}=1}^((\tfrac {N}{2))-1}{\tilde {x))_{ijl}(n_{1},n_{2},n_{3})\cos(\varphi (2k_{1}+i)\cos(\varphi (2k_{2}+j)\cos(\varphi (2k_{3}+l))}$

where

${\displaystyle {\tilde {x))_{ijl}(n_{1},n_{2},n_{3})={\tilde {x))(n_{1},n_{2},n_{3})+(-1)^{l}{\tilde {x))\left(n_{1},n_{2},n_{3}+{\frac {n}{2))\right)}$
${\displaystyle +(-1)^{j}{\tilde {x))\left(n_{1},n_{2}+{\frac {n}{2)),n_{3}\right)+(-1)^{j+l}{\tilde {x))\left(n_{1},n_{2}+{\frac {n}{2)),n_{3}+{\frac {n}{2))\right)}$
${\displaystyle +(-1)^{i}{\tilde {x))\left(n_{1}+{\frac {n}{2)),n_{2},n_{3}\right)+(-1)^{i+j}{\tilde {x))\left(n_{1}+{\frac {n}{2))+{\frac {n}{2)),n_{2},n_{3}\right)}$
${\displaystyle +(-1)^{i+l}{\tilde {x))\left(n_{1}+{\frac {n}{2)),n_{2},n_{3}+{\frac {n}{3))\right)}$
${\displaystyle +(-1)^{i+j+l}{\tilde {x))\left(n_{1}+{\frac {n}{2)),n_{2}+{\frac {n}{2)),n_{3}+{\frac {n}{2))\right){\text{ where ))i,j,l=0{\text{ or ))1.}$
##### Arithmetic complexity

The whole 3-D DCT calculation needs ${\displaystyle ~[\log _{2}N]~}$ stages, and each stage involves ${\displaystyle ~{\tfrac {1}{8))\ N^{3}~}$ butterflies. The whole 3-D DCT requires ${\displaystyle ~\left[{\tfrac {1}{8))\ N^{3}\log _{2}N\right]~}$ butterflies to be computed. Each butterfly requires seven real multiplications (including trivial multiplications) and 24 real additions (including trivial additions). Therefore, the total number of real multiplications needed for this stage is ${\displaystyle ~\left[{\tfrac {7}{8))\ N^{3}\ \log _{2}N\right]~,}$ and the total number of real additions i.e. including the post-additions (recursive additions) which can be calculated directly after the butterfly stage or after the bit-reverse stage are given by[110] ${\displaystyle ~\underbrace {\left[{\frac {3}{2))N^{3}\log _{2}N\right]} _{\text{Real))+\underbrace {\left[{\frac {3}{2))N^{3}\log _{2}N-3N^{3}+3N^{2}\right]} _{\text{Recursive))=\left[{\frac {9}{2))N^{3}\log _{2}N-3N^{3}+3N^{2}\right]~.}$

The conventional method to calculate MD-DCT-II is using a Row-Column-Frame (RCF) approach which is computationally complex and less productive on most advanced recent hardware platforms. The number of multiplications required to compute VR DIF Algorithm when compared to RCF algorithm are quite a few in number. The number of Multiplications and additions involved in RCF approach are given by ${\displaystyle ~\left[{\frac {3}{2))N^{3}\log _{2}N\right]~}$ and ${\displaystyle ~\left[{\frac {9}{2))N^{3}\log _{2}N-3N^{3}+3N^{2}\right]~,}$ respectively. From Table 1, it can be seen that the total number

TABLE 1 Comparison of VR DIF & RCF Algorithms for computing 3D-DCT-II
Transform Size 3D VR Mults RCF Mults 3D VR Adds RCF Adds
8 × 8 × 8 2.625 4.5 10.875 10.875
16 × 16 × 16 3.5 6 15.188 15.188
32 × 32 × 32 4.375 7.5 19.594 19.594
64 × 64 × 64 5.25 9 24.047 24.047

of multiplications associated with the 3-D DCT VR algorithm is less than that associated with the RCF approach by more than 40%. In addition, the RCF approach involves matrix transpose and more indexing and data swapping than the new VR algorithm. This makes the 3-D DCT VR algorithm more efficient and better suited for 3-D applications that involve the 3-D DCT-II such as video compression and other 3-D image processing applications.

