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Walsh Transform

Walsh Transform. ELE5430 Pattern Recognition W.K. Cham Professor Department of Electronic Engineering The Chinese University of Hong Kong. Walsh Transform ( 2 ). Historical Note Dyadic Symmetry Some properties of Dyadic Symmetry Generation of Walsh matricws

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Walsh Transform

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  1. Walsh Transform ELE5430 Pattern Recognition W.K. Cham Professor Department of Electronic Engineering The Chinese University of Hong Kong

  2. Walsh Transform (2) • Historical Note • Dyadic Symmetry • Some properties of Dyadic Symmetry • Generation of Walsh matricws • Conversion between matrices of different ordering • Dyadic Decomposition • Fast Walsh Transform • Ongoing Works

  3. Historical Note We have explained how a vector can be represented as a weighted sum of orthogonal basis vectors. The representation of a signal as a weighted sum of a set of functions has many applications and a long history. Many people have contributed in this work and one of the most important contribution is by this man in 1807. in 1922, Rademacher devised an incomplete set of orthogonal functions. The Rademacher functions are defined within the [0,1) and take the values +1 and -1. Reademacher, H., 'Einige Satze von allgemeinen Orthogonal-funktionen," Math. Annalen 87, pp.122-138, 1922. (3)

  4. Rademacher functions R0(t) = 1, t[0,1); For i  1, Ri(½ t) = Ri-1(t); Ri(½ t + ½)= - Ri-1(t) , t[0,1); Walsh functions In 1923, J.L. Walsh add more new functions and formed a complete orthonormal set of functions, now known as Walsh functions. Walsh, J.L., 'A closed set of orthogonal functions," American J. Of the Mathematics, Vol.45, pp.5-24, 1923. (4)

  5. Walsh Transform However, Walsh function did not receive much attention. Fourier function which is the eigenfunction of convolution attracted all attention. It also lead to the development of FFT and a complete theory for discrete system. In 1969, Pratt and others used the Walsh transform (WT), developed from the Walsh functions, in place of the FFT for image coding. Pratt, W.K., Kane, J., Andrews, H.C.,'Hadamard Transform Image Coding,' Proc. IEEE, Vol.57, No.1, pp.58-68, Jan. 1969. In the early 1970's, the simplicity of WT resulted in a wide range of applications and interest. Symposium on Applications of Walsh Functions, Washington, D.C., 1970. Symposium on Applications of Walsh Functions, Washington, D.C., 1971. Symposium on Applications of Walsh Functions, Washington, D.C., 1972. Symposium on Applications of Walsh Functions, Washington, D.C., 1973. (5)

  6. y(t) = a +b +c +d +e +f +g +h Walsh Transform The effectiveness of most of these applications, especially filtering and coding, depends on the ability of the transform to pack signal energy into a few transform coefficients. The WT is inferior to DFT in that aspect. Later, people discovered the Discrete Cosine Transform (DCT) which has even better energy packing ability. The interest in WT diminished quickly. Today Walsh transform is mainly used in multiplexing which is to send several data simultaneously. It does not require high energy packing ability. Multiplexing To transmit data a, b, c, d, e, f, g, h at the same time over a period, the following signal will be sent. To decode a, b, c, d, e, f, g and h , perform the dot product between y(t) and the corresponding Walsh function. (6)

  7. Nomenclature During the development, different researchers adopted different nomenclatures. In fact, it is not unified even today. Def: Hadamard matrix is a square matrix of only plus and minus one whose rows (and columns) are orthogonal to one another. Hadamard functions, the counterpart in the continuous case, are also called Walsh-like functions. Walsh functions and transform are particular cases of Hadamard functions and transforms. (7)

