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EE 5407 Part II: Spatial Based Wireless
Communications
Instructor: Prof. Rui Zhang
E-mail: rzhang@i2r.a-star.edu.sg
Website: http://www.ece.nus.edu.sg/stfpage/elezhang/
Lecture IV: MIMO Systems
March 21, 2011
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Introduction to MIMO Systems
• So far, channel fading is treated as a “disadvantageous” factor for
reliable transmission.
• From now on, we will look at how to make use of channel fading, rather
than “compensating” for it.
• Consider a MIMO system with t > 1 transmit antennas and r > 1
receive antennas: there are in total t × r physical links (very likely over
different channels due to independent fading)
• Intuitively, if each transmit antenna sends one (possibly independent)
data stream, the total transmission rate can be increased dramatically
over the SISO system: a technique so-called spatial multiplexing.
• However, each receive antenna receives the combinations of transmitted
signals from all transmit antennas. Also, each modulated symbol can be
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transmitted over all or part of transmit antennas. Thus there may exist
cross-talk among different data streams at the receiver.
• One major challenge for MIMO spatial multiplexing is the transceiver
design, whereby the transmitted symbols from multiple data streams can
be recovered at the receiver with reasonable complexity and yet
reasonably good performance.
• We will see that whether the CSI is known at the transmitter can lead to
very different transceiver designs for MIMO spatial multiplexing.
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Outline
• Review of capacity results for SISO AWGN (additive white Gaussian
noise) and fading channels
• MIMO AWGN channel
– CSIT-known case: capacity; transceiver design (eigenmode
transmission)
– CSIT-unknown case: capacity; transceiver design (horizontal
encoding, linear vs. nonlinear receivers)
• MIMO fading channel
– Ergodic capacity with or without CSIT
– Outage capacity with or without CSIT
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Channel Capacity Analysis
• Channel capacity describes the maximum information rate that can be
reliably transmitted over a channel subject to a given transmit power
constraint.
• The capacity for AWGN channels was derived by Claude Shannon in
1948.
• The capacity of SISO fading channels has been studied by Goldsmith
and Varaiyaa , Caire and Shamaib , et al.
a
A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Transactions on Information Theory, vol.43, No.6, pp.1986-1992, November
1997.
b
G. Caire and S. Shamai (Shitz), “On the capacity of some channels with channel state
information,” IEEE Transactions on Information Theory, vol.45, No.6, pp.2007-2019, September 1999.
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• The capacity of MIMO fading channels has been studied by Telatarc ,
Foshini and Gansd , et al.
c
I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Transactions
on Telecommunications, vol.10, No.6, pp.585-595, November 1999.
d
G. J. Foshini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol.6, No.3,
pp.311-335, March 1998.
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Capacity of SISO AWGN Channel
• Consider the following SISO AWGN channel:
y(n) = hx(n) + z(n), n = 1, . . . , N
(1)
– h ∈ C denotes the channel, which is constant and assumed known at
the receiver
– E[|x(n)|2 ] ≤ P , where P is the transmit power constraint
– z(n) ∼ CN (0, σz2 ), is independent over n (AWGN)
• Channel capacity measures the maximum information rate that can be
reliably transmitted over the channel with any arbitrarily small
probability of decoding error as the code length N goes to infinity.
• Claude Shannon proved that the capacity of the SISO AWGN channel is
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equal to the mutual information between x and y, which is given by
I(x; y) = h(y) − h(y|x) = h(y) − h(z)
(2)
where h(X) is the differential entropy of a RV X (measuring the amount
of uncertainty for X), and h(Y |X) denotes the conditional differential
entropy of a pair of RVs X and Y (measuring the amount of uncertainty
for Y conditional on knowing X).
2
2
• For a CSCG RV X ∼ CN (0, σX
), h(X) = log2 πeσX
bits.
• Since z(n) ∼ CN (0, σz2 ), we have h(z) = log2 πeσz2 .
• The mutual information is then written as
I(x; y) = h(y) − log2 πeσz2
(3)
• The capacity is defined as the maximum mutual information over all
possible distributions of x subject to E[|x(n)|2 ] ≤ P .
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• Note that E[|y(n)|2 ] ≤ |h|2 P + σz2 .
2
• For a RV X with zero-mean and variance σX
, h(X) is maximized when
X is Gaussian distributed.
• To maximize h(y), we need that y(n) is Gaussian distributed such that
h(y) = log2 πe(|h|2 P + σz2 ).
• This suggests that x also needs to be Gaussian distributed with
E(|x(n)|2 ) = P .
• Thus it follows that the capacity of the SISO AWGN channel is achieved
when x(n) ∼ CN (0, P ) (i.e., x(n) is drawn from a Gaussian codebook),
and is given by
2
|h| P
(4)
C = log2 πe(|h|2 P + σz2 ) − log2 πeσz2 = log2 1 +
2
σz
where the capacity unit is bits/second/Hz (bps/Hz).
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• Note that the capacity is equal to log2 (1 + γ), where γ =
receiver SNR.
|h|2 P
σz2
is the
• At the asymptotically low-SNR regime, i.e., γ → 0, since log(1 + x) ≈ x
as x → 0, we have
γ
(5)
C(γ → 0) ≈
log 2
Thus the capacity doubles for every 3dB increase in SNR.
• At the asymptotically high-SNR regime, i.e., γ → ∞, since
log(1 + x) ≈ log x as x → ∞, we have
C(γ → ∞) ≈ log2 γ
(6)
Thus the capacity increases by 1 bps/Hz for every 3dB increase in SNR.
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Capacity of SISO Fading Channel
• Consider the following SISO fading channel similarly as defined for the
SISO AWGN channel (with the symbol index n dropped):
y = hx + z
(7)
whereas the channel h ∈ C is now a RV, which is constant during each
transmission block, but can change from one block to the other (i.e.,
block-fading); it is assumed that the instantaneous channel h is known at
the receiver.
• Each transmission block consists of N symbols.
• The instantaneous mutual information (IMI) between x and y
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conditional on channel state h is given by
2
|h| P
I(x; y|h) = log2 1 +
σz2
(8)
• Note that the IMI is a RV, the PDF of which is determined by the
distribution of |h|2 .
• Two types of fading channel capacities are defined as follows:
– Ergodic capacity is defined as the statistical average of the IMI:
2
|h| P
(9)
Cerg = E log2 1 +
σz2
where the expectation is taken over |h|2 .
– Outage capacity is defined as the information rate, below which the
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IMI falls with a prescribed probability q%:
2
|h| P
Pr log2 1 +
< Cout,q% = q%
σz2
(10)
where Cout,q% is called q% outage capacity.
• Ergodic capacity has two important applications:
– In the case where the instantaneous CSI on |h| is unknown at the
transmitter, the transmitter can employ a long code that spans over
different fading states of the channel to achieve reliable
communication, provided that the information rate of the code is less
than Cerg of the fading channel. This technique is known as coded
diversity. In this case, N needs not to be large.
– In the CSIT-known case, the transmitter can use different codes for
different channel fading states. As long as the selected code for each
fading state has a rate smaller than the IMI, reliable communication
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is ensured. The maximum achievable average rate over all different
fading states is then given by Cerg of the fading channel. This
technique is known as adaptive coding. In this case, N needs to be
sufficiently large.
• According to Jensen’s inequalitya we have
2
2
|h| P
E[|h| ]P
≤
log
1
+
E log2 1 +
2
σz2
σz2
$
(11)
Thus, the ergodic capacity of a fading channel is no larger than that of
an AWGN channel with a constant channel gain, which is equal to the
average channel gain of the fading channel.
• Outage capacity is usually applicable to data traffic with stringent delay
a
Jensen’s inequality says that for a concave function f (x) where x is a RV, it holds that
E[f (x)] ≤ f (E[x]). A function f (x) is concave iff for any two arbitrary values of x, x1 and x2 ,
and positive number λ between 0 and 1, we have λf (x1 )+(1−λ)f (x2 ) ≤ f (λx1 +(1−λ)x2 ).
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requirements (e.g., voice, real-time video), where the information of each
transmission block has a constant rate, and needs to be decoded at the
receiver with a block decoding error probability less than the prescribed
outage probability target. In this case, N needs to be sufficiently large to
