An Operator-Based Approach to the

Unfolding Transformation

Marvi Teixeira

Polytechnic University of Puerto Rico

IEEE MWSCAS 2008

Outline

� Definitions

� Applications

� Motivation /Objectives� Motivation /Objectives

� Mathematical Background

� The Role of Stride Permutations in Unfolding Data Flow Graphs of Cyclic Shifts

� An Operator-Based Approach to Unfold DFGs of Cyclic Shifts

� Unfolding DFGs of non-Cyclic shifts.

� Example

� Conclusions/Future Work

Definitions

� Unfolding can be seen as a technique that maps a

DSP program to a new program that describes two

or more iterations of the original program. or more iterations of the original program.

� The number of such iterations is called the unfolding

factor.

� The unfolding transformation can be applied to the

program’s data-flow graph (DFG).

Applications of Unfolding

� Design of high-speed and low-power VLSI architectures.

� Design of parallel architectures with implications regarding throughput increases or power consumption reductions.

� Others

Motivation

� Unfolding algorithms, even if proven to be

correct, appear to be developed ad hoc, may correct, appear to be developed ad hoc, may

have no clear theoretical origins and lack a

simple mathematical notation suitable for

further manipulation.

Objectives

� Ultimate Goal: Provide, within a unified theoretical

framework, an operator-based approach to unfold

DSP programs.DSP programs.

� Tackle first the problem of unfolding data flow graphs

of cyclic and non-cyclic shifts.

� Tackle more complicated DSP programs as future

work.

� The Cyclic Shift Operator:

0 0 0 ... 0 1 0 1 0 0 0 0 x x

Mathematical Background (a)

0 0 0 ... 0 1

1 0 0 ... 0 0

0 1 0 ... 0 0

0 0 1 ... 0 0

. . . .. . . .

0 0 0 ... 1 0

SN

=

0 1

1 2

2 3

3 4

4 5

5 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1 0 0 0 0 0

56

x x

x x

x x

x x

x x

x x

S x

= =

Length-N, order 1, Cyclic Shift

Operator

Length-6 order 5 Cyclic Shift

Mathematical Background (b)

Data Flow Graph of a Cyclic Shift

0 1

1 2

2 3

3 4

4 5

5 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1 0 0 0 0 0

56

x x

x x

x x

x x

x x

x x

S x

= =

x0

x1

x2

x3

x4

x5y5 = x0

y0 = x1

y1 = x2

y2 = x3

y3 = x4

y4 = x5

xi yj

Mathematical Background (c)

Blocking Circulant Matrices (a):� Circulant matrices of composite size can be reordered in

blocked form by using stride permutations. The result is a blocked form by using stride permutations. The result is a

Block Pseudocirculant Matrix.

xPPHPyP rNrNNrNrN 0000 ,

1

,,,

−=

1

,, 000

−= rNNrNpr PHPH

�The matrix associated to the cyclic shift operator is circulant and therefore amenable to be written in blocked form.

Mathematical Background (d)

�Blocking Circulant Matrices (b): Example

0 0

'1 2 ' 10

y y

y y yP y P y P H P P x

−

= = = = =

0 3 2 1 0h h h h x

h h h h x

1 2 ' 10

4 ,2 4 ,2 4 ,2 4 4 ,2 4 ,2'2 1 1

3 3

y y yP y P y P H P P x

y y y

y y

− = = = = =

1 0 3 2 11

4 ,2 4 ,2 4 ,2

2 1 0 3 2

3 2 1 0 3

h h h h xP P P

h h h h x

h h h h x

−

=

0 2 3 1 0

'2 0 1 3 2 0 2 1 '0

2'1 3 0 2 1 1 0 1

3 1 2 0 3

p

h h h h x

h h h h x H S H xH x

h h h h x H H x

h h h h x

= = =

1

4,2 4,2

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

P P−

= =

The Role of Stride Permutations in Unfolding the DFG of Cyclic Shifts (a)

{'

' 5 1 '

