2012 05-10 kaiser

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Optimization of an SDR multiOptimization of an SDR multi--standard system standard system using graph theoryusing graph theory

UNIVERSITE LIBANAISE

using graph theoryusing graph theory

Patricia KAISER, Yves LOUET, Amine EL SAHILI.

• SCEE/IETR, Supelec, France.• Lebanese University– Hadath.

Contact : [email protected]

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Presentation OutlineUNIVERSITE LIBANAISE

0. Context.1 Th i l i i1. Theoretical prerequisites.2. A theoretical graph model of the multi-standard

system.3. A cost function of a multi-standard system.4. Complexity of our optimization problem.5. Exclusion of some options of implementation.6 The basic idea of the Minimum Cost Design algorithm6. The basic idea of the Minimum Cost Design algorithm. 7. Conclusions and perspectives.8. References

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The design of an SDR architecture ranges b/w :• Velcro Approach : using self contained complex communication components


communication components.

• Very fine Grain approach: manipulating small size operators to support different standards.

There may be another choice design:

• Intermediate granularity explorationg y pFormalization at an intermediate granularity (PARAMETRIZATION APPROACH)

Our aim is to find the best trade-off between:

“Efficiency and Flexiblity”

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• Identifying an optimal level of granularity in which components can be considered as COs and the behaviour is controlled by

Parametrization approachUNIVERSITE LIBANAISE

a set of parameters.

Parametrisation Techniques can be tackled from two approaches:

1. Pragmatic Approach : practical approach in creating and developing COs.

2. Theoretical Approach : graphical approach to model an SDR multi-

standard system which provides all the possible options of implementation.

Afterwards, by optimizing a cost function, the appropriate common operators

will be identified.

Our thesis foundation:Theoretical approach of parametrization : Searching for possible mathematical aspects, particularly GRAPH THEORY tools, to solve our optimization problem.

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Directed hypergraphsUNIVERSITE LIBANAISE

H ))(),(( HEHV)(HV)(HE )(HV

A directed hypergraph is a pair where: • = non-empty set of vertices• = set of ordered pairs of subsets of , called set of hyperarcs

Let be a hyperarc in . = tail of ; = head of

Let . = forward star of ; = out-degree o f . = backward star of ; = in-degree of .

)( EVHExample: ; V set of vertices and E set of hyperarcs

))(),(( EHETE )(HE)(ET E )(EH E

)(HVv)}(),({)( ETvHEEvFS v)}(),({)( EHvHEEvBS v

)(|)(| vdvFS v)(|)(| vdvBS v

),( EVH

2x

3x

4x

5x

6x

1E

2E}),{},,({

}){},,({

: where},{

},,,,,{

65432

3211

21

654321

xxxxE

xxxE

EEE

xxxxxxV

1x

Example: ; V set of vertices and E set of hyperarcs.

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Weight of a pathUNIVERSITE LIBANAISEUNIVERSITE LIBANAISE

),( EXH

),,.....,,,,,( 132211 nvEvEvErvP qiqii

qjEHvETv ijjijj ,....,1 )(and )( 1

In a directed hypergraph , a path from r to n (r , n X) is defined by a sequence of nodes and hyperarcs where:

1x

2x3x 5x

7x

8x

9x1E 2E

E

4E

5E

6E

3

2

53

6 5

4Path Q

qjijjijj , ,)()( 1

4x 6x3E3

We’ll define the weight of a path P by:

)(

))(()(PEE

ijP

ij

EBFwPw

Example:

36632}){},({}){},({}){},({)( 877332 xxwxxwxxwQw

),,,,,,( 8572312 xExExExQ

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The multi-standard SDR system was represented as a graph with many different layers. Two node dependencies, the “OR” and the “AND” dependencies, were

essential to clearly illustrate the implementation needs of each block and describe th t f ti b t bl k f hi h l l d th i l l l

“AND” and “OR” dependencies


ANDOR

level n

level n-1

A

B C

A

B C

PE A needs:Either B OR C

PE A needs:Both B AND C

the type of connection between blocks of higher levels and others in lower levels.

