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Low-Complexity Decoding Schemes for MIMO systems Ghaya Rekaya-Ben Othman [email protected] June 2017

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Low-Complexity Decoding Schemes for MIMO systems

Ghaya Rekaya-Ben Othman [email protected]

June 2017

Patenting : my vision as an inventor

Patent Factory : Feedback

Sequential decoding

2

2

2

1

2

3

Outline

June 2017 1/41G. Rekaya-Ben Othman

2

4 Recursive Block decoding

Patenting : My Vision as an inventor

Patents ?

•

It’s too complicated.

It’s for compagnies not for academics.

It’s a waste of time as it is never valorized.

The patenting process is too long.

It blocks research work.

We are not used to do patents.

June 2017 2/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Who talks about innovation and patents?

June 2017 G. Rekaya-Ben Othman 3/41

•

G. Rekaya-Ben Othman

Lawyer

Financial

Ingenier

Manger

Economist

Politicien

But, what about Inventor ??

Vision PF MIMO Sequential decoder Block decoder

Inventor Vision

•

Everybody can make patents.

It’s not limited to compagnies, and especially big companies.

Patents are not just for GREAT IDEAS.

Making a patent must be a reflex to protect its know-how.

A patent is not an end in itself, but must be part of a personal or collective approach to innovation and creativity.

June 2017 4/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

•

An inventor needs to be accompanied :

Employer : help defining an approach to innovation and creativity.

Lawyer : why and how to proceed.

Patent engineer: report (ID), write the patent, inventor meetings, process follow-up.

Inventor Vision !

June 2017 5/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

•

Why patenting ?

•

By obligation (Standardization or Product Issues)

For the patent premium

Professional progress

Future Valorization project

For Academics

Combining theoretical and applied research : A First step for Innovation.

June 2017 6/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

• •

Life stages of an inventor

•

Stage 1 First Patents

How to detect the idea!

How to make a patent?

Steps to make patents

Stage 2 5 to 10 Patents

Become an Inventor

Identify missing results

Enriching Patent Groups

Methodology for creating a patent family

Stage 3 >10 Patents

Become an Expert

Structure patents by classes or groups

Identify technologies

Identify compagnies

On the way to Valorization

June 2017 7/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Patent Factory : MIMO Decoders

Optical fiber communications

Sink node

Remote Control Center

Sensor nodes

Wireless sensor network

Multiple antennas systems

MIMO channel

Multi-user communications

Vision PF MIMO Sequential decoder Block decoder

Examples of applications of MIMO technologies

June 2017 8/41G. Rekaya-Ben Othman

Proposed solutions are implementable in wireless standards products such as:

WiFi (IEEE 802.11n) :

• MIMO spatial streams up to 8x8 MIMO configurations.

• Commonly used MIMO configurations are 2 x 2, 2 x3 and 3 x 2, with high

density modulations (up to 256-QAM).

LTE and LTE-advanced : • Used MIMO configurations are : in the Downlink 4x4 and 8x8, in the Uplink

2x4 and 4x4.

• MU-MIMO configurations are also considered.

5G

• Deployable systems must support MIMO and MU-MIMO communications.

• Massive MIMO

MIMO Decoding

Potential Standars

June 2017 9/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

CHALLENGE: meet the target QoS specifications (fixed complexity/performance) for any application implementable in any MIMO configuration.

MIMO Decoding

MIMO transmission chain

June 2017 10/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Inputs

5 Patents

Outputs

14 patents, 1 in progress

8 Publications, 2 journals in preparation

Start discussions with some companies for valorisation

Patent Factory results

June 2017 11/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Family 1: Preprocessing

Family 2: Sequential decoding

Family 3: Fixed

Complexity

Family 4: Block

Decoding

Family 5: General Tools

New decoding schemes: 5 Families

June 2017 12/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Problem

Proposed SolutionsNew techniques to improve the quality of the channel matrix before decoding:

• Algebraic reduction • Augmented LLL reduction• Reordering of channel matrix to control zero localization

Patents Outcomes

3 filed patents

Worst channel realization can induce very high decoding complexity.

MIMO Decoding

Family 1: Preprocessing

June 2017 13/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Patents Outcomes 6 filed patents

Problem

The complexity of the tree search phase increases function of the number of transmit and receive antennas and the constellation size.

Proposed SolutionsNew decoding methods allowing the generation of limited tree search. • SB-Stack (Hard output and Soft output) • Zig-zag Stack • Enhanced initial radius selection methods• Parameterized sequential decoding (level, block-dependent bais parameter)

1 idea in progress

Family 2: Sequential decoding

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

14/41

Patents Outcomes 2 filed patents

Problem For real time applications, a fixed decoding complexity is required.

Proposed SolutionsNew methods allowing to have fixed decoding time with guaranteed performance.

• Stack reordering for early termination • Anticipated termination

Family 3: Fixed Complexity

June 2017 15/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Patents Outcomes

5 filed patents

Problem For very large decoding systems dimension, a parallelization of some decoding process could be a very good hardware solution.

Proposed Solutions A judicious division of the decoding system is proposed based on variant parameters and criterions.

• Block division to ensure an order of diversity • Block division to reduce error propagation • Block division to reduce decoding complexity for each block • Semi-exhaustive recursive decoding

Family 4: Block Decoding

June 2017 16/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Patents Outcomes

3 Filed patents

Problem

Reduce the overall decoding complexity, by considering all the decoding chain.

Proposed Solutions

• Adaptive decoding • MAP decoding using augmented lattice • Design criterion for low-complexity decodable Space-Time Codes.

We propose new tools/methods to enhance the decoding chain and offer the best complexity/performance tradeoff. These methods are available for all the presented ideas in the four patent groups.

Family 5: General ideas

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

17/41

Example 1 of a decoding chain:

Example 2 of a decoding chain:

Algebraic Reduction

SB-Stack

or

Zig-Zag Stack

Anticipated Termination

or

Early Termination

Block DivisionParametrized SB-Stack

for each Block

Decoding Chain: Families association

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

18/41

Theoretical Validation

Numerical validation

Some results are validated by a theoretical study, through the derivation of the error probability.

Formulas of some parameters and criterion are derived.

Most of the ideas are validated by simulation, through a program C simulator.Some others are under validation.

Complexity is counted as the number of multiplications.

Numerical simulations give a very reliable complexity evaluation. Results are mature for practical implementation.

Maturity and Validation

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

19/41

Patent application Corresponding publication

1 G. Rekaya-Ben Othman, R. Ouertani and J.-C. Belfiore, "Procédé de Décodage d’un Signal Transmis Dans un Système Multi-antenne", French Application filed February 2008 No. FR 08/50690, International Application No. PCT 2009/098178 and US 2011/058617 A1.

R. Ouertani, G. Rekaya-Ben Othman and J-C. Belfiore, "An Adaptive MIMO decoder", IEEE VTC, Barcelona, Spain, April 2009.

2 G. Rekaya-Ben Othman, A. Salah and S. Guillouard, "Procédé de Décodage d’un Signal Mettant en Œuvre une Construction Progressive d’un Arbre de Décodage, Produit Programme d’ordinateur et Signal Correspondants", French patent application filed May 2008 FR 08/52985, Extended PCT 2009/135850 and US 2011/0122004 A1

- R. Ouertani, G. Rekaya-Ben Othman and A. Salah, "The Spherical Bound Stack Decoder", IEEE International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob), Avignon, France, October 2008.

- A. Salah, G.Rekaya-Ben Othman, R. Ouertani and S. Guillouard, "New Soft Stack Decoder for MIMO Channel", Asilomar Conference on Signals, Systems and Computers, California, USA, October 2008.

3 G. Rekaya-Ben Othman, L. Luzzi et J-C. Belfiore, "Procédé de Décodage d’un Signal Ayant Subi un Codage Espace-Temps Avant Emission, Dans un Système Multi-Antennaires", French Application filed September 2008 No. FR 08/55882.

- L. Luzzi, G.Rekaya-Ben Othman and J-C Belfiore, "Algebraic Reduction for Space-Time Codes Based on Quaternion Algebras", Advances in Mathematics of Communications (AMC), vol. 6, n° 1, February 2012.

- G. Rekaya-Ben Othman, L. Luzzi and J-C. Belfiore, "Algebraic Reduction for the Golden Code", IEEE International Conference on Communications (ICC), Dresden, Germany, June 2009.

