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

MIMO 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  

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

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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 +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