Relations between DSTVIII and DSTVII

Contents

Definitions

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

DSTVIII matrix definition

$${\mathbf{DSTVII}}{\mathbf{I}}\mathrm{{=}}{\left\{{\sin\left({\frac{\mathit{\pi}}{{n}\mathrm{{-}}\frac{1}{2}}\left({{k}\mathrm{{+}}\frac{1}{2}}\right)\left({{l}\mathrm{{+}}\frac{1}{2}}\right)}\right)}\right\}}_{k\mathrm{,}l}$$

N1=9; N=N1;
k=0:N1-1;  l=0:N1-1;
DST8=sin(pi/(N-1/2)*(k+1/2)'*(l+1/2))       % display DCTV matrix

DST8 =
  Columns 1 through 7
    0.0923    0.2737    0.4457    0.6026    0.7390    0.8502    0.9325
    0.2737    0.7390    0.9830    0.9325    0.6026    0.0923   -0.4457
    0.4457    0.9830    0.7390   -0.0923   -0.8502   -0.9325   -0.2737
    0.6026    0.9325   -0.0923   -0.9830   -0.4457    0.7390    0.8502
    0.7390    0.6026   -0.8502   -0.4457    0.9325    0.2737   -0.9830
    0.8502    0.0923   -0.9325    0.7390    0.2737   -0.9830    0.6026
    0.9325   -0.4457   -0.2737    0.8502   -0.9830    0.6026    0.0923
    0.9830   -0.8502    0.6026   -0.2737   -0.0923    0.4457   -0.7390
    1.0000   -1.0000    1.0000   -1.0000    1.0000   -1.0000    1.0000
  Columns 8 through 9
    0.9830    1.0000
   -0.8502   -1.0000
    0.6026    1.0000
   -0.2737   -1.0000
   -0.0923    1.0000
    0.4457   -1.0000
   -0.7390    1.0000
    0.9325   -1.0000
   -1.0000    1.0000

DSTVIII in terms of Tschebyshev polynomials

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

$${\mathbf{DSTVIII}}\mathrm{{=}}{\mathbf{D}}_{DSTVIII}\mathrm{\cdot}{\mathrm{\left[{{W}_{l}\left({{\mathrm{\alpha}}_{k}}\right)}\right]}}_{k\mathrm{,}l}$$

where

$${\mathrm{\alpha}}_{k}\mathrm{{=}}\cos\left({\frac{\left({{k}\mathrm{{+}}\frac{1}{2}}\right)\mathit{\pi}}{{N}\mathrm{{-}}\frac{1}{2}}}\right){\mathrm{,}}{k}\mathrm{{=}}{0}\mathrm{\ldots}{N}\mathrm{{-}}{1}$$

are roots of polynomial

$${U}_{DSTVIII}\mathrm{{=}}\mathrm{\left({{x}{+}{1}}\right)}{V}_{{n}\mathrm{{-}}{1}}$$

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

alpha=sort([roots(TschebyshevV(N-1)); -1],'descend');
DST8t=zeros(N1);
for l=0:N1-1,
    DST8t(:,l+1)=polyval(TschebyshevW(l),alpha)';
end
Da8=diag(sin(1/2*pi/(N-1/2)*(k+1/2)));
DST8t=Da8*DST8t                    % display DCTV matrix

% compare DST8 and DST8t matrices (show that both definitions above are equivalent)
max(max(abs(DST8-DST8t)))

DST8t =
  Columns 1 through 7
    0.0923    0.2737    0.4457    0.6026    0.7390    0.8502    0.9325
    0.2737    0.7390    0.9830    0.9325    0.6026    0.0923   -0.4457
    0.4457    0.9830    0.7390   -0.0923   -0.8502   -0.9325   -0.2737
    0.6026    0.9325   -0.0923   -0.9830   -0.4457    0.7390    0.8502
    0.7390    0.6026   -0.8502   -0.4457    0.9325    0.2737   -0.9830
    0.8502    0.0923   -0.9325    0.7390    0.2737   -0.9830    0.6026
    0.9325   -0.4457   -0.2737    0.8502   -0.9830    0.6026    0.0923
    0.9830   -0.8502    0.6026   -0.2737   -0.0923    0.4457   -0.7390
    1.0000   -1.0000    1.0000   -1.0000    1.0000   -1.0000    1.0000
  Columns 8 through 9
    0.9830    1.0000
   -0.8502   -1.0000
    0.6026    1.0000
   -0.2737   -1.0000
   -0.0923    1.0000
    0.4457   -1.0000
   -0.7390    1.0000
    0.9325   -1.0000
   -1.0000    1.0000
ans =
   1.6198e-13

