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
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 +n1 y2 = h12x1 +h22x2 +n2
y = Hx +n
x1
x2 y2
y1
Antenna Space
M-antenna node receives in M-dimensional space
y1 y2
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&= h11 h12
!
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&x1+ h21 h22
!
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&x2 + n1 n2
!
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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
Multi-Input Multi-Output Systems (MIMO)
• Channel Model for MIMO
• MIMO Decoding
• MIMO Gains
• Multi-User MIMO Systems
Zero-Forcing Decoding (algebra)
y1 y2
!
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&= h11 h12
!
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&x1+ h21 h22
!
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&x2 + n1 n2
!
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* h22 * -‐ h21
+ )
y1h22 − y2h21 = (h11h22 − h12h21)x1 x1 = y1h22 − y2h21
h11h22 − 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 x
1, decode vector y on the direc?on orthogonal to x
2• 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 +n1 y2 = 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
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
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 +n1 y2 = h2x +n2
Receive Diversity
• 1 x 2 example
SNR= P(2X)
P(n1 +n2),(where(P(refers(to(the(power
= E[(2X)2] E[n12 +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(X2) ( h1 2 + h2 2)σ 2
= ( h1 2 + h2 2)E(X2) σ2
y1 = h1x + n1 y2 = 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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SNRsin gle = E[( h1 2 X)2] E[(h1*n1)2]
= h1 4 E(X2) ( h1 2)σ 2
= h1 2 E(X2) σ 2
gain = ( h1 2 + h2 2) h1 2
x = h1*y1+ h2*y2 h1 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 = x
10
0 x
1!
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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 = x
1−x
2*x
2x
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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 = x
1−x
2*x
2x
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space
y(t) = h
1x
1+ h
2x
2+ n
1y(t +1) = h
2x
1
*
− h
1x
2*+ n
2Multiplexing-Diversity Tradeoff
h11 h12 h21
h22 x1
x2 y2
y1
x = x
1−x
2*x
2x
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x = x
10 0 x
1!
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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
Multi-Input Multi-Output Systems (MIMO)
• Channel Model for MIMO
• MIMO Decoding
• MIMO Gains
• Multi-User MIMO Systems
h
1α x +h
2β x = 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
β x α x
!!
€
h
1!!
€
h
2(h
1aα + h
2 aβ )x ≠ 0 (h
1bα + h
2bβ )x ≠ 0
Bob
!!
€
⇒ Nulling : !h
1α = −h
2β
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 h12 h21 h22 h31 h32
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Zero-Forcing Beamforming
yk = ( Pk hkwk )sk + ( Pjhkwj)
j≠k
∑
sj + nk(w(wA1B1, w, wA2B2, w, wA3B3) x) xAB√P√PAB (wC1, wC2, wC3) xC√PC
AP
Chris Alice
Bob
xA
xB xC
Null the interference:
Pjhkwj
j≠k
∑
= 0y = HWPx + n
Let 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 SNR
ZFMU-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