The main consideration in choosing a fast algorithm is to avoid computational and structural complexities. As the technology of computers and DSPs advances, the execution time of arithmetic operations (multiplications and additions) is becoming very fast, and regular computational structure becomes the most important factor.[111] Therefore, although the above proposed 3-D VR algorithm does not achieve the theoretical lower bound on the number of multiplications,[112] it has a simpler computational structure as compared to other 3-D DCT algorithms. It can be implemented in place using a single butterfly and possesses the properties of the Cooley–Tukey FFT algorithm in 3-D. Hence, the 3-D VR presents a good choice for reducing arithmetic operations in the calculation of the 3-D DCT-II, while keeping the simple structure that characterize butterfly-style Cooley–Tukey FFT algorithms.

Two-dimensional DCT frequencies from the JPEG DCT

The image to the right shows a combination of horizontal and vertical frequencies for an 8 × 8 ${\displaystyle (~N_{1}=N_{2}=8~)}$ two-dimensional DCT. Each step from left to right and top to bottom is an increase in frequency by 1/2 cycle. For example, moving right one from the top-left square yields a half-cycle increase in the horizontal frequency. Another move to the right yields two half-cycles. A move down yields two half-cycles horizontally and a half-cycle vertically. The source data ( 8×8 ) is transformed to a linear combination of these 64 frequency squares.

### MD-DCT-IV

The M-D DCT-IV is just an extension of 1-D DCT-IV on to M dimensional domain. The 2-D DCT-IV of a matrix or an image is given by

${\displaystyle X_{k,\ell }=\sum _{n=0}^{N-1}\;\sum _{m=0}^{M-1}\ x_{n,m}\cos \left(\ {\frac {\,(2m+1)(2k+1)\ \pi \,}{4N))\ \right)\cos \left(\ {\frac {\,(2n+1)(2\ell +1)\ \pi \,}{4M))\ \right)~,}$
for ${\displaystyle ~~k=0,\ 1,\ 2\ \ldots \ N-1~~}$ and ${\displaystyle ~~\ell =0,\ 1,\ 2,\ \ldots \ M-1~.}$

We can compute the MD DCT-IV using the regular row-column method or we can use the polynomial transform method[113] for the fast and efficient computation. The main idea of this algorithm is to use the Polynomial Transform to convert the multidimensional DCT into a series of 1-D DCTs directly. MD DCT-IV also has several applications in various fields.

## Computation

Although the direct application of these formulas would require ${\displaystyle ~{\mathcal {O))(N^{2})~}$ operations, it is possible to compute the same thing with only ${\displaystyle ~{\mathcal {O))(N\log N)~}$ complexity by factorizing the computation similarly to the fast Fourier transform (FFT). One can also compute DCTs via FFTs combined with ${\displaystyle ~{\mathcal {O))(N)~}$ pre- and post-processing steps. In general, ${\displaystyle ~{\mathcal {O))(N\log N)~}$ methods to compute DCTs are known as fast cosine transform (FCT) algorithms.

The most efficient algorithms, in principle, are usually those that are specialized directly for the DCT, as opposed to using an ordinary FFT plus ${\displaystyle ~{\mathcal {O))(N)~}$ extra operations (see below for an exception). However, even "specialized" DCT algorithms (including all of those that achieve the lowest known arithmetic counts, at least for power-of-two sizes) are typically closely related to FFT algorithms – since DCTs are essentially DFTs of real-even data, one can design a fast DCT algorithm by taking an FFT and eliminating the redundant operations due to this symmetry. This can even be done automatically (Frigo & Johnson 2005). Algorithms based on the Cooley–Tukey FFT algorithm are most common, but any other FFT algorithm is also applicable. For example, the Winograd FFT algorithm leads to minimal-multiplication algorithms for the DFT, albeit generally at the cost of more additions, and a similar algorithm was proposed by (Feig, Winograd & July 1992) for the DCT. Because the algorithms for DFTs, DCTs, and similar transforms are all so closely related, any improvement in algorithms for one transform will theoretically lead to immediate gains for the other transforms as well (Duhamel & Vetterli 1990).

While DCT algorithms that employ an unmodified FFT often have some theoretical overhead compared to the best specialized DCT algorithms, the former also have a distinct advantage: Highly optimized FFT programs are widely available. Thus, in practice, it is often easier to obtain high performance for general lengths N with FFT-based algorithms.[a] Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the 8 × 8 DCT-II used in JPEG compression, or the small DCTs (or MDCTs) typically used in audio compression. (Reduced code size may also be a reason to use a specialized DCT for embedded-device applications.)

