%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% Matrix A is the top left 16x512 submatrix of OEIS A195467 (consecutive Gray code permutations) %%%%%%%%% and shows the 16 different powers of the 9-bit gray code permutation: g8=[1 0 0 0 0 0 0 0; 0 1 0 0 0 0 0 0; 0 0 0 1 0 0 0 0; 0 0 1 0 0 0 0 0; 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 1 0; 0 0 0 0 1 0 0 0; 0 0 0 0 0 1 0 0]; g16=[g8 zeros(8); zeros(8) rot90(g8)']; g32=[g16 zeros(16); zeros(16) rot90(g16)']; g64=[g32 zeros(32); zeros(32) rot90(g32)']; g128=[g64 zeros(64); zeros(64) rot90(g64)']; g256=[g128 zeros(128); zeros(128) rot90(g128)']; g512=[g256 zeros(256); zeros(256) rot90(g256)']; A = [ [0:511]; [0:511]*g512; [0:511]*g512^2; [0:511]*g512^3; [0:511]*g512^4; [0:511]*g512^5; [0:511]*g512^6; [0:511]*g512^7; [0:511]*g512^8 [0:511]*g512^9; [0:511]*g512^10; [0:511]*g512^11; [0:511]*g512^12; [0:511]*g512^13; [0:511]*g512^14; [0:511]*g512^15 ]; %%% whos A %%% Name Size Bytes Class Attributes %%% A 16x512 65536 double %%% A(:,1:32) = %%% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 %%% 0 1 3 2 6 7 5 4 12 13 15 14 10 11 9 8 24 25 27 26 30 31 29 28 20 21 23 22 18 19 17 16 %%% 0 1 2 3 5 4 7 6 10 11 8 9 15 14 13 12 20 21 22 23 17 16 19 18 30 31 28 29 27 26 25 24 %%% 0 1 3 2 7 6 4 5 15 14 12 13 8 9 11 10 30 31 29 28 25 24 26 27 17 16 18 19 22 23 21 20 %%% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 %%% 0 1 3 2 6 7 5 4 12 13 15 14 10 11 9 8 25 24 26 27 31 30 28 29 21 20 22 23 19 18 16 17 %%% 0 1 2 3 5 4 7 6 10 11 8 9 15 14 13 12 21 20 23 22 16 17 18 19 31 30 29 28 26 27 24 25 %%% 0 1 3 2 7 6 4 5 15 14 12 13 8 9 11 10 31 30 28 29 24 25 27 26 16 17 19 18 23 22 20 21 %%% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 %%% 0 1 3 2 6 7 5 4 12 13 15 14 10 11 9 8 24 25 27 26 30 31 29 28 20 21 23 22 18 19 17 16 %%% 0 1 2 3 5 4 7 6 10 11 8 9 15 14 13 12 20 21 22 23 17 16 19 18 30 31 28 29 27 26 25 24 %%% 0 1 3 2 7 6 4 5 15 14 12 13 8 9 11 10 30 31 29 28 25 24 26 27 17 16 18 19 22 23 21 20 %%% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 %%% 0 1 3 2 6 7 5 4 12 13 15 14 10 11 9 8 25 24 26 27 31 30 28 29 21 20 22 23 19 18 16 17 %%% 0 1 2 3 5 4 7 6 10 11 8 9 15 14 13 12 21 20 23 22 16 17 18 19 31 30 29 28 26 27 24 25 %%% 0 1 3 2 7 6 4 5 15 14 12 13 8 9 11 10 31 30 28 29 24 25 27 26 16 17 19 18 23 22 20 21 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% Matrix B0 is the top left 16x512 submatrix of OEIS A197819 %%%%%%%%% and shows 16 binary Walsh functions w(A001317) of length 512: %%% H is the binary Walsh matrix of order 512: H = (hadamard(512)-1)/(-2) ; %%% whos H %%% Name Size Bytes Class Attributes %%% H 512x512 2097152 double %%% H(1:16,1:16) = %%% 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %%% 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 %%% 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 %%% 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 %%% 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 %%% 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 %%% 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 %%% 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 %%% 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 %%% 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 %%% 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 %%% 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 %%% 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 %%% 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 %%% Define binary Walsh functions w(A001317) of lengh 512: % The first 16 elements of OEIS A001317 (Sierpinski triangle rows read like binary numbers) are: % 1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535 % As these numbers exceed the size of H % they are replaced by their modulo 512 values: % 1, 3, 5, 15, 17, 51, 85, 255, 257, 259, 261, 271, 273, 307, 341, 511 % As Walsh functions are counted from 0 and MATLAB counts indexes from 1 % the 1 has to be added to each of these numbers. B0 = cell2mat( { H( 1 + 1 , : ) ; H( 3 + 1 , : ) ; H( 5 + 1 , : ) ; H( 15 + 1 , : ) ; H( 17 + 1 , : ) ; H( 51 + 1 , : ) ; H( 85 + 1 , : ) ; H( 255 + 1 , : ) ; H( 257 + 1 , : ) ; H( 259 + 1 , : ) ; H( 261 + 1 , : ) ; H( 271 + 1 , : ) ; H( 273 + 1 , : ) ; H( 307 + 1 , : ) ; H( 341 + 1 , : ) ; H( 511 + 1 , : ) } ) ; %%% B0 is repeatedly streched: % function [y] = strech(x) % y = zeros(16,512) ; % for k=1:256 % y(1:16,2*k-1) = x(1:16,k) ; % y(1:16,2*k) = x(1:16,k) ; % end % end B1 = strech(B0) ; B2 = strech(B1) ; B3 = strech(B2) ; B4 = strech(B3) ; B5 = strech(B4) ; B6 = strech(B5) ; B7 = strech(B6) ; B8 = strech(B7) ; %%% whos B0 B1 B2 %%% Name Size Bytes Class Attributes %%% B0 16x512 65536 double %%% B1 16x512 65536 double %%% B2 16x512 65536 double %%% etc. %%% B0(:,1:32) = %%% 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 %%% 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 %%% 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 %%% 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 %%% 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 %%% 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 %%% 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 %%% 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 %%% 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 %%% 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 %%% 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 %%% 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 %%% 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 %%% 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 %%% 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 %%% 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 %%% B1(:,1:32) = %%% 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 %%% 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 %%% 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 %%% 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 %%% 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 %%% 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 %%% 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 %%% 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 %%% 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 %%% 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 %%% 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 %%% 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 %%% 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 %%% 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 %%% 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 %%% 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 %%% B2(:,1:32) = %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 %%% 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 %%% 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 %%% etc. %%% The entries of B0,B1,... are the binary digits of the entries of Bsum: Bsum = B0 + 2.*B1 + 4.*B2 + 8.*B3 + 16.*B4 + 32.*B5 + 64.*B6 + 128.*B7 + 256.*B8 ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% The matrices A and Bsum are equal: isequal(A,Bsum) ans = 1 pFad - Phonifier reborn

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