Low-Complexity Decoding Schemes for MIMO systems
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1 Low-Complexity Decoding Schemes for MIMO systems Ghaya Rekaya-Ben Othman June 2017
2 June 2017 G. Rekaya-Ben Othman 1/41 Outline 1 2 Patenting : my vision as an inventor 2 2 Patent Factory : Feedback 3 2 Sequential decoding 2 4 Recursive Block decoding
3 Patenting : My Vision as an inventor
4 June 2017 G. Rekaya-Ben Othman 2/41 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.
5 June 2017 G. Rekaya-Ben Othman 3/41 Who talks about innovation and patents? Lawyer Financial Ingenier Manger Economist Politicien But, what about Inventor??
6 June 2017 G. Rekaya-Ben Othman 4/41 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.
7 June 2017 G. Rekaya-Ben Othman 5/41 Inventor Vision! 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.
8 June 2017 G. Rekaya-Ben Othman 6/41 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.
9 June 2017 G. Rekaya-Ben Othman 7/41 Life stages of an inventor Stage 1 First Patents How to detect the idea! How to make a patent? Stage 2 5 to 10 Patents Become an Inventor Identify missing results Enriching Patent Groups Stage 3 >10 Patents Become an Expert Structure patents by classes or groups Identify technologies Identify compagnies Steps to make patents Methodology for creating a patent family On the way to Valorization
10 Patent Factory : MIMO Decoders
11 Examples of applications of MIMO technologies MIMO channel Multiple antennas systems Optical fiber communications Sensor nodes Sink node Wireless sensor network Remote Control Center Multi-user communications June 2017 G. Rekaya-Ben Othman 8/41
12 June 2017 G. Rekaya-Ben Othman 9/41 MIMO Vision Decoding PF MIMO Sequential decoder Block decoder Potential Standars Proposed solutions are implementable in wireless standards products such as: WiFi (IEEE n) : 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
13 June 2017 G. Rekaya-Ben Othman 10/41 MIMO Vision Decoding PF MIMO Sequential decoder Block decoder MIMO transmission chain CHALLENGE: meet the target QoS specifications (fixed complexity/ performance) for any application implementable in any MIMO configuration.
14 June 2017 G. Rekaya-Ben Othman 11/41 Patent Factory results Inputs 5 Patents Outputs 14 patents, 1 in progress 8 Publications, 2 journals in preparation Start discussions with some companies for valorisation
15 June 2017 G. Rekaya-Ben Othman 12/41 New decoding schemes: 5 Families Family 1: Preprocessing Family 5: General Tools Family 2: Sequential decoding Family 4: Block Decoding Family 3: Fixed Complexity
16 June 2017 G. Rekaya-Ben Othman 13/41 MIMO Vision Decoding PF MIMO Sequential decoder Block decoder Family 1: Preprocessing Problem Worst channel realization can induce very high decoding complexity. Proposed Solutions New 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
17 June 2017 G. Rekaya-Ben Othman 14/41 Problem Family 2: Sequential decoding The complexity of the tree search phase increases function of the number of transmit and receive antennas and the constellation size. Proposed Solutions New 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) Patents Outcomes 6 filed patents 1 idea in progress
18 June 2017 G. Rekaya-Ben Othman 15/41 Family 3: Fixed Complexity Problem For real time applications, a fixed decoding complexity is required. Proposed Solutions New methods allowing to have fixed decoding time with guaranteed performance. Stack reordering for early termination Anticipated termination Patents Outcomes 2 filed patents
19 Family 4: Block Decoding 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 Patents Outcomes 5 filed patents June 2017 G. Rekaya-Ben Othman 16/41
20 June 2017 G. Rekaya-Ben Othman 17/41 Family 5: General ideas Problem Reduce the overall decoding complexity, by considering all the decoding chain. Proposed Solutions 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. Adaptive decoding MAP decoding using augmented lattice Design criterion for low-complexity decodable Space-Time Codes. Patents Outcomes 3 Filed patents
21 June 2017 G. Rekaya-Ben Othman 18/41 Decoding Chain: Families association Example 1 of a decoding chain: Block Division Parametrized SB-Stack for each Block Example 2 of a decoding chain: Anticipated Termination or Early Termination Algebraic Reduction SB-Stack or Zig-Zag Stack
22 June 2017 G. Rekaya-Ben Othman 19/41 Maturity and Validation Theoretical Validation Some results are validated by a theoretical study, through the derivation of the error probability. Formulas of some parameters and criterion are derived. Numerical validation 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.
