15/11/02
Capacity of multiple-input multiple-output
(MIMO) systems in wireless communications
Bengt Holter
Department of Telecommunications
Norwegian University of Science and Technology
NTNU
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Outline
15/11/02
• Introduction
• Channel capacity
– Single-Input Single-Output (SISO)
– Single-Input Multiple-Output (SIMO)
– Multiple-Input Multiple-Output (MIMO)
– MIMO capacity employing space-time block coding (STBC)
• Outage capacity
– SISO
– SIMO
– MIMO employing STBC
• Summary
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Introduction
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MIMO = Multiple-Input Multiple-Output
• Initial MIMO papers
– I. Telatar, ”Capacity of multi-antenna gaussian channels,” AT&T Technical
Memorandum, jun. 1995
– G. J. Foschini, ”Layered space-time architecture for wireless communication in
a fading environment when using multi-element antennas”, Bell Labs Technical
Journal, 1996
• MIMO systems are used to (dramatically) increase the capacity and
quality of a wireless transmission.
• Increased capacity obtained with spatial multiplexing of transmitted
data.
• Increased quality obtained by using space-time coding at the transmitter.
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Entropy
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• For a discrete random variable X with alphabet X and distributed
according to the probability mass function p(x), the entropy is defined
as
1
1
H(X) =
log2
log2 p(x) · p(x) = E log2
.
· p(x) = −
p(x)
p(x)
x∈X
x∈X
(1)
• The entropy of a random variable is a measure of the uncertainty of
the random variable; it is a measure of the amount of information
required on the average to describe the random variable.
• With a base 2 logarithm, entropy is measured in bits.
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Gamma distribution
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• X follows a gamma distribution with shape parameter α > 0 and scale
parameter β > 0 when the probability density function (PDF) of X is
given by
xα−1 e−x/β
fX (x) =
.
(2)
β α Γ(α)
∞
where Γ(·) is the gamma function (Γ(α) = 0 e−t tα−1dt [(α) > 0]).
• The short hand notation X ∼ G(α, β) is used to denote that X follows
a gamma distribution with shape parameter α and scale parameter β.
• Mean: E{X} = µx = α · β.
• Variance: E{X 2 } = σx2 − µ2x = α · β 2.
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SISO
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SISO = Single-input Single-output
• Representing the input and and output of a memoryless wireless channel with the random variables X and Y respectively, the channel capacity is defined as
C = max I(X; Y ).
(3)
p(x)
• I(X; Y ) denotes the mutual information between X and Y and it
is measure of the amount of information that one random variable
contains about another random variable.
• According to the definition in (3), the mutual information is maximized
with respect to all possible transmitter statistical distributions p(x).
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SISO cont’d
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• The mutual information between X and Y can also be written as
I(X; Y ) = H(Y ) − H(Y |X).
(4)
• From the equation above, it can be seen that mutual information
can be described as the reduction in the uncertainty of one random
variable due to the knowledge of the other.
• The mutual information between X and Y will depend on the properties of the wireless channel used to convey information from the
transmitter to the receiver.
• For a SISO flat fading wireless channel, the input/output relations
(per channel use) can be modelled by the complex baseband notation
y = hx + n
(5)
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SISO cont’d
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• y represents a single realization of the random variable Y (per channel
use).
• h represents the complex channel between the transmitter and the
receiver.
• x represents the transmitted complex symbol.
• n represents complex additive white gaussian noise (AWGN).
• Note that in previous lectures by Prof. Alouini, the channel gain |h|
was denoted α. In this presentation, α is used as the shape parameter
of a gamma distributed random variable.
• Based on different communication scenarios, |h| may be modelled by
various statistical distributions.
• Common multipath fading models are Rayleigh, Nakagami-q (Hoyt),
Nakagami-n (Rice), and Nakagami-m.
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SISO cont’d
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Capacity with a transmit power constraint
• With an average transmit power constraint PT , the channel capacity
is defined as
C=
max
p(x):P ≤PT
I(X; Y ).
(6)
• If each symbol per channel use at the transmitter is denoted by x, the
average power constraint can be expressed as P = E{|x|2 } ≤ PT .