DSTVII matrix definition

$${\mathbf{DSTVII}}\mathrm{{=}}{\left\{{\sin\left({\frac{\mathit{\pi}}{{n}\mathrm{{+}}\frac{1}{2}}\left({{k}\mathrm{{+}}\frac{1}{2}}\right)\left({{l}\mathrm{{+}}{1}}\right)}\right)}\right\}}_{k\mathrm{,}l}$$

N2=8; N=N2;
k=0:N2-1;  l=0:N2-1;
DST7=sin(pi/(N+1/2)*(k+1/2)'*(l+1))       % display DCTV matrix

DST7 =
  Columns 1 through 7
    0.1837    0.3612    0.5264    0.6737    0.7980    0.8952    0.9618
    0.5264    0.8952    0.9957    0.7980    0.3612   -0.1837   -0.6737
    0.7980    0.9618    0.3612   -0.5264   -0.9957   -0.6737    0.1837
    0.9618    0.5264   -0.6737   -0.8952    0.1837    0.9957    0.3612
    0.9957   -0.1837   -0.9618    0.3612    0.8952   -0.5264   -0.7980
    0.8952   -0.7980   -0.1837    0.9618   -0.6737   -0.3612    0.9957
    0.6737   -0.9957    0.7980   -0.1837   -0.5264    0.9618   -0.8952
    0.3612   -0.6737    0.8952   -0.9957    0.9618   -0.7980    0.5264
  Column 8
    0.9957
   -0.9618
    0.8952
   -0.7980
    0.6737
   -0.5264
    0.3612
   -0.1837

DSTVII in terms of Tschebyshev polynomials

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

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

where

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

are roots of polynomial

$${U}_{DSTVII}\mathrm{{=}}{V}_{n}$$

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

beta=sort([roots(TschebyshevV(N))],'descend');
DST7t=zeros(N);
for l=0:N2-1,
    DST7t(:,l+1)=polyval(TschebyshevU(l),beta)';
end
Db7=diag(sin(pi/(N+1/2)*(k+1/2)));
DST7t=Db7*DST7t                    % display DCTV matrix

% compare DST7 and DST7t matrices (show that both definitions above are equivalent)
max(max(abs(DST7-DST7t)))

DST7t =
  Columns 1 through 7
    0.1837    0.3612    0.5264    0.6737    0.7980    0.8952    0.9618
    0.5264    0.8952    0.9957    0.7980    0.3612   -0.1837   -0.6737
    0.7980    0.9618    0.3612   -0.5264   -0.9957   -0.6737    0.1837
    0.9618    0.5264   -0.6737   -0.8952    0.1837    0.9957    0.3612
    0.9957   -0.1837   -0.9618    0.3612    0.8952   -0.5264   -0.7980
    0.8952   -0.7980   -0.1837    0.9618   -0.6737   -0.3612    0.9957
    0.6737   -0.9957    0.7980   -0.1837   -0.5264    0.9618   -0.8952
    0.3612   -0.6737    0.8952   -0.9957    0.9618   -0.7980    0.5264
  Column 8
    0.9957
   -0.9618
    0.8952
   -0.7980
    0.6737
   -0.5264
    0.3612
   -0.1837
ans =
   1.4100e-13

Finding relations

Because there exist relation

$${W}_{l}\mathrm{{=}}{U}_{l}\mathrm{{+}}{U}_{{l}\mathrm{{-}}{1}}$$

and

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

we can express DSTVIII through DSTVII

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

where

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

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=[B; (-1).^(0:size(B,2)-1)];

Check expression of DSTVIII through DSTVII

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

DST8a=Da8*blkdiag(inv(Db7)*DST7,1)*B
% compare DST8 and DST8a matrices (show correctness of representation of DSTVIII through DSTVII)
max(max(abs(DST8-DST8a)))

