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EE 5407 Part II: Spatial Based Communications

Instructor: Prof. Rui Zhang E-mail: [email protected] Website: http://www.ece.nus.edu.sg/stfpage/elezhang/ Lecture III: Transmit & Transmit Diversity

March 4, 2011

& 1 % ' Transmit 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 (CSIT) is known, transmit beamforming can be used

& 2 % ' 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

& 3 % ' $ System Model

Consider the following MISO channel: • y(n) = hT x(n) + z(n) (1)

T – x(n) = [x1(n),...,xt(n)] ; t denotes the number of transmit antennas

T jθi 2 – h = [h ,...,ht] ; hi = √βie ,i =1,...,t, where βi = hi and 1 | | 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 k k ≤ over all transmit antennas – z(n) (0, σ2) ∼ CN z

& 4 % ' Transmit Beamforming with CSIT $

Transmitted signal: x(n) = ws(n) • T t×1 – Transmit beamforming vector: w = [w ,...,wt] C 1 ∈ – Information signal: s(n) C ∈

& 5 % ' $

Sum-power constraint • – E[ s(n) 2]=1 | | – w 2 P k k ≤ – E[ x(n) 2] = w 2E[ s(n) 2] P 1 = P k k k k | | ≤ × Received signal is • y(n) = hT ws(n) + z(n) (2) t

= hiwi s(n) + z(n) (3) i=1 ! X

& 6 % 'The instantaneous receiver SNR is defined as $ • E t 2 i=1 hiwi s(n) γ = (4) h E[ z(n) 2] i P | | t 2 i=1 hiwi = 2 (5) σz P It is desirable to choose w to maximize γ subject to w 2 P . • k k ≤ 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

& 7 % ' 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)

& 8 % '– 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 ∈{ } 2 index of the transmit antenna that has the largest βj = hj . Then, the | | transmit beamforming weights for AS are

√ AS P if i = j wi = i =1,...,t (6)  0 otherwise  The resulted receiver SNR is • max(β1,...,βt)P γAS = 2 (7) σz If CSI feedback is used, for AS, the receiver only needs to send back • transmit antenna index j to the transmitter.

& 9 % '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

P−EGC P −jθi wi = e , i =1,...,t (8) r t The resulted receiver SNR is • t 2 √βi P γ = i=1 (9) P−EGC tσ2 P z  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

& 10 % ' $ 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 . k k ≤ First, we have the following inequalities: • t 2 t t t 2 2 2 hiwi hi wi hi P (10) ≤ | | | | ≤ | | ! i=1 i=1 i=1 i=1 X X X X

where the first inequality is due to Cauchy-Schwarz inequality, and holds ∗ with equality iff wi = c hi , i, with c being any complex number, while · ∀ t 2 the second inequality holds with equality when wi = P . i=1 | | To make the above two equalities hold at the sameP time, we have •

& 11 % ' t ch∗ 2 = P , yielding $ i=1 | i | P P c = (11) t 2 hi s i=1 | | Thus we obtain the transmit beamformingP weights for P-MRC as • P wP−MRC = h∗, i =1,...,t (12) i t 2 i hi s i=1 | | The maximum receiver SNR achievedP by the P-MRC transmit • beamforming is

t 2 t hi P ( β )P γ = i=1 | | = i=1 i (13) P−MRC σ2 σ2 P z  P z If CSI feedback is applied, for P-MRC, the receiver needs to send back • the instantaneous channels h1,...,ht to the transmitter.

& 12 % ' $

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 2 γ = (β1P )/σz . 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.

& 13 % ' 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

2 wi P ,i =1,...,t (14) | | ≤ 0

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?)

& 14 % ' $ Transmit Diversity Without CSIT

Consider the simplest case of t = 2. • In the case without CSIT, how to achieve the MISO channel (with iid h • 1 and h2) diversity gain over a SISO channel (with only h1 or h2)? Two heuristic schemes: • – Power Splitting

P x(n) = 2 s(n), n (15)  q P  ∀ 2  q  – Alternate Transmission

& 15 % ' $ √P x(n) = s(n), n =1, 3, 5, ...  0    0 x(n) = s(n), n =2, 4, 6, ... (16)  √P    For power splitting, the received signal can be written as •

T P y(n) = h x(n) + z(n) = (h1 + h2) s(n) + z(n) (17) r 2 Assuming that h1 and h2 are iid CSCG RVs with zero mean and variance • 2 σh, 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)

& 16 % '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.