• Compared to the original definition in (3), the capacity of the channel
is now defined as the maximum of the mutual information between the
input random variable X and the output random variable Y over all
statistical distributions on the input that satisfy the power constraint.
• Since both x and y are continuous upon transmission and reception,
the channel is modelled as an amplitude continuous but time discrete
channel.
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SISO cont’d
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Assumptions
• Perfect channel knowledge at the receiver.
• X is independent of N .
• N ∼ N (0, σn2)
Mutual information: With hd (·) denoting differential entropy (entropy of
a continuous random variable), the mutual information may be expressed
as
I(X; Y ) =
=
=
=
hd (Y ) − hd (Y |X)
hd (Y ) − hd (hX + N |X)
hd (Y ) − hd (N |X)
hd (Y ) − hd (N )
(7)
(8)
(9)
(10)
• (9) follows from the fact that since h is assumed perfectly known by
the receiver, there is no uncertainty in hX conditioned on X.
• (10) follows from the fact that N is assumed independent of X, i.e.,
there is no information in X which reduces the uncertainty of N .
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SISO cont’d
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Noise differential entropy
• Since N already is assumed to be a complex gaussian random variable,
i.e., the noise PDF is given by
1 − nσ22
e n
fN (n) =
2
πσn
(11)
• Differential entropy
hd (N ) = −
fN (n) log2 fN (n)dn
(12)
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SISO cont’d
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• Inserting the noise PDF into (12)
2
2
n log2 e
hd (N ) = − fN (n) −
− log2 πσn dn
σn2
log2 e
2
2
=
n
f
(n)dn
+
log
πσ
f (n)dn
2
n
σn2
2
E N2
=
log
e
+
log
2
2 πσn
σn2
= log2 e + log2 πσn2
2
= log2 πeσn ,
where E{N 2} = σn2.
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SISO cont’d
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Received signal power
• Since hd (N ) is given, the mutual information I(X; Y ) = hd (Y ) − hd (N )
is maximized by maximizing hd (Y ).
• Since the normal distribution maximizes the entropy over all distributions with the same covariance, I(X; Y ) is maximized when Y is
assumed gaussian, i.e., hd (Y ) = log2 (πeσy2), where E{Y 2} = σy2.
• Assuming the optimal gaussian distribution for X, the received average
signal power σy2 may be expressed as
E{Y 2} = E{(hX + N )(h∗X ∗ + N ∗)}
= σx2|h|2 + σn2.
(13)
(14)
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SISO cont’d
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SISO fading channel capacity
C
= hd (Y ) − hd (N )
= log2 (πe(σx2|h|2 + σn2)) − log2(πeσn2)
σx2 2
= log2 1 + 2 |h|
σn
PT 2
= log2 1 + 2 |h| ,
σn
(15)
(16)
(17)
(18)
where it is assumed that σx2 = PT .
• Denoting the total received signal-to-noise ratio (SNR) γt =
the SISO fadig channel capacity is given by
PT
|h|2,
σn2
C = log2 (1 + γt)
• Note that since γt is a random variable, the capacity also becomes a
random variable.
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SISO cont’d
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Nakagami-m fading ⇒ Gamma distributed SNR
• With the assumption that the fading amplitude |h| is a Nakagami-m
distributed random variable, the PDF is given by
2mm |h|2m−1
m|h|2
fα(α) =
(19)
exp
Ωm Γ(m)
Ω
where Ω = E{|h|2} and m is the Nakagami-m fading parameter which
ranges from 1/2 (half Gaussian model) to ∞ (AWGN channel).
• Using transformation of random variables, it can be shown that the
overall received SNR γt is a gamma distributed random variable G(α, β),
γtm−1 e−γt/β
,
fγt (γt ) = m
β Γ(m)
γt
where α = m and β = γ t/m. In short γt ∼ G m, m where γ t =
(20)
PT Ω
.
