Decode input using sphere decoder
The Sphere Decoder
System
object™ decodes the symbols sent over N_{T} antennas
using the sphere decoding algorithm.
To decode input symbols using a sphere decoder:
Define and set up your sphere decoder object. See Construction.
Call step
to decode input symbols
according to the properties of comm.SphereDecoder
.
The behavior of step
is specific to each object in
the toolbox.
Starting in R2016b, instead of using the step
method
to perform the operation defined by the System
object, you can
call the object with arguments, as if it were a function. For example, y
= step(obj,x)
and y = obj(x)
perform
equivalent operations.
H = comm.SphereDecoder
creates a System
object, H
. This object uses the sphere decoding algorithm to find the
maximumlikelihood solution for a set of received symbols over a MIMO channel with
N_{T} transmit antennas and
N_{R} receive antennas.
H = comm.SphereDecoder(
creates a sphere decoder object, Name
,Value
)H
, with the
specified property name set to the specified value. Name must appear
inside single quotes (''). You can specify several namevalue pair
arguments in any order as Name1,Value1,…,NameN,ValueN.
H = comm.SphereDecoder(
creates
a sphere decoder object, CONSTELLATION
,BITTABLE
)H
, with the Constellation
property set to CONSTELLATION
, and the BitTable
property
set to BITTABLE
.

Signal constellation per transmit antenna Specify the constellation as a complex column vector containing
the constellation points to which the transmitted bits are mapped.
The default setting is a QPSK constellation with an average power
of 

Bit mapping used for each constellation point. Specify the bit mapping for the symbols that the The matrix size must be 

Initial search radius of the decoding algorithm. Specify the initial search radius for the decoding algorithm
as either When you set this property to When you set this property to 

Specify the decoding decision method as either When you set this property to When you set this property to 
step  Decode received symbols using sphere decoding algorithm 
Common to All System Objects  

