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1 ISSN , CODEN RUXUEW Jounal of Softwae, Vol17, No5, May 26, pp DOI: 1136/jos Tel/Fax: by Jounal of Softwae All ights eseved Walsh 1, 2 2, 1 (, 4374) 2 (, 4374) Walsh Matix s Copy Geneation and Its Compute Images CHEN Ning-Tao 1, WANG Neng-Chao 2, SHI Bao-Chang 2 1 (School of Compute Science and Technology, Huazhong Univesity of Science and Technology, Wuhan 4374, China) 2 (Paallel Computing Institute, Huazhong Univesity of Science and Technology, Wuhan 4374, China) Coesponding autho: Phn: , nt_chen@263net, Chen NT, Wang NC, Shi BC Walsh matix s copy geneation and its compute images Jounal of Softwae, 26,17(5): Abstact: Walsh functions ae widely used in many aeas such as signal and image pocessing, digital communication etc Walsh function system is othogonal and completed Among many methods to geneate Walsh functions, Swick s copy theoy is famous This method takes Walsh function s ode numbes as the copy infomation and can geneate an abitay function given cetain ode numbe, which in fact is the vecto pocessing method and is not applicable fo such 2D signal pocessing applications as fast tansfoms Walsh function system can be expessed by Walsh matix W k In this pape, ow copy and block copy methods ae put fowad based on W k Copy opeatos ae designed based on symmeties and a new odeing is discoveed (named Walsh-Like odeing) Afte the iteative fomulas of 6 odeing Walsh matices ae deduced by Konecke poduct, the compute images of these matices ae illustated It is poved that the poposed method is moe advanced than the fome The late can achieve highe pefomance and is applicable to the fast tansfoms designing The fouth symmetic odeing of Walsh function system is discoveed (Walsh-Like odeing), and the conjectue that the new odeing is likely to be the evese fom of Walsh odeing is made by analyzing and compaing with these images Key wods: : Walsh function; copy; evolution; symmety; factal Walsh Walsh,,Swick, Walsh, Walsh, Walsh Walsh W k W k, ( Walsh ) Konecke 6 Walsh,, 25 Suppoted by the National Natual Science Foundation of China unde Gant No ( ) Received ; Accepted

2 1242 Jounal of Softwae Vol17, No5, May 26,, Walsh 4 : Walsh, Walsh Walsh : Walsh ; ; ; ; : TP391 : A Walsh Walsh E k ±1 E k,walsh 2 k W k, Walsh,Walsh Walsh W k, [1] Walsh Swick [2] 1969 Walsh Walsh [3], Paley Hadamad Walsh, Walsh, X Swick, Walsh, Walsh,, [4], [5], 6 Walsh,Walsh 3 :Paley Hadamad Walsh :Paley Hadamad, Paley Hadamad ;Walsh Gay Walsh, Walsh [6] Gay : Gay Gay Gay Walsh Walsh ; Gay Walsh Walsh, Gay Gay [7] ( X ),X Walsh, Walsh X T [1] Walsh Walsh, [8,9],, Walsh, Walsh, Walsh 1 Walsh 2 4, 6 Walsh 3 3 C, 6 Walsh 1 11 Walsh 1923, JL Walsh Walsh : W ( x) = i k 1 = k sgn[cos j 2 πx], x < 1, i =,1,2,,2 1, j = :k,k Walsh 2 k Walsh ;sgn,sgn[x] x> 1,x< 1; j 1 j,j i j Walsh Walsh : k 1 = (1) j, j k 1 =i k 1,=~k 1, Paley Walsh (2) j Gay, j k 1 =i k 1, j k 1 =i k 1 i k,=~k 1, Walsh Walsh JLWalsh Walsh (3) j, j k 1 =i,=~k 1, Paley Walsh Hadamad, Hadamad (4) j Gay, i Gay, i, j =i k 1, j =i k i k 1,=1~k 1, Walsh Walsh j 2

3 :Walsh 1243 (5) j Gay, i, Gay i, j k 1 =i, j k 1 =i i 1,=1~k 1, Walsh Walsh, X 1 k=3 Walsh 8 Walsh,j Gay 1, 1,Walsh,, Swick 12 Fig1 Eight Walsh odeing functions 1 Walsh 8 Walsh 1969,DA Swick Walsh Walsh, Walsh 1983, Swick ( ) Walsh, Walsh i,, Walsh 1 Walsh 8 Walsh, j Gay, (, 1, 1 ) 13 Walsh Table 1 Ode code copy technique 1 Ode numbe Gay code (j 2 j 1 j ) Initial Step 1(j 2 ) Step 2 (j 1 ) Step 3 (j ) Walsh ±1, Walsh, Walsh Walsh, [1] E =[,1], E 1 : E 1 =E 1 () E 1 (1), E 1 () =[,1/2], E 1 () =[1/2,1], E 1, E 2 : E 2 =[,1/4] [1/4,1/2] [1/2,3/4] [3/4,1],, k E k 2 k ( 2 )

