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class="vector-toc-numb">3.5</span> <span>DST V–VIII</span> </div> </a> <ul id="toc-DST_V–VIII-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverse_transforms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverse_transforms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Inverse transforms</span> </div> </a> <ul id="toc-Inverse_transforms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Computation</span> </div> </a> <ul id="toc-Computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Discrete sine transform</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>discrete sine transform (DST)</b> is a <a href="/wiki/List_of_Fourier-related_transforms" title="List of Fourier-related transforms">Fourier-related transform</a> similar to the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (DFT), but using a purely <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with <a href="/wiki/Even_and_odd_functions" title="Even and odd functions">odd</a> <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and/or output data are shifted by half a sample. </p><p>The DST is related to the <a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">discrete cosine transform</a> (DCT), which is equivalent to a DFT of real and <i>even</i> functions. See the DCT article for a general discussion of how the boundary conditions relate the various DCT and DST types. Generally, the DST is derived from the DCT by replacing the <a href="/wiki/Neumann_boundary_condition" title="Neumann boundary condition">Neumann condition</a> at <i>x</i>=0 with a <a href="/wiki/Dirichlet_condition" class="mw-redirect" title="Dirichlet condition">Dirichlet condition</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Both the DCT and the DST were described by <a href="/wiki/Nasir_Ahmed_(engineer)" title="Nasir Ahmed (engineer)">Nasir Ahmed</a>, T. Natarajan, and <a href="/wiki/K.R._Rao" class="mw-redirect" title="K.R. Rao">K.R. Rao</a> in 1974.<sup id="cite_ref-pubDCT_2-0" class="reference"><a href="#cite_note-pubDCT-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ahmed_3-0" class="reference"><a href="#cite_note-Ahmed-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The type-I DST (DST-I) was later described by <a href="/wiki/Anil_K._Jain_(electrical_engineer,_born_1946)" title="Anil K. Jain (electrical engineer, born 1946)">Anil K. Jain</a> in 1976, and the type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=1" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>DSTs are widely employed in solving <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> by <a href="/wiki/Spectral_methods" class="mw-redirect" title="Spectral methods">spectral methods</a>, where the different variants of the DST correspond to slightly different odd/even boundary conditions at the two ends of the array. </p> <div class="mw-heading mw-heading2"><h2 id="Informal_overview">Informal overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=2" title="Edit section: Informal overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:DST-symmetries.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/DST-symmetries.svg/350px-DST-symmetries.svg.png" decoding="async" width="350" height="518" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/DST-symmetries.svg/525px-DST-symmetries.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/DST-symmetries.svg/700px-DST-symmetries.svg.png 2x" data-file-width="560" data-file-height="828" /></a><figcaption>Illustration of the implicit even/odd extensions of DST input data, for <i>N</i>=9 data points (red dots), for the four most common types of DST (types I–IV).</figcaption></figure> <p>Like any Fourier-related transform, discrete sine transforms (DSTs) express a function or a signal in terms of a sum of <a href="/wiki/Sinusoid" class="mw-redirect" title="Sinusoid">sinusoids</a> with different <a href="/wiki/Frequencies" class="mw-redirect" title="Frequencies">frequencies</a> and <a href="/wiki/Amplitude" title="Amplitude">amplitudes</a>. Like the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (DFT), a DST operates on a function at a finite number of discrete data points. The obvious distinction between a DST and a DFT is that the former uses only <a href="/wiki/Sine_function" class="mw-redirect" title="Sine function">sine functions</a>, while the latter uses both cosines and sines (in the form of <a href="/wiki/Complex_exponential" class="mw-redirect" title="Complex exponential">complex exponentials</a>). However, this visible difference is merely a consequence of a deeper