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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
- Conversion between matrices of different ordering
- Dyadic Decomposition
- Fast Walsh Transform
- Ongoing Works

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)

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)

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)

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)

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)

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)

Natural-ordered Walsh transform An order 2N

transform can generated using an order N

transform as follows.

(9)

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)

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)

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)

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, s1 for even symmetry and s-1 for odd

symmetry.

(13)

Some Properties of Dyadic Symmetry (14)

Thm 1 If a 2m vector has dyadic symmetry S1, 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)

Independent and Dependent Dyadic Symmetry

Def The r dyadic symmetries S1, S2, ..., Sr are

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)

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)

Generation of Walsh Matrices (17)

Thm 3 Every basis vector of an order-2m Walsh

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 )

1st odd DS?

(17)

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)

Binary Walsh matrix

A binary 2m x 2m Walsh 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)

b(i,j) j S-1 iT where i and j are 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)

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) j S-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 the jth element in the ith row

depends on the types of the m independent dyadic

symmetries.

The sign of the jth 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)

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) j S-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)

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) j S-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) r

iT

(23)

b(i,j) j S-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)

e.g. We use dyadic symmetries (001) (011) and

(111) to generate an order-8 sequency-ordered

Walsh matrix.

b(i,j) j S-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)

b(i,j) j S-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)

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)

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-1 idT Z N-1 inT idT D Z-1 izT

D N-1 inT inT N Z-1 izT N D-1

idT

b(i,j) j S-1 iT

Proof b(i,j) j N-1 inT j Z-1 izT

j D-1 id for all j.

(28)

natural-ordering dyadic-ordering

sequency-ordering

in

id

iz

e.g. Conversion from dyadic-ordering to

sequency-ordering izT Z D-1 idT

(29)

natural-ordering dyadic-ordering

sequency-ordering

in

id

iz

e.g. Conversion from natural-ordering to

sequency-ordering izT Z N-1 inT

(30)

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