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 dont care minterms of
  Boolean function F.

3

• 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.

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
s2 2 (a d) - 2n 2(11) 4 0
28

• 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
s1 2 (a d) - 2n 2(11) 4 0
29

• 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

30

• 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.

31

• 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.

34

• Thus in the final formulas describing all
  spectral coefficients the number of dont 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 dont
  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 dont 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 dont care minterms of
  Boolean function F.

• 1
• 1 1

0 0 0 0
0 0 0 0

• 1
• 1 1

a8
b0
c0 d8
47

• si 2 (a d) - 2n
• 2 (a d) - (a b c d)
• (a d) - (b c) when i 0.

• 1
• 1 1

0 0 0 0
si (a d) - (b c) 16
0 0 0 0

• 1
• 1 1

b0
a8
c0 d8
48

• si 2 (a d) - 2n
• 2 (a d) - (a b c d)
• (a d) - (b c) when i 0.

• 1
• 1 1

0 0 0 0
si (a d) - (b c) 16
0 0 0 0

• 0
• 1 1

Ones inside
zeros inside
Ones outside
Zeros outside
b1
a7
c0 d8
(78 ) - (10) 14
49

• si 2 (a d) - 2n
• 2 (a d) - (a b c d)
• (a d) - (b c) when i 0.

• 1
• 1 1

0 0 0 0
si (a d) - (b c) -16

• 1
• 1 1

0 0 0 0
b8
a0
c8 d0
50

• si 2 (a d) - 2n
• 2 (a d) - (a b c d)
• (a d) - (b c) when i 0.

• 1
• 1 1

0 0 0 0
si (a d) - (b c) 0

• 1
• 1 1

0 0 0 0
b4
a4
c4 d4
51

• si 2 (a d) - 2n
• 2 (a d) - (a b c d)
• (a d) - (b c) when i 0.

• 1
• 1 1

0 0 0 0
si (a d) - (b c) 0

• 1
• 1 1

0 0 0 0
b4
a4
c4 d4
52

• si 2 (a d) - 2n
• 2 (a d) - (a b c d)
• (a d) - (b c) when i 0.

• 1
• 1 1

0 0 0 0
si (a d) - (b c) (34) (44) -1

• 0
• 1 1

0 0 0 0
b4
a3
c4 d4
53
Problems for students

• Butterfly for Hadamard
• Software for Hadamard
• Matrices for Hadamard
• Hadamard coefficients
• Basic Orthogonal functions and their meanings.
  Use Kmaps.