σn2
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SISO cont’d
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Ergodic channel capacity of SISO channel with Rayleigh fading
7
Capacity [bit/s/Hz]
6
5
4
3
2
1
0
0
3
6
9
12
SNR [dB]
15
18
21
24
Ergodic capacity of a Rayleigh fading SISO channel (dotted line) compared to the
Shannon capacity of a SISO channel (solid line)
3dB increase in SNR ⇒ 1 bit/s/Hz capacity increase
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SIMO
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SIMO = Single-Input Multiple-Output
• For a SIMO flat fading wireless channel, the input/output relations
(per channel use) can be modelled by the complex baseband notation
y = hx + n
(21)
• y represents a single realization of the multivariate random variable Y
(array repsonse per channel use).
• h represents the complex channel vector between a single transmit
antenna and nR receive antennas, i.e., h = [h11, h21 , . . . , hnR 1 ]T.
• x represents the transmitted complex symbol per channel use.
• n represents a complex additive white gaussian noise (AWGN) vector.
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SIMO
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Mutual information
• With hd (·) denoting differential entropy (entropy of a continuous random variable), the mutual information may be expressed as
I(X; Y) =
=
=
=
hd (Y) − hd (Y|X)
hd (Y) − hd (hX + N|X)
hd (Y) − hd (N|X)
hd (Y) − hd (N)
(22)
(23)
(24)
(25)
• It will be assumed that N ∼ N (0, Kn ), where Kn = E{NNH } is the
noise covariance matrix.
• Since the normal distribution maxmizes the entropy over all distributions with the same covariance (i.e. the power constraint), the mutual
information is maximized when Y represents a multivariate Gaussian
random variable, i.e., Y = N (0, Ky ) where Ky = E{YYH} is the covariance matrix of the desired signal.
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SIMO cont’d
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Desired signal covariance matrix
• For a complex gaussian vector Y, the differential entropy is less than
or equal to log2 det(πeKy ), with equality if and only if y is a circularly
symmetric complex Gaussian with E{YYH} = Ky .
• With the assumption that the signal X is uncorrelated with all elements
in N, the received covariance matrix Ky may be expressed as
E{YYH} = E{(hX + N)(hX + N)H}
= σx2hhH + Kn
(26)
(27)
where σx2 = E{X 2 }.
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SIMO cont’d
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SIMO fading channel capacity
C
hd (Y) − hd (N)
log2[det(πe(σx2hhH + Kn ))] − log2[det(πeKn)]
log2[det(σx2hhH + Kn)] − log2 [det Kn ]
log2[det((σx2hhH + Kn)(Kn )−1 )]
log2[det(σx2hhH(Kn)−1 + InR )]
log2[det(InR + σx2(Kn)−1 hHh)]
PT
1 + 2 ||h||2 · det(InR )
= log2
σn
PT
= log2 1 + 2 ||h||2
σn
=
=
=
=
=
=
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
where it is assumed that Kn = σn2InR and σx2 = PT .
• Note that for the SISO fading channel, Kn = σn2.
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SIMO cont’d
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• The capacity formula for the SIMO fading channel could also have
been found by assuming maximum ratio combining at the receiver.
• With perfect channel knowledge at the receiver, the optimal weights
are given by
wopt = (Kn )−1 h.
(36)
• Using these weights together with the assumption that Kn = σn2InR ,
the overall (instantaneous) SNR γt for the current observed channel h
is equal to
γt =
PT
||h||2.
2
σn
(37)
• Thus, since γt in this case represents the maximum available SNR, the
capacity can be written as
C = log2 (1 + γt ) = log2 (1 +
PT
2
||
h
||
).
σn2
(38)
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SIMO cont’d
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Nakagami-m fading ⇒ Gamma distributed SNR
• With the assumption that all channel gains in the channel vector h are
independent and indentically distributed (i.i.d.) Nakagami-m random
variables (i.e. ml = m), then the overall SNR γt is a gamma distributed
random variable with shape parameter α = nR · m and scale parameter
β = γ l /m)
• In short, γt ∼ G(nR · m, γ l /m).
• γ l represents the average SNR per receiver branch (assumed equal for
all branches in this case)
• Coefficient of variation τ =
σγt
µ γt
=
√ 1 .
nR ·m
• Effective diversity order [Nabar,02]: Ndiv =
1
τ2
= nR · m.