distinction: a DST implies different <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a> than the DFT or other related transforms. </p><p>The Fourier-related transforms that operate on a function over a finite <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>, such as the DFT or DST or a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>, can be thought of as implicitly defining an <i>extension</i> of that function outside the domain. That is, once you write a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> as a sum of sinusoids, you can evaluate that sum at any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, even for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> where the original <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> was not specified. The DFT, like the Fourier series, implies a <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> extension of the original function. A DST, like a <a href="/wiki/Sine_and_cosine_transforms" title="Sine and cosine transforms">sine transform</a>, implies an <a href="/wiki/Even_and_odd_functions" title="Even and odd functions">odd</a> extension of the original function. </p><p>However, because DSTs operate on <i>finite</i>, <i>discrete</i> sequences, two issues arise that do not apply for the continuous sine transform. First, one has to specify whether the function is even or odd at <i>both</i> the left and right boundaries of the domain (i.e. the min-<i>n</i> and max-<i>n</i> boundaries in the definitions below, respectively). Second, one has to specify around <i>what point</i> the function is even or odd. In particular, consider a sequence (<i>a</i>,<i>b</i>,<i>c</i>) of three equally spaced data points, and say that we specify an odd <i>left</i> boundary. There are two sensible possibilities: either the data is odd about the point <i>prior</i> to <i>a</i>, in which case the odd extension is (−<i>c</i>,−<i>b</i>,−<i>a</i>,0,<i>a</i>,<i>b</i>,<i>c</i>), or the data is odd about the point <i>halfway</i> between <i>a</i> and the previous point, in which case the odd extension is (−<i>c</i>,−<i>b</i>,−<i>a</i>,<i>a</i>,<i>b</i>,<i>c</i>) </p><p>These choices lead to all the standard variations of DSTs and also <a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">discrete cosine transforms</a> (DCTs). 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 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2\times 2\times 2=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2\times 2\times 2=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf51a34d0fc10d80d3df826cbeced2bf8065ea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.594ex; height:2.176ex;" alt="{\displaystyle 2\times 2\times 2\times 2=16}"></span> possibilities. Half of these possibilities, those where the <i>left</i> boundary is odd, correspond to the 8 types of DST; the other half are the 8 types of DCT. </p><p>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 <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> by <a href="/wiki/Spectral_method" title="Spectral method">spectral methods</a>, the boundary conditions are directly specified as a part of the problem being solved. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=3" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formally, the discrete sine transform is a <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a>, invertible <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <i>F</i> : <b>R</b><sup><i>N</i></sup> <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">-></span> <b>R</b><sup><i>N</i></sup> (where <b>R</b> denotes the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>), or equivalently an <i>N</i> × <i>N</i> <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a>. There are several variants of the DST with slightly modified definitions. The <i>N</i> real numbers <i>x</i><sub>0</sub>,...,<i>x</i><sub><i>N</i> − 1</sub> are transformed into the <i>N</i> real numbers <i>X</i><sub>0</sub>,...,<i>X</i><sub><i>N</i> − 1</sub> according to one of the formulas: </p> <div class="mw-heading mw-heading3"><h3 id="DST-I">DST-I</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=4" title="Edit section: DST-I"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Discrete_sine_transform.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Discrete_sine_transform.svg/300px-Discrete_sine_transform.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Discrete_sine_transform.svg/450px-Discrete_sine_transform.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Discrete_sine_transform.svg/600px-Discrete_sine_transform.