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
MIMO 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
To guarantee the full rank of H, antenna spacing at the transmiCer and receiver must exceed half of the wavelength
MIMO Decoding (antenna space)
• Zero forcing
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 direcGon 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 equaGons, 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 EsGmate h11, h12 EsGmate h21, h22
MIMO Gains
• Multiplex Gain
Exploit antenna diversity to deliver multiple streams concurrently
• Diversity Gain
Exploit antenna diversity to increase the SNR of a single stream
Diversity Gain
• 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
Diversity Gain
• 1 x 2 example
SNR= P(2X)
P(n1+n2),(where(P(refers(to(the(power
= E[(2X)2] E[n12 +n22]
= 4E[X2]
2σ2 ,(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
y1 = h1x +n1 y2 = h2x +n2
Diversity Gain
SNRdiversity = E[(( h1 2 + h2 2)X)2] E[(h1*n1+h2*n2)2]
= ( h1 2 + h2 2)2E(X2) ( h1 2 + h2 2)σ2
= ( h1 2 + h2 2)E(X2) σ2
y1 = h1x +n1 y2 = h2x +n2
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MulGply 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 + h2 2)X)2] E[(h1*n1 +h*2n2)2]
= h1 4E(X2) ( h1 2)σ 2
= h1 2E(X2) σ2
gain = ( h1 2 + h2 2) h1 2
Trade off
• 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
Degree of Freedom
• For N x M MIMO channel
Degree of Freedom (DoF): min {N,M}
Maximum diversity: NM
Space-Time Code Examples: 2 £ 1 Channel
Repetition Scheme:
X = x 0 0 x
time
space 1
1
diversity: 2
data rate: 1/2 sym/s/Hz
Alamouti Scheme:
X =
time
space
x -x * x x2
1 2
1*
diversity: 2
data rate: 1 sym/s/Hz
Space-Time Code Examples: 2 £ 2 Channel
Repetition Scheme:
X = x 0 0 x
time
space 1
1
diversity: 4
data rate: 1/2 sym/s/Hz
Alamouti Scheme:
X =
time
space
x -x * x x2
1 2
1*
diversity: 4
data rate: 1 sym/s/Hz
But the 2 £ 2 channel has 2 degrees of freedom!
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
2aβ ) x ≠ 0 ( h
1bα + h
2bβ ) x ≠ 0
Bob
!!
€
⇒ Nulling : !h
1α = −h
2β
Homework
• Say there exist a 3x2 link, which has a channel
How can a three-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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Interference Alignment
N-antenna node can only decode N signals wanted signal I
1I
2If I
1and I
2are aligned,
à appear as one interferer
à 2-antenna receiver can decode the wanted signal
2-antenna receiver
Interference Alignment
If I
1and I
2are aligned,
à appear as one interferer
à 2-antenna receiver can decode the wanted signal N-antenna node can only decode N signals
2-antenna receiver
I
1+ I
2wanted signal
Rotate Signal
1. Transmitter can rotate the received signal
To rotate received signal y to y’ = Ry,
transmitter multiplies its transmitted signal by the same rotation matrix R
y y’
2-antenna receiver