Relations between DCTV and DSTV

Contents

Definitions

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

DCTV matrix definition

$${\bf DCTV}={\left\{{\cos\left({{{\pi}\over{n-{{1}\over{2}}}}kl}\right)}\right\}}_{k,l}$$

N1=9; N=N1;
k=0:N1-1;  l=0:N1-1;
DCT5=cos(pi/(N-1/2)*k'*l)       % display DCTV matrix

DCT5 =
  Columns 1 through 7
    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
    1.0000    0.9325    0.7390    0.4457    0.0923   -0.2737   -0.6026
    1.0000    0.7390    0.0923   -0.6026   -0.9830   -0.8502   -0.2737
    1.0000    0.4457   -0.6026   -0.9830   -0.2737    0.7390    0.9325
    1.0000    0.0923   -0.9830   -0.2737    0.9325    0.4457   -0.8502
    1.0000   -0.2737   -0.8502    0.7390    0.4457   -0.9830    0.0923
    1.0000   -0.6026   -0.2737    0.9325   -0.8502    0.0923    0.7390
    1.0000   -0.8502    0.4457    0.0923   -0.6026    0.9325   -0.9830
    1.0000   -0.9830    0.9325   -0.8502    0.7390   -0.6026    0.4457
  Columns 8 through 9
    1.0000    1.0000
   -0.8502   -0.9830
    0.4457    0.9325
    0.0923   -0.8502
   -0.6026    0.7390
    0.9325   -0.6026
   -0.9830    0.4457
    0.7390   -0.2737
   -0.2737    0.0923

DCTV in terms of Tschebyshev polynomials

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

$${\mathbf {DCTV}} = \left[ {T_l \left( {\alpha _k } \right)} \right]_{k,l} $$

where

$${\rm \alpha}_{k}=\cos\left({{{\left({k+1}\right){\rm \pi}}\over{N-{{1}\over{2}}}}}\right),k=-1\ldots N-2$$

are roots of polynomial

$$U_{DCTV}=\left({x-1}\right)W_{n-1}$$

alpha=sort([roots(TschebyshevW(N-1)); 1],'descend');
DCT5t=zeros(N1);
for l=0:N1-1,
    DCT5t(:,l+1)=polyval(TschebyshevT(l),alpha)';
end
Da5=diag(cos(0*pi/(N-1/2)*k));
DCT5t=Da5*DCT5t                    % display DCTV matrix

% compare DCT5 and DCT5t matrices (show that both definitions above are equivalent)
max(max(abs(DCT5-DCT5t)))

DCT5t =
  Columns 1 through 7
    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
    1.0000    0.9325    0.7390    0.4457    0.0923   -0.2737   -0.6026
    1.0000    0.7390    0.0923   -0.6026   -0.9830   -0.8502   -0.2737
    1.0000    0.4457   -0.6026   -0.9830   -0.2737    0.7390    0.9325
    1.0000    0.0923   -0.9830   -0.2737    0.9325    0.4457   -0.8502
    1.0000   -0.2737   -0.8502    0.7390    0.4457   -0.9830    0.0923
    1.0000   -0.6026   -0.2737    0.9325   -0.8502    0.0923    0.7390
    1.0000   -0.8502    0.4457    0.0923   -0.6026    0.9325   -0.9830
    1.0000   -0.9830    0.9325   -0.8502    0.7390   -0.6026    0.4457
  Columns 8 through 9
    1.0000    1.0000
   -0.8502   -0.9830
    0.4457    0.9325
    0.0923   -0.8502
   -0.6026    0.7390
    0.9325   -0.6026
   -0.9830    0.4457
    0.7390   -0.2737
   -0.2737    0.0923
ans =
   2.5746e-13

DSTV matrix definition

$$ {\mathbf{DSTV}}\mathrm{{=}}\mathrm{\left\{{\cos\left({\frac{\mathrm{\pi}}{{n}{+}\frac{1}{2}}\left({{k}{+}{1}}\right)\left({{l}{+}{1}}\right)}\right)}\right\}} $$

