EE 5407 Part II: Spatial Based Wireless Communications

Transcription

1 EE 5407 Part II: Spatial Based Wireless Communications Instructor: Prof. Rui Zhang Website: Lecture II: Receive Beamforming March 4,

2 Outline System model: SIMO channel Receive beamforming schemes Selection combining (SC) Equal-gain combining (EGC) Maximal-ratio combining (MRC) Diversity order Array gain Correlated noise 2

3 System Model Consider the SIMO channel modeled by y(n) = hx(n) + z(n) (1) y(n) = [y 1 (n),...,y r (n)] T ; r denotes the number of receive antennas h = [h 1,..., h r ] T ; h i = β i e jθ i, i = 1,...,r, where β i = h i 2 and θ i = h i ; assume that the instantaneous Channel State Information at the Receiver (CSIR), h, is perfectly known, via pilot-based channel estimation E[ x(n) 2 ] = P, where P denotes the transmission power z(n) = [z 1 (n),...,z r (n)] T ; assume that z i (n) s are independent over i (for the time being), and z i (n) CN(0, σ 2 i ), i {1,...,r}, where σ 2 i denotes the noise power at the ith receive antenna 3

4 Receive Beamforming Receive beamforming vector: w = [w 1,..., w r ] T C r 1 4

5 Combiner output after applying receive beamforming is ŷ(n) = w T y(n) (2) ( r r = w i h i )x(n) + w i z i (n) (3) The instantaneous combiner output SNR is defined as [ E ( r w ih i )x(n) 2] γ = [ E r w iz i (n) 2] (4) = r w ih i 2 P r w i 2 σ 2 i (5) It is desirable to choose w to maximize γ 5

6 Selection Combining (SC) The instantaneous SNR at the ith receive antenna (before combining) is γ i = E[ h ix(n) 2 ] E[ z i (n) 2 ] = β ip σ 2 i, i = 1,..., r (6) SC simply selects the signal from the antenna with the largest γ i as the combiner output, thus the SNR of the combiner output is γ SC = max(γ 1,..., γ r ) (7) Let j denote the index of the antenna with its SNR equal to γ SC. Then the receive beamforming weights for SC are 1 if i = j wi SC = i = 1,...,r (8) 0 otherwise 6

7 Assume that h h w (iid Rayleigh fading), i.e., h i s are independent CSCG random variables (RVs) each with zero mean and variance σ 2 h. Thus, β i s are iid exponential RVs with E[β i ] = σ 2 h, i. Assume also that σ i = σ z, i. Then the average SNR over all different fading channel realizations for each of the receive antennas is [ ] βi P Γ = E[γ i ] = E = σ2 h P (9) σz 2 Thus, γ i s are iid exponential RVs with the PDFs written as σ 2 i f i (γ i ) = 1 Γ e γ i Γ, γi 0 (10) and the cumulative density function (CDF) of γ i is given by F i (γ i ) = γi 0 f i (x)dx = γi 0 1 Γ e x Γ dx = 1 e γ i Γ, γi 0 (11) For a given SNR target at the receiver, denoted γ, the probability that 7

8 the ith receive antenna instantaneous SNR γ i falls below γ is given by Pr(γ i < γ) = F i ( γ) = 1 e γ Γ (12) Then the probability that the SNR of SC output falls below γ is given by r ( Pr(γ SC < γ) = Pr(γ i < γ) = 1 e Γ) γ r (13) This probability is called outage probability, p out, for the given SNR target γ. A lower outage probability means a better diversity combining performance at the receiver. Consider a communication system over the block-fading channel where different transmission blocks are independently encoded and decoded. The system adopts a modulation and coding scheme (MCS) designed for the AWGN channel, which achieves a block decoding error probability, p G e, with a constant receiver SNR γ. Suppose that the outage probability 8

