Spectral Techniques

in Logic Design

Claudio Moraga

University of Dortmund Germany

Radomir S. Stanković

Technical University of Niš Serbia

Tempus Working Group on Logic Design

Tempus Project CD-JEP-16160-2001; Athens, April 21st. 2003

 

Introduction

B = {0,1}

(B, ) : Boolean Algebra

(B, · ) : Galois Field GF(2), Boolean Ring, (with x x’ = 1)

Extensions to Bn

Logic design:

Problem Circuit

“AND-OR-NOT” Logic ;

Reed-Muller expressions

Zhegalkin polynomials

Truth table Normal Forms Minimization

Fourier transform of functions Bn B Walsh transform

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“AND-EXOR” Logic

Symmetries and Co-Symmetries

Let C(i,k) denote the context of z with respect to the i-th and k-th

components.

C(i,k) = { z Bn| zi=a, zk=b, a,b {0, 1} }. The elements of C(i,k)

will be simply denoted by z(ab). Similarly for the context of w.

Definition: f : Bn {0, 1} has:

E{zi, zk} f(z(11)) = f(z(00)) EC{zi, zk} f(z(11)) = f ’(z(00))

N{zi, zk} f(z(10)) = f(z(01)) NC{zi, zk} f(z(10)) = f ’(z(01))

P{zi (zk)} f(z(11)) = f(z(01)) PC{zi (zk)} f(z(11)) = f ’(z(01))

P{zi (z’k)} f(z(10)) = f(z(00)) PC{zi (z’k)} f(z(10)) = f ’(z(00))

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Spectral characterization of Symmetries

f: Bn {0, 1} has:

E{zi, zj} S(w(10)) = -S(w(01)) EC{zi, zj} S(w(00)) = -S(w(11))

N{zi, zj} S(w(10)) = S(w(01)) NC{zi, zj} S(w(00)) = S(w(11))

M{zi, zj} S(w(10)) = S(w(01))

=0

MC{zi, zj} S(w(00)) = S(w(11)) = 0

P{zi (zj)} S(w(11)) = S(w(10)) PC{zi (zj)} S(w(00)) = S(w(01))

P{zi (z’j)} S(w(11)) = - S(w(10)) PC{zi (z’j)} S(w(00)) = - S(w(01))

Notation: f S. S denotes the Walsh spectrum of f

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Test of Partial Symmetries [P{zi(zj‘) } S(w(11)) = (-)S(w(10))]

f (z) z1 z0 16S (w) w1 w0 00 01 10 11 00 01 10 11

00 1 0 1 0 00 - 4 - 4 0 0 z3 z2 01 0 1 1 1 w3 w2 01 - 4 - 4 0 0 10 1 1 1 1 10 0 0 4 4 11 1 0 0 0 11 8 -8 - 4 - 4

i=1, j=2, P{z1(z’2)}

i=1, j=0, P{z1(z0)}

w2 = w1 = 1 w1 = w0 = 1 w 6 7 14 15 3 7 11 15

w 2j 2 3 10 11 2 6 10 14

2n S(w) 0 0 - 4 - 4 0 0 4 - 4

2n S(w 2j) 0 0 4 4 0 0 4 - 4

1 2 30 4 5 6 7 8 9 10 11

12 13 14 15

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Using (Partial) Symmetries

P{z1(z’2)} f(z) | = f(z)| z1=1 z1=0

z2=0 z2=0

* if (z2 = 0) then f(z) indep. of z1

P{z1(z0)} f(z) | = f(z) | z1=1 z1=0

z0=1 z0=1

* if (z0 = 1) then f(z) indep. of z1

* if (z2 = 0) or (z0 = 1) then f(z) is indep. of z1

* if (z2 = 0) or (z0 = 1) and z1 =0 then z1 :=1 (else z1 = z1)

OR

z0

z1

z2

z3

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Example

f (z) z1 z0 00 01 10 11

00 1 0 1 0 f(z) = z’2z’0 z’3z1z’0 z’3z2z0 z3 z2 01 0 1 1 1 z3z’1z’0 z3z’2 10 1 1 1 1

11 1 0 0 0 c = (19, 2)

(max fan-in 5)

c = (20, 3) (max fan-in 4)

f(z) = z2 z3 z’0z’2(z1 z3) z’0z’1

c = (14, 3) (max fan-in 3)

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g (z) z1 z0 00 01 10 11

00 1 0 z3 z2 01 0 1 1 10 1 1

11 1 0 0

z0 z1 z2 z3

OR

g

f(z)

g (z) z1 z0 00 01 10 11

00 0 0 1 0 z3 z2 01 0 0 1 1 10 1 1 1 1 11 1 1 0 0

z0 z1 z2 z3

& OR

f(z)

&

&

&

OR

(max fan-in 4)c = (18, 3)