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]

$$ {\mathbf{DSTV}}\mathrm{{=}}{\mathbf{D}}_{DSTV}\mathrm{\cdot}{\mathrm{\left[{{U}_{l}\left({{\mathit{\beta}}_{k}}\right)}\right]}}_{k\mathrm{,}l} $$

where

$${\rm \beta}_{k}=\cos\left({{{\left({k+1}\right){\rm \pi}}\over{N+{{1}\over{2}}}}}\right),k=0\ldots N-1 $$

are roots of polynomial

$${U}_{DCTVI}\mathrm{{=}}{W}_{n}$$

$${\mathbf{D}}_{DSTV}\mathrm{{=}}{diag}{\mathrm{\left\{{\sin\left({\left({{k}{+}{1}}\right)\frac{\mathrm{\pi}}{\left({{N}{+}\frac{1}{2}}\right)}}\right)}\right\}}}_{k}$$

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

$${T}_{l}\mathrm{{=}}\frac{{U}_{l}\mathrm{{+}}{U}_{{l}\mathrm{{-}}{2}}}{2}$$

and

$$ \begin{array}{l} {{\mathrm{\alpha}}_{k}\mathrm{{=}}{\mathrm{\beta}}_{k}{\mathrm{,}}{k}\mathrm{{=}}{0}\mathrm{\ldots}{N}\mathrm{{-}}{1}}\\ {{\mathrm{\alpha}}_{N}\mathrm{{=}}{1}} \end{array} $$

we can express DCTV through DSTV

$$ \begin{array}{l} {{\mathbf{DCTV}}\mathrm{{=}}{\left[{{T}_{l}\left({{\mathit{\alpha}}_{k}}\right)}\right]}_{k\mathrm{,}l}\mathrm{{=}}{\left[{\begin{array}{cc}{1}&{}\\{}&{{U}_{l}\left({{\mathit{\beta}}_{k}}\right)}\end{array}}\right]}_{k\mathrm{,}l}\mathit{\cdot}{\mathbf{B}}\mathrm{{=}}\mathrm{\left[{\begin{array}{cc}{1}&{}\\{}&{{\left({{\mathbf{D}}_{DSTV}}\right)}^{{-}{1}}\mathit{\cdot}{\mathbf{D}}_{DSTV}\mathit{\cdot}{\left[{{U}_{l}\left({{\mathit{\beta}}_{k}}\right)}\right]}_{k,l}}\end{array}}\right]}\mathit{\cdot}{\mathbf{B}}\mathrm{{=}}}\\ {\mathrm{{=}}\mathrm{\left[{\begin{array}{cc}{1}&{}\\{}&{{\left({{\mathbf{D}}_{DSTV}}\right)}^{{-}{1}}\cdot{\mathbf{DSTV}}}\end{array}}\right]}\mathrm{\cdot}{\mathbf{B}}} \end{array}$$

where

$${\mathbf{B}}\mathrm{{=}}\frac{1}{2}\left[{\begin{array}{cccccc}{2}&{2}&{2}&{\mathrm{\cdots}}&{2}&{2}\\{2}&{}&{\mathrm{{-}}{1}}&{}&{}&{}\\{}&{1}&{}&{\mathrm{\ddots}}&{}&{}\\{}&{}&{1}&{\mathrm{\ddots}}&{\mathrm{{-}}{1}}&{}\\{}&{}&{}&{\mathrm{\ddots}}&{}&{\mathrm{{-}}{1}}\\{}&{}&{}&{}&{1}&{\mathrm{{-}}{1}}\end{array}}\right]$$

B=toeplitz([0.5 zeros(1,N-1)]',[0.5           0        -0.5 zeros(1,N-length([0.5           0        -0.5]))]);
B(1,1)=1;
B=[B [0; B(1:end-1,end)]];
B(8,9)=-0.5;
B=[ones(1,size(B,2)); B];