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MIMO
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MIMO = Multiple-Input Multiple-Output
• For a MIMO flat fading wireless channel, the input/output relations
(per channel use) can be modelled by the complex baseband notation
y = Hx + n
(39)
• x is the (nT × 1) transmit vector.
• y is the (nR × 1) (array response) receive vector.
• H is the (nR × nT ) channel matrix.
• n is the (nR × 1) additive white Gaussian noise (AWGN) vector.

h11
 h21
H=
 ...
hnR 1
···
···
...
···

h1nT
h2nT 
... 

hnR nT
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MIMO cont’d
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Mutual information
• With hd (·) denoting differential entropy (entropy of a continuous random variable), the mutual information may be expressed as
I(X; Y) =
=
=
=
hd (Y) − hd (Y|X)
hd (Y) − hd (HX + N|X)
hd (Y) − hd (N|X)
hd (Y) − hd (N)
(40)
(41)
(42)
(43)
• Assuming N ∼ N (0, Kn ).
• Since the normal distribution maxmizes the entropy over all distributions with the same covariance (i.e. the power constraint), the mutual
information is maxmized when Y represents a multivariate Gaussian
random variable.
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MIMO cont’d
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Desired signal covariance matrix
• With the assumption that X and N are uncorrelated, the received
covariance matrix Ky may be expressed as
E{YYH } = E{(HX + N)(HX + N)H}
= HKx HH + Kn
(44)
(45)
where Kx = E{XXH}.
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MIMO cont’d
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MIMO fading channel capacity
C
=
=
=
=
=
=
hd (Y) − hd (N)
log2 [det(πe(HKx HH + Kn ))] − log2[det(πeKn)]
log2 [det(HKx HH + Kn)] − log2 [det Kn ]
log2 [det((HKx HH + Kn)(Kn )−1 )]
log2 [det(HKx HH (Kn)−1 + InR )]
log2 [det(InR + (Kn)−1 HKx HH )]
(46)
(47)
(48)
(49)
(50)
(51)
• When the transmitter has no knowledge of the channel, it is optimal to
evenly distribute the available power PT among the transmit antennas,
i.e., Kx = PnTT InT .
• Assuming that the noise is uncorrelated between branches, the noise
covariance matrix Kn = σn2InR .
• The MIMO fading channel capacity can then be written as
C = log2
PT
H
det InR +
HH
.
nT σn2
(52)
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MIMO cont’d
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• By the law of large numbers, the term n1T HHH ⇒ InR as nT gets large
and nR is fixed. Thus the capacity in the limit of large nT is
PT
(53)
C = nR · log2 1 + 2
σn
SISO capacity
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MIMO cont’d
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• Further analysis of the MIMO channel capacity is possible by diagonalizing the product matrix HHH either by eigenvalue decomposition
or singular value decomposition.
• Eigenvalue decomposition of the matrix product HHH = EΛEH :
PT
(54)
C = log2 det InR + 2 EΛEH
σ n nT
where E is the eigenvector matrix with orthonormal columns and Λ is
a diagonal matrix with the eigenvalues on the main diagonal.
• Singular value decomposition of the channel matrix H = UΣVH :
PT
(55)
C = log2 det InR + 2 UΣΣH UH
σ n nT
where U and V are unitary matrices of left and right singular vectors
respectively, and Σ is a diagonal matrix with singular values on the
main diagonal.
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MIMO cont’d
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• Using the singular value decomposition approach, the capacity can
now be expressed as
PT
C = log2 det InR + 2 UΣΣH UH
(56)
σ n nT
PT
(57)
= log2 det InT + 2 UHUΣ2
σ n nT
PT
= log2 det InT + 2 Σ2
(58)
σ n nT
PT 2
PT 2
PT 2
= log2
1 + 2 σ1
1 + 2 σ2 · · · 1 + 2 σk
(59)
σ n nT
σ n nT
σ n nT
k PT 2
=
1 + 2 σi
(60)
σ
n
T
n
i=1
where k = rank{H} ≤ min{nT , nR }, Σ is a real matrix, and det(IAB +
AB) = det(IBA + BA)
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MIMO cont’d
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• Using the same approach with an eigenvalue decomposition of the
matrix product HHH, the capacity can also be expressed as
k PT
C=
1 + 2 λi
(61)
σ n nT
i=1
where λi are the eigenvalues of the matrix Λ.