23 June 2017 G. Rekaya-Ben Othman 20/41 Portfolio and corresponding publications 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/ and US 2011/ A1. 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/ and US 2011/ A1 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/ R. Ouertani, G. Rekaya-Ben Othman and J-C. Belfiore, "An Adaptive MIMO decoder", IEEE VTC, Barcelona, Spain, April 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 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 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 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.
24 Portfolio and correspondinf publications 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/ 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 , September L. Luzzi, G. Rekaya Ben Othman and J-C. Belfiore, "Augmented Lattice Reduction for Low Complexity MIMO Decoding", IEEE PIMRC, Istanbul, Turqey, September A.Mejri et G. Rekaya Ben Othman, "Méthode de Décodage MAP par Réseau de Points Augmenté", French Application filed October 2013 No. FR13/ A. Mejri and G. Rekaya Ben Othman, " MAP Decoder for Physical-Layer Network Coding Using Lattice Sphere Decoding", IEEE 21 st International Conference on Telecommunications ICT, Lisbon, Portugal, May A. Mejri and G. Rekaya Ben Othman, "Efficient Decoding Algorithms for the Compute-and-Forward Strategy", IEEE Transactions on Communications, June 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, EP A.Mejri and G.Rekaya-BenOthman, «Reduced Complexity Stack Decoder for MIMO Systems», IEEE Vehicular Technology Conference, VTC, Glasgow, UK, May June 2017 G. Rekaya-Ben Othman 21/41
25 June 2017 G. Rekaya-Ben Othman 22/41 Portfolio and corresponding publications Patent application 7 G. Rekaya-Ben Othman and A. Mejri, "Tree Search-Based Decoding", European patent application, February 2015, EP G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, "Space- Time Coding for Communication Systems", European patent Application April 2015, EP Corresponding publication A. Mejri and G. Rekaya-Ben Othman, "Reduced-Complexity Lattice Spherical Decoding", IEEE Twelfth International Symposium on Wireless Communication Systems, ISWCS, Brussels, Belgium, August 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 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 G. Rekaya Ben Othman and Asma Mejri, "Sequential Decoding With Stack Reordering", European patent application, June 2015, EP 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. - 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.
26 June 2017 G. Rekaya-Ben Othman 23/41 Portfolio and corresponding publications Patent application Corresponding publication 11 G. Rekaya Ben Othman and Asma Mejri, «Parameterized Sequential Decoding», European patent application, November 2015, EP G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «Semi- Exhaustive Recursive Block Decoding Method and Device», November 2015, EP M.A. Ksiba and G. Rekaya-Ben Othman, «Semi-Exhaustive Reduced-complexity Recursive Block Decoding for MIMO Systems», IEEE International Conference on Communication, October G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «RECURSIVE SUB-BLOCK DECODING», 2015, EP M.A. Ksiba and G. Rekaya-Ben Othman, «Semi-Exhaustive Recursive Multi-Block Decoding for MIMO Systems», Submitted to IEEE PIMRC G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «REORDERED SUB-BLOCK DECODING», 2015, EP G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «WEIGHTED SEQUENTIAL DECODING», 2015, EP
27 June 2017 G. Rekaya-Ben Othman 24/41 Portfolio and corresponding publications Patent application 16 G. Rekaya-Ben Othman and M-A. Khsiba, «METHODS AND DEVICES FOR SEQUENTIAL SPHERE DECODING», 2016, EP Corresponding publication M.A. Ksiba and G. Rekaya-Ben Othman, «Dichotomic Sphere Decoder», WCNC G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «METHODS AND DEVICES FOR DECODING DATA SIGNALS», 2016, EP G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «METHODS AND DEVICES FOR SUB-BLOCK DECODING DATA SIGNALS»,2016, EP G. Rekaya-Ben Othman, A. Mejri and M-A. Khsiba, «METHODS AND DEVICES FOR SUB-BLOCK DECODING DATA SIGNALS»,2016, EP
28 Sequential decoding
29 June 2017 G. Rekaya-Ben Othman 25/41 Classification of MIMO decoders Decoders Sub-optimal optimal Linear Non-linear Sequential decoding Exhaustive Search ZF MMSE ZF-DFE MMSE-DFE Depth First Search Best First Search SE SD Stack Fano
30 June 2017 G. Rekaya-Ben Othman 26/41 MIMO system Complex-valued received matrix: Real-valued vectorized system: Y nr T = H nr n t X nt T + Z nr T. y 2nr T = H eq2nr T 2Ÿ s 2Ÿ + z 2nr T where: y = vec(y), z = nd = # (s ) (s ) ( vec(z) and s 2Ÿ = # Ÿ(s 1 ) (s 1 ) Ÿ(s Ÿ ) (s Ÿ ) $ t # QR decomposition of Heq: Heq = QR, where R is an upper triangular, Q is an orthogonal matrix. ML metric is : m( s) =Î y H eq s Î 2 =Î Q t y R s Î 2
31 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
32 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
33 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
34 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
35 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
36 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
37 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
38 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
39 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
40 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
41 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
42 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s SD searches the ML solution under a sphere of a given radius centered on the received point.
