0
                                             0
                                             0
                                             0
                                             0
                                             0

Search distance            [wHat] =     1.9670

# of expansions              [nv] =          7

# of flops (approx.)         [nf] =        384

Verify solution          [zHatML] =          0
                                             1
                                             0
                                             0
                                             0
                                             0
                                             0
                                             0

Verify squared distance  [wHatML] =     1.9670

</pre><h2>References<a name="5"></a></h2>
      <p>The author would like to acknowledge the following work on the design and optimization of spherical LAST codes by means of
         which the sample 2x2x2 code provided along with this distribution in <tt>optimizedLASTcode.mat</tt> was generated:
      </p>
      <p>I. Berenguer, X. Wang, N. Prasad, J. Wang, and M. Madihian, <b>Design of minimium-error-rate lattice (space-time) codes via stochastic optimization and gradient estimation</b>, IEEE Global Communications Conference (Globecom), St. Louis, MO, Dec. 2005.
      </p>
      <p>For more details on the <tt>SEAL_det</tt> algorithm and boundary control for spherical LAST decoding, please see the following paper:
      </p>
      <p>K. Su, I. Berenguer, I.J. Wassell, X. Wang, <b>Efficient maximum-likelihood decoding of spherical lattice space-time codes</b>, IEEE International Conference on Communications (ICC), Istanbul, Turkey, June 2006.
      </p>
      <p>and its companion website:</p>
      <p><a href="http://www.comm.utoronto.ca/~karen/seqdecLs.php">A Depth-First (Space-Time) Lattice Decoder with Spherical Boundary Control</a>.</p>
      <p class="footer"><br>
         Published with MATLAB&reg; 7.0.1<br></p>
      <!--
##### SOURCE BEGIN #####
%% SEAL_det
% A stack-based sequential depth-first decoder that returns Maximum-Likelihood
% solutions to spherical LAST coded MIMO system-type problems, i.e., a lattice
% decoder with spherical boundary control.  In such problems, the depth of the
% search tree is known but the number of children per node is not fixed. The
% number of children and their associated symbol decisions are governed by the
% structure of the spherical LAST code.  |SEAL_det| enforces boundary control
% by decomposing the code generator matrix in a manner that is like a dual to
% the standard sequential decoding procedure itself.
%
% Note that inputs to |SEAL_det| must be real.  The depth-first search is
% effected by applying the *Schnorr-Euchner (Adaptive radius)* enumeration to
% explore the nodes of the tree.  Its operation is identical to that of a
% sphere decoder based on this enumeration.
%
% Please send comments and bug reports to karen.su@utoronto.ca.

%% Basic Operation
% This real-valued depth-first stack-based sequential decoding algorithm uses a
% decoding tree of |m+1| levels where each node has an arbitrary and unknown
% number of children.  At each stage, the node under consideration is expanded
% if its weight is less than the squared distance to nearest currently known
% lattice point.  This distance threshold is initially set to infinity.
%
% Because it is a depth-first traversal, we expand a node by computing its
% first child.  If it is a leaf node, clearly it cannot be further expanded.
% In this case, we will have found a closer lattice point than that previously
% known. Therefore we can adaptively reduce the distance threshold to reflect
% this new discovery.
%
% If the weight of the node under consideration is larger than this distance
% threshold, then the current search path is terminated because it cannot
% possibly lead to a closest lattice point. Upon path termination, the next
% node to be considered is the next sibling of its parent.
%
% Note that it is possible for a node to have no children, and also for the
% next sibling to be invalidated by means of boundary control.

%% Solution Validation
% In some cases, the LAST codebook may be punctured, i.e., not all lattice
% points lying inside the shaping region are in fact valid codewords. Some of
% the so-called outer shell codewords, i.e., those requiring the greatest
% transmitted power, may be removed the codebook.  Unfortunately the
% depth-first nature of the search performed by |SEAL_det| does not lend itself
% easily to solution validation. If it is an important feature for the desired
% application, please consider the priority first LAST decoder |LAST_dec|,
% which is also available via the Matlab File Exchange.

%% Detailed Parameter Specification
% Complex to real conversion by stacking should be done prior to calling 
% |SEAL_det|. In addition, if either lattice reduction assistance or MMSE
% pre-processing are desired, these operations should also be applied in 
% advance.
%
% Notes:
%
% - |size(H,1)| is expected to be equal to |length(y)|.
%
% - |size(H,1)| is expected to be greater than or equal to |size(H,2)|.
%
% - |size(H,2)| is expected to be equal to |m|.
%
% - The solution |zHat| is an integer vector; be sure to apply any necessary 
%   scaling prior to calling |SEAL_det| so that this solution is appropriate.

%% Usage Example
mPhy  = 2;                   % 2x2 MIMO system
L     = 2;                   % code spans 2 time periods
m     = 2*mPhy*L;            % code matrix size is governed by the number of 
                             % real space-time dimensions
y     = randn(m,1);          % generate random target vector
HPhyC = randn(mPhy,mPhy)+i*randn(mPhy,mPhy);
HPhyR = [real(HPhyC) -imag(HPhyC); imag(HPhyC) real(HPhyC)];
H     = kron(eye(L),HPhyR);  % construct space-time x space-time channel

load optimizedLASTcode;      % get LAST code specification: G,u,D
[zHat,wHat,nv,nf] = SEAL_det(mPhy,y,G,u,D,H,L);

fprintf('Solution                   [zHat] = ');
disp([zHat]);
fprintf('Search distance            [wHat] = ');
disp([wHat]);
fprintf('# of expansions              [nv] = ');
disp([nv]);
fprintf('# of flops (approx.)         [nf] = ');
disp([nf]);

wHatML = Inf;
nwords = size(COORDINATES,2);
for jj = 1:nwords
  zHatTest = COORDINATES(:,jj);
  wHatTest = norm(y-H*G*zHatTest)^2;
  if wHatTest < wHatML
    wHatML = wHatTest;
    zHatML = zHatTest;
  end
end

fprintf('Verify solution          [zHatML] = ');
disp([zHatML]);
fprintf('Verify squared distance  [wHatML] = ');
disp([wHatML]);

%% References
% The author would like to acknowledge the following work on the design and
% optimization of spherical LAST codes by means of which the sample 2x2x2 code
% provided along with this distribution in |optimizedLASTcode.mat| was
% generated:
%
% I. Berenguer, X. Wang, N. Prasad, J. Wang, and M. Madihian, *Design of
% minimium-error-rate lattice (space-time) codes via stochastic optimization
% and gradient estimation*, IEEE Global Communications Conference (Globecom),
% St. Louis, MO, Dec. 2005.
%
% For more details on the |SEAL_det| algorithm and boundary control for
% spherical LAST decoding, please see the following paper:
%
% K. Su, I. Berenguer, I.J. Wassell, X. Wang, *Efficient maximum-likelihood
% decoding of spherical lattice space-time codes*, IEEE International Conference
% on Communications (ICC), Istanbul, Turkey, June 2006.
%
% and its companion website:
%
% http://www.comm.utoronto.ca/~karen/seqdecLs.php

##### SOURCE END #####
-->
   </body>
</html>