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Polynomial Matrix Decompositions UPTEC F10 059 Examensarbete 20 p November 2010 Polynomial Matrix Decompositions Evaluation of Algorithms with an Application to Wideband MIMO Communications Rasmus Brandt Abstract Polynomial Matrix Decompositions: Evaluation of Algorithms with an Application to Wideband MIMO Communications Rasmus Brandt Teknisk- naturvetenskaplig fakultet UTH-enheten The interest in wireless communications among consumers has exploded since the introduction of the ''3G'' cell phone standards. One reason for their success is the Besöksadress: increasingly higher data rates achievable through the networks. A further increase in Ångströmlaboratoriet Lägerhyddsvägen 1 data rates is possible through the use of multiple antennas at either or both sides of Hus 4, Plan 0 the wireless links. Postadress: Precoding and receive filtering using matrices obtained from a singular value Box 536 751 21 Uppsala decomposition (SVD) of the channel matrix is a transmission strategy for achieving the channel capacity of a deterministic narrowband multiple-input multiple-output Telefon: (MIMO) communications channel. When signalling over wideband channels using 018 – 471 30 03 orthogonal frequency-division multiplexing (OFDM), an SVD must be performed for Telefax: every sub-carrier. As the number of sub-carriers of this traditional approach grow 018 – 471 30 00 large, so does the computational load. It is therefore interesting to study alternate means for obtaining the decomposition. Hemsida: http://www.teknat.uu.se/student A wideband MIMO channel can be modeled as a matrix filter with a finite impulse response, represented by a polynomial matrix. This thesis is concerned with investigating algorithms which decompose the polynomial channel matrix directly. The resulting decomposition factors can then be used to obtain the sub-carrier based precoding and receive filtering matrices. Existing approximative polynomial matrix QR and singular value decomposition algorithms were modified, and studied in terms of decomposition quality and computational complexity. The decomposition algorithms were shown to give decompositions of good quality, but if the goal is to obtain precoding and receive filtering matrices, the computational load is prohibitive for channels with long impulse responses. Two algorithms for performing exact rational decompositions (QRD/SVD) of polynomial matrices were proposed and analyzed. Although they for simple cases resulted in excellent decompositions, issues with numerical stability of a spectral factorization step renders the algorithms in their current form purposeless. For a MIMO channel with exponentially decaying power-delay profile, the sum rates achieved by employing the filters given from the approximative polynomial SVD algorithm were compared to the channel capacity. It was shown that if the symbol streams were decoded independently, as done in the traditional approach, the sum rates were sensitive to errors in the decomposition. A receiver with a spatially joint detector achieved sum rates close to the channel capacity, but with such a receiver the low complexity detector set-up of the traditional approach is lost. Summarizing, this thesis has shown that a wideband MIMO channel can be diagonalized in space and frequency using OFDM in conjunction with an approximative polynomial SVD algorithm. In order to reach sum rates close to the capacity of a simple channel, the computational load becomes restraining compared to the traditional approach, for channels with long impulse responses. Handledare: Mats Bengtsson Ämnesgranskare: Mikael Sternad Examinator: Tomas Nyberg ISSN: 1401-5757, UPTEC F10 059 Popul¨arvetenskaplig sammanfattning p˚asvenska Tr˚adl¨os kommunikation ¨ar ett omr˚adevars popul¨aritet har ¨okat de senaste ˚aren. Ett sk¨al till "3G-internets" framg˚ang ¨ar de h¨oga datatakter som ¨ar m¨ojliga. Datatakten i en tr˚adl¨os l¨ank beror p˚asignalens bandbredd samt s¨andeffekten, och genom att ¨oka endera erh˚allsh¨ogre datatakter. B˚adebandbredd och s¨andeffekt ¨ar dock dyra resurser, eftersom deras anv¨andande ofta ¨ar reglerat av nationella och internationella myndigheter. Ett annat s¨att att ¨oka datatakten i en tr˚adl¨os l¨ank kan vara att l¨agga till fler anten- ner p˚as¨andar- och mottagarsidan, ett s.k. MIMO-system. Ett s˚adant system kan ses som en upps¨attning av enkelantennl¨ankar med inb¨ordes p˚averkan och kan beskrivas av en matris. Da- tatakten f¨or flerantennl¨anken kan maximeras genom att skicka flera parallella datastr¨ommar ¨over MIMO-kanalen. Eftersom de olika uts¨anda signalerna samsas om radiokanalen kommer de att blandas. Varje mottagarantenn kommer d¨arf¨or att ta emot en kombination av de uts¨anda signalerna fr˚ande olika s¨andarantennerna. F¨or att undvika att signalerna blandas m˚astede kodas. Det visar sig att genom att koda de s¨anda signalerna med en speciell matris, samt avkoda de mottagna signalerna med en annan matris, s˚atransformeras kanalen till en upps¨attning av parallella virtuella kanaler. P˚a