Portfolio and corresponding publications

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

20/41

Patent application Corresponding publication

4 L. Luzzi, G. Rekaya Ben Othman and J-C. Belfiore, "Méthode de Décodage par Réseau de Points Augmenté pour Système Multi-source", French Application filed December 2009 No. FR 09/59680, International extension September 2015 PCT/EP2015/072280.

- L. Luzzi, G. Rekaya-Ben Othman and J-C. Belfiore, "Augmented Lattice Reduction for MIMO decoding", IEEE Transactions on Wireless Communications, vol. 9, n° 9, pp. 2853-2859, September 2010.

- L. Luzzi, G. Rekaya Ben Othman and J-C. Belfiore, "Augmented Lattice Reduction for Low Complexity MIMO Decoding", IEEE PIMRC, Istanbul, Turqey, September 2010.

5 A.Mejri et G. Rekaya Ben Othman, "Méthode de Décodage MAP par Réseau de Points Augmenté", French Application filed October 2013 No. FR13/ 59497

- A. Mejri and G. Rekaya Ben Othman, " MAP Decoder for Physical-Layer Network Coding Using Lattice Sphere Decoding", IEEE 21st International Conference on Telecommunications ICT, Lisbon, Portugal, May 2014.

- A. Mejri and G. Rekaya Ben Othman, "Efficient Decoding Algorithms for the Compute-and-Forward Strategy", IEEE Transactions on Communications, June 2015.

6 G.Rekaya-Ben Othman and A.Mejri, » Methods and Systems for Decoding a Data Signal Based on the Generation of a Decoding Tree", European patent application, September 2014, EP14306517.5

A.Mejri and G.Rekaya-BenOthman, « Reduced Complexity Stack Decoder for MIMO Systems », IEEE Vehicular Technology Conference, VTC, Glasgow, UK, May 2015.

Portfolio and correspondinf publications

June 2017 21/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Patent application Corresponding publication

7 G. Rekaya-Ben Othman and A. Mejri, "Tree Search-Based Decoding", European patent application, February 2015, EP 15305255.0

A. Mejri and G. Rekaya-Ben Othman, "Reduced-Complexity Lattice Spherical Decoding", IEEE Twelfth International Symposium on Wireless Communication Systems, ISWCS, Brussels, Belgium, August 2015.

8 G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, "Space-Time Coding for Communication Systems", European patent Application April 2015, EP 15305677.5

- A. Mejri, M-A. Khsiba and G. Rekaya Ben Othman, "Reduced-Complexity ML Decodable STBCs: Revisited Design Criteria", IEEE Twelfth International Symposium on Wireless Communication Systems, ISWCS, Brussels, Belgium, August 2015.

- A. Mejri, M-A. Khsiba and G. Rekaya Ben Othman, "Revisited Design Criteria for STBCs With Reduced Complexity ML Decoding", to be submitted to IEEE Transactions on Wireless Communications.

9 G. Rekaya Ben Othman and Asma Mejri, "Anticipated Termination for Sequential Decoders", European patent application, June 2015, EP 15305907.6

- A. Mejri, G. Rekaya Ben Othman and M.A. Ksiba, "Early Termination Techniques for MIMO Lattice Sequential Decoders", IEEE International Conference on Communications and Networking, November 2015, Tunisia.

10 G. Rekaya Ben Othman and Asma Mejri, "Sequential Decoding With Stack Reordering", European patent application, June 2015, EP 15305910.0

- A. Mejri, G. Rekaya Ben Othman and M.A. Ksiba, "Early Termination Techniques for MIMO Lattice Sequential Decoders", IEEE International Conference on Communications and Networking, November 2015, Tunisia.

Portfolio and corresponding publications

June 2017 22/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Patent application Corresponding publication

11 G. Rekaya Ben Othman and Asma Mejri, « Parameterized Sequential Decoding », European patent application, November 2015, EP 15306847.3

12 G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, « Semi-Exhaustive Recursive Block Decoding Method and Device », November 2015, EP 15306808.5

M.A. Ksiba and G. Rekaya-Ben Othman, « Semi-Exhaustive Reduced-complexity Recursive Block Decoding for MIMO Systems » , IEEE International Conference on Communication, October 2015.

13 G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, « RECURSIVE SUB-BLOCK DECODING », 2015, EP 15307153.5

M.A. Ksiba and G. Rekaya-Ben Othman, « Semi-Exhaustive Recursive Multi-Block Decoding for MIMO Systems », Submitted to IEEE PIMRC 2017.

14 G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, « REORDERED SUB-BLOCK DECODING », 2015, EP 15307154.3

15 G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, « WEIGHTED SEQUENTIAL DECODING », 2015, EP 15307155.0

Portfolio and corresponding publications

June 2017 23/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Patent application Corresponding publication

16 G. Rekaya-Ben Othman and M-A. Khsiba, « METHODS AND DEVICES FOR SEQUENTIAL SPHERE DECODING », 2016, EP 16305401.8

M.A. Ksiba and G. Rekaya-Ben Othman, « Dichotomic Sphere Decoder», WCNC 2017

17 G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, « METHODS AND DEVICES FOR DECODING DATA SIGNALS», 2016, EP 16305417.4

18 G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «METHODS AND DEVICES FOR SUB-BLOCK DECODING DATA SIGNALS»,2016, EP 16306758.0

19 G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «METHODS AND DEVICES FOR SUB-BLOCK DECODING DATA SIGNALS»,2016, EP 16306728.3

Portfolio and corresponding publications

June 2017 24/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sequential decoding

Exhaustive

Search

Decoders

Sub-optimal optimal

Linear Non-linear Sequential decoding

Depth First

Search

Best First

Search

SDSE Stack Fano

ZF MMSE ZF-DFE MMSE-DFE

Classification of MIMO decoders

June 2017 25/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

• Complex-valued received matrix:

• ︎ Real-valued vectorized system:

where:

• QR decomposition of Heq: Heq = QR, where R is an upper triangular, Q is an orthogonal matrix.

• ML metric is :

Decoding algorithms in MIMO systems State of the art of early termination techniques 1st proposed technique 2nd proposed technique Conclusion

System model

I Complex-valued received matrix: Yn

r

◊T = Hnr

◊nt

Xn

t

◊T + Znr

◊T .

I Real-valued vectorized system: y2nr

T

= Heq2n

r

T◊2Ÿs̃2Ÿ + z2nr T

where: y = ˜vec(Y), z = ˜vec(Z) and s̃2Ÿ =#

Ÿ(s1) ⁄(s1) · · · Ÿ(sŸ) ⁄(sŸ)$

t

I QR decomposition of Heq

: Heq

= QR

R is upper triangular.

I ML metric: m(s̃) =Î y ≠ Heq

s̃ Î2=Î Qty ≠ Rs̃ Î2

4 / 20

Decoding algorithms in MIMO systems State of the art of early termination techniques 1st proposed technique 2nd proposed technique Conclusion

System model

I Complex-valued received matrix: Yn

r

◊T = Hnr

◊nt

Xn

t

◊T + Znr

◊T .

I Real-valued vectorized system: y2nr

T

= Heq2n

r

T◊2Ÿs̃2Ÿ + z2nr T

where: y = ˜vec(Y), z = ˜vec(Z) and s̃2Ÿ =#

Ÿ(s1) ⁄(s1) · · · Ÿ(sŸ) ⁄(sŸ)$

t

I QR decomposition of Heq

: Heq

= QR

R is upper triangular.

I ML metric: m(s̃) =Î y ≠ Heq

s̃ Î2=Î Qty ≠ Rs̃ Î2

4 / 20

Decoding algorithms in MIMO systems State of the art of early termination techniques 1st proposed technique 2nd proposed technique Conclusion

System model

I Complex-valued received matrix: Yn

r

◊T = Hnr

◊nt

Xn

t

◊T + Znr

◊T .

I Real-valued vectorized system: y2nr

T

= Heq2n

r

T◊2Ÿs̃2Ÿ + z2nr T

where: y = ˜vec(Y), z = ˜vec(Z) and s̃2Ÿ =#

Ÿ(s1) ⁄(s1) · · · Ÿ(sŸ) ⁄(sŸ)$

t

I QR decomposition of Heq

: Heq

= QR

R is upper triangular.

