Relations between DCTVI and DSTV

Definitions

Transform matrix is defined for operating on column-vectors y=T*x, where y, x are column-vectors, T is transform matrix

DCTVI matrix definition

N1=9; N=N1;
k=0:N1-1;  l=0:N1-1;
DCT6=cos(pi/(N-1/2)*k'*(l+1/2))       % display DCTV matrix

DCT6 =
Columns 1 through 7
1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
0.9830    0.8502    0.6026    0.2737   -0.0923   -0.4457   -0.7390
0.9325    0.4457   -0.2737   -0.8502   -0.9830   -0.6026    0.0923
0.8502   -0.0923   -0.9325   -0.7390    0.2737    0.9830    0.6026
0.7390   -0.6026   -0.8502    0.4457    0.9325   -0.2737   -0.9830
0.6026   -0.9325   -0.0923    0.9830   -0.4457   -0.7390    0.8502
0.4457   -0.9830    0.7390    0.0923   -0.8502    0.9325   -0.2737
0.2737   -0.7390    0.9830   -0.9325    0.6026   -0.0923   -0.4457
0.0923   -0.2737    0.4457   -0.6026    0.7390   -0.8502    0.9325
Columns 8 through 9
1.0000    1.0000
-0.9325   -1.0000
0.7390    1.0000
-0.4457   -1.0000
0.0923    1.0000
0.2737   -1.0000
-0.6026    1.0000
0.8502   -1.0000
-0.9830    1.0000


DCTVI in terms of Tschebyshev polynomials

The DCTVI matrix can be expressed in terms of Tschebyshev polynomials [1]

where

are roots of polynomial

alpha=sort([roots(TschebyshevW(N-1)); 1],'descend');
DCT6t=zeros(N1);
for l=0:N1-1,
DCT6t(:,l+1)=polyval(TschebyshevV(l),alpha)';
end
Da6=diag(cos(1/2*pi/(N-1/2)*k));
DCT6t=Da6*DCT6t                    % display DCTV matrix

% compare DCT6 and DCT6t matrices (show that both definitions above are equivalent)
max(max(abs(DCT6-DCT6t)))

DCT6t =
Columns 1 through 7
1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
0.9830    0.8502    0.6026    0.2737   -0.0923   -0.4457   -0.7390
0.9325    0.4457   -0.2737   -0.8502   -0.9830   -0.6026    0.0923
0.8502   -0.0923   -0.9325   -0.7390    0.2737    0.9830    0.6026
0.7390   -0.6026   -0.8502    0.4457    0.9325   -0.2737   -0.9830
0.6026   -0.9325   -0.0923    0.9830   -0.4457   -0.7390    0.8502
0.4457   -0.9830    0.7390    0.0923   -0.8502    0.9325   -0.2737
0.2737   -0.7390    0.9830   -0.9325    0.6026   -0.0923   -0.4457
0.0923   -0.2737    0.4457   -0.6026    0.7390   -0.8502    0.9325
Columns 8 through 9
1.0000    1.0000
-0.9325   -1.0000
0.7390    1.0000
-0.4457   -1.0000
0.0923    1.0000
0.2737   -1.0000
-0.6026    1.0000
0.8502   -1.0000
-0.9830    1.0000
ans =
1.6165e-13


DSTV matrix definition

N2=8; N=N2;
k=0:N2-1;  l=0:N2-1;
DST5=sin(pi/(N+1/2)*(k+1)'*(l+1))       % display DCTV matrix

DST5 =
Columns 1 through 7
0.3612    0.6737    0.8952    0.9957    0.9618    0.7980    0.5264
0.6737    0.9957    0.7980    0.1837   -0.5264   -0.9618   -0.8952
0.8952    0.7980   -0.1837   -0.9618   -0.6737    0.3612    0.9957
0.9957    0.1837   -0.9618   -0.3612    0.8952    0.5264   -0.7980
0.9618   -0.5264   -0.6737    0.8952    0.1837   -0.9957    0.3612
0.7980   -0.9618    0.3612    0.5264   -0.9957    0.6737    0.1837
0.5264   -0.8952    0.9957   -0.7980    0.3612    0.1837   -0.6737
0.1837   -0.3612    0.5264   -0.6737    0.7980   -0.8952    0.9618
Column 8
0.1837
-0.3612
0.5264
-0.6737
0.7980
-0.8952
0.9618
-0.9957


DSTV in terms of Tschebyshev polynomials

The DSTV matrix can be expressed in terms of Tschebyshev polynomials [1]

where

are roots of polynomial

beta=sort([roots(TschebyshevW(N))],'descend');
DST5t=zeros(N);
for l=0:N2-1,
DST5t(:,l+1)=polyval(TschebyshevU(l),beta)';
end
Db5=diag(sin(pi/(N+1/2)*(k+1)));
DST5t=Db5*DST5t                    % display DCTV matrix

% compare DST5 and DST5t matrices (show that both definitions above are equivalent)
max(max(abs(DST5-DST5t)))