achieve reliable transmission for each block.
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• For a given outage probability target q%, the effective capacity is defined
as the maximum average rate of the transmitted information successfully
decoded at the receiver
Ceff,q% = (1 − q%)Cout,q%
(12)
• The outage probability has different meanings for the CSIT-known and
CSIT-unknown cases
– In the CSIT-known case, the transmitter knows the instantaneous
channel and thus the IMI. Thus, if an outage event will occur at the
receiver, this is known at the transmitter, and is thus called
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transmitter-aware outage.
– In the CSIT-unknown case, an outage event is only known at the
receiver but not at the transmitter, and is thus called receiver-aware
outage.
2
• Let γ = |h|σ2P denote the instantaneous receiver SNR. Suppose that the
z
transmitter employs an optimal Gaussian codebook with information
rate equal to Cout,q% . Then from (10) it follows that an outage event
occurs iff
γ < 2Cout,q% − 1 , γmin
(13)
• However, practical MCS cannot perform as well as the optimal Gaussian
code. As a result, there is usually a positive SNR “gap” between the
theoretical minimum SNR γmin and the actual required operating SNR γ̄,
where γ̄ > γmin . Thus the outage probability is generally defined as
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pout = Pr(γ < γ̄)
(14)
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SISO Fading Channel: Ergodic Capacity
Assume iid Rayleigh fading with σh2 = E[|h|2 ] = 1, and σz2 = 1.
7
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Capacity (bps/Hz)
5
AWGN Channel Capacity
Fading Channel Ergodic Capacity
4
3
2
1
0
−10
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−5
0
5
SNR (dB)
10
15
20
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SISO Fading Channel: Outage Capacity
Assume iid Rayleigh fading with σh2 = E[|h|2 ] = 1, σz2 = 1, and P = 5.
Outage Probability
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
3.5
4
4.5
5
Effective Capacity (bps/Hz)
Outage Capacity (bps/Hz)
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1
0.5
0
0
0.5
1
1.5
2
2.5
3
Outage Capacity (bps/Hz)
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Capacity of MIMO AWGN Channel
• Consider the following r × t MIMO channel (with symbol index n
dropped):
y = Hx + z
(15)
– H ∈ Cr×t is a constant matrix, which is known at the receiver
– Assume Gaussian signals, i.e., x ∼ CN (0, S x ), where S x , E[xxH ] is
the covariance matrix for the transmitted signal vector
– If Rank(S x ) = 1, it is called “beamforming mode”; if Rank(S x ) > 1, it
is called “spatial multiplexing mode”
– Tr(S x ) = E[kxk2 ] ≤ P , where P is the sum-power constraint at the
transmitter
– z ∼ CN (0, σz2 I r ): spatially white Gaussian noise
• The capacity of the MIMO AWGN channel for a given S x is equal to the
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mutual information between x and y, which is given by
C = I(x; y) = h(y) − h(y|x) = h(y) − h(z)
(16)
• For a CSCG random vector x ∼ CN (0, S x ), h(x) = log2 πedet(S x ) bits.
• Note that y ∼ CN (0, HS x H H + σz2 I r ) and z ∼ CN (0, σz2 I r ).
• For square matrices A and B, det(A)/det(B) = det(AB −1 ). Thus we
have
C = log2 πedet(HS x H H + σz2 I r ) − log2 πedet(σz2 I r )
1
= log2 det I r + 2 HS x H H
σz
(17)
where the unit is bits/second/Hz (bps/Hz).
• Next, we study the design of S x to maximize the MIMO AWGN channel
capacity for the CSIT-known case and CSIT-unknown case, respectively,
as well as their respective MIMO transceiver designs.
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MIMO AWGN Channel With CSIT
• Assume that H is perfectly known at the transmitter.
• We want to find the optimal S x to maximize the MIMO channel
capacity subject to Tr(S x ) ≤ P .
• First, let the truncated SVD of H denoted by
H = Ũ Λ̃Ṽ
H
(18)
H
where Ũ ∈ Cr×m with m = Rank(H), and Ũ Ũ = I m ; Ṽ ∈ Ct×m with
H
Ṽ Ṽ = I m ; and Λ̃ is a m × m strictly positive diagonal matrix with the
diagonal elements (singular values) given by λ1 ≥ . . . ≥ λm > 0.
• Then the MIMO AWGN channel capacity given in (17) is written as
1
H
H
C = log2 det I r + 2 Ũ Λ̃Ṽ S x Ṽ Λ̃Ũ
(19)
σz
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H
• Let S̃ x = Ṽ S x Ṽ . Using the fact that det(I + AB) = det(I + BA),
we have
1 2
(20)
C = log2 det I m + 2 Λ̃ S̃ x
σz
• The Hadamard’s inequality states that the determinant of any positive
semi-definite matrix A is less than the product of all its diagonal
elements, i.e.,
Y
[A]i,i
(21)
det(A) ≤
i
with equality iff A is a diagonal matrix.
• Thus we have
C≤
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m
X
i=1
log2
λ2i [S̃ x ]i,i
1+
σz2
!
(22)
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with equality iff S̃ x is a diagonal matrix, i.e.,
H
S̃ x = Ṽ S x Ṽ = Σ
(23)
where Σ is a m × m non-negative diagonal matrix.
• Then we conclude that the optimal transmit covariance to maximize the
MIMO channel capacity is in the form of
S opt
x = Ṽ ΣṼ
H
(24)
• Let the diagonal elements of Σ be denoted by p1 , . . . , pm . Then the
resulted MIMO channel capacity is given by
m
X
λ2i pi
log2 1 + 2
Copt =
(25)
σ
z
i=1
• Consider now the power constraint:
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Tr(S opt
x ) ≤ P
(26)
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Since Tr(AB) = Tr(BA), we have
H
H
Tr(S opt
x ) = Tr(Ṽ ΣṼ ) = Tr(ΣṼ Ṽ ) = Tr(Σ) =
m
X
i=1
pi ≤ P
(27)
• One method to set pi ’s is the “equal-power (EP)” allocation:
pi =
P
, i = 1, . . . , m
m
(28)
And the resultant capacity is
CEP =
m
X
i=1
&
2
λP
log2 1 + 2i
σz m
(29)
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Eigenmode Transmission
• According to S opt
x , we can design the transmitted signal vector as
1
x = Ṽ Σ 2 s
(30)
where
– Ṽ is called “precoding” matrix
– Σ is called “power allocation” matrix
– s ∈ Cm×1 is the information signal vector, s ∼ CN (0, I m )
– Check power constraint:
P
1
1
H
E[kxk2 ] = E[xH x] = E[sH Σ 2 Ṽ Ṽ Σ 2 s] = E[sH Σs] = m
i=1 pi
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• Substituting this form of x into the MIMO channel (15) yields
1
2
y = H Ṽ Σ s + z
H
1
2
= Ũ Λ̃Ṽ Ṽ Σ s + z
1
= Ũ Λ̃Σ 2 s + z
(31)
• Suppose that the receiver multiplies the received signal vector by the
H
“decoding” matrix Ũ . We thus have
H
ỹ = Ũ y
H
1
2
= Ũ Ũ Λ̃Σ s + z̃
1
= Λ̃Σ 2 s + z̃
(32)
H
where z̃ = Ũ z ∼ CN (0, σz2 I m )
• Let ỹ = [ỹ1 , . . . , ỹm ]T , s = [s1 , . . . , sm ]T , and z̃ = [z̃1 , . . . , z̃m ]T .
1
2
• Since Λ̃Σ is a diagonal matrix, we see from (32) that the MIMO
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channel has been decomposed into a set of m parallel SISO channels
represented by
√
(33)
ỹi = λi pi si + z̃i , i = 1, . . . , m
$
• The receiver SNR for the ith decomposed SISO channel is given by
λ2i pi
γi = 2
σz
(34)
and the channel capacity is given by
2
λ pi
Ci = log2 (1 + γi ) = log2 1 + i 2
σz
(35)
• Then the sum capacity over m SISO channels is given by
m
m
X
X
λ2i pi
Ci =
log2 1 + 2
= Copt
σ
z
i=1
i=1
(36)
• Thus, the joint deployment of linear precoders Ṽ and linear decoders Ũ
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achieves the MIMO AWGN channel capacity in the case of known CSIT.
• The transceiver design based on the SVD of the MIMO channel matrix is
usually called “eigenmode transmission”.
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Water-Filling Power Allocation
• We can further optimize pi ’s subject to
given in (25).
Pm
i=1
pi ≤ P to maximize Copt
• This yields the following optimization problem:
m
X
λ2i pi
Maximize
log2 1 + 2
{pi }
σz
i=1
Subject to
m
X
i=1
pi ≤ P
pi ≥ 0, i = 1, . . . , m
(37)
• Applying the Lagrange multiplier method to solve this problem, the cost
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function is written as
J(p1 , . . . , pm ) =
m
X
i=1
log2 1 +
λ2i pi
σz2
−υ
m
X
pi
(38)
i=1
where υ ≥ 0 is the Lagrange multiplier associated with the sum-power
constraint.
• The cost function is maximized when the derivatives of J(·) over
pi , i = 1, . . . , m, are all equal to zero. Thus we have
1
• Let µ =
&
1
.
υ log 2
λ2i
σz2
λ2i pi
+ σ2
z
= υ log 2, i = 1, . . . , m
(39)
We have
σz2
pi = µ − 2 , i = 1, . . . , m
λi
(40)
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• Furthermore, using the fact that pi ≥ 0 yields the optimal power
allocation as
2 +
σ
pi = µ − z2
, i = 1, . . . , m
λi
(41)
where (x)+ , max(0, x).
• The constant µ can be determined from the equality
m 2 +
X
σ
=P
µ − z2
λi
i=1
(42)
• The above power allocation rule is called “water-filling (WF)” where
– µ: fixed water-level for all sub-channels
–
σz2
:
λ2i
normalized (to sub-channel power gain) noise power of the ith
sub-channel
– pi : water poured (power allocated) to the ith sub-channel
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• The capacity of the MIMO AWGN channel with WF power allocation is

