Download Capacity of Multi-antenna Guassian Channels

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[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.
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
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
Independent Parallel Gaussian Channel
Capacity of deterministic channel:
• Maximize Mutual information
Power constraint
• 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.
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
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 separationindependent 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)It
The Capacity is calculated to be
m= min{r,t} & n=max{r,t}, Lji are Laquerre polynomials
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.
(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.
IN 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.