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Fast Spectral Transforms and Logic Synthesis

DoRon Motter

August 2, 2001

Introduction

• Truth Table Representation– Value provides complete information for one

combination of input variables

– Provides no information about other combinations

• Spectral Representation– Value provides some information about the behavior of

the function at multiple points

– Does not contain complete information about any single point

Spectral Transformation

• Synthesis– Many algorithms proposed leveraging fast

transformation

• Verification– Correctness may be checked more efficiently

using a spectral representation.

Review – Linear Algebra

• Let M be a real-valued square matrix.– The transposed matrix Mt is found by interchanging

rows and columns– M is orthogonal if

MMt = MtM = I

– M is orthogonal up to the constant k if MMt = MtM = kI

– M-1 is the inverse of M if MM-1 = M-1M = I

– M has its inverse iff the column vectors of M are linearly independent

Review – Linear Algebra

• Let A and B be (n n) square matrices

• Define the Kronecker product of A and B as

BBB

BBB

BBB

BA

nnnn

n

n

aaa

aaa

aaa

21

22221

11211

Spectral Transform

• General Transform Idea– Consider the 2n output values of f as a column

vector F– Find some transformation matrix T(n) and

possibly its inverse T-1(n)– Produce RF, the spectrum of F, as a column

vector by• RF = T(n)F• F = T-1(n)RF

Walsh-Hadamard Transform

• The Walsh-Hadamard Transform is defined:

1)0( T

)1()1(

)1()1()(

nn

nnn

TT

TTT

Walsh-Hadamard Matrices

1111

1111

1111

1111

(2)

11

11)1(

T

Τ

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

(3)T

Walsh-Hadamard Example

1

0

0

0

0

1

1

0

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

321321321321 ),,( xxxxxxxxxxxxf


1

0

0

0

0

1

1

0

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111


3


3

1

1

1

1

1

1

3

1

0

0

0

0

1

1

0

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111



6

1

1

1

2

2

2

2

1

1

1

1

1

1

1

1

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111



• Why is it useful?– Each row vector of T(n) has a ‘meaning’

1

x1

x2

x1 x2

– Since we take the dot product of the row vector with F we find the correlation between the two

1111

1111

1111

1111

(2)T


Constant, call it x0

x1

x2

x1 x2

x3

x1 x3

x2 x3

x1 x2 x3

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

(3)T


• Alternate Definition

• Recursive Kronecker Structure gives rise to DD/Graph Algorithm

11

11)(

1

n

inT


• Alternate Definition

• Note, T is orthogonal up to the constant 2:

in

ixn 211)(

1

T

IΤΤ 220

02

11

11

11

11)1()1(

t

Decision Diagrams

• A Decision Diagram has an implicit transformation in its function expansion– Suppose

– This mapping defines an expansion of f

)1()(1TT

n

in

1

01 )1()1(f

ff TT

Binary Decision Diagrams

• To understand this expansion better, consider the identity transformation

• Symbolically,

101

0

1

01 )1()1(

fxfxf

fxxf

f

ff

iiii

II

ii xxn

10

01)(T

Binary Decision Diagrams

• The expansion of f defines the node semantics

• By using the identity transform, we get standard BDDs

• What happens if we use the Walsh Transform?

Walsh Transform DDs

in

ixn 211)(

1

T

1010

1

0

1

01

212

1

11

11211

2

1

)1()1(

ffxfff

f

fxf

f

ff

i

i

TT

Walsh Transform DDs

1010 212

1ffxfff i •

102

1ff 102

1ff

1 ix21

Walsh Transform DDs

• It is possible to convert a BDD into a WTDD via a graph traversal.

• The algorithm essentially does a DFS

Applications: Synthesis

• Several different approaches (all very promising) use spectral techniques– SPECTRE – Spectral Translation– Using Spectral Information as a heuristic– Iterative Synthesis based on Spectral

Transforms

Thornton’s Method

• M. Thornton developed an iterative technique for combination logic synthesis

• Technique is based on finding correlation with constituent functions

• Needs a more arbitrary transformation than Walsh-Hadamard– This is still possible quickly with DD’s

Thornton’s Method

• Constituent Functions– Boolean functions whose output vectors are the

rows of the transformation matrix– If we use XOR as the primitive, we get the

rows for the Walsh-Hadamard matrix• Other functions are also permissible

Thornton’s Method

1. Convert the truth table F from {0, 1} to {1, -1}

2. Compute transformation matrix T using constituent functions {Fc(x)}

• Constituent functions are implied via gate library

3. Compute spectral coefficients

4. Choose largest magnitude coefficient

5. Realize constituent function Fc(x) corresponding to this coefficient

Thornton’s Method

6. Compute the error e(x) = Fc(x) F with respect to some operator,

7. If e(x) indicates w or fewer errors, continue to 8. Otherwise iterate by synthesizing e(x)

8. Combine intermediate realizations of chosen {Fc(x)} using and directly realize e(x) for the remaining w or fewer errors

Thornton’s Method

• Guaranteed to converge

• Creates completely fan-out free circuits

• Essentially a repeated correlation analysis

• Extends easily to multiple gate libraries

Thornton’s Method: Example

• Step 1: Create the truth table using {-1, 1}322132131)( xxxxxxxxxxf

1111

1111

1111

1111

1111

1111

1111

1111321

Fxxx


• Step 2: Compute transformation matrix T using constituent functions.

• In this case, we’ll use AND, OR, XOR


1

x1

x2

x3

x1 x2

x1 x3

x2 x3

x1 x2 x3

x1 + x2

x1 + x3

x2 + x3

x1 + x2 + x3

x1x2

x1x3

x2x3

x1x2x3

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111


• Step 3: Compute spectral coefficients

• S[Fc(x)]= [-2, -2, 2, -2, 2, -2, 2, -6, 2,

-2, 2, 2, -2, -2, -2, -4]

• Step 4: Choose largest magnitude coefficient.– In this case, it corresponds to x1 x2 x3


• Step 5: Realize the constituent function Fc

– In this case we use XNOR since the coefficient is negative


• Step 6: Compute the error function e(x)– Use XOR as a robust error operator

111111

111111

111111

111111

111111

111111

111111

111111321

eFxxx cF


• Step 7: Since there is only a single error, we can stop, and realize the final term directly

Conclusion

• The combination of spectral transforms and implicit representation has many applications

• Many ways to leverage the spectral information for synthesis