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Multi-Input Multi-Output Systems (MIMO)

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(1)

Multi-Input Multi-Output Systems (MIMO)

•  Channel Model for MIMO

•  MIMO Decoding

•  MIMO Gains

•  Multi-User MIMO Systems

(2)

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

"

#

$$

$$

$

%

&

'' '' '

(3)

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  

(4)

Antenna Space

M-antenna node receives in M-dimensional space  

y1 y2

!

"

##

$

%

&

&= h11 h12

!

"

##

$

%

&

&x1+ h21 h22

!

"

##

$

%

&

&x2 + n1 n2

!

"

##

$

%

&

&

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

(5)

MIMO Decoding (algebra)

y1 y2

!

"

##

$

%

&

&= h11 h12

!

"

##

$

%

&

&x1+ h21 h22

!

"

##

$

%

&

&x2 + n1 n2

!

"

##

$

%

&

&

 *  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  

 

(6)

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  

(7)

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  

(8)

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

(9)

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

(10)

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

(11)

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

!

"

#

$#

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 +h*2n2)2]

= h1 4E(X2) ( h1 2)σ 2

= h1 2E(X2) σ2

gain = ( h1 2 + h2 2) h1 2

(12)

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

(13)

Degree of Freedom

•  For N x M MIMO channel

„  Degree of Freedom (DoF): min {N,M}

„  Maximum diversity: NM

(14)

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

(15)

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!

(16)

h

1

α x +h

2

β 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

1a

α + h

2a

β ) x ≠ 0 ( h

1b

α + h

2b

β ) x ≠ 0

Bob

!!

⇒ Nulling : !h

1

α = −h

2

β

(17)

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

"

#

$$

$$

%

&

'' ''

(18)

Interference Alignment

N-antenna node can only decode N signals wanted signal I

1

I

2

If I

1

and I

2

are aligned,

à appear as one interferer

à 2-antenna receiver can decode the wanted signal

2-antenna receiver

(19)

Interference Alignment

If I

1

and I

2

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

I

1

+ I

2

wanted signal

(20)

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  

(21)

Rotate Signal

β x

α x y

1

= (h

11

α + h

21

β )x y

2

= (h

12

α + h

22

β )x

y = (h

11

+ h

21

, h

12

+ h

22

) y' = (u,v)

(h

11

α + h

21

β ) = u (h

12

α + h

22

β ) = v

How  to  align  the  signal  along  the  interference?  

à  Find  the  direcGon  of  the  interference    

and  rotate  the  signal  to  that  direcGon  

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