In fact, even the DCT algorithms using an ordinary FFT are sometimes equivalent to pruning the redundant operations from a larger FFT of real-symmetric data, and they can even be optimal from the perspective of arithmetic counts. For example, a type-II DCT is equivalent to a DFT of size ${\displaystyle ~4N~}$ with real-even symmetry whose even-indexed elements are zero. One of the most common methods for computing this via an FFT (e.g. the method used in FFTPACK and FFTW) was described by Narasimha & Peterson (1978) and Makhoul (1980), and this method in hindsight can be seen as one step of a radix-4 decimation-in-time Cooley–Tukey algorithm applied to the "logical" real-even DFT corresponding to the DCT-II.[b] Because the even-indexed elements are zero, this radix-4 step is exactly the same as a split-radix step. If the subsequent size ${\displaystyle ~N~}$ real-data FFT is also performed by a real-data split-radix algorithm (as in Sorensen et al. (1987)), then the resulting algorithm actually matches what was long the lowest published arithmetic count for the power-of-two DCT-II (${\displaystyle ~2N\log _{2}N-N+2~}$ real-arithmetic operations[c]).

A recent reduction in the operation count to ${\displaystyle ~{\tfrac {17}{9))N\log _{2}N+{\mathcal {O))(N)}$ also uses a real-data FFT.[114] So, there is nothing intrinsically bad about computing the DCT via an FFT from an arithmetic perspective – it is sometimes merely a question of whether the corresponding FFT algorithm is optimal. (As a practical matter, the function-call overhead in invoking a separate FFT routine might be significant for small ${\displaystyle ~N~,}$ but this is an implementation rather than an algorithmic question since it can be solved by unrolling or inlining.)

## Example of IDCT

An example showing eight different filters applied to a test image (top left) by multiplying its DCT spectrum (top right) with each filter.

Consider this 8x8 grayscale image of capital letter A.

Original size, scaled 10x (nearest neighbor), scaled 10x (bilinear).
Basis functions of the discrete cosine transformation with corresponding coefficients (specific for our image).
DCT of the image = ${\displaystyle {\begin{bmatrix}6.1917&-0.3411&1.2418&0.1492&0.1583&0.2742&-0.0724&0.0561\\0.2205&0.0214&0.4503&0.3947&-0.7846&-0.4391&0.1001&-0.2554\\1.0423&0.2214&-1.0017&-0.2720&0.0789&-0.1952&0.2801&0.4713\\-0.2340&-0.0392&-0.2617&-0.2866&0.6351&0.3501&-0.1433&0.3550\\0.2750&0.0226&0.1229&0.2183&-0.2583&-0.0742&-0.2042&-0.5906\\0.0653&0.0428&-0.4721&-0.2905&0.4745&0.2875&-0.0284&-0.1311\\0.3169&0.0541&-0.1033&-0.0225&-0.0056&0.1017&-0.1650&-0.1500\\-0.2970&-0.0627&0.1960&0.0644&-0.1136&-0.1031&0.1887&0.1444\\\end{bmatrix))}$.

Each basis function is multiplied by its coefficient and then this product is added to the final image.

On the left is the final image. In the middle is the weighted function (multiplied by a coefficient) which is added to the final image. On the right is the current function and corresponding coefficient. Images are scaled (using bilinear interpolation) by factor 10×.

## Notes

1. ^ Algorithmic performance on modern hardware is typically not principally determined by simple arithmetic counts, and optimization requires substantial engineering effort to make best use, within its intrinsic limits, of available built-in hardware optimization.
2. ^ The radix-4 step reduces the size ${\displaystyle ~4N~}$ DFT to four size ${\displaystyle ~N~}$ DFTs of real data, two of which are zero, and two of which are equal to one another by the even symmetry. Hence giving a single size ${\displaystyle ~N~}$ FFT of real data plus ${\displaystyle ~{\mathcal {O))(N)~}$ butterflies, once the trivial and / or duplicate parts are eliminated and / or merged.
3. ^ The precise count of real arithmetic operations, and in particular the count of real multiplications, depends somewhat on the scaling of the transform definition. The ${\displaystyle ~2N\log _{2}N-N+2~}$ count is for the DCT-II definition shown here; two multiplications can be saved if the transform is scaled by an overall ${\displaystyle {\sqrt {2))}$ factor. Additional multiplications can be saved if one permits the outputs of the transform to be rescaled individually, as was shown by Arai, Agui & Nakajima (1988) for the size-8 case used in JPEG.

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text:
### Home > A2C > Chapter 7 > Lesson 7.3.2 > Problem7-174

7-174.

Use your knowledge of shifting parent graphs to graph each equation below.

