Multi-Input Multi-Output Systems (MIMO) -...

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

h13 h21

h22 h23 h31 h32

h33

HN×M =

h11 h12 h13 h21 h22 h23 h31 h2 h33

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Channel Model (2x2)

•  Can be extended to N x M systems

h11

h12

h21

h22

y1 =h11x1 +h21x2 +n1y2 =h12x1 +h22x2 +n2

y =Hx+n

x1

x2 y2

y1

Antenna Space M-antenna node receives in M-dimensional space

y1y2

!

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h11h12

!

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&&x1 +

h21h22

!

"

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&&x2 +

n1n2

!

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&&

y =h1x1 +

h2x2 +

n

h2 = (h21,h22 )

x2

antenna 1

antenna 2 h1 = (h11,h12 )x1

2 x 2

y = (y1, y2 )

antenna 1

antenna 2

antenna 3

Zero-Forcing Decoding (algebra)

y1y2

!

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h11h12

!

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&&x1 +

h21h22

!

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&&x2 +

n1n2

!

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* h22

* -‐ h21 + )

y1h22 − y2h21 = (h11h22 − h12h21)x1

x1 =y1h22 − y2h21h11h22 − h12h21

Orthogonal vectors

Given x1, solve x2 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)

h2 = (h21,h22 )

x2

antenna 1

antenna 2 h1 = (h11,h12 )x1

y = (y1, y2 )

x’1

‖x’1‖< ‖x1‖

•  To decode x1, decode vector y on the direc?on 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

y1 =h11x1 +h21x2 +n1y2 =h12x1 +h22x2 +n2

Two equa?ons, but four unknowns

Antenna 1 at Tx

Antenna 2 at Tx

h11

h12

h21

h22

x1

x2 y2

y1

Access code 1

Access code 2

Stream 1

Stream 2

Es?mate h11, h12 Es?mate h21, h22

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

�  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

h1

h2 x

y2

y1 y1 =h1x+n1y2 =h2x+n2

Receive Diversity •  1 x 2 example

SNR= P(2X)

P(n1 +n2 ),(where(P(refers(to(the(power

=E[(2X)2 ]E[n1

2 +n22 ]

=4E[X2 ]2σ 2 , (where(σ (is(the(variance(of(AWGN

= 2*SNRsingle(antenna

•  Increase SNR by 3dB •  Especially beneficial for the low SNR link

h1

h2 x

y2

y1

Receive Diversity Maximal Ratio Combining (MRC)

SNRMRC =E[(( h1

2+ h2

2 )X)2 ]E[(h1

*n1 + h2*n2 )

2 ]

=( h1

2+ h2

2 )2E(X 2 )( h1

2+ h2

2 )σ 2

=( h1

2+ h2

2 )E(X 2 )σ 2

y1 = h1x + n1y2 = h2x + n2

!"#

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Mul?ply each y with the conjugate of the channel

h1*y1 = h1

2 x + h1*n1

h2*y2 = h2

2 x + h2*n2

!

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$#

SNRsingle =E[( h1

2 X)2 ]E[(h1

*n1)2 ]

=h1

4 E(X 2 )( h1

2 )σ 2

=h1

2 E(X 2 )σ 2

gain =( h1

2+ h2

2 )h1

2

x = h1*y1 + h2

*y2h1

2+ h2

2

Transmit Diversity

•  Deliver a symbol twice in two consecutive time slots

•  Repetitive code

h1

h2

t y(t) y(t+1)

x =x1 00 x1

!

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?me

space

x1 0

t+1

0 x1

•  Diversity: 2 •  Data rate: 1/2 symbols/s/Hz

Transmit Diversity

•  Alamouti code (space-time block code)

x =x1 −x2

*

x2 x1*

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space

h1

h2

x1 -‐x2*

x2 x1*

t t+1 y(t) y(t+1)

•  Diversity: 2 •  Data rate: 1 symbols/s/Hz

Transmit Diversity •  Alamouti code (space-time block code)

x =x1 −x2

*

x2 x1*

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#

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?me

space

y(t) = h1x1 + h2x2 + n1y(t +1) = h2x1

* − h1x2* + n2

Multiplexing-Diversity Tradeoff h11

h12

h21

h22

x1

x2 y2

y1

x =x1 −x2

*

x2 x1*

"

#

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x =x1 00 x1

!

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Repe??ve scheme Alamou? scheme

Diversity: 4 Data rate: 1/2 sym/s/Hz

Diversity: 4 Data rate: 1 sym/s/Hz

But 2x2 MIMO has 2 degrees of freedom

h1αx+h2βx = 0

Interference Nulling

•  Signals cancel each other at Alice’s receiver

•  Signals don’t cancel each other at Bob’s receiver

�  Because channels are different

Alice

βxαx

!!

€

h1

!!

€

h2(h1aα + h2aβ)x ≠ 0(h1bα + h2bβ)x ≠ 0

Bob

!!

€

⇒Nulling : !h1α = −h2β

Quiz

•  Say there exist a 3x2 link, which has a channel How can a 3-antenna transmitter transmit a signal x, but null its signal at two antennas of a two-antenna receiver?

H3×2 =

h11 h12h21 h22h31 h32

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Zero-Forcing Beamforming

yk = ( Pk hkwk )sk + ( Pj hkwj )j≠k∑ sj + nk

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

AP

Chris Alice Bob

xA xB xC

Null the interference: Pj hkwj

j≠k∑ = 0

y =HWPx+nLet W =H† =H*(HH*)−1, then y = Px+n

MU-MIMO Bit-Rate Selection

AP

Chris Alice Bob

hA hB hC

ant. 2

ant. 1

Alice

Bob

ant. 2

Alice

Chris

ant. 1

ant. 2

ant. 1

Bob

Chris

Select a proper rate based on SNRZF

MU-MIMO User Selection

AP

Chris Alice Bob

hA hB hC

ant. 2

ant. 1

Alice

Bob

ant. 2

Alice

Chris

ant. 1

ant. 2

ant. 1

Bob

Chris

Pairing different clients as concurrent receivers

results in different sum-‐rates

Multi-Input Multi-Output Systems (MIMO) -...

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