Either B OR C Both B AND C

• Performing the required tasks of block A using blocks in level n (block A itself) instead of blocks in level n-1, needs less time but has much higher cost.

• Realizing the functionalities of block A using blocks of lower levels (blocks in level n-1) will decrease the cost but increase the execution time of the system.

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Graph Modeling

WiFi #1 WiFi #2 WiFi #3 WiMAXVelcro


Scrambler/Randomiser

Convolutional Coder

InterleaverConstellation

MapperFFT‐N

RS Encoder

ButterflyLUTa a’ b b’

Increasing Granularity

NAND NOT XOR AND OR

Adder MultiplierLFSR

Very Fine Grain

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Theoretical representation

The graph structure of a multi-standard system is defined theoretically as a directed hypergraph H=(V,E) where:


• The blocks (functions and operators) present in the figure represent the set of vertices V.

• A hyperarc e in E is defined such that:1. a parent node constitutes the only tail node.

2. The necessary descendent node(s) capable of realizing this same parent node form the head node(s) of e.

A

B C D FFig. 2

Example:

2. Fig. of hyperarcs theare {F})({A}, }),,,{},({

},,,,{

DCBA

FDCBAV

( )

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A pictorial view of one option of implementation

S T

A generated graph provides a pictorial view of each possible of implementation

The operators chosen to install in the design are plot with out-degree zero, along with all the operators that they build, step by step, until they reach the functionalities of the top level standards


A1 A2 A3

B1 B2 B3 B4

C1 C2 C3

D1 D2 D3 D4

S T

A1 A2 A3

B2 B3 B4

C1 C2

functionalities of the top level standards

C1 C2

D2 D3 D4

The break-down of two standards S and T up to four lower levels.

Chosen COs: D2, D3, D4, C1, & B3

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0. Context.1 Th i l i i1. Theoretical prerequisites. 2. A theoretical graph model of the multi-standard


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Cost parameters

The graph structure of the multi-standard system provides all the options capable of implementing the standards to be supported. Each option has a certain cost to be paid. This

cost is calculated via a certain suggested cost function. The parameters used in the cost function were:


function were:

Parameters associated with vertices:• Building Cost (BC) : paid once during the useful life of a radio independently of the

number of times in which the block is going to be called (can be in terms of gates/LUTs/number of cycles,…)

• Computational Cost (CC) : considered to be the time taken by a PE to compute a function, paid every time a component is brought into play (can be in terms of time of

execution, number of multiplications,… )

Parameters associated with hyperarcs:• Number of Calls (NoC) : the number of times a higher level PE calls a lower one.

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Cost function

The cost function yields the cost paid to implement the multi-standard system after choosing some common operators (COs) capable of this performance

The cost function is defined as:


The objective is to optimize this cost function to its minimum cost possible and thus solving the optimization problem that finds balance between economy

d i ffi i

block installed standard block installed path- P

)( ))()((xy x yx

xBCPwxCCCF

and computing efficiency.

Simulated annealing algorithm and Genetic algorithm were used to solve our optimization problem by previous PhD students.

We propose a new algorithm which solves our optimization problem, called the Minimum Cost Design algorithm, using graph theory.

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A decision problem is said to be an NP-problem if it can be solved by a polynomial nondeterministic algorithm where:

G i t

NP - problemsUNIVERSITE LIBANAISE

Non deterministic algorithm

Polynomial if there exists a polynomial p s.t, for every yes-instance I, there is some guess S that leads the deterministic checking stage to respond "yes" for Iand S within

time p(Length(I))

Guessing stage

Checking stage

Summary: a decision problem is proved to be an NP-problem if for a given instance Iand after guessing a certain solution S, we can verify if the answer for I and S is "yes"

in polynomial time.

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Description of our optimization problem

Instance: A multi-standard graph structure H with all the associated necessary entities containing L levels.


Question: Find the set of operators which implements the design and which has the minimum cost.