6 ,3 6 ,3 6 6 ,3 6 ,3

x

y P y P S P P x x−

= = =

p 3

p 3

S

S142 43

0 0

1 3 '

y y

y yy

1 3 '

0

2 1' ' 5 1

6 ,3 6 ,3 1 6 ,3 6 6 ,3 6 ,3

3 4 '

2

4 2

5 5

0

1

21

6 ,3 6 ,3 6 ,3

3

4

5

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1 0 0 0 0 0

y yy

y yy P y P y P S P P x

y yy

y y

y y

x

x

xP P P

x

x

x

−

−

= = = = =

=

The Role of Stride Permutations in Unfolding DFGs of Cyclic Shifts (b)

0

3 '

2 0

1' ' '

2 1 p3

0 0 1 0 0 0

0 0 0 1 0 00 0

0 0 0 0 1 00 0

x

xI x

xy I x x

= = =

S2 1 p3

4 '

2 2

2

5

0 00 0 0 0 0 1

0 00 1 0 0 0 0

1 0 0 0 0 0

y I x xx

S xx

x

= = =

S

( ) ( ) ( )5 1 26 ,3 6 6 ,3 3 2 2 2 23

' ' ' 'p

y P S P x x S I S I I x−

= = = ⊗ ⊕ ⊕S

' '

2 2 0 2 0

' ' ' '

2 2 1 2 1 p3

' '

2 2 2 2 2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

22 2 2 3

3 2

I S x I x

y I I x I x x

I I x S x

S I IS I p

= = =

⊕ ⊕⊗

S

S14424431442443 1442443

The Role of Stride Permutations in Unfolding DFGs of Cyclic Shifts (c)

' '

2 2 0 2 0

' ' ' '

2 2 1 2 1 p3

' '

2 2 2 2 2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

22 2 2 33 2

I S x I x

y I I x I x x

I I x S x

S I IS I p

= = =

⊕ ⊕⊗

S

S14424431442443 1442443

x0

x3

x1

x4

x2

x5

x1

x4

x2

x5

x3

x0

2 2 2S I I⊕ ⊕ 2

3 2S I⊗

'

0y

'

1y

'

2y

'

0x

'

1x

'

2x

1

2 2S S=

2

2 2I S=

2

2 2I S=

22 2 2 33 2

S I IS I p⊕ ⊕

⊗S

( ) ( ) ( )5 1 26 ,3 6 6 ,3 3 2 2 2 23

' ' ' 'p

y P S P x x S I S I I x−

= = = ⊗ ⊕ ⊕S

An Operator Based Approach to Unfold DFGs of Cyclic Shifts (a)

(( )) 1

, ,' ' '

M N

p r N r N N ro o oy x P S P x

−= =S

S

14243p ro

S

( )1 /

(( ))

/0

(( ))

' '/

ro N roM r ooN rr oo i o

i MIn t

ry S I x

N rS

−

=

+

= ⊗

⊕

For M=5, N=6 and r0=3 this equation gives back the equation from previous example.

An Operator Based Approach to Unfold DFGs of Cyclic Shifts (b)

� The operator-based formulation in previous

equation effectively embodies an algorithm to

r0-unfold the DFG of an order-M cyclic shift r0-unfold the DFG of an order-M cyclic shift

acting on a length-N signal.

� A detailed proof by induction will be

published elsewhere.

Unfolding DFGs of non-Cyclic Shifts (a)

� Example: Unfold the DFG of a delay by five.

� y[n] = x[n-5]

� y[3k] = x[3k -5]= x[3(k-2) +1]

y[3k+1]= x[3k+1 -5]= x[3k -4]= x[3(k-2 )+2]� y[3k+1]= x[3k+1 -5]= x[3k -4]= x[3(k-2 )+2]

� y[3k+2] = x[3k+2 -5]= x[3k -3]=x[3(k-1)]

x y5D

'

0x

'

1x

'

2x

'

0y

'

1y

'

2y

1D

2D

2D

Unfolding DFGs of non Cyclic Shifts (b)

� By disregarding the modularity constraint, brought about by the circular reordering imposed by the cyclic shift, and by dropping the tensor formulation we obtain an operator-based expression suitable to unfold DFGs of non-cyclic shifts.suitable to unfold DFGs of non-cyclic shifts.