release  Allow System object property value changes 
This object implements a softoutput maxlog a posteriori probability (APP) MIMO detector by means of a softoutput SchnorrEuchner sphere decoder (SESD), implemented as single tree search (STS) tree traversal. The algorithm assumes the same constellation and bit table on all of the transmit antennas. Given as inputs, the received symbol vector and the estimated channel matrix, the algorithm outputs the loglikelihood ratios (LLRs) of the transmitted bits.
The algorithm assumes a MIMO system model with N_{T} transmit antennas and N_{R} receive antennas where N_{T} symbols are simultaneously sent, expressed as:
y = Hs + n.
where y is the received symbols, H is the MIMO channel matrix, s is the transmitted symbol vector, and n is the thermal noise.
The MIMO detector seeks the maximumlikelihood (ML) solution, , such that:
$${\widehat{s}}_{{}_{ML}}=\underset{s\in o}{{\displaystyle {}_{\mathrm{arg}\mathrm{min}}}}{\Vert yHs\Vert}^{2}$$
where O is the complexvalued constellation from which the N_{T} elements of s are chosen.
Soft detection also computes a loglikelihood ratio (LLR) for each bit that serves as a measure of the reliability of the estimate for each bit. The LLR is calculated as using the maxlog approximation:
$$L\left({x}_{j,b}\right)=\underset{{\lambda}^{ML}}{\underbrace{\underset{s\in {x}_{j,b}^{(0)}}{\mathrm{min}}{\Vert yHs\Vert}^{2}}}\underset{{\lambda}_{j,b}^{\overline{ML}}}{\underbrace{\underset{s\in {x}_{j,b}^{(1)}}{\mathrm{min}}{\Vert yHs\Vert}^{2}}}$$
where
L(x_{j,b}) is the LLR estimate for each bit.
$${x}_{j,b}$$ is each sent bit, the bth bit of the jth symbol.
$${x}_{j,b}^{(0)}$$ and $${x}_{j,b}^{(1)}$$ are the disjoint sets of vector symbols that have the bth bit in the label of the jth scalar symbol equal to 0 and 1, respectively. The two λ symbols denotes the distance calculated as norm squared., specifically:
$${\lambda}^{ML}$$ is the distance $${\widehat{s}}_{ML}$$.
$${\lambda}_{j,b}^{{}^{\overline{ML}}}$$ is the distance to the counterhypothesis, which denotes the binary complement of the bth bit in the binary label of the jth entry of $${\widehat{s}}_{ML}$$, specifically the minimum of the symbol set $${x}_{j,b}^{\left({x}_{j,b}^{\overline{ML}}\right)}$$, which contains all of the possible vectors for which the bth bit of the jth entry is flipped compared to the same entry of $${\widehat{s}}_{ML}$$.
Based on whether $${x}_{j,b}^{\left({x}_{j,b}^{ML}\right)}$$ is 0
or 1
, the LLR estimate for bit $${x}_{j,b}$$ is computed as follows:
$$L({x}_{j,b})=\{\begin{array}{cc}{\lambda}^{ML}{\lambda}_{j,b}^{\overline{ML}},\text{\hspace{0.17em}}& {x}_{j,b}^{ML}=\text{\hspace{0.17em}}0\\ {\lambda}_{j,b}^{\overline{ML}}{\lambda}^{ML},& {x}_{j,b}^{ML}=\text{\hspace{0.17em}}1\end{array}$$
The design of a decoder strives to efficiently find $${\widehat{s}}_{ML}$$, $${\lambda}^{ML}$$, and $${\lambda}_{j,b}^{\overline{ML}}$$.
This search can be converted into a tree search by means of the sphere decoding algorithms. To this end, the channel matrix is decomposed into $$H=QR$$ by means of a QR decomposition. Leftmultiplying y by Q^{H}, the problem can be reformulated as:
$$\begin{array}{l}{\lambda}^{ML}=\underset{s\in o}{{\displaystyle \text{\hspace{0.17em}}\mathrm{arg}\text{\hspace{0.17em}}\mathrm{min}}}{\Vert \overline{y}Rs\Vert}^{2}\\ {\lambda}_{j,b}^{\overline{ML}}{=}_{s\in {x}_{j,b}^{\left({x}_{j,b}^{\overline{ML}}\right)}}^{\mathrm{arg}\mathrm{min}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Vert \overline{y}Rs\Vert}^{2}}\end{array}$$
Using this reformulated problem statement, the triangular structure of R can be exploited to arrange a tree structure such that each of the leaf nodes corresponds to a possible s vector and the partial distances to the nodes in the tree can be calculated cumulatively adding to the partial distance of the parent node.
In the STS algorithm, the $${\lambda}^{ML}$$ and $${\lambda}_{j,b}^{\overline{ML}}$$ metrics are searched concurrently. The goal is to have a list containing the metric $${\lambda}^{ML}$$, along with the corresponding bit sequence $${x}^{ML}$$ and the metrics $${x}_{j,b}^{\left({x}_{j,b}^{ML}\right)}$$ of all counterhypotheses. The subtree originating from a given node is searched only if the result can lead to an update of either $${\lambda}^{ML}$$ or $${\lambda}_{j,b}^{\overline{ML}}$$.
The STS algorithm flow can be summarized as:
If when reaching a leaf node, a new ML hypothesis is found $$\left(d\left(x\right)<{\lambda}^{ML}\right)$$, all $${\lambda}_{j,b}^{\overline{ML}}$$ for which $${x}_{j,b}={x}_{j,b}^{\overline{ML}}$$ are set to $${\lambda}^{ML}$$ which now turns into a valued counterhypothesis. Then, $${\lambda}^{ML}$$ is set to the current distance, d(x).
If $$d(x)\ge {\lambda}^{ML}$$, only the counterhypotheses have to be checked. For all j and b for which $$\left(d\left(x\right)<{\lambda}^{ML}\right)$$ and $${x}_{j,b}={x}_{j,b}^{\overline{ML}}$$, the decoder updates $${\lambda}_{j,b}^{\overline{ML}}$$ to be d(x).
A subtree is pruned if the partial distance of the node is bigger than the current $${\lambda}_{j,b}^{\overline{ML}}$$ which may be affected when traversing the subtree.
The STS concludes once all of the tree nodes have been visited once or pruned.
The output LLR values are not scaled by the noise variance. For coded links employing iterative coding (LDPC or turbo) or MIMO OFDM with Viterbi decoding, the output LLR values should be scaled by the channel state information to achieve better performance.
[1] Studer, C., A. Burg, and H. Bölcskei. “SoftOutput Sphere Decoding: Algorithms and VLSI Implementation”. IEEE Journal of Selected Areas in Communications. Vol. 26, No. 2, February 2008, pp. 290–300.
[2] Cho, Y. S., et.al. "MIMOOFDM Wireless communications with MATLAB," IEEE Press, 2011.
[3] Hochwald, B.M., S. ten Brink. “Achieving nearcapacity on a multipleantenna channel”, IEEE Transactions on Communications, Vol. 51, No. 3, Mar 2003, pp. 389399.
[4] Agrell, E., T. Eriksson, A. Vardy, K. Zeger. “Closest point search in lattices”, IEEE Transactions on Information Theory, Vol. 48, No. 8, Aug 2002, pp. 22012214.
Sphere Decoder  comm.LTEMIMOChannel
 comm.MIMOChannel
 comm.OSTBCCombiner