Multiple-Input-Multiple Output (MIMO)
Systems
Wireless Communications
8/1/2006
MIMO-1
MIMO : Features

A MIMO system has 3 basic features:
 Beam-forming (diversity gain),
 Diversity (diversity gain),
 Spatial multiplexing (increase data rates)
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Research in MIMO
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A combination of the 3 features is also possible depending on the
number of TX and Rx elements and number of independent data
streams
Increasing the capacity of MIMO systems
Developing systems that can operate close to the capacity
Primary applications
4-th generation wireless and High capacity LANs ( 802.11…)
 3GPP release 7 and 8 (LTE)

Wireless Communications
8/1/2006
MIMO-2
MIMO Channel Model
Transmitter
With Nt TX Antennas
Total TX Power = P
Receiver
With Nr RX Antennas
X
 h11
h
21
H 
 

hNt 1
h12
h22

hN t 2
 h1N r 
 h2 N r 

 

 hN t N r 
Y  HX  n

Channel can be deterministic, or slowly changing

Channel state information (CSI) may be



Unknown to both TX and RX
Known to RX only
Known to both RX and TX
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MIMO-3
Parallel Decomposition of MIMO Channels
~
Y  U *T Y  [ HX  n]
 U *T [UV *T X  n]
 (U *T U )V *T X  U *T n]
~
 X  n~ ,
~
~
where X  V *T X ; X  VX
n~  U *T n
[Note : var n~  var n ]
i
N
i
C   log 2 [1   i2 i ]; i 
i 1
~
~
Y  X  n~
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8/1/2006
Pi
2
~ ~
E{ X *T X }  E[ X *T X ]  diag [ Pi ];
MIMO-4
Modeling of MIMO Fading Channel

MIMO channel response on a
slow fading model
Channel Time-variance
Time delayspread
With suitable choices of array geometry and
antenna element patterns,
H( ) = H which is an Mr x Mt matrix with
complex Gaussian i. i. d random variables
Accurate for NLOS rich-scattering
environments, with sufficient antenna
spacing at transmitter and receiver with all
elements identically polarized
Wireless Communications
8/1/2006
a and b are transmit and
receive array factor vectors
respectively. S is the complex
gain that is dependant on
direction and delay. g(t) is the
transmit and receive pulse
shaping impulse response
MIMO-5
MIMO Design Criterion

MIMO Systems can provide two types of gain
Spatial Multiplexing Gain
Diversity Gain
• Maximize transmission rate
(optimistic approach)
• Minimize Pe (conservative
approach)
• Use rich scattering/fading to
advantage
• Go for Reliability / QoS etc
• Counter fading

As expected, there is a tradeoff

System designs are carried out to achieve a little bit of both.
Wireless Communications
8/1/2006
MIMO-6
MIMO Diversity Gain : Beamforming

Beamforming takes advantage
directionality of the array.

Beamformer controls the phase and relative amplitude of the signal at
TX
At the Rx side, information from different sensors are combined to a
preferentially observed radiation pattern
Beam formers are usually smart antennas:



of
interference
to
change
the
Phased Array Systems (Switched Beamforming) with a finite number of fixed
predefined patterns.

Adaptive Array Systems (AAS) (Adaptive Beamforming) with an infinite
number of patterns adjusted to the scenario in real time.
Wireless Communications
8/1/2006
MIMO-7
MIMO Diversity Gain : Beamforming

Beamforming provides diversity gain by coherent combining of the
multiple signal paths.
y=u*Hvx + u*n

If H is known, the received SNR is optimized by choosing u and v as the
principal left and right singular vectors of the channel matrix H.

Capacity for with beamforming is given as
Wireless Communications
8/1/2006
MIMO-8
Diversity in MIMO

Each pair of transmit-receive antennas provides a signal path
from transmitter to receiver. By sending the SAME information
through different paths, multiple independently-faded replicas of
the data symbol can be obtained at the receiver end.

A diversity gain d implies that in the high SNR region, Pe decays
at a rate of 1/SNRd as opposed to 1/SNR for a SISO system

The maximal diversity gain dmax is the total number of
independent signal paths that exist between the transmitter and
receiver

For an (MR,MT) system, the total number of signal paths is MRMT
1 ≤ d ≤ dmax= MRM
The higher the diversity gain, the lower the Pe
Wireless Communications
8/1/2006
MIMO-9
Spatial multiplexing in MIMO: Equivalent Channel Model

We can convert a general MIMO model to an equivalent diagonal
model of some rank r ≤ min( MT,MR)

For the sake of simplicity, let us assume that H is M by M and is of full
rank ; Then
H  UV T * , where  is a diagonal matrix of singular v alues  i andU T *U  I etc