1. $y = −2\left(x − 3\right)^{2} + 4$

1. $y = \frac { 1 } { 2 } ( x + 2 ) ^ { 3 } - 3$

1. $y = 2\left|x − 5\right|$

1. $y = \sqrt { x - 2 } - 3$

If you are stuck, you can always graph by making a table.
Be sure you know what the parent graph looks like so that you get a complete graph.

format: plain


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# Powers of Two

## In lexicographic ordering

A recent link on lobste.rs caught my attention. Take the first 10 powers of 2 (starting from zero), sorted in lexicographic order, $1, 128, 16, 2, 256, 32, 4, 512, 64, 8$ Now, put a decimal point after the first digit in each, $1.0, 1.28, 1.6, 2.0, 2.56, 3.2, 4.0, 5.12, 6.4, 8.0$ At first glance, it’s hard to see any patterns in this, but if we take $$10^{n/10}$$, for $$n = 0,\ldots,9$$, we get the suspiciously similar sequence $1.0, 1.259, 1.585, 1.995, 2.512, 3.162, 3.981, 5.012, 6.310, 7.943$ One might ask why this happens intuitively. I believe that the most intuitive explanation stems from the simple fact that $$\log_{10} 2 \approx 0.3$$.

How does this come into play? Well let’s look at the sequence of shifted powers of two. Shifting a number so that the decimal point sits after the first non-zero digit is the same as dividing by the highest power of 10 less than that number. For a given $$n$$, that power of 10 is given by $10^{\lfloor \log_{10} n \rfloor}$ So, terms in our sequence can all be written in the form, \begin{aligned} \frac{2^n}{10^{\lfloor \log_{10} 2^n \rfloor}} &= \frac{10^{n \log_{10} 2}}{10^{\lfloor n \log_{10} 2 \rfloor}} \\ &= 10^{n \log_{10} 2 - \lfloor n \log_{10} 2 \rfloor} \\ &= 10^{\{ n \log_{10} 2 \}} \end{aligned} where $$\{x\}$$ denotes the fractional part of $$x$$, i.e. $$\{ x \} = x - \lfloor x \rfloor$$. Evaluating this for $$n = 0, \ldots, 9$$, gives in sequence, $1.0, 2.0, 4.0, 8.0, 1.6, 3.2, 6.4, 1.28, 2.56, 5.12$ Okay, they’re shifted, but not lexicographically sorted. In fact, the lexicographic ordering is a red herring, we can achieve the same sequence by first shifting the decimal place and then sorting in ascending order!

So where does the correspondence between this and $$10^{n/10}$$ appear? Well, because $$\log_{10} 2 \approx 0.3$$, we have that our terms in our sequence can be approximated as \begin{aligned} 1 &\approx 10^{\{ 0.3 \cdot 0 \}} = 10^0 = 1 \\ 2 &\approx 10^{\{ 0.3 \cdot 1 \}} = 10^{0.3} = 1.995 \\ 4 &\approx 10^{\{ 0.3 \cdot 2 \}} = 10^{0.6} = 3.981 \\ 8 &\approx 10^{\{ 0.3 \cdot 3 \}} = 10^{0.9} = 7.943 \\ 1.6 &\approx 10^{\{ 0.3 \cdot 4 \}} = 10^{0.2} = 1.585 \\ 3.2 &\approx 10^{\{ 0.3 \cdot 5 \}} = 10^{0.5} = 3.162 \\ 6.4 &\approx 10^{\{ 0.3 \cdot 6 \}} = 10^{0.8} = 6.310 \\ 1.28 &\approx 10^{\{ 0.3 \cdot 7 \}} = 10^{0.1} = 1.259 \\ 2.56 &\approx 10^{\{ 0.3 \cdot 8 \}} = 10^{0.4} = 2.512 \\ 5.12 &\approx 10^{\{ 0.3 \cdot 9 \}} = 10^{0.7} = 5.012 \\ \end{aligned} Similarly, $1024 = 2^{10} = 10^{10 \log_{10} 2} \approx 10^{0.3 \cdot 10} = 10^3 = 1000$ A relation that should be second nature to anybody with an interest in computers.

© 2019 Joe Davis.


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# Courses Memorizeing steps and math?

1. Dec 4, 2017

### Apple_Mango

Hello, I am almost done with Algebra 1. I am curious to know how many steps do higher levels math like Calculus three have. I keep thinking how higher levels maths has 150 steps to memorize just for one equation. I hope my speculation isn't true. How many steps does an equation have once I got to higher levels maths like Cal 3? Are the amount of steps needed to memorize in higher level math as numerous as Algebra 2?