+ 
2
σz
2 +
2
m
m µ
−
λ
X
X
2
i
λi
µλi


CWF =
log2 1 +
(43)
log2
=
2
2
σ
σ
z
z
i=1
i=1
&
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Optimality of Beamforming
• First, consider the r × 1 SIMO AWGN channel:
y = hx + z
(44)
• The (truncated) SVD of h is
h=
h
khk1
khk
(45)
• Thus the optimal precoder is trivially 1, and the optimal decoder is
hH
u =
khk
H
(46)
• This is same as MRC receive beamforming.
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• The receiver SNR after applying MRC beamforming is
γSIMO
khk2 P
=
σz2
(47)
• The SIMO AWGN channel capacity is
P H
1
H
CSIMO = log2 det I r + 2 hP h
= log2 1 + 2 h h
σz
σz
2
khk P
= log2 (1 + γSIMO )
= log2 1 +
2
σz
(48)
• Thus, MRC receive beamforming achieves the SIMO AWGN channel
capacity.
• Next, consider the following 1 × t MISO channel:
&
y = hT x + z
(49)
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• The (truncated) SVD of hT is
hT
h = 1khk
khk
T
(50)
• Thus the optimal decoder is trivially 1, and the optimal precoder is
h∗
v=
khk
(51)
• This is same as P-MRC transmit beamforming (without power gain
√
P ).
• The optimal transmit covariance matrix is
H
S opt
=
vP
v
=
x
P
∗ T
h
h
2
khk
(52)
• Note that Rank(S opt
x ) = 1. Thus “beamforming mode” is optimal.
• The SNR of the MISO channel after applying P-MRC transmit
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beamforming is
γMISO
khk2 P
=
σz2
(53)
• The MISO AWGN channel capacity is
4
1
P khk
∗
=
log
1
+
CMISO = log2 1 + 2 hT S opt
h
2
x
σz
σz2 khk2
khk2 P
= log2 1 +
= log2 (1 + γMISO )
2
σz
(54)
• Thus, P-MRC transmit beamforming achieves the MISO AWGN channel
capacity.
• Last, consider the strongest eigenmode beamforming (SEB) scheme for
the MIMO AWGN channel:
&
y = Hx + z
(55)
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• Recall that the precoder for this scheme is given by
√
opt
wt = P v 1
(56)
where v 1 is the first column of V (Recall that the SVD of H is U ΛV H ).
• Thus the transmit covariance matrix for SEB is
S SEB
= P v1vH
x
1
(57)
) = 1, thus beamforming mode is used.
Note that Rank(S SEB
x
• With this transmit covariance, the maximum achievable rate is
P
H
RSEB = log2 det I r + 2 Hv 1 v H
H
1
σz
P
P
= log2 1 + 2 kHv 1 k2 = log2 1 + 2 ku1 λ1 k2
σz
σz
2
λP
= log2 1 + 1 2
σz
&
(58)
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where u1 is the first column of U .
= u1 yields the receiver SNR as
• Applying the decoder wopt
t
γSEB
• Then it follows that
2
2
Hv
|
P
|uH
λ
1
1
1P
=
= 2
2
σz
|uH
z|
1
RSEB = log2 (1 + γSEB ) = log2 1 +
(59)
λ21 P
σz2
(60)
Thus, the SEB achieves the maximum rate over the AWGN MIMO
(i.e.,
channel with rank-one transmit covariance matrix S SEB
x
beamforming mode is used).
• However, in general the capacity of the MIMO AWGN channel is
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achieved by spatial multiplexing mode with Rank(S opt
x ) ≥ 1:

