EE 5407 Part II: Spatial Based Wireless Communications · EE 5407 Part II: Spatial Based Wireless Communications Instructor: Prof. Rui Zhang ... • Assuming that h1 and h2 are iid

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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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