FastXFormSynths.ppt

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Fast Spectral Transforms andLogic Synthesis

DoRon Motter

August 2, 2001


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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 singlepoint


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Spectral Transformation

Synthesis

Many algorithms proposed leveraging fast

transformation

Verification

Correctness may be checked more efficiently

using a spectral representation.


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ReviewLinear Algebra

Let M be a real-valued square matrix.

The transposed matrix Mt is found by interchangingrows and columns

M is orthogonal if

MMt = MtM = I

M is orthogonal up to the constant kif

MMt = MtM = kI

M-1 is the inverse ofM if

MM

-1

= M-1

M = IM has its inverse iff the column vectors ofM are

linearly independent


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ReviewLinear Algebra

Let A and B be (nn) square matrices

Define the Kronecker product of A and B as

BBB

BBB

BBB

BA

nnnn

n

n

aaa

aaaaaa

21

22221

11211


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Spectral Transform

General Transform Idea

Consider the 2n output values of f as a columnvectorF

Find some transformation matrix T(n) andpossibly its inverse T-1(n)

Produce RF, the spectrum ofF, as a columnvector by

RF = T(n)F F = T-1(n)RF


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Walsh-Hadamard Transform

The Walsh-Hadamard Transform is defined:

1)0( T

)1()1(

)1()1()(

nn

nnn

TT

TTT


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Walsh-Hadamard Matrices

1111

1111

1111

1111

(2)

11

11)1(

T

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

(3)T


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Walsh-Hadamard Example

1

00

0

0

1

1

0

11111111

1111111111111111

11111111

11111111

11111111

11111111

11111111

321321321321 ),,( xxxxxxxxxxxxf


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1

00

0

0

1

1

0

11111111

1111111111111111

11111111

11111111

11111111

11111111

11111111

321321321321 ),,( xxxxxxxxxxxxf 3


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3

11

1

1

1

1

3

1

00

0

0

1

1

0

11111111

1111111111111111

11111111

11111111

11111111

11111111

11111111



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6

11

1

2

2

2

2

1

11

1

1

1

1

1

11111111

1111111111111111

11111111

11111111

11111111

11111111

11111111



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Why is it useful?

Each row vector ofT(n) has a meaning

1x1

x2

x1x2

Since we take the dot product of the row vector

with F we find the correlation between the two

1111

1111

11111111

(2)T


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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


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Alternate Definition

Recursive Kronecker Structure gives rise to

DD/Graph Algorithm

11

11)(

1

n

inT


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Alternate Definition

Note, T is orthogonal up to the constant 2:

in

i

xn 211)(1

T

I 22002

1111

1111)1()1(

t


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Decision Diagrams

A Decision Diagram has an implicit

transformation in its function expansion

Suppose

This mapping defines an expansion of f

)1()(1

TT

n

i

n

1

01

)1()1( f

f

fTT


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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


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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?


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Walsh Transform DDs

in

ixn 211)(

1

T

1010

1

0

1

01

212

1

11

11211

2

1

)1()1(

ffxfff

f

fxf

fff

i

i

TT


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Walsh Transform DDs

1010 212

1ffxfff i

1021

ff 1021

ff

1 ix21


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Walsh Transform DDs

It is possible to convert a BDD into a

WTDD via a graph traversal.

The algorithm essentially does a DFS


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Applications: Synthesis

Several different approaches (all very

promising) use spectral techniques

SPECTRESpectral Translation

Using Spectral Information as a heuristic

Iterative Synthesis based on Spectral

Transforms


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Thorntons 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 DDs


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Thorntons Method

Constituent Functions

Boolean functions whose output vectors are the

rows of the transformation matrix

If we use XOR as the primitive, we get therows for the Walsh-Hadamard matrix

Other functions are also permissible


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Thorntons 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 library3. Compute spectral coefficients

4. Choose largest magnitude coefficient

5. Realize constituent functionFc(x) correspondingto this coefficient


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Thorntons Method

6. Compute the errore(x) =Fc(x) F withrespect to some operator,

7. If e(x) indicates w or fewer errors,

continue to 8. Otherwise iterate bysynthesizing e(x)

8. Combine intermediate realizations ofchosen {F

c(x)} using and directly realize

e(x) for the remaining w or fewer errors


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Thorntons Method

Guaranteed to converge

Creates completely fan-out free circuits

Essentially a repeated correlation analysis

Extends easily to multiple gate libraries


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Thorntons Method: Example

Step 1: Create the truth table using {-1, 1}

322132131)( xxxxxxxxxxf

1111

1111

1111

1111

1111

1111

1111

1111

321

Fxxx


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Step 2: Compute transformation matrix T

using constituent functions.

In this case, well use AND, OR, XOR


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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


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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


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Step 5: Realize the constituent function FcIn this case we use XNOR since the coefficient

is negative


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Step 6: Compute the error function e(x)

Use XOR as a robust error operator

111111

111111

111111

111111

111111

111111

111111

111111

321

eFxxx cF


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Step 7: Since there is only a single error, we

can stop, and realize the final term directly


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Conclusion

The combination of spectral transforms and

implicit representation has many

applications

Many ways to leverage the spectralinformation for synthesis

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