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

## Presentation on theme: "Capacity of Multi-antenna Guassian Channels Introduction: Single user with multiple antennas at transmitter and receiver. Higher data rate Limited bandwidth."— Presentation transcript:

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.

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

Similar presentations