43 June 2017 G. Rekaya-Ben Othman 27/41 Sphere Decoder (SD) Depth-first search strategy s 2 s 1 s root s s ŝ =[1, 3] t SD searches the ML solution under a sphere of a given radius centered on the received point.
44 June 2017 G. Rekaya-Ben Othman 28/41 Spherical-Bound Stack Decoder (SB-Stack) Best-first search strategy s root s 2 Empty Stack s 1 SB-Stack combines the search strategy of the Stack decoder and the search region of the SD 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.
45 June 2017 G. Rekaya-Ben Othman 28/41 Spherical-Bound Stack Decoder (SB-Stack) Best-first search strategy s root Empty Stack s 2 s 1 SB-Stack combines the search strategy of the Stack decoder and the search region of the SD 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.
46 June 2017 G. Rekaya-Ben Othman 28/41 Spherical-Bound Stack Decoder (SB-Stack) Best-first search strategy Empty Stack decreasing metric s root SB-Stack combines the search strategy of the Stack decoder and the search region of the SD s 2 s 1 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.
47 June 2017 G. Rekaya-Ben Othman 28/41 Spherical-Bound Stack Decoder (SB-Stack) Best-first search strategy s root Empty Stack decreasing metric SB-Stack combines the search strategy of the Stack decoder and the search region of the SD s 2 s 1 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.
48 June 2017 G. Rekaya-Ben Othman 28/41 Spherical-Bound Stack Decoder (SB-Stack) Best-first search strategy Empty Stack leaf node s root s 2 s 1 ŝ =[1, SB-Stack combines the search strategy of the Stack decoder and the search region of the SD 3] t 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.
49 June 2017 G. Rekaya-Ben Othman 29/41 Spherical-Bound Stack Decoder (SB-Stack) 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
50 Block Decoding
51 June 2017 G. Rekaya-Ben Othman 30/40 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?
52 June 2017 G. Rekaya-Ben Othman 31/41 Recursive Block Decoding : 2 blocks 3: Detect s (2) y (2) R 2 B s (2) n-p z (2) = * + 2. Feedback y (1) 0 R 1 s(1) p z (1) 1: Detect s (1) ŝ = argmin k y 0 Rsk 2 (s (1),s (2) )2A p A n p = argmin (s (1),s (2) )2A p A n p k y (2) R 2 s (2) Bs (1) k 2 + k y (1) R 1 s (1) k 2
53 Recursive Block Decoding : 2 blocks 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 June 2017 G. Rekaya-Ben Othman 32/41
54 Recursive Block Decoding : 2 blocks Empirical distribution of the Occurrence of 0 Compx semisbstack1/4 +ML (SE) 3/ candidates SemiExhaustive in ML the snr=6 solution SemiExhaustive ML snr=3 SemiExhaustive ML snr= Compx semisbstack1/4 +ML (SE) 3/ Sys13CP4ClassiqueSEX0Y0 ZFDFE S ys22list50minrepetition20radius2nse Sys13CP4ClassiqueSEX0Y0 6 x Se 104 ZFDFE ZFDFE S ys22list50minrepetition20radius2nsegm S ys22list100minrepetition20radius2ns ZFDFE ZFDFE S ys22list100minrepetition20radius2nsegm S ys22list100minrepetition50radius2ns SemiExhaustive ML snr=6 ZFDFE ZFDFE S ys22list100minrepetition50radius2nsegm S ys22list50minrepetition1radius2nseg ZFDFE S ys22list50minrepetition1radius2nsegma 6 x 5 Semi 104 ZFDFE S ys22list100minrepetition1radius2nse ZFDFE ZFDFE S ys22list100minrepetition1radius2nsegm S ys31list50minrepetition20radius2nse ZFDFE S ys31list50minrepetition20radius2nsegm S ys31list100minrepetition20radius2ns ZFDFE S ys31list100minrepetition20radius2nsegm ys31list100minrepetition50radius2ns S S ys31list100minrepetition50radius2nsegm 10 6 ZFDFE ys31list50minrepetition1radius2nseg