I ML metric: m(s̃) =Î y ≠ Heq

s̃ Î2=Î Qty ≠ Rs̃ Î2

4 / 20

System model

I Complex-valued received matrix: Yn

r

◊T = Hnr

◊nt

Xn

t

◊T + Znr

◊T .

I Real-valued vectorized system: y2nr

T

= Heq2n

r

T◊2Ÿs̃2Ÿ + z2nr T

where: y = ˜vec(Y), z = ˜vec(Z) and s̃2Ÿ =#

Ÿ(s1) ⁄(s1) · · · Ÿ(sŸ) ⁄(sŸ)$

t

I QR decomposition of Heq

: Heq

= QR

R is upper triangular.

I ML metric: m(s̃) =Î y ≠ Heq

s̃ Î2=Î Qty ≠ Rs̃ Î2

4 / 20

MIMO system

June 2017 26/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

19.8

34.3

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

Depth-first search strategy

Sphere Decoder (SD)

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

0

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

0

14.5

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

0

14.5

2.5

4.9

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

0

14.5

2.5

4.9

7.4

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

0

14.5

2.5

4.9

7.4

0

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

0

14.5

2.5

4.9

7.4

0

2.5

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

0

14.5

2.5

4.9

7.4

0

2.5

0.4

0.2

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

SD searches the ML solution under a sphere of

a given radius centered on the received point.

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (2/3): case of the Sphere Decoder

s2

s1

⇥ s2

s1

sroot

14.5

19.8

34.3

4.9

19.4

0

14.5

2.5

4.9

7.4

0

2.5

0.2

0.6

0.4

ŝ = [1,�3]t

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 8 / 18

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Sphere Decoder (SD)

SD searches the ML solution under a sphere of

a given radius centered on the received point.

27/41

Depth-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (3/3): case of the SB-Stack decoder

Empty Stack

s2

s1

sroot

ŝ = [1,�3]t

Using the same initial radius, the SB-Stack achieves same ML performance asthe Sphere Decoder with complexity gain of 30%.

Choice of initial radius C is fundamental for both decoders.

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 9 / 18

• Using the same initial radius, the SB-Stack achieves same ML performance as the Sphere Decoder with complexity gain of 30%.

• Choice of initial radius C is fundamental for both decoders.

SB-Stack combines the search strategy of the

Stack decoder and the search region of the SD

Spherical-Bound Stack Decoder (SB-Stack)

June 2017 28/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Best-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (3/3): case of the SB-Stack decoder

Empty Stack

3 0.4

2 2.5

1 14.5

s2

s1

sroot

14.5 2.5 0.4

1 2 3

ŝ = [1,�3]t

Using the same initial radius, the SB-Stack achieves same ML performance asthe Sphere Decoder with complexity gain of 30%.

Choice of initial radius C is fundamental for both decoders.

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 9 / 18

• Using the same initial radius, the SB-Stack achieves same ML performance as the Sphere Decoder with complexity gain of 30%.

• Choice of initial radius C is fundamental for both decoders.

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Spherical-Bound Stack Decoder (SB-Stack)

SB-Stack combines the search strategy of the

Stack decoder and the search region of the SD

28/41

Best-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (3/3): case of the SB-Stack decoder

Empty Stack

1 14.5

2 2.5

3 0.4

decreasing

metric

s2

s1

sroot

14.5 2.5 0.4

1 2 3

ŝ = [1,�3]t

Using the same initial radius, the SB-Stack achieves same ML performance asthe Sphere Decoder with complexity gain of 30%.

Choice of initial radius C is fundamental for both decoders.

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 9 / 18

• Using the same initial radius, the SB-Stack achieves same ML performance as the Sphere Decoder with complexity gain of 30%.

• Choice of initial radius C is fundamental for both decoders.

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Spherical-Bound Stack Decoder (SB-Stack)

SB-Stack combines the search strategy of the

Stack decoder and the search region of the SD

28/41

Best-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (3/3): case of the SB-Stack decoder

Empty Stackdecreasing

metric

1 14.5

2 2.5

4 0.6

s2

s1

sroot

14.5 2.5 0.4

1 2 3

0.2

40.6

ŝ = [1,�3]t

Choice of initial radius C is fundamental for both decoders.

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 9 / 18

• Choice of initial radius C is fundamental for both decoders.

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Spherical-Bound Stack Decoder (SB-Stack)

SB-Stack combines the search strategy of the

Stack decoder and the search region of the SD

28/41

Best-first search strategy

System model and assumptions Initial Radius Selection Conclusion

Example (3/3): case of the SB-Stack decoder

Empty Stack

1 14.5

2 2.5

4 0.6

leaf node

s2

s1

sroot

14.5 2.5 0.4

1 2 3

0.2

4

0.2

0.6

0.4

ŝ = [1,�3]t

Choice of initial radius C is fundamental for both decoders.

G. Rekaya (Telecom-ParisTech) ISWCS August 28th, 2015 9 / 18

• Choice of initial radius C is fundamental for both decoders.

June 2017 G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Spherical-Bound Stack Decoder (SB-Stack)

SB-Stack combines the search strategy of the

Stack decoder and the search region of the SD

28/41

Best-first search strategy

• SB-Stack

• Hard output: ML Solution is returned

• Soft output: ML solution and its neighborhoods are returned, LLR (Log

Likelihood Ratio) are calculated using returned list of candidates.

• Zig-zag Stack

• Is a variant of SB-Stack

• Using the search strategy of the Stack decoder, the algorithm builds the

search tree by zigzagging around projection of each tree point.

• Less complexity than SB-Stack

June 2017 29/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Spherical-Bound Stack Decoder (SB-Stack)

Block Decoding

• Commonly, a whole decoding of a MIMO system is processed.

• Another way to do is to divide the MIMO system into blocks and to make recursive block decoding.

• Only few results exists in the literature :

- for specific codes, but performance are sub-otipmal

- for algebraic space-time codes, optimal performance but high complexity (close to exhaustive search.

Is block decoding really promising in term of performance and complexity ?

How to get the best block division ?

How to decode the different blocks?

Block Decoding

June 2017 30/40G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

0

R2

B

R1

* +=

y(2)

y(1) s(1)

s(2) z(2)

z(1)p

n-p

2. Feedback

1: Detect s(1)

3: Detect s(2)

Fig. 1. Block division of the R matrix

Based on this division, (5) can be rewritten as:

ŝ = argmin(s(1),s(2))2Ap⇥An�p

k y0 �R s k2

= argmin(s(1),s(2))2Ap⇥An�p

k y(2) �R2 s(2) �B s(1) k2 +

k y(1) �R1 s(1) k2 (6)

III. RELATED WORK

Reducing complexity while maintaining a good errorperformance and full diversity has been the object of manystudies in the literature. We will focus on recursive signalset detection based works. Two main approaches are studiedhere.The first approach is based on the division of the channelmatrix in 2 blocks as in Fig.1. In [5] an ML decodingscheme is performed on the first block of size p, then adecision feedback equalizer (namely ZF-DFE) is appliedto the remaining system given the output of the first MLdecoding (i.e by subtracting the first ML output from thereceived signal). It was shown that this scheme is able toincrease the diversity order for the worst sub-channel from 1to p. An ordering scheme could be also applied to give thebest decoding to the worst sub-channels, thus it is shown thatan SNR gain equal to the number of transmitting antennascan be obtained.The second approach, space-time coded systems orientedand compatible with sphere decoder [2], consists in splittingthe received signal into L � 2 subsets each of cardinality �.A conditional maximization of the likelihood function withrespect to one signal set point given another is performed.Informally:

1) Exhaustive search for one sub-set.2) Remove interference of all possible values of the first

sub-set from remaining L� 1 sub-sets.3) Decode L� 1 sub-sets with a ZF decoder for each

decoded point of the first block.4) Select optimal solution overall calculated solutions.

