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Clément Dudal – Séminaire 25/11/10
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Sphere Decoding
Application to a MC-CSK system
Clément Dudal – Séminaire 25/11/10
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Sphere Decoding
Mathematical approachMinimization of a cost function f(x1, . . . , xK) with respect to its K arguments taking value in a discrete set of cardinality L
Digital communications approachMinimization of the distance between the received symbol and the possible transmitted symbols in order to recover the information in a multi-user system
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Clément Dudal – Séminaire 25/11/10
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Sphere Decoding Computation of the distance based on the optimum
Maximum Likelihood operator absolute minimum
Smart hypotheses leading to a smaller complexity than classic ML algorithm
Main idea: searching over the noiseless possible received signals that lie within a hypersphere of radius R around the actual received signal
Multi-user detection
Clément Dudal – Séminaire 25/11/10
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Principle of Sphere Decoding
The distance is seen as a sum of nonnegative functions with an increasing number of arguments
d = f(x1, . . . , xK) = h(xK) + h(xK,xK-1) +…+ h(xK,…,x1)
Graphically, the process is a (K)-level tree graph with one uppermost node (level K) and LK-1 leaves (level 1)
Each branch corresponds to an intermediate distance
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Clément Dudal – Séminaire 25/11/10
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Graphical approach
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Clément Dudal – Séminaire 25/11/10
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Algorithm
2 stages Initialization Pruning
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Clément Dudal – Séminaire 25/11/10
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Other methods
Sum of the K values gives a first minimum of the function, the starting radius μ0
Algorithm
2 stages Initialization
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At level K smallest intermediate value leads to L next nodes at level K-1.
At level K-1 smallest value among the L nodes and so on until the level 1.
Clément Dudal – Séminaire 25/11/10
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Algorithm
2 stages Pruning
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Exploration of the other branches
Each branch which will certainly give a higher μ0 is pruning out
If a leaf is reached with a smaller sum than μ0, μ0 is updated with that new value
The process continues until all branches have been explored or pruned out
Clément Dudal – Séminaire 25/11/10
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Exemple: K=2, L=4
Initialization
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Clément Dudal – Séminaire 25/11/10
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Exemple: K=2, L=4
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Initialization
Clément Dudal – Séminaire 25/11/10
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Exemple: K=2, L=4
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Pruning
Clément Dudal – Séminaire 25/11/10
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Exemple: K=2, L=4
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Pruning (first level)
Clément Dudal – Séminaire 25/11/10
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Exemple: K=2, L=4
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Pruning (second level)
Clément Dudal – Séminaire 25/11/10
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Exemple: K=2, L=4
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Pruning (second level)
Clément Dudal – Séminaire 25/11/10
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The complexity is smaller than a classic ML algorithm when the SNR grows as the pruning is more and more efficient
The algorithm gives the absolute minimum of the cost function
Conclusion
The information of all the users is recovered in one time
Clément Dudal – Séminaire 25/11/10
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Application to a MC-CSK system
Multiple Carrier Code Shift Keying
Clément Dudal – Séminaire 25/11/10
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Reminder
Spectral efficiencyηs = Rb/B (bit.Hz-1.s-1)
Power efficiencyηp = d²/Eb
Trade-off between ηs and ηp
Clément Dudal – Séminaire 25/11/10
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MC-CSK system
Broadcast synchronous system
K flows assimilated to K ‘‘users’’
Each user
L spreading sequences of length N (Gold sequences)
Coding log2(L) bits
Multi-user detection, not interference limited
Clément Dudal – Séminaire 25/11/10
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MC-CSK system
…0010…abcdbcdacdabdabc
CSK : code shift keying (N=4, L=4)
a b c d c d a b
code shifts
S1 S2
…0010…
abcd
-a -b -c -d -a -b -c -d a b c d -a -b -c -d
CDM : code division multiplexing (N=4)
Spectral efficiency x log2(L)
Potentially good power efficiency
f1f2f3f4f1 f2 f3 f4 f1 f2 f3 f4
MC-CSK :
Easier synchro
Clément Dudal – Séminaire 25/11/10
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MC-CSK systemC1 = matrix containing all the sequences of user 1
C = [C1 C2 … CK] = generation matrix
g1 vector of length L containing one ‘1’ and L-1 ‘0’. The ‘1’ points out the transmitted sequence for user 1
g = [g1 g2 … gK]T
Received signal: z = C.g + wn
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Clément Dudal – Séminaire 25/11/10
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MC-CSK system and Sphere Decoding
In reception Estimation of g = [g1 g2 … gK]T
ĝ = arg min ||z - Cg||2² (cost function)
QR decomposition of C C = Q*R Q unitary, R upper triangular
ĝ = arg min ||y - Rg||2², y = QT*.z21
g1,…,gK
g1,…,gK
Clément Dudal – Séminaire 25/11/10
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R =
d = ||y - Rg||2² = h(gK) + h(gK,gK-1) + … + h(gK,…,g1)
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MC-CSK system and Sphere Decoding
Clément Dudal – Séminaire 25/11/10
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Initialization ĝK = arg min h(gK)
ĝK-1 = arg min h(ĝK,gK-1)
ĝ1 = arg min h(ĝK,…, g1)
μ0 = h(ĝK)+h(ĝK,ĝK-1)+…+h(ĝK,ĝK-1,…,ĝ1)
Pruning
gK
g1
gK-1
MC-CSK system and Sphere Decoding
Clément Dudal – Séminaire 25/11/10
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Efficiency : 1/N
Flow #1
Channel
11011Flow #K
Seq #K Seq #K
decoder
Efficiency : K/N
00101
spreading
Seq #1 (N chips)
despreading
Seq #1
decoder
Why using MC-CSK and Sphere Decoding?
L-ary CodeShift
KeyingSphere
Decoding
Flow #1
Flow #KSpectral
Efficiency:K*log2(L)/N
16QAM16-FSK
5 dB
IF orthogonality of sequences & permutations MC-CSK power efficiency
= FSK power efficiency !
Clément Dudal – Séminaire 25/11/10
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Why using MC-CSK and Sphere Decoding?
System not interference limited
Optimal decoding with lower complexity
Scalable to (small) overloaded systems
Improving in the same time spectral and power efficiency
Clément Dudal – Séminaire 25/11/10
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Results
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Clément Dudal – Séminaire 25/11/10
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Clément Dudal – Séminaire 25/11/10
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Limitations
For higher loads (>15%), the complexity
and time simulation are too important
Need of another decoder to reach
interesting spectral efficiency
LASSO
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Clément Dudal – Séminaire 25/11/10
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Brief presentation of the LASSO Least Absolute Shrinkage and Selection
Operator – (Tibshirani, 1994)
Working on the hypothesis of the sparsity of vector g
Using a penalty on the L1 norm and the residual sum of squares
min || y – Cg ||2² s.t. || g ||1² <= t
Clément Dudal – Séminaire 25/11/10
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Thank you for your attention
Questions?