MIMO CHANNEL CAPACITY
Ochi Laboratory
Contents
Introduction
Review of information theory
Fixed MIMO channel
Fading MIMO channel
Summary and Conclusions
1. Introduction
The use of multiple antennas can provide gain due to
Antenna gain
More receive antenna
more power is harvested
Interference gain
Interference nulling by beam-forming (array gain)
Interference averaging (to zero) due to independent observation
Diversity again against fading
Receive diversity
Transmit diversity
Information theoretic model of MIMO channel is
consider
MIMO channel model
Assume
transmit and
receive antennae
Called
⨯
MIMO
system
Fading radio channels
modeled as freq-flat:
Fixed
Time-varying
Know both/either in the
transmitter and/or receiver
Perfect
channel state
information (CSI)
A priori unknown
2. Review of information theory
Information theory (IT) has its origins in
analyzing the limits communication.
Information theory answers two fundamental
questions in communication theory:
What is the ultimate data compression rate?
Answer: entropy.
What is the ultimate data transmission rate?
The entropy of a binary variable as a
function of the probability
=
( )
Basic concepts
Assume a discrete valued
random variable
(RV) X with probability
mass function p(x)
The average information or
entropy
of RV X:
= −
log
= −
log
(∗)
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Note
: Because information is measured in bit, the logarithm function here is base-2
Function E(x) is the expected value of variable X
Measure the expected uncertainty
in RV X
Approximately how much
information we learn on average
from one instance of the RV X
How many bits are needed, on the
average, to convey the information
obtain in RV X
Basic concepts
Joint entropy of RV’s X and Y
,
= −
log ( , )
Measuring how much uncertainty in
the two RV X and Y taken together
Conditional entropy of RV Y
given X=x
=
=
Measuring of how much uncertainty
remains about the RV Y when know
RV X
Chain rule:
,
=
+
Note: because information is measured in bit, the logarithm function here is base-2
Mutual information:
is the relative entropy between the joint
distribution and product distribution:
Measuring the mutual independence of
two RVs
Channel capacity
Shannon proved that reliable (virtual error-free)
communication is possible at rates C up to:
= max
( )
( ; )
The distribution
∗
( )
that C achieves the
maximum is called the optimal input distribution
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Gaussian channel
Mutual Information
- P =
is the power
constraint [J/symbol]
-
=
is the noise
variance
Gaussian
Channel
Capacity
(bit per
transmission)
Gaussian band-limited channel
Common model for communication
over a radio network or a telephone line
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W
-W
Assume noise power-spectral density is
⟹
Noise power:
⨯ 2
=
Energy per sample of T second:
=
⇨
Channel Capacity
There are
2W
samples per second
⇨
W→∞
Example:
Telephone channel, W=3.3KHz, if
=
= 40
= 10
Parallel Gaussian channels
Capacity:
=
1
2
log 1 +
Optimal transmission:
~
0,
,
, … ,
,
⇨ water-filling (do not
3. Fixed MIMO Gaussian channel
Signal
( )
is transmitted at
time interval
n
from antenna
( = 1,2, … ,
)
Signal
( )
is received at time
interval
n
from antenna
j
( = 1,2, … ,
)
=
ℎ
+
( )
Where
ℎ ( )
is the complex
channel gain with
ℎ ( )
= 1
Matrix formulation MIMO
channel
The signal received at all antennae
=
+
(1)
where:
=
( ) …
( )
∈
=
( ) …
( )
∈
=
ℎ ( )
⋯
ℎ
( )
⋮
⋱
⋮
ℎ
( ) ⋯
ℎ
( )
∈
×
=
( ) …
( )
∈
Noise and power constraint
The noise vector
=
( )
…
( )
∈
With
( )~
(0,
)
The transmitted signal satisfied the average power
constraint:
( ) =
( )
≤
Since the noise power is normalized to unity, we
commonly refer to the power constraint
P as the SNR
Singular value decomposition
The MIMO model is a special case of parallel
Gaussian channels
For every
∈
×
, we can write as
=
∧
Where
∈
×
,
V ∈
×
are unitary matrices and
∧
∈
×
Equivalent channel model
=
,
=
,
=
Since U and V are unitary matrices the channel model
=
+
(1)
⟹ Equivalent channel model
=
+
( )
∧
⁄
is diagonal matrix of size
×
, we have
decomposed the correlated parallel channels into
independent parallel channels
Equivalent channel model
⟹
Independent parallel Gaussian channels
∧
∧
Derivation of channel capacity
The rank of matrix H is
≤ min
,
The number of positive singular values is
rank(H)
The capacity of MIMO AWGN channel:
=
log 1 +
( )
,
=
MIMO channel capacity for full-rank
channel matrix
No CSI at the transmitter (and full-rank H)
= log det
+
CSI at the transmitter (and full-rank H)
= max log det
+
Where Q is the covariance matrix of the input vector
x satisfying the power constraint
≤
MIMO channel characteristics
Number of antennae vs.
capacity of the channel
MIMO channel capacity vs.
SNR
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Ref. form
http://www.mathworks.com/matlabcentral/fx_files/30588/1/untitled.jpg
Ref. form
http://ars.els-cdn.com/content/image/1-s2.0-S0166531609001096-gr10.jpg
4. Fading MIMO channels
The channels are usually assumed to be ergodic
Fading is fast enough and gets all realizations so many
time that
The sample average equals the theoretical mean
Fading channel mode with perfect receiver CSI
Assuming that the channel is memoryless
(independent channel state for each
transmission), the capacity equals the mean of
the mutual information
=
log det
+
Non-ergodic channels
The channels are not always ergodic: fading can be slow that it
undergoes only some realizations.
⟹ this random process becomes non-ergodic
In no-ergodic channel,
the channel capacity ≠
the average
maximum mutual information
⟹ to measure the capacity of this channel:
using probability of
5. Summary and conclusion
AWGA MIMO channels are an extension of
parallel Gaussian channels
Parallel channels: channels on different
frequencies
The linear capacity increase becomes natural
= log det
+
Fading AWGN MIMO channel
Ergodic channels:
Channel experiences all its states several times
No delay constraints and/or fast fading
Capacity equals the average mutual information:
=
log det
+
Capacity increases linearly with
=
Non-ergodic channels
Capacity does not equal the average mutual information
Capacity versus outage probability is applied to measure
the non-ergodic channels capacity
Expected value
The expected value (or expectation,
mathematical expectation, EV, mean,
the first moment) of random variable is
the weighted average of all possible
values that this random variable can
take on.
For example: Let
X
represent the outcome
of a roll of a six-sided
die
. More
specifically,
X
will be the number
of
pips
showing on the top face of
the
die
after the toss. The possible values
for
X
are 1, 2, 3, 4, 5, 6, all equally likely
(each having the probability of 1/6). The
expectation of
X
is: (1)
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