# MIMO Capacity in a Fading Environment

The Shannon Capacity of a channel is the data rate that can be achieved over a given bandwidth (BW) and at a particular signal to noise ratio (SNR) with diminishing bit error rate (BER). This has been discussed in an earlier post for the case of SISO channel and additive white Gaussian noise (AWGN). For a MIMO fading channel the capacity with channel not known to the transmitter is given as (both sides have been normalized by the bandwidth [1]):

Shannon Capacity of a MIMO Channel

where NT is the number of transmit antennas, NR is the number of receive antennas, γ is the signal to interference plus noise ratio (SINR), INR is the NRxNR identity matrix and H is the NRxNT channel matrix. Furthermore, hij, an element of the matrix H defines the complex channel coefficient between the ith receive antenna and jth transmit antenna. It is quite obvious that the channel capacity (in bits/sec/Hz) is highly dependent on the structure of matrix H. Let us explore the effect of H on the channel capacity.

Let us first consider a 4×4 case (NT=4, NR=4) where the channel is a simple AWGN channel and there is no fading. For this case hij=1 for all values of i and j. It is found that channel capacity of this simple channel for an SINR of 10 dB is 5.36bits/sec/Hz. It is further observed that the channel capacity does not change with number of transmit antennas and increases logarithmically with increase in number of receive antennas. Thus it can be concluded that in an AWGN channel no multiplexing gain is obtained by increasing the number of transmit antennas.

We next consider a more realistic scenario where the channel coefficients hij are complex with real and imaginary parts having a Gaussian distribution with zero mean and variance 0.5. Since the channel H is random the capacity is also a random variable with a certain distribution. An important metric to quantify the capacity of such a channel is the Complimentary Cumulative Distribution Function (CCDF). This curve basically gives the probability that the MIMO capacity is above a certain threshold.

It is obvious (see figure above) that there is a very high probability that the capacity obtained for the MIMO channel is significantly higher than that obtained for an AWGN channel e.g. for an SINR of 9 dB there is 90% probability that the capacity is greater than 8 bps/Hz. Similarly for an SINR of 12 dB there is a 90% probability that the capacity is greater than 11 bps/Hz. For a stricter threshold of 99% the above capacities are reduced to 7.2 bps/Hz and 9.6 bps/Hz.

In a practical system the channel coefficients hij would have some correlation which would depend upon the antenna spacing. Lower the antenna spacing higher would be the antenna correlation and lower would be the MIMO system capacity. This would be discussed in a future post.

The MATLAB code for calculating the CCDF of channel capacity of a MIMO channel is given below.

```clear all
close all

Nr=4;
Nt=4;
I=eye(Nr);
g=15.8489;

for n=1:10000
H=sqrt(1/2)*randn(Nr,Nt)+j*sqrt(1/2)*randn(Nr,Nt);
C(n)=log2(det(I+(g/Nt)*(H*H')));
end

[a,b]=hist(real(C),100);
a=a/sum(a);
plot(b,1-cumsum(a));
xlabel('Capacity (bps/Hz)')
ylabel('Probability (Capacity > Abcissa)')
grid on```

[1] G. J. Foschini and M. J. Gans,”On limits of Wireless Communications in a Fading Environment when Using Multiple Antennas”, Wireless Personal Communications 6, pp 311-335, 1998.

# Average Cell Throughput Calculations for LTE

Average Cell Throughput requires the following simulation results

• Average SINR distribution table (system level result), which provides the SINR probability

• Average throughput or spectral efficiency versus average SINR table (link level result)

For urban channel model and a fixed inter-site distance of 1732m,downlink throughput for LTE for different values of SINR is shown below.

MCS vs SINR

Average Cell Throughput=Σ(Pi*Ri)

where

Pi=Probability of occurrence of a specific SINR value at cell edge obtained using simulations
Ri=Average throughput corresponding to SINR range

Let us consider the following distribution for the SINR at the cell edge:

P1=0.5 (SINR=1.50-3.50 dB)
P2=0.25 (SINR=3.50-7.00 dB)
P3=0.15 (SINR=7.00-9.50 dB)
P4=0.10 (SINR=9.50-11.50 dB)

Cell Throughput=(0.50*4)+(0.25*6)+(0.15*8)+(0.10*12)=5.9 Mbps

Note: This throughput is much less than the theoretical maximum which assumes that the full 20 MHz bandwidth is being utilized in a 4×4 MIMO configuration.

# WCDMA Capacity (Mbps)

The capacity of any wireless communication channel is given by the well known Shannon Capacity Theorem:

C=B*log2(1+SNR)

or

C=B*log2(1+P/(NoB))

where C is the capacity of the channel in bits/sec, P in the noise power in Watts, No is the noise power spectral density in Watts/Hz and B is the channel bandwidth in Hz. It is obvious that the channel capacity increases with increase in signal power. However, the relationship with bandwidth is a bit complicated. The increase in bandwidth decreases the SNR (keeping the signal power and noise power spectral density same). Therefore the capacity does not increase linearly with bandwidth.

It is assumed that the signal power is -80dBm and the noise power spectral density is -170dBm/Hz.

Let us now consider the case of a WCDMA system where the noise component consists of the AWGN noise as well Multiple Access Interference (MAI). The above capacity formula is then modified as:

C=B*log2(1+SINR)

or

C=B*log2(1+P/(NoB+(U-1)*P))

where U is the number of users.

It is assumed that WCDMA system has a bandwidth of 5MHz, signal power of -80dBm and noise power spectral density of -170dBm/Hz. It is observed that the capacity of the WCDMA system decreases exponentially with increase in number of users (all users are assumed to have equal power at the receiver). The capacity of a 5MHz channel drops from about 38Mbps for a single user to about 370kbps for 20 users (the combined capacity of 20 users is 7.4Mbps which is much lesser than the capacity of a single user in AWGN only).

Note:

1. In some references the signal power is adjusted by the processing gain but I think that this is not correct for capacity calculations because in that case the bandwidth should be adjusted as well.

2. The capacity in fading environments would be less than AWGN capacity. However, multi-antenna systems allow for higher capacities.