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Discrete sine transform (<a rel="nofollow" class="external free" href="https://www.desmos.com/calculator/r0od93dfgp">https://www.desmos.com/calculator/r0od93dfgp</a>).</figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}X_{k}&=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N+1}}(n+1)(k+1)\right]&k&=0,\dots ,N-1\\X_{k-1}&=\sum _{n=1}^{N}x_{n-1}\sin \left[{\frac {\pi nk}{N+1}}\right]&k&=1,\dots ,N\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi>k</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>n</mi> <mi>k</mi> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi>k</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X_{k}&=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N+1}}(n+1)(k+1)\right]&k&=0,\dots ,N-1\\X_{k-1}&=\sum _{n=1}^{N}x_{n-1}\sin \left[{\frac {\pi nk}{N+1}}\right]&k&=1,\dots ,N\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/219e4eea1bb20a68f1e63c6b282b7f0fac410d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.588ex; margin-bottom: -0.25ex; width:63.651ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}X_{k}&=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N+1}}(n+1)(k+1)\right]&k&=0,\dots ,N-1\\X_{k-1}&=\sum _{n=1}^{N}x_{n-1}\sin \left[{\frac {\pi nk}{N+1}}\right]&k&=1,\dots ,N\end{aligned}}}"></span></dd></dl> <p>The DST-I matrix is <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> (up to a scale factor). </p><p>A DST-I is exactly equivalent to a DFT of a real sequence that is odd around the zero-th and middle points, scaled by 1/2. For example, a DST-I of <i>N</i>=3 real numbers (<i>a</i>,<i>b</i>,<i>c</i>) is exactly equivalent to a DFT of eight real numbers (0,<i>a</i>,<i>b</i>,<i>c</i>,0,−<i>c</i>,−<i>b</i>,−<i>a</i>) (odd symmetry), scaled by 1/2. (In contrast, DST types II–IV involve a half-sample shift in the equivalent DFT.) This is the reason for the <i>N</i> + 1 in the denominator of the sine function: the equivalent DFT has 2(<i>N</i>+1) points and has 2π/2(<i>N</i>+1) in its sinusoid frequency, so the DST-I has π/(<i>N</i>+1) in its frequency. </p><p>Thus, the DST-I corresponds to the boundary conditions: <i>x</i><sub><i>n</i></sub> is odd around <i>n</i> = −1 and odd around <i>n</i>=<i>N</i>; similarly for <i>X</i><sub><i>k</i></sub>. </p> <div class="mw-heading mw-heading3"><h3 id="DST-II">DST-II</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=5" title="Edit section: DST-II"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)(k+1)\right]\quad \quad k=0,\dots ,N-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>N</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)(k+1)\right]\quad \quad k=0,\dots ,N-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec4a97c9d25b6266d25cd0d66623125a8732469" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:60.018ex; height:7.343ex;" alt="{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)(k+1)\right]\quad \quad k=0,\dots ,N-1}"></span> </p><p>Some authors further multiply the <i>X</i><sub><i>N</i> − 1</sub> term by 1/<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> (see below for the corresponding change in DST-III). This makes the DST-II matrix <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> (up to a scale factor), but breaks the direct correspondence with a real-odd DFT of half-shifted input. </p><p>The DST-II implies the boundary conditions: <i>x</i><sub><i>n</i></sub> is odd around <i>n</i> = −1/2 and odd around <i>n</i> = <i>N</i> − 1/2; <i>X</i><sub><i>k</i></sub> is odd around <i>k</i> = −1 and even around <i>k</i> = <i>N</i> − 1. </p> <div class="mw-heading mw-heading3"><h3 id="DST-III">DST-III</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=6" title="Edit section: DST-III"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}={\frac {(-1)^{k}}{2}}x_{N-1}+\sum _{n=0}^{N-2}x_{n}\sin \left[{\frac {\pi }{N}}(n+1)\left(k+{\frac {1}{2}}\right)\right]\quad \quad k=0,\dots ,N-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>N</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}={\frac {(-1)^{k}}{2}}x_{N-1}+\sum _{n=0}^{N-2}x_{n}\sin \left[{\frac {\pi }{N}}(n+1)\left(k+{\frac {1}{2}}\right)\right]\quad \quad k=0,\dots ,N-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf69b9150d88dbb7ad408958aefaa7b3338ca50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:74.298ex; height:7.343ex;" alt="{\displaystyle X_{k}={\frac {(-1)^{k}}{2}}x_{N-1}+\sum _{n=0}^{N-2}x_{n}\sin \left[{\frac {\pi }{N}}(n+1)\left(k+{\frac {1}{2}}\right)\right]\quad \quad k=0,\dots ,N-1}"></span> </p><p>Some authors further multiply the <i>x</i><sub><i>N</i> − 