  8. Many methods were developed to generate Walsh transforms. Depending on the row orders, they can be grouped into the following 3: Natural-ordered Walsh transform Pratt, W.K., Kane, J., Andrews, H.C.,'Hadamard Transform Image Coding,' Proc. IEEE, Vol.57, No.1, pp.58-68, Jan. 1969. Dyadic-ordered Walsh transform Shanks, J.L., 'Computation of the fast Walsh-fourier Transform,' IEEE Trans. Vol.18, May 1969, pp.457-459. Sequency-ordered Walsh transform Harmuth, H.F.,'A Generalized Concept of Frequency and some Applications,' IEEE Trans. on Information Theory, Vol.14, No.3, May 1968, pp.375-382. (8)

  9. -Wn2 -Wn4 Natural-ordered Walsh transform An order 2N transform can generated using an order N transform as follows. (9)

  10. Sequency-ordered Walsh transform Conversion from natural-ordered Walsh transform e.g. Let F be the vector of natural-ordered Walsh transform coefficients and C be the vector of sequency-ordered Walsh transform coefficients. 000 001 010 011 100 101 110 111  000  111 011 100 001 110  010  011 Compute F (with coefficients in natural order)which is then converts into C (with coefficients in sequency order). (10)

  11. Dyadic Symmetry (11) Def: A particular type of even symmetry is said to exist in a vector of n elements if and only if the n elements can be divided into n/2 paris of elements of the same value. Def: A particular type of odd symmetry is said to exist in a vector of n elements if and only if the n elements can be divided into n/2 paris of elements of the same magnitude and opposite sign. e.g. (a, b, c, d) has 3 ways to be paired up and so 3 types of even symmetry. (a, b, c, d), (a, b, c, d), (a, b, c, d) e.g. (a, b, c, d, e, f, g, h) has 105 ways to be paired up and so 105 types of even symmetry. (a, b, c, d , e, f, g, h), (a, b, c, d , e, f, g, h), (a, b, c, d , e, f, g, h), .... (11)

  12. Dyadic Symmetry Def: A vector of 2m elements ( a(0), a(1), a(2), ..., a(2m-1) ) is said to have the i th dyadic symmetry if a(j) = s  a(j  i) where (i)  is the exclusive or operator, (ii) j  [0, 2m-1] and i [0, 2m], (iii) s = 1 when the symmetry is even and s = -1 when the symmetry is odd. e.g. (a, a, b, b, c, c, d, d) has the 1st even dyadic symmetry. (a, b, a, b, c, d, c, d) has the 2nd even dyadic symmetry. (a, b, b, a, c, d, d, c) has the 3rd even dyadic symmetry. (a, b, c, d, a, b, c, d ) has the 4th even dyadic symmetry. (a, b, c, d, b,a, d, c) has the 5th even dyadic symmetry. (a, b, c, d, c, d, a, b) has the 6th even dyadic symmetry. (a, b, c, d, d, c, b, a) has the 7th even dyadic symmetry. (12)

  13. 1 (a0a1a2a3a4a5 a6 a7) 2 (a0a1a2a3a4 a5a6a7) 3 (a0a1 a2 a3a4 a5 a6a7) 4 (a0a1a2 a3a4a5 a6a7) 5 (a0a1a2 a3a4a5a6a7) 6 (a0a1a2a3a4a5a6a7) 7 (a0a1a2a3 a4a5a6a7) e.g. dyadic symmetry Def: A vector of 2m elements (a(0), a(1), a(2), ..., a(2m-1)) is said to have the ith dyadic symmetry if a(j) = s  a(j  i) where  is the exclusive or operator, j[0, 2m-1] and i[0, 2m], s=1 for even symmetry and s=-1 for odd symmetry. (13)

  14. Some Properties of Dyadic Symmetry (14) Thm 1: If a 2m vector has dyadic symmetryS1, S2, ..., Sr, this vector also has dyadic symmetry Sk where Sk = S1 S2  ... Sr. e.g. If an order-8 vector (a, b, c, d, e, f, g, h) has the 1st & 2nd even dyadic symmetry, then it also has the 3rd even dyadic symmetry. It has the 1st even dyadic symmetry so (a, a, b, b, e, e, g, g ). It has the 2nd even dyadic symmetry so (a, b, a, b, e, f, e, f ). Hence, a= b and e= f. The vector is (a, a, a, a, e, e, e, e ) and so it has the 3rd even dyadic symmetry. (14)

