Multi-Input Multi-Output Systems (MIMO)

• Channel Model for MIMO • MIMO Decoding • MIMO Gains • Multi-User MIMO Systems 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

h11 " % h12 h21 h11 h12 h13  h13 $ ' h h h h $ 21 22 23  ' 22 HN×M = $ ' h h31 23 h31 h2 h33  h32 $ ' h33 #$     &' … … Channel Model (2x2)

h11 x1 y1 h12

h21 x y 2 h22 2

y1 = h11x1 +h21x2 +n1

y2 = h12x1 +h22x2 +n2

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 $ ! h $ ! h $ ! n $ # 1 & = # 11 &x +# 21 &x +# 1 & # & # & 1 # & 2 # & " y2 % " h12 % " h22 % " n2 %     y = h1x1 +h2x2 +n

 y = (y , y )  1 2 antenna 2 h2 = (h21,h22 ) antenna 2 x2  h1 = (h11,h12 ) x1 antenna 1 antenna 3 antenna 1 Multi-Input Multi-Output Systems (MIMO)

• Channel Model for MIMO • MIMO Decoding • MIMO Gains • Multi-User MIMO Systems Zero-Forcing Decoding (algebra)

Orthogonal vectors

! $ ! $ ! $ ! $ y h h n * h22 # 1 & = # 11 &x +# 21 &x +# 1 & # y & # h & 1 # h & 2 # n & + ) " 2 % " 12 % " 22 % " 2 % * - h21

y1h22 − y2h21 = (h11h22 − h12h21)x1

y1h22 − y2h21 x1 = h11h22 − h12h21

Given x1, solve x2 by successive interference cancellaon (SIC) To guarantee the full rank of H, antenna spacing at the transmier and receiver must exceed half of the wavelength

Zero-Forcing Decoding (antenna space)

  y = (y1, y2 ) h2 = (h21,h22 ) antenna 2 x2  h1 = (h11,h12 ) x1 antenna 1

x’1

‖x’1‖< ‖x1‖

• To decode x1, decode vector y on the direcon orthogonal to x2 • To improve the SNR, re-encode the ﬁrst detected signal, subtract it from y, and decode the second signal Channel Estimation • Estimate N x M matrix H

h11 x1 y1 h12 y1 = h11x1 +h21x2 +n1

h21 y2 = h12x1 +h22x2 +n2

x2 h y2 22 Two equaons, but four unknowns

Antenna 1 at Tx Access code 1 Stream 1

Antenna 2 at Tx Access code 2 Stream 2

Esmate h11, h12 Esmate h21, h22 Multi-Input Multi-Output Systems (MIMO)

• Channel Model for MIMO • MIMO Decoding • MIMO Gains • Multi-User MIMO Systems MIMO Gains • Multiplex Gain

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 Receive Diversity • 1 x 2 example

h1 x y1 h2 y1 = h1x +n1 y = h x +n 2 2 2 y2

Decode the SNR of (y1 + y2) Uncorrelated whit Gaussian noise with zero mean

Packet can be delivered through at least one of the many diver paths Receive Diversity • 1 x 2 example

h1 y x 1 h2

y2 P(2X) SNR= ,(where(P(refers(to(the(power P(n +n ) 1 2 • Increase SNR by 3dB E[(2X)2 ] = • Especially beneﬁcial for [ 2 2 ] E n1 +n2 the low SNR link 4E[X2 ] = , (where(σ (is(the(variance(of(AWGN 2σ 2

= 2 *SNRsingle(antenna Receive Diversity Maximal Ratio Combining (MRC)

Mulply each y with the conjugate of the channel

! ! * 2 * * * y h x n h1 y1 = h1 x + h1 n1 h y + h y # 1 = 1 + 1 # x 1 1 2 2 " " = 2 2 * 2 * h + h # y2 = h2 x + n2 # h y = h x + h n 1 2 $ $ 2 2 2 2 2

2 2 2 2 2 E[(( h1 + h2 )X) ] E[( h1 X) ] SNRMRC = * * 2 SNRsin gle = * 2 E[(h1 n1 + h2n2 ) ] E[(h1 n1) ] 2 2 4 ( h + h )2 E(X 2 ) h E(X 2 ) = 1 2 = 1 2 2 2 2 2 ( h1 + h2 )σ ( h1 )σ 2 2 2 2 2 2 2 ( h1 + h2 ) ( h1 + h2 )E(X ) h1 E(X ) gain = 2 = 2 = 2 σ σ h1 Transmit Diversity

t t+1 h1 y(t) y(t+1) x1 0

0 x1 h2

• Deliver a symbol twice in two consecutive time slots • Repetitive code me ! $ • Diversity: 2 x1 0 x = # & • Data rate: 1/2 symbols/s/Hz 0 x "# 1 %& space Transmit Diversity

t t+1 h1 y(t) y(t+1) x1 -x2*

x2 x1* h2

• Alamouti code (space-time block code) me " x x* % $ 1 − 2 ' • Diversity: 2 x = • $ x x* ' Data rate: 1 symbols/s/Hz # 2 1 & space Transmit Diversity • Alamouti code (space-time block code) me " * % x −x y(t) = h1x1 + h2 x2 + n1 x = $ 1 2 ' * y(t +1) = h x* − h x* + n $ x x ' 2 1 1 2 2 # 2 1 & space Multiplexing-Diversity Tradeoff

h11 x1 y1 h12

h21 x y 2 h22 2

Repeve scheme Alamou scheme ! $ " * % x1 0 x −x x = # & x = $ 1 2 ' # 0 x & $ x x* ' " 1 % # 2 1 & Diversity: 4 Diversity: 4 Data rate: 1/2 sym/s/Hz 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 h1αx +h2βx = 0 !!h1 !!⇒ Nulling : !h1α = −h2β αx (h α + h β)x ≠ 0 Bob !!h2 1a 2a x β (h1bα + h2bβ)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 h % $ 11 12 ' H h h 3×2 = $ 21 22 ' $ ' h h #$ 31 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

(wC1, wC2, wC3) xC√PC (wB1, wB2, wB3) xB√PB (wA1, wA2, wA3) xA√PA

xC xA xB Alice Chris Bob Null the interference: yk = ( Pk hkwk )sk + ( Pj hkw j )sj + nk ∑ Pj hkw j = 0 j≠k ∑ j≠k y = HWPx + n Let W = H† = H*(HH* )−1, then y = Px + n MU-MIMO Bit-Rate Selection

hC hA hB Select a proper rate Alice Chris based on SNRZF Bob

ant. 1 ant. 1 ant. 1

Alice Alice

Bob Bob

Chris ant. 2 ant. 2 Chris ant. 2 MU-MIMO User Selection

hC Pairing diﬀerent clients hA hB as concurrent receivers Alice Chris results in diﬀerent sum-rates Bob

ant. 1 ant. 1 ant. 1

Alice Alice

Bob Bob

Chris ant. 2 ant. 2 Chris ant. 2