)(HVS

This is a minimization problem. This optimization problem can be converted to the decision problem version as follows:

Instance: • A multi-standard graph structure H with all the associated necessary entities with L levels

0B

B

with L levels.• a constant

Question: Can we find a set of operators which implements the design and whose cost is ? (YES-NO questions)

)(HVS

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THEOREM: The above described decision problem is an NP-problem on condition that the number of levels L of a multi-standard graph structure is upper bounded by a constant


upper bounded by a constant..

Proof: We consider an instance I and a certain guess solution S for this instance, then derive a polynomial equation for the number of operations required to check if the answer for I and S is "yes". A guess is a set , suppose .

We have to check three points:

1 Verify if the guess can implement the design : at most operations

kS ||)(HVS

)1(|)(| LHE1. Verify if the guess can implement the design : at most operations.

Worst case : the k blocks of the guess S occupy the lowest level passing by all the elements in E(H) at most L-1 until we reach the top level

standards.

2. Calculate the cost : Find the maximum number of multiplications and additions.

)1(|)(| LHE

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Worst case : The k chosen blocks are in the lowest level getting longest paths An “AND” connection between any block v and all the blocks which occupy a


lower level than v getting maximum number of paths.

2a blocks

3a blocks

La blocks....

k guessed blocks occupying the lowest level

2 blocks

In fact, this is a case worse than the worst case 21

iLi

as,....2

max

Let

Let number of paths reaching a lowest level block = )( 1 LsOn


Let number of paths reaching a lowest level block = . total number of paths reaching all the k lowest level blocks is .

Total number of multiplications is Total number of additions is

)(1 sOn1kn

1kLn11 kn

Operations for second point are less than: )1( 11 knkLn

3. comparing the cost found in 2 with B : just a matter of one operation.

If we add the three operations which were required to verify the answer to the question of our decision problem having guessed a certain guess, we get a total of :

1)]1([)1(|)(| 11 knkLnLHE

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Let: |))(|,,max( HEksr


)( LrO

)( 10rO

Equation belongs to .If L is upper bounded by a constant, for example L less than 10, we get that equation belongs to

polynomial equation.

So our optimization problem is an NP-problem if . 10L

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S T

When searching for minimum cost designs

Excluding some generated graphsfor minimum cost designs

S T


A1 A2 A3

B2 B3 B4

C1 C2

D2 D3 D4

B4

A1 A2 A3

B1 B2 B3 B4

C1 C2 C3

D1 D2 D3 D4

Chosen COs: D2, D3, D4, C1, B3, & B4.

This option represents an alternative in which certain lower level blocks are installed in the

design, together with higher level ones which can be built by these of lower level

The break-down of two standards S and T up to four lower levels.

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An idea of the exclusion proof

CC

C


E FD

A

IH

A

H

E FD

A

E FD

A

IH IH

LKJ

HIH

IH

LKJ

The duplicated part 26

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Path 1 reaching DP

Path 2 reaching DP

Path 1 reaching DP

Path 2 reaching DP

Path 1 reaching DP

Path 2 reaching DP

Generalization


DUP DUP DUP DUPDUPB

BPBP

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The basic idea

The MCD algorithm operates as follows:

1. Select an option of implementation (a generated graph)

2. Calculate its cost

3. Compare the calculated cost with all the previously calculated ones. If it was the minimum found so far, the cost will be updated.

4. Generate new options from the selected option in hand to add them to the set of options.

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Generate the options ofimplementation


In this algorithm, the options are generated in a step by step manner, generating options from others.

As an initialization, the option chosen is the generated graph containing all the top level vertices representing the standards, but no hyperarcs (Velcro approach).

Selecting a certain option of implementation in hand, new options can be emerged from it as follows:

1. Select a vertex in the generated graph of this selected option.2. Search for a hyperarc in the original input directed hypergraph representation.3. Add this new hyperarc to the option in hand, we get a new generated graph option.