T T Let y = [y(n),y(n + 1)] and zn = [z(n), z(n + 1)] . Then the • n 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.

& 17 % 'Thus we can apply MRC receive beamforming to obtain the maximum $ • instantaneous receiver SNR as 2 2 ( h1 + h2 )P γ = | | 2| | (21) σz Comparing γ with the receiver SNR for a SISO system (say, using the 2 2|h1| P first antenna) with repetition coding, which is 2 , a diversity order σz 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?

& 18 % ' Alamouti Code $

Transmitted signals at symbol time n =1, 2 are • P s(1) P s∗(2) x(1) = , x(2) = − (22) r 2  s(2)  r 2  s∗(1)      Similar for n =3, 4, n =5, 6,... • & 19 % 'Received signals are $ • P y(1) = [h1s(1) + h2s(2)] + z(1) r 2 P ∗ ∗ y(2) = [ h1s (2) + h2s (1)] + z(2) (23) r 2 − 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 × P y = Hs + z (24) r 2 where

h1 h2 H = (25)  h∗ h∗  2 − 1   Assuming that h and h are perfectly known at the receiver (CSIR), the • 1 2

& 20 % ' $ received signals are pre-multiplied by HH , yielding

P yˆ = HH y = HH Hs + HH z (26) r 2 Notice that • 2 2 h1 + h2 0 HH H = | | | | (27)  0 h 2 + h 2  | 1| | 2|   Thus for yˆ = [ˆy , yˆ ]T , we have • 1 2

P 2 2 yˆ1 = h1 + h2 s(1) +z ˆ1 r 2 | | | |  P 2 2 yˆ2 = h1 + h2 s(2) +z ˆ2 (28) r 2 | | | |  wherez ˆ = h∗z(1) + h z∗(2) andz ˆ = h∗z(1) h z∗(2). 1 1 2 2 2 − 1

& 21 % 'Note that $ • E[ zˆ 2] = E[ zˆ 2] = ( h 2 + h 2)σ2 (29) | 1| | 2| | 1| | 2| z Then the receiver SNR is given by • 2 P 2 2 2 2 2 ( h1 + h2 ) ( h + h )P γ = | | | | = | 1| | 2| (30) AC q( h 2 + h 2)σ2  2σ2 | 1| | 2| z z Thus Alamouti code achieves a diversity order of 2. • Furthermore, the array gain for Alamouti code is • E[γAC] αAC = 2 = 1 (31) σhP 2 σz 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).

& 22 % ' Joint Transmit and Receive Beamforming $

Consider the following MIMO system: • y(n) = Hx(n) + z(n) (32) – y(n) Cr×1; H Cr×t; x(n) Ct×1; and z(n) Cr×1 ∈ ∈ ∈ ∈ 2 – z(n) (0, σ Ir) ∼ CN z t×1 – x(n) = wts(n), where wt C denotes the transmit beamforming ∈ 2 vector, which satisfies the sum-power constraint: wt P k k ≤ – Assume both known CSIT and CSIR

r×1 The receive beamforming vector is denoted by wr C . After • ∈ applying wr to the received signal vector, the resultant receiver output is (the symbol index n is dropped by brevity)

H H H yˆ = wr y = wr Hwts + wr z (33)

& 23 % ' $ The receiver output SNR is then defined as • E H 2 H 2E 2 H 2 [ wr Hwts ] wr Hwt [ s ] wr Hwt γ = | H 2 | = | H | H | | = | 2 2| (34) E[ w z ] w E[zz ]wr wr σ | r | r k k z

It is desirable to jointly choose wt and wr to maximize γ subject to • 2 wt P . k k ≤ 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. •