  15. Independent and Dependent Dyadic Symmetry Def: The r dyadic symmetries S1, S2, ..., Srare said to be dependent if there exist r elements k1, k2, ..., kr not all zero, such that k1S1  k2S2  ... krSr = 0. Otherwise, the r dyadic symmetries are said to be independent. e.g. Dyadic symmetries ( 001 ), ( 010 ), (011) are dependent because (001)  (010)  (011) = 0 k1S1  k2S2 k3S3 = 0 e.g. Dyadic symmetries ( 001 ), ( 010 ), (100) are independent. (15)

  16. Independent and Dependent Dyadic Symmetry Thm 2: If a vector has r independent dyadic symmetries, this vector has also 2r-1 dyadic symmetries. e.g. Suppose a vector has dyadic symmetries ( 001 ), ( 011 ), (111). By Theorem 1, it also has dyadic symmetries which are linear combination of these 3 dyadic symmetries. Hence, it has dyadic symmetries (010) = (001)  (011) (110) = (001)  (111) (100) = (011)  (111) (101) = (001)  (011)  (111) (16)

  17. 2ndeven/odd DS? 3rdeven/odd DS? Generation of Walsh Matrices (17) Thm 3: Every basis vector of an order-2mWalsh matrix has all the 2m-1 dyadic symmetries. The following is not a row of the Walsh matrix. What dyadic symmetries does it have? ( 1 -1 1 -1 1 -1 -1 1 ) 1stodd DS? (17)

  18. j i Generation of Walsh matrix using independent Dyadic Symmetry We use dyadic symmetries (100)(010) and (100) to generate an order-8 natural-ordered Walsh matrix. (100) (010) (001) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 (18)

  19. 00 01 10 11 00 01 10 11 A binary 2m x 2mWalsh matrix can be generated using j i b(i,j) = jS-1 iT where i and jare m-vectors representing (i,j) and S is a 1xmmxmmx1matrix whose rows are m independent dyadic symmetries. e.g. Binary Walsh matrix A binary 2m x 2mWalsh matrix was defined from a Walsh matrix as follows: w(i,j) = 1 b(i,j) = 0; w(i,j) = -1 b(i,j) = 1; (19)

  20. b(i,j) = jS-1 iTwhere i and jare m-vectors and S is a matrix whose rows are m independent dyadic symmetries. e.g. (100) (010) (001) i = (i(1) i(2) i(3) ) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 j = (j(1) j(2) j(3) ) 000 001 010 011 100 101 110 111 The elements of i = (i(1) i(2) ... i(m)) determines the type (odd or even) of dyadic symmetry of the ith row. (20)

  21. b(i,3)= i(2)  i(3) b(i,5)= i(1)  i(3) The elements of i = (i(1) i(2) ... i(m)) determines the type (odd or even) of dyadic symmetry of the ith row. b(i,j) = jS-1 iT j = (j(1) j(2) j(3) ) 000 001 010 011 100 101 110 111 (100) (010) (001) i = (i(1) i(2) i(3) ) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 The sign of thejth element in the ith row depends on the types of the m independent dyadic symmetries. The sign of thejth element in the ith row depends on the types of the m independent dyadic symmetries. The actual relationship can be found by expressing j = (j(1) j(2) ... j(m)) w.r.t. the a basis formed by the m independent dyadic symmetries. (21)

  22. b(i,3)= i(2)  i(3) The elements of i = (i(1) i(2) ... i(m)) determines the type (odd or even) of dyadic symmetry of the ith row. b(i,j) = jS-1 iT j = (j(1) j(2) j(3) ) 000 001 010 011 100 101 110 111 (100) (010) (001) i = (i(1) i(2) i(3) ) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 Let r = (r(1) r(2) ... r(m)) s.t. j = (r(1)S1 r(2)S2  ...  r(m)Sm). If r(m)=1, then b(i,j) depends on Sm. If r(m)=0, then b(i,j) is independent of Sm or r = j S-1. (22)

