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EE 5407 Part II: Spatial Based Wireless
Communications
Instructor: Prof. Rui Zhang
E-mail: [email protected]
Website: http://www.ece.nus.edu.sg/stfpage/elezhang/
Lecture III: Transmit Beamforming & Transmit Diversity
March 4, 2011
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Transmit Antenna Processing
• When multiple antennas are available at the receiver, the transmission
quality can be improved through exploiting receive beamforming. This
method is useful for uplink (from mobile terminal to base station) as the
base station can usually be equipped with multiple antennas.
• Equipping multiple antennas at the mobile terminal side may not be
practical due to size limitation and complexity constraint. Thus it is
difficult to improve the performance of downlink (from base station to
mobile terminal) using receive beamforming.
• In this case, we may consider using the base station antennas to improve
the downlink performance.
– Transmit beamforming: When the channel state information at
the transmitter (CSIT) is known, transmit beamforming can be used
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to achieve diversity gain as well as array gain.
– Transmit diversity: When CSIT is not available, transmit diversity
technique can be used to achieve diversity gain.
• If mobile terminal is also equipped with multiple antennas, transmit
beamforming/diversity can be jointly deployed with receive beamforming
to further improve the uplink and downlink performance.
• Outline of this lecture:
– System model: MISO channel
– Transmit beamforming with CSIT
– Transmit diversity without CSIT: Alamouti code
– Joint transmit and receive beamforming for MIMO channel with both
CSIT and CSIR
– Joint transmit diversity and receive beamforming for MIMO channel
with CSIR only
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System Model
• Consider the following MISO channel:
y(n) = hT x(n) + z(n) (1)
– x(n) = [x1(n), . . . , xt(n)]T ; t denotes the number of transmit antennas
– h = [h1, . . . , ht]T ; hi =
√βie
jθi , i = 1, . . . , t, where βi = |hi|2 and
θi = ∠hi
– Assume a sum-power constraint at the transmitter (for the time
being): E[‖x(n)‖2] ≤ P , where P denotes the sum-power constraint
over all transmit antennas
– z(n) ∼ CN (0, σ2z)
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Transmit Beamforming with CSIT
• Transmitted signal: x(n) = ws(n)
– Transmit beamforming vector: w = [w1, . . . , wt]T ∈ C
t×1
– Information signal: s(n) ∈ C
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• Sum-power constraint
– E[|s(n)|2] = 1
– ‖w‖2 ≤ P
– E[‖x(n)‖2] = ‖w‖2E[|s(n)|2] ≤ P × 1 = P
• Received signal is
y(n) = hT ws(n) + z(n) (2)
=
(
t∑
i=1
hiwi
)
s(n) + z(n) (3)
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• The instantaneous receiver SNR is defined as
γ =E
[
∣
∣
(∑t
i=1 hiwi
)
s(n)∣
∣
2]
E[|z(n)|2] (4)
=
∣
∣
∑t
i=1 hiwi
∣
∣
2
σ2z
(5)
• It is desirable to choose w to maximize γ subject to ‖w‖2 ≤ P .
• It is also desirable to require only partial knowledge of h at the
transmitter to design w, because in practice the CSI is usually difficult
to obtain at the transmitter side.
• There are two commonly adopted methods to obtain CSIT for a link
that has two-way communications:
– If the link employs a time-division-duplex (TDD) method to switch
transmissions between two nodes over the same frequency band, the
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channel from one node to the other can be highly correlated with that
in the reverse transmit direction, a phenomenon so-called channel
reciprocity. Thus, one node can obtain the channel to the other by
estimating the reverse channel from the other node to itself.
– If the link employs a frequency-division-duplex (FDD) method to
allow both nodes to transmit at the same time slot but over different
frequency bands, the channel over which one node transmits to the
other can be different from that over which it receives from the other.
Thus, the channel reciprocity may not hold and the first method to
obtain CSIT may fail. However, one node can help the other node
obtain CSIT by estimating the channel from the other node and then
sending it back to the other node, a technique so-called CSI feedback.