+ 
2
σz
2
m
µ
−
λ
X
i
λ2i


log2 1 +
CWF =

2
σ
z
i=1
(61)
• Clearly, RSEB is equal to CWF iff
σz2
µ− 2 = P
λ1
σz2
µ − 2 ≤ 0, i = 2, . . . , m
λi
(62)
(63)
• Since λ2 ≥ λ3 ≥ . . . ≥ λm , the conditions in (62) and (63) are satisfied iff
2
σz2
σz
σz2 σz2
P+ 2 ≤ 2 ⇒P ≤ 2 − 2
(64)
λ1
λ2
λ2 λ1
• Thus the SEB is capacity optimal only when P is sufficiently small.
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Asymptotic Capacity Analysis
• We have shown that for the asymptotically low-power regime, i.e.,
P → 0, the MIMO AWGN channel capacity is achieved by beamforming
mode and behaves as
λ21 P
λ21 P
(65)
≈
CMIMO (P → 0) = log2 1 + 2
σz
(log 2)σz2
• Recall that the SISO AWGN channel capacity at the asymptotically
low-power regime behaves as
|h|2 P
CSISO (P → 0) ≈
(log 2)σz2
(66)
• Thus the capacity gain of the MIMO system over the SISO system for
&
40
%
$
'
the asymptotically low-power regime is given by
λ21
CMIMO (P → 0)
= 2
CSISO (P → 0)
|h|
(67)
which is the achievable by the strongest eigenmode beamforming (SEB).
• On the other hand, consider the asymptotically high-power regime, i.e.,
P → ∞, the MIMO AWGN channel capacity is achieved by spatial
multiplexing mode and behaves as (lower-bounded by equal-power
allocation)
m
2
X
λi P
CMIMO (P → ∞) ≥
≈ c + m log2 P
log2 1 + 2
(68)
σ
m
z
i=1
where c =
λ2i
i=1 log2 σz2 m
Pm
is a constant independent of P .
• Notice that the MIMO AWGN channel capacity with the optimal WF
power allocation is upper-bounded by (considering each sub-channel is
&
41
%
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'
allocated with full-power P )
CMIMO (P → ∞) ≤
where c̃ =
λ2i
i=1 log2 σz2
Pm
m
X
i=1
log2 1 +
λ2i P
σz2
≈ c̃ + m log2 P
(69)
is still a constant independent of P .
• With upper and lower capacity bounds, the asymptotic ratio between
the MIMO channel capacity and the logarithm (base-2) of the
transmission power as the power goes to infinity, so-called “spatial
multiplexing gain”, is given by
CMIMO
=m
P →∞ log2 P
(70)
lim
• Note that for very large values of P , the MIMO channel capacity (using
either EP or WF power allocation) increases by m bps/Hz for every 3dB
increase in P .
&
42
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'
• Since for the SISO system, the channel capacity with P → ∞ behaves as
where ĉ =
CSISO (P → ∞) ≈ ĉ + log2 P
(71)
CMIMO (P → ∞)
=m
CSISO (P → ∞)
(72)
|h|2
log2 σ2 .
z
• Thus it follows that
• Note that the spatial multiplexing gain is equal to m, the rank of the
MIMO channel matrix, and is not necessarily equal to min(t, r) unless
the channel matrix is full-rank, which occurs with probability one for the
case of iid Rayleigh fading MIMO channel. Thus, independent channel
fading is an advantageous factor for maximizing the spatial multiplexing
gain of MIMO channel.
&
$
43
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$
MIMO AWGN Channel Capacity With CSIT
Assume t = r = 2, H = [1 ρ; ρ 1] with ρ = 0.5, and σz = 1.
10
9
Achievable Rate (bps/Hz)
8
7
SISO Capacity
MISO/SIMO Capacity
MIMO: Strongest Eigenmode Beamforming (SEB)
MIMO: Eigenmode Transmission with EP
MIMO Capacity: Eigenmode Transmission with WF
6
5
4
3
2
1
0
−10
&
−5
0
5
10
15
20
SNR (dB)
44
%
'
$
MIMO AWGN Channel Without CSIT
• For the CSIT-known case, we have shown that the linear precoders and
decoders based on the MIMO channel SVD (eigenmode transmission)
together with WF power allocation achieve the MIMO channel capacity.
Furthermore, the MIMO channel is decomposed into parallel SISO
AWGN channels, which greatly simplifies the transceiver design.
• Now consider the case where the channel H is only known at the
receiver, but is unknown at the transmitter.
• Assume that the transmitter employs an “isotropic transmission” with
the “white” signal covariance matrix given by
S (w)
x =
P
It
t
(73)
• Note that Tr(S (w)
x ) = P.
&
45
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• With the white transmit covariance, the capacity of the MIMO AWGN
channel is given by
1
H
C (w) = log2 det I r + 2 HS (w)
x H
σz
P
= log2 det I r + 2 HH H
σz t
P
H
H
= log2 det I r + 2 Ũ Λ̃Ṽ Ṽ Λ̃Ũ
σz t
P 2
= log2 det I m + 2 Λ̃
σz t
m
X
λ2i P
=
log2 1 + 2
(74)
σ
t
z
i=1
• Note that C (w) is identical to CEP in the case of known CSIT with
equal-power allocation if t = m, i.e., t ≤ r and H is full-rank.
&
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• It follows from (74) that the spatial multiplexing gain for the MIMO
AWGN channel with the white transmit covariance is m, which is the
rank of the MIMO channel matrix, like in the CSIT-known case.
$
• However, we will see soon that the transceiver design in the
CSIT-unknown case to achieve C (w) is in general far more complex as
compared with the eigenmode transmission in the CSIT-known case.
• Before that, let’s take a look at the special case with t = 2 and r = 1.
Recall that in Lecture III we have introduced the Alamouti code, for
P
which the transmit covariance matrix can be shown to be S (w)
=
I .
x
2 2
• In this case, from (74) we see that the MISO AWGN channel capacity
with the white transmit covariance is given by
2
khk P
(75)
= log2 (1 + γAC )
C (w) = log2 1 +
2
2σz
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'
where γAC =
2
khk P
2σz2
$
is the SNR after decoding the Alamouti code.
• Thus Alamouti code achieves the MISO AWGN channel capacity with
t = 2.
• In Lecture III, we have also introduced a heuristic scheme (which
achieves the same diversity order of 2 as the Alamouti code) using
alternate transmission and repetition coding over two transmit antennas.
The maximum achievable rate for this scheme is given by
2
1
khk P
R = log2 1 +
(76)
2
2
σz
where the factor 12 in front of the log function is due to repetition coding.
1
x
• Since log 1 + 2 ≥ 2 log(1 + x), ∀x ≥ 0, we can show that for any P ≥ 0
&
R ≤ C (w)
(77)
48
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• As P → ∞, we have
khk
1
log2 (P ) + 12 log2 σ2
R
2
z
lim (w) = lim
2
P →∞ C
P →∞
khk
log2 (P ) + log2 2σ2
2
=
1
2
(78)
z
Thus, the achievable rate of the heuristic scheme is half of that of the
Alamouti code (the channel capacity) at the high-power regime, due to
repetition coding.
• On the other hand, as P → 0, we have
khk
2
P
(2 log 2)σz2
R
= lim
=1
lim
2
P →0 C (w)
P →0 khk P
(79)
(2 log 2)σz2
Thus, the achievable rate of the heuristic scheme is equal to that of the
Alamouti code (the channel capacity) at the low-power regime.
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• Next, consider the case of t = 2 and r ≥ 2. In Lecture III, we have
introduced the joint transmit diversity (using Alamouti code) and
receiver beamforming (using MRC) scheme to achieve the full diversity
order tr for the iid Rayleigh fading MIMO channel.
• Assume that H is full-rank and thus m = min(t, r) = 2. The maximum
achievable rate for this scheme is given by
Ωsum P
(λ21 + λ22 )P
R = log2 1 +
= log2 1 +
(80)
2
2
2σz
2σz
P
=
I is given by
• The capacity of the MIMO channel with S (w)
x
2 2
2
2
λ1 P
λ2 P
(w)
C = log2 1 +
+ log2 1 +
2σz2
2σz2
&
(81)
50
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'
• As P → ∞, we have
λ21 +λ22
log2 2σ2
z
log2 (P ) +
R
= lim
lim
P2
P →∞ C (w)
P →∞
2 log (P ) +
2
λ2i
i=1 log2 2σz2
1
=
2
(82)
Thus, the achievable rate of this scheme is half of the channel capacity at
the high-power regime, although having the same S (w)
x .
• This is because the Alamouti code on the average transmits only one
data stream with two transmit antennas.
• As a counterpart, as P → 0, we have
R
=
lim
P →0 C (w)
(λ21 +λ22 )P
(2 log 2)σz2
lim 2 2
P →0 (λ1 +λ2 )P
(2 log 2)σz2
=1
(83)
Thus, the achievable rate of this scheme is equal to the channel capacity
at the low-power regime.
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Horizontal Encoding
• For general MIMO system configurations, how to achieve/approach the
channel capacity C (w) with the white transmit covariance by practical
transceivers is a challenging task in the CSIT-unknown case.
• A practical transmitter design for spatial multiplexing is “Horizontal
Encoding”, where the transmitted signal vector is given by
r
P
x=
[s1 , . . . , st ]T