4 1244 Jounal of Softwae Vol17, No5, May 26, E k 2 k, E k 2 k, 2 k W k, Walsh k 2 1 () i () i i i 1 Ek = U Ek, Ek =, k k i= 2 2 [ ) E E 1 E 2 E 3 [ )[ ) [ )[ )[ )[ ) [ )[ )[ )[ )[ )[ )[ )[ ) Fig2 Bisection sets 2 14 [5] Walsh Walsh ;Paley Walsh, :, : 1( ) f(x) [,1), f(1/2 x)=f(1/2x), x<1/2;, f(1/2 x)= f(1/2x), x<1/2, f(x) 2( ) f(x) [,1],, f(x)=f(x 1/2),1/2 x<1;,, f(x)= f(x 1/2),1/2 x<1, f(x) 15, [1],Walsh Walsh,Paley Paley [1] (-1 ), Walsh Paley ;Walsh Paley,W k W k 1, [5] 12 13,Hadamad Walsh,W k W k 1, [1],Walsh Paley, W k 1 W k,w k W k 1 4, [1,5], ( 1 ), (-1 ), [5],, [1] [5], :, ;,, Walsh ; Paley ; Paley ; Walsh, :, Walsh ;, [6] Gay Swick Walsh,Walsh 3 : Walsh Walsh Walsh Walsh, Walsh? Paley Paley, Walsh Walsh,

5 :Walsh Walsh 21 [1,8] Konecke Walsh W k 1 Walsh W k [1] H G : e e H =, G = e e e e e e H,G 4,,e, e, e e Konecke, A,H A H A A H A (m1) (n1), A An A An A An A An H A =, Am Amn Am Amn Am Amn Am Amn A An An A A An An A G A =, Am Amn Amn Am Am Amn Amn Am A A n A An A An An A A m Amn Am A mn A m Amn Amn Am H A =, G A = A A n A An A An An A Am Amn Am Amn Am Amn Amn Am G:,G,, G Walsh Walsh,G,, Walsh Walsh G, Walsh G G, G G : e e G =, e e G = e e e e, e, e e, A An A An A An An A A m Amn Am A mn A G A =, m Amn Amn Am G A = Am Amn Am Amn Am Amn Amn Am A An A An A An An A Walsh H k k H Konecke, 6 Walsh 22 6 Walsh (1) Paley : Paley W W k :W k =H W k 1 =H 2 W k 2 = =H k W,W =[],,, W, H W Konecke k, Paley W k Paley W 3

6 1246 Jounal of Softwae Vol17, No5, May 26 3(a), (2) Walsh :W k =G W k 1 =G 2 W k 2 = =G k W, 3(b) (3) Walsh :W k =G W k 1 = G 2 W k 2 = =G k W, 3(c) (4) Walsh :W k = G W k 1 = G 2 Wk 2= = G k W, 3(d) (5) Walsh :W k =G W k 1 = G (6) Paley :W k =H W k 1 = H 2 2 W k 2 = = G W k 2 = = H k k W, 3(e) W, 3(f) (a) Hadamad odeing (c) X T odeing (b) X odeing (d) Walsh-Like odeing (e) Walsh odeing Fig3 3 Walsh (f) Paley odeing Copy geneation of six odeings of Walsh matix 3 6 Walsh 31 :OS Windows 2 Pofessional,CPU 24GHz,256M,VC6 1 (1) A[2 k ][2 k ] Walsh, 1 1; (2) Walsh,, B[2 k ][k],b, ; (3) B, A ( Paley, ) C ( vecto_width ): fo (i=;i<2 k ;i)a[i][]=1; /* A 1 1, B ;

7 :Walsh 1247 fo (i=;i<2 k ;i){ } vecto_width=1; /*, 1 fo (j=;j<k;j){ } stat=; /* A fo (step=;step<vecto_width;step){ /*step A if (B[i][k j 1]==)/*, else A[i][vecto_widthstep]=A[i][stat]; /* A[i][vecto_widthstep]= A[i][stat]; /* }vecto_width<<=1; /* B, 2 ( ) (1) A[2 k ][2 k ], Walsh, A[][]=1; (2) A ( 3 ) C ( matix_width ): (a) ( Walsh ) A[][]=1; /* A 1 1 matix_width=1; /* A 1 fo (i=;i<k;i){ } /*, fo (ow=matix_width 1;ow>=;ow ) fo (col=;col<matix_width;col) A[2*ow1][col]=A[2*ow][col]=A[ow][col]; /*, fo (ow=;ow<matix_width;ow=2) fo (col=;col<matix_width;col){ A[ow][colmatix_width]=A[ow][matix_width col 1]; A[ow1][colmatix_width]= A[ow][matix_width col 1]; } matix_width<<=1; /*, (b) ( Walsh ): A[][]=1;matix_width=1; /* A 1 1, A 1 fo (i=;i<k;i){ /* 4, fo (ow=;ow<matix_width;ow) } fo (col=;col<matix_width;col){ A[ow][colmatix_width]=A[ow][matix_width col 1]; /* A[owmatix_width][col]=A[matix_width ow 1][col]; /* /*, fo (ow=;ow<matix_width;ow)