Last edited by a moderator: Dec 4, 2017
2. Dec 4, 2017

I have found in general that most of the more advanced mathematics really doesn't tax the memory all that much if you understand the principles that it is using. Learning the fundamentals is important, and it is important to have a solid foundation in algebra, geometry, and trigonometry to be ready for calculus. Calculus builds a lot off of these more basic subjects, and if you learn them well, calculus should not be too terribly difficult to understand. If anything, the part of the memory that calculus uses is being able to remember all the algebra, geometry, and trigonometry that you have previously learned. $\\$ Editing: One additional tip that may be helpful: If you forget some of the algebra, geometry, and trigonometry a few weeks or months after you have learned it, don't be afraid to go back and review it, and relearn it, if necessary. You are likely to find after relearning it a couple of times, that it really becomes committed to memory.

3. Dec 4, 2017

### Staff: Mentor

Even as an EE, I do a lot of manual math in my R&D work. There just don't seem to be too many automated computer tools that can help me do it, so I just work through the problems the best that I can. Here are parts of a calculation from a unique electronic filter design from today...You can see that in the end the solution is pretty obvious, but it takes many steps to get to the final simple solution.

https://i.stack.imgur.com/ZaOE1.png

4. Dec 4, 2017

### symbolipoint

No memorizing steps! You will need to UNDERSTAND. You will pick the steps in a problem as you go. Your knowledge and skills in Algebra will allow you to decide on any sequence of steps in a solution process.

5. Dec 4, 2017

### symbolipoint

STEPS MAY VARY.

Skills among any group of Algebra 2 students will also vary. Some students can make small mental step leaps in their heads better than other students in the group and therefore may need to use fewer steps on paper.

6. Dec 5, 2017


text:
Is Your Box of Celebrations Chocolates Biased?

We surely all know the feeling of disappointment when the best chocolates in the Celebrations box are all gone and there’s only the rubbish ones left in the tub. Myself being particularly fond of the Malteasers and Galaxy chocolates, I’ve often felt a little suspicious that there might not be as many of them in a box than the other chocolates. But, I’ve never had a chance to put my suspicions to the test…

This all changed a couple of days ago, as we came into the possession of a 2.4 kg box of celebrations(!). So, for fun, my partner and I decided to count up the number of each type of chocolate in the box. (You might take this as a sign of social distancing-induced boredom, but no, this kind of behaviour is pretty much the norm for us).

And with that, we finally had the data we’d been longing for:

There were 267 chocolates in total in the box, and 8 types of chocolate. If there were equal numbers of each type of chocolate, that would mean 33.375 chocolates of each type - or, to be more realistic, somewhere between 32-34 of each type.

But in our box, there were a whopping 52 Mars Bars, and 44 Milky Ways… and only around 20 Galaxy and Galaxy Caramels. Unfortunate, since neither of us like Mars Bars or Milky Way all that much.

Now, the distribution of chocolates within our box got me ever more suspicious. Could Mars Inc be overfilling Celebrations with certain types of chocolate at the expense of others?

Nothing that a good bit of statistics and probability can’t help us answer!

Stats Time!

Let’s first suppose that the company aren’t hand-picking every single chocolate to ensure the same number of chocolate types in each day.

Instead, let’s suppose the company produce equal amounts of each type of chocolate, and a random selection goes into each box.

In this scenario, even if usually, the number of each type of chocolate should turn out roughly equal, we’d expect there to sometimes be boxes with a surprisingly larger number of some types of chocolates over others.

But how likely would we be to see the distribution of chocolate types as unequal as what we observed in our own Celebrations box?

Since our data consists of counts of categorical data, we can answer this question with a chi-squared test comparing our observed frequency to the expected average frequency (i.e. equal numbers of each type of chocolate).

chisq.test(table, p = rep(0.125, 8))
##
##  Chi-squared test for given probabilities
##
## data:  table
## X-squared = 27.142, df = 7, p-value = 0.0003142

With a p-value of 0.0003, our observed frequency is looking pretty unlikely, if the chocolates really are selected randomly from an equal number of each chocolate type.

Next, focussing just on the Mars Bars - since they’re the most frequent (and most suspicious) chocolate in our box by far… if the chocolate selection is made at random, from a pool of chocolate types of equal number, what’s the probability of ending up with 52 Mars Bars in a box of 267?

Here, we have a binary outcome: Mars, or not Mars. This therefore becomes a classic binomial problem. If each type of chocolate is equally likely to be selected, the probability p of picking a Mars Bar is 1/8, or p = 0.125. The probability of picking something other than a Mars Bar is 1 - p = 0.875.

Then we can calculate the probability of picking 52 Mars Bars out of 267 using this formula:

P(x) = N! / k!(N-k)! . pk(1-p)N - k

where N is the total number and k is the number of successful outcomes (in this case, the number of Mars Bars).