9 with the SNR target γ over the fading channel is p out. Assume that p G e p out (e.g., p G e = 10 5, p out = 10 2 ). Then, the block decoding error probability using the same MCS over the fading channel, p F e, is upper-bounded by p F e (1 p out ) p G e + p out 1 = p G e + (1 p G e )p out p G e + p out p out Thus, p F e is upper-bounded (well approximated) by p out. Two lessons are then drawn: The MCS for AWGN channel can be used for slow-fading channel; The resulted block decoding error probability over fading channel can be approximately measured by the corresponding outage probability. Consider an iid Rayleigh fading SIMO channel with r receive antennas. Suppose that the average SNR for each receive antenna is Γ = 10, and the target SNR for the SC output is γ = 10. Then if r = 1, the outage probability is p out = 1 e 10/10 = 63.2%. For r = 4, 9

10 p out = (1 e 10/10 ) 4 = 16%. For r = 10, p out = (1 e 10/10 ) 10 = 1%. From (13), we obtain the CDF for the SC combiner output SNR as ( ) F SC (γ SC ) = 1 e γ SC r Γ, γsc 0 (14) The PDF of γ SC is then obtained as f SC (γ SC ) = F SC(γ SC ) γ SC = r Γ e γsc Γ ( ) 1 e γ SC r 1 Γ, γsc 0 (15) Thus the average SNR for the SC output is E[γ SC ] = + 0 xf SC (x)dx = Γ r 1 i (16) The average SNR is a relevant measure of the receiver long-term performance when channel coding is implemented across many consecutive transmission blocks (over different fading channels). 10

11 Equal-Gain Combining (EGC) For selection combining, only one receive antenna is used at a given time, thus the signal powers from the other receive antennas are not utilized. One way to improve on this is to coherently combine the signals from all receive antennas. For EGC, the signal at each received antenna is multiplied by a complex phaser which is matched to the phase of the corresponding channel. The resultant signals from all receive antennas have zero phase, and thus can be constructively added up. The receive beamforming weights for EGC are w EGC i = e jθ i, i = 1,...,r (17) 11

12 From (5), it follows that the combiner output SNR for EGC is γ EGC = ( r r σ2 i βi ) 2 P (18) EGC may or may not outperform selection combining (SC) in terms of instantaneous combiner output SNR for a given set of channel power β i s. For example, consider r = 2 and assume σ i = σ z, i. If β 1 = β 2, then it follows from (7) and (18) that γ EGC /γ SC = (2P/σz)/(P/σ 2 z) 2 = 2, i.e., EGC achieves a 3dB SNR gain over SC. However, if β 1 = 100β 2, we have γ EGC /γ SC = (121P/(2σz))/(100P/σ 2 z) 2 0.6, which is a 2.2dB SNR loss. In general, there are no closed-form expressions for the PDF/CDF of γ EGC, and thus so is the outage probability Pr(γ EGC < γ) in the case of EGC, even for the iid Rayleigh fading SIMO channel (where β i s are iid exponential RVs). 12

13 Assume that σ i = σ z, i. Then we can bound γ EGC as follows: γ EGC = ( r rσz 2 βi ) 2 P ( r β i)p rσ 2 z = r γ i r (19) Thus, the combiner output SNR for EGC is no less than the arithmetic mean of the SNRs of all individual receive antennas. Furthermore, assume that h h w (iid Rayleigh fading). Thus, β i s are iid exponential RVs with E[β i ] = σ 2 h, i and β i s are iid Rayleigh π distributed RVs with E[ β i ] = σ h, i. Then we can obtain the average 2 SNR for EGC in terms of individual receive-antenna average-snr Γ = σ2 h P σ 2 z as E[γ EGC ] = ( ) rσh 2 + (r 1)r πσ2 h P 4 rσ 2 z = ( 1 + (r 1) π ) Γ (20) 4 13

14 Maximal-Ratio Combining (MRC) EGC applies equal-amplitude weights to all receive antennas regardless of their instantaneous SNR values, thus dose not perform well when one receive antenna has a considerably lower SNR than the others. For MRC, the signals from different receive antennas are first weighted properly according to individual noise powers and instantaneous channel powers and then coherently combined. The receive beamforming weights for MRC are w MRC i = h i σ 2 i, i = 1,..., r (21) 14

15 From (5), it follows that the combiner output SNR for MRC is γ MRC = = = ( r r β i σ 2 i ) 2 P β i σ 2 i (22) r β i P σi 2 (23) r γ i (24) Thus, the output SNR of MRC is equal to the sum of the SNRs of all individual receive antennas. Next, we show that γ MRC is the maximum achievable SNR for all possible receive beamforming schemes (e.g., SC, EGC), i.e., wi MRC s are the optimal combining weights to maximize γ defined in (5). 15

