Exploiting Partial Channel Knowledge at the Transmitter in MISO and MIMO Wireless

Transcription

1 Exploiting Partial Channel Knowledge at the Transmitter in MISO and MIMO Wireless SPAWC 2003 Rome, Italy June 18, 2003 E. Yoon, M. Vu and Arogyaswami Paulraj Stanford University Page 1

2 Outline Introduction Perfect CSI-Tx Models for Partial CSI Exploiting Instantaneous-CSI Exploiting Correlation-CSI Exploiting Parametric/Selection-CSI Summary Page 2

3 Exploiting Available Knowledge at Tx In MIMO/MISO system channel knowledge at Tx (CSI-Tx) can be used for Improving Capacity / Rate Reducing BER / Enhancing Diversity SNR Tools we have include ST Coding Power Control / Adaptive Modulation We assume perfect CSI-Rx. Page 3

4 Sources of Channel Error at Tx Using reciprocity in duplexed systems we need TDD : δ t << T c (Coherence Time) FDD : δ f << B c (Coherence BW) Feedback from Rx δ lag < T c where δ lag in the delay in feedback loop. Page 4

5 Problem Taxonomy Performance Criterion Nature of Channel Knowledge - Instantaneous Capacity - Ergodic Capacity - Error Rate - SNR Coding? - Instantaneous - Statistical - Parametric/Selection Signal & Receiver Power Constraints - Alamouti, SM,.. - ML, MMSE,.. Channel Model - Sum Power - Per Antenna - Average or Peak - Time Selective/Flat - Frequency Selective/Flat Page 5

6 ST Coding Using Available CSI-Tx General Structure Transmitter Channel Receiver N Outer Encoder ST Coding H Decode X S Y Available CSI Y = HS + N The core problem is to design S(Codeword) to maximize some performance criterion using the available CSI-Tx. Different approaches are possible. S = WX (W : Linear Pre-filter) (e.g. Modal Beamforming) S {W(X) More general map} (e.g. Alamouti) We choose W or W( ) based on channel knowledge. Page 6

7 MIMO Wireless Channel n 1 Tx s 1 H n N y 1 Rx s M y N Signal Model : y = Hs + n Capacity : C = ( max log 2 det I M + ρ R ss : Tr(R ss )=M M HR ssh H) (bps/hz) BER(PEP) : P (S (i) S (j) ) 1 det(i M + ρ 4M E{EH i,j HH HE i,j }) Page 7

8 Outline Introduction Perfect CSI-Tx Perfect CSI-Tx : MIMO Perfect CSI-Tx : MISO Models for Partial CSI Exploiting Instantaneous-CSI Exploiting Correlation-CSI Exploiting Parametric/Selection-CSI Summary Page 8

9 Perfect CSI-Tx : MIMO ỹ i = Es M T σ i s i + ñ i, i = 1, 2,, r = rank(h) Channel decouples into independent SISO channels. Capacity increases multiplicatively by min{m t, M r } and an additive term. Page 9

10 Perfect CSI-Tx : MIMO -(ctd) Capacity given H : C = max P Mk=1 γ k =M T rx i=1 log ρ «γ i λ i, M T γ opt i = µ M «T, i = 1,, r ρλ i + rx i=1 γ opt i = M T, where λ i = σ i 2, γ i = E{ s i 2 }, ρ = E s N 0. Optimal power allocation (special water pouring) maximizes capacity. Page 10

11 Perfect CSI-Tx : MISO x w Tx Rx n y = hwx + n Capacity given H : C = log ( 1 + ρ ) hww H h H M T Capacity is maximized when w = hh h. Capacity increases additively by log(m T ) at high SNR. Page 11

12 Outline Introduction Perfect CSI-Tx Models for Partial CSI Exploiting Instantaneous-CSI Exploiting Correlation-CSI Exploiting Parametric/Selection-CSI Summary Page 12

13 Partial CSI : Instantaneous-CSI We are given Ĥ modeled as H = Ĥ + ɛ H N (Ĥ, αi) Perfect Estimate : α = 0 No Estimate : Ĥ = 0 Quality Factor β = Ĥ 2 α Other error models are possible. Page 13

14 Partial CSI : Correlation-CSI Simplified model for correlated channels H = R 1 2 r H ω R 1 2 t If R = E{vec(H)vec(H) H } then R = R t R r Example with Tx correlation H = H ω R 1 2 t (R r = I) Channel information known is R t only. R t = EΛE H = i λ i e i e H i Page 14