Although we can calculate this by hand, r provides a handy function, dbinom() to calculate binomial probabilities.

dbinom(52, size = 267, prob = 0.125)
## [1] 0.0003406686

So, the probability of obtaining 52 Mars Bars in a box of 267 Celebrations, if all types of chocolate are equally likely to be picked, is 0.0003.

Now, this is only the point probability , aka the probability of obtaining exactly 52 Mars Bars in the box, but we can also calculate the probability of obtaining 52 or more Mars Bars in our Celebrations box to give us a p-value:

# with lower.tail set to FALSE, pbinom gives P(X > x) so I have set n equal to 51
pbinom(51, size = 267, prob = 0.125, lower.tail = FALSE)
## [1] 0.0007809731

Not looking much better.

To make even clearer how unlikely this is, here’s where our outcome sits on a probability distribution:

Data Time!

At this point, it’s looking more and more like the selection of chocolates in the box is biased.

However, I am making my inferring on the basis of just a single sample. To be more certain, I’d need to do this with more Celebrations boxes… but with 2.4 kg of chocolate, it doesn’t look like we’ll be buying more anytime soon!

In the absence of more samples, I decided to collect anecdotal evidence from google. My google search revealed to me that other people have had similar experiences to ours:

First, I found some stellar reporting from the Liverpool Echo here, confirming an abundance of Mars Bars in their sample.

Then, I found this hero who also counted up the number of chocolates in his Celebrations box, and even got 13 more people involved! And what would you know, Mars Bars were again the clear winner.

But why is this going on? Why are our Celebrations boxes overfilled with Mars Bars?

Now, a simple answer could be that Mars Bars are in fact the most popular chocolate type, and the company are merely this catering to the tastes of the nation.

It turns out, YouGov have conducted a poll on this subject.

The results possibly won’t shock you:

Malteasers are the nation’s favourite, across all age groups - and also my personal favourites, so I’m clearly basic af.

Mars on the otherhand, is the nation’s least favourite…

Clearly this isn’t a case of the company catering to the nation’s tastes… so what else could explain this? Is it an attempt to change public opinion? Are Mars Bars cheaper to produce? Or is it all designed to encourage us to buy more boxes, in the hope that the next one is the one, the mythical box with 52 Malteasers?

We may never know. But one thing I do know is that I won’t be sharing my Maltesers.

DISCLAIMER

This blogpost is all made in the name of fun and probability - I have no beef with Mars Inc -


text:
## Rocky Mountain Journal of Mathematics

### On the existence of ground states of nonlinear fractional Schrödinger systems with close-to-periodic potentials

#### Abstract

We are concerned with the nonlinear fractional Schrödinger system $$\begin{cases}(-\Delta )^{s} u+V_{1}(x)u=f(x,u)+\Gamma (x)|u|^{q-2}u|v|^{q} &\mbox {in } \mathbb {R}^{N},\\ (-\Delta )^{s} v+V_{2}(x)v=g(x,v)+\Gamma (x)|v|^{q-2}v|u|^{q} &\mbox {in } \mathbb {R}^{N},\\ u,v\in H^{s}(\mathbb {R}^{N}), \end{cases}$$ where $(-\Delta )^{s}$ is the fractional Laplacian operator, $s\in (0,1)$, $N>2s$, $4\leq 2q\lt p\lt 2^{\ast }$, $2^{\ast }={2N}/({N-2s})$. $V_{i}(x)=V^{i}_{per }(x)+V^{i}_{loc }(x)$ is closed-to-periodic for $i=1,2$, $f$ and $g$ have subcritical growths and $\Gamma (x)\geq 0$ vanishes at infinity. Using the Nehari manifold minimization technique, we first obtain a bounded minimizing sequence, and then we adopt the approach of Jeanjean-Tanaka (2005) to obtain a decomposition of the bounded Palais-Smale sequence. Finally, we prove the existence of ground state solutions for the nonlinear fractional Schrödinger system.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1647-1683.

Dates
First available in Project Euclid: 19 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1539936040

Digital Object Identifier
doi:10.1216/RMJ-2018-48-5-1647

Mathematical Reviews number (MathSciNet)
MR3866563

Zentralblatt MATH identifier
06958796

Subjects
Primary: 35A15: Variational methods 35J60: Nonlinear elliptic equations

#### Citation

Wei, Gongming; Duan, Xueliang. On the existence of ground states of nonlinear fractional Schrödinger systems with close-to-periodic potentials. Rocky Mountain J. Math. 48 (2018), no. 5, 1647--1683. doi:10.1216/RMJ-2018-48-5-1647. https://projecteuclid.org/euclid.rmjm/1539936040

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text:
# 8.2: Bioinorganic chemistry

Many biological reactions are known to involve metal ions. There are also metals recognized as essential elements, although their roles in living organisms are not clear. Bioinorganic chemistry, the study of the functions of metals in biological systems using the knowledge and methods of inorganic chemistry, has progressed remarkably in recent years.

