Diversity and Degrees of Freedom in Wireless Communications

Transcription

1 1 Diversity and Degrees of Freedom in Wireless Communications Mahesh Godavarti Altra Broadband Inc., Alfred O. Hero-III Dept. of EECS, University of Michigan hero Outline 1. MIMO systems and motivating examples 2. Link degrees-of-freedom (DOF) vs. diversity (DIV) 3. Error exponent analysis

2 2 Acknowledgments This work was supported in part by a Graduate Fellowship to the first author from the Department of Electrical Engineering and Computer Science at the University of Michigan and by the Department of Defense Research & Engineering (DDR&E) Multidisciplinary University Research Initiative (MURI) on Low Energy Electronics Design for Mobile Platforms managed by the Army Research Office (ARO) under ARO grant DAAH

3 3 Rank One Specular Component Pre-Processing 1 Post-Processing 1 Pre-Processing 2 Post-Processing 2 Information to be Transmitted Received Information Pre-Processing M Post-Processing N Independent Paths (Possibly over time) M-Transmit Antennas N-Receive Antennas Figure 1: Diagram of a multiple antenna communication system

4 4 Source T x M matrix a(t) A/D demux Space Time Encoder S(t) RF Mod RF Mod Transmitter RF Demod RF Demod Codeword Estimator ^ S(t) Space Time Decoder D/A mux ^a(t) RF Demod Receiver T x N matrix Figure 2: Space-time transmitter/receiver.

5 5 Diversity and Degrees-of-Freedom Folk definitions: Degrees-of-Freedom (DOF): Aka multiplexing gain, transmit diversity DOF is number of independent channels exploited by the transmitter Diversity(DIV): Aka processing gain, receive diversity DIV is number of independent channels exploited by receiver

6 x l Ô ρhs l w l C T log ρµ large ρµ logp e T log ρµ large ρµ 6 Example: SISO system Received signal l 1 T Channel capacity for energy-constrained signaling and informed receiver (IR) is: µ DOF advantage is T Probability of decoding error for BPSK with ML decoding: µ DIV advantage is T

7 x l Ô ρhs w l C T log ρµ large ρµ logp e NT log ρµ large ρµ 7 Example: SIMO system Received signal l 1 T Channel capacity for energy-constrained signaling and informed receiver (IR) is: µ DOF advantage is NT Probability of decoding error for BPSK with ML decoding: µ DIV advantage is NT

8 8 Towards Formal Definition of DOF/DIV 1. Fix a rate-r coder and min-p e decoder: denoted C 2. Define Definition 1 DIV M N T C µ lim ρ logp e ρ;m N T C µ loginf C P e ρ;1 1 1 C µ (1) and DOF M N T C µ lim ρ R ρ;m N T C µ sup C R ρ;1 1 1 C µ (2) Note: These definitions depend on C

9 9 C-independent definitions of DOF/DIV of a channel: Definition 2 DIV M N T C µ lim ρ inf C logp e ρ;m N T C µ inf C logp e ρ;1 1 1 C µ (3) and DOF M N T C µ lim ρ sup C R ρ;m N T C µ sup C R ρ;1 1 1 C µ lim C ρ;m N (4) Tµ C ρ;1 1 1µ ρ Note: for IR Rayleigh fading channels, consistent with Heath and Paulraj s DOF (2001): asymptotic rate of minimum Euclidean distance in ρ Zheng and Tse s DOF (2002): asymptotic rate of C outage in logρ Tarokh et. al. s DIV (1998): asymptotic exponent of min-p e in 1 ρ

10 10 Application: Rayleigh fading MIMO system ρ X SH W M Ö X: T N; S: T M; andh: M N T : time interval over which H remains constant H i.i.d. Gaussian known to receiver (IR) W i.i.d. AWGN Constraint on C: tr E SS TM.

11 C TE logdet I N ρh Hµ TE logdet I M ρhh µ which is attained by S N 0 ci T Æ IM µ DOF M N T µ T min M Nµ DIV M N T C µ N min M T µ DOF: Channel capacity is Hence: 2: DIV: Let transmitter adopt the space-time QPSK signal adopted by Tarokh etal (1999) and consider MAP receiver Find:

12 P e ρ M N Tµ exp LE U P Sµ ρµµ E 0 γ P G Sµ ρµ Ð ßÞ 12 DOF-DIV via error exponent approach Gallager s Random Coding Error Exponents Establishes bound on probability of error P e for any communication system having information rate R and transmitted P Sµ distribution Error exponent: E U P Sµ ρµ 1 γ max γ¾ 0 1 max P¾P γr ln X¾X S¾S p X Sµµ1 1 γµ dp Sµ dx R o ρ M N Tµ max P¾P ln X¾X S¾S p X Sµµ1 2 dp Sµ 2 dx Special case γ 1 involves cutoff rate R o : P e exp R o Rµ

13 13 DIV and Cutoff Rate R o Hµ max P S H ln S 1 S 2 ¾Cl T MdP S H S 1µdP S H S 2 µ e ND S 1 S 2 µ 1. T/R Informed cutoff rate: H known to both T/R D S 1 S ρ 2 µ H 4 tr S 1 S S 2 µ 1 S 2 DIV Nmin T Mµ µ µh 2. R informed cutoff rate: H known to R only D S 1 S 2 ln I ρ µ T 4 S 1 S 2 µ S 1 S 2 µ DIV Nmin T Mµ µ 3. Uninformed cutoff rate: H unknown to either T/R D S 1 S 2 µ ln I ρ T 2 S 1S 1 S 2S 2 µ I T ρs 1 S 1 I T ρs 2 S Ö 2 µ DIV N min T 2Mµ min T Mµµ

14 logp e ρ M N Tµ ßÞ Ð 14 Error Exponent: DOF-DIV Tradeoff Analysis For IR and Gaussian P Sµ satisfying E tr S S MT Hence: E 0 γ P G Sµ ρµ loge det I 1 M 1 γ ρ HH γt M DIV M N T µ E 0 γ P Sµ ρµ Tµ γr ρ;m N DOF M N T Ð µ ßÞ where Tµ R ρ M N E 0 γ P Sµ ρµ γ

15 15 5 M = 3, N = 1, T = DOF DIV Figure 3: DOF(C γ )asafunctionofdiv(c γ ). The y-intercept is T min M Nµ and the x-intercept is N min M T µ

16 16 Conclusions DOF and DIV are convenient encapsulations of link performance General system-dependent and -independent definitions for DOF and DIV given Our definitions specialize to previous definitions for Rayleigh channels Random coding exponents can be used to study design tradeoffs between DOF and DIV