( )1

(( ))

0' '

roM r oo

ro i

i MInt

ry S xZ

−

=

+−

=

⊕

By letting M=5 and r0 = 3 we can 3-unfold the DFG of a non-cyclic delay by 5.

1 ' 2 '

0 0

' 2 ' 2 ' '

1 1 p3

2 ' 1 '

2 2

0 1 0 0 0 0 0

0 0 1 0 0 0 0

1 0 0 0 0 0 0

2 1 2 233

z x z x

y z x z x x

z x z x

S z z z p

− −

− −

− −

= = =

− − −⊕ ⊕

Z

Z

14243 1442443 1442443

( )1 /

(( ))

/0

(( ))

' '/

ro N roM r ooN rr oo i o

i MInt

ry S I x

N rS

−

=

+

= ⊗

⊕

Unfolding DFGs of non Cyclic Shifts (c)

1 ' 2 '

0 0

' 2 ' 2 ' '

1 1 p3

2 ' 1 '

2 2

0 1 0 0 0 0 0

0 0 1 0 0 0 0

1 0 0 0 0 0 0

z x z x

y z x z x x

z x z x

− −

− −

− −

= = =

Z

14243 1442443 1442443

( )1

(( ))

0' '

roM r oo

ro i

i MInt

ry S xZ

−

=

+−

=

⊕

x y5D

'

0x

'

1x

'

2x

'

0y

'

1y

'

2y

1D

2D

2D

2 21 0 0 0 0 0 0

2 1 2 233

z x z x

S z z z p

− − −⊕ ⊕

Z

14243 1442443 1442443

Unfolding DFGs of non Cyclic Shifts (d)

( )1

(( ))' '

roM r oo

r

i MInt

ry S xZ

−

+−

= ⊕

� The algorithm embodied by this formula was known to the community, but it was nor derived neither formulated in operator notation. Its relation to the problem of unfolding DFGs within the circular context was not made.

( )0

' 'oro i

y S xZ=

=

⊕

Example 2

� 2-unfold a length-6, order-5 cyclic shiftsuch as the one depicted previously:

Using r0=2, N = 6 and M = 5 in our expression Using r0=2, N = 6 and M = 5 in our expression gives:

{(( 5 )) 16

6 ,2 6 ,2 6 6 ,2 6 ,2

2

'

'p

y P y P S P P x

x

−= =

S

14243( )

16 / 2

(( 5 ))26 / 22

0

5(( ))

2' '

6 /2i

iInt

y S I xS=

+ = ⊗

⊕

( ) ( )1 12 0 2

3 3 3 32 3 2 3' ( ) ' ( ) 'y S I S S x S I S I x= ⊗ ⊕ = ⊗ ⊕

Example 2, cont’

' ' '2330 0 03

00 0'

IIy x xSy

= = =

( ) ( )1 12 0 2

3 3 3 32 3 2 3' ( ) ' ( ) 'y S I S S x S I S I x= ⊗ ⊕ = ⊗ ⊕

330 0 032' ' '

3 3 31 1 1

00 0'

0 0 0

IIy x xSy

I I Sy x x

= = =

x0

x2

x4

x1

x3

x5x0

x1

x3

x5

x2

x4

1

2 3S I⊗2

3 3S I⊕

2

3S

3I

'

0x

'

1x

'

0y

'

1y

Example 2, cont’: the non-cyclic case

' ' '2 30 0 0

' ' '3 21 1 1

0 1 0 0'

1 0 0 0

y x xz zy

y x xz z

− −

− −

= = =

2D

3D

'

0x

x y5D

'

1x

'

0y

'

1y

Conclusions and Future Work

� The new operator-based formulations

provide an unified framework and embody an

unfolding algorithm, which quickly leads to unfolding algorithm, which quickly leads to

unfolded DFGs of cyclic and non-cyclic shifts.

� DSP programs of higher complexity will be

the subject of future work.