16 Let w i = σ i w i, i and h i = h i /σ i, i. Then γ given in (5) can be rewritten as γ = r w 2 i h i P r w (25) i 2 According to Cauchy-Schwarz inequality, we have r 2 r r w i hi w i 2 h i 2 (26) where equality holds iff w i = c h i, i, with c being any complex number. Without loss of generality, we set c = 1. Thus we have γ r w i 2 r h i 2 P r w i 2 = r h i 2 P = r h i 2 P σ 2 i = r γ i = γ MRC 16

17 where equality holds iff w i = h i, i w i = h i σ 2 i = w MRC i, i (27) Thus, MRC weights achieve the maximum output SNR γ MRC. Assume that h h w (iid Rayleigh fading) and σ i = σ z, i. With Γ = σ2 h P denoting the average SNR of each receive antenna, the PDF of σz 2 γ MRC follows a chi-square distribution with 2r degrees of freedom: γ f MRC (γ MRC ) = γr 1 MRC e MRC Γ, γ (r 1)! Γ r MRC 0 (28) Note that γ MRC is the sum of r iid exponential RVs each with mean Γ. Then the outage probability with which the instantaneous output SNR 17

18 of MRC falls below a given SNR target γ is given by Pr(γ MRC < γ) = γ 0 f MRC (x)dx = 1 e γ Γ r ( γ Γ) i 1 (i 1)! (29) The average output SNR for MRC is given by E[γ MRC ] = r E[γ i ] = rγ (30) 18

19 Outage Probability vs. Average Per-Antenna SNR Assume iid Rayleigh fading channel, σ h = σ z = 1, γ = 1, and r = 1 or Outage Probability (log scale) r=1 r=4, SC r=4, EGC r=4, MRC Average Per Antenna SNR, Γ (db) 19

20 Diversity Order Consider the high-snr regime, where the average SNR per receive antenna Γ and the resultant outage probability p out (Γ) 0 for any given finite SNR target γ. For many practical fading channels, as Γ, p out (Γ) behaves as p out (Γ) cγ d (31) where c is a positive constant independent of Γ, and d is called diversity order, which can be obtained as d = lim Γ log (p out (Γ)) log(γ) (32) If we plot p out vs. Γ in a log-log scale, then d is approximately equal to the negative slope of the plot when Γ. 20

21 The diversity order is used to compare the asymptotic (transmit power goes to infinity) outage probabilities of different receive beamforming schemes for the SIMO fading channel under the same output SNR target. Now consider the case of iid Rayleigh fading SIMO channel with iid Gaussian noise at the receiver. For selection combining, from (13) it follows that p SC out(γ) = ( 1 e γ Γ) r (33) Since e x 1 + x around x = 0, with x = γ we have Γ ( ) 1 e γ r ( ( Γ 1 1 γ )) r ( γ ) r = (34) Γ Γ as Γ and thus γ Γ d SC = lim Γ log 0. Thus, the diversity order for SC is (( γ r ) Γ) r log( γ) r log(γ) = lim = r (35) log(γ) Γ log(γ) 21

22 For maximal-ratio combining, from (29) it follows that p MRC out (Γ) = 1 e γ Γ r ( γ Γ) i 1 (i 1)! Note that the Taylor series for e x around x = 0 is With x = γ Γ and p MRC out (Γ) 1 e γ Γ 1 + x1 1! + x2 2! + x3 3! +... and as Γ, we thus have ( r γ i 1 ( ( γ Γ) (i 1)! e γ Γ Γ r! ( ( γ e γ Γ Γ r! ) r ) = e γ Γ ( γ Γ) r r! ) r ) ( 1 γ Γ)( γ Γ r! ) r (36) (37) ( γ r Γ) r! 22

23 The diversity order for MRC is thus obtained as ( ) ( γ Γ) r d MRC = lim Γ log r! log(γ) = r (38) For equal-gain combining, we do not have the closed-form expression for p EGC out (Γ). However, since γ EGC γ MRC, we have for the same γ that p EGC out (Γ) p MRC out (Γ) (39) On the other hand, from (19) it follows that γ EGC (γ MRC /r). Thus, it can be shown that and p EGC out (Γ) p MRC out (Γ/r) (40) ( log p MRC out (Γ/r) ) lim Γ log(γ) = r (41) 23