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g (z) z1 z0 00 01 10 11

00 1 0 z3 z2 01 0 1 1 10 1 1 11 1 0 0

z0 z1 z2 z3

OR

g

f(z)

g (z) z1 z0 00 01 10 11

00 1 1 1 0 z3 z2 01 0 0 1 1 10 1 0 1 1 11 1 1 0 0

OR

&

OR

f(z) z0

z1

z2

z3

c = (12, 4) (max fan-in 3)© cm

Test of Partial Symmetries [P{zi(zj‘) } S(w(11)) = (-)S(w(10))]

f (z) z1 z0 16S (w) w1 w0 00 01 10 11 00 01 10 11

00 1 0 1 0 00 - 4 - 4 0 0 z3 z2 01 0 1 1 1 w3 w2 01 - 4 - 4 0 0 10 1 1 1 1 10 0 0 4 4 11 1 0 0 0 11 8 -8 - 4 - 4

S(1111) = S(1110) possibly P{z1(z0)}, P{z2(z0)}, P{z3(z0)}

S(1111) = -S(1011) possibly P{z1(z‘2)}, P{z0(z‘2)}, P{z3(z‘2)},

|S(1111)| |S(1101)| impossible: P{z3(z1)}, P{z2(z1)}, P{z0(z1)},

P{z3(z‘1)}, P{z2(z‘1)}, P{z0(z‘1)}

|S(1111)| |S(0111)| impossible: P{z2(z3)}, P{z1(z3)}, P{z0(z3)},

P{z2(z‘3)}, P{z1(z‘3)}, P{z0(z‘3)}

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Spectral Logic Design (Here: restricted to the Walsh Xform.)

1. Fundamentals

The Walsh functions, properties

The Walsh transform; fast algorithms

Spectral decision diagrams

Walsh spectrum of logic functions, properties

2. Spectral Logic Design based on symmetries and co-symmetries

3. Spectral Logic Design based on autocorrelations

4. Spectral Logic Design using minterm exchange

5. Spectral Logic Design based on linearizations

(1 Semester, 2 hrs a week)

Version including Reed Muller, Haar and Arithmetic transforms: 1 Semester, 4 hrs a week

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Walsh (Hadamard) Functions

jj

1- n

1j

n zw zw, let z w,

B

1n

1jjjjj

1n

1j

zwzww.z

w 1)(1)((-1)]zw,)1exp[((z) πχ

1n

1j

zwjj

1n

1j

jj1)()zw,h(zwh

Definition:

h : B {1, -1}; x B h(x) = (-1)x = 1 – 2x

h(0) = 1 h(1) = -1

Furthermore, h : (B, ) ({1, -1}, · )

x, y B h(x y) = h(x)·h(y)

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Matrix representation of Walsh functions

Define X(n) = [ i(j)] , where i,j {0, 1, ..., 2n-1}

X (1) 1 1

1 1

and X(2)

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

Then:

i.e. X(2) = X(1) X(1) X(n) = X(n-1) X(1) = X(1) X(n-1)

Moreover: X(1) = 2 (X(1))-1 X(n) = 2n (X(n))-1

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Walsh spectrum of Boolean functions

Let f: Bn {0, 1}. z, w B holds the following:

S(w) = 2 -n hf(z) w(z) zBn

f(z) = h-1 S(w) w(z) wBn

R(w) = 2 -n f(z) w(z) zBn

f(z) = R(w) w(z) wBn

f S

f R

S = 2-n X(n) hF

hF = X(n) S .

R = 2-n X(n) F F = X(n) R

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The ”Butterfly“ algorithms

f(1)

f(0)

1-1

1 1 =

f(1)

f(0)(1)X

f(0) f(0) + f(1) f(1) f(0) - f(1)

f(3)

f(2) –

f(1)

f(0)f(3)

f(2) +

f(1)

f(0)

=

f(3)

f(2)

f(1)

f(0)

=

f(3)

f(2)

f(1)

f(0)

(1)(1)

(1)(1)

(1)(1)(2)

XX

XX

XXX

f(0) f(0) + f(1) + f(2) + f(3) = R(0)

f(1) f(0) – f(1) + f(2) – f(3) = R(1)

f(2) f(0) + f(1) – f(2) – f(3) = R(2)

f(3) f(0) – f(1) – f(2) + f(3) = R(3)

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Properties

Let L be a non singular matrix in (Bn, , · ).

Moreover let f,g: Bn {0, 1}; w, z, b Bn; f Sf and g Sg

P1: (f(z))’ - Sf

P2: f(L · z) Sf (w · L-1)

P3: f(z 2k) w(2k) Sf (w) (0 < k < n)

P4: f(z) zj Sf(w 2j)

P5: f(z) g(z) (Sf * Sg )(w)

P6: h(zj) 2j (z)

P7: h: ( L, ) ({w }, · )

with L = {£:Bn {0, 1} | b Bn, £(z) = <b,z>}

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