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MIMO cont’d
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Ergodic channel capacity of a MIMO fading channel
66
60
Capacity [bit/s/Hz]
54
48
42
36
30
24
18
12
6
0
−9
−6
−3
0
3
6
9
12 15 18
SNR [dB]
21
24
27
30
33
36
39
The Shannon capacity of a SISO channel (dotted line) compared to the ergodic
capacity of a Rayleigh fading MIMO channel (solid line) with nT = nR = 6
3dB increase in SNR ⇒ 6 bits/s/Hz capacity increase!
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MIMO with STBC
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Transmit diversity
• Antenna diversity techniques are commonly utilized at the base stations due to less constraints on both antenna space and power. In
addition, it is more economical to add more complex equipment to
the base stations rather than at the remote units.
• To increase the quality of the transmission and reduce multipath fading
at the remote unit, it would be beneficial if space diversity also could
be utilized at the remote units.
• In 1998, S. M. Alamouti published a paper entitled ”A simple transmit
diversity technique for wireless communications”. This paper showed
that it was possible to generate the same diversity order traditionally obtained with SIMO system with a Multiple-Input Single-Output
(MISO) system.
• The generalized transmission scheme introduced by Alamouti has later
been known as Space-Time Block Codes (STBC).
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MIMO with STBC
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Alamouti STBC
• With the Alamouti space-time code [Alamouti,1998], two consecutive
symbols {s0, s1} are mapped into a matrix codeword S according to
the following mapping:
s1 s2
S=
,
(62)
−s∗2 s∗1
• The individual rows represent time diversity and the individual columns
space (antenna) diversity.
• Assuming a block fading model, i.e., the channel remains constant
for at least T channel uses, the received signal vector x (array response/per channel use) may be expressed as
xk = Hsk + nk , k = 1, . . . , T.
(63)
[Alamouti,1998] S. M. Alamouti, ”A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Comm., Vol.16, No.8, October 1998
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MIMO with STBC
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xk = Hsk + nk , k = 1, . . . , T.
• xk ∈ C nR denotes the received signal vector per channel use.
• sk ∈ C nT denotes the transmitted signal vector (a single row from the
matrix codeword S transposed into a column vector).
• H ∈ C nR ×nT denotes the channel matrix with (possibly correlated) zeromean complex Gaussian random variable entries.
• nk ∈ C nR denotes the additive white Gaussian noise where each entry
of the vector is a zero-mean complex Gaussian random variable.
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MIMO with STBC
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• For T consecutive uses of the channel, the received signal may be
expressed as
X = HS + N,
(64)
where X = [x1, x2, · · · , xT ] (T consecutive array responses ⇒ time responses in nR branches), S = [s1, s2, · · · , sT ], and N = [n1, n2, · · · , nT ].
• For notational simplicity [Hassibi,2001], the already introduced matrices X, S, and N may be redefined as X = [x1, x2 , · · · , xT ]T, S =
[s1, s2, · · · , sT ]T, and N = [n1, n2, · · · , nT ]T.
• With this new definition of the matrices X, S, and N, time runs vertically and space runs horizontally and the received signal for T channel
uses may now be expressed as
X = SHT + N.
(65)
[Hassibi,2001] B. Hassibi, B. M. Hochwald, ”High-rate codes that are linear in space and
time,” 2001
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MIMO with STBC
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• In [Hassibi,2001], the transpose notation on H is omitted and H is just
redefined to have dimension nT × nR .
