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?

#### Author: Yasir Ahmed (aka John)

More than 20 years of experience in various organizations in Pakistan, the USA, and Europe. Worked as a Research Assistant within the Mobile and Portable Radio Group (MPRG) of Virginia Tech and was one of the first researchers to propose Space Time Block Codes for eight transmit antennas. The collaboration with MPRG continued even after graduating with an MSEE degree and has resulted in 12 research publications and a book on Wireless Communications. Worked for Qualcomm USA as an Engineer with the key role of performance and conformance testing of UMTS modems. Qualcomm is the inventor of CDMA technology and owns patents critical to the 4G and 5G standards.

Raj: What this expression is doing is essentially introducing a phase shift before the transmission such that after passing through the channel the signal has zero phase shift. The expression (conj(h1)./abs(h1)) introduces the phase shift at the Tx and multiplication at the Rx in the expression h1.*sr1 cancels that phase shift. Note we only adjusted the phase and not the amplitude.

Hope this makes sense!

how can the following expression be used to fedback the channel state information (CSI) to the transmitter? : sr1=(1/sqrt(2))*s.*(conj(h1)./abs(h1)); suggest links if any for further references. thanks