• Three transmit beamforming schemes are considered:
– Antenna selection (AS)
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– Pre-equal-gain-combining (P-EGC)
– Pre-maximal-ratio-combining (P-MRC)
• For AS, only the antenna with the largest instantaneous channel gain is
selected for transmission at each time. Let j ∈ 1, . . . , t denote the
index of the transmit antenna that has the largest βj = |hj|2. Then, the
transmit beamforming weights for AS are
wASi =
√P if i = j
0 otherwisei = 1, . . . , t (6)
• The resulted receiver SNR is
γAS =max(β1, . . . , βt)P
σ2z
(7)
• If CSI feedback is used, for AS, the receiver only needs to send back
transmit antenna index j to the transmitter.
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• For Pre-EGC, the transmitter equally splits the power to all transmit
antennas, and pre-compensates for each transmit antenna the channel
phase shift such that the signals from all transmit antennas are
coherently added up at the receiver. Thus, the transmit beamforming
weights for P-EGC are
wP−EGCi =
√
P
te−jθi , i = 1, . . . , t (8)
• The resulted receiver SNR is
γP−EGC =
(∑t
i=1
√βi
)2P
tσ2z
(9)
• If CSI feedback is used, for P-EGC, the receiver needs to send back the
channel phase shifts θ1, . . . , θt to the transmitter.
• For P-MRC, the transmitter allocates the power to transmit antennas
based on their instantaneous channel gains, and also compensates for
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their channel phase shifts such that the transmitted signals are
coherently combined at the receiver.
• The transmit beamforming weights for P-MRC are designed to maximize
the receiver SNR γ given in (5) subject to ‖w‖2 ≤ P .
• First, we have the following inequalities:
∣
∣
∣
∣
∣
t∑
i=1
hiwi
∣
∣
∣
∣
∣
2
≤t∑
i=1
|hi|2t∑
i=1
|wi|2 ≤(
t∑
i=1
|hi|2)
P (10)
where the first inequality is due to Cauchy-Schwarz inequality, and holds
with equality iff wi = c · h∗i ,∀i, with c being any complex number, while
the second inequality holds with equality when∑t
i=1 |wi|2 = P .
• To make the above two equalities hold at the same time, we have
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∑t
i=1 |ch∗i |2 = P , yielding
c =
√
P∑t
i=1 |hi|2(11)
• Thus we obtain the transmit beamforming weights for P-MRC as
wP−MRCi =
√
P∑t
i=1 |hi|2h∗
i , i = 1, . . . , t (12)
• The maximum receiver SNR achieved by the P-MRC transmit
beamforming is
γP−MRC =
(∑t
i=1 |hi|2)
P
σ2z
=(∑t
i=1 βi)P
σ2z
(13)
• If CSI feedback is applied, for P-MRC, the receiver needs to send back
the instantaneous channels h1, . . . , ht to the transmitter.
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• The outage probability, diversity order, and array gain analysis for the
MISO channel with AS, P-EGC, and P-MRC transmit beamforming is
similar to that for the SIMO channel with SC, EGC, and MRC receive
beamforming given in Lecture II, respectively. Thus we omit the details
here (while it is still worthwhile verifying them by yourself !!!).
• Example: Consider an iid Rayleigh fading MISO channel with h ∼ hw.
If t = 1 (i.e., a SISO system), the instantaneous receiver SNR is
γ = (β1P )/σ2z . Using (13) and the results we have derived for the MRC
receive beamforming in the SIMO channel case, it can be easily shown
that the diversity order and array gain for P-MRC transmit
beamforming are both equal to t.
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Per-Antenna Power Constraint
• So far, we have studied transmit beamforming for the MISO channel
subject to the sum-power constraint at the transmitter. In practice, the
per-antenna-based transmit power constraint is usually more relevant
than the sum-power constraint, due to the fact that each transmit
antenna has its own power amplifier which operates properly only when
the transmit power is below a predesigned threshold.
• Under the per-antenna power constraint, the transmit beamforming
weights need to satisfy
|wi|2 ≤ P0, i = 1, . . . , t (14)
where P0 denotes the transmit power constraint for each antenna.
• What are the optimal transmit beamforming weights in this case?