)(vFSE0)(s.t vdv

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Example of options generation

S T

A1 A2 A3

INPUT:S T

A1 A2 A3

S T

A1 A2 A3

S6 7 8 3

T

A1 A2 A3


Selected option at some point in

the algorithm

B4

B4 B4B1

S T

A1 A2 A3

B4B2 B3

Th i t di t d

150/11

100/10

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2 3

5 4

1020 30

A1 A2 A3

B1 B2 B3 B4

C1 C2 C3

D1 D2 D3 D4

S T

A1 A2 A3

B4

S T

A1 A2 A3

B4

C2

The input directed hypergraph representation S T

A1 A2 A3

B4

C331

For the calculation of the cost, we introduce a vector assigned to each block in the graph. The components of this vector represent the weights of the paths from all the top level standards to and is equal to the number of

vkvk

vkdim


Calculate the cost of an option

v

vp p qsuch paths.

v

Initialization: we set vector for all top level block in the chosen optionand vector otherwise.

)1(vk

k

0vk

Aft d t t filli th ‘ i l f th hi h t l l t d d

v

vkAfterwards, we start filling the ‘s recursively from the highest level standards going down by searching all the paths from the standards and each time multiplying by the weight of the BF-reduction traversed along.

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The components of the vectors are developed as follows:1. We select the vertices in the generated graph option recursively from the highest

level vertices to the lowest level ones. Suppose we fall on vertex .2 S l t h i th ti i h d h th t b l


xk

vE E )(FS2. Select a hyperarc in the option in hand such that belongs .

3. For each belongs , multiply each of the components of the vector by the weight on the BF-reduction to form new components to be added to the vector

S6 7 8

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3

T

A1 A2 A33

)1( )1(

)0( )0()0(

)6( )7()8( )3,8(

E E )(vFSh )(Eh vk

}){},({ hvhk

4

2

B4

C2

20 10

D2 D3 D4

10

)0(

)0(

)0(

)0()0(

)24( )21,24( )15,40,21,24(

)30,80,42,48(

)300,800,420,480( )600,1600,840,960(

)300,800,420,480(

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Finally to attain the cost of the selected option of implementation:

1. we multiply the components of by the CC of for all installed block (i.e) and add all the attained values. This will correspond to the total

computational cost of the option in hand.

2. The BC of each installed block is added once.

vk v v0)( vd

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Conclusion.• Our optimization problem is a complex problem (NP under a constraint).

S


• Propose a new algorithm that can optimize the SDR multi-standard system using graph theory aspects after having excluded some particular options

of implementation, thus extracting the most appropriate Common Operators from the most convenient granularity levels for an optimal design.

• Advantages : it’s an optimal algorithm, not near optimal, like the previously used ones.

• Disadvantages : it’s an exponential algorithm and thus will need more computing effortcomputing effort.

PERSPECTIVES:

• Explore the performance of our proposed algorithm in comparison with other used algorithms.

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International Journals:• Patricia Kaiser, Yves Louet, Amine El Sahili, “Complexity of the optimization problem of the SDR multi-standard system”, Journal of comlpexity , in preparation.• Patricia Kaiser, Amine El Sahili, Yves Louet, “Optimization of the SDR multi-standard system using graph theory,” Frequenz 2012, in preparation.

Conferences:• Patricia Kaiser, Yves Louet, Amine El Sahili, ”A cost function expression for SDR multi-standard systems design using directed hypergraphs”, XXXth URSI General Assembly and Scientific Symposium , Istanbul, Turkey, August, 2011.• Patricia Kaiser, Amine El Sahili, Yves Louet, “ An upper bound for the total number of options to implement an SDR multi standard system ” ICT Lebanon April 2012

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options to implement an SDR multi-standard system, ICT, Lebanon April 2012

Workshops:• Patricia Kaiser, Sufi Tabassum Gul, Christophe Moy, Yves Louet "Graph theory approach for optimization of Multi-standard software Defined Radio equipments", ERRT 6th karlsruhe, Mainz, Germany, June 2010.• Patricia Kaiser, Amine El Sahili, Yves Louet, “ An algorithm proposal for a minimum cost SDR multi-standard system using graph theory”, 7th karlsruhe workshop, 2012.

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Thanks for your attention.

Questions?

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2012 05-10 kaiser

Technology