We saw that equal gain combining and maximal ratio combining result in tremendous improvement in bit error rate performance in a Rayleigh fading channel. These are receive diversity schemes i.e. schemes that work with multiple receive antennas. Now let us turn our attention to schemes that work with multiple transmit antennas. We know that the main aim of a combining scheme is to coherently add the signals. If the same signal is transmitted from multiple transmit antennas the resulting signals would not add up coherently when they arrive at the receiver (remember that each path introduces a random phase shift). One solution to this problem is that the channel state information (CSI) be fed back to the transmitter. So if this done quickly enough, before the channel state changes, the phase of the signals at the transmit side could be pre-adjusted so that when these signals arrive at the receiver they combine constructively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function[ber]=err_rate5(l,EbNo)
si=2*(round(rand(1,l))-0.5);
sq=2*(round(rand(1,l))-0.5);
s=si+j*sq;
n=(1/sqrt(2*10^(EbNo/10)))*(randn(1,l)+j*randn(1,l));
h1=(1/sqrt(2))*((randn(1,l))+j*(randn(1,l)));
h2=(1/sqrt(2))*((randn(1,l))+j*(randn(1,l)));
sr1=(1/sqrt(2))*s.*(conj(h1)./abs(h1));
sr2=(1/sqrt(2))*s.*(conj(h2)./abs(h2));
r=h1.*sr1+h2.*sr2+n;
si_=sign(real(r));
sq_=sign(imag(r));
ber1=(l-sum(si==si_))/l;
ber2=(l-sum(sq==sq_))/l;
ber=mean([ber1 ber2]);
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is observed that above scheme has exactly the same bit error rate performance as equal gain combining. The reason for this is that in the above scheme the noise at the receiver is halved (single receiver means single noise component) but the transmit power is also halved from each of the transmit antennas (to keep the total transmit power same). Thus it does not matter whether the phase adjustment happens at the receiver or the transmitter. But the important question is that can the channel state information be fed back to the transmitter quickly enough?

We just saw the advantage an equal gain combiner (a combining scheme that just adds the signals after co-phasing them) provides in a Rayleigh fading channel. Lets now look at a variant of this scheme called maximal ratio combining (MRC). In MRC the signals arriving at the receivers are weighted by the channel gains i.e. a stronger signal is weighted more than a weaker signal before combining. It must be noted that in an actual system the received signals are both scaled and phase shifted thus an MRC receiver multiplies the received signals by the complex conjugate of the channel coefficients before addition.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function[ber]=err_rate4(l,EbNo)
si=2*(round(rand(1,l))-0.5);
sq=2*(round(rand(1,l))-0.5);
s=si+j*sq;
n1=(1/sqrt(2*10^(EbNo/10)))*(randn(1,l)+j*randn(1,l));
h1=(1/sqrt(2))*((randn(1,l))+j*(randn(1,l)));
n2=(1/sqrt(2*10^(EbNo/10)))*(randn(1,l)+j*randn(1,l));
h2=(1/sqrt(2))*((randn(1,l))+j*(randn(1,l)));
r1=abs(h1).*s+n1;
r2=abs(h2).*s+n2;
r=abs(h1).*r1+abs(h2).*r2;
si_=sign(real(r));
sq_=sign(imag(r));
ber1=(l-sum(si==si_))/l;
ber2=(l-sum(sq==sq_))/l;
ber=mean([ber1 ber2]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

OR (using the complex channel coefficient)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function[ber]=err_rate4(l,EbNo)
si=2*(round(rand(1,l))-0.5);
sq=2*(round(rand(1,l))-0.5);
s=si+j*sq;
n1=(1/sqrt(2*10^(EbNo/10)))*(randn(1,l)+j*randn(1,l));
h1=(1/sqrt(2))*((randn(1,l))+j*(randn(1,l)));
n2=(1/sqrt(2*10^(EbNo/10)))*(randn(1,l)+j*randn(1,l));
h2=(1/sqrt(2))*((randn(1,l))+j*(randn(1,l)));
r1=h1.*s+n1;
r2=h2.*s+n2;
r=conj(h1).*r1+conj(h2).*r2;
si_=sign(real(r));
sq_=sign(imag(r));
ber1=(l-sum(si==si_))/l;
ber2=(l-sum(sq==sq_))/l;
ber=mean([ber1 ber2]);
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We see that there is an incremental improvement in BER using MRC instead of EGC (1dB can sometimes be significant).

Note:

1. The performance of MRC is the same using both the techniques given above.

2. Phase rotation of the noise component does not effect the BER performance.

When wireless signals travel from a single transmit antenna to multiple receive antennas they experience different fading conditions. While signal from one path may experience a deep fade the signal from another path may be stronger. Therefore selecting the stronger of the two signals (selection combining, threshold combining) or adding the signals (equal gain combining, maximal ratio combining) would always yield much better results (lower bit error rate). However, there must be sufficient spacing between the different receive antennas for the received signals to be dissimilar (uncorrelated). In the simulation below we consider a 1-Tx, 2-Rx scenario. The signals arriving at the two receive antennas are added together before detection.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function[ber]=err_rate4(l,EbNo)
si=2*(round(rand(1,l))-0.5);
sq=2*(round(rand(1,l))-0.5);
s=si+j*sq;
n1=(1/sqrt(2*10^(EbNo/10)))*(randn(1,l)+j*randn(1,l));
h1=(1/sqrt(2))*((randn(1,l))+j*(randn(1,l)));
n2=(1/sqrt(2*10^(EbNo/10)))*(randn(1,l)+j*randn(1,l));
h2=(1/sqrt(2))*((randn(1,l))+j*(randn(1,l)));
r1=abs(h1).*s+n1;
r2=abs(h2).*s+n2;
r=r1+r2;
si_=sign(real(r));
sq_=sign(imag(r));
ber1=(l-sum(si==si_))/l;
ber2=(l-sum(sq==sq_))/l;
ber=mean([ber1 ber2]);
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Note:

1. Not only the signals on the two paths experience uncorrelated fading but the the noise at the receiver front ends is also uncorrelated.

2. In reality the signals over both the paths would also experience random phase shifts but these can be removed before the combining process at the receiver.