multi-input multi-output systems (mimo)hsinmu/courses/_media/wn_11fall/mim… · multi-input...
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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
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…
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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$
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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
MIMO Decoding (algebra)
y1y2
!
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h11h12
!
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$
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&&x1 +
h21h22
!
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$
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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
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 x1, decode vector y on the direcGon 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 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+n1y2 =h2x+n2
Diversity Gain • 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 y1 =h1x+n1y2 =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 )σ 2
=( h1
2+ h2
2 )E(X2 )σ 2
y1 =h1x+n1y2 =h2x+n2
!"#
$#
MulGply each y with the conjugate of the channel
h1*y1 = h1
2 x + h1*n1
h2*y2 = h2
2 x + h2*n2
!
"#
$#
SNRsingle =E[(( h1
2+ h2
2 )X)2 ]E[(h1
*n1 +h2*n2 )
2 ]
=h1
4E(X2 )( h1
2 )σ 2
=h1
2E(X2 )σ 2
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: 2data rate: 1/2 sym/s/Hz
Alamouti Scheme:
X =
time
space
x -x *
x x2
1 2
1*
diversity: 2data rate: 1 sym/s/Hz
Space-Time Code Examples: 2£ 2 Channel
Repetition Scheme:
X = x 0
0 x
time
space
1
1
diversity: 4data rate: 1/2 sym/s/Hz
Alamouti Scheme:
X =
time
space
x -x *
x x2
1 2
1*
diversity: 4data rate: 1 sym/s/Hz
But the 2£ 2 channel 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
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 h12h21 h22h31 h32
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Interference Alignment
N-antenna node can only decode N signals
wanted signal I1 I2
If I1 and I2 are aligned,
à appear as one interferer
à 2-antenna receiver can decode the wanted signal
2-antenna receiver
Interference Alignment
If I1 and I2 are 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 I1 + I2
wanted 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
= Ry
Rotate Signal
βxαx y1 = (h11α +h21β)x
y2 = (h12α +h22β)x
y = (h11 +h21,h12 +h22 )y ' = (u, v)
(h11α +h21β) =u(h12α +h22β) = v
How to align the signal along the interference? à Find the direcGon of the interference and rotate the signal to that direcGon