Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems

MIMO Each node has multiple antennas Capable of transmitting (receiving) multiple streams concurrently Exploit antenna diversity to increase the capacity h 11 h12 h 13 h 21 h 31 h 32 h 33 h 22 h 23 " $ $ H N M = $ $ # $ h 11 h 12 h 13 h 21 h 22 h 23 h 31 h2 h 33 % ' ' ' ' &'

Channel Model (2x2) h 11 h 12 h 21 h22 y 1 x 2 y 2 y 1 = h 11 +h 21 x 2 +n 1 y 2 = h 12 +h 22 x 2 +n 2 y = Hx +n Can be extended to N x M systems

Antenna Space M-antenna node receives in M-dimensional space 2 x 2! # " y 1 y 2 $! & = h $! 11 # % " h 12 & x + h 21 1 # % " h 22 y = h 1 + h 2 x 2 + n $! & x + n 1 2 # % " n 2 $ & % h 2 = (h 21,h 22 ) y = (y 1, y 2 ) antenna 2 antenna 2 x 2 h 1 = (h 11,h 12 ) antenna 1 antenna 3 antenna 1

Zero-Forcing Decoding (algebra) Orthogonal vectors + )! # " y 1 y 2 $! & = h 11 # % " h 12 $! & + h 21 # % " h 22 $! & x 2 + n 1 # % " n 2 $ & % * h 22 * - h 21 y 1 h 22 y 2 h 21 = (h 11 h 22 h 12 h 21 ) = y 1 h 22 y 2 h 21 h 11 h 22 h 12 h 21 Given, solve x 2 by successive interference cancella?on (SIC) To guarantee the full rank of H, antenna spacing at the transmiier and receiver must exceed half of the wavelength

Zero-Forcing Decoding (antenna space) antenna 2 h 2 = (h 21,h 22 ) x 2 y = (y 1, y 2 ) h 1 = (h 11,h 12 ) antenna 1 x 1 x 1 < To decode, decode vector y on the direc?on orthogonal to x 2 To improve the SNR, use SIC decoding, which re- encodes the first detected signal, subtracts it from y, and decodes the second signal

Channel Estimation Estimate N x M matrix H h 11 h 12 y 1 y 1 = h 11 +h 21 x 2 +n 1 h 21 h22 y 2 = h 12 +h 22 x 2 +n 2 x 2 y 2 Two equa?ons, but four unknowns Antenna 1 at Tx Access code 1 Stream 1 Antenna 2 at Tx Access code 2 Stream 2 Es?mate h 11, h 12 Es?mate h 21, h 22

Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems

Multiplex Gain MIMO Gains Exploit antennas to deliver multiple streams concurrently Diversity Gain Exploit antenna diversity to increase the SNR of a single stream Receive diversity and transmit diversity

Degree of Freedom For N x M MIMO channel Degree of Freedom (DoF): min {N,M} Maximum diversity: NM

Multiplexing-Diversity Tradeoff Tradeoff between diversity gain and multiplex gain Say we have a N x N system Degree of freedom: N The transmitter can transmit k streams concurrently, where k <= N The optimal value of k is determined by the tradeoff between the diversity gain and multiplex gain

1 x 2 example Receive Diversity x h 1 h 2 y 1 y 1 = h 1 x +n 1 y 2 y 2 = h 2 x +n 2 Decode the SNR of (y 1 + y 2 ) Uncorrelated white Gaussian noise with zero mean Packet can be delivered through at least one of the many diver paths

1 x 2 example Receive Diversity x h 1 h 2 y 1 SNR= P(2X), where P refers to the power P(n 1 + n 2 ) = E[(2X)2 ] E[(n 1 + n 2 ) 2 ] = E[(2X)2 ] E[n 1 2 + n 2 2 ] = 4E[X 2 ], where σ is the variance of AWGN 2σ 2 = 2 * SNR single antenna y 2 Increase SNR by 3dB Especially beneficial for the low SNR link

