Capacity of Multi-antenna Guassian Channels

1. Capacity of Multi-antenna Guassian Channels Introduction: Single user with multiple antennas at transmitter and receiver. Higher data rate Limited bandwidth and power resources

2. Channel Model: • y = Hx + n (linear model) • H is a r x t complex matrix, y is a r x 1 received matrix & x is t x 1 tx matrix • n- circularly symmetric gaussian noise vector with zero mean and E[nnt] = Ir • E[xtx] ≤ P, where P is the total power • yi =∑hij xj + ni, i = 1,….,r (the received signal is a linear combination of tx signals.) • hij- gains of each transmission path( from j to i) • Component xj is the elementary signal of vector x transmitted from from antenna j.

3. Multiple Antenna System

4. Channel State Information(CSI): • Determined by the values taken by H • Crucial factor for performance of transmission. • Estimate of fading gains fedback to transmitter(pilot signals). H matrix • Deterministic • Random • Random but fixed when chosen.

5. Deterministic Channel Using Singular value decomposition Where U and V are unitary and D is diagonal. Componentwise form: It can be seen that the channel now is equivalent to a set of min{r,t} parallel channels

6. Independent Parallel Gaussian Channel

8. Capacity of deterministic channel: • Maximize Mutual information Power constraint

9. Each subchannel contributes to the total capacity through log2(λiµ)+. • More power is allocated to subchannels with higher SNR. • If λiµ≥1 the subchannel provides an effective mode of transmission. • We’ve used water-filling technique based on the assumption that the transmitter has complete knowledge of the channel.

10. Inference: If t=r=m, & H=Im Transmission occurs over m parallel AWGN channels each with SNR p/m and capacity log2(1+p/m) Therefore C = mlog2(1+p/m) Capacity is proportional to transmit/receive antennas As m inf, the capacity tends to the limiting value C = plog2e

11. Independent Rayleigh Fading Channel Assumptions: • H is a random matrix. Each channel use corresponds to an independent realization of H & this is known only to the receiver. • Entries of H are independent zero mean gaussian with real and imaginary parts having variance ½. • Each entry of H has uniformly distributed phase and Rayleigh distributed magnitude(antenna separation-independent fading) • H is independent of x and n.

12. Capacity: The output of the channel is (y,H) = (Hx+n,H) Mutual Information between i/p and the o/p is given by: The MI is maximized by complex circularly symmetric gaussian distribution with mean zero and covariance (P/t)It The Capacity is calculated to be m= min{r,t} & n=max{r,t}, Lji are Laquerre polynomials

13. Inference: (i)If t=1 and r=n(r>>t), C = log2(1+rp) (ii)If t>1 & r inf, C = t log2(1+(p/t)r). (iii) If r=1, t=n(t>>r) C = log2(1+p) (iv) If r>1 and t inf, C = r log2(1+p) The capacity increases only logarithmically in i and iii.

14. (v) If r=t i.e m=n=r the capacity plot is as below(for various values of ‘p’ b/w 0 & 35db)

15. Non-Ergodic Channels: • H is chosen randomly at the beginning and held fixed for all transmission. • Avg Channel capacity has no meaning. • Outage probability- probability that the tx rate increases the MI. IN is the instantaneous MI & R is the tx rate in bits/channel use

16. Inference: As r and t grow • The instantaneous MI tends to a gaussian r.v in distribution. • The channel tends to an ergodic channel

17. Multi-access Channel • Number of tx eaxh with multiple tx antennas and each subject to a power constraint P. • Single receiver • Received signal y

18. The achievable rate vector is given by: Where the C(a,b,P) is the single user a receiver b transmitter capacity under power constraint P

19. Conclusion: Use of multiple antennas increases achievable rates on fading channels if • Channel parameters can be estimated at Rx • Gains between different antenna pairs behave idependently.