Now the channel can be modeled as
~
Y  U *T [ HX  n]  U *T [UV *T X  n]  (U *T U )V *T X  U *T n]
~
~
 X  n~ , where X  V *T X ; n~  U *T n [Note : var n~i  var ni ]

Equivalent form tells us that an (MT,MR) MIMO channel opens up
m = min (MT,MR) independent SISO channels between the
transmitter andNthe receiver The capacity equation becomes
P
C   log 2 [1   i2  i ];  i  i2

i 1
~ ~
E{ X *T X }  E[ X *T X ]  diag [ Pi ];
N
Wireless Communications
8/1/2006
Pi : Power allocated to the - ith channel; P   Pi
i 1
MIMO-10
MIMO System: A practical approach

A MIMO system schematic is given below:
1
2
R bits/symbol
Channel
coding
Symbol
mapping
SpaceTime
Coding
.
.
MT
Redundancy in time
Coding rate = rc
rs : number of different
symbols N transmitted in T
symbol periods
rs = N/T
Non-redundant
portion of symbols
Space- time redundancy over T
symbol periods
Spatial multiplexing gain = rs
Spectral efficiency = (R*rc info bits/symbol)(rs)(Rs symbols/sec)
w
= Rrcrs bits/s/Hz assuming Rs = w
rs is the parameter of concern : 0 ≤ rs ≤ MT
If rs = MT, - spatial multiplexing mode (max transmission
rate)
If rs ≤ 1, - diversity mode
Wireless Communications
8/1/2006
MIMO-11
MIMO Channel Capacity


Static Channels and Fading Channels
Static Channels





Capacity is given in terms of mutual information b/w channel input vector x
and output vector y as C= max p(x) I(X;Y)=max p(x) [H(Y)-H(Y/X)]
Mutual information of y depends on covariance matrix of y given as
Ry=E [yyH]=HRxHh + IMr
Entropy of y is maximized when Ry is maximized which in turn means that
Rx has to be maximized
[6] shows that Rx is maximized when x is a ZMCSCG random vector
MIMO capacity obtained by maximizing Rx is given as
C=max Rx:Tr(Rx)=p Blog2det[Imr + HRxHH]
Wireless Communications
8/1/2006
MIMO-12
MIMO Channel Capacity
(Static Channels)

Channel Known at Transmitter: Water filling

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


H is known at the transmitter
MIMO capacity with CSIT and CSIR is C=max pi:∑ipi≤pBlog2(1+σi2pi)
p=P/σn2.Expressing capacity in terms of power Pi to i-th parallel
channel
C=max pi:∑ipi≤pBlog2(1+ σi2 P/σn2 ) =max pi:∑ipi≤pBlog2(1+ µi2 Pi/P )
where µi= σi2P/ σn2 ( SNR associated with ith channel at full power)
Solving above gives the water-filling power allocation for the MIMO
channel Pi/P = 1/µ0-1/µi for µi > µ0 and 0 otherwise
The resulting capacity is then C =∑i: µi > µ0 Blog (µi/ µ0)
Thus CSIT converts MIMO channel into non-interfering SISO
channels
Wireless Communications
8/1/2006
MIMO-13
MIMO Channel Capacity (Static Channels) contd..

Channel Unknown at Tx: Uniform Power Allocation
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Tx does not know channel information which means Tx cannot optimize its
power allocation or input covariance structure across antennas
If the distribution of H follows ZMSW distribution (zero mean and identity
covariance matrix) then allocate equal power to each transmit antenna
Input covariance matrix will be scaled identity matrix Rx=p/Mt IMt [2]
I= Blog2 det[IMr + p/Mt HHH]
Using SVD of H, I= ∑i Blog2(1+µi/Mt)
Average mutual information depends on the distribution of the singular
values of H
Wireless Communications
8/1/2006
MIMO-14
MIMO Channel Capacity (Static Channels) contd..
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In a static channel if the average mutual information is not known then the
rate of transmission is unknown.
In such case capacity is given by Outage capacity ( probability that
transmitted data will not be received correctly)
Pout=P(H: Blog2det[IMr + p/Mt HHH] < C)
For ZMSW, according to law of large numbers
Mt -> ∞ 1/Mt HHH =IMr., C= MrBlog2(1+p) or C= MBlog2(1+p) where M=min (Mr,
Mt )
Capacity grows linearly with M for large M [3]
As SNR increases the capacity grows linearly with M [4]
Thus even without CSIT there is a linear growth in capacity
Capacity scales with number of Rx antennas and not Tx antennas [2]
Cost of linear growth of capacity: Demodulation complexity
If not ZMSW, then beamforming can achieve channel capacity
Wireless Communications
8/1/2006
MIMO-15
MIMO Channel Capacity (Fading Channels)