Answer: P-EGC with transmit power P0 at all antennas (Why?)
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Transmit Diversity Without CSIT
• Consider the simplest case of t = 2.
• In the case without CSIT, how to achieve the MISO channel (with iid h1
and h2) diversity gain over a SISO channel (with only h1 or h2)?
• Two heuristic schemes:
– Power Splitting
x(n) =
√
P2
√
P2
s(n), ∀n (15)
– Alternate Transmission
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x(n) =
√P
0
s(n), n = 1, 3, 5, ...
x(n) =
0√
P
s(n), n = 2, 4, 6, ... (16)
• For power splitting, the received signal can be written as
y(n) = hT x(n) + z(n) = (h1 + h2)
√
P
2s(n) + z(n) (17)
• Assuming that h1 and h2 are iid CSCG RVs with zero mean and variance
σ2h, thus (h1 + h2)/
√2 is also a CSCG RV with zero mean and the same
variance. Thus the received signal is statistically equivalent to that over
the following SISO channel
y(n) = h1
√Ps(n) + z(n) (18)
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• Therefore, power splitting does not provide any diversity gain over the
SISO system.
• For alternate transmission, the received signal is given by
y(n) = h1
√Ps(n) + z(n), n = 1, 3, 5, ...
y(n) = h2
√Ps(n) + z(n), n = 2, 4, 6, ... (19)
• Suppose that s(n) = s(n + 1), n = 1, 3, 5, ..., i.e., the information signal
is repeated over two consecutive transmitted symbols, a technique known
as repetition coding.
• Let yn = [y(n), y(n + 1)]T and zn = [z(n), z(n + 1)]T . Then the
equivalent system model for alternate transmission becomes
yn = hs(n) + zn, n = 1, 3, 5, ... (20)
which is an equivalent SIMO system with r = 2.
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• Thus we can apply MRC receive beamforming to obtain the maximum
instantaneous receiver SNR as
γ =(|h1|2 + |h2|2)P
σ2z
(21)
Comparing γ with the receiver SNR for a SISO system (say, using the
first antenna) with repetition coding, which is 2|h1|2P
σ2z
, a diversity order
gain of 2 is achieved.
• However, notice that with repetition coding the same information symbol
is transmitted twice, and thus the spectral efficiency is reduced by half.
• How to characterize the tradeoff between diversity performance and
spectral efficiency?
Answer: capacity analysis (To be given in Lecture IV).
• Is there a transmission scheme for the case of unknown CSIT to achieve
the diversity order of 2 without losing spectral efficiency?
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Alamouti Code
• Transmitted signals at symbol time n = 1, 2 are
x(1) =
√
P
2
s(1)
s(2)
, x(2) =
√
P
2
−s∗(2)
s∗(1)
(22)
• Similar for n = 3, 4, n = 5, 6,...
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• Received signals are
y(1) =
√
P
2[h1s(1) + h2s(2)] + z(1)
y(2) =
√
P
2[−h1s
∗(2) + h2s∗(1)] + z(2) (23)
• Let y = [y(1), y∗(2)]T , s = [s(1), s(2)]T , and z = [z(1), z∗(2)]T . The
equivalent 2 × 2 MIMO channel becomes
y =
√
P
2Hs + z (24)
where
H =
h1 h2
h∗2 −h∗
1
(25)
• Assuming that h1 and h2 are perfectly known at the receiver (CSIR), the
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received signals are pre-multiplied by HH , yielding
y = HHy =
√
P
2HHHs + HHz (26)
• Notice that
HHH =
|h1|2 + |h2|2 0
0 |h1|2 + |h2|2
(27)
• Thus for y = [y1, y2]T , we have
y1 =
√
P
2
(
|h1|2 + |h2|2)
s(1) + z1
y2 =
√
P
2
(
|h1|2 + |h2|2)
s(2) + z2 (28)
where z1 = h∗1z(1) + h2z
∗(2) and z2 = h∗2z(1) − h1z
∗(2).