• For a 2 × 2 MIMO channel, equation (65) becomes [Hassibi,2001]
x11 x12
s1
s2
h11 h12
n11 n12
=
+
(66)
x21 x22
−s∗2 −s1
h21 h22
n21 n22
• x11 and x12 represent the received symbols at antenna element no.1
and 2 at time index t and likewise x21 and x22 represent the received
symbols at antenna element no.1 and 2 at time index t + Ts
• This can be reorganized [Alamouti,1998] and written as






x11
h11 h21
n
11
∗
∗
∗
 x21 
 h21 −h11  s1
 n∗21 
+
 x = h
h22  s2
n12 
12
12
x∗22
h∗22 −h∗12 n∗22
s
x
H
(67)
n
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MIMO with STBC
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• With matched filtering at the receiver (perfect channel knowledge):
y = HH x
= HH Hs + HH n
= ||H||2F s + HHn.
(68)
where ||H||2F represents the squared Frobenius norm of the matrix H.
H H =
H
=

h∗11
h∗21
|h11
h21 h∗12
−h11 h∗22
|2
+ |h12
|2
h22
−h12
h11
 h∗21
 h
12
h∗22
|2
|2
+ |h21
0
+ |h22

h21
−h∗11 
h22 
−h∗12
|h11
|2
+ |h12
|2
0
+ |h21 |2 + |h22|2
= ||H||2F · I2 .
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MIMO with STBC
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• This means that the received signals after matched filtering are decoupled and they can be written individually as
y1
y2
= ||H||2F s1 + HHn
= ||H||2F s2 + HHn
(69)
(70)
• In general, the effective channel induced by space-time block coding of
complex symbols (before detection) can be represented as [Sandhu,2000]
yk = ||H||2F sk + HHn.
(71)
[Sandhu,2000] S. Sandhu, A. Paulraj, ”Space-Time Block Codes: A Capacity Perspective,” IEEE Comm. Letter, Vol.4, No.12, December 2000.
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MIMO with STBC
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• The overall SNR before detection of each symbol is equal to
γtmimo
where P =
||H||4F PnTT
||H||4F |sk |2
2
=
=
P
||
H
||
=
F.
E{|HHn|2}
||H||2F σn2
(72)
PT
.
σn2 nT
• For each transmitted symbol, the effective channel is a scaled AWGN
channel with SNR= P ||H||2F .
• The capacity of a MIMO fading channel using STBC can then be
written as
PT
K
(73)
· log2 1 + 2 ||H||2F .
C=
T
σ n nT
where
K
T
in front of the equation denotes the rate of the STBC.
• With the Alamouti STBC, two symbols (K = 2) are transmitted in
two time slots (T = 2), i.e., the Alamouti code is a full rate STBC.
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MIMO with STBC
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• Assuming uncorrelated channels and that all channel envelopes are
i.i.d. Nakagami-m distributed random variables with equal average
power E{|hij |2} = Ω, the overall SNR may be expressed as a gamma
distributed random variable:
PT
· ||H||2F
2
nT σ n
(74)
||H||2F ∼ G(nT · nR , Ω)
(75)
γtmimo ∼ G(N · m, γ l /m)
(76)
γtmimo =
where N = nT · nR and γ l =
PT Ω
.
σn2 nT
• Effective diversity order Ndiv =
1
τ2
= N · m.
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MIMO with STBC
15/11/02
Capacity summary
• Note that the capacity formulas given below are obtained with the
assumption of an average power constraint PT at the transmitter, uncorrelated equal noise power σn2 in all branches, perfect channel knowledge at the receiver and no channel knowledge at the transmitter.
• SISO: C = log2 1 +
PT
|h|2
σn2
• SIMO: C = log2 1 +
.
PT
||h||2
σn2
• MIMO: C = log2 InR +
.
PT
HHH
σn2 nT
.
• MIMO with STBC: C = log2 1 +
PT
||H||2F
σn2 nT
.
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MIMO with STBC
15/11/02
STBC - a capacity perspective
• STBC arec useful since they are able to provide full diversity over the
coherent, flat-fading channel.
• In addition, they require simple encoding and decoding.
• Although STBC provide full diversity at a low computational cost, it
can be shown that they incur a loss in capacity because they convert
the matrix channel into a scalar AWGN channel whose capacity is
smaller than the true channel capacity.
S. Sandu, A. Paulraj,”Sapce-time block codes: A capacity perspective,” IEEE Communications Letters, Vol.4, No.12, December 2000.
