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

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[nn t ] = I r E[x t x] ≤ P, where P is the total power y i =∑h ij x j + n i, i = 1,….,r (the received signal is a linear combination of tx signals.) h ij - gains of each transmission path( from j to i) Component x j is the elementary signal of vector x transmitted from from antenna j.

Multiple Antenna System

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.

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

Independent Parallel Gaussian Channel

Capacity of deterministic channel: Maximize Mutual information Power constraint

Each subchannel contributes to the total capacity through log 2 (λ i µ) +. More power is allocated to subchannels with higher SNR. If λ i µ≥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.

Inference: If t=r=m, & H=I m pp Transmission occurs over m parallel AWGN channels each with SNR p/m and capacity log 2 (1+p/m) p Therefore C = mlog 2 (1+p/m) Capacity is proportional to transmit/receive antennas As m inf, the capacity tends to the limiting value p C = plog 2 e

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.

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)I t The Capacity is calculated to be m= min{r,t} & n=max{r,t}, L j i are Laquerre polynomials

Inference: (i)If t=1 and r=n(r>>t), p C = log 2 (1+rp) (ii)If t>1 & r inf, p C = t log 2 (1+(p/t)r). (iii) If r=1, t=n(t>>r) p C = log 2 (1+p) (iv) If r>1 and t inf, p) C = r log 2 (1+p) The capacity increases only logarithmically in i and iii.

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

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. I N is the instantaneous MI & R is the tx rate in bits/channel use

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

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

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

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