Start with a base station consisting of a uniform linear array (ULA) with 16 antennas, and two users having receiver ULA arrays with 8 and 4 antennas, respectively. Show that using block diagonalization-based precoding and combining weights achieves spatial multiplexing, where the received signal at each user can be decoded without interference from the other user. Specify two data streams for each user.

Specify the transmitter location in txpos and two user receiver locations in rxpos1 and rxpos2. Array elements are spaced one-half wavelength apart.

txpos = (0:15)*0.5;
rxpos1 = (0:7)*0.5;
rxpos2 = (0:3)*0.5;

Create the channel matrix cell array using scatteringchanmtx and then compute the beamforming weights wp and wc. Each channel corresponds to a user. Assume that the channels have 10 scatterers. Each channel has two subchannels specified by the vector ns.

chanmat = {scatteringchanmtx(txpos,rxpos1,10), ...
    scatteringchanmtx(txpos,rxpos2,10)};
ns = [2 2];
[wp,wc] = blkdiagbfweights(chanmat,ns);

The weights diagonalize the channel matrices for each user.

For channel 1:

disp(wp*chanmat{1}*wc{1})

   8.2269 - 0.0000i   0.0000 - 0.0000i
   0.0000 + 0.0000i   6.1371 - 0.0000i
   0.0000 - 0.0000i  -0.0000 + 0.0000i
   0.0000 - 0.0000i   0.0000 + 0.0000i

For channel 2:

disp(wp*chanmat{2}*wc{2})

  -0.0000 + 0.0000i  -0.0000 + 0.0000i
  -0.0000 + 0.0000i  -0.0000 + 0.0000i
   8.7543 - 0.0000i   0.0000 - 0.0000i
   0.0000 + 0.0000i   4.4372 + 0.0000i

First create four subchannels to carry the data streams: two subchannels per channel. Each data stream contains 20 samples of $\pm 1$. Precode the input streams and combine the streams to produce the recovered signals.

x = 2*round(rand([20,4])) - 1;
xp = x*wp;
y1 = xp*chanmat{1} + 0.1*randn(20,8);
y2 = xp*chanmat{2} + 0.1*randn(20,4);
y = [y1*wc{1},y2*wc{2}];

Overlay stem plots of the input and recovered signals to show that the received user signals are the same as the transmitted signals.

for m = 1:4
    subplot(4,1,m)
    s = stem([x(:,m) 2*((real(y(:,m)) > 0) - 0.5)]);
    s(1).LineWidth = 2;
    s(2).MarkerEdgeColor = 'none';
    s(2).MarkerFaceColor = 'r';
    ylabel('Signal')
    title(sprintf('User %d Stream %d',ceil(m/2),rem(m-1,2) + 1))
    if m==1
        legend('Input','Recovered','Location','best')
    end
end
xlabel('Samples')

Start with a base station consisting of a uniform linear array (ULA) with 16 antennas, and two users having receiver ULA arrays with 8 and 5 antennas, respectively. Show how to use three-dimensional arrays of channel matrices to handle two subcarriers. Then, the channel matrix for the first user takes the form 2-by-16-by-8 and the channel matrix for the second users takes the form 2-by-16-by-5. Also assume that there are two data streams for each user.

Specify the transmitter location in txpos and two user receiver locations in rxpos1 and rxpos2. Array elements are spaced one-half wavelength apart.

nr1 = 8;
nr2 = 5;
txpos = (0:15)*0.5;
rxpos1 = (0:(nr1-1))*0.5;
rxpos2 = (0:(nr2-1))*0.5;

Create the channel matrices using scatteringchanmtx and put them in a cell array. To create a second subchannel for each receiver, duplicate each channel matrix. Assume 10 point scatterers in computing the channel matrix.

smtmp1 = scatteringchanmtx(txpos,rxpos1,10);
smtmp2 = scatteringchanmtx(txpos,rxpos2,10);
sm1 = zeros(2,16,8);
sm2 = zeros(2,16,5);
sm1(1,:,:) = smtmp1;
sm1(2,:,:) = smtmp1;
sm2(1,:,:) = smtmp2;
sm2(2,:,:) = smtmp2;
chanmat = {sm1,sm2};

Specify that there are two data streams for each user.

ns = [2 2];

Specify the transmitted powers for each subcarrier.

pt = [1.0 1.5];

Compute the beamforming weights.

[wp,wc] = blkdiagbfweights(chanmat,ns,pt);

Show that the channels are diagonalized for the first subcarrier.

ksubcr = 1;
wpx = squeeze(wp(ksubcr,:,:));
chanmat1 = squeeze(chanmat{1}(ksubcr,:,:));
chanmat2 = squeeze(chanmat{2}(ksubcr,:,:));
wc1 = squeeze(wc{1}(ksubcr,:,:));
wc2 = squeeze(wc{2}(ksubcr,:,:));
wpx*chanmat1*wc1

ans = 4×2 complex

   8.2104 + 0.0000i  -0.0000 - 0.0000i
   0.0000 - 0.0000i   5.9732 - 0.0000i
   0.0000 - 0.0000i  -0.0000 - 0.0000i
   0.0000 - 0.0000i   0.0000 - 0.0000i

wpx*chanmat2*wc2

ans = 4×2 complex

  -0.0000 + 0.0000i   0.0000 + 0.0000i
  -0.0000 + 0.0000i   0.0000 + 0.0000i
   8.8122 + 0.0000i  -0.0000 + 0.0000i
   0.0000 + 0.0000i   4.8186 - 0.0000i

Propagate the signals to each user and then decode. Generate four streams of random data containing -1's and +1's and having two columns for each user. Each stream is a subchannel.

x = 2*(round(rand([20 4]))) - 1;

Precode the data streams.

xp = x*wpx;
y1 = xp*chanmat1 + 0.1*randn(20,8);
y2 = xp*chanmat2 + 0.1*randn(20,5);

Decode the data streams.

y = [y1*wc1,y2*wc2];

Overlay stem plots of the input and recovered signals to show that the received user signals are the same as the transmitted signals.

for m = 1:4
    subplot(4,1,m)
    s = stem([x(:,m) 2*((real(y(:,m)) > 0) - 0.5)]);
    s(1).LineWidth = 2;
    s(2).MarkerEdgeColor = 'none';
    s(2).MarkerFaceColor = 'r';
    ylabel('Signal')
    title(sprintf('User %d Stream %d',ceil(m/2),rem(m-1,2) + 1))
    if m==1
        legend('Input','Recovered','Location','best')
    end
end
xlabel('Samples')