Polar decoder and associated method
An optimization-based decoder with improved performance and cyclic redundancy check is provided. An optimization problem is generated and solved. The optimization problem includes a function implementing constraints on variables. Main variables correspond to nodes of the encoding graph, including input variables corresponding to output nodes and output variables corresponding to input nodes. Ancillary variables correspond to an additional output a logical operations in the encoding graph. Constraints are configured to generate cost penalties, for instance when the main and ancillary variables of a candidate solution disagree with conditions of the encoding graph, and to correspond with the cross-entropy between each codeword symbol and the corresponding input variable. A candidate solution is obtained by solving the optimization problem, wherein the output variables in the candidate solution correspond to decoded symbols.
1 . A method for decoding a noisy codeword, the noisy codeword comprising a plurality of codeword symbols corresponding to a plurality of input symbols encoded via an encoder and transmitted via a noisy channel, the method comprising:
generating an optimization problem comprising an objective function, the objective function being evaluated according to variables of candidate solutions, said variables comprising:
a plurality of main variables, each corresponding to one of a plurality of nodes in an encoding graph implemented by the encoder, said main variables comprising input variables at output nodes of the encoding graph and output variables at input nodes of the encoding graph, and
a plurality of ancillary variables, each corresponding to an additional output of one of a plurality of logical operations in the encoding graph,
wherein the objective function comprises an aggregation of constraints, the constraints comprising:
node constraints configured to generate a first cost penalty when the main and ancillary variables of a candidate solution disagree with conditions of the encoding graph implemented by the encoder, and
receiver constraints configured to generate a second cost penalty corresponding to a cross-entropy between each codeword symbol and a corresponding one of the input variables; and
solving the optimization problem to obtain a candidate solution, wherein the output variables in the candidate solution correspond to decoded symbols.
2 . The method of claim 1 , wherein the encoder is a polar encoder.
3 . The method of claim 1 , wherein the input symbols and codeword symbols are binary symbols, and wherein the logical operations are XOR operations.
4 . The method of claim 1 , wherein the input symbols comprise information symbols and frozen symbols, and the main variables exclude input nodes of the encoding graph that correspond to frozen symbols.
5 . The method of claim 4 , wherein the main and ancillary variables exclude uninformative nodes at intermediate layers of the encoding graph.
6 . The method of claim 1 , wherein the input symbols comprise information symbols and frozen symbols, further wherein the objective function additionally comprises frozen constraints configured to generate a third cost penalty when input variables corresponding to frozen symbols are not equal to a predetermined value.
7 . The method of claim 1 , wherein the input symbols further comprise cyclic redundancy check (CRC) symbols, and the objective function additionally comprises CRC constraints configured to generate a fourth cost penalty when the main and ancillary variables of the candidate solution fail a CRC verification corresponding to a CRC function used to generate the CRC symbols.
8 . The method of claim 7 , wherein the CRC verification is represented as a plurality of logical operations on the main and ancillary variables, and the CRC constraints correspond to a plurality of constraints that are minimized when the plurality of logical operations are respected.
9 . The method of claim 1 , wherein the optimization problem is a quadratic unconstrained binary optimization problem.
10 . The method of claim 9 , wherein solving the optimization problem comprises using one of quantum annealing and simulated annealing.
11 . A system for decoding a noisy codeword, the system comprising:
a receiver configured for receiving the noisy codeword, wherein the noisy codeword comprises a plurality of codeword symbols corresponding to a plurality of input symbols encoded via an encoder and transmitted via a noisy channel;
memory for storing variable values; and
a decoder comprising:
an objective function creation module configured to:
generate an optimization problem comprising an objective function, the objective function being evaluated according to variables of candidate solutions, said variables comprising:
a plurality of main variables, each corresponding to one of a plurality of nodes in an encoding graph implemented by the encoder, said main variables comprising input variables at output nodes of the encoding graph and output variables at input nodes of the encoding graph, and
a plurality of ancillary variables, each corresponding to an additional output of one of a plurality of logical operations in the encoding graph,
wherein the objective function comprises an aggregation of constraints, the constraints comprising:
node constraints configured to generate a first cost penalty when the main and ancillary variables of a candidate solution disagree with conditions of the encoding graph implemented by the encoder, and
receiver constraints configured to generate a second cost penalty corresponding to a cross-entropy between each codeword symbol and a corresponding one of the input variables; and
a solver configured for solving the optimization problem to obtain a candidate solution, wherein the output variables in the candidate solution correspond to decoded symbols.
12 . The system of claim 11 , wherein the encoder is a polar encoder.
13 . The system of claim 11 , wherein the input symbols and codeword symbols are binary symbols, and wherein the logical operations are XOR operations.
14 . The system of claim 11 , wherein the input symbols comprise information symbols and frozen symbols, and wherein the objective function creation module is configured to exclude input nodes of the encoding graph that correspond to frozen symbols from the main variables.
15 . The system of claim 14 , wherein objective function creation module is configured to exclude uninformative nodes from the main and ancillary variables at intermediate layers of the encoding graph.
16 . The system of claim 11 , wherein the input symbols comprise information symbols and frozen symbols, and wherein the objective function creation module is configured to generate frozen constraints for aggregation in the objective function, the frozen constraints configured to generate a third cost penalty when input variables corresponding to frozen symbols are not equal to a predetermined value.
17 . The system of claim 11 , wherein the input symbols further comprise cyclic redundancy check (CRC) symbols, and wherein the objective function creation module is configured to generate CRC constraints for aggregation in the objective function, the CRC constraints configured to generate a fourth cost penalty when the main and ancillary variables of the candidate solution fail a CRC verification corresponding to a CRC function used to generate the CRC symbols.
18 . The system of claim 17 , wherein the CRC verification is represented as a plurality of logical operations on the main and ancillary variables, and the CRC constraints correspond to a plurality of constraints that are minimized when the plurality of logical operations are respected.
19 . The system of claim 11 , wherein the optimization problem is a quadratic unconstrained binary optimization problem, and wherein the solver is configured to perform one of quantum annealing and simulated annealing.
20 . A non-transitory computer-readable medium storing thereon computer executable instructions for decoding a noisy codeword, the noisy codeword comprising a plurality of codeword symbols corresponding to a plurality of input symbols encoded via an encoder and transmitted via a noisy channel, the instructions, when executed by one or more processors of a computing system, causing the computing system to:
generate an optimization problem comprising an objective function, the objective function being evaluated according to variables of candidate solutions, said variables comprising:
a plurality of main variables, each corresponding to one of a plurality of nodes in an encoding graph implemented by the encoder, said main variables comprising input variables at output nodes of the encoding graph and output variables at input nodes of the encoding graph, and
a plurality of ancillary variables, each corresponding to an additional output of one of a plurality of logical operations in the encoding graph,
wherein the objective function comprises an aggregation of constraints, the constraints comprising:
node constraints configured to generate a first cost penalty when the main and ancillary variables of a candidate solution disagree with conditions of the encoding graph implemented by the encoder, and
receiver constraints configured to generate a second cost penalty corresponding to a cross-entropy between each codeword symbol and a corresponding one of the input variables; and
solve the optimization problem to obtain a candidate solution, wherein the output variables in the candidate solution correspond to decoded symbols.