S ZFDFE S ys31list50minrepetition1radius2nsegma S 8000 ZFDFE ML S ys31list50minrepetition1radius2nsegma S ys22list50minrepetition20radius2nsegma ML ML S ys22list50minrepetition20radius2nsegma S ys22list100minrepetition20radius2nsegm ML ML S ys22list100minrepetition20radius2nsegma S ys22list100minrepetition50radius2nsegm ML ML S ys22list100minrepetition50radius2nsegma S ys22list50minrepetition1radius2nsegma ML ML S ys22list50minrepetition1radius2nsegma S ys22list100minrepetition1radius2nsegma 0 ML ML S ys22list100minrepetition1radius2nsegma S ys31list50minrepetition20radius2nsegma ML S ys31list50minrepetition20radius2nsegma ML S ys31list100minrepetition20radius2nsegm ML ML S ys31list100minrepetition20radius2nsegma S ys31list100minrepetition50radius2nsegm ML ys31list100minrepetition50radius2nsegma ML S S ys31list50minrepetition1radius2nsegma ML ys31list50minrepetition1radius2nsegma ML S 0 5 S ys31list100minrepetition1radius2nsegma ML ys31list100minrepetition1radius2nsegma S x 104 SemiExhaustive ML snr=9 3.5 x 105 SemiExhaustive ML snr=12 4 x 105 Sem x SemiExhaustive ML snr= x SemiE Fig. 5. Complexity Comparison chooses the block having the largest determinanttotobe b June 2017 G. Rekaya-Ben Othman x /41 SemiExhaustive ML snr=
55 Semi-exhaustive Block Decoding : 2 blocks 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. June 2017 G. Rekaya-Ben Othman 34/41
56 June 2017 G. Rekaya-Ben Othman 35/41 g g g Semi-exhaustive Block g Decoding g : 2 g blocks 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: P ef apple ( p 2, r 2 2 ) 2 ( p 2 ) + I + A n p 1+ d2 min 4 2 n 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: ( p 2, r2 2 ) 2 ( p 2 ) apple 2apple 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.
57 June 2017 G. Rekaya-Ben Othman 36/41 Semi-exhaustive Block Decoding : 2 blocks 4x4 MIMO system using 4-QAM constellation and spatial multiplexing
58 June 2017 G. Rekaya-Ben Othman 37/41 Semi-exhaustive Block Decoding : n blocks ŝ = argmin k y Rsk 2 s2a n = argmin (s (1),...,s (k) ) 2 Q k j=1 Ap j kx j=1 k y (j) R j s (j) B j 1 s (j 1,1) k 2 (6
59 June 2017 G. Rekaya-Ben Othman 38/41 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.
60 June 2017 G. Rekaya-Ben Othman 39/41 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: P ef apple 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: I + A p k n + 1+ d2 min 4 2 This goes back to find some for each positive block constant i the minimum that controls threshold ther i SN, as function of the SNR and the block size p i. X kx 1 i=1 ( P P Semi-exhaustive Block Decoding : n blocks P i j=1 p j 2, r2 i 2 ) P 2 i j=1 ( p j 2 ) quation (20) shows that the diversity order that co P i j=1 ( p j ( 2, r2 i 2 ) P 2 i j=1 p j 2 ) apple 2d, i 2 1,...,k 1
61 June 2017 G. Rekaya-Ben Othman 40/41 Family 4: Block Decoding 8x8 MIMO system using 4-QAM constellation and spatial multiplexing
62 June 2017 G. Rekaya-Ben Othman 41/41 Block division Given a matrix R, the technical challenge we are answering is how to choose the optimal number of blocks and their sizes to get the best Performance/complexity tradeoff. We have defined 3 metrics, function of: - Sparsity: hard, soft and weighted sparsity defined function of number and position of zero (and almost zero) entries of matrix R. - Orthogonality: for block interference regions. - Zero positions: to allow block decoding parallelization.
63 Questions?
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