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Calculating Spectral Coefficients for Walsh Transform using Butterflies

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Title: Calculating Spectral Coefficients for Walsh Transform using Butterflies

1
Calculating Spectral Coefficients for Walsh
Transform using Butterflies

Marek Perkowski
September 21, 2005

2
Symbols

a -- the number of true minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
1
b -- the number of false minterms of Boolean
function F, where the function F has the logical
value 0 and the standard trivial function has the
logical value 1
c -- the number of true minterms of Boolean
function F, where the function F has the logical
value 1 and the standard trivial function has the
logical value 0
d -- the number of false minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
0, and e be the number of don't care minterms of
Boolean function F.

Then, for completely specified Boolean functions
having n variables, this formula holds
a b c d 2n
Accordingly, for incompletely specified Boolean
functions, having n variables, holds
a b c d e 2n

4
X Y
X Y
0
1
0
1
0
0
1
1
1
1
Standard Trivial function for XOR of input
variables
Data Function
X Y
X Y
0
1
X Y
0
1
0
1
0
0
0
1
1
1
Standard Trivial function for input variable Y
Standard Trivial function for input variable X
Standard Trivial Function for whole map
5

The spectral coefficients for completely
specified Boolean function can be defined in the
following way
s0 2n 2 a
si 2 (a d) - 2n, when i ? 0.

X Y
0
1
a -- the number of true minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
1
0
1
1
1
d -- the number of false minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
0
a 2, d 2, n 2
s0 2n 2 a 4 2 2 0 s3 2 (a d)
- 2n 2(22) 4 4 best correlation
6

si 2 (a d) - 2n, when i ? 0.

Negation of the previous function
X Y
0
1
1
a -- the number of true minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
1
0
1
1
d -- the number of false minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
0
a 2, d 2, n 2
s3 2 (a d) - 2n 2(00) 4 - 4 worst
correlation
7

s0 2n 2 a 4 2 4 - 4

Other functions
X Y
0
1
1
1
a -- the number of true minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
1
0
1
1
1
X Y
0
1
0
0
0
s0 2n 2 a 4 2 0 4
0
0
1
8
Butterflies in S encoding
S encoding
0
0