Check expression of DCTV through DSTV

$$ {\mathbf{DCTV}}\mathrm{{=}}\mathrm{\left[{\begin{array}{cc}{1}&{}\\{}&{{\left({{\mathbf{D}}_{DSTV}}\right)}^{{-}{1}}\cdot{\mathbf{DSTV}}}\end{array}}\right]}\mathrm{\cdot}{\mathbf{B}} $$

DCT5a=Da5*blkdiag(1,inv(Db5)*DST5)*B
% compare DCT5 and DCT5a matrices (show correctness of representation of DCTV through DSTV)
max(max(abs(DCT5-DCT5a)))

DCT5a =
  Columns 1 through 7
    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
    1.0000    0.9325    0.7390    0.4457    0.0923   -0.2737   -0.6026
    1.0000    0.7390    0.0923   -0.6026   -0.9830   -0.8502   -0.2737
    1.0000    0.4457   -0.6026   -0.9830   -0.2737    0.7390    0.9325
    1.0000    0.0923   -0.9830   -0.2737    0.9325    0.4457   -0.8502
    1.0000   -0.2737   -0.8502    0.7390    0.4457   -0.9830    0.0923
    1.0000   -0.6026   -0.2737    0.9325   -0.8502    0.0923    0.7390
    1.0000   -0.8502    0.4457    0.0923   -0.6026    0.9325   -0.9830
    1.0000   -0.9830    0.9325   -0.8502    0.7390   -0.6026    0.4457
  Columns 8 through 9
    1.0000    1.0000
   -0.8502   -0.9830
    0.4457    0.9325
    0.0923   -0.8502
   -0.6026    0.7390
    0.9325   -0.6026
   -0.9830    0.4457
    0.7390   -0.2737
   -0.2737    0.0923
ans =
   1.9429e-15

Check expression of DSTV through DCTV

$$\left[{\begin{array}{cc}{1}&{}\\{}&{\mathbf{DSTV}}\end{array}}\right]\mathrm{{=}}\mathrm{\left[{\begin{array}{cc}{1}&{}\\{}&{{\mathbf{D}}_{DSTV}}\end{array}}\right]}\mathrm{\cdot}{\mathbf{DCTV}}\mathrm{\cdot}{\mathbf{B}}^{\mathrm{{-}}{1}}$$

DST5a=blkdiag(1,Db5)*inv(Da5)*DCT5*inv(B);
DST5a=DST5a(2:end,2:end)
% compare DST5 and DST5a matrices (show correctness of representation of DSTV through DCTV)
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 =
   1.5543e-15

Check computation of DCTV transform

x=randn(N1,1);
disp('x''=');disp(x');
y=DCT5*x;                      % true result
disp('y''=');disp(y');
y1=Da5*blkdiag(1,inv(Db5)*DST5)*B*x;            % compute DCTV using DSTV transform
disp('y1''=');disp(y1');

x'=
  Columns 1 through 7
   -0.1241    1.4897    1.4090    1.4172    0.6715   -1.2075    0.7172
  Columns 8 through 9
    1.6302    0.4889
y'=
  Columns 1 through 7
    6.4922    1.0315    1.6055   -2.3749   -2.9026    2.0955   -1.6228
  Columns 8 through 9
   -1.7966   -0.3369
y1'=
  Columns 1 through 7
    6.4922    1.0315    1.6055   -2.3749   -2.9026    2.0955   -1.6228
  Columns 8 through 9
   -1.7966   -0.3369

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(Da5)*DCT5*inv(B)*[0;x];            % compute DSTV using DCTV transform
disp('y1''=');disp(y1');

x'=
  Columns 1 through 7
    1.0347    0.7269   -0.3034    0.2939   -0.7873    0.8884   -1.1471
  Column 8
   -1.0689
y'=
  Columns 1 through 7
    0.0359    2.2056    0.4258    2.7480   -1.2166    2.3001   -1.0191
  Column 8
   -1.8927
y1'=
  Columns 1 through 7
   -0.0000    0.0359    2.2056    0.4258    2.7480   -1.2166    2.3001
  Columns 8 through 9
   -1.0191   -1.8927

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.


Published with MATLAB® 7.14