1</sub> term by <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> (see above for the corresponding change in DST-II). This makes the DST-III matrix <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> (up to a scale factor), but breaks the direct correspondence with a real-odd DFT of half-shifted output. </p><p>The DST-III implies the boundary conditions: <i>x</i><sub><i>n</i></sub> is odd around <i>n</i> = −1 and even around <i>n</i> = <i>N</i> − 1; <i>X</i><sub><i>k</i></sub> is odd around <i>k</i> = −1/2 and odd around <i>k</i> = <i>N</i> − 1/2. </p> <div class="mw-heading mw-heading3"><h3 id="DST-IV">DST-IV</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=7" title="Edit section: DST-IV"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)\left(k+{\frac {1}{2}}\right)\right]\quad \quad k=0,\dots ,N-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>N</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)\left(k+{\frac {1}{2}}\right)\right]\quad \quad k=0,\dots ,N-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d240979de077d6ef8b3b352dd1b2da2158fd1c88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.466ex; height:7.343ex;" alt="{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)\left(k+{\frac {1}{2}}\right)\right]\quad \quad k=0,\dots ,N-1}"></span> </p><p>The DST-IV matrix is <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> (up to a scale factor). </p><p>The DST-IV implies the boundary conditions: <i>x</i><sub><i>n</i></sub> is odd around <i>n</i> = −1/2 and even around <i>n</i> = <i>N</i> − 1/2; similarly for <i>X</i><sub><i>k</i></sub>. </p> <div class="mw-heading mw-heading3"><h3 id="DST_V–VIII"><span id="DST_V.E2.80.93VIII"></span>DST V–VIII</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=8" title="Edit section: DST V–VIII"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>DST types I–IV are equivalent to real-odd DFTs of even order. In principle, there are actually four additional types of discrete sine transform (Martucci, 1994), corresponding to real-odd DFTs of logically odd order, which have factors of <i>N</i>+1/2 in the denominators of the sine arguments. However, these variants seem to be rarely used in practice. </p> <div class="mw-heading mw-heading3"><h3 id="Inverse_transforms">Inverse transforms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=9" title="Edit section: Inverse transforms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The inverse of DST-I is DST-I multiplied by 2/(<i>N</i> + 1). The inverse of DST-IV is DST-IV multiplied by 2/<i>N</i>. The inverse of DST-II is DST-III multiplied by 2/<i>N</i> (and vice versa). </p><p>As for the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFT</a>, 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 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {2/N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {2/N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc67f411ae2805d323fa48a91e852adce10c0fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.712ex; height:3.343ex;" alt="{\textstyle {\sqrt {2/N}}}"></span> so that the inverse does not require any additional multiplicative factor. </p> <div class="mw-heading mw-heading2"><h2 id="Computation">Computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=10" title="Edit section: Computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although the direct application of these formulas would require O(<i>N</i><sup>2</sup>) operations, it is possible to compute the same thing with only O(<i>N</i> log <i>N</i>) complexity by factorizing the computation similar to the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT). (One can also compute DSTs via FFTs combined with O(<i>N</i>) pre- and post-processing steps.) </p><p>A DST-III or DST-IV can be computed from a DCT-III or DCT-IV (see <a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">discrete cosine transform</a>), respectively, by reversing the order of the inputs and flipping the sign of every other output, and vice versa for DST-II from DCT-II. In this way it follows that types II–IV of the DST require exactly the same number of arithmetic operations (additions and multiplications) as the corresponding DCT types. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=11" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A family of transforms composed of sine and <a href="/w/index.php?title=Sine_hyperbolic&action=edit&redlink=1" class="new" title="Sine hyperbolic (page does not exist)">sine hyperbolic</a> functions exists; these transforms are made based on the <a href="/wiki/Natural_frequency" title="Natural frequency">natural</a> <a href="/wiki/Vibration" title="Vibration">vibration</a> of thin square plates with different <a href="/wiki/Boundary_conditions" class="mw-redirect" title="Boundary conditions">boundary conditions</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_sine_transform&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBritanakYipRao2010" class="citation book cs1">Britanak, Vladimir; Yip, Patrick C.; <a href="/wiki/K._R._Rao" title="K. R. Rao">Rao, K. R.</a> (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA35"><i>Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations</i></a>. <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a>. pp. 35–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780080464640" title="Special:BookSources/9780080464640"><bdi>9780080464640</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Cosine+and+Sine+Transforms%3A+General+Properties%2C+Fast+Algorithms+and+Integer+Approximations&rft.pages=35-6&rft.pub=Elsevier&rft.date=2010&rft.isbn=9780080464640&rft.aulast=Britanak&rft.aufirst=Vladimir&rft.au=Yip%2C+Patrick+C.&rft.au=Rao%2C+K.+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiRlQHcK-r_kC%26pg%3DPA35&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+sine+transform" class="Z3988"></span></span> </li> <li id="cite_note-pubDCT-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-pubDCT_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhmedNatarajanRao1974" class="citation cs2"><a href="/wiki/N._Ahmed" class="mw-redirect" title="N. Ahmed">Ahmed, Nasir</a>; Natarajan, T.; Rao, K. R. 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A. Martucci, "Symmetric convolution and the discrete sine and cosine transforms," <i>IEEE Trans. Signal Process.</i> <b>SP-42</b>, 1038–1051 (1994).</li> <li>Matteo Frigo and <a href="/wiki/Steven_G._Johnson" title="Steven G. Johnson">Steven G. Johnson</a>: <i>FFTW</i>, <a rel="nofollow" class="external text" href="http://www.fftw.org/">FFTW Home Page</a>. A free (<a href="/wiki/GNU_General_Public_License" title="GNU General Public License">GPL</a>) C library that can compute fast DSTs (types I–IV) in one or more dimensions, of arbitrary size. Also M. Frigo and S. G. Johnson, "<a rel="nofollow" class="external text" href="http://fftw.org/fftw-paper-ieee.pdf">The Design and Implementation of FFTW3</a>," <i>Proceedings of the IEEE</i> <b>93</b> (2), 216–231 (2005).</li> <li>Takuya Ooura: General Purpose FFT Package, <a rel="nofollow" class="external text" href="http://www.kurims.kyoto-u.ac.jp/~ooura/fft.html">FFT Package 1-dim / 2-dim</a>. Free C & FORTRAN libraries for computing fast DSTs in one, two or three dimensions, power of 2 sizes.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=621">"Section 12.4.1. Sine Transform"</a>, <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.), New York: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88068-8" title="Special:BookSources/978-0-521-88068-8"><bdi>978-0-521-88068-8</bdi></a>, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html#pg=621">the original</a> on 2011-08-11<span class="reference-accessdate">, retrieved <span class="nowrap">2011-08-13</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+12.4.1.+Sine+Transform&rft.btitle=Numerical+Recipes%3A+The+Art+of+Scientific+Computing&rft.place=New+York&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-88068-8&rft.aulast=Press&rft.aufirst=WH&rft.au=Teukolsky%2C+SA&rft.au=Vetterling%2C+WT&rft.au=Flannery%2C+BP&rft_id=http%3A%2F%2Fapps.nrbook.com%2Fempanel%2Findex.html%23pg%3D621&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+sine+transform" class="Z3988"></span>.</li> <li>R. Chivukula and Y. Reznik, "<a rel="nofollow" class="external text" href="http://www.reznik.org/papers/ADIP2011_DST67.pdf">Fast Computing of Discrete Cosine and Sine Transforms of Types VI and VII</a>," <i>Proc. SPIE</i> Vol. 8135, 2011.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Discrete_sine_transform&oldid=1234149098">https://en.wikipedia.org/w/index.php?title=Discrete_sine_transform&oldid=1234149098</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Discrete_transforms" title="Category:Discrete transforms">Discrete transforms</a></li><li><a href="/wiki/Category:Fourier_analysis" title="Category:Fourier analysis">Fourier analysis</a></li><li><a href="/wiki/Category:Indian_inventions" title="Category:Indian inventions">Indian inventions</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 12 July 2024, at 20:51<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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