15 Partial CSI : Parametric/Selection-CSI (A) In Ricean channel, H can be modeled as K H = 1 + K H K H ω }{{}}{{} fixed component variable component Channel information available is K. (B) Demmel Condition Number of channel matrix = κ Only κ is known about the channel. ( ) κ 2 = H 2 F λ min Page 15

16 Outline Introduction Perfect CSI-Tx Models for Partial CSI Exploiting Instantaneous-CSI Exploiting Instantaneous CSI : Beamforming vs Alamouti Exploiting Instantaneous CSI : Optimality of Beamforming Exploiting Correlation-CSI Exploiting Parametric/Selection-CSI Summary Page 16

17 Exploiting Instantaneous CSI : Beamforming vs Alamouti y = Es 2 hs + n Find S to minimize PEP based on channel estimate quality factor β = ĥ 2 α being known. For β = 0, S = x 0 x 1 x 1 x 0 (i.e. Standard Alamouti Coding) For β =, S = [ 2 h H h 2 F x 0 2 hh h 2 F x 1 ] (i.e. MRC Beamforming) Page 17

18 Exploiting Instantaneous CSI : Optimality of Beamforming When does w = h (Beamforming) achieve capacity? Beamforming Optimal 1.1 β Beamforming NOT optimal SNR Assuming channel knowledge quality factor β is estimated perfectly. Page 18

19 Outline Introduction Perfect CSI-Tx Models for Partial CSI Exploiting Instantaneous-CSI Exploiting Correlation-CSI Exploiting Correlation-CSI : Maximum Rate Exploiting Correlation-CSI : Minimum Error Rate Exploiting Parametric/Selection-CSI Summary Page 19

20 Exploiting Correlation-CSI : Maximum Rate We assume H = H w R 1/2 t where R t is known. Rate optimization is possible only in a statistical sense. Ergodic capacity assuming S = WX and R XX = I { C = log 2 det Optimum Pre-filter max E W 2 =M ( I MR + ρ M HWWH H H)}. W opt = Q Rt Λ 1/2 w Q Rt : eigenvector matrix of R t (i.e., R t = Q Rt Λ Rt Q H R t ) Λ w : diagonal power allocation matrix with Tr(Λ w ) = M Page 20

21 Exploiting Correlation-CSI : Maximum Rate -(ctd) For maximum rate, we should transmit along the eigenvectors of R t Finding Λ w (optimal power allocation matrix) in a closed form is an open problem Attempts to characterize the solution Using dominant eigenmode of H if rank of R t is one Stochastic waterpouring on the weighted eigenmodes of R t instead of eigenmodes of H Page 21

22 Exploiting Correlation-CSI : Simulation Results 10 9 Stochastic Water Pouring Unknown Channel Optimal Water Pouring with Channel Knowledge 8 Ergodic Capacity (bps/hz) SNR (db) Comparison of ergodic capacity (4 4 channel) R t = Page 22

23 Exploiting Correlation-CSI : Minimum Error Rate We assume H = H w R 1/2 t where R t is known. [y 1 y 2 y T ] = }{{} Y (i) Es M H ωr 1 2 t W [x 1 x 2 x T ] +n }{{} X (i) We optimize the Average Pairwise Error Probability (PEP) given by P (S (i) S (j) ) ( ) M 1 det ( I M + ρ ). 4M EH i,j WH R t WE i,j where E i,j = X (i) X (j) (M T T ) is an error between two codewords. Page 23

24 Exploiting Correlation-CSI : Minimum Error Rate -(ctd) Λ w is a diagonal matrix whose diagonal elements can be computed using waterpouring. Λ opt w = arg max W 2 =M M log k=1 ( 1 + ρ ) 4M λ(k) W λ(k) R t λ (k) E i,j The optimal pre-filter W opt satisfies W opt = Q Rt Λ 1 2 w(opt) Q H E i,j For OSTBC with E i,j E H i,j = d2 min I M W opt OST BC = Q R t Λ 1 2 w(opt) which implies that we can signal on the modes of R t. Page 24

25 Exploiting Correlation-CSI : Simulation Results 10 0 No precoding Precoding 10 1 Symbol Error Rate SNR (db) Precoding for Alamouti coding with R t improves performance. (2 2 channel) R t = Page 25

26 Outline Introduction Perfect CSI-Tx Models for Partial CSI Exploiting Instantaneous-CSI Exploiting Correlation-CSI Exploiting Parametric/Selection-CSI Selection-CSI : SM vs Alamouti Coding Selection-CSI : Tx Antenna Selection Summary Page 26