24 Since we have log lim Γ ( p MRC out (Γ/r) ) log(γ) ( log p EGC out (Γ) ) lim Γ log(γ) ( log p MRC out (Γ) ) lim Γ log(γ) and both the upper and lower bounds in the above are equal to r, it follows that the diversity order for EGC is d EGC = r. To summarize, the diversity orders for SC, EGC, and MRC are all equal to r for the iid Rayleigh fading SIMO channel with iid receiver noise. Caveat: This may not be true for other fading distributions! 24

25 Array Gain In general, the array gain defines the increment of average SNR for a multi-antenna system as compared to that for a SISO system. Assume that for the SIMO channel, each receive antenna has the same average SNR, Γ, which is independent of r. The array gain is then defined as the ratio between the average output SNR with r 2 receive antennas to that with r = 1. Let γ avg (r) denote the average output SNR of the SIMO channel, with γ avg (1) = Γ. The array gain as a function of r is then defined as α = γ avg(r) Γ (42) 25

26 For SC, from (16) it follows that the array gain is given by α SC = Γ r Γ 1 i = r 1 i (43) For ECG, from (20) it follows that the array gain is given by α EGC = ( 1 + (r 1) π 4) Γ Γ = 1 + (r 1) π 4 (44) For MRC, from (30) it follows that the array gain is given by α MRC = rγ Γ = r (45) It can be verified that for r 2 α MRC > α EGC > α SC > 1 (46) 26

27 Correlated Noise In the previous study, we have assumed that the receiver noise is independent over receive antennas. In practice, this may not be true due to antenna coupling non-white alien interference In general, the receiver noise z can be modeled by a CSCG random vector with zero mean and covariance matrix, S z, where S z has the size of r r and is full-rank. Next, we derive the optimal receive beamforming vector for the SIMO channel with correlated noise to maximize the receiver output SNR. Thus we need to find a 1 r vector w H (note that here we use w H 27

28 instead of w T to represent the combining operation) to maximize γ = E[ wh hx(n) 2 ] E[ w H z(n) 2 ] = wh h 2 P w H S z w (47) (48) Let w = S 1 2 z w. Thus, w = S 1 2 z w. Then γ can be rewritten as According to Cauchy-Schwarz inequality, we have 1 2 γ = wh S z h 2 P (49) w 2 w H S 1 2 z h 2 w 2 S 1 2 z h 2 (50) where equality holds iff w = cs 1 2 z h, with c being any complex number. Without loss of generality, we set c = 1. 28

29 Thus we have 1 2 γ w 2 S z h 2 P w 2 where equality holds iff = S 1 2 z h 2 P = h H S 1 z hp γ max (51) w = S 1 2 z h w = S 1 2 z (S 1 2 z h) = S 1 z h w opt (52) Thus, the maximum output SNR γ max is achievable by the optimal receiver beamforming vector w opt. The optimal receive beamforming vector w H opt can be decomposed as w H opt = h H S 1 z = (h H S 1 2 z )S 1 2 z (53) where S 1 2 z is called noise whitening filter and h H S 1 2 z matched filter. is called The received signal y(n) is first multiplied by the noise whitening filter 29

30 to obtain ỹ(n) = S 1 2 z y(n) = S 1 2 z hx(n) + z(n) (54) where z(n) = S 1 2 z z(n) has the covariance matrix E[ z(n) z H (n)] = E[S 1 2 z z(n)z H (n)s 1 2 z ] = S 1 2 z E[z(n)z H (n)]s 1 2 z = S 1 2 z S z S 1 2 z = I (55) Thus the equivalent noise z(n) has a white spectrum (iid distribution). Consider the system model in (54) with an equivalent SIMO channel S 1 2 z h. Then the optimal MRC receive beamforming filter for this channel with the iid unit-variance noise is (S 1 2 z matched filter. h) H, which is also called Thus the optimal receive beamforming filter in the case of correlated noise can be considered as the cascading of a noise whitening filter and a matched filter (MRC receive beamforming). 30