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MIMO with STBC
15/11/02
PT
C = log2 InR + 2 HHH
σ n nT
k PT 2
= log2
1 + 2 σi
σ n nT
i=1


k
<i
<i
<i
i
i
k
1
2
1
2
3
2
2
2
2
3
2
2
2
k
= log2 1 + P
σi + P
σi1 σi2 + P
σi1 σi2 σi3 + · · · + P
σi2

i=1
= log2 1 + P ||H||2F + P 2
2
≥ log2 1 + P ||H||F
K
≥
· log2 1 + P ||H||2F
T
i1 =i2
i1 =i2 =i3
i
1 <i2
i1 <i
2 <i3
i1 =i2
σi21 σi22 + P 3
i=1
σi21 σi22 σi23 + · · · + P k
i1 =i2 =i3
k
σi2
i=1
• The capacity difference is a function of the channel singular values.
This can used to determine under which conditions STBC is optimal
in terms of capacity.
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
MIMO with STBC
15/11/02
• When the channel matrix is a rank one matrix, there is only a single
non-zero singular value, i.e., a space-time block code is optimal (with
respect to capacity) when it is rate one (K = T ) and it is used over
a channel of rank one [Sandhu,2000].
• For the i.i.d. Rayleigh channel with nR > 1, the rank of the channel
matrix is greater than one, thus a space- time block code of any rate
used over the i.i.d. Rayleigh channel with multiple receive antennas
always incurs a loss in capacity.
• A full rate space-time block code used over any channel with one
receive antenna is always optimal with respect to capacity.
• Essentially, STBC trades off capacity benefits for low complexity encoding and decoding.
• Note that with spatial multiplexing, the simplification is opposite of
STBC. It trades of diversity benefits for lower complexity.
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Outage capacity
15/11/02
Outage capacity
• Defined as the probability that the instantaneous capacity falls below
a certain threshold or target capacity Cth
Cth
Pout (Cth ) = Prob[C ≤ Cth ] =
fC (C)dC = PC (Cth)
(77)
0
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Outage capacity - SISO
SISO capacity
C = log2
15/11/02
PT
1 + 2 · |h|2
σn
= log2 1 + γtsiso .
(78)
• Assuming that |h| is Nakagami-m distributed random variable,
• γtsiso is a Gamma distributed random variable with shape parameter
α = m and scale parameter β = γ l /m.
}=
• γ l = E{γtsiso } = E{ PTσ|h|
2
2
n
PT Ω
.
σn2
• E{|h|2 } = Ω.
• γtsiso ∼ G(m, γ l /m).
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Outage capacity - SISO
15/11/02
Transformation of random variables
• Let X and Y be continuous random variables with Y = g(X). Suppose g is one-to-one, and both g and its inverse function, g −1, are
continuously differentiable. Then
−1
dg
(y)
−1
.
fY (y) = fX [g (y)] (79)
dy • Let C = g(γtsiso) = log2(1 + γtsiso ).
• Then γtsiso = g −1(C) = 2C − 1.
• Capacity PDF
(2C − 1)m−1 e−(2
fC (C) = fγtsiso (2 − 1) · 2 ln 2 =
β m Γ(m)
C
C
C
−1)/β
· 2C ln 2
(80)
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Outage capacity - SISO
15/11/02
• The SISO outage capacity can be obtained by solving the integral
Cth C
C
(2 − 1)m−1 e−(2 −1)/β
C
Pout(Cth ) =
ln 2 · dC
(81)
·
2
m Γ(m)
β
0
(2Cth − 1)m
= 1 − Q m,
(82)
γl
• Q(·, ·) is the normalized complementary incomplete gamma function
defined as
Q(a, b) =
• Γ(a, b) =
∞
b
Γ(a, b)
Γ(a)
(83)
e−t ta−1 dt.
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Outage capacity - SIMO
15/11/02
SIMO capacity
PT
C = log2 1 + 2 · ||h||2 = log2 1 + γtsimo .
σn
(84)
• Assuming that every channel gain in the vector h, |hl |, is a Nakagami-m
distributed random variable with the same m parameter.
• γtsimo is a Gamma distributed random variable with shape parameter
α = nR · m and scale parameter β = γ l /m.
