# Calculation of DST_IV using DST_III (alternative)

## 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_III matrix definition

DST3=sin(pi/N*[(0:N-1)+1/2]'*[(0:N-1)+1])

DST3 =
0.1951    0.3827    0.5556    0.7071    0.8315    0.9239    0.9808    1.0000
0.5556    0.9239    0.9808    0.7071    0.1951   -0.3827   -0.8315   -1.0000
0.8315    0.9239    0.1951   -0.7071   -0.9808   -0.3827    0.5556    1.0000
0.9808    0.3827   -0.8315   -0.7071    0.5556    0.9239   -0.1951   -1.0000
0.9808   -0.3827   -0.8315    0.7071    0.5556   -0.9239   -0.1951    1.0000
0.8315   -0.9239    0.1951    0.7071   -0.9808    0.3827    0.5556   -1.0000
0.5556   -0.9239    0.9808   -0.7071    0.1951    0.3827   -0.8315    1.0000
0.1951   -0.3827    0.5556   -0.7071    0.8315   -0.9239    0.9808   -1.0000


## Finding relations

We will base our derivation on already existing relations between DCT_IV and DCT_III transforms

where

Applying relations between dual transforms

where

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_III through DST_IV

Check DSTIII matrix

inv(D)*DST4*J*B*J

ans =
0.1951    0.3827    0.5556    0.7071    0.8315    0.9239    0.9808    1.0000
0.5556    0.9239    0.9808    0.7071    0.1951   -0.3827   -0.8315   -1.0000
0.8315    0.9239    0.1951   -0.7071   -0.9808   -0.3827    0.5556    1.0000
0.9808    0.3827   -0.8315   -0.7071    0.5556    0.9239   -0.1951   -1.0000
0.9808   -0.3827   -0.8315    0.7071    0.5556   -0.9239   -0.1951    1.0000
0.8315   -0.9239    0.1951    0.7071   -0.9808    0.3827    0.5556   -1.0000
0.5556   -0.9239    0.9808   -0.7071    0.1951    0.3827   -0.8315    1.0000
0.1951   -0.3827    0.5556   -0.7071    0.8315   -0.9239    0.9808   -1.0000


Check computation of DSTIII transform

x=randn(1,N)
y=x*DST3                % true result
y1=x*inv(D)*DST4*J*B*J  % compute DSTIII using DSTIV transform

x =
1.0668    0.0593   -0.0956   -0.8323    0.2944   -1.3362    0.7143    1.6236
y =
-0.7635   -0.1033    2.4214   -0.9373    3.4933   -1.7795    1.3049    2.4655
y1 =
-0.7635   -0.1033    2.4214   -0.9373    3.4933   -1.7795    1.3049    2.4655


## Check expression of DST_IV through DST_III

D*DST3*J*inv(B)*J

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*D*DST3*J*inv(B)*J  % compute DSTIV using DSTIII transform

y =
0.8971   -2.2108    2.9958   -0.6407    2.1065    0.9202   -1.0888    2.4288
y1 =
0.8971   -2.2108    2.9958   -0.6407    2.1065    0.9202   -1.0888    2.4288


## 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.