The Effect of Spatial Correlations on MIMO Capacity: A (not so) Large N Analytical Approach: Aris Moustakas 1, Steven Simon 1 & Anirvan Sengupta 1,2

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1 The Effect of Spatial Correlations on MIMO Capacity: A (not so) Large N Analytical Approach: Aris Moustakas 1, Steven Simon 1 & Anirvan Sengupta 1, 1, Rutgers University

2 Outline Aim: Calculate statistics of MIMO Capacities with Correlated Channels and Interference Use partial knowledge of channel (at transmitter) to optimize throughput Method (large antenna N analysis) Results/Examples Gaussian character of capacity Applications

3 Mutual Information with Correlations & Interference I log = + det ( + + ν + GQG + HH ) ( ) det ν + HH G : n R x n T Channel matrix H : n I xn T Interferer matrix (known to receiver) ν : n R x n R Noise (due to interference with channel unknown to receiver) Q : Transmitted Signal covariance G (and thus I) are random: Treat them statistically Assumption: Temporal Average = Spatial Average over channel realizations 3

4 Statistical Treatment of G Small parameter λ/l = Wavelength/Mean Free Path <= Leading correction in systematic expansion in λ/l : E[G G + ] since E[G]=0 (diffuson approximation) G, H: Gaussian random (not generally iid) [ ] * s s s G = T ij ab E G E ia jb R n T [ ] * I I I H H = T ij ab ia jb R n I s = S / I = I / N N In general not separable: e.g. MUD or multi-keyholes [ ] * G jb E G n k k k ia = T ij R ab k T 4

5 Correlation Matrix R (and analogously T) Response of Antenna α to Incoming Wave k of Amplitude A(k) (k) α χ r α = d kˆ A (k) χ α (k) R = E r * dk ˆ αβ χ α *( k ) χ β ( k ) w ( k [ r ] = α β w(k) wave weight: depends on angle-spread & other channel parameters ) A(k) w ( ϕ ) ϕ = exp δ Based on antenna/array and channel properties can determine R ab correlated antennas: rank R = 1; (one big antenna) beamforming uncorrelated antennas: eigs of R equal no interference (each antenna sees different field) 5

6 Statistics of Mutual Information Aim: Calculate Statistical Properties of I Method: Large (but finite) Number Antenna Approximation Calculate Moments of I (e.g. E[ I ], Var[ I ], Sk[ I ] ): Quantities of Interest: E[ I ] Outage Capacity (Solve for Prob(I>I 0 )=p 0 ) Optimize Transmission given partial channel information C = max Q E[I Q] (versus C = E[ max Q (I Q) ]) Optimization based on quasi-static channel statistics T, R: Statistical Waterpouring C is Realistic closed loop capacity 6

7 Gaussianity of the PDF(I) Approximation: PDF ( I ) N, ( E [] I Var [] I ) Open-Loop and Closed-Loop Capacity distributions for n T =n R =3 Open Loop; correlated Tx Closed Loop; correlated Tx True Closed Loop; correlated Tx SNR = 0dB Probability that Capacity<x-axis Nominally valid for large antenna numbers Surprisingly accurate even for n T =n R =3 and correlated channels Angle spread 5 o Linear array with d min =λ C=max Q E[ I Q] C=E[max Q I Q] Statistical Waterpouring gives good results Simulated Distribution Simulated Distribution Analytic Dis tribution Capacity (bps/hz) 7

8 Gaussianity of the PDF(I) Approximation: PDF ( I ) N, ( E [] I Var [] I ) Nominally valid for large antenna numbers Accurate even with small n T =n R =n I =, Open Loop Capacity distribution for n T =n I =n R =n = & 3; All channels iid Analytic; n = E[ I ] = 5.75; Var( I ) = 1.38 Simulated; n= E[ I ] = 5.84; Var( I ) =1.40 S NR = s = 10dB I = 1 Analytic; n=3 E I ] = 8.6; Var( I ) = 1.38 Simulated; n=3 E[ I ] = 8.68; Var( I ) = 1.38 Probability that Capacity<x-axis Simulated Distribution Analytic Distribution Capacity (bps/hz) 8

9 Why is N(E[I],Var[I]) so accurate? Main Reason: E[I], Var[I]: For finite n T, n R, n I they are finite and n dependent : the leading terms capture (most) n dependence Higher moments (Skewness etc): O(1/n) Also correction to E[I] = O(1/n), Var[I]=O(1/n ) Additional small parameter (for not too large SIR)? 9