(84)
t
where s1 , . . . , st correspond to independently encoded data streams with
information rates R1 , . . . , Rt ; and E[|si |2 ] = 1, i = 1, . . . , t.
Pt
• Then the transmission sum-rate is Rsum = i=1 Ri .
• The above spatial multiplexing scheme is also popularly known as “Bell
Labs Layered Space Time (BLAST)” transmission scheme.
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• For horizontal encoding, each receive antenna receives the combined
signal of all transmitted data streams. Thus spatial signal processing
over all receive antennas is needed to recover each data stream by
suppressing the interference from other data streams.
$
• There are in general two classes of receivers for horizontal encoding
– Linear receiver: zero-forcing (ZF) receiver;
minimum-mean-squared-error (MMSE) receiver
– Nonlinear receiver with successive interference cancelation (SIC):
ZF-SIC receiver; MMSE-SIC receiver
• Note that another possible scheme for MIMO spatial multiplexing is
“Vertical Encoding”, where s1 , . . . , st are jointly encoded rather than
independently encoded as for horizontal encoding. At the receiver,
iterative (soft) MIMO detection and channel decoding based on the
maximum likelihood (ML) principle are applied to decode the
information from all data streams.
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$
Linear Receiver
• Consider the MIMO AWGN channel:
y = Hx + z
with S (w)
x =
P
I,
t t
(85)
and z ∼ CN (0, σz2 I r ). Assume that H is full-rank.
• Linear receiver first applies a decoding matrix T ∈ Ct×r to extract the
signals from different data streams and then decodes them separately.
ỹ = T y = T Hx + T z
(86)
• Two design criteria are commonly adopted for linear receivers
– ZF: T is designed to satisfy T H = I t , i.e., the interference in each
received data stream due to all other data streams is removed.
– MMSE: T is designed to minimize the MSE between ỹ and x:
E[kỹ − xk2 ].
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ZF Receiver
• For ZF receiver, in order to have a feasible T such that T H = I t , the
channel matrix H needs to be square or tall, i.e., r ≥ t.
• If r = t and H is a square matrix, the only candidate for T to make
T H = I t is
T = H −1
(87)
• However, if r > t and H is a tall matrix, there are more than one
candidates for T to make T H = I t .
• In this case, one commonly adopted choice for T is the pseudo-inverse of
H given by
−1 H
H
H , H†
(88)
T ZF = H H
Note that T ZF becomes H −1 if r = t.
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• The reason for choosing H † as T ZF in the case of r > t is given next.
• Suppose that there is no noise at the receiver, i.e., z = 0. Thus, the
MIMO channel becomes
y = Hx
(89)
• The above MIMO system consists of r linear equations with t < r
unknowns, and is thus over-determined.
• Let x̂ be an estimate of x. It is desirable to minimize the squared error
between H x̂ and y, denoted by J(x̂) = kH x̂ − yk2 .
• Then we have
J(x̂) = (H x̂ − y)H (H x̂ − y)
&
= x̂H H H H x̂ − y H H x̂ − x̂H H H y + y H y
(90)
56
%
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• For any complex vector x, we have
dxH Ax
dxH y
dy H x
= Ax,
= y,
=0
∗
∗
∗
dx
dx
dx
(91)
• In order to find x̂ to minimize the squared error, we have
dJ(x̂)
H
H
∗ = 0 ⇒ H H x̂ = H y
dx̂
(92)
• Thereby we obtain the so-called “least-square (LS)” estimate of x as
−1 H
H
x̂LS = H H
H y = T ZF y
(93)
• Thus the LS estimation matrix is identical to the ZF decoding matrix.
• For r ≥ t and H is full-rank, the truncated SVD of H is given by
H
H = Ũ Λ̃V H , where Ũ ∈ Cr×t with Ũ Ũ = I t ; V ∈ Ct×t with
V H V = V V H = I t ; and Λ̃ is a t × t diagonal matrix with the diagonal
elements (singular values) given by λ1 ≥ . . . ≥ λt > 0.
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• Then it can be verified that
−1
H
H
−1
H H = V Λ̃Ũ Ũ Λ̃V H
V Λ̃Ũ
T ZF = H H H
−2
= (V Λ̃ V H )V Λ̃Ũ
H
−1
= V Λ̃ Ũ
H
(94)
• Substituting the above alternative expression for T ZF into (86) yields
ỹ = x + z̃
−1
H
where z̃ = V Λ̃ Ũ z ∼
CN (0, σz2 V
−2
(95)
H
−2
Λ̃ V ), V Λ̃ V
H
H
= H H
−1
.
• For the above equivalent channel, it is observed that the total signal
Pt σz2
2
2
power is E[kxk ] = P , and the total noise power is E[kz̃k ] = i=1 λ2 .
i
Thus, the average SNR is given by
ZF
γavg
&
E[kxk2 ]
P
=
=
Pt
E[kz̃k2 ]
i=1
σz2
λ2i
λ2t P
≤ 2
σz
(96)
58
%
'
where the upper bound is determined by the smallest singular value λt .
$
• This disadvantageous phenomenon for ZF receiver is so-called “noise
enhancement” due to the fact that the LS estimator in fact ignores the
noise effect at the receiver.
• Let x = [x1 , . . . , xt ]T , z̃ = [z̃1 , . . . , z̃t ]T , and vij = [V ]i,j , i = 1, . . . , t,
j = 1, . . . , t.
• The SNR of the ith data stream with ZF receiver is
γiZF
P
E[kxi k2 ]
t
=
=
P
E[kz̃i k2 ]
σz2 tj=1
|vij |2
λ2j
=
σz2 [(H
P
t
H
H)−1 ]i,i
, i = 1, . . . , t
(97)
• Thus, the achievable sum-rate for ZF receiver is
ZF
=
Rsum
&
t
X
log2 (1 + γiZF )
(98)
i=1
59
%
'
$
MMSE Receiver
• MMSE receiver is the optimal linear receiver that maximizes the receiver
SNR for each data stream, where the noise includes both the additive
noise and the interference from all other data streams.
• In general, MMSE receiver works for any t and r. However, in order to
achieve spatial multiplexing gain equal to t, we need that r ≥ t.
• Let the ith row of the MMSE decoding matrix T MMSE be tH
i , i = 1, . . . , t,
where ti ∈ Cr×1 .
• The total MSE to be minimized by T MMSE is expressed as
t
X
H
2
2
E |ti y − xi |
E kT MMSE y − xk =
(99)
i=1
• Thus the total MSE can be minimized by independently optimizing ti ’s.
&
60
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2
• Let Ji (ti ) = E[|tH
i y − xi | ], i = 1, . . . , t, be the MSE for the ith data
stream, which can be further expressed as
$
H
∗
Ji (ti ) = E[(tH
y
−
x
)(y
t
−
x
i
i
i
i )]
H
H
2
= tH
i Ryy ti − r xy ti − ti r xy + E[|xi | ]
(100)
where
Ryy
r xy
P
= E[yy ] = E[(Hx + z)(Hx + z) ] = HH H + σz2 I r
t
P
= E[yx∗i ] = E[(Hx + z)x∗i ] = hi
(101)
t
H
H
• Note that H = [h1 , . . . , ht ].
• The MMSE estimator ti is then obtained as
&
dJi (ti )
= 0 ⇒ Ryy ti = r xy
∗
dti
(102)
61
%
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• Thus we have
ti = R−1
yy r xy =
where Γ =
HH H +
1
Ir
Γ
−1
hi , i = 1, . . . , t
(103)
P
.
tσz2
• Substituting ti = R−1
yy r xy into (100) yields the minimum MSE for the ith
data stream as
−1
R
Jimin = E[|xi |2 ] − r H
xy yy r xy
• Furthermore, we have
T MMSE
−1
1
H
H
HH + I r
=H
Γ
(104)
(105)
• As Γ → ∞, i.e., the noise effect can be safely ignored, it can be shown
that T MMSE → T ZF .
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• Let H {−i} be the matrix H with the ith column being deleted,
i = 1, . . . , t.
• Using the matrix inversion lemma,
(A − BD −1 C)−1 = A−1 + A−1 B(D − CA−1 B)−1 CA−1
(106)
H
+
(1/Γ)I
,
B
=
h
,
C
=
h
with A = H {−i} H H
r
i
{−i}
i , and D = −1, we
have
−1
−1
1
A−1 hi hH
i A
H
−1
HH + I r
=A −
(107)
H −1
Γ
1 + hi A hi
• Thus, from (103), we obtain an alternative expression for ti as
ti =
1
A−1 hi
1 + βi
(108)
−1
where βi = hH
i A hi .
• Using this new expression for ti , from (104) it follows that the minimum
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MSE is given by
1
P
P
−1
− hH
A
hi
i
t
t
1 + βi
P βi
P
P 1
= −
=
t
t 1 + βi
t 1 + βi
Jimin = E[|xi |2 ] − r H
xy ti =
(109)
• Now consider the equivalent SIMO channel for decoding the ith data
stream given by
y = hi xi + z i
(110)
P
where z i = j6=i hj xj + z denotes the effective noise for the ith data
stream including both the interference from all other data streams and
the additive noise (SIMO channel with correlated noise studied in
Lecture II).
• Note that z i ∼ CN (0, (P/t)A).
&
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• Let ỹ = [ỹ1 , . . . , ỹt ]T . Applying ti given in (108) to y yields
ỹi =
tH
i y
=
tH
i hi xi
2
where z̃i = tH
z
∼
CN
(0,
σ
i
z̃ ).
i
H
• Note that σz̃2 = tH
i E[z i z i ]ti =
+
1
1+βi
tH
i zi
2
βi
=
xi + z̃i
1 + βi
−1
P H −1
h
A
AA
hi
t i
=
• Thus the MMSE receiver SNR for the ith data stream is
2
2
βi
βi
P
2
E[|x
|
]
i
1+βi
1+βi
