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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)

Multi-Input Multi-Output Systems (MIMO)

•  Channel Model for MIMO

•  MIMO Decoding

•  MIMO Gains

•  Multi-User MIMO Systems

(3)

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

"

#

$$

$$

$

%

&

'' '' '

(4)

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  

(5)

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

(6)

Multi-Input Multi-Output Systems (MIMO)

•  Channel Model for MIMO

•  MIMO Decoding

•  MIMO Gains

•  Multi-User MIMO Systems

(7)

Zero-Forcing 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  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  

 

(8)

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  

(9)

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  

(10)

Multi-Input Multi-Output Systems (MIMO)

•  Channel Model for MIMO

•  MIMO Decoding

•  MIMO Gains

•  Multi-User MIMO Systems

(11)

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

(12)

Degree of Freedom

•  For N x M MIMO channel

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

„  Maximum diversity: NM

(13)

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

(14)

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

(15)

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  

(16)

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

!

"

#

$#

Mul?ply  each  y  with  the  conjugate  of  the  channel  

h1*y1 = h1 2 x + h1*n1 h2*y2 = h2 2 x + h2*n2

!

"

#

$#

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

(17)

Transmit Diversity

•  Deliver a symbol twice in two consecutive time slots

•  Repetitive code

h1  

h2   t  

y(t)      y(t+1)  

x = x

1

0

0 x

1

!

"

# #

$

%

&

&

?me  

space   x1                    0  

t+1  

0                x1    

•  Diversity:  2  

•  Data  rate:  1/2  symbols/s/Hz    

(18)

Transmit Diversity

•  Alamouti code (space-time block code)

x = x

1

−x

2*

x

2

x

1*

"

#

$ $

%

&

' '

?me  

space  

h1  

h2   x1              -­‐x2*  

x2            x1*     t   t+1  

y(t)      y(t+1)  

•  Diversity:  2  

•  Data  rate:  1  symbols/s/Hz    

(19)

Transmit Diversity

•  Alamouti code (space-time block code)

x = x

1

−x

2*

x

2

x

1*

"

#

$ $

%

&

' '

?me  

space  

y(t) = h

1

x

1

+ h

2

x

2

+ n

1

y(t +1) = h

2

x

1

*

− h

1

x

2*

+ n

2

(20)

Multiplexing-Diversity Tradeoff

h11   h12   h21  

h22   x1  

x2   y2  

y1  

x = x

1

−x

2*

x

2

x

1*

"

#

$ $

%

&

' '

x = x

1

0 0 x

1

!

"

# #

$

%

&

&

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    

(21)

Multi-Input Multi-Output Systems (MIMO)

•  Channel Model for MIMO

•  MIMO Decoding

•  MIMO Gains

•  Multi-User MIMO Systems

(22)

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

2 a

β )x ≠ 0 (h

1b

α + h

2b

β )x ≠ 0

Bob

!!

⇒ Nulling : !h

1

α = −h

2

β

(23)

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

"

#

$$

$$

%

&

'' ''

(24)

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

= 0

y = HWPx + n

Let W = H = H*(HH*)−1, then y = Px + n

(25)

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

ZF  

(26)

MU-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  

Pairing  different  clients    

as  concurrent  receivers  

results  in  different  sum-­‐rates  

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