8 1248 Jounal of Softwae Vol17, No5, May 26 } fo (col=;col<matix_width;col) A[owmatix_width][colmatix_width]= A[owmatix_width][matix_width col 1]; matix_width<<=1; /*,,,,, A, 4 3, k 1, 4, 4 CPU time (ms) Ode code copy Row copy Block copy k Fig4 Pefomance compaison of the thee copy methods 4 3 Walsh, [1] Hadamad ( H), H, ;, 1,, 51/2 1,, H 2 ( ) Walsh,,5, 8 8 ;8 8, 5 5(1) 5(2) Hadamad [11] Hadamad ( 5(1)) Walsh ( 5(9)) Paley ( 5(11)), :Hadamad (nested);walsh ;Paley Walsh,Paley ( 5(11) 5(12)), Walsh ( 5(9) 5(1)) : Walsh Walsh, ; Walsh Walsh Walsh Walsh Walsh Walsh?, Walsh,Walsh 2 4,Walsh 2 4 H 2

9 :Walsh 1249 (1) Hadamad odeing (W 5 ) (2) Hadamad odeing (W 8 ) (3) X odeing (W 5 ) (4) X odeing (W 8 ) (5) X T odeing (W 5 ) (6) X T odeing (W 8 ) (7) Walsh-Like odeing (W 5 ) (8) Walsh-Like odeing(w 8 ) (9) Walsh odeing (W 5 ) (1) Walsh odeing (W 8 ) (11) Paley odeing (W 5 ) (12) Paley odeing (W 8 ) Fig5 Images of six odeings of Walsh matix 5 6 Walsh Table 2 Fou symmetic Walsh matices 2 4 Walsh Symmety Copy mode Shift symmety Mio image symmety Row copy Paly odeing Walsh odeing Block copy Hadamad odeing Walsh-Like odeing 4 Walsh, ( Walsh ), 6, Walsh, Walsh Walsh Walsh 3 Walsh Walsh ( 2) Swick,, ;,, Walsh, Walsh (FWT)Walsh FWT, Walsh Hadamad Walsh, Walsh, Walsh Walsh [8,9] Walsh Walsh 4,

10 125 Jounal of Softwae Vol17, No5, May 26 Refeences: [1] Wang NC The mathematical beauty of Walsh functions (I), (II) Jounal of Yunnan Univesity (Natual Science), 1997, 19(Supplement): (in Chinese with English abstact) [2] Swick DA Walsh function geneation IEEE Tans on Infomation Theoy, 1969,IT-15(1): [3] Li ZH, Zhang QS Odeing of Walsh function IEEE Tans on Electomagnetic Compatibility, 1983,EMC-25(2): [4] Wang NC Bisection techniques of synchonous paallel algoithms design Science in China (Seies A), 1995,25(2): (in Chinese with English abstact) [5] Wang NC Walsh functions geneated with evolution China Jounal of Image and Gaphics, 1996,1(3): (in Chinese with English abstact) [6] Lu XP, He NZ, Wang NC The evese foms of Gay-code Jounal of Yunnan Univesity (Natual Science), 1997,19(Supplement): (in Chinese with English abstact) [7] Zhang QS Signal s Copy Theoy and Its Applications Beijing: People s Posts & Telecommunications Publishing House, 21 (in Chinese) [8] Guo WB, Wang NC, Shi BC Fast algoithms fo the Walsh tansfom in M-odeing Mini-Mico System, 23,24(2): (in Chinese with English abstact) [9] Li Q, Zhou ML, Shi BC, Wang NC Fast algoithms fo the Walsh tansfom in contay Walsh-odeing Science Geneal Jounal, 1998,43(5): (in Chinese with English abstact) [1] Qi DX Factal and its Compute Geneation Beijing: Science Pess, 1996 (in Chinese) [11] Wolfam S A New Kind of Science Champaign: Wolfam Media, Inc, 22 : [1] Walsh (I) (II) ( ),1997,19( ): [4] (A ),1995,25(2): [5] Walsh,1996,1(3): [6],, Gay ( ),1997,19( ): [7] :,21 [8],, M Walsh,23,24(2): [9],,, Walsh Walsh,1998,43(5): [1] :,1996 (1976 ),,,,, (1959 ),,,,, (1937 ),,,,,