Transcript An introduction to multiple antenna systems

Promises of Wireless
MIMO Systems
Mattias Wennström
Uppsala University
Sweden
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Systems Group
Outline
•
•
•
•
Introduction...why MIMO??
Shannon capacity of MIMO systems Telatar, AT&T 1995
The ”pipe” interpretation
To exploit the MIMO channel
– BLAST Foschini, Bell Labs 1996
– Space Time Coding Tarokh, Seshadri & Calderbank 1998
– Beamforming
• Comparisons & hardware issues
• Space time coding in 3G & EDGE
Release ’99
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Why multiple antennas ????
• Frequency and time processing are at limits
• Space processing is interesting because it
does not increase bandwidth
outdoor
”Specular” ”Scattering”
channels
channels
indoor
Phased array
range extension,
interference reduction
Adaptive Antennas
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interference cancellation
MIMO
Systems
(diversity)
Initial Assumptions
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•
•
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•
•
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Flat fading channel (Bcoh>> 1/ Tsymb)
Slowly fading channel (Tcoh>> Tsymb)
nr receive and nt transmit antennas
Noise limited system (no CCI)
Receiver estimates the channel perfectly
We consider space diversity only
”Classical” receive diversity
H11
H21

PT
*
C  log2 det I  2 HH 
 σ nt

= log2[1+(PT/s2)·|H|2]
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[bit/(Hz·s)]
Capacity increases logarithmically
with number of receive antennas...
H = [ H11 H21]
Transmit diversity / beamforming
H11
H12
Cdiversity = log2(1+(PT/2s2)·|H|2)
[bit/(Hz·s)]
Cbeamforming = log2(1 +(PT/s2 )·|H|2)
[bit/(Hz·s)]
• 3 dB SNR increase if transmitter knows H
• Capacity increases logarithmically with nt
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Multiple Input Multiple Output systems
H
H   11
 H 21
H11
H12
H21
H12 
H 22 
H22
Cdiversity = log2det[I +(PT/2s2 )·HH†]=
Where the i are the
eigenvalues to HH†
P
P




 log2 1  T 2 1   log2 1  T 2 2 
 2s

 2s

Interpretation:
1
Transmitter
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2
m=min(nr, nt) parallel channels,
equal power allocated to each ”pipe”
Receiver
MIMO capacity in general
H unknown at TX
H known at TX


P
C  log2 det  I  2T HH *  
 s nt

m


P
  log2 1  2T i 
i 1
 s nt 
 p 
C   log2 1  i 2 i 
 s 
i 1
m
Where the power distribution over
”pipes” are given by a water filling
solution

1
PT   pi     
i 
i 1
i 1 
m  min(nr , nt )
p1
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p2
p3
p4
m
1
2
3
4
m

The Channel Eigenvalues
Orthogonal channels HH† =I, 1= 2= …= m= 1


P
Cdiversity   log2 1  2T i   min(nt , nr )  log2 (1  PT / s 2 nt )
i 1
 s nt 
m
• Capacity increases linearly with min( nr , nt )
• An equal amount of power PT/nt is allocated
to each ”pipe”
Transmitter
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Receiver
Random channel models and
Delay limited capacity
• In stochastic channels,
the channel capacity becomes a random
variable
Define : Outage probability Pout = Pr{ C < R }
Define : Outage capacity R0 given a outage
probability Pout = Pr{ C < R0 }, this is the delay
limited capacity.
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Outage probability approximates the
Word error probability for coding blocks of approx length100
Example : Rayleigh fading channel
Hij CN (0,1)
Ordered eigenvalue
distribution for
nr= nt = 4 case.
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nr=1
nr= nt
To Exploit the MIMO Channel
Bell Labs Layered
Space Time Architecture
Time
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s1 s1 s1 s1 s1 s1
s2 s2 s2 s2 s2 s2
s3 s3 s3 s3 s3 s3
V-BLAST
s0 s1 s2 s0 s1 s2
s0 s1 s2 s0 s1
s0 s1 s2 s0
D-BLAST
• nr  nt required
• Symbol by symbol detection.
Using nulling and symbol
cancellation
• V-BLAST implemented -98
by Bell Labs (40 bps/Hz)
• If one ”pipe” is bad in BLAST
we get errors ...
{G.J.Foschini, Bell Labs Technical Journal 1996 }
Space Time Coding
• Use parallel channel to obtain diversity not
spectral efficiency as in BLAST
• Space-Time trellis codes : coding and diversity
gain (require Viterbi detector)
• Space-Time block codes : diversity gain
(use outer code to get coding gain)
• nr= 1 is possible
• Properly designed codes acheive diversity of nr nt
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*{V.Tarokh, N.Seshadri, A.R.Calderbank
Space-time codes for high data rate wireless communication:
Performance Criterion and Code Construction
, IEEE Trans. On Information Theory March 1998 }
Orthogonal Space-time Block Codes
Block of T
symbols
Constellation
mapper
Data in
STBC
Block of K
symbols
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nt transmit
antennas
• K input symbols, T output symbols T K
• R=K/T is the code rate
• If R=1 the STBC has full rate
• If T= nt the code has minimum delay
• Detector is linear !!!
*{V.Tarokh, H.Jafarkhani, A.R.Calderbank
Space-time block codes from orthogonal designs,
IEEE Trans. On Information Theory June 1999 }
STBC for 2 Transmit Antennas
 c0
[ c0 c1 ]  
 c1
 c1* 