Fading Channels

Channel Gain matrix experiences flat fading (H varies with time)
 Channel known at Tx: Water-filling


Capacity is given in terms of ergodic capacity
Power allocation under ergodic capacity has two possibilities

Short term power constraint (Power associated with each channel realization must equal
average power constraint P )

C=EH [max Rx:Tr(Rx)=p Blog2det[Imr + HRxHH]] = EH [max pi:∑ipi≤pBlog2(1+ µi2 Pi/P ) ]
Long term power constraint (Different powers for different channel realizations)
C=max pH:E(pH)=p EH [ max Rx: Tr(Rx)=pH Blog2 det[IMr + HRxHH]
Short-term power constraint give rise to water-filling in space
Long-term power constraint allows for two-dimensional water-filling across both space
and time

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Wireless Communications
8/1/2006
MIMO-16
MIMO Channel Capacity (Fading Channels)

Channel Unknown at Tx: Ergodic Capacity

Ergodic Capacity: Optimization problem at Tx (Finding optimum input
covariance matrix)
 For scalar channels, Rx=p/Mt IMt
 C=EH[Blog2 det[IMr + p/Mt HHH]]
 Like static channels, ergodic capacity also linearly scales with M for ZMSW
channel
Wireless Communications
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MIMO-17
MIMO Channel Capacity (Fading Channels)


Channel Unknown at Tx: Capacity with Outage
Outage capacity can be improved by not allotting power to one or
more Tx antennas, especially when outage probability is high
Wireless Communications
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MIMO-18
MIMO Channel Capacity
(Fading Channels)

No CSI at the Tx or Rx

Linear growth in capacity as a function of number of antennas disappear
 Capacity depends on the underlying channel model
 Capacity depends on the structure of the fading process[5]
 For general fading process, no multiplexing gain associated with multiple
antennas when there is no Tx or Rx CSI
Wireless Communications
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MIMO-19
MIMO Channel Capacity (Fading Channels)


In the case of flat fading channel ergodic capacity or the outage
capacity is used
Frequency selective fading capacity (more degrees of freedom)

Capacity Calculations are complex
Wireless Communications
8/1/2006
MIMO-20
MIMO APPLICATION: V-BLAST

Vertical Bell Labs Layered Space-Time Architecture (BLAST) is a
transmitter-receiver architecture used to implement multiplexing in
MIMO.

Transmitter: Split data into MT streams  maps to symbols  send. This
process can be considered to be the encoding of the serial data into a
vertical vector.
Coder
Modulator
Data
Source


S/P
Stream 1
Channel
Coder
Modulator
Stream 2
Coder
Modulator
Stream M
Receiver: successive cancellation to recover signals.
This architecture can achieve at most a diversity order of Mr, since
each coded symbol is transmitted from one antenna and received by
Mr antennas.
Wireless Communications
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MIMO-21
D-BLAST

In V-BLAST, there is no coding across these sub-channels: outage
therefore occurs whenever one of these sub-channels is in a deep
fade and cannot support the rate of the stream using that sub-channel.

receiver first estimates x11 obtained without interference  MMSE
estimate of x12 is obtained by suppressing the interference from x21
combine x11 and x12 to decode the x1  x1 is cancelled and the
process restarts with x2

By coding across the sub-channels, D-BLAST can average over the
randomness of the individual sub-channels and get better outage
performance. The D-BLAST scheme suffers from a rate loss because
in the initialization phase some of the antennas have to be kept silent.
Wireless Communications
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MIMO-22
Space-Time Coding

Receiver-diversity: maximum ratio combining & selection
combining
Transmitter- diversity: space-time coding

Space-Time Codes can be designed in two different ways:
(1) Space-Time Block Code or STBC;
(2) Space-Time Trellis Code or STTC.
The first code is the easiest way to achieve Spatial Diversity and
is widely used. The second code is more complex and
expensive nowadays.

STBC is designed such that the vectors transmitted in any pair
of transmitters are orthogonal. The result of this is simple,
linear, optimal decoding at the receiver.
Wireless Communications
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MIMO-23
Space-Time Coding contd..

The Alamouti Space-Time Block Code for 2 Tx antennas

It is the only orthogonal STBC that achieves rate 1. That is to say that
it is the only STBC that can achieve its full diversity gain without
needing to sacrifice its data rate.
For more than two antennas there are several Pseudo-Alamouti Codes.
They can not achieves rate 1.