The choice of the signal set to decode first is crucial for theperformance of the algorithm i.e to guarantee a maximaldesired diversity order. Thus empirical [6–8] and analytical[9] set selection criteria on the equivalent channel matrixare derived.In [6] (and [7], [8]), authors examined the cases of GoldenCode [12] (and 3⇥3, 4⇥4 perfect codes [13] and any n⇥nalgebraic space-time code respectively). In these works, the

main set selection criteria considered are the determinantof covariance matrices of the sub-channels. This quantitymeasures the instantaneous SNR of the corresponding linearsystem and thus should be large. Another criterion was alsostudied which is the condition number of this covariancematrix which measures the accuracy of the zero-forcingapproximation and thus should be small. Then, the ratio ofthese quantities should be maximized. It was experimentallyfound that in the case of Golden Code, the sub-channel whosecondition number is the smallest has the biggest determinantand thus a determinant based criterion is sufficient. In PerfectCode case, the condition number based criterion makes theperformance slightly better and thus we can obviate theneed to compute it too, taking into account the additionalcondition number computation complexity to be added inthis case.

In [9], inspired from the above mentioned works, au-thors introduce two new low complexity decoders namelyACZF (Adaptive Conditional Zero-Forcing)and ACZF-SIC(Adaptive Conditional Zero-Forcing with SuccessiveInterference Cancellation) where they give 2 equivalentsufficient conditions based on STBC characteristics to getfull diversity with these decoders. One sufficient conditionis the full rank of at least one of the L sub-matrices.

IV. THE PROPOSED RECURSIVE BLOCK DECODING

In this section we present the main result of the paperwhich is semi-exhaustive recursive block decoding. In[9](Table 1), it is shown that for some ST codes, decodingcomplexity is slightly reduced using the proposed recursiveblock decoder compared to known ML decoding complex-ities of these codes. This is due to the exhaustive searchperformed in the first step (decoding of first block). Ourproposed decoder solves this issue by reducing the numberof candidates kept in the first step compared to the exhaustivesearch. In addition, it offers a flexibility on the diversityorder (impacting the overall complexity) by choosing a targetdiversity order less or equal to the full diversity imposedin the above mentioned works. The control of the diversityorder is obtained through the choice of decoding parameters(like: block size, block order, list size or equivalently astopping radius). Parameters such as SNR and constellationsize are also taken into account.We divide the R matrix into 2 sub-blocks and we split,accordingly, the n real information symbols contained in s .Then, 4 steps are performed:

1) Generate a list containing the ML solution and someof its neighborhoods as an output of the decoding ofthe first block .

2) Subtract the interference of the decoded block (for eachlist point) from the remaining system.

3) ML or ZF-DFE decoding of the second block for eachcandidate of the list.

4) Select the solution that minimizes the overall ML metricwith respect to (4)

For the first stage, we propose 2 possibilities to generate thelist:

• by looking for all points inside a sphere centered on MLsolution adapted for a decoding using Sphere Decoder.

• by looking to construct a list of fixed size adapted fora decoding using Stack Decoder.

2

where � 2 CntT⇥ntT corresponds to the coding matrix of the underlying code [11]. For simplicity, given that bothuncoded and coded schemes result in a same real-valued lattice representation, we consider in the remaining of thiswork the spatial multiplexing scheme. Let n = 2nt.

2.1 ML detection

In the coherent case where H is considered known at the receiver side, ML decoder finds the information vector sminimizing

ŝ = argmins2An

k y �H s k2 (4)

where A represents the M-ary QAM constellation to which belong the real and imaginary parts of information symbols.This system can be resolved by using lattice decoders based on tree-search algorithms [2]. To get the tree structure, aQR decomposition is applied on the lattice generator matrix H. Equation (4) is equivalent to:

ŝ = argmins2An

k y0 �R s k2 (5)

where Q is an orthogonal matrix, R an upper triangular one and y0 = Qty.

2.2 Block division

Figure 1: Block division of the R matrix

We will consider a block division of matrix R to proceed to a recursive block decoding. The considered blockdivision is depicted in Fig.1.R1 2 Rp⇥p going to be decoded first is an upper triangular matrix. R2 2 Rn�p⇥n�p an upper triangular matrix is goingto be decoded in the second stage. B 2 Rn�p⇥p is a rectangular feedback matrix. s(1) and s(2) are the correspondingsymbol vectors of size p and n� p respectively. Based on this division, (5) can be rewritten as:

ŝ = argmin(s(1),s(2))2Ap⇥An�p

k y0 �R s k2

= argmin(s(1),s(2))2Ap⇥An�p

k y(2) �R2 s(2) �B s(1) k2 +

k y(1) �R1 s(1) k2 (6)

3 Related work

Reducing complexity while maintaining a good error performance and full diversity has been the object of many studiesin the literature. We will focus on recursive signal set detection based works. Two main approaches are studied here.The first approach is based on the division of the channel matrix in 2 blocks as in Fig.1. In [5] an ML decoding schemeis performed on the first block of size p, then a decision feedback equalizer (namely ZF-DFE) is applied to the remainingsystem given the output of the first ML decoding (i.e by subtracting the first ML output from the received signal). Itwas shown that this scheme is able to increase the diversity order for the worst sub-channel from 1 to p. An orderingscheme could be also applied to give the best decoding to the worst sub-channels, thus it is shown that an SNR gain

2

Recursive Block Decoding : 2 blocks

June 2017 31/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Existing algorithms :

1. Select the first subset function of its determinant.2. Exhaustive search for selected sub-set.3. Remove interference of all possible values of the first decoded sub-set from

remaining sub-set. 4. Decode second sub-set with a ZF decoder for each decoded point of the first sub-set.5. Select optimal solution overall calculated solutions.

• Advantage: ML performance

• Inconvenient: Complexity close to exhaustive search

Recursive Block Decoding : 2 blocks

June 2017 32/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Empirical distribution of the Occurrence of candidates in the ML solution

Recursive Block Decoding : 2 blocks

June 2017 33/41G. Rekaya-Ben Othman

0 5 10 150

500

1000

1500

2000

2500SemiExhaustive ML snr=0

0 5 10 150

500

1000

1500

2000

2500

3000

3500

4000

4500SemiExhaustive ML snr=3

0 5 10 150

2000

4000

6000

8000

10000

12000

14000SemiExhaustive ML snr=6

0 5 10 150

1

2

3

4

5

6x 104 SemiExhaustive ML snr=9

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5x 105 SemiExhaustive ML snr=12

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 105 SemiExhaustive ML snr=15

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 105 SemiExhaustive ML snr=24

Fig. 3. Empirical distribution of the Occurrence Of candidates in the MLsolution

0 2 4 6 8 10 12 14 16 1810−5

10−4

10−3

10−2

10−1

100Perf semiSBstack1/4 +ML (SE) 3/4

Sys13CP4ClassiqueSEX0Y0ZFDFESys22List50MinRepetition20Radius2nsegmaZFDFESys22List100MinRepetition20Radius2nsegmaZFDFESys22List100MinRepetition50Radius2nsegmaZFDFESys22List50MinRepetition1Radius2nsegmaZFDFESys22List100MinRepetition1Radius2nsegmaZFDFESys31List50MinRepetition20Radius2nsegmaZFDFESys31List100MinRepetition20Radius2nsegmaZFDFESys31List100MinRepetition50Radius2nsegmaZFDFESys31List50MinRepetition1Radius2nsegmaZFDFESys31List50MinRepetition1Radius2nsegmaMLSys22List50MinRepetition20Radius2nsegmaMLSys22List100MinRepetition20Radius2nsegmaMLSys22List100MinRepetition50Radius2nsegmaMLSys22List50MinRepetition1Radius2nsegmaMLSys22List100MinRepetition1Radius2nsegmaMLSys31List50MinRepetition20Radius2nsegmaMLSys31List100MinRepetition20Radius2nsegmaMLSys31List100MinRepetition50Radius2nsegmaMLSys31List50MinRepetition1Radius2nsegmaMLSys31List100MinRepetition1Radius2nsegma

Fig. 4. Performance Comaprison

0 2 4 6 8 10 12 14 16 18104

105

106

107Compx semiSBstack1/4 +ML (SE) 3/4

Sys13CP4ClassiqueSEX0Y0ZFDFESys22List50MinRepetition20Radius2nsegmaZFDFESys22List100MinRepetition20Radius2nsegmaZFDFESys22List100MinRepetition50Radius2nsegmaZFDFESys22List50MinRepetition1Radius2nsegmaZFDFESys22List100MinRepetition1Radius2nsegmaZFDFESys31List50MinRepetition20Radius2nsegmaZFDFESys31List100MinRepetition20Radius2nsegmaZFDFESys31List100MinRepetition50Radius2nsegmaZFDFESys31List50MinRepetition1Radius2nsegmaZFDFESys31List50MinRepetition1Radius2nsegmaMLSys22List50MinRepetition20Radius2nsegmaMLSys22List100MinRepetition20Radius2nsegmaMLSys22List100MinRepetition50Radius2nsegmaMLSys22List50MinRepetition1Radius2nsegmaMLSys22List100MinRepetition1Radius2nsegmaMLSys31List50MinRepetition20Radius2nsegmaMLSys31List100MinRepetition20Radius2nsegmaMLSys31List100MinRepetition50Radius2nsegmaMLSys31List50MinRepetition1Radius2nsegmaMLSys31List100MinRepetition1Radius2nsegma

Fig. 5. Complexity Comparison

chooses the block having the largest determinant to be the1st block to be decoded. It was shown in [12] and [15] thatbased on this criteria, the ZF-DFE is able to achieve fulldiversity.Simulation are reported for an extended version of this paper.

