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Decode input using sphere decoder


The comm.SphereDecoder System object™ decodes the symbols sent over NT antennas using the sphere decoding algorithm.

This object accepts variable-size inputs. After the object is locked, you can change the size of each input channel, but you cannot change the number of channels. For more information, see Variable-Size Signal Support with System Objects.

To decode input symbols using a sphere decoder:

  1. Define and set up your sphere decoder object. See Construction.

  2. 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 maximum-likelihood solution for a set of received symbols over a MIMO channel with NT transmit antennas and NR receive antennas.

H = comm.SphereDecoder(Name,Value) creates a sphere decoder object, H, with the specified property name set to the specified value. Name must appear inside single quotes (''). You can specify several name-value pair arguments in any order as Name1,Value1,…,NameN,ValueN.

H = comm.SphereDecoder(CONSTELLATION,BITTABLE) creates a sphere decoder object, 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 1. The length of the vector must be a power of two. The object assumes that each transmit antenna uses the same constellation.


Bit mapping used for each constellation point.

Specify the bit mapping for the symbols that the Constellation property specifies as a numerical matrix. The default is [0 0; 0 1; 1 0; 1 1], which matches the default Constellation property value.

The matrix size must be [ConstellationLength bitsPerSymbol]. ConstellationLength represents the length of the Constellation property. bitsPerSymbol represents the number of bits that each symbol encodes.


Initial search radius of the decoding algorithm.

Specify the initial search radius for the decoding algorithm as either Infinity | ZF Solution. The default is Infinity.

When you set this property to Infinity, the object sets the initial search radius to Inf.

When you set this property to ZF Solution, the object sets the initial search radius to the zero-forcing solution. This calculation uses the pseudo-inverse of the input channel when decoding. Large constellations and/or antenna counts can benefit from the initial reduction in the search radius. In most cases, however, the extra computation of the ZF Solution will not provide a benefit.


Specify the decoding decision method as either Soft | Hard. The default is Soft.

When you set this property to Soft, the decoder outputs log-likelihood ratios (LLRs), or soft bits.

When you set this property to Hard, the decoder converts the soft LLRs to bits. The hard-decision output logical array follows the mapping of a zero for a negative LLR and one for all other values.


stepDecode received symbols using sphere decoding algorithm
Common to All System Objects

Allow System object property value changes


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Modulate a set of bits using 16-QAM constellation. Transmit the signal as two parallel streams over a MIMO channel. Decode using a sphere decoder with perfect channel knowledge.

Specify the modulation order, the number of transmitted bits, the Eb/No ratio, and the symbol mapping.

bps = 4; % Bits per symbol
M = 2^bps; % Modulation order
nBits = 1e3*bps;
ebno = 10;
symMap = [11 10 14 15 9 8 12 13 1 0 4 5 3 2 6 7];

Generate and display the symbol mapping of the 16-QAM modulator by using the qammod function and the custom symbol map.

sym = qammod(symMap(1:M)',M,symMap, ...

Convert the decimal value of the symbol map to binary bits using the left bit as the most significant bit (msb). The M-by-bps matrix bitTable is used by the sphere decoder.

bitTable = int2bit(symMap,bps)';

Create a 2x2 MIMO Channel System object with PathGainsOutputPort set to true to use the path gains as a channel estimate. To ensure the repeatability of results, set the object to use the global random number stream.

mimo = comm.MIMOChannel( ...
    'PathGainsOutputPort',true, ...
    'RandomStream','Global stream');

Create an AWGN Channel System object.

awgnChan = comm.AWGNChannel('EbNo',ebno,'BitsPerSymbol',bps);

Create a Sphere Decoder System object that processes bits using hard-decision decoding. Configure using the custom bit table and symbol map.

sphDec = comm.SphereDecoder('Constellation',sym, ...

Create an error rate System object.

berRate = comm.ErrorRate;

Set the global random number generator seed.


Generate a random data stream.

data = randi([0 1],nBits,1);

Modulate the data and reshape it into two streams to be used with the 2x2 MIMO channel.

modData = qammod(data,M,symMap, ...
modData = reshape(modData,[],2);

Pass the modulated data through the MIMO fading channel and add AWGN.

[fadedSig,pathGains] = mimo(modData);
rxSig = awgnChan(fadedSig);

Decode the received signal using pathGains as a perfect channel estimate.

decodedData = sphDec(rxSig,squeeze(pathGains));

Convert the decoded hard-decision data, which is a logical matrix, into a double column vector to enable the calculation of error statistics. Calculate and display the bit error rate and the number of errors.

dataOut = double(decodedData(:));
errorStats = berRate(data,dataOut);
ans = 2×1



This object implements a soft-output max-log a posteriori probability (APP) MIMO detector by means of a soft-output Schnorr-Euchner 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 log-likelihood ratios (LLRs) of the transmitted bits.

The algorithm assumes a MIMO system model with NT transmit antennas and NR receive antennas where NT 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 maximum-likelihood (ML) solution, s^ML, such that:


where O is the complex-valued constellation from which the NT elements of s are chosen.

Soft detection also computes a log-likelihood 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 max-log approximation:



  • L(xj,b) is the LLR estimate for each bit.

  • xj,b is each sent bit, the bth bit of the jth symbol.

  • xj,b(0) and xj,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:

    • λML is the distance s^ML.

    • λj,bML¯ is the distance to the counter-hypothesis, which denotes the binary complement of the bth bit in the binary label of the jth entry of s^ML, specifically the minimum of the symbol set xj,b(xj,bML¯), which contains all of the possible vectors for which the bth bit of the jth entry is flipped compared to the same entry of s^ML.

Based on whether xj,b(xj,bML) is 0 or 1, the LLR estimate for bit xj,b is computed as follows:


The design of a decoder strives to efficiently find s^ML, λML, and λj,bML¯.

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. Left-multiplying y by QH, the problem can be reformulated as:


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 λML and λj,bML¯ metrics are searched concurrently. The goal is to have a list containing the metric λML, along with the corresponding bit sequence xML and the metrics xj,b(xj,bML) of all counter-hypotheses. The sub-tree originating from a given node is searched only if the result can lead to an update of either λML or λj,bML¯.

The STS algorithm flow can be summarized as:

  1. If when reaching a leaf node, a new ML hypothesis is found (d(x)<λML), all λj,bML¯ for which xj,b=xj,bML¯ are set to λML which now turns into a valued counter-hypothesis. Then, λML is set to the current distance, d(x).

  2. If d(x)λML, only the counter-hypotheses have to be checked. For all j and b for which (d(x)<λML) and xj,b=xj,bML¯, the decoder updates λj,bML¯ to be d(x).

  3. A sub-tree is pruned if the partial distance of the node is bigger than the current λj,bML¯ which may be affected when traversing the subtree.

  4. 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.

Selected Bibliography

[1] Studer, C., A. Burg, and H. Bölcskei. “Soft-Output 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., "MIMO-OFDM Wireless communications with MATLAB," IEEE Press, 2011.

[3] Hochwald, B.M., S. ten Brink. “Achieving near-capacity on a multiple-antenna channel”, IEEE Transactions on Communications, Vol. 51, No. 3, Mar 2003, pp. 389-399.

[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. 2201-2214.

Extended Capabilities

Version History

Introduced in R2013a