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<HTML><HEAD><TITLE>Newmat09 - fast trig transforms</TITLE></HEAD><BODY><H2>Fast trigonometric transforms</H2><A HREF="nric.html"> next</A> - <A HREF="nric.html"> skip</A> - <A HREF="refer.html"> up</A> - <A HREF="index.html"> start</A><P>These are the sin and cosine transforms as defined by Charles Van Loan(1992) in <I>Computational frameworks for the fast Fourier transform</I>published by SIAM. See page 229. Some other authors use slightlydifferent conventions.All the functions call the <A HREF="fft.html">fast Fourier transforms</A>and require an eventransform length, denoted by <I>m</I> in these notes. As with the FFT<I>m</I> should be the product of numbers less than about 10 for fastexecution. <P>The functions I define are <PRE> DCT(U,V); // U, V are ColumnVectors, length <I>m+1</I> DCT_inverse(U,V); // inverse of DCT DST(U,V); // U, V are ColumnVectors, length <I>m+1</I> DST_inverse(U,V); // inverse of DST DCT_II(U,V); // U, V are ColumnVectors, length <I>m</I> DCT_II_inverse(U,V); // inverse of DCT_II DST_II(U,V); // U, V are ColumnVectors, length <I>m</I> DST_II_inverse(U,V); // inverse of DST_II</PRE>where <TT>U</TT> is the input and <TT>V</TT> is the output. <TT>V = U</TT> is OK.The length of <TT>V</TT> is set by the functions.<P>Here are the formulae:<P><H3>DCT</H3><PRE> m-1 k v[k] = u[0]/2 + SUM { u[j] cos (pi jk/m) } + (-) u[m]/2 j=1</PRE>for <TT>k = 0...m</TT>, where <TT>u[j]</TT> and <TT>v[k]</TT> are stored in<TT>U(j+1)</TT> and <TT>V(k+1)</TT>.<P><H3>DST</H3><PRE> m-1 v[k] = SUM { u[j] sin (pi jk/m) } j=1</PRE>for <TT>k = 1...(m-1)</TT>, where <TT>u[j]</TT> and <TT>v[k]</TT> are storedin <TT>U(j+1)</TT> and <TT>V(k+1)</TT>.<TT>u[0]</TT> and <TT>u[m]</TT> are ignored and <TT>v[0]</TT> and <TT>v[m]</TT>are set to zero.<P><H3>DCT_II</H3><PRE> m-1 v[k] = SUM { u[j] cos (pi (2j+1)k/m) } j=0</PRE>for <TT>k = 0...(m-1)</TT>, where <TT>u[j]</TT> and <TT>v[k]</TT> are storedin <TT>U(j+1)</TT> and <TT>V(k+1)</TT>.<P><H3>DST_II</H3><PRE> m v[k] = SUM { u[j] sin (pi (2j-1)k/m) } j=1</PRE>for <TT>k = 1...m</TT>, where <TT>u[j]</TT> and <TT>v[k]</TT> are stored in<TT>U(j)</TT> and <TT>V(k)</TT>.<P>Note that the relationship between the subscripts in the formulae and thoseused in <I>newmat</I> is different for DST_II (and DST_II_inverse).<P><A HREF="nric.html"> next</A> - <A HREF="nric.html"> skip</A> - <A HREF="refer.html"> up</A> - <A HREF="index.html"> start</A><P></BODY></HTML>⌨️ 快捷键说明
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