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• Note that
E[|z1|2] = E[|z2|2] = (|h1|2 + |h2|2)σ2z (29)
• Then the receiver SNR is given by
γAC =
(√
P2
(|h1|2 + |h2|2))2
(|h1|2 + |h2|2)σ2z
=(|h1|2 + |h2|2)P
2σ2z
(30)
• Thus Alamouti code achieves a diversity order of 2.
• Furthermore, the array gain for Alamouti code is
αAC =E[γAC]
σ2
hP
σ2z
= 1 (31)
Thus Alamouti code does not provide array gain over the SISO channel.
• How about the case of t > 2 without CSIT? Answer: space-time coding
(no further investigation in this course).
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Joint Transmit and Receive Beamforming
• Consider the following MIMO system:
y(n) = Hx(n) + z(n) (32)
– y(n) ∈ Cr×1; H ∈ C
r×t; x(n) ∈ Ct×1; and z(n) ∈ C
r×1
– z(n) ∼ CN (0, σ2zIr)
– x(n) = wts(n), where wt ∈ Ct×1 denotes the transmit beamforming
vector, which satisfies the sum-power constraint: ‖wt‖2 ≤ P
– Assume both known CSIT and CSIR
• The receive beamforming vector is denoted by wr ∈ Cr×1. After
applying wr to the received signal vector, the resultant receiver output is
(the symbol index n is dropped by brevity)
y = wHr y = wH
r Hwts + wHr z (33)
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• The receiver output SNR is then defined as
γ =E[|wH
r Hwts|2]E[|wH
r z|2] =|wH
r Hwt|2E[|s|2]wH
r E[zzH ]wr
=|wH
r Hwt|2‖wr‖2σ2
z
(34)
• It is desirable to jointly choose wt and wr to maximize γ subject to
‖wt‖2 ≤ P .
• Possible schemes:
– AS at Tx and SC at Rx
– AS at Tx and MRC at Rx
– P-MRC at Tx and SC at Rx
– ...
• Next, we find the optimal wt and wr to maximize γ.
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• According to Cauchy-Schwarz inequality, we have for any given wt
γ ≤ ‖wr‖2‖Hwt‖2
‖wr‖2σ2z
=‖Hwt‖2
σ2z
(35)
where equality holds iff wr = cHwt, with c being any complex number.
For convenience, we set c to make ‖wr‖ = 1, thus c = 1/‖Hwt‖.• The above SNR upper bound is maximized when wt maximizes ‖Hwt‖2
subject to ‖wt‖2 ≤ P .
• Let the singular-value decomposition (SVD) of H be denoted by
H = UΛV H (36)
where U ∈ Cr×r with UUH = UHU = Ir; V ∈ C
t×t with
V V H = V HV = I t; and Λ is a r × t matrix, the elements of which are
all zeros, except that [Λ]i,i = λi > 0, i = 1, . . . , m with
m = Rank(H) ≤ min(t, r). It is assumed that λ1 ≥ . . . ≥ λm > 0.
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• Then the maximization of ‖Hwt‖2 becomes equivalent to
wHt HHHwt = wH
t V ΛHUHUΛV Hwt = wHt V ΛHΛV Hwt (37)
• Let wt = V Hwt and thus wt = V wt. The power constraint for wt
becomes ‖wt‖2 = wHt V V Hwt = ‖wt‖2 ≤ P .
• The maximization problem then becomes equivalent to
Maximize ‖Λwt‖2 subject to : ‖wt‖2 ≤ P (38)
• It is easy to verify that the optimal solution for wt is
woptt =
√P [1, 0, . . . , 0]T , and the maximum objective value is λ2
1P .
• Let V = [v1, . . . , vt] and U = [u1, . . . , ur].
• Then it follows that the optimal transmit beamforming vector is
woptt = V w
optt =
√Pv1 (39)
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and the optimal receive beamforming vector is
woptr =
Hwoptt
‖Hwoptt ‖ =
λ1
√Pu1
λ1
√P
= u1 (40)
• From (35), the resultant maximum receiver SNR is
γopt =‖Hw
optt ‖2
σ2z
=λ2
1P
σ2z
(41)
• Substituting woptt and wopt
r into (33) yields
y = (woptr )HHw
optt s + z = uH
1 UΛV Hv1
√Ps + z = λ1
√Ps + z (42)
where z = uH1 z ∼ CN (0, σ2
z).
