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# Architecture for Descaling and Inverse Discrete Cosine Transforming

IP.com Disclosure Number: IPCOM000104132D
Original Publication Date: 1993-Mar-01
Included in the Prior Art Database: 2005-Mar-18
Document File: 2 page(s) / 57K

IBM

Feig, E: AUTHOR

## Abstract

Disclosed is a method to descale and take inverse discrete cosine transform (DCT) on blocks of sizes 8 times 8, which allows for very efficient pipeline realization. The invention is based on the following factorization for C sub 8, the DCT on 8 points.

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Architecture for Descaling and Inverse Discrete Cosine Transforming

Disclosed is a method to descale and take inverse discrete
cosine transform (DCT) on blocks of sizes 8 times 8, which allows for
very efficient pipeline realization.  The invention is based on the
following factorization for C sub 8, the DCT on 8 points.

C sub 8 %% =  %%  S sub <2,4> % S sub <2,3> % S sub
<2,2> % S sub <2,1> % M %  S sub <1,3> % S sub <1,2> % S sub <1,1> %
P % D
where D is the 8 times 8 diagonal matrix whose (1,1) entry is
sqrt<2>/4 and whose (j,j)-th entry for j = 1,2,....,7 is (1/2) cos
(pi j /16), and M is the 9 times 9 diagonal matrix

M %% = %% Diag %% left lbrace %% 1% , % 1 % , % 2 sqrt<2> %
,% 1 % , % 2 sqrt<2>  % , % 1, % a sub 7 % , % a sub 8 % , % a sub 9
% % right rbrace
where a sub 7 = -2  left lbrk % cos ( pi / 8 )  +  cos ( 3 pi /  8 )
right rbrk , %% a sub 8 %% here = %% -2 % cos ( pi / 8 ), %% a sub 9
= -2 left lbrk cos ( pi / 8 )  - cos ( 3 pi  / 8 ) right rbrk.

The 2-dimensional DCT on 8 times 8 points, C sub <8 ctimes 8>,
is the tensor product of the 1-dimensional DCT on 8 points with
itself, namely, C sub <8 ctimes 8> %% = %% C sub 8 % ctimes % C sub
8.  Invoking a standard theorem in algebra that the tensor product of
a product of matrices equals the product of the tensor products of
the various factors, one obtains a factorization for C sub <8 ctimes
8>.  The factor P % D % ctimes P % D will be absorbed into the
descaling; hence the computation of its product together with the
descaling requires 64 multiplications, which may be pipelined.  This
part of the computation will be called the "descaling" stage.

The product by S sub <1,2> % S sub <1,1> % ctimes % S sub <1,2>
% S sub <1,1> is done in row-column fashion with 8 times 16 = 128
additions.  Then product by S sub <1,3>  % ctimes % S sub <1,3> is
done in row-column fashion.  Since S sub <1,3> is a 9 times 8 matrix,
afte...