& 24 % ' $ According to Cauchy-Schwarz inequality, we have for any given wt • 2 2 2 wr Hwt Hwt γ k k k 2 2 k = k 2 k (35) ≤ wr σ σ k k z z 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 . k k k k 2 The above SNR upper bound is maximized when wt maximizes Hwt • 2 k k subject to wt P . k k ≤ Let the singular-value decomposition (SVD) of H be denoted by • H = UΛV H (36)

r×r H H t×t where U C with UU = U U = Ir; V C with H ∈ H ∈ VV = V V = It; 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 λ . . . λm > 0. ≤ 1 ≥ ≥

& 25 % ' 2 $ Then the maximization of Hwt becomes equivalent to • k k H H H H H H H H H wt H Hwt = wt V Λ U UΛV wt = wt V Λ ΛV wt (37)

H Let w˜ t = V wt and thus wt = V w˜ t. The power constraint for w˜ t • 2 H H 2 becomes w˜ t = w VV wt = wt P . k k t k k ≤ The maximization problem then becomes equivalent to • 2 2 Maximize Λw˜ t subject to : w˜ t P (38) k k k k ≤

It is easy to verify that the optimal solution for w˜ t is • opt T 2 w˜ t = √P [1, 0,..., 0] , and the maximum objective value is λ1P .

Let V = [v ,..., vt] and U = [u ,..., ur]. • 1 1 Then it follows that the optimal transmit beamforming vector is • opt opt wt = V w˜ t = √P v1 (39)

& 26 % 'and the optimal receive beamforming vector is $

opt opt Hwt λ1√P u1 wr = = = u1 (40) Hwopt λ √P k t k 1 From (35), the resultant maximum receiver SNR is • opt 2 2 Hwt λ1P γopt = k 2 k = 2 (41) σz σz

Substituting wopt and wopt into (33) yields • t r opt H opt H H √ √ yˆ = (wr ) Hwt s +ˆz = u1 UΛV v1 Ps +ˆz = λ1 Ps +ˆz (42) wherez ˆ = uH z (0, σ2). 1 ∼ CN 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)”.

& 27 % '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 λ2. Unfortunately, for the • 1 iid Rayleigh fading MIMO channel case, there is no closed-form expression for the PDF/CDF of λ2, if m 2. 1 ≥ Nevertheless, we know that the sum of squared absolute values of all • elements in H, i.e., t r 2 Ω = hij (43) sum | | i=1 j=1 X X follows a chi-square distribution with 2tr degrees of freedom (see Slide 17 of Lecture II).

& 28 % 'Furthermore, since m λ2 = Ω , we have $ • i=1 i sum Ω P sum λ2 Ω (44) m ≤ 1 ≤ sum 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 • E E 2 [γopt] [λ1] αopt = 2 = (46) σhP 2 2 σh σz Using the fact that E[Ω ] = trσ2 and (44), it follows that • sum h tr α tr max(t,r) α tr (47) m ≤ opt ≤ ⇒ ≤ opt ≤ The upper and lower bounds become equal when t = 1 or r = 1.

& 29 % 'Joint Transmit Diversity and Receive Beamforming $

Consider the following r 2 MIMO system: • × y(n) = Hx(n) + z(n) (48)

r×2 T 2×1 – H C = [h ,..., hr] , where hi C ,i =1,...,r ∈ 1 ∈ – y(n) Cr×1; x(n) C2×1; and z(n) Cr×1 ∈ ∈ ∈ 2 – z(n) (0, σ Ir) ∼ CN z – E[ x(n) 2] P k k ≤ – 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

& 30 % 'From (30), the per-receive-antenna output SNR is $ • 2 hi P γi = k k2 , i =1,...,r (49) 2σz MRC combiner output SNR is • r Ω P γ = γ = sum , i =1,...,r (50) i 2σ2 i=1 z X 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 • 2 E[Ωsum]P 2rσhP 2 2 2σz 2σz α = 2 = 2 = r (51) σhP σhP 2 2 σz σz

& 31 % ' 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

& 32 % ' $

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 ≤

& 33 %