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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

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2 Outline Taylor Expansion Diagram TED-based Factorization DSP example Results Conclusions

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3 Taylor Expansion Diagram Graph based representation of arithmetical expression. Based on Taylor Series Expansion: f x f(0)f ’(0)f ’’(0)/2 1 x x2x2

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4 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)f x ’(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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5 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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6 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) for j in 0 to N-1 loop temp:=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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7 TED-based Factorization, Example DCT - Direct implementation: Y=M*X 12 Additions 16 Multiplications

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8 TED-based Factorization TED for the DCTII size 4 These nodes and associated sub-graphs are shared by Y1, Y3. x0-x3 x1-x2

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9 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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10 TED-based Factorization S2=x1-x2 S3=x1+x2 Continue with next substitutions:

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11 TED-based Factorization S0=x0-x3; S1=x0+x3; S2=x1-x2; S3=x1+x2; Y0=S3+S1; Y1=A*S0+B*S2 Y2=C*(S1-S3); Y3=-A*S2+B*S0 No more candidates can be found for common sub-expression elimination Each sub expression S n in this graph is represented by an adder The expressions can be rewritten as: 8 Additions 5 Multiplications

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12 TED-based Factorization Algorithm

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13 Results

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14 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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15 TEDify: a tool to optimize mathematical expressions using TEDs Available at: Software: TEDify

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16 Thanks Any questions ?

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17 Results Transform:Original # ADDOriginal # MPY# ADD after TED# MPY after TEDTime WHT 4x ,08 WHT 8x ,09 WHT 16x ,211 WHT 32x ,768 WHT 64x ,158 DCT 4x ,084 DCT 8x ,097 DCT 16x ,182 DCT 32x ,210 DCT 64x ,035 DCT128x DHT 4x ,092 DHT 8x ,094 DHT 16x ,195 DHT32x ,386 DHT 64x ,98 DHT 128x DHT 256x

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