Editing Discrete sine transform (section)

{{Short description|Transform in mathematics}}
In [[mathematics]], the '''discrete sine transform (DST)''' is a [[List of Fourier-related transforms|Fourier-related transform]] similar to the [[discrete Fourier transform]] (DFT), but using a purely [[real number|real]] [[matrix (mathematics)|matrix]]. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with [[even and odd functions|odd]] [[symmetry]] (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.

The DST is related to the [[discrete cosine transform]] (DCT), which is equivalent to a DFT of real and ''even'' 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 [[Neumann boundary condition|Neumann condition]] at ''x''=0 with a [[Dirichlet condition]].<ref>{{cite book |last1=Britanak |first1=Vladimir |last2=Yip |first2=Patrick C. |last3=Rao |first3=K. R. |author3-link=K. R. Rao |title=Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations |date=2010 |publisher=[[Elsevier]] |isbn=9780080464640 |pages=35–6 |url=https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA35}}</ref> Both the DCT and the DST were described by [[Nasir Ahmed (engineer)|Nasir Ahmed]], T. Natarajan, and [[K.R. Rao]] in 1974.<ref name="pubDCT">{{Citation |first1=Nasir |last1=Ahmed |author1-link=N. Ahmed |first2=T. |last2=Natarajan |first3=K. R. |last3=Rao |title=Discrete Cosine Transform |journal=IEEE Transactions on Computers |date=January 1974 |volume=C-23 |issue=1 |pages=90–93 |doi=10.1109/T-C.1974.223784 |s2cid=149806273 |url=https://www.ic.tu-berlin.de/fileadmin/fg121/Source-Coding_WS12/selected-readings/Ahmed_et_al.__1974.pdf}}</ref><ref name="Ahmed">{{cite journal |last=Ahmed |first=Nasir |author-link=N. Ahmed |title=How I Came Up With the Discrete Cosine Transform |journal=[[Digital Signal Processing (journal)|Digital Signal Processing]] |date=January 1991 |volume=1 |issue=1 |pages=4–5 |doi=10.1016/1051-2004(91)90086-Z |bibcode=1991DSP.....1....4A |url=https://www.scribd.com/doc/52879771/DCT-History-How-I-Came-Up-with-the-Discrete-Cosine-Transform|url-access=subscription }}</ref> The type-I DST (DST-I) was later described by [[Anil K. Jain (electrical engineer, born 1946)|Anil K. Jain]] in 1976, and the type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.<ref>{{cite journal |last1=Dhamija |first1=Swati |last2=Jain |first2=Priyanka |title=Comparative Analysis for Discrete Sine Transform as a suitable method for noise estimation |journal=International Journal of Computer Science |date=September 2011 |volume=8 |issue=5 |pages=162–164 |url=https://www.researchgate.net/publication/267228857 |access-date=4 November 2019 |via=ResearchGate}}</ref>