VIII. CONCLUSION

..............

REFERENCES[1] Gerard J. Foschini. Layered space-time architecture for

wireless communication in a fading environment when usingmulti-element antennas. Bell Labs Technical Journal, 1(2):41–59, Autumn 1996.

[2] E. Viterbo and E. Biglieri. A universal decoding algorithmfor lattice codes. In Quatorzieme colloque GRETSI, 1993.

[3] C. P. Schnorr and M. Euchner. Lattice Basis Reduction:Improved Practical Algorithms and Solving Subset SumProblems. In Math. Programming, pages 181–191, 1993.

[4] B. Hassibi and H. Vikalo. On the sphere-decoding algorithmi. expected complexity. Signal Processing, IEEE Transactionson, 53(8):2806–2818, Aug 2005.

[5] Byonghyo Shim and Insung Kang. Sphere decoding with aprobabilistic tree pruning. Signal Processing, IEEE Transac-tions on, 56(10):4867–4878, Oct 2008.

[6] Tao Cui, Shuangshuang Han, and C. Tellambura. Probability-distribution-based node pruning for sphere decoding. Vehicu-lar Technology, IEEE Transactions on, 62(4):1586–1596, May2013.

[7] Kwan wai Wong, Chi ying Tsui, R.S.-K. Cheng, and Wai-Ho Mow. A vlsi architecture of a k-best lattice decodingalgorithm for mimo channels. In Circuits and Systems, 2002.ISCAS 2002. IEEE International Symposium on, volume 3,pages III–273–III–276 vol.3, 2002.

[8] Qingwei Li and Zhongfeng Wang. Improved k-best sphere de-coding algorithms for mimo systems. In Circuits and Systems,2006. ISCAS 2006. Proceedings. 2006 IEEE InternationalSymposium on, pages 4 pp.–1162, May 2006.

[9] Chung-An Shen and A.M. Eltawil. An adaptive reducedcomplexity k-best decoding algorithm with early termination.In Consumer Communications and Networking Conference(CCNC), 2010 7th IEEE, pages 1–5, Jan 2010.

[10] R. Gowaikar and B. Hassibi. Statistical pruning for near-maximum likelihood decoding. Signal Processing, IEEETransactions on, 55(6):2661–2675, June 2007.

[11] Won-Joon Choi, R. Negi, and J.M. Cioffi. Combined mland dfe decoding for the v-blast system. In Communications,2000. ICC 2000. 2000 IEEE International Conference on,volume 3, pages 1243–1248 vol.3, 2000.

5

0 5 10 150

500

1000

1500

2000

2500SemiExhaustive ML snr=0

0 5 10 150

500

1000

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2000

2500

3000

3500

4000

4500SemiExhaustive ML snr=3

0 5 10 150

2000

4000

6000

8000

10000

12000

14000SemiExhaustive ML snr=6

0 5 10 150

1

2

3

4

5

6x 104 SemiExhaustive ML snr=9

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5x 105 SemiExhaustive ML snr=12

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 105 SemiExhaustive ML snr=15

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 105 SemiExhaustive ML snr=24

Fig. 3. Empirical distribution of the Occurrence Of candidates in the MLsolution

0 2 4 6 8 10 12 14 16 1810−5

10−4

10−3

10−2

10−1

100Perf semiSBstack1/4 +ML (SE) 3/4

Sys13CP4ClassiqueSEX0Y0ZFDFESys22List50MinRepetition20Radius2nsegmaZFDFESys22List100MinRepetition20Radius2nsegmaZFDFESys22List100MinRepetition50Radius2nsegmaZFDFESys22List50MinRepetition1Radius2nsegmaZFDFESys22List100MinRepetition1Radius2nsegmaZFDFESys31List50MinRepetition20Radius2nsegmaZFDFESys31List100MinRepetition20Radius2nsegmaZFDFESys31List100MinRepetition50Radius2nsegmaZFDFESys31List50MinRepetition1Radius2nsegmaZFDFESys31List50MinRepetition1Radius2nsegmaMLSys22List50MinRepetition20Radius2nsegmaMLSys22List100MinRepetition20Radius2nsegmaMLSys22List100MinRepetition50Radius2nsegmaMLSys22List50MinRepetition1Radius2nsegmaMLSys22List100MinRepetition1Radius2nsegmaMLSys31List50MinRepetition20Radius2nsegmaMLSys31List100MinRepetition20Radius2nsegmaMLSys31List100MinRepetition50Radius2nsegmaMLSys31List50MinRepetition1Radius2nsegmaMLSys31List100MinRepetition1Radius2nsegma

Fig. 4. Performance Comaprison

0 2 4 6 8 10 12 14 16 18104

105

106

107Compx semiSBstack1/4 +ML (SE) 3/4

Fig. 5. Complexity Comparison

chooses the block having the largest determinant to be the1st block to be decoded. It was shown in [12] and [15] thatbased on this criteria, the ZF-DFE is able to achieve fulldiversity.Simulation are reported for an extended version of this paper.

VIII. CONCLUSION

..............

REFERENCES[1] Gerard J. Foschini. Layered space-time architecture for

wireless communication in a fading environment when usingmulti-element antennas. Bell Labs Technical Journal, 1(2):41–59, Autumn 1996.

[2] E. Viterbo and E. Biglieri. A universal decoding algorithmfor lattice codes. In Quatorzieme colloque GRETSI, 1993.

[3] C. P. Schnorr and M. Euchner. Lattice Basis Reduction:Improved Practical Algorithms and Solving Subset SumProblems. In Math. Programming, pages 181–191, 1993.

[4] B. Hassibi and H. Vikalo. On the sphere-decoding algorithmi. expected complexity. Signal Processing, IEEE Transactionson, 53(8):2806–2818, Aug 2005.

[5] Byonghyo Shim and Insung Kang. Sphere decoding with aprobabilistic tree pruning. Signal Processing, IEEE Transac-tions on, 56(10):4867–4878, Oct 2008.

[6] Tao Cui, Shuangshuang Han, and C. Tellambura. Probability-distribution-based node pruning for sphere decoding. Vehicu-lar Technology, IEEE Transactions on, 62(4):1586–1596, May2013.

[7] Kwan wai Wong, Chi ying Tsui, R.S.-K. Cheng, and Wai-Ho Mow. A vlsi architecture of a k-best lattice decodingalgorithm for mimo channels. In Circuits and Systems, 2002.ISCAS 2002. IEEE International Symposium on, volume 3,pages III–273–III–276 vol.3, 2002.

[8] Qingwei Li and Zhongfeng Wang. Improved k-best sphere de-coding algorithms for mimo systems. In Circuits and Systems,2006. ISCAS 2006. Proceedings. 2006 IEEE InternationalSymposium on, pages 4 pp.–1162, May 2006.

[9] Chung-An Shen and A.M. Eltawil. An adaptive reducedcomplexity k-best decoding algorithm with early termination.In Consumer Communications and Networking Conference(CCNC), 2010 7th IEEE, pages 1–5, Jan 2010.

[10] R. Gowaikar and B. Hassibi. Statistical pruning for near-maximum likelihood decoding. Signal Processing, IEEETransactions on, 55(6):2661–2675, June 2007.