DST5t =
Columns 1 through 7
0.3612    0.6737    0.8952    0.9957    0.9618    0.7980    0.5264
0.6737    0.9957    0.7980    0.1837   -0.5264   -0.9618   -0.8952
0.8952    0.7980   -0.1837   -0.9618   -0.6737    0.3612    0.9957
0.9957    0.1837   -0.9618   -0.3612    0.8952    0.5264   -0.7980
0.9618   -0.5264   -0.6737    0.8952    0.1837   -0.9957    0.3612
0.7980   -0.9618    0.3612    0.5264   -0.9957    0.6737    0.1837
0.5264   -0.8952    0.9957   -0.7980    0.3612    0.1837   -0.6737
0.1837   -0.3612    0.5264   -0.6737    0.7980   -0.8952    0.9618
Column 8
0.1837
-0.3612
0.5264
-0.6737
0.7980
-0.8952
0.9618
-0.9957
ans =
1.4100e-13


Finding relations

Because there exist relation

and

we can express DCTVI through DSTV

where

B=toeplitz([1 zeros(1,N-1)]',[1 -1 zeros(1,N-length([1 -1]))]);
B=[B [0; B(1:end-1,end)]];
B(8,9)=-2;
B=[ones(1,size(B,2)); B];


Check expression of DCTVI through DSTV

DCT6a=Da6*blkdiag(1,inv(Db5)*DST5)*B
% compare DCT6 and DCT6a matrices (show correctness of representation of DCTVI through DSTV)
max(max(abs(DCT6-DCT6a)))

DCT6a =
Columns 1 through 7
1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
0.9830    0.8502    0.6026    0.2737   -0.0923   -0.4457   -0.7390
0.9325    0.4457   -0.2737   -0.8502   -0.9830   -0.6026    0.0923
0.8502   -0.0923   -0.9325   -0.7390    0.2737    0.9830    0.6026
0.7390   -0.6026   -0.8502    0.4457    0.9325   -0.2737   -0.9830
0.6026   -0.9325   -0.0923    0.9830   -0.4457   -0.7390    0.8502
0.4457   -0.9830    0.7390    0.0923   -0.8502    0.9325   -0.2737
0.2737   -0.7390    0.9830   -0.9325    0.6026   -0.0923   -0.4457
0.0923   -0.2737    0.4457   -0.6026    0.7390   -0.8502    0.9325
Columns 8 through 9
1.0000    1.0000
-0.9325   -1.0000
0.7390    1.0000
-0.4457   -1.0000
0.0923    1.0000
0.2737   -1.0000
-0.6026    1.0000
0.8502   -1.0000
-0.9830    1.0000
ans =
1.7208e-15


Check expression of DSTV through DCTVI

DST5a=blkdiag(1,Db5)*inv(Da6)*DCT6*inv(B);
DST5a=DST5a(2:end,2:end)
% compare DST5 and DST5a matrices (show correctness of representation of DSTV through DCTVI)
max(max(abs(DST5-DST5a)))

DST5a =
Columns 1 through 7
0.3612    0.6737    0.8952    0.9957    0.9618    0.7980    0.5264
0.6737    0.9957    0.7980    0.1837   -0.5264   -0.9618   -0.8952
0.8952    0.7980   -0.1837   -0.9618   -0.6737    0.3612    0.9957
0.9957    0.1837   -0.9618   -0.3612    0.8952    0.5264   -0.7980
0.9618   -0.5264   -0.6737    0.8952    0.1837   -0.9957    0.3612
0.7980   -0.9618    0.3612    0.5264   -0.9957    0.6737    0.1837
0.5264   -0.8952    0.9957   -0.7980    0.3612    0.1837   -0.6737
0.1837   -0.3612    0.5264   -0.6737    0.7980   -0.8952    0.9618
Column 8
0.1837
-0.3612
0.5264
-0.6737
0.7980
-0.8952
0.9618
-0.9957
ans =
1.9984e-15


Check computation of DCTVI transform

x=randn(N1,1);
disp('x''=');disp(x');
y=DCT6*x;                      % true result
disp('y''=');disp(y');
y1=Da6*blkdiag(1,inv(Db5)*DST5)*B*x;            % compute DCTVI using DSTV transform
disp('y1''=');disp(y1');

x'=
Columns 1 through 7
-0.8637    0.0774   -1.2141   -1.1135   -0.0068    1.5326   -0.7697
Columns 8 through 9
0.3714   -0.2256
y'=
Columns 1 through 7
-2.2120   -2.0540   -0.4309    2.3145   -0.0095   -3.0319   -0.2648
Columns 8 through 9
0.2902   -2.5875
y1'=
Columns 1 through 7
-2.2120   -2.0540   -0.4309    2.3145   -0.0095   -3.0319   -0.2648
Columns 8 through 9
0.2902   -2.5875


Check computation of DSTV transform

x=randn(N2,1);
disp('x''=');disp(x');
y=DST5*x;                      % true result
disp('y''=');disp(y');
y1=blkdiag(1,Db5)*inv(Da6)*DCT6*inv(B)*[0;x];            % compute DSTV using DCTVI transform
disp('y1''=');disp(y1');
% Reference

x'=
Columns 1 through 7
1.1174   -1.0891    0.0326    0.5525    1.1006    1.5442    0.0859
Column 8
-1.4916
y'=
Columns 1 through 7
2.3113   -1.8069   -1.2896    3.4160   -0.3740    3.5372    0.3434
Column 8
1.3075
y1'=
Columns 1 through 7
0.0000    2.3113   -1.8069   -1.2896    3.4160   -0.3740    3.5372
Columns 8 through 9
0.3434    1.3075