DST8a =
  Columns 1 through 7
    0.0923    0.2737    0.4457    0.6026    0.7390    0.8502    0.9325
    0.2737    0.7390    0.9830    0.9325    0.6026    0.0923   -0.4457
    0.4457    0.9830    0.7390   -0.0923   -0.8502   -0.9325   -0.2737
    0.6026    0.9325   -0.0923   -0.9830   -0.4457    0.7390    0.8502
    0.7390    0.6026   -0.8502   -0.4457    0.9325    0.2737   -0.9830
    0.8502    0.0923   -0.9325    0.7390    0.2737   -0.9830    0.6026
    0.9325   -0.4457   -0.2737    0.8502   -0.9830    0.6026    0.0923
    0.9830   -0.8502    0.6026   -0.2737   -0.0923    0.4457   -0.7390
    1.0000   -1.0000    1.0000   -1.0000    1.0000   -1.0000    1.0000
  Columns 8 through 9
    0.9830    1.0000
   -0.8502   -1.0000
    0.6026    1.0000
   -0.2737   -1.0000
   -0.0923    1.0000
    0.4457   -1.0000
   -0.7390    1.0000
    0.9325   -1.0000
   -1.0000    1.0000
ans =
   4.8850e-15

Check expression of DSTVII through DSTVIII

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

DST7a=blkdiag(Db7,1)*inv(Da8)*DST8*inv(B);
DST7a=DST7a(1:end-1,1:end-1)
% compare DST7 and DST7a matrices (show correctness of representation of DSTVII through DSTVIII)
max(max(abs(DST7-DST7a)))

DST7a =
  Columns 1 through 7
    0.1837    0.3612    0.5264    0.6737    0.7980    0.8952    0.9618
    0.5264    0.8952    0.9957    0.7980    0.3612   -0.1837   -0.6737
    0.7980    0.9618    0.3612   -0.5264   -0.9957   -0.6737    0.1837
    0.9618    0.5264   -0.6737   -0.8952    0.1837    0.9957    0.3612
    0.9957   -0.1837   -0.9618    0.3612    0.8952   -0.5264   -0.7980
    0.8952   -0.7980   -0.1837    0.9618   -0.6737   -0.3612    0.9957
    0.6737   -0.9957    0.7980   -0.1837   -0.5264    0.9618   -0.8952
    0.3612   -0.6737    0.8952   -0.9957    0.9618   -0.7980    0.5264
  Column 8
    0.9957
   -0.9618
    0.8952
   -0.7980
    0.6737
   -0.5264
    0.3612
   -0.1837
ans =
   1.7208e-15

Check computation of DSTVIII transform

x=randn(N1,1);
disp('x''=');disp(x');
y=DST8*x;                      % true result
disp('y''=');disp(y');
y1=Da8*blkdiag(inv(Db7)*DST7,1)*B*x;            % compute DSTVIII using DSTVII transform
disp('y1''=');disp(y1');

x'=
  Columns 1 through 7
    0.9642    0.5201   -0.0200   -0.0348   -0.7982    1.0187   -0.1332
  Columns 8 through 9
   -0.7145    1.3514
y'=
  Columns 1 through 7
    1.0025   -0.4754    1.6154    0.9415    2.1413   -2.1092    3.9088
  Columns 8 through 9
   -0.8884    0.5748
y1'=
  Columns 1 through 7
    1.0025   -0.4754    1.6154    0.9415    2.1413   -2.1092    3.9088
  Columns 8 through 9
   -0.8884    0.5748

Check computation of DSTVII transform

x=randn(N2,1);
disp('x''=');disp(x');
y=DST7*x;                      % true result
disp('y''=');disp(y');
y1=blkdiag(Db7,1)*inv(Da8)*DST8*inv(B)*[x;0];            % compute DSTVII using DSTVIII transform
disp('y1''=');disp(y1');
% Reference

x'=
  Columns 1 through 7
   -0.2248   -0.5890   -0.2938   -0.8479   -1.1201    2.5260    1.6555
  Column 8
    0.3075
y'=
  Columns 1 through 7
    2.2859   -3.8946   -0.4126    3.0927   -3.5857    0.8359    2.0049
  Column 8
   -1.3812
y1'=
  Columns 1 through 7
    2.2859   -3.8946   -0.4126    3.0927   -3.5857    0.8359    2.0049
  Columns 8 through 9
   -1.3812   -0.0000


Published with MATLAB® 7.14