We have already seen in previous posts that the BER of BPSK increases significantly when the channel changes from a simple AWGN channel to a fading channel. One solution to this problem, that was proposed by Alamouti, was to use Transmit Diversity i.e. multiple transmit antennas transmit the information over multiple time slots increasing the likelihood of receiving the information. We have considered the simplest case of two transmit antennas and BPSK modulation (QPSK modulation would give the same BER with twice the throughput). Given below is the Python code for this, feel free to modify it and run it from the console given below.

So we have seen that multiple transmit antennas provide the same gain as multiple receive antennas if the channel state information can be fed back to the transmitter. But what if the channel state information cannot be fed back to the transmitter (or it can be done but not quickly enough). The solution to this problem is the so called “Alamouti Scheme”. In this scheme two symbols are simultaneously transmitted from two transmit antennas and in the next time slot phase shifted versions of these two symbols are transmitted over the two transmit antennas. The channel is assumed to be quasi static i.e. it is static over the duration of two time slots but then changes for the next two time slots. A combining scheme is used at the receiver which separates the two symbols.

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% l: Length of symbol sequence
% EbNo: Energy per bit to noise power spectral density
% ber: Output bit error rate
% Copyright www.raymaps.com
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function[ber]=err_rate7(l,EbNo)
s1i=2*(round(rand(1,l))-0.5);
s1q=2*(round(rand(1,l))-0.5);
s1=s1i+j*s1q;
s2i=2*(round(rand(1,l))-0.5);
s2q=2*(round(rand(1,l))-0.5);
s2=s2i+j*s2q;
n1=(1/sqrt(2*10^(EbNo/10)))*(randn(1,l)+j*randn(1,l));
n2=(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)));
r1=h1.*(sqrt(1/2)*s1)+h2.*(sqrt(1/2)*s2)+n1;
r2=-h1.*conj(sqrt(1/2)*s2)+h2.*conj(sqrt(1/2)*s1)+n2;
s1_=conj(h1).*r1+h2.*conj(r2);
s2_=conj(h2).*r1-h1.*conj(r2);
s1i_=sign(real(s1_));
s1q_=sign(imag(s1_));
ber1=(l-sum(s1i==s1i_))/l;
ber2=(l-sum(s1q==s1q_))/l;
ber=mean([ber1 ber2]);
return
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It is observed that this scheme is 3dB worse than MRC (and transmit diversity with CSI). This reason for this is that unlike MRC the signals are transmitted from two transmit antennas thus the power is halved at each transmit antenna (this scheme is also 3dB worse than transmit diversity with CSI at transmitter because although both schemes transmit half the power from each source but in this scheme the noise power is doubled due to the combining scheme working over two time slots).