10 Method: Calculate generating function: g ( g ( µ ) [ ] 3 µ I µ µ exp µ E [] I + Var [] I Sk [] I L µ ) E e µ = + E det µ ( ν + GQG + HH ) ( ) + µ det ν + HH = 6 Replica Trick: Generate µ replicas of determinants Then analytically continue µ to zero (Sengupta & Mitra) (Parisi) 10

11 Method: [ ( ) ( ) ] + + µ + + µ det ν + GQG + HH ν + g ( µ ) = E det HH To average over G use identity: ( ) X ( + GQG + HH ) X det GQG HH = ν ν + + dx e To average over H tricky Need to combine positive and negative powers of dets: ( ) HH + + da e A ( + HH ) A det + = ν ν Thus A have to be Grassman variables: da = 0; AdA = 1; AB = BA ; A = 0 11

12 Method: Represent determinant by integrals Integrate out G, H (correlated Gaussian) Introduce auxiliary µ x µ matrix variables {t, r} to represent g(µ): { dt, dr } exp [ S ( t, ) ] g ( µ ) = r Analytically continue µ to zero and find saddle point of S for large n t αβ t δ αβ + δ t αβ ; α, β = 1, L µ Saddle point S 0 gives ergodic capacity Corrections (δt αβ ) k+ give systematically higher moments, e.g. Var, Sk Need to solve few algebraic equations to find saddle point 1

13 Equations to Solve. Example: (n I = 0) [ ] * s G = T ab E G ia jb ij R n T Given [ + tt ] + Tr log [ 1 + rr ] n tr E [ I ] = Tr log1 s s T t r = = 1 n T 1 n T Tr 1 Tr 1 R + rr T + tt s s s s Solve for t, r Answer 13

14 Results: Ergodic Capacity E[I]: Open Loop (Q=1) Large N approach valid For n T > n T dependence of Capacity due to antenna correlations = Real SNR=100 Antenna Spacing = λ/ 3D : Capacity ~ N /3 Only Antennas on the Surface Resolve Incoming Directions 14

15 Results: Ergodic Capacity E[I] Open Loop vs. Closed Loop: 3 Open and Closed Loop Capacity for 16 Tx & 16 Rx antenna arrays. Angle Spread =5 o Calculate C = max Q E[ I Q] Determine optimal Q analytically.5 i.i.d. channel Closed Loop with Covariance Feedback Open Loop SNR = 10dB Capacity per antenna(bps/hz) No Interferers Result: SNR dependent High SNR=10: Low gain Closed Loop with Covariance Feedback Open Loop i.i.d case Antenna separation d min /λ 15

16 Results: Ergodic Capacity E[I] Open Loop vs. Closed Loop: 1 Open and Closed Loop Capacity for 16 Tx & 16 Rx antenna arrays. Angle Spread =5 o. Calculate C = max Q E[ I Q] 0.9 No Interferers 0.8 Result: SNR dependent Low SNR=1: High gain Beamforming beneficial at low SNR Closed Loop with Covariance Feedback i.i.d. channel SNR = 0dB Open Loop Capacity per antenna(bps/hz) Antenna separation d min /λ 16

17 Results: Ergodic Capacity E[I] Gain of knowing the channel of the interferer at the receiver: Knowing H vs. ν (E(HH*)) Open Loop Capacity for n T = n R = 16 system in the presence of n I interferers with partially known channel SINR = 10dB Gain is substantial if channel is known accurately (INR= I ) Capacity per antenna (bps/hz) Result is Angle-Spread (δ) dependant ν 0 and n I <n T results to: C If n I =n T (=n R ) and ν 0 : C = n T log 1 + [ SNR ] 4 3 n = 8 I n I = 16 R I = 1 δ I = 0 o Percent of noise with known channel I /(1+ I ) 17

18 Results: Higher Moments: Var[I], Sk[I], Variance: Calculate using O(1) in n T, n R etc (O(1/N) for n T <<n R (or vice-versa)) Diverges (logarithmically) at large SNR (small parameter?) Large N result gives very accurate prediction Skewness (and correction to E[I]): From corrections O(1/N) (for n T ~n R etc) Higher moments: Cum ( δt ) 3 αβ k = O ( N k ( δ t αβ ) ; α, β = 1, L µ ) I N ( E [ I ], Var [ I ]) 18

19 Summary Applications Powerful method to calculate MIMO capacities with even few antennas with correlated channels/noise and interference applicable to other cases (MUD etc) Straightforward method to find optimal transmission scheme based on partial channel information. Gaussian approximation of capacity accurate even for few antennas. simplifies System Level Analysis. analytic results for system level capacity scheduling Hochwald et al allows Outage Capacity calculation. 19