t
MMSE
=
=
= βi
γi
2
σz̃2
1
P
β
1+βi
t i
(111)
1
1+βi
2
P
β.
t i
(112)
• Notice that for the SIMO channel given in (110) with correlated noise,
we have shown in Lecture II that the optimal receive beamforming
−1
P
hi , which is identical to the MMSE decoding
vector is wopt = t A
vector ti in (108) if ignoring the multiplication constants.
&
65
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• Furthermore, wopt maximizes the receiver SNR as (see (51) of Lecture II)
−1
P
P
H
−1
hi = hH
A
hi = βi
(113)
A
γmax = hi
i
t
t
which is same as that achieved by the MMSE decoder.
• The achievable sum-rate for MMSE receiver is
MMSE
Rsum
=
t
X
log2 (1 + γiMMSE )
(114)
i=1
where the achievable rate for the ith data stream is
RiMMSE = log2 (1 + γiMMSE ) = log2 (1 + βi ).
• It can be shown that RiMMSE is indeed the capacity of the SIMO channel
given in (110) for the ith data stream.
&
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Nonlinear Receiver
• Nonlinear receiver applies successive interference cancelation (SIC) to
decode different data streams.
• Assume that the decoding order for the SIC is given by the reverse of the
transmit antenna index, i.e., the data stream from the 1st transmit
antenna is decoded last, the data stream from the 2nd transmit antenna
is decoded second last, ...., the data stream from the tth transmit
antenna is decoded first.
• Then the SIC is described as follows.
– Step 1: Apply a linear decoder tt to the received signal y, and extract
the data stream from the tth transmit antenna
&
ỹt = tH
t y
(115)
67
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– Step 2: Decode from ỹt the tth data stream information st , and
reconstruct xt
– Step 3: Subtract ht xt from y, and yield
y {−t} = y − ht xt = H {−t} x{−t} + z
(116)
where H {−t} is H with the tth column deleted, and x{−t} is x with
the tth element deleted.
– Step 4: Return to Step 1, extract and decode the (t − 1)-th data
stream from y {−t} , and then subtract the corresponding signal
component from y {−t} , until all t data streams are decoded.
• For each iteration of Step 1, if ti , i = t, t − 1, . . . , 2, is designed using the
ZF criterion, the nonlinear receiver is called ZF-SIC; while if the MMSE
criterion is used, it is called MMSE-SIC. Note that t1 is given by the
receive MRC for both cases of ZF-SIC and MMSE-SIC.
&
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$
ZF-SIC
• For each iteration of SIC, the ZF decoding vector tH
i , i = 1, . . . , t, can be
obtained as the last row of the matrix H †[i] , where
H [i] = [h1 , . . . , hi ]
(117)
• Alternatively, the decoding vectors for ZF-SIC can be derived from the
QR decomposition of the MIMO channel matrix.
• Assume that r ≥ t and H is full-rank, the truncated QR decomposition
of H is given by
H = QR
(118)
where Q ∈ Cr×t satisfies that QH Q = I t , and R ∈ Ct×t is an
upper-triangular matrix with φij = [R]i,j = 0, ∀i > j.
• The QR decomposition of H can be obtained using the “Gram-Schmidt”
&
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procedure as presented next.
• Let Q = [e1 , . . . , et ], where kei k = 1, i = 1, . . . , t and eH
i ej = 0, ∀i 6= j.
• First, since h1 = φ11 e1 , we obtain immediately
e1 =
h1
, φ11 = kh1 k
kh1 k
(119)
• Second, since h2 = φ12 e1 + φ22 e2 , we multiply both left- and right-hand
sides by eH
1 to obtain
φ12 = eH
1 h2
(120)
Then by letting u2 = h2 − φ12 e1 , we obtain
φ22 = ku2 k, e2 =
u2
ku2 k
• Thus in general for any i ∈ {1, . . . , t} with hi =
&
(121)
P
j<i
φji ej + φii ei , we
70
%
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'
can obtain first
φji = eH
j hi , ∀j < i
and then ui = hi −
P
j<i
(122)
φji ej , and finally
ui
φii = kui k, ei =
kui k
(123)
• The decoding matrix for ZF-SIC is given by
T ZF−SIC = QH
(124)
H
• Note that the ith row of T ZF−SIC is tH
i = ei , i = 1, . . . , t.
• Applying T ZF−SIC to y yields
ỹ = T ZF−SIC y = QH QRx + QH z = Rx + z̃
(125)
where z̃ = QH z ∼ CN (0, σz2 I t ).
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• Thus we have the equivalent channel for decoding the ith data stream as
X
ỹi = φii xi +
φij xj + z̃i , i = 1, . . . , t
(126)
j>i
• Since for decoding the ith data stream, all data streams with index j > i
have already been decoded and thus can be subtracted from ỹi . Thus,
the effective channel for the ith data stream becomes
X
ŷi = ỹi −
φij xj = φii xi + z̃i , i = 1, . . . , t
(127)
j>i
• Thus the SNR for the ith data stream by ZF-SIC is given by
γiZF−SIC
&
φ2ii Pt
= 2
σz
(128)
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• The achievable sum-rate for ZF-SIC is given by
ZF−SIC
=
Rsum
t
X
log2 (1 + γiZF−SIC )
(129)
i=1
• Next, we show that the precoders obtained by the channel QR
decomposition are identical to those obtained by the last rows of H †[i] ’s.
• Without loss of generality, consider i = t and thus H [t] = H.
• We thus have
H † = (H H H)−1 H H = (RH QH QR)−1 RH QH = R† QH = R−1 QH
• The last row of H † is thus given by
[R
−1
]t,t eH
t
1 H
=
e
φtt t
(130)
where we have used the fact that for an upper-triangular matrix R, R−1
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is also upper-triangular, and [R]ii and [R−1 ]ii are inverses of each other.
$
H
• This row vector is thus identical to tH
t = et by ignoring the
multiplication constant 1/φtt .
• Furthermore, we obtain an alternative expression for the SNR of the tth
data stream by the linear ZF receiver as
γtZF = γtZF−SIC
φ2tt Pt
= 2
σz
(131)
• Comparing this with that given in (97), we have
1
2
=
φ
tt
H
[(H H)−1 ]t,t
(132)
• In general, it can be shown that
&
γiZF < γiZF−SIC , 1 ≤ i < t
(133)
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• Thus we have
ZF
ZF−SIC
< Rsum
Rsum
(134)
• So far, we have assumed that the decoding order for ZF-SIC is fixed as
the inverse of the antenna index.
• Notice that in total there are t! different decoding orders for ZF-SIC,
each of which corresponds to a different channel QR decomposition and
ZF−SIC
thus different φii ’s, γiZF−SIC ’s, as well as Rsum
.
• Let P be a t × t permutation matrix (each row/column has one element
equal to one and zeros elsewhere), which specifies the decoding order.
• Then consider the following QR decomposition:
HP = Qp Rp
(135)
• Note that H = Qp Rp P −1 = Qp Rp P since P −1 = P .
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• By letting T ZF−SIC = QH
p , we obtain
H
Q
R
P
x
+
Q
ỹ = T ZF−SIC y = QH
p
p
p
p z = Rp xp + z̃
(136)
2
z
∼
CN
(0,
σ
where xp = P x is a permuted version of x and z̃ = QH
p
z I t ).
• In the case r = t and for the asymptotically high-power region, i.e.,
P → ∞, we have
!
t
t
P
2
X
Y
φ
P
ZF−SIC
Rsum
=
log2 1 + ii2t ≈ t log2 2 + log2
φ2ii
σz
σz t
i=1
i=1
P
= t log2 2 + log2 |det(H)|2
σz t
2
H
2
(137)
2
since |det(H)| = det(H H) = det(R ) = (det(R)) =
Qt
i=1
φ2ii .
• Thus the achievable rate of ZF-SIC is independent of the decoding order
for the high-power regime.
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MMSE-SIC
• For MMSE-SIC, the effective SIMO channel for decoding the ith data
stream with data streams of index j > i already decoded and subtracted
from y is given by
ŷ i = hi xi + z i
(138)
P
where z i = j<i hj xj + z denotes the total noise including both the
interference from data streams with index j < i and the additive noise.
• From our previous study, we know that the optimal MMSE decoding
vector for the channel given in (138) is
ti =
&
1
A−1
i hi
1 + αi
(139)
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where
Ai =
X
hj hH
j
j<i
−1
αi = hH
i Ai hi
X
1
tσz2
H
+ Ir =
hj hj +
Ir
Γ
P
j<i
(140)
(141)
• The resultant SNR for the ith data stream is given by
γiMMSE−SIC = αi
(142)
• The achievable sum-rate for MMSE-SIC is given by
MMSE−SIC
Rsum
=
t
X
log2 (1 + γiMMSE−SIC )
(143)
i=1
MMSE−SIC
is indeed equal to the MIMO channel
• Next, we show that Rsum
capacity C (w) with the white transmit covariance as given in (74), i.e.,
the MMSE-SIC receiver is capacity-optimal.
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• First, we express the achievable rate for the ith data stream, i > 1, as
$
RiMMSE−SIC
= log2 (1 + αi )

