Multiple Antennas Have a Big Multi- User Advantage in Wireless Communications Bertrand Hochwald (Bell Labs)

Goal: Talk with Multiple Users Simultaneously Multiple antennas at Access Point Downlink transmission Single antenna at terminal Rayleigh scattering channel environment Transmitter knows channel Simple receiver processing Near-capacity performance

3 Access Point Talking with Multiple Users Urban scattering environment Users moving slowly so transmitter can learn channel

4 System Model Access Point transmits x on M antennas to K users Terminals receive y h1h1 h4h4 h2h2 h3h D is positive diagonal and has unit trace (unit transmit power) w is additive noise, variance s 2

5 Properties of Sum Capacity Linear in min ( M,K ) H has complex- Gaussian entries (all users are statistically the same) MK

6 Attaining Downlink Capacity Until now: Optimal strategies require dirty paper methods Hierarchy of transmissions where users 1,…,k -1 are viewed as interference for user k (not currently practical) Sub-optimal strategies approximate dirty paper Lattice techniques (difficult) Circumvents dirty paper Dirty paper methods assume transmitter knows interference from users 1,…, k -1 at user k, but cannot control it. But we have control of interference! Our method:

7 Synopsis where  are positive constants  normalizes power l is a vector of integers 4 dB 30 bits/channel use 12 bits/channel use

Channel Inversion We want to transmit separate messages to K users simultaneously. If then we can use: where u is the K -dimensional data desired for the users and  is a normalizing constant The channel multiplies by H and the terminals receive:

9 Performance Depends on  Behavior of  Let. Choose  as either: Let’s choose this normalization Then This sum-rate is up to 80% of sum-capacity, but for the sum-rate is not linear in M. We are most interested in. Then s = H -1 u. Performance difference small

10 Properties of  when  When M=K and we are transmitting unit-variance Gaussian u, This density has infinite mean! The large eigenvalues of ( HH * ) -1 are to blame. Let  be the largest eigenvalue: This density also has infinite mean, and eigenvalues Inner products of u and eigenvectors

11 Sum-Rate for Channel Inversion Limit of this curve Huge gap

12 How to Make  Smaller  Two possible methods: Regularize inverse: “Perturb” u so that it points along the eigenvectors of ( HH * ) -1 corresponding to small eigenvalues We use a combination of both where the positive scalar  introduces a small amount of interference between users

13 Regularizing the Inverse  >0 makes the inverse more well-behaved. After passing through the channel, the signal becomes: The desired term for user k is: Q is matrix of eigenvectors of HH *  is matrix of eigenvalues of HH * Unfortunately, there is also interference from other users. Remember: this is the “unnormalized” signal

14 Maximize Signal-to-Interference+Noise Average desired signal over the eigenvectors Q Average undesired signal over the eigenvectors yields Amazingly, maximized for  =K  2, independently of ’s

15 Regularization Gives Linear Growth Linear growth but slope is wrong We need something more! Sum-capacity Plain channel inversion Regularized inversion Huge gap Not so huge gap

16 Vector Perturbation Goal: Form a v from u such that s = H -1 v has norm (much) smaller than H -1 u and v can still be decoded as u Use an idea from Tomlinson-Harashima precoding: v = u +  l where l is a vector of integers and  is a positive scalar. With a good choice of l we can steer v to the space of smaller eigenvalues of H -1. With a good choice of  we can recover u at the receivers, independently of l.

17 Connection with Tomlinson- Harashima Precoding Our goal is not to cancel interference. Instead, we choose a vector of integers l to minimize THP is a scalar technique used to mitigate interference at the transmitter. History (mid-1970’s): Suppose an interfering signal i is added at the receiver to the transmitted signal. Then the transmitter can compensate for i by sending s =( u - i ) mod  This is the same as sending s = u - i +  l. The receiver sees u+  l and can remove l with  =4 u e {-1,1} This vector perturbation gives a dramatic (orders of magnitude) decrease in  versus H -1 u Example Shortest vector in a lattice problem, with several known algorithms---“sphere encoder” We use the same f  ( y ) at the receivers

18 Brief Analysis of Minimization K=10, 16-QAM signals,  = v = u +  l Results of minimizing  k are eigenvalues of ( HH * ) -1 q k are eigenvectors of ( HH * ) -1 v aligned more closely to eigenvectors corresponding to small eigenvalues All eigenvalues contribute approximately the same to 

19 Combining Regularization and Perturbation Use “sphere encoder” to solve: Transmit: QR-based method (not described) Optimal  generally smaller than regularization alone

20 Combine with Turbo Codes Log-likelihood ratio for deciding between +1 and -1 Traditional Gaussian ratio (not used) Ratio formed from mod- function and likelihood: Turbo codes get “soft” decisions from this likelihood

21 Performance with UMTS Turbo Code 4 dB Sum-rate = 20 bits/channel use for K =10 Sum-rate = 30 bits/channel use for K =10

22 Getting Even Closer to Capacity? Match the Turbo Code to the channel Transmit at higher rates to users whose channels are best Sum-capacity not necessarily achieved with equal rates Compute and overcome mod penalty at receiver Analyze combined regularization/perturbation Optimal  Compare with capacity when only transmitter knows channel Receiver really only needs to know only E , not the actual channel But transmitter-only capacity seems to be intractable

23 Other Applications CDMA The matrix H is the correlation matrix of spreading codes sent through a channel with delay spread DSL The matrix H represents the coupling between twisted pairs of different users Full paper: