Methods and apparatus for lattice-based signal modulation using a generalization of polar codes
A method includes receiving a bit string at a processor, performing an error correction, and causing transmission of a modulated signal. The error correction includes identifying a set of binary strings based on the bit string, mapping each binary string from the set of binary strings to a first abelian group element from a set of first abelian group elements, and applying a generalization of polar codes to the set of first abelian group elements to produce a set of second abelian group elements. The error correction also includes mapping each of the second abelian group elements to an in-phase/quadrature (I/Q) point from a set of I/Q points and identifying real-valued points based on the set of I/Q points, each of the real-valued points representing an I/Q point from the set of I/Q points. The modulated signal has a modulation that is based on the real-valued points.
1. A non-transitory, processor-readable medium storing instructions that, when executed by a processor, cause the processor to:
receive a bit string;
perform an error correction by:
identifying a plurality of binary strings based on the bit string,
mapping each binary string from the plurality of binary strings to a first abelian group element from a plurality of first abelian group elements,
applying a generalization of polar codes to the plurality of first abelian group elements to produce a plurality of second abelian group elements,
mapping each second abelian group element from the plurality of second abelian group elements to an in-phase/quadrature (I/Q) point from a plurality of I/Q points; and
identifying a plurality of real-valued points based on the plurality of I/Q points, each real-valued point from the plurality of real-valued points representing an I/Q point from the plurality of I/Q points; and
cause transmission of a signal having a modulation based on the plurality of real-valued points.
2. The non-transitory, processor-readable medium of claim 1 , wherein the plurality of I/Q points is included in a lattice-based signal constellation.
3. The non-transitory, processor-readable medium of claim 1 , wherein the modulation is a Quadrature Amplitude Modulation (QAM).
4. The non-transitory, processor-readable medium of claim 1 , further storing instructions to cause the processor to apply at least one of a permutation or a bijection to each binary string from the plurality of binary strings prior to performing the mapping of the second abelian group elements to the plurality of I/Q points.
5. The non-transitory, processor-readable medium of claim 4 , wherein the at least one of the permutation or the bijection includes at least one of a gray code or a reverse gray code.
6. The non-transitory, processor-readable medium of claim 1 , wherein the plurality of I/Q points is a lattice-based signal constellation, the non-transitory, processor-readable medium further storing instructions to cause the processor to reduce an order of an effective constellation associated with the lattice-based signal constellation to a subgroup using partial freezing of at least one binary string from the plurality of binary strings.
7. The non-transitory, processor-readable medium of claim 1 , wherein the generalization of polar codes includes a systematic code.
8. The non-transitory, processor-readable medium of claim 1 , wherein the plurality of first abelian group elements is associated with a first bit length and the plurality of second abelian group elements is associated with a second bit length different than the first bit length.
9. A method comprising:
receiving a bit string;
identifying a plurality of binary strings based on the bit string;
mapping each binary string from the plurality of binary strings to a first abelian group element from a plurality of first abelian group elements;
applying a generalization of polar codes to the plurality of first abelian group elements to produce a plurality of second abelian group elements;
mapping each second abelian group element from the plurality of second abelian group elements to an in-phase/quadrature (I/Q) point from a plurality of I/Q points;
identifying a plurality of real-valued points based on the plurality of I/Q points, each real-valued point from the plurality of real-valued points representing an I/Q point from the plurality of I/Q points; and
transmitting a signal having a modulation based on the plurality of real-valued points.
10. The method of claim 9 , wherein the plurality of I/Q points is included in a lattice-based signal constellation.
11. The method of claim 9 , further comprising applying at least one of a permutation or a bijection to each binary string from the plurality of binary strings prior to performing the mapping of the second abelian group elements to the plurality of I/Q points.
12. The method of claim 11 , wherein the at least one of the permutation or the bijection includes at least one of a gray code or a reverse gray code.
13. The method of claim 9 , wherein the plurality of I/Q points is a lattice-based signal constellation, the method further comprising reducing an order of an effective constellation associated with the lattice-based signal constellation to a subgroup using partial freezing of at least one binary string from the plurality of binary strings.
14. The method of claim 9 , wherein the plurality of first abelian group elements is associated with a first bit length and the plurality of second abelian group elements is associated with a second bit length different than the first bit length.
15. An apparatus comprising:
a communication interface configured to transmit and receive wireless signals; and
a processor configured to:
receive a bit string;
identify a plurality of binary strings based on the bit string,
map each binary string from the plurality of binary strings to a first abelian group element from a plurality of first abelian group elements,
apply a generalization of polar codes to the plurality of first abelian group elements to produce a plurality of second abelian group elements,
map each second abelian group element from the plurality of second abelian group elements to an in-phase/quadrature (I/Q) point from a plurality of I/Q points;
identify a plurality of real-valued points based on the plurality of I/Q points, each real-valued point from the plurality of real-valued points representing an I/Q point from the plurality of I/Q points; and
cause the communication interface to transmit a signal having a modulation based on the plurality of real-valued points.
16. The apparatus of claim 15 , wherein the plurality of I/Q points is included in a lattice-based signal constellation.
17. The apparatus of claim 15 , wherein the processor is further configured to apply at least one of a permutation or a bijection to each binary string from the plurality of binary strings prior to performing the mapping of the second abelian group elements to the plurality of I/Q points.
18. The apparatus of claim 17 , wherein the at least one of the permutation or the bijection includes at least one of a gray code or a reverse gray code.
19. The apparatus of claim 15 , wherein the plurality of I/Q points is a lattice-based signal constellation, the processor further configured to reduce an order of an effective constellation associated with the lattice-based signal constellation to a subgroup using partial freezing of at least one binary string from the plurality of binary strings.
20. The apparatus of claim 15 , wherein the plurality of first abelian group elements is associated with a first bit length and the plurality of second abelian group elements is associated with a second bit length different than the first bit length.