c0* 
Full rate and
minimum delay
Antenna
Time
Assume 1 RX antenna:
Received signal at time 0
Received signal at time 1
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r0  h1c0  h2c1  n0
r1  h1c1*  h2c0*  n1
r  Hc  n
 r0 
r   * ,
r1 
 h1
H *
h2
h2 
,
*
 h1 
n0 
n   * ,
n1 
c0 
c 
 c1 
2
*
*
*
~
~
r  H r  H Hc  H n  H F c  n
Diagonal matrix due to orthogonality
The MIMO/ MISO system is in fact
transformed to an equivalent SISO system
with SNR
SNReq = || H ||F2 SNR/nt
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|| H ||F2 = 1 2
1 2
The existence of Orthogonal STBC
• Real symbols :
For nt =2,4,8 exists delay optimal
full rate codes.
For nt =3,5,6,7,>8 exists full rate
codes with delay (T>K)
• Complex symbols : For nt =2 exists delay optimal
full rate codes.
For nt =3,4 exists rate 3/4 codes
For nt > 4 exists (so far)
rate 1/2 codes
Example: nt =4, K=3, T=4
 R=3/4 s1 s2
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 s1
 0
s3  
  *
  s2
 *
 s3
0
s2
s1
s3*
 s3
s1*
 s2
0
 s3 
s2* 
0 

s1* 
Outage capacity of STBC
CSTBC
 SNR
 log2 1 
H
n
t



F

2
Cdiversity
 SNR

 log2 det  I 
HH 
n
t


Optimal capacity
 STBC is optimal
wrt capacity if
HH† = || H ||F2
which is the case for
• MISO systems
• Low rank channels
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Performance of the STBC…
(Rayleigh faded channel)
The PDF of
|| H ||F2 = 1 2 ..  m
Assume BPSK modulation
BER is then given by
 1 
Pb  

 4 SNR 
nr nt
 2nr nt  1


 nr nt 
Diversity gain
nrnt which is
same as for
orthogonal
channels
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nt=4 transmit antennas and
nr is varied.
MIMO With Beamforming
Requires that channel H is known at the transmitter
Is the capacity-optimal transmission strategy if
1
2

1
1
 SNR
Which is often true for line of sight (LOS) channels
Only one ”pipe” is used
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Cbeamforming = log2(1+SNR·1)
[bit/(Hz·s)]
Comparisons...
2 * 2 system. With specular component (Ricean fading)
One dominating
eigenvalue. BF puts
all energy into
that ”pipe”
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Correlated channels / Mutual coupling ...
When angle spread (D)
is small, we have a
dominating eigenvalue.
The mutual coupling
actually
improves the performance
of the STBC by making the
eigenvalues ”more equal”
in magnitude.
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WCDMA Transmit diversity concept
(3GPP Release ’99 with 2 TX antennas)
•2 modes
Open loop mode is exactly the
2 antenna STBC  s0  s1* 
• Open loop (STTD)

 s1
• Closed loop (1 bit / slot feedback)
• Submode 1 (1 phase bit)
• Submode 2 (3 phase bits / 1 gain bit)

s0* 
The feedback bits (1500 Hz) determines the beamformer weights
Submode 1 Equal power and bit chooses phase between
{0,180} / {90/270}
Submode 2 Bit one chooses power division {0.8 , 0.2} / {0.2 , 0.8}
and 3 bits chooses phase in an 8-PSK constellation
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GSM/EDGE Space time coding proposal
• Frequency selective channel …
• Require new software in terminals ..
• Invented by Erik Lindskog
Time Reversal Space Time Coding
(works for 2 antennas)
Block
S1(t)
Time reversal
Complex conjugate
Time reversal
Complex conjugate
S(t)
S2(t)
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-1
”Take- home message”
• Channel capacity increases linearly
with min(nr, nt)
• STBC is in the 3GPP WCDMA proposal
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