0 0 1
0
2
-
1 1 -1

0
0
-

2 1 -1
-2
4
-
3 0 1
-
XOR
9
Butterflies in S encoding
S encoding
0
0

0 0 -1
0
-2
-
1 1 1

0
0
-

2 1 1
2
-4
-
3 0 -1
-
XNOR
10
Butterflies in S encoding
S encoding
0
2

0 0 -1
-2
-2
-
1 1 1

-2
2
-

2 1 1
0
-2
-
3 0 1
-
X Y
11
Butterflies in S encoding
S encoding
0
2

0 0 1
2
2
-
1 1 -1

-2
2
-

2 1 1
0
2
-
3 0 1
-
X Y
12
Butterflies in S encoding
S encoding
2
2

0 0 1
-2
0
-
1 1 1

2
0
-

2 1 -1
-2
2
-
3 0 1
-
X Y
13
Butterflies in S encoding
S encoding
2
2

0 0 1
2
0
-
1 1 1

2
0
-

2 1 1
2
-2
-
3 0 -1
-
X Y
14
Butterflies in S encoding
S encoding
-2
0

0 0 -1
0
0
-
1 1 -1

-4
2
-

2 1 1
0
0
-
3 0 1
-
X
15
Butterflies in S encoding
S encoding
2
0

0 0 1
0
0
-
1 1 1

4
-2
-

2 1 -1
0
0
-
3 0 -1
-
X
16
Butterflies in S encoding
S encoding
0
0

0 0 1
4
2
-
1 1 -1

0
0
-

2 1 1
2
0
-
3 0 -1
-
Y
17
Butterflies in S encoding
S encoding
0
0

0 0 -1
-4
-2
-
1 1 1

0
0
-

2 1 -1
-2
0
-
3 0 1
-
Y
18
Butterflies in S encoding
S encoding
-2
-4

0 -1
0
0
-
1 -1

0
-2
-

2 -1
0
0
-
3 -1
-
Constant 1
19
Butterflies in S encoding
S encoding
2
4

0 0 1
0
0
-
1 1 1

0
2
-

2 1 1
0
0
-
3 0 1
-
Constant 0
20
Butterflies in R encoding
R encoding
2
0

0 0
0
0
-
1 0

0
2
-

2 0
0
0
-
3 0
-
Constant 0
21
Butterflies in R encoding
R encoding
2
4

0 1
0
0
-
1 1

0
2
-

2 1
0
0
-
3 1
-
Constant 1
22
Butterflies in R encoding
R encoding
1
2

0 0
0
-1
-
1 1

0
1
-

2 1
1
-2
-
3 0
-
XOR
23
Butterflies in R encoding
R encoding
1
2

0 1
0
1
-
1 0

0
1
-

2 0
-1
2
-
3 1
-
XNOR
24
Butterflies in R encoding
R encoding
1
3

0 0
-1
-1
-
1 1

-1
2
-

2 1
0
-1
-
3 1
-
X OR Y
25
Butterflies in S encoding
S encoding
0
-2

0 1
2
2
-
1 -1

2
-2
-

2 -1
0
2
-
3 -1
-
X OR Y
26
Butterflies in S encoding
S encoding
-2
-2

0 -1
2
0
-
1 -1

-2
0
-

2 1
2
-2
-
3 -1
-
X OR Y
Y
X
1
1
1
27

si 2 (a d) - 2n, when i ? 0.

si 2 (a d) - 2n, when i ? 0.

Since each standard function has the same number
of true and false minterms that is equal to 2n-1,
then we can have alternative definitions of
spectral coefficients.
Please note that only for the spectral
coefficient s0, is the above rule not valid and
the appropriate standard function is a tautology,
i.e. the logical function that is true for all
its minterms.
Thus, we have
a b c d 2n-1
and
si 2 (c b) - 2n

or
si 2 (a d) - 2n 2 (a 2n-1 - c) -
2n
or
si 2 (a d) - 2n
2 (a d) - (a b c d)
(a d) - (b c), when i ? 0.
and
s0 2n 2 a a b c d 2 a
b c d - a b - a,
since for s0, c and d are always equal to 0.

The spectral coefficients for incompletely
specified Boolean function, having n variables,
can be defined in the following way
s0 2n 2 a - e
and
si 2 (a d) e - 2n, when i ? 0.
As we can see, for the case when e 0, i.e. for
completely specified Boolean function, the above
formulas reduce to the formulas presented
previously.

32
All but S0 coefficients

And again, by easy mathematical transformations,
we can define all but s0 spectral coefficients in
the following way
si 2 (a d) e - 2n
2 (ad) e - (abcde)
(ad) - (bc), when i? 0

33
s0 spectral coefficient

Simultaneously, the s0 spectral coefficient can
be rewritten in the following way
s0 2n - 2a e abcde - 2a e
bcd-a b - a, since for s0, c and d are always
equal to 0.

Thus, in the final formulas, describing all
spectral coefficients, the number of don't care
minterms e can be eliminated from them.
Moreover, the final formulas are exactly the same
as the ones for completely specified Boolean
function.
Of course, it does not mean, that the spectral
coefficients for incompletely specified Boolean
function do not depend on the number of don't
care minterms.
They do depend on those numbers, but the problem
is already taken into account in the last two
formulas themselves.
Simply, the previously stated formula for the
numbers a, b, c, d, and e bonds all these values
together.

35
Properties of Transform Matrices

The transform matrix is complete and orthogonal,
and therefore, there is no information lost in
the spectrum S, concerning the minterms in
Boolean function F.
Only the Hadamard-Walsh matrix has the recursive
Kronecker product structure
and for this reason is preferred over other
possible variants of Walsh transform known in the
literature as Walsh-Kaczmarz, Rademacher-Walsh,
and Walsh-Paley transforms.
Only the Rademacher-Walsh transform is not
symmetric
all other variants of Walsh transform are
symmetric,
so that, disregarding a scaling factor, the same
matrix can be used for both the forward and
inverse transform operations.