The following list shows typical bioactive substances containing metals.

1. Electron carriers. Fe: cytochrome, iron-sulfur protein. Cu : blue copper protein.
2. Metal storage compound. Fe : ferritin, transferrin. Zn : metallothionein.
3. Oxygen transportation agent. Fe: hemoglobin, myoglobin. Cu: hemocyanin.
4. Photosynthesis. Mg: chlorophyll.
5. Hydrolase. Zn: carboxypeptidase. Mg: aminopeptidase.
6. Oxidoreductase. Fe: oxygenase, hydrogenase. Fe, Mo: nitrogenase.
7. Isomerase. Fe: aconitase. Co: vitamin B12 coenzyme.

The basis of chemical reactions of metalloenzymes are

1. Coordinative activation (coordination form, electronic donating, steric effect),
2. Redox (metal oxidation state),
3. Information communication, and, in many cases, reaction environments are regulated by biopolymers such as proteins, and selective reactions are performed.

Examples of actions of metals other than by metalloenzymes include

1. Mg: MgATP energy transfer
2. Na/K ion pumping,
3. Ca: transfer of hormone functions, muscle contraction, nerve transfer, blood coagulation, are some of the important roles of metals.

## (a) Oxidation

Oxidation reactions in living systems are fundamental to life, and many studies of these systems have been performed. In particular, the mechanisms of oxygen gas transportation by hemoglobin and mono-oxygen oxidation by the iron porphyrin compounds named P-450 have been studied at length. Oxygen gas transportation, which has been studied for many years, is described below. Iron porphyrins hemoglobin and myoglobin and the copper compound hemocyanin are involved in the transportation of oxygen gas in air to cells in living organisms. The basis of this function is reversible bonding and dissociation of dioxygen to iron or copper ions. In order to perform these functions, metals must be in oxidation states and coordination environments suitable for the reversible coordination of dioxygen. The iron porphyrin compound hemoglobin is found in red bloods of human beings and other animals.

Hemoglobin has the structure of heme iron with four iron porphyrin units combined with a globin protein. Dioxygen is transported in blood by being coordinated to ferrous ions in the hem iron unit. The Fe (II) ion is penta-coordinate with four nitrogen atoms of porphyrin and a nitrogen atom of the polypeptide histidine, and becomes hexa-coordinate when a dioxygen coordinates to it. The spin state of Fe (II) changes from high spin to low spin upon the coordination of dioxygen. The high spin Fe(II) is above the plane of porphyrin because it is too large to fit in the available space. When the Fe(II) ion becomes low spin upon dioxygen coordination, the size of the iron ion decreases and it just fits into the hole of the porphyrin molecule.

This molecular-level movement has attracted interest as an allosteric effect because it affects the whole protein through the histidine coordinate bond and governs the specific bond of a dioxygen molecule. Oxidation of the Fe(II) ion of a hem molecule is prevented by a macromolecular protein, and if the hem iron is taken out of the prote in, Fe(II) ion is oxidized to Fe(III), and two porphyrin rings are bridged by a peroxide $$\mu$$-O22-, which finally changes to a bridging $$\mu$$-O2-structure.

When the hem is in this state, it loses the ability to coordinate to the dioxygen molecule. Based on this phenomenon, a synthetic porphyrin that is able reversibly to coordinate to a dioxygen by suppressing dimerization of the iron porphyrin has been developed, and was named the picket fence porphyrin after its three dimensional form.

## (b) Nitrogen fixation

The reaction which converts the nitrogen in air into ammonia is basic to all life. Nitrogen fixation, the reaction to fix atmospheric nitrogen to form ammonia, is carried out by Rhizobium in the roots of legumes or by bacteria in algae in an anaerobic atmosphere. All animals and plants, including mankind, were depended on biological nitrogen fixation as a source of nitrogen for protein and other compounds containing nitrogen before the invention of the Harber-Bosch process.