Some Quasi-Orthogonal STBC improves the data rate but permits
some Inter-Symbol-Interferences (ISI). Despite this, the bit error rate
(BER) is still within the tolerance range. None of these codes are able
to achieve full code rate like Alamouti and the receiver become much
more complex.
Wireless Communications
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MIMO-24
MIMO Standards

Overview of all current MIMO standards and their technologies.

Spatial multiplexing techniques makes the receivers very
complex, and therefore it is typically combined with OFDM,
where the problems created by multi-path channel are handled
efficiently.
Wireless Communications
8/1/2006
MIMO-25
MIMO Standards contd..

The IEEE 802.16e standard incorporates MIMO-OFDMA. The
IEEE 802.11n standard, which is expected to be finalized soon,
recommends MIMO-OFDM. MIMO is also planned to be used in
Mobile radio telephone standards such as recent 3GPP and
3GPP2 standards. In 3GPP, High-Speed Packet Access plus
(HSPA+) and Long Term Evolution (LTE) standards take MIMO
into account. Moreover, to fully support cellular environments
MIMO research consortia including IST-MASCOT propose to
develop advanced MIMO techniques, i.e., multi-user MIMO (MUMIMO).
Wireless Communications
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MIMO-26
MIMO and OFDM

Both of MIMO & OFDM need precoding (a matrix channel into a set of
parallel independent sub-channels). In the OFDM setting, the matrix
channel is given by the circular matrix C, defined by the ISI channel
together with the cyclic prefix added onto the input symbols. In fact,
the decomposition in C  Q 1Q is the SVD decomposition of a circular
matrix C, with U  Q 1 and V = Q.

The important difference between the ISI channel and the MIMO
channel is that, for the former, the U and V matrices (DFTs) do not
depend on the specific realization of the ISI channel while for the latter,
they do depend on the specific realization of the MIMO channel.
Wireless Communications
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MIMO-27
MIMO and OFDM contd..

OFDM is adapted for multi-path propagation in wireless systems. The
length of the OFDM-frames is determined by the Guard Interval (GI).
This Gurad Interval restricts the maximum path delay and therefore
the expansion of the network area. MIMO also uses the multi-path
propagation.

OFDM is a wideband system with many narrowband sub-carriers. The
mathematical MIMO channel model (see chapter 2) is based on a
narrow band non-frequency selective channel. The latter is supported
by OFDM as well. Fading effects in wideband systems normally occur
only at particular frequencies and interfere with few sub-carriers. The
data is spread over all carriers, so that only a small amount of bits get
lost, and these can be repaired by a forward error correction (FEC).
OFDM provides a robust multi-path system suitable for MIMO. At the
same time OFDM provides high spectral efficiency and a degree of
freedom in spreading the time dimension of Space-Time Block Codes
over several sub-carriers. This results in a stronger system based on
the principle described previously.
Wireless Communications
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MIMO-28
MIMO-OFDM
From: Tetsushi Abe, Takahiro Asai and Hirohito Suda, “A Practical Throughput Comparison
of MIMO-CDMA and MIMO-OFDM”, Vehicular Technology Conference, 2004.
Wireless Communications
8/1/2006
MIMO-29
MIMO Capacity: References
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[1] Andrea Goldsmith “Wireless Communication” text book
[2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T-Bell Labs Internal
Memo., pp. 585–595,June 1995.
[3] A.L. Moustakas, S.H.Simon, A.M. Sengupta, “MIMO capacity through correlated
channels in the presence of correlated interferers and noise: a (not so) large N analysis,”
IEEE Trans. Inform. Theory, vol. 48, pp. 2545
- 2561, Oct. 2003.
[4] G. J. Foschini, “Layered space-time architecture for wireless communication in fading
environments when using multi-element antennas,” Bell Labs Techn. J., pp. 41–59,
Autumn 1996.
[5] A. Lapidoth and S. Moser, “On the fading number of multi-antenna systems over flat
fading channels with memory and incomplete side information,” Proc. Intl. Symp. Inform.
Theory, p. 478, July 2002.
[6] E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. on
Telecomm. ETT, vol. 10,pp. 585–596, Nov. 1999.
[7] Goldsmith. A, Jafar S.A, Jindal N, Vishwanathan.S “Capacity limits of MIMO
channels", Selected Areas in Communication IEEE Journal, vol. 21, pp 684-702, June
2003
[8] Prof.Sam Shanmugan’s charts on MIMO and MUD
[9] R. U. Nabar A. J. Paulraj, D. A. Gore and H. B¨olcskei, “An overview of MIMO
communications—a key to gigabit wireless,” Proceedings of the IEEE, vol. 92, no. 2, pp.
198–218, Feb. 2004.
Wireless Communications
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MIMO-30