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<h1>SEA_det</h1>
<introduction>
<p>A stack-based sequential depth-first decoder that returns Maximum-Likelihood solutions to M-QAM modulated MIMO system-type
problems, i.e., a lattice decoder with optional justified rectangular boundary control. In such problems, the depth of the
search tree is known and the number of children per node is also fixed.
</p>
<p>Real or complex inputs are permitted. The depth-first search is effected by applying the <b>Schnorr-Euchner (Adaptive radius)</b> enumeration to traverse the nodes of the tree.
</p>
<p>Please send comments and bug reports to <a href="mailto:karen.su@utoronto.ca">karen.su@utoronto.ca</a>.
</p>
</introduction>
<h2>Contents</h2>
<div>
<ul>
<li><a href="#1">Basic Operation</a></li>
<li><a href="#2">Detailed Parameter Specification</a></li>
<li><a href="#3">Usage Example</a></li>
<li><a href="#4">References</a></li>
</ul>
</div>
<h2>Basic Operation<a name="1"></a></h2>
<p>The real-valued version of this stack-based sequential decoding algorithm is based on a decoding tree of <tt>m+1</tt> levels with each node having <tt>2*xMax</tt> 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.
</p>
<p>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.
</p>
<p>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.
</p>
<h2>Detailed Parameter Specification<a name="2"></a></h2>
<p>If <tt>xMax == 0</tt>, we do not apply (rectangular, or any) boundary control. In other words, <tt>SEA_det</tt> behaves as a lattice decoder.
</p>
<p>For more sophisticated operation, <tt>xMax</tt> may also be a vector of length <tt>m</tt>. Then each node at the beginning of stage <tt>j</tt> in the tree, where the root node is at the beginning of stage <tt>m</tt> and the leaf nodes are found at the end of stage <tt>1</tt>, has <tt>2*xMax(j)</tt> children. Equivalently, symbol <tt>xHat(j)</tt> is drawn from <tt>{-xMax(j)+1,..,-1,0,1,..,xMax(j)}</tt>.
</p>
<p>If <tt>cplx == 1</tt>, we consider a tree of <tt>2*m+1</tt> levels with each node still having <tt>2*xMax</tt> children. In addition, either <tt>xMax</tt> should be a complex-valued vector, or <tt>imag(xMax)</tt> will be taken to be equal to <tt>real(xMax)</tt>, i.e., a square QAM constellation will be assumed by default.
</p>
<p>If either lattice reduction assistance or MMSE pre-processing are desired, these operations should be applied in advance of
calling <tt>SEA_det</tt>.
</p>
<p>Notes:</p>
<p>- <tt>size(H,1)</tt> is expected to be equal to <tt>length(y)</tt>.
</p>
<p>- <tt>size(H,1)</tt> is expected to be greater than or equal to <tt>size(H,2)</tt>.
</p>
<p>- <tt>size(H,2)</tt> is expected to be equal to <tt>m</tt>.
</p>
<p>- <tt>length(xMax)</tt> is expected to be equal to either <tt>1</tt> or <tt>m</tt>.
</p>
<p>- The solution <tt>xHat</tt> is an integer vector; be sure to apply any necessary scaling prior to calling <tt>SEA_det</tt> so that this solution is appropriate.
</p>
<h2>Usage Example<a name="3"></a></h2><pre class="codeinput">m = 4; <span class="comment">% 4x4 MIMO system</span>
xMax = 2; <span class="comment">% 16-QAM modulation</span>
cplx = 1;
y = 3*(randn(m,1)+i*randn(m,1)); <span class="comment">% generate random complex target vector</span>
H = randn(m,m)+i*randn(m,m); <span class="comment">% generate random complex channel</span>
[xHat,wHat,nv,nf,nl] = SEA_det(m,xMax,y,H,cplx);
xMaxV = [3 3 2 1]; <span class="comment">% various square modulations</span>
[xHatV,wHatV,nvV,nfV,nlV] = SEA_det(m,xMaxV,y,H,cplx);
fprintf(<span class="string">'Solutions [xHat xHatV] = '</span>);
disp([xHat xHatV]);
fprintf(<span class="string">'Search distances [wHat wHatV] = '</span>);
disp([wHat wHatV]);
fprintf(<span class="string">'# of expansions [nv nvV] = '</span>);
disp([nv nvV]);
fprintf(<span class="string">'# of flops (approx.) [nf nfV] = '</span>);
disp([nf nfV]);
fprintf(<span class="string">'# of leafs visited [nl nlV] = '</span>);
disp([nl nlV]);
wHatML = Inf;
wHatMLV = Inf;
xMaxT = max(xMax,xMaxV);
<span class="keyword">for</span> jj = -xMaxT(1)+1:xMaxT(1)
<span class="keyword">for</span> kk = -xMaxT(2)+1:xMaxT(2)
<span class="keyword">for</span> ll = -xMaxT(3)+1:xMaxT(3)
<span class="keyword">for</span> mm = -xMaxT(4)+1:xMaxT(4)
<span class="keyword">for</span> jjc = -xMaxT(1)+1:xMaxT(1)
<span class="keyword">for</span> kkc = -xMaxT(2)+1:xMaxT(2)
<span class="keyword">for</span> llc = -xMaxT(3)+1:xMaxT(3)
<span class="keyword">for</span> mmc = -xMaxT(4)+1:xMaxT(4)
xHatT = [jj+i*jjc;kk+i*kkc;ll+i*llc;mm+i*mmc];
wHatT = norm(y-H*xHatT)^2;
<span class="keyword">if</span> wHatT < wHatMLV & sum([real(xHatT);imag(xHatT)]>=-repmat(xMaxV,1,2).'+1) == 8 & sum([real(xHatT);imag(xHatT)]<=repmat(xMaxV,1,2).') == 8
wHatMLV = wHatT;
xHatMLV = xHatT;
<span class="keyword">end</span>
<span class="keyword">if</span> wHatT < wHatML & sum([real(xHatT);imag(xHatT)]>=-xMax+1) == 8 & sum([real(xHatT);imag(xHatT)]<=xMax) == 8
wHatML = wHatT;
xHatML = xHatT;
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="keyword">end</span>
fprintf(<span class="string">'Verify solutions [xHatML xHatMLV] = '</span>);
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