[11] Won-Joon Choi, R. Negi, and J.M. Cioffi. Combined mland dfe decoding for the v-blast system. In Communications,2000. ICC 2000. 2000 IEEE International Conference on,volume 3, pages 1243–1248 vol.3, 2000.

5

0 5 10 150

500

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1500

2000

2500SemiExhaustive ML snr=0

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4x 105 SemiExhaustive ML snr=15

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 105 SemiExhaustive ML snr=24

Fig. 3. Empirical distribution of the Occurrence Of candidates in the MLsolution

0 2 4 6 8 10 12 14 16 1810−5

10−4

10−3

10−2

10−1

100Perf semiSBstack1/4 +ML (SE) 3/4

Fig. 4. Performance Comaprison

0 2 4 6 8 10 12 14 16 18104

105

106

107Compx semiSBstack1/4 +ML (SE) 3/4

Fig. 5. Complexity Comparison

chooses the block having the largest determinant to be the1st block to be decoded. It was shown in [12] and [15] thatbased on this criteria, the ZF-DFE is able to achieve fulldiversity.Simulation are reported for an extended version of this paper.

VIII. CONCLUSION

..............

REFERENCES[1] Gerard J. Foschini. Layered space-time architecture for

wireless communication in a fading environment when usingmulti-element antennas. Bell Labs Technical Journal, 1(2):41–59, Autumn 1996.

[2] E. Viterbo and E. Biglieri. A universal decoding algorithmfor lattice codes. In Quatorzieme colloque GRETSI, 1993.

[3] C. P. Schnorr and M. Euchner. Lattice Basis Reduction:Improved Practical Algorithms and Solving Subset SumProblems. In Math. Programming, pages 181–191, 1993.

[4] B. Hassibi and H. Vikalo. On the sphere-decoding algorithmi. expected complexity. Signal Processing, IEEE Transactionson, 53(8):2806–2818, Aug 2005.

[5] Byonghyo Shim and Insung Kang. Sphere decoding with aprobabilistic tree pruning. Signal Processing, IEEE Transac-tions on, 56(10):4867–4878, Oct 2008.

[6] Tao Cui, Shuangshuang Han, and C. Tellambura. Probability-distribution-based node pruning for sphere decoding. Vehicu-lar Technology, IEEE Transactions on, 62(4):1586–1596, May2013.

[7] Kwan wai Wong, Chi ying Tsui, R.S.-K. Cheng, and Wai-Ho Mow. A vlsi architecture of a k-best lattice decodingalgorithm for mimo channels. In Circuits and Systems, 2002.ISCAS 2002. IEEE International Symposium on, volume 3,pages III–273–III–276 vol.3, 2002.

[8] Qingwei Li and Zhongfeng Wang. Improved k-best sphere de-coding algorithms for mimo systems. In Circuits and Systems,2006. ISCAS 2006. Proceedings. 2006 IEEE InternationalSymposium on, pages 4 pp.–1162, May 2006.

[9] Chung-An Shen and A.M. Eltawil. An adaptive reducedcomplexity k-best decoding algorithm with early termination.In Consumer Communications and Networking Conference(CCNC), 2010 7th IEEE, pages 1–5, Jan 2010.

[10] R. Gowaikar and B. Hassibi. Statistical pruning for near-maximum likelihood decoding. Signal Processing, IEEETransactions on, 55(6):2661–2675, June 2007.

[11] Won-Joon Choi, R. Negi, and J.M. Cioffi. Combined mland dfe decoding for the v-blast system. In Communications,2000. ICC 2000. 2000 IEEE International Conference on,volume 3, pages 1243–1248 vol.3, 2000.

5

0 5 10 150

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4x 105 SemiExhaustive ML snr=15

0 5 10 150

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4x 105 SemiExhaustive ML snr=24

Fig. 3. Empirical distribution of the Occurrence Of candidates in the MLsolution

0 2 4 6 8 10 12 14 16 1810−5

10−4

10−3

10−2

10−1

100Perf semiSBstack1/4 +ML (SE) 3/4

Fig. 4. Performance Comaprison

0 2 4 6 8 10 12 14 16 18104

105

106

107Compx semiSBstack1/4 +ML (SE) 3/4

Fig. 5. Complexity Comparison

VIII. CONCLUSION

..............

REFERENCES[1] Gerard J. Foschini. Layered space-time architecture for

5

Vision PF MIMO Sequential decoder Block decoder

Proposed recursive decoding:

1. Generate a list containing the ML solution and some of its neighbors as an output of the decoding of the first block .

2. Subtract the interference of the decoded block (for each list point) from the remaining system.

3. ML decoding of the second block for each candidate of the list. 4. Select the solution that minimizes the overall ML.

• Semi-exhaustive search for the first stage:

- Using Sphere decoder: once the ML solution is found, search all points inside a sphere centered on ML point.

- Using SB-Stack decoder: once the ML solution is found, the search is continued to construct a list of fixed size.

Semi-exhaustive Block Decoding : 2 blocks

June 2017 34/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Now from (13),

Pr (esc) = 1� Pr (es)

= 1�⇣1� Pr

⇣gs

(1)

c⌘⌘⇣1� Pr

⇣gs

(2)

c| gs(1)

⌘⌘

⇡ Pr⇣gs

(1)

c⌘+ Pr

⇣gs

(2)

c| gs(1)

⌘(19)

since Pr⇣gs

(1)

c⌘and Pr

⇣gs

(2) | gs(1)⌘are small at high SNR.

Combining (12),(19),(18)and(15), the FER in (8) is upper-bounded by:

Pef �(p

2

,

r2

2�2 )

�(p2

)+

|I|+ |A|n�p⇣1 +

d2min4�2

⌘n (20)

Equation (20) shows that the diversity order that could be achieved by this decoding scheme is controlled by the firstterm given that the second one achieves full diversity. Therefore, to guarantee an overall diversity order of at least ,the first term (function of the block size p, the noise term �2 and the metric threshold r) should decrease at the orderof �2. This goes back, for a given fixed high SNR (or small �2) and block size p, to find the minimum threshold r thatsatisfies:

�(p2

,

r2

2�2 )

�(p2

) ��2 (21)

for some positive constant � that controls the SNR gain. This inequality on r is solved numerically in simulations witha margin of error as small as possible.

5.2 FER Analysis for ZF-DFE decoding in the 2nd stage

In this section, we show that the overall diversity order provided by this decoding scheme for block decoding wheren� p � 2 is limited by the sub-optimal decoding in the second stage i.e max = 1.Same as in 5.1, we derive Pr

⇣gs

(2)

c| gs(1)

⌘= 1� Pr

⇣gs

(2) | gs(1)⌘in the case where a ZF-DFE decode with no channel

ordering is used for the 2nd decoding stage given that s(1) was visited(i.e decoding inside the framed sub-tree in Fig.2).

Now the ZF-DFE decoder is looking for the estimate ds(2) with respect to the following system:

y(2) = R2s(2) +Bs(1) + z(2) (22)

After SIC operation using the transmitted sub-vector s(1) (i.e no error propagation), the system is rewritten as:

y0(2) = y(2) �Bs(1) = R2s(2) + z(2) (23)

Given assumptions on noise statistics and that no channel ordering is performed, ZF-DFE decoder, applied on suchsystem, is known to provide a maximum diversity order of max = 1 which controls then the overall diversity order(even if an exhaustive search in the first stage is performed).

Note that if n� p = 1 i.e only one symbol left to detect, the ZF-DFE decoder coincides with ML decoder and thusdiversity order follows the same rule as in previous section.

6 Simulation Results

The proposed decoder has been validated by numerical simulations considering spatial multiplexing for nt = nr = 4using a 4-QAM modulation. Block dimensions are p = n � p = 4 (n = 8 in real valued system). We provide herenumerical results evaluating the Symbol Error Rate SER and the average decoding complexity(computed as the overallnumber of multiplications) averaging over 2⇥ 106 channel realizations.We compare diversity order in Fig.3 and complexity in Fig.4 of Sphere Decoder with 4 possible configurations of ourdecoder scheme by varying the minimum target diversity order and the SNR gain factor �.

In the first scenario (black line), we set the target diversity = 4 (i.e full diversity). We observe that this diversityorder is indeed achieved with a noticeable complexity reduction compared to the SD.