dessa virtuella kanaler kan sedan oberoende datastr¨ommar skickas. Kodningsmatriserna ges av en s.k. singul¨arv¨ardesuppdelning av den ursprungliga kanalmatrisen. F¨or ett enkelantennsystem med h¨og bandbredd kommer radiokanalen att p˚averka de olika frekvenskomponenterna i signalen olika. Om inte systemet tar h¨ansyn till den effekten kommer dess prestanda att p˚averkas. Ett s¨att att undvika denna frekvensselektivitet ¨ar att signalera ¨over kanalen med s.k. OFDM. Genom OFDM-systemet delas den ursprungliga signalen upp i flera signaler med l˚agbandbredd. Genom att skicka dessa smalbandiga signaler p˚aolika delar av frekvensbandet s˚ap˚averkar de inte varandra. Den frekvensselektiva kanalen har s˚aledes delats upp i ett antal icke frekvensselektiva parallella subkanaler. Genom att skicka en bredbandig signal ¨over ett OFDM-baserat MIMO-system kan ¨annu h¨ogre datatakter ˚astadkommas. Dock m˚astekodningsmatriserna ber¨aknas f¨or varje parallell subkanal i frekvensbandet, vilket inneb¨ar att m˚angaber¨akningsoperationer kr¨avs. Det h¨ar examensarbetet har unders¨okt en ny upps¨attning algoritmer f¨or att erh˚allaapproximatio- ner av de kodningsmatriser som beh¨ovs. Kvaliteten p˚ade approximativa kodningsmatriserna j¨amf¨ordes med de exakta matriserna och antalet n¨odv¨andiga ber¨akningsoperationer m¨attes. Det visade sig att de nya algoritmerna kan producera kodningsmatriser av god kvalitet, men med fler n¨odv¨andiga ber¨akningsoperationer ¨an det traditionella s¨attet att erh˚allakodnings- matriserna. Kodningsmatriserna fr˚ande nya algoritmerna simulerades ocks˚ai ett kommunikationssy- stem. Med de nya matriserna kan man uppn˚adatatakter som ¨ar n¨ara den teoretiska maxka- paciteten f¨or en enkel radiokanal om en avancerad dekoder anv¨ands p˚amottagarsidan. Om ist¨allet en upps¨attning av enkla dekodrar anv¨ands, som i det traditionella systemet, blir systemprestanda lidande. Sammanfattningsvis s˚ahar det h¨ar examensarbetet visat att kodningsmatriserna erh˚allna fr˚ande nya algoritmerna kan anv¨andas i ett bredbandigt MIMO-system f¨or att maximera datatakten. Dock s˚akr¨aver de fler ber¨akningsoperationer, och en mer avancerad dekoder, ¨an det traditionella systemet. De nya algoritmerna ¨ar s˚aledesinte konkurrenskraftiga j¨amf¨ort med det traditionella systemet. Acknowledgements This diploma work was performed at the Signal Processing Laboratory at the School of Electrical Engineering at Kungliga Tekniska H¨ogskolan in Stockholm, and will lead to a degree of Master of Science in Engineering Physics from Uppsala University. First and foremost, I would like to thank my supervisor Mats Bengtsson for proposing the thesis topic and taking me on as a MSc thesis worker. His advice and guidance has helped me considerably during the course of this work. My ¨amnesgranskare Mikael Sternad at the Division for Signals and Systems at Uppsala university also deserves my gratitude; his comments have been very valuable to the final version of this thesis. My family has always been supporting my endeavours, and for that I am endlessly grateful. Finally, thank you Melissa for being so lovely and cheerful, and for moving to Sweden to be with me. Contents 1 Introduction 1 1.1 Wireless Communications . .1 1.2 Multiple Antennas and Wideband Channels . .3 1.3 Problem Formulation and Contributions . .3 1.4 Thesis Outline . .4 2 Preliminaries 5 2.1 Complex Polynomials . .5 2.1.1 Addition and Subtraction . .6 2.1.2 Multiplication . .6 2.2 Polynomial Matrices . .7 2.2.1 Givens Rotations . .7 2.2.2 Decompositions . .9 2.2.3 Coefficient Truncation . .9 2.3 Computational Complexity . 10 3 MIMO Channels and Multipath Propagation 11 3.1 Propagation and Modeling . 11 3.1.1 Propagation . 11 3.1.2 Channel Modeling . 13 3.1.3 MIMO Channels . 14 3.2 Channel Capacity and Achievable Rate . 15 3.3 Equalization Techniques . 16 3.4 Summary . 17 4 Polynomial Decomposition Algorithms: Coefficient Nulling 18 4.1 Performance Criteria . 18 4.2 PQRD-BC: Polynomial QR Decomposition . 20 4.2.1 Convergence and Complexity . 21 4.2.2 Discussion . 22 4.3 MPQRD-BC: Modified PQRD-BC . 22 4.3.1 Convergence and Complexity . 23 4.3.2 Simulations . 25 4.3.3 Discussion . 29 4.4 PSVD by PQRD-BC: Polynomial Singular Value Decomposition . 29 4.4.1 Convergence and Complexity . 31 i 4.4.2 Discussion . 32 4.5 MPSVD by MPQRD-BC: Modified PSVD . 32 4.5.1 Convergence and Complexity . 33 4.5.2 Simulations . 34 4.5.3 Discussion . 39 4.6 Sampled PSVD vs. SVD in DFT Domain . 39 4.6.1 Frequency Domain Comparison . 39 4.6.2 Computational Load Comparison, Set-Up Phase . 39 4.6.3 Computational Load, Online Phase . 41 4.6.4 Discussion . 42 4.7 Summary . 43 5 Rational Decomposition Algorithms: Polynomial Nulling 44 5.1 Rational Givens Rotation . 44 5.2 PQRD-R: Rational QR Decomposition . 45 5.2.1 Simulations . 46 5.2.2 Discussion . 47 5.3 PSVD-R by PQRD-R: Rational Singular Value Decomposition . 50 5.3.1 Simulations . 50 5.3.2 Discussion . 51 5.4 Summary . 51 6 Polynomial SVD for Wideband Spatial Multiplexing 55 6.1 Generic System Model .
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