• γtsimo ∼ G(nR · m, γ l /m).
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Outage capacity - SIMO
15/11/02
Transformation of random variables
• Let C = g(γtsimo) = log2(1 + γtsimo).
• Then γtsimo = g −1 (C) = 2C − 1.
• Capacity PDF
(2C − 1)nR m−1e−(2 −1)/β
C
·
2
ln 2 (85)
fC (C) = fγtsimo (2 − 1) · 2 ln 2 =
β nR ·m Γ(nR · m)
C
C
C
• The SIMO outage capacity can be obtained by solving the integral
Cth C
C
(2 − 1)nR m−1e−(2 −1)/β
C
Pout (Cth) =
ln 2 · dC
(86)
·
2
nR ·m Γ(n · m)
β
R
0
(2Cth − 1)m
= 1 − Q nR · m,
(87)
γl
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Outage capacity - MIMO with STBC
15/11/02
MIMO with STBC
K
PT
K
2
mimo
C=
.
log2 1 + 2 · ||H||F =
log2 1 + γt
T
σn
T
(88)
• Assuming that every channel gain in the matrix H, |hij |, is a Nakagamim distributed random variable with the same m parameter.
• γtmimo is a Gamma distributed random variable with shape parameter
α = N · m (N = nT · nR ) and scale parameter β = γ l /(nT m).
• γtmimo ∼ G(N · m, γ l /(nT m)).
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Outage capacity - MIMO with STBC
15/11/02
Transformation of random variables
• Let C = g(γtmimo) =
K
T
log2(1 + γtmimo).
• Then γtmimo = g −1(C) = 2(C·T )/K − 1.
• Capacity PDF
fC (C) = fγtmimo (2(C·T )/K − 1) · 2(C·T )/K
K
ln 2
T
(89)
• The MIMO outage capacity can be obtained by solving the integral
Cth (C·T )/K
(C·T )/K
−1)/β
(2
− 1)N m−1e−(2
(C·T )/K
Pout(Cth ) =
ln 2 · dC
·
2
N ·m Γ(N · m)
β
0
(2(Cth ·T )/K − 1)m · nT
= 1 − Q N · m,
(90)
γl
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Outage capacity - MIMO
15/11/02
MIMO capacity
• Recall that C =
PT
i=1 log2 1 + σ 2 λi .
k
n
• With the assumption that all eigenvalues are i.i.d random variables
and nT = nR , the maximum capacity can be expressed as C = nT ·
log2(1 + PσT2 λ).
n
• Let C = g(λ) = nT · log2 (1 +
• Then λ = g −1 (C) =
PT
λ).
σn2
2C/nT −1
.
PT /σn2
• Capacity PDF
fC (C) = fλ
2C/nT − 1
PT /σn2
2
C/nT nT σn
·2
PT
ln 2.
(91)
• Need to know the PDF of λ to obtain the capacity PDF.
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Outage capacity
15/11/02
Capacity CDF at 10dB SNR
1
1x1
3x3
1x8
10x10
0.9
Prob. capacity ≤ abscissa
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
Capacity in bits/s/Hz
25
30
35
Outage capacity of i.i.d. Rayleigh fading channels at 10dB branch SNR
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Outage capacity
15/11/02
Capacity CDF at 1dB SNR
1
0.9
2x2
2x2(STBC)
Prob. capacity ≤ abscissa
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
Capacity in bits/s/Hz
3.5
4
4.5
5
Outage capacity of a 2x2 MIMO Rayleigh fading channel using the Alamouti STBC at
the transmitter at 1dB branch SNR
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Summary
15/11/02
• The capacity formulas of SISO, SIMO and MIMO fading channels have
been derived based on maximizing the mutual information between the
transmitted and received signal.
• The Alamouti space-time block code has been presented. Although
capable of increasing the diversity benefits, the use of STBC trades
off capacity for low complexity encoding and decoding.
• By using transformation of random variables, closed-form expressions
for the outage capacity for SISO, SIMO and MIMO (STBC at the
transmitter) i.i.d. Nakagami-m fading channels were derived.
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