Calculation of DST_IV using DST_II (alternative)
Contents
Definitions
Result of transform is y=x*T, where y, x are row-vectors T is transform matrix
DST_IV matrix definition
N=8; DST4=sin(pi/N*[(0:N-1)+1/2]'*[(0:N-1)+1/2])
DST4 =
0.0980 0.2903 0.4714 0.6344 0.7730 0.8819 0.9569 0.9952
0.2903 0.7730 0.9952 0.8819 0.4714 -0.0980 -0.6344 -0.9569
0.4714 0.9952 0.6344 -0.2903 -0.9569 -0.7730 0.0980 0.8819
0.6344 0.8819 -0.2903 -0.9952 -0.0980 0.9569 0.4714 -0.7730
0.7730 0.4714 -0.9569 -0.0980 0.9952 -0.2903 -0.8819 0.6344
0.8819 -0.0980 -0.7730 0.9569 -0.2903 -0.6344 0.9952 -0.4714
0.9569 -0.6344 0.0980 0.4714 -0.8819 0.9952 -0.7730 0.2903
0.9952 -0.9569 0.8819 -0.7730 0.6344 -0.4714 0.2903 -0.0980
DST_II matrix definition
DST2=sin(pi/N*((0:N-1)+1)'*((0:N-1)+1/2))
DST2 =
0.1951 0.5556 0.8315 0.9808 0.9808 0.8315 0.5556 0.1951
0.3827 0.9239 0.9239 0.3827 -0.3827 -0.9239 -0.9239 -0.3827
0.5556 0.9808 0.1951 -0.8315 -0.8315 0.1951 0.9808 0.5556
0.7071 0.7071 -0.7071 -0.7071 0.7071 0.7071 -0.7071 -0.7071
0.8315 0.1951 -0.9808 0.5556 0.5556 -0.9808 0.1951 0.8315
0.9239 -0.3827 -0.3827 0.9239 -0.9239 0.3827 0.3827 -0.9239
0.9808 -0.8315 0.5556 -0.1951 -0.1951 0.5556 -0.8315 0.9808
1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000
Finding relations
We will base our derivation on already existing alternative relations between DST_IV and DST_III transforms
where
Applying relation between DST_III and DST_II matrix
we will get
B=diag(ones(1,N))+diag(ones(1,N-1),1);
B(1,1)=2; B=B/2;
D=diag(cos(pi/2/N*([0:N-1]+1/2)));
% K=diag((-1).^(0:N-1));
J=rot90(eye(N));
Check expression of DST_II through DST_IV
Check DSTII matrix
J*B'*J*DST4*inv(D)
ans =
0.1951 0.5556 0.8315 0.9808 0.9808 0.8315 0.5556 0.1951
0.3827 0.9239 0.9239 0.3827 -0.3827 -0.9239 -0.9239 -0.3827
0.5556 0.9808 0.1951 -0.8315 -0.8315 0.1951 0.9808 0.5556
0.7071 0.7071 -0.7071 -0.7071 0.7071 0.7071 -0.7071 -0.7071
0.8315 0.1951 -0.9808 0.5556 0.5556 -0.9808 0.1951 0.8315
0.9239 -0.3827 -0.3827 0.9239 -0.9239 0.3827 0.3827 -0.9239
0.9808 -0.8315 0.5556 -0.1951 -0.1951 0.5556 -0.8315 0.9808
1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000
Check computation of DSTII transform
x=randn(1,N) y=x*DST2 % true result y1=x*J*B'*J*DST4*inv(D) % compute DSTII using DSTIV transform
x = -1.6041 0.2573 -1.0565 1.4151 -0.8051 0.5287 0.2193 -0.9219 y = -0.6885 -0.3088 -2.5156 -0.6767 -1.6931 1.2588 -4.2247 -2.0199 y1 = -0.6885 -0.3088 -2.5156 -0.6767 -1.6931 1.2588 -4.2247 -2.0199
Check expression of DST_IV through DST_II
J*inv(B')*J*DST2*D
ans =
0.0980 0.2903 0.4714 0.6344 0.7730 0.8819 0.9569 0.9952
0.2903 0.7730 0.9952 0.8819 0.4714 -0.0980 -0.6344 -0.9569
0.4714 0.9952 0.6344 -0.2903 -0.9569 -0.7730 0.0980 0.8819
0.6344 0.8819 -0.2903 -0.9952 -0.0980 0.9569 0.4714 -0.7730
0.7730 0.4714 -0.9569 -0.0980 0.9952 -0.2903 -0.8819 0.6344
0.8819 -0.0980 -0.7730 0.9569 -0.2903 -0.6344 0.9952 -0.4714
0.9569 -0.6344 0.0980 0.4714 -0.8819 0.9952 -0.7730 0.2903
0.9952 -0.9569 0.8819 -0.7730 0.6344 -0.4714 0.2903 -0.0980
Check computation of DSTIV transform
y=x*DST4 % true result y1=x*J*inv(B')*J*DST2*D % compute DSTIV using DSTII transform
y = -0.5464 0.2416 -2.0110 -0.4914 -1.9794 1.2821 -0.3356 -4.4742 y1 = -0.5464 0.2416 -2.0110 -0.4914 -1.9794 1.2821 -0.3356 -4.4742
Reference
[1] Markus Pueschel, Jose M.F. Moura. The Algebraic Approach to the Discrete Cosine and Sine Transforms and their Fast Algorithms SIAM Journal of Computing 2003, Vol. 32, No. 5, pp. 1280-1316.