2012 05-10 kaiser

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Optimization of an SDR multiOptimization of an SDR multi--standard systemstandard 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 : patricia.kaiser@supelec.fr
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Presentation Outline
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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0. Context.
system.
8. References
3
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 p
Formalization 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 approach
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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0. Context.
system.
8. References
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Directed hypergraphs
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 path
UNIVERSITE 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
2x
3x 5x
7x
8x
9x
1E 2E
E
4E
5E
6E
3
2
5
3
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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0. Context.
system.
8. References
9
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 WiMAX
Velcro
Scrambler/R
andomiser
Convolutional
Coder
Interleaver
Constellation
Mapper
FFT‐N
RS Encoder
ButterflyLUTa a’ b b’
Increasing Granularity
NAND NOT XOR AND OR
Adder Multiplier
LFSR
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 F
Fig. 2
Example:
2.Fig.ofhyperarcstheare{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.
system.
8. References
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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
   
blockinstalledstandard blockinstalled 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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0. Context.
system.
8. References
17
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 - problems
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
0B
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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11
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
i
Li
as
,....2
max

Let
Let number of paths reaching a lowest level block = )( 1
 L
sOn
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  sOn
1kn
1kLn
11 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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12
Let: |))(|,,max( HEksr 
)( L
rO
)( 10
rO
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 .10L
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0. Context.
system.
8. References
24

13
S T
When searching for
minimum cost designs
Excluding some generated graphs
for 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.
25
An idea of the exclusion proof
C
C
C
E FD
A
IH
A
H
E FD
A
E FD
A
IH IH
LKJ
H
IH
IH
LKJ
The duplicated part 26

14
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
BP
BP
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0. Context.
system.
8. References
28

15
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 of
implementation
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.
)(vFSE
0)(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
S
6 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
4
5
2 3
5 4
10
20
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
vk
vk
vkdim
Calculate the cost of an option
v
vp p q
such paths.
v
Initialization: we set vector for all top level block in the chosen option
and vector otherwise.
)1(vk
k
0vk
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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17
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
v
E 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
S
6 7 8
4
5
3
T
A1 A2 A3
3
)1( )1(
)0( )0(
)0(
)6( )7(
)8( )3,8(
E E )(vFS
h )(Eh vk
}){},({ hv
hk
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 v
0)( 
vd
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18
0. Context.
system.
8. References
35
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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0. Context.
system.
8. References
37
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?
39

2012 05-10 kaiser

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2012 05-10 kaiser