Receive Diversity Maximal Ratio Combining (MRC) Mul?ply each y with the conjugate of the channel! # " $# y 1 = h 1 x + n 1 y 2 = h 2 x + n 2! # " # $ h 1 * y 1 = h 1 2 h 2 * y 2 = h 2 2 x + h 1 * n 1 x + h 2 * n 2 x = h 1 * y 1 + h 2 * y 2 h 1 2 + h 2 2 SNR MRC = E[(( h 2 2 1 + h 2 )X) 2 ] E[(h * 1 n 1 + h * 2 n 2 ) 2 ] = ( h 2 2 1 + h 2 ) 2 E(X 2 ) 2 2 ( h 1 + h 2 )σ 2 = ( h 1 2 + h 2 2 )E(X 2 ) σ 2 SNR sin gle = E[( h 1 X) 2 ] E[(h * 1 n 1 ) 2 ] 4 2 = h 1 E(X 2 ) 2 ( h 1 )σ 2 = h 1 2 E(X 2 ) σ 2 gain = ( h 1 2 + h 2 2 ) h 1 2

Transmit Diversity t t+1 0 h 1 y(t) y(t+1) 0 h 2 Deliver a symbol twice in two consecutive time slots Repetitive code! x = # "# space 0 0 $ & %&?me Diversity: 2 Data rate: 1/2 symbols/s/hz

Transmit Diversity t t+1 - x 2 * h 1 y(t) y(t+1) x 2 * h 2 Alamouti code (space-time block code) x = " * $ x 2 $ * # x 2 space?me % ' ' & Diversity: 2 Data rate: 1 symbols/s/hz

Transmit Diversity Alamouti code (space-time block code) x = " * $ x 2 $ * # x 2 space?me % ' ' & y(t) = h 1 + h 2 x 2 + n 1 y(t +1) = h 2 x 1 * h 1 x 2 * + n 2

Multiplexing-Diversity Tradeoff h 11 h 12 h 21 h22 y 1 x 2 y 2 Repe??ve scheme! x = # "# 0 0 $ & %& Diversity: 4 Data rate: 1/2 sym/s/hz Alamou? scheme " x = $ $ # x 2 * x 2 * % ' ' & Diversity: 4 Data rate: 1 sym/s/hz But 2x2 MIMO has 2 degrees of freedom

Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems

Interference Nulling Alice Bob αx βx!! h 1!! h 2 h 1 αx +h 2 βx = 0!! Nulling :!h α = h β 1 2 (h 1a α + h 2a β)x 0 (h 1b α + h 2b β)x 0 Signals cancel each other at Alice s receiver Signals don t cancel each other at Bob s receiver Because channels are different

Quiz Say there exist a 3x2 link, which has a channel " $ H 3 2 = $ $ # $ h 11 h 12 h 21 h 22 h 31 h 32 % ' ' ' &' How can a 3-antenna transmitter transmit a signal x, but null its signal at two antennas of a two-antenna receiver?

Zero-Forcing Beamforming (w C1, w C2, w C3 ) x C P C (w B1, w B2, w B3 ) x B P B (w A1, w A2, w A3 ) x A P A AP Alice x A x B Bob x C Chris j k y k = ( P k h k w k )s k + ( P j h k w j ) s j + n k y = HWPx + n Null the interference: P j h k w j = 0 Let W = H = H * (HH * ) 1, then y = Px + n j k

Interference Alignment 2-antenna receiver I 2 I 1 wanted signal N-antenna node can only decode N signals If I 1 and I 2 are aligned, à appear as one interferer à 2-antenna receiver can decode the wanted signal

Interference Alignment 2-antenna receiver I 1 + I 2 wanted signal N-antenna node can only decode N signals If I 1 and I 2 are aligned, à appear as one interferer à 2-antenna receiver can decode the wanted signal

Rotate Signal 1. Transmitter can rotate the received signal 2-antenna receiver y y = Ry To rotate received signal y to y = Ry, transmitter multiplies its transmitted signal by the same rotation matrix R

Rotate Signal αx βx y 1 = (h 11 α +h 21 β)x y 2 = (h 12 α +h 22 β)x y = (h 11 +h 21,h 12 +h 22 ) y' = (u, v) (h 11 α +h 21 β) = u (h 12 α +h 22 β) = v How to align the signal along the interference? à Find the direc?on of the interference and rotate the signal to that direc?on

MU-MIMO Bit-Rate Selection AP Alice h A h B Bob h C Chris Select a proper rate based on SNR ZF ant. 1 ant. 1 ant. 1 Alice Alice Bob Bob ant. 2 Chris ant. 2 Chris ant. 2

MU-MIMO User Selection Alice h A h B Bob AP h C Chris Pairing different clients as concurrent receivers results in different sum- rates ant. 1 ant. 1 ant. 1 Alice Alice Bob Bob ant. 2 Chris ant. 2 Chris ant. 2