36
Properties of Transform Matrices

When the classical matrix multiplication method
is used to generate the spectral coefficients for
different Walsh transforms, then the only
difference is the order in which particular
coefficients are created.
The values of all these coefficients are the same
for every Walsh transform.
Each spectral coefficient sI gives a correlation
value between the function F and a standard
trivial function eI corresponding to this
coefficient.
The standard trivial functions for the spectral
coefficients are, respectively,
for the coefficient s0 ( dc coefficient ) - the
universe of the Boolean function denoted by e0,
for the coefficient si ( first order coefficient
) the variable xi of the Boolean function
denoted by ei,

37
Properties of Transform Matrices cont

for the coefficient sij ( second order
coefficient ) - the exclusive-or function between
variables xi and xj of the Boolean function
denoted by eij,
for the coefficient sijk ( third order
coefficient ) - the exclusive-or function between
variables xi, xj, and xk of the Boolean function
denoted by eijk, etc.

38
Properties of Transform Matrices

The sum of all spectral coefficients sI of
spectrum S for any completely specified Boolean
function is ? 2n.
The sum of all spectral coefficients sI of
spectrum S for any incompletely specified Boolean
function is not ? 2n.

39
Properties of Transform Matrices

The maximum value of any individual spectral
coefficient sI in spectrum S is ? 2n.
This happens when the Boolean function is equal
to either a standard trivial function eI ( sign
) or to its complement ( sign - ).
In either case, all the remaining spectral
coefficients have zero values because of the
orthogonality of the transform matrix T.
Each but e0 standard trivial function eI
corresponding to n variable Boolean function has
the same number of true and false minterms equal
to 2n-1.

40
Properties of Transform Matrices

The spectrum S of each true minterm of n variable
Boolean function is given by s0 2n - 2, and all
remaining 2n - 1 spectral coefficients sI are
equal to ? 2.

S encoding
-2
0
0

m0 -1
0
0
0
-
m1 -1

-4
- 8
2
-

m2 1
0
0
0
-
m3 1

-
-2
0
0

m4 -1
-
0
0
0
-
-
m5 -1

0
-4
2
-
-

m6 1
0
0
0
-
-
m7 1
-
41
Properties of Transform Matrices

The spectrum S of each true minterm of n variable
Boolean function is given by s0 2n - 2, and all
remaining 2n - 1 spectral coefficients sI are
equal to ? 2.

S encoding
6 23 - 2
0
2

m0 -1
- 2
-2
-2
-
m1 1

- 2
-2
2
-

m2 1
- 2
0
-2
-
m3 1

-
2
- 2
4

m4 1
-
- 2
0
0
-
-
m5 1

- 2
0
2
-
-

m6 1
- 2
0
0
-
-
m7 1
-
42
Properties of Transform Matrices

The spectrum S of each true minterm of n variable
Boolean function is given by s0 2n - 2, and all
remaining 2n - 1 spectral coefficients sI are
equal to ? 2.
The spectrum S of each don't care minterm of n
variable Boolean function is given by s0 2n -
1,
and all remaining 2n - 1 spectral coefficients sI
are equal to ? 1.
The spectrum S of each false minterm of n
variable Boolean function is given by sI 0.

43
What we achieved?

1. If the function is known to be linear or
affine, by measuring once we can distinguish
which one is the linear function in the box. We
cannot distinguish a linear function from its
negation. They differ by sign that is lost in
measurement.
2. If function is a constant (zero for
satisfiability and one for tautology) we can find
with high probability that it is constant.
Thus we can solve SAT with high probability but
without knowing which input minterm satisfies. (a
single one in a Kmap of zeros)
Thus we can solve Tautology with high probability
but without knowing which input minterm fails (a
single zero in kmap of ones).

44
What we achieved?

3. We can find the highest spectral coefficients
by generating them randomly.
In terms of signals and images it gives the basic
harmonics or patterns in signal, such as
textures.
4. Instead of using Hadamard gates in the
transform we can use V to find a separation of
some Boolean functions.
But we were not able to extend this to more
variables.

45
What we achieved?

5. Knowing some high coefficients, we can
calculate their values exactly, one run
(measurement) for each spectral coefficient.

46
Symbols

a -- the number of true minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
1
b -- the number of false minterms of Boolean
function F, where the function F has the logical
value 0 and the standard trivial function has the
logical value 1
c -- the number of true minterms of Boolean
function F, where the function F has the logical
value 1 and the standard trivial function has the
logical value 0
d -- the number of false minterms of Boolean
function F, where both the function F and the
standard trivial function have the logical values
0, and e be the number of don't care minterms of
Boolean function F.