= log2 1 +
hH
i

(144)
X
j<i
P H

= log2 1 + 2 hi
tσz

= log2 det I r +
&

hj hH
j
tσz2
Ir
+
P
!−1
P X
H
h
h
j
j + Ir
tσz2 j<i
P X
H
h
h
j
j + Ir
2
tσz j<i
X
P
H
h
h
= log2 det 
j
j + Ir
tσz2 j<i
!−1

hi 
!−1
!−1
(145)

hi 
(146)

P
H
h
h
i
i
tσz2
!
P X
H

h
h
j
j + Ir
2
tσz j≤i
(147)
(148)
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• Since det(B −1 A) =
det(A)
,
det(B )
$
we have
RiMMSE−SIC
= log2 det
,Ii − Ii−1
• Thus we have
MMSE−SIC
=
Rsum
P X
H
h
h
j
j + Ir
2
tσz j≤i
t
X
!
RiMMSE−SIC = I1 +
i=1
= log2 det
− log2 det
t
P X
H
h
h
i
i + Ir
tσz2 i=1
t
X
i=2
!
P X
H
h
h
j
j + Ir
2
tσz j<i
!
(149)
(Ii − Ii−1 ) = It
= log2 det
P
H
HH
+ Ir
tσz2
= C (w)
• Since the above proof holds regardless of the decoding order, the
achievable sum-rate for MMSE-SIC is independent of the decoding order.
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• However, different decoding orders will result in different transmission
MMSE−SIC
rates allocated over the t data streams, to make Rsum
= C (w) .
• Question: Do we need r ≥ t in the above proof for MMSE-SIC?
• In general, for ZF-based receivers (linear ZF or ZF-SIC), it needs that
r ≥ t; for MMSE-based receivers (linear MMSE or MMSE-SIC), r can be
smaller than t.
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MIMO AWGN Channel Capacity Without CSIT
$
Assume t = r = 2, H = [1 ρ; ρ 1] with ρ = 0.5, and σz = 1.
18
16
Achievable Rate (bps/Hz)
14
12
Capacity/MMSE−SIC
ZF−SIC
MMSE
ZF
10
8
6
4
2
0
0
&
5
10
15
20
25
30
SNR (dB)
82
%
'
MIMO AWGN Channel Capacity Without CSIT
$
Assume t = r = 2, H = [1 ρ; ρ 1] with ρ = 0.9, and σz = 1.
14
Achievable Rate (bps/Hz)
12
10
Capacity/MMSE−SIC
ZF−SIC
MMSE
ZF
8
6
4
2
0
0
&
5
10
15
20
25
30
SNR (dB)
83
%
'
$
Capacity of MIMO Fading Channel
• Consider the following r × t MIMO fading channel:
y = Hx + z
(150)
– H is constant during each transmission block, but can change from
one block to the other (i.e., block-fading); assume that the
instantaneous channel H is known at the receiver.
– x ∼ CN (0, S x ), where the transmit covariance matrix S x is constant
over all transmission blocks if the channel is unknown at the
transmitter, but may change over transmission blocks according to
the instantaneous channel if known at the transmitter.
– Tr(S x ) ≤ P for any transmission block.
– z ∼ CN (0, σz2 I r ).
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• The IMI of the MIMO fading channel for a given pair of H and S x is
given by
1
I(x; y|H) = log2 det I r + 2 HS x H H
(151)
σz
• The ergodic capacity of the MIMO fading channel is then defined as
Cerg = E [I(x; y|H)]
(152)
where the expectation is taken over H.
• The q% outage capacity of the MIMO fading channel is denoted by
Cout,q% , which is defined as
Pr (I(x; y|H) < Cout,q% ) = q%
&
(153)
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MIMO Fading Channel With CSIT
• In the CSIT-known case, S x for each transmission block can be
optimized based upon the instantaneous channel H.
• In the ergodic capacity case, S x for each transmission block should be
chosen to maximize the IMI, which is equivalent to the MIMO AWGN
channel capacity for the given H. Thus, the optimal S x is designed
using eigenmode transmission and WF power allocation.
• In the outage capacity case, if we can find the minimum outage
probability q% for a constant transmission rate R such that
Pr (I(x; y|H) < R) ≤ q%
(154)
Then we can claim that R is the q% (transmitter-aware) outage capacity.
• Thus maximizing the outage capacity for a given outage probability
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target is equivalent to minimizing the outage probability for a given
constant rate target, i.e., Cout,q% and q% have one-to-one correspondence.
$
• In order to minimize the outage probability for a given constant
transmission rate, it is desirable to maximize the IMI for each
transmission block based on the instantaneous channel H given P .
• Thus the optimal S x for each transmission block is given by the
eigenmode transmission and WF power allocation, same as that for the
ergodic capacity case.
• However, for transmission blocks with superior channel conditions, S x
that satisfies the power constraint and results in an IMI larger than the
given rate target R may not be unique. For such cases, it is usually
desirable to find the optimal S x to minimize the transmit power given
that the resultant IMI is equal to R.
• For example, consider a SISO fading channel with constant transmission
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rate R. A transmission outage will not occur if log2 1 +
Thus if |h|2 >
(2R −1)σz2
,
P
any transmit power between
P |h|2
σz2
(2R −1)σz2
|h|2
≥ R.
$
and P can
make a non-outage transmission, while the minimum power is
(2R −1)σz2
.
|h|2
• For MIMO fading channels, the transmit power to support a given rate
R is minimized by eigenmode transmission with WF power allocation:
2 +
σz
, i = 1, . . . , m
(155)
pi = µ − 2
λi
where the water-level µ in this case should be chosen such that the rate
constraint is satisfied with equality, i.e.,
2 +
m X
µλi
log2
=R
(156)
2
σz
i=1
Pm
• If i=1 pi > P , transmitter-aware outage occurs; otherwise, no outage
occurs.
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$
MIMO Fading Channel Without CSIT
• In the CSIT-unknown case, a constant transmit covariance matrix S x is
used for all transmission blocks.
• In the ergodic capacity case, the optimal S x should maximize the
statistical average of the IMI given by
1
(157)
E log2 det I r + 2 HS x H H
σz
if the distribution of H is known at the transmitter.
• In the case of H ∼ H w , it can be shown that the optimal transmit
covariance is the white covariance given by
&
S (w)
x =
P
It
t
(158)
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• From (74), it follows that the resultant ergodic capacity is given by
m
2
X
λi P
(159)
Cerg =
E log2 1 + 2
σ
t
z
i=1
• Thus the ergodic capacity is determined by the distribution of the
squared singular values of the random MIMO channel matrix.
• Note that λ2i ’s are the eigenvalues of the matrix H H H if r ≥ t = m,
since with the truncated SVD of H, we have
H
2
H H H = V Λ̃Ũ Ũ Λ̃V H = V Λ̃ V H
(160)
• Similarly, it can be shown that λ2i ’s are the eigenvalues of HH H if
t > r = m.
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• Define the following matrix

