Factorization of DSP Transforms using Taylor Expansion Diagram

Jeremie Guillot, E. Boutillon

M.Ciesielski *, D. Gomez-Prado*, Q.Ren*, S. Askar*

LESTER Lab, Université de Bretagne SUD

*VLSI CAD Lab, University of Massachusetts, Amherst

2

Outline

• Taylor Expansion Diagram

• TED-based Factorization

• DSP example

• Results

• Conclusions

3

Taylor Expansion Diagram

...)0(2

1)0()0()(

!

1)( "2'

000

fxfxfxf)x(xk

xf kk

k

• Graph based representation of arithmetical expression.

• Based on Taylor Series Expansion:

f

x

f(0) f ’(0) f ’’(0)/2

1x x2

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Your First TED

• Example:

f(x,y)=5x+3y+5xy-3

• Taylor decomposition:• f(x,y)= (3y-3) + x*(5y+5)

• g(y) = -3+y*(3)

• h(y) = 5+y*(5)

• Representation used by the tool:(^0 -3) means an (additive) edge

with power 0 and weight -3

f(0)=g(y) fx’(0)=h(y)

f(x,y)x

g(y) h(y)

f(x,y)x

y y

one

^1 5^0 5

^0 -3^1 3

^0 1 ^1 1

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• After normalization:

• And more…

Properties:– Acyclic and oriented graph.

– Compact representation of linear expression.

– When the graph is reduced, ordered and normalized, it is canonical.

– For a given functionality, there exists only one representation

• useful for verification, equivalence checking…)

– Handles word-level & bit-level.

Your First TED, cont’d

^0 -3

f(x,y)

x

y y

ONE

^1 5

^0 5

^1 3

^0 1^1 1

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Discrete Cosine Transform,

one of the main block in JPEG/MPEG compression

TED-based Factorization, Example

• DCT can be expressed as follows:

• A direct implementation: (N=4)

)2

1(cos)(

1

0

njN

xjyN

nn

for j in 0 to N-1 looptemp:=0;for n in 0 to N-1 loop

temp:=temp+x(n)*cosine(n,j);end loop;y(j)<=temp;

end loop;

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TED-based Factorization, Example• DCT - Direct implementation: Y=M*X

BAAB

CCCC

ABBA

DDDD

M

)8

5cos()

8cos()

8

7cos()

8

3cos(

)4

7cos()

4

5cos()

4

3cos()

4cos(

)8

7cos()

8

5cos()

8

3cos()

8cos(

1111

3*2*1*0*3

3*2*1*0*2

3*2*1*0*1

3*2*1*0*0

XBXAXAXBY

XCXCXCXCY

XAXBXBXAY

XDXDXDXDY

12 Additions

16 Multiplications

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TED-based Factorization• TED for the DCTII size 4

These nodes and associated sub-graphs are shared by Y1, Y3.

x0-x3

x1-x2

).4

cos( ,)8

3cos( ),

8cos(

CBAwith

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S0=x0-x3

S1=x0+x3

TED-based Factorization

• Changing variable order helps identify candidates for CSE.

• Reuse sub-expressions by creating new variables:

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TED-based Factorization

S2=x1-x2

S3=x1+x2

• Continue with next substitutions:

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TED-based Factorization

S0=x0-x3; S1=x0+x3; S2=x1-x2; S3=x1+x2;Y0=S3+S1;Y1=A*S0+B*S2Y2=C*(S1-S3);Y3=-A*S2+B*S0

• No more candidates can be found for common sub-expression elimination

• Each sub expression Sn in this graph is represented by an adder

• The expressions can be rewritten as:

8 Additions 5 Multiplications

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TED-based Factorization Algorithm

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Results

WH

T 4

x4W

HT

8x8

WH

T 1

6x16

WH

T 3

2x32

WH

T 6

4x64

DC

T 4

x4D

CT

8x8

DC

T 1

6x16

DC

T 3

2x32

DC

T 6

4x64

DC

T12

8x12

8D

HT

4x4

DH

T 8

x8D

HT

16x

16

DH

T32

x32

DH

T 6

4x64

DH

T 1

28x1

28

DH

T 2

56x2

56M

EA

N

0,00%10,00%20,00%30,00%40,00%50,00%60,00%70,00%80,00%90,00%

100,00%

Optimization Rate

Optimization Rate ADD Optimization Rate MPY

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Conclusions

• TED makes the CSE process straightforward.

• It extracts the functionality from the specification and reduces computation.

• Other factorization schemes are currently under development (Radix Decomposition, etc.).

• Applications:• High Level Synthesis.

• Compilation

• Mathematical software…

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• TEDify: a tool to optimize mathematical expressions using TEDs

• Available at: http://tango.ecs.umass.edu/TED/Doc/html/index.html

Software: TEDify

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Thanks• Any questions ?

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ResultsTransform: Original # ADD Original # MPY # ADD after TED # MPY after TED Time

WHT 4x4 12 16 8 0 0,08

WHT 8x8 56 64 24 0 0,09

WHT 16x16 240 256 64 0 0,211

WHT 32x32 992 1024 160 0 1,768

WHT 64x64 4032 4096 384 0 27,158

DCT 4x4 12 16 8 5 0,084

DCT 8x8 56 64 34 21 0,097

DCT 16x16 240 256 126 85 0,182

DCT 32x32 992 1024 454 341 1,210

DCT 64x64 4032 4096 1654 1365 16,035

DCT128x128 16256 16384 6166 5461 468

DHT 4x4 12 16 8 0 0,092

DHT 8x8 56 64 32 4 0,094

DHT 16x16 240 256 112 28 0,195

DHT32x32 992 1024 360 140 1,386

DHT 64x64 4032 4096 1200 620 17,98

DHT 128x128 16256 16384 4016 2604 340

DHT 256x256 65280 65536 14000 10668 10756