# Calculation of DCT_IV using DCT_III

## Definitions

Result of transform is y=x*T, where y, x are row-vectors T is transform matrix

## DCT_IV matrix definition

N=8;
DCT4=cos(pi/N*[(0:N-1)+1/2]'*[(0:N-1)+1/2])

DCT4 =

0.9952    0.9569    0.8819    0.7730    0.6344    0.4714    0.2903    0.0980
0.9569    0.6344    0.0980   -0.4714   -0.8819   -0.9952   -0.7730   -0.2903
0.8819    0.0980   -0.7730   -0.9569   -0.2903    0.6344    0.9952    0.4714
0.7730   -0.4714   -0.9569    0.0980    0.9952    0.2903   -0.8819   -0.6344
0.6344   -0.8819   -0.2903    0.9952   -0.0980   -0.9569    0.4714    0.7730
0.4714   -0.9952    0.6344    0.2903   -0.9569    0.7730    0.0980   -0.8819
0.2903   -0.7730    0.9952   -0.8819    0.4714    0.0980   -0.6344    0.9569
0.0980   -0.2903    0.4714   -0.6344    0.7730   -0.8819    0.9569   -0.9952



## DCT_III matrix definition

DCT3=cos(pi/N*[[(0:N-1)+1/2]'*(0:N-1)])

DCT3 =

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



## Finding relations

From [1] we know that DCTIV matrix can be expressed in terms of Tschebyshev polynomials

DCTIII matrix can be analogously expressed as

Because there exist relation

we can express DCTIII through DCTIV

where

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)));


## Check expression of DCT_III through DCT_IV

Check DCTIII matrix

inv(D)*DCT4*B

ans =

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



Check computation of DCTIII transform

x=randn(1,N);
y=x*DCT3            % true result
y1=x*inv(D)*DCT4*B  % compute DCTIII using DCTIV transform

y =

3.2983    2.8450   -0.8779   -1.4889    1.2293   -0.1254   -1.0177    2.3680

y1 =

3.2983    2.8450   -0.8779   -1.4889    1.2293   -0.1254   -1.0177    2.3680



## Check expression of DCT_IV through DCT_III

D*DCT3*inv(B)

ans =

0.9952    0.9569    0.8819    0.7730    0.6344    0.4714    0.2903    0.0980
0.9569    0.6344    0.0980   -0.4714   -0.8819   -0.9952   -0.7730   -0.2903
0.8819    0.0980   -0.7730   -0.9569   -0.2903    0.6344    0.9952    0.4714
0.7730   -0.4714   -0.9569    0.0980    0.9952    0.2903   -0.8819   -0.6344
0.6344   -0.8819   -0.2903    0.9952   -0.0980   -0.9569    0.4714    0.7730
0.4714   -0.9952    0.6344    0.2903   -0.9569    0.7730    0.0980   -0.8819
0.2903   -0.7730    0.9952   -0.8819    0.4714    0.0980   -0.6344    0.9569
0.0980   -0.2903    0.4714   -0.6344    0.7730   -0.8819    0.9569   -0.9952



Check computation of DCTIV transform

y=x*DCT4            % true result
y1=x*D*DCT3*inv(B)  % compute DCTIV using DCTIII transform

y =

3.3152    1.2507   -2.0502    0.0787    1.0839   -1.2442    0.6869    2.1754

y1 =

3.3152    1.2507   -2.0502    0.0787    1.0839   -1.2442    0.6869    2.1754



## 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.