 HHH r ≥ t
W =
 HH H r < t
(161)
• The distribution law of W when H ∼ H w is called “Wishart
distribution”, for which the joint distribution of the eigenvalues λ2i ’s are
known (details are omitted here).
• With the Wishart distribution for W , it can be shown that the spatial
multiplexing gain for the ergodic capacity of MIMO fading channel is
given by
Cerg
= min(t, r)
P →∞ log2 P
lim
(162)
• This is consistent with the spatial multiplexing gain we have obtained for
the MIMO AWGN channel case, which is shown equal to the rank of the
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given MIMO channel matrix, since for H ∼ H w , the rank of the random
matrix H is min(t, r) with probability one.
• In the CSIT-unknown case, Cerg can be achieved by horizontal encoding
at the transmitter and MMSE-SIC decoding at the receiver, similarly as
for the MIMO AWGN channel case. However, it is worth noting that in
the MIMO fading channel case, each data stream of horizontal encoding
needs to span over all different fading states (i.e., to achieve coded
diversity), and the transmission rate of each data stream needs to be set
appropriately according to the channel distribution as well as the
decoding order at the receiver.
• Next, consider the outage capacity of the MIMO fading channel without
CSIT. In this case, it is desirable to find a constant transmit covariance
S x that minimizes the outage probability for a given constant
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transmission rate R:
1
Pr log2 det I r + 2 HS x H H < R
σz
(163)
by assuming that the channel distribution is known at the transmitter.
• For the case of H ∼ H w , it has been conjectured that the optimal S x is
in the following form
Sx =
P
diag(1, . . . , 1, 0 . . . , 0 )
| {z } | {z }
k
k ones
(164)
t−k zeros
i.e., the transmitter selects (arbitrarily) a subset of k out of t transmit
antennas for transmission with the white covariance.
• For example, consider the MISO fading channel with t = 2, r = 1, and
h ∼ hw . We know that if k = 1 and thus S x = P diag(1, 0), i.e., only the
first transmit antenna is used and the MISO fading channel becomes a
SISO fading channel, the resultant outage probability for a constant
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transmission rate R is
2
P |h1 |
<R
Pr log2 1 +
σz2
(165)
Since |h1 |2 is exponentially distributed with mean σh2 , we obtain the
outage probability as
(1)
pout = 1 − e−γ
where γ =
(2R −1)σz2
.
2P
σh
(166)
On the other hand, if k = t = 2 and thus
S x = (P/2)I 2 (e.g., using Alamouti code), the outage probability for the
MISO fading channel with the same rate R is given by
2
2
P (|h1 | + |h2 | )
Pr log2 1 +
<R
(167)
2
2σz
Since |h1 |2 + |h2 |2 is chi-square distributed with 4 degrees of freedom, we
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obtain the outage probability as
(2)
pout = 1 − e−2γ (1 + 2γ)
(1)
(168)
(2)
Thus, we conclude that if eγ > (1 + 2γ), then pout < pout , i.e., using single
transmit antenna results in a lower outage probability than using both
antennas, and vice versa. Alamouti code can be capacity-suboptimal!
• In general, with other parameters fixed, the smaller the rate target R is,
the larger is the optimal number of active antennas, k. This is because
smaller R corresponds to smaller outage probability, which is achieved by
more diversity or more transmit antennas.
• For general MIMO systems without CSIT, the outage capacity of the
MIMO fading channel with the (truncated) white transmit covariance is
achieved by vertical encoding (e.g., space-time code) and iterative
receiver.
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• If horizontal encoding is used along with linear/nonlinear receivers, a
practical design is to assign all data streams/transmit antennas the same
transmission rate R/t.
• However, horizontal encoding with equal-rate allocation in general does
not achieve the outage capacity even with the MMSE-SIC receiver and
r ≥ t.
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Summary
• MIMO AWGN channel
– CSIT-known case: eigenmode transmission and WF power allocation
achieve the capacity
– CSIT-unknown case: horizontal encoding and nonlinear MMSE-SIC
receiver achieve the capacity
– For both cases, spatial multiplexing gain is equal to the rank of
MIMO channel matrix
• MIMO fading channel
– Ergodic capacity
∗ CSIT-known case: eigenmode transmission and WF power
allocation based on instantaneous CSI achieve the capacity
∗ CSIT-unknown case: horizontal encoding and nonlinear MMSE-SIC
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receiver across different fading states achieve the capacity
– Outage capacity
∗ CSIT-known case: eigenmode transmission and WF power
allocation based on instantaneous CSI achieve the capacity
∗ CSIT-unknown case: vertical encoding (with transmit antenna
selection) and iterative receiver achieve the capacity
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