• Thus, the MIMO channel is converted to an equivalent SISO channel by
joint transmit and receive beamforming, which is usually called
“strongest eigenmode beamforming (SEB)”.
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• Sanity check: what is the optimal transmit/receive beamforming vector
when t = 1 or r = 1?
• Last, we investigate the diversity order and array gain of the strongest
eigenmode beamforming for the iid Rayleigh fading MIMO channel case
(H ∼ Hw). In this case, m = min(t, r) with probability one.
• Clearly, we need to study the distribution of λ21. Unfortunately, for the
iid Rayleigh fading MIMO channel case, there is no closed-form
expression for the PDF/CDF of λ21, if m ≥ 2.
• Nevertheless, we know that the sum of squared absolute values of all
elements in H , i.e.,
Ωsum =t∑
i=1
r∑
j=1
|hij|2 (43)
follows a chi-square distribution with 2tr degrees of freedom (see Slide 17
of Lecture II).
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• Furthermore, since∑m
i=1 λ2i = Ωsum, we have
Ωsum
m≤ λ2
1 ≤ Ωsum (44)
• Following the proof of the diversity order for the EGC in Lecture II, we
obtain that the diversity order in this case is
dopt = tr (45)
• Using (41), the array gain is obtained as
αopt =E[γopt]
σ2
hP
σ2z
=E[λ2
1]
σ2h
(46)
• Using the fact that E[Ωsum] = trσ2h and (44), it follows that
tr
m≤ αopt ≤ tr ⇒ max(t, r) ≤ αopt ≤ tr (47)
The upper and lower bounds become equal when t = 1 or r = 1.
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Joint Transmit Diversity and Receive Beamforming
• Consider the following r × 2 MIMO system:
y(n) = Hx(n) + z(n) (48)
– H ∈ Cr×2 = [h1, . . . , hr]
T , where hi ∈ C2×1, i = 1, . . . , r
– y(n) ∈ Cr×1; x(n) ∈ C
2×1; and z(n) ∈ Cr×1
– z(n) ∼ CN (0, σ2zIr)
– E[‖x(n)‖2] ≤ P
– Assume known CSIR only (unknown CSIT)
• Transmitter scheme: Alamouti code (AC)
• Receiver scheme: independently decodes the AC at each receive antenna
and then applies MRC to combine the decoded output signals from all
receive antennas
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• From (30), the per-receive-antenna output SNR is
γi =‖hi‖2P
2σ2z
, i = 1, . . . , r (49)
• MRC combiner output SNR is
γ =r∑
i=1
γi =ΩsumP
2σ2z
, i = 1, . . . , r (50)
• For H ∼ Hw, γ has a chi-square distribution with 4r degrees of freedom
• Thus the proposed joint transmit diversity and receive beamforming
scheme achieves a diversity order of 2r
• The array gain in this case is
α =
E[Ωsum]P2σ2
z
σ2
hP
σ2z
=
2rσ2
hP
2σ2z
σ2
hP
σ2z
= r (51)
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Summary
• Transmit beamforming with CSIT
– antenna selection (AS)
– pre-equal-gain-combining (P-EGC)
– pre-maximal-ratio-combining-(P-MRC)
– sum-power constraint vs. per-antenna power constraint
• Transmit diversity without CSIT
– Alamouti code (AC)
• Joint transmit and receiver beamforming with both CSIT and CSIR
– strongest eigenmode beamforming (SEB)
• Joint transmit diversity and receive beamforming with CSIR only
– AC at Tx and MRC at Rx
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• Expected diversity order and array gain for different antenna
configurations (assuming known CSIR, iid Rayleigh fading channel, and
iid receiver noise) are summarized as follows:
Configuration Diversity Order Array Gain
SIMO r r
MISO (w/ CSIT) t t
MISO (w/o CSIT) ≤ t 1
MIMO (w/ CSIT) tr ≥ max(t, r),≤ tr
MIMO (w/o CSIT) ≤ tr r
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