31 Sanity check: Let S z be a diagonal matrix with diagonal elements denoted by σ 2 i, i = 1,...,r. Note that in this case, the receiver noise is independent over receive antennas. Thus we have γ max = h H S 1 z hp = r h i 2 P σ 2 i = r γ i = γ MRC (56) [ w opt = S 1 h1 z h =,..., h ] T r = w σ1 2 σr 2 MRC w H opt = w T MRC (57) Thus, the maximum SNR and optimal receive beamforming become those for the MRC in the case of independent noise. Last, we investigate the diversity order and array gain for the case of correlated noise. For convenience, we consider r = 2 and the following 31

32 symmetric noise covariance matrix S z = 1 ρ ρ 1 σ 2 z (58) where 0 ρ < 1 for S z to be a valid covariance matrix with full-rank. The eigenvalue decomposition (EVD) of the correlation matrix is 1 ρ = 1 + ρ ρ ρ where 1 + ρ and 1 ρ are non-negative eigenvalues UΛU H In order to find the diversity order and array gain, we need to study the distribution of h H S 1 z h. Assume that h h w, i.e., h CN(0, σ 2 h I r). Then the random vector h Uh has the same distribution as h since 32

33 h is CSCG, so is a linear transformation of h, i.e., h = Uh. ] E[ h = UE [h] = U 0 = 0 E[ h hh ] = UE [ hh H] U H = σ 2 h UI ru H = σ 2 h I r Thus, the distribution of h H S 1 z h is the same as that of h H S 1 z h. Then we have h H S 1 z h = h H U H S 1 z Uh = h H U H (UΛ 1 U H )Uh = h H Λ 1 h = β 1 (1 + ρ)σ 2 z + β 2 (1 ρ)σ 2 z where β 1 and β 2 are iid exponential RVs with mean σ 2 h. (59) To summarize, the maximum SNR γ max has the same distribution as γ max = h H S 1 z hp = β 1P (1 + ρ)σ 2 z + β 2P (1 ρ)σ 2 z 33 (60)

34 Note that (β 1 + β 2 )P (1 + ρ)σ 2 z γ max (β 1 + β 2 )P (1 ρ)σ 2 z Define the outage probability for γ max to fall below a given target γ as p out (Γ), where Γ = (Pσ 2 h )/σ2 z denotes the per-receive-antenna average SNR. (61) Thus it can be shown that p MRC out ( Γ 1 ρ ) p out (Γ) p MRC out ( ) Γ 1 + ρ (62) where p MRC out denotes the outage probability for the MRC under the same SNR target γ in the case of iid noise. Then we have lim Γ log ( p MRC out log(γ) ( )) Γ 1+ρ lim Γ log (p out (Γ)) log(γ) lim Γ log ( ( )) p MRC Γ out 1 ρ log(γ) 34

35 Both the above upper and lower bounds can be shown equal to 2. Thus the diversity order in the case of iid Rayleigh fading SIMO channel with r = 2 and correlated noise is 2, the same as that in the case of iid noise. From (60), it follows that the maximum average SNR in the case of correlated noise is E[ γ max ] = E[β 1]P + E[β ( ) 2]P 1 = Γ (1 + ρ)σz 2 (1 ρ)σz 2 (1 + ρ) + 1 (1 ρ) (63) Thus the array gain is derived as α = E[ γ max] Γ = 1 (1 + ρ) + 1 (1 ρ) = 2 1 ρ 2 (64) 2 The array gain in the case of correlated noise (ρ 0) is larger than that in the case of iid noise (ρ = 0). What happens if ρ = 1? 35

36 Summary Receive beamforming for SIMO channel with perfect CSIR Selection combining (SC), equal gain combining (EGC), and maximal-ratio combining (MRC) Complexity: SC < EGC < MRC Instantaneous SNR: SC < MRC, EGC < MRC, SC EGC Diversity order: SC = EGC = MRC (iid Rayleigh fading, iid noise) Array gain: SC < EGC < MRC (iid Rayleigh fading, iid noise) Correlated noise Diversity order: same as that in the case of iid noise Array gain: larger than that in the case of iid noise (the worst-case noise is iid) 36