7

Now from (13),

Pr (esc) = 1� Pr (es)

= 1�⇣1� Pr

⇣gs

(1)

c⌘⌘⇣1� Pr

⇣gs

(2)

c| gs(1)

⌘⌘

⇡ Pr⇣gs

(1)

c⌘+ Pr

⇣gs

(2)

c| gs(1)

⌘(19)

since Pr⇣gs

(1)

c⌘and Pr

⇣gs

(2) | gs(1)⌘are small at high SNR.

Combining (12),(19),(18)and(15), the FER in (8) is upper-bounded by:

Pef �(p

2

,

r2

2�2 )

�(p2

)+

|I|+ |A|n�p⇣1 +

d2min4�2

⌘n (20)

Equation (20) shows that the diversity order that could be achieved by this decoding scheme is controlled by the firstterm given that the second one achieves full diversity. Therefore, to guarantee an overall diversity order of at least ,the first term (function of the block size p, the noise term �2 and the metric threshold r) should decrease at the orderof �2. This goes back, for a given fixed high SNR (or small �2) and block size p, to find the minimum threshold r thatsatisfies:

�(p2

,

r2

2�2 )

�(p2

) ��2 (21)

for some positive constant � that controls the SNR gain. This inequality on r is solved numerically in simulations witha margin of error as small as possible.

5.2 FER Analysis for ZF-DFE decoding in the 2nd stage

In this section, we show that the overall diversity order provided by this decoding scheme for block decoding wheren� p � 2 is limited by the sub-optimal decoding in the second stage i.e max = 1.Same as in 5.1, we derive Pr

⇣gs

(2)

c| gs(1)

⌘= 1� Pr

⇣gs

(2) | gs(1)⌘in the case where a ZF-DFE decode with no channel

ordering is used for the 2nd decoding stage given that s(1) was visited(i.e decoding inside the framed sub-tree in Fig.2).

Now the ZF-DFE decoder is looking for the estimate ds(2) with respect to the following system:

y(2) = R2s(2) +Bs(1) + z(2) (22)

After SIC operation using the transmitted sub-vector s(1) (i.e no error propagation), the system is rewritten as:

y0(2) = y(2) �Bs(1) = R2s(2) + z(2) (23)

Given assumptions on noise statistics and that no channel ordering is performed, ZF-DFE decoder, applied on suchsystem, is known to provide a maximum diversity order of max = 1 which controls then the overall diversity order(even if an exhaustive search in the first stage is performed).

Note that if n� p = 1 i.e only one symbol left to detect, the ZF-DFE decoder coincides with ML decoder and thusdiversity order follows the same rule as in previous section.

6 Simulation Results

The proposed decoder has been validated by numerical simulations considering spatial multiplexing for nt = nr = 4using a 4-QAM modulation. Block dimensions are p = n � p = 4 (n = 8 in real valued system). We provide herenumerical results evaluating the Symbol Error Rate SER and the average decoding complexity(computed as the overallnumber of multiplications) averaging over 2⇥ 106 channel realizations.We compare diversity order in Fig.3 and complexity in Fig.4 of Sphere Decoder with 4 possible configurations of ourdecoder scheme by varying the minimum target diversity order and the SNR gain factor �.

In the first scenario (black line), we set the target diversity = 4 (i.e full diversity). We observe that this diversityorder is indeed achieved with a noticeable complexity reduction compared to the SD.

7

where, r is the sphere radius, p is the first block size, n is the system size.

• The diversity order that could be achieved by this decoding scheme is controlled by the first term given that the second one achieves full diversity.

• To guarantee an overall diversity order of at least K, the first term should be upper bounded by:

Diversity Order Analysis

• Considering recursive decoding, with semi-exhaustive search for the first block and ML decoding for the second block, the Frame Error probability is upper-bounded by:

• This goes back, for a given fixed SNR and block size p, to find the minimum r that satisfies the upper bound —> Could be done numerically.

June 2017 35/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Semi-exhaustive Block Decoding : 2 blocks

In the second and third scenario, we set the minimum target diversity = 2 with 2 variants for SNR gain factor �.We validate in Fig.3 that SER curves (blue and magenta lines) have a same slope of 2 with di↵erent SNR gains. Thelast scenario(in green) depicts the case where the minimum target diversity order is set to = 1.

We can observe that at low SNR regime, the second,third and fourth scenarios (blue, magenta and green curves)have higher complexity than the first considered scenario. This behavior is due to the choice, at this regime, of avariable delta function of SNR in the first scenario, which seems to be the best way for the choice of delta. Thisbehavior could be explained by the fact for small values of ⇢, �⇢�n ! 1 which gives too small radius (due to propertiesof normalized upper gamma function), thus no lattice point could be found, causing the algorithm to restart andincrease complexity.

Figure 3: Symbol Error Rate for nt = nr = 4 using 4-QAM constellation and spatial multiplexing

Figure 4: Total computational complexity for nt = nr = 4 using 4-QAM constellation and spatial multiplexing

7 Conclusion

This work was dedicated to semi-exhaustive recursive block decoding implementable in linear communication systemsincluding MIMO systems. A complexity reduction coupled with a flexibility on desired diversity order, betweenminimum and full diversity for a given system is achieved. In this work, the case of 2 blocks of the same size was studied,in future work we are considering, among others, the case of division into more than 2 sub-blocks with di↵erent sizes.

8

In the second and third scenario, we set the minimum target diversity = 2 with 2 variants for SNR gain factor �.We validate in Fig.3 that SER curves (blue and magenta lines) have a same slope of 2 with di↵erent SNR gains. Thelast scenario(in green) depicts the case where the minimum target diversity order is set to = 1.

We can observe that at low SNR regime, the second,third and fourth scenarios (blue, magenta and green curves)have higher complexity than the first considered scenario. This behavior is due to the choice, at this regime, of avariable delta function of SNR in the first scenario, which seems to be the best way for the choice of delta. Thisbehavior could be explained by the fact for small values of ⇢, �⇢�n ! 1 which gives too small radius (due to propertiesof normalized upper gamma function), thus no lattice point could be found, causing the algorithm to restart andincrease complexity.

Figure 3: Symbol Error Rate for nt = nr = 4 using 4-QAM constellation and spatial multiplexing

Figure 4: Total computational complexity for nt = nr = 4 using 4-QAM constellation and spatial multiplexing

7 Conclusion

This work was dedicated to semi-exhaustive recursive block decoding implementable in linear communication systemsincluding MIMO systems. A complexity reduction coupled with a flexibility on desired diversity order, betweenminimum and full diversity for a given system is achieved. In this work, the case of 2 blocks of the same size was studied,in future work we are considering, among others, the case of division into more than 2 sub-blocks with di↵erent sizes.

8

4x4 MIMO system using 4-QAM constellation and spatial multiplexing

June 2017 36/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Semi-exhaustive Block Decoding : 2 blocks

June 2017 37/41G. Rekaya-Ben Othman

Fig. 1. Block division of the R matrix

can be rewritten as:

ŝ = argmins2An

k y �R s k2

= argmin(s(1),...,s(k))2Qk

j=1 Apj

kX

j=1

k y(j) �Rj s(j) �Bj�1 s(j�1,1) k2

(6)

Where s(j,1) is the vector composed of [s(j), . . . , s(1)]t andB0 = 0 .

III. RELATED WORKReducing complexity while maintaining a good error

performance and full diversity has been the object of manystudies in the literature. We will focus on recursive signalset detection based works. Two main approaches are studiedhere.The first approach is based on the division of the channelmatrix in 2 blocks. In [5] an ML decoding scheme isperformed on the first block of size p1, then a decisionfeedback equalizer (namely ZF-DFE) is applied to theremaining system given the output of the first ML decoding(i.e by subtracting the first ML output from the receivedsignal). It was shown that this scheme is able to increasethe diversity order for the worst sub-channel from 1 to p1.An ordering scheme could be also applied to give the bestdecoding to the worst sub-channels, thus it is shown that anSNR gain equal to the number of transmitting antennas canbe obtained.The second approach, Space-Time coded systems orientedand compatible with sphere decoder [2], consists in splittingthe received signal into L � 2 subsets each of cardinality �.A conditional maximization of the likelihood function withrespect to one signal set point given another is performed.Informally:

1) Exhaustive search for one sub-set.2) Remove interference of all possible values of the first

sub-set from remaining L� 1 sub-sets.3) Decode L� 1 sub-sets with a ZF decoder for each

decoded point of the first block.4) Select optimal solution overall calculated solutions.

