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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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)

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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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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

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Thank you for your attention

Questions?