27 Selection-CSI : SM vs Alamouti Coding (AC) Keeping transmission rate same, we wish to choose between SM and AC. We can show that Demmel condition number of channel (κ 2 = H 2 F λ min ) can be used to minimize PEP. We choose between S 1 or S 2 based on κ η (threshold) S 1 = 4 x 0 x 2 x 1 x 3 5 (SM), S 2 = 4 x 0 x 1 x 1 x (AC) Page 27

28 Selection-CSI : Simulation Results Symbol error rate MIMO Diversity Spatial multiplexing Optimal selection SNR (db) Comparison of switched (SM,AC) transmission with fixed AC and SM (2 2 channel) Page 28

29 Selection-CSI : Tx Antenna Selection System with M T transmit antennas and P transmit RF chains: get best performance with fixed RF hardware. A total of ( M T P ) distinct choices which we index using i. Maximum Information Rate Our objective is to maximize C = max i,r ss log 2 det ( I M + ρ P H ir ss H H i ), with Tr(R ss ) = P and where R ss (P P ) is covariance matrix Page 29

30 Selection-CSI : Minimum SER with Alamouti Coding We assume an OSTBC transmission over the M R P link. The received SNR η η = ρ P H i 2 F The P columns of H that maximizes H i 2 F are the optimal antenna subset. It can be shown that ρ P H 2 F η opt ρ P P M T H 2 F. This shows that selection provides the same diversity order M T M R Page 30

31 Antenna Selection Performance Ergodic capacity(bps/hz) P SNR (db) 8 10 Ergodic Capacity with transmit antenna selection as a function of selected antennas P and SNR. M T = 4 Page 31

32 Outline Introduction Perfect CSI-Tx Models for Partial CSI Exploiting Instantaneous-CSI Exploiting Correlation-CSI Exploiting Parametric/Selection-CSI Summary Capacity with Instantaneous-CSI Capacity with Transmit Correlation-CSI Problem Taxonomy Conclusion Page 32

33 Summary : Capacity with Instantaneous-CSI For i.i.d zero mean Gaussian channels, at high SNR and large number of antennas Inst. CSI quality β = (full CSI) min{m t, M r } Asymptotic capacity [ C SISO + log ( max{mt,m r } min{m t,m r } )] β = 0 (no CSI) min{m t, M r } [ { C SISO + max 0, log ( M r ) }] M t Page 33

34 Summary : Capacity with Instantaneous-CSI Capacity gain with instantaneous Tx channel knowledge Full Tx CSI No Tx CSI No Tx CSI + gain 70 Ergodic capacity (bps/hz) Fixed antenna ratio M r M t SNR in db = 1 8, M r = [1, 2, 4, 8] going from bottom to top pairs. Page 34

35 Summary : Capacity with Transmit Correlation-CSI For H = H ω R 1 2 t, at high SNR and large number of transmit antennas C MIMO = r [ { ( Mr )}] C SISO + max 0, log + k where r = min{m r, k} and k = effective rank(r t ), k M t. r log(λ i ) i=1 Correlation MISO MIMO R t = I (i.i.d.) min(m t, M r ) C [ { SISO C SISO + max 0, log ( M r ) }] M t R t = qq H (rank 1) C SISO + log(λ max ) C SISO + log(λ max ) + log(m r ) Page 35

36 Summary : Capacity with Transmit Correlation-CSI 10 Ergodic capacity gain by knowing the transmit correlation 9 8 Ergodic capacity (bps/hz) Rt known Rt unknown Rt unknown + gain Number of transmit antennas R t rank one, SNR = 10dB, and ratio M r M t = 16, 8, 4, 2 from top to bottom pairs. Page 36

37 Available CSI-Tx Steers Energy in Preferred Directions No CSI Tx: Encoder No preferred direction e 1 Some CSI Tx: e 1 e 1, e 2 are preferred directions Encoder e 2 (modal beamforming) e 2 Page 37

38 Problem Taxonomy Performance Criterion Nature of Channel Knowledge - Instantaneous Capacity - Ergodic Capacity - Error Rate - SNR Coding? - Instantaneous - Statistical - Parametric/Selection Signal & Receiver Power Constraints - Alamouti, SM,.. - ML, MMSE,.. Channel Model - Sum Power - Per Antenna - Average or Peak - Time Selective/Flat - Frequency Selective/Flat Page 38

39 Time and Frequency Selective Fading Analogy between space and time/frequency domains Spatial modes frequency tones / time slots Exploiting time and frequency dimensions in MIMO Richer problem than time and frequency flat channels New tools: power control, space-time or space-frequency coding Page 39

40 Conclusion In MISO and MIMO wireless any CSI-Tx can improve performance. Practical systems usually have some CSI-Tx. Important research area for emerging wireless systems with many open questions. Page 40