The choice of the signal set to decode first is crucial for theperformance of the algorithm i.e to guarantee a maximaldesired diversity order. Thus empirical [6–8] and analytical

[9] set selection criteria on the equivalent channel matrixare derived.In [6] (and [7], [8]), authors examined the cases of GoldenCode [13] (and 3⇥3, 4⇥4 perfect codes [14] and any n⇥nalgebraic Space-Time code respectively). In these works, themain set selection criteria considered are the determinantof covariance matrices of the sub-channels. This quantitymeasures the instantaneous SNR of the corresponding linearsystem and thus should be large. Another criterion was alsostudied which is the condition number of this covariancematrix which measures the accuracy of the zero-forcingapproximation and thus should be small. Then, the ratio ofthese quantities should be maximized. It was experimentallyfound that in the case of Golden Code, the sub-channel whosecondition number is the smallest has the biggest determinantand thus a determinant based criterion is sufficient. In PerfectCode case, the condition number based criterion makes theperformance slightly better and thus we can obviate theneed to compute it too, taking into account the additionalcondition number computation complexity to be added inthis case.

In [9], inspired from the above mentioned works, au-thors introduce two new low complexity decoders namelyACZF (Adaptive Conditional Zero-Forcing)and ACZF-SIC(Adaptive Conditional Zero-Forcing with SuccessiveInterference Cancellation) where they give 2 equivalentsufficient conditions based on STBC characteristics to getfull diversity with these decoders. One sufficient conditionis the full rank of at least one of the L sub-matrices.

IV. THE PROPOSED RECURSIVE BLOCK DECODINGIn this section we present the main result of the paper

which is semi-exhaustive recursive block decoding. In[9](Table 1), it is shown that for some ST codes, decodingcomplexity is slightly reduced using their proposed recursiveblock decoder compared to known ML decoding complex-ities of these codes. This is due to the exhaustive searchperformed in the first step (decoding of first block). Ourproposed decoder solves this issue by reducing the numberof candidates kept in the first step compared to the exhaustivesearch. In addition, it offers a flexibility on the diversityorder (impacting the overall complexity) by choosing a targetdiversity order less or equal to the full diversity imposedin the above mentioned works. The control of the diversityorder is obtained through the choice of decoding parameters(like: block size, block order, list size or equivalently astopping radius). Parameters such as SNR and constellationsize are also taken into account.We formalize our proposed decoder in Algorithm 1. Itconsists in 3 main steps:

• Preprocessing: Setting of a division scheme i.e choosingthe number of blocks k and their sizes (p1, . . . , pk).Then find a set of radii (r1, . . . , rk�1) representingthreshold on candidates weight coming from eachblock such that a targert minimum diversity order d isachieved. Refer to V for derivation method.

• From lines 5 to 15: starting from the first to thelast-but-one block, generate a list of candidates foreach block (after having removed interference fromprevious blocks). Build progressively a set Spotential ofNbTotal candidates ((n� pk)-size vectors) and theircorresponding weight Wpotential.

2

Vision PF MIMO Sequential decoder Block decoder

Semi-exhaustive Block Decoding : n blocks

Proposed recursive decoding:

1. Set a division scheme by choosing the number an seize of the blocks

2. Calculate a set of Radii representing threshold on list candidates of each block for a target minimum diversity.

3. Starting from the last block, generate a list of candidates for each block, after removing the interference of the decoded blocks.

4. For the first Block, sort the list of potential candidates function of their weights, and so ML decoding of the second block for each candidate of the list.

5.Select the solution that minimizes the overall ML.

June 2017 38/41G. Rekaya-Ben Othman

Vision PF MIMO Sequential decoder Block decoder

Semi-exhaustive Block Decoding : n blocks

where, ri are the sphere radii, pj are the block sizes, n is the system size.

• The diversity order that could be achieved by this decoding scheme is controlled by the second term given that the first one achieves full diversity.

• To guarantee an overall diversity order of at least K, each term of the sum must be upper bounded by:

Diversity Order Analysis

• Considering recursive decoding, with semi-exhaustive search (n-1) blocks and ML decoding for the first block, the Frame Error probability is upper-bounded by:

This goes back to find for each block i the minimum threshold ri , as function of the SNR and the block size pi.

June 2017 39/41G. Rekaya-Ben Othman

Now from (13),

Pr (esc) = 1� Pr (es)

= 1� (1� Ek)i�1Y

j=1

(1� Ei) (18)

At high SNR, Ei, i 2 1 . . . k are small, hence (18) could bewritten as:

Pr (esc) ⇡ 1�

0

@1�kX

j=1

Ei

1

A

⇡ |A|pk

⇣1 +

d2min4�2

⌘n +k�1X

i=1

�(Pi

j=1 pj2 ,

r2i2�2 )

�(Pi

j=1 pj2 )

(19)

Finally, combining (12), (19), the FER in (8) is upper-bounded by:

Pef |I|+ |A|pk⇣1 +

d2min4�2

⌘n +k�1X

i=1

�(Pi

j=1 pj2 ,

r2i2�2 )

�(Pi

j=1 pj2 )

(20)

Equation (20) shows that the diversity order that could beachieved by this decoding scheme is controlled by the secondterm given that the first one achieves full diversity. Therefore,to guarantee an overall diversity order of at least d, eachmember of the second term should decrease at the order of�

2d. This goes back to find for each block i the minimumthreshold ri ,as function of the SNR (or small �2) andblock sizes (p1, p2, . . . , pk�1), such that:

�(Pi

j=1 pj2 ,

r2i2�2 )

�(Pi

j=1 pj2 )

��2d , i 2 1, . . . , k � 1 (21)

for some positive constant � that controls the SNR gain.This inequality on ri is solved numerically in simulationswith a margin of error as small as possible.

VI. SIMULATION RESULTS

The proposed decoder has been validated by numericalsimulations considering spatial multiplexing for nt = nr = 8using a 4-QAM modulation. We provide results for a divisionin 4 blocks but using different dimensions p1, p2, p3, p4(n = 16 in real valued system). In fig.3 we show that acomplexity reduction of almost 50% compared to SphereDecoeder is achieved. We also show, by plotting the SymbolError Rate SER in Fig.2, that our decoder is able to achieveall desired diversity orders. The legend in both plots is writtenas follows: target diversity order d / SNR gain � / block sizesp1, p2, p3, p4. The average decoding complexity(computed asthe overall number of multiplications) and SER are averagedover 107 channel realizations.We compare our results to Sphere Decoder by varying theminimum target diversity order d and the SNR gain factor�.In the first scenario (black lines), we set the same targetdiversity d = 8 (i.e full diversity) and the same snr gain� = 10. We observe that this diversity order is indeedachieved with a noticeable complexity reduction comparedto the SD. We also notice that error performance as wellas complexity are sensitive to a change of block sizes ,forexample(p1, p2, p3, p4) = (4, 4, 4, 4) or (7, 4, 3, 2). A slightcomplexity reduction and an additive SNR gain are observed

between black curves in fig.3 and fig.2 respectively.In the second and third scenario, we set the minimum targetdiversity d = 6 (blue curves) and d = 2(cyan curves). Forthe same block size division (p1, p2, p3, p4) = (4, 4, 4, 4),we set 2 variants for SNR gain factor �. We see that for bothtarget diversity orders, in Fig.2 SER curves (blue and cyanlines) have the desired slope (diversity) but with differentSNR gains. Finally, we can observe that at low SNR regime,we have a higher complexity than Sphere Decoder. Thisbehavior is due to the fact that the quantity ��2d in (21)is higher than 1, which is not suitable for a probabilitycomputation. Hence, numerical solver of (21) gives a toosmall values of radii ri (due to properties of normalizedupper gamma function), thus no lattice point could be found,causing the algorithm to restart and increase complexity.

Fig. 2. Symbol Error Rate for nt = nr = 8 using 4-QAM constellationand spatial multiplexing

Fig. 3. Total computational complexity for nt = nr = 4 using 4-QAMconstellation and spatial multiplexing

VII. CONCLUSIONThis work was dedicated to semi-exhaustive recursive

block decoding implementable in linear communicationsystems including MIMO systems. A complex