$N_{2} + 8 H^{+} + 8 e^{-} + 16 MgATP \rightarrow 2 NH_{3} + H_{2} + 16 MgADP + 16 Pi \quad \text{(where Pi is an inorganic phosphate)}$

An enzyme named nitrogenase catalyzes this reaction. Nitrogenase contains iron-sulfur and iron-molybdenum sulfur proteins, and reduces dinitrogen by coordination and cooperative proton and electron transfers, while using MgATP as an energy source. Because of the importance of this reaction, attempts to clarify the structure of nitrogenase and to develop artificial catalysts for nitrogen fixation have continued for many years. Recently, the structure of an active center in nitrogenase called iron-molybdenum cofactor was clarified by single crystal X-ray analysis (Figure $$\PageIndex{2}$$). According to this analysis, its structure has Fe3MoS4 and Fe4S4 clusters connected through S.

It is believed that dinitrogen is activated by coordination between the two clusters. On the other hand, the portion called P cluster consists of two Fe4S4 clusters. The roles and reaction mechanism of both parts are not yet clear.

## (c) Photosynthesis

The formation of glucose and dioxygen by the reaction of carbon dioxide and water is a skillful reaction using photoenergy and in which chlorophyll (Figure $$\PageIndex{3}$$), which is a magnesium porphyrin and a manganese cluster complex, plays the central role. A chloroplast contains photosystem I (PSI) and photosystem II (PSII), which use light energy to reduce carbon dioxide and to oxidize water.

Chlorophyll is a fundamental component of PSI. Chlorophyll is a porphyrin complex of magnesium and is responsible for the green colors of leaves. It plays an important role in receiving light energy and transferring it to redox reaction systems. Chlorophyll is excited from the singlet ground state to the singlet excited state by light, the energy of the excited state is transferred to an acceptor within 10 ps, and the resultant energy reduces an iron-sulfur complex and is finally used for reduction of carbon dioxide in subsequent dark reactions. Since charge separation by photochemical excitation is the most important first stage, studies on photoinduced electron transfer are have been actively performed using various kinds of porphyrin compounds as models of chlorophylls. PSI, which obtains oxidizing energy by electron transfer, converts ADP to ATP.

On the other hand, the oxidized form of PSII oxidizes water through a chain of redox reactions of oxo cluster complexes of manganese, and generates oxygen. Since four electrons shift in the reduction of Mn(IV) to Mn (II) in this reaction, at least two manganese species are involved. Probably, a cluster complex which contains two Mn(II) and two Mn(IV) species mediates the electron transfer via four step reactions. However, the details of this reaction are as yet unclear because it is very difficult to isolate this cluster and to analyze its structure. The electron transfer stage is being studied at present by using various manganese complexes as model systems.

Photosynthesis is a very interesting research theme in bioinorganic chemistry as it involves a few metal ions, a porphyrin, sulfide and oxide clusters that constitute a cycle of subtle electron transfer and redox reactions, and generate oxygen gas by photolysis of water and produce carbohydrates from carbon dioxide by reductive dark reactions,. Recently, the reaction center of a photosynthetic bacteria was crystallized and J. Deisenhofer and his colleagues won a Nobel prize for its structural analysis (1988).

Exercise $$\PageIndex{2}$$

Give examples of small molecules that are fundamentally important for living things.


text:
fn: while
[contents]

#### Syntax

The syntax for while loops is:

f++:
while{options}(condition)
{
//loop code-block
}

n++:
@while{options}(condition)
{
//loop code-block
}

Note: Single-line code blocks do not need to be enclosed in parentheses.

#### Description

The while function takes a single parameter specifying a loop condition, then loops through the code-block following the call while the loop condition does not evaluate to 0 (for while loops the loop condition is evaluated before each iteration of the loop code-block).

Note: f++ is used for while conditions, even for n++. If you accidentally use n++ for the condition it will most often run without any syntax or semantic errors anyway.

Note: If not writing to the output file Nift will skip to the first non-whitespace (ie. to the first character that is not a space, tab or newline) after a while loop and inject it to the output file where the call started. If you want to prevent Nift from doing this put a '!' after the loop, eg.:

@while{!o}(condition)
{
# block
}!

#### Options

The following options are available for while loops:

option description
f++ parse while loop code-block with f++
n++ parse while loop code-block with n++
!\n do not add newline between each loop iteration
\n add double newline between each loop iteration
eob="value" add value between each loop iteration
option description

#### f++ example

Examples of while being used with f++:

int i=0
while(i < 10)
console("i: i+=1")

int i=0
while{!s, !o}(i+=1; i<10)
console("i: ", i)

#### n++ example

Examples of while being used with n++:

@int i=0
@while{\n}(i < 10)
i:  @++(i)

@int i=0
@while{f++, !o, !s}(i+=1; i < 10)
{
//f++ code for scripting
}