  23. b(i,3)= i(2)  i(3) The elements of i = (i(1) i(2) ... i(m)) determines the type (odd or even) of dyadic symmetry of the ith row. b(i,j) = jS-1 iT j = (j(1) j(2) j(3) ) 000 001 010 011 100 101 110 111 (100) (010) (001) i = (i(1) i(2) i(3) ) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 r = j S-1. If r(m)=1, then b(i,j) depends on Sm ; Otherwise not. = j S-1 iT. b(i,j) = r(1)i(1)  r(2)i(2)  ...  r(m)i(m) = riT (23)

  24. b(i,j) = jS-1 iT Def: The matrix S in the natural-ordered binary Walsh matrix is Def: The matrix S in the dyadic-ordered binary Walsh matrix is Def: The matrix S in the sequency-ordered binary Walsh matrix is (24)

  25. j = ( j(1) j(2) j(3) ) 000 001 010 011 100 101 110 111 i=(i(1) i(2) i(3) ) 0 0 0 0 1 0 0 1 2 0 1 0 3 0 1 1 4 1 0 0 5 1 0 1 6 1 1 0 7 1 1 1 e.g. We use dyadic symmetries (001)(011) and (111) to generate an order-8 sequency-ordered Walsh matrix. b(i,j) = jS-1 iT Hence, b(0,j)=0; b(1,j)=j(1); b(2,j) = j(1)j(2); b(3,j) = j(2); b(4,j) = j(2)j(3); b(7,j) = j(3); b(5,j) = j(1)j(2)j(3); b(6,j) = j(1j(3); (25)

  26. b(i,j) = jS-1 iT e.g. We use dyadic symmetries (001)(010) and (001) to generate an order-8 natural-ordered Walsh matrix. = i(1)j(1)  i(2)j(2)  i(3)j(3) e.g. We use dyadic symmetries (100)(010) and (100) to generate an order-8 dyadic-ordered Walsh matrix. = i(1)j(3)  i(2)j(2)  i(3)j(1) (26)

  27. Conversion between matrices of different ordering (27) Sequency-ordered Walsh transform coefficients from natural-ordered Walsh transform e.g. Let F be the vector of natural-ordered Walsh transform coefficients and C be the vector of sequency-ordered Walsh transform coefficients. 000 001 010 011 100 101 110 111  000  111 011 100 001 110  010  011 Compute F (with coefficients in natural order)which is then converts into C (with coefficients in sequency order). (27)

  28. j i natural-ordering dyadic-ordering sequency-ordering in id iz Conversion between natural-ordering, dyadic-ordering and sequency-ordering can be obtained using the following equations where in , id ,and iz are the corresponding indices. izT = Z D-1idT = Z N-1inT idT = D Z-1izT = D N-1inT inT = N Z-1izT = N D-1idT b(i,j) = jS-1 iT Proof: b(i,j) =jN-1 inT=jZ-1 izT=j D-1id for all j. (28)

  29. j i 000 001 010 011 100 101 110 111  000  001 011 010 111 110  100  101 id = (id(1) id(2) id(3) ) = = iz = (iz(1) iz(2) iz(3) ) natural-ordering dyadic-ordering sequency-ordering in id iz e.g. Conversion from dyadic-ordering to sequency-ordering izT = Z D-1idT (29)

  30. j i 000 001 010 011 100 101 110 111  000  111 011 100 001 110  010  101 in = (in(1) in(2) in(3) ) = = iz = (iz(1) iz(2) iz(3) ) natural-ordering dyadic-ordering sequency-ordering in id iz e.g. Conversion from natural-ordering to sequency-ordering izT = Z N-1inT (30)

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