Design of robust and universal polar codes

Abstract

Aspects of the disclosure relate to polar coding. A polar codeword may be generated by sorting a plurality of synthetic channels utilized for transmission of the polar codeword over an air interface in order of reliability utilizing a convex combination of the mutual information calculated for each synthetic channel based on an Additive White Gaussian Noise (AWGN) channel and the mutual information calculated for each synthetic channel based on a binary erasure channel. A polar codeword may further be generated by sorting the plurality of synthetic channels in order of reliability utilizing cumulative sums calculated for each synthetic channel. Each cumulative sum may be calculated from a binary representation of a position of the synthetic channel within the plurality of synthetic channels.

What is claimed is: 
     
         1 . A method of polar coding at a transmitting wireless communication device, comprising:
 calculating a respective cumulative sum for each synthetic channel of a set of synthetic channels;   sorting the set of synthetic channels in order of reliability based on the respective cumulative sums to produce an order of synthetic channels;   identifying K best synthetic channels of the set of synthetic channels in accordance with the order of synthetic channels;   placing information bits within information bit locations of an information block corresponding to the K best synthetic channels;   placing frozen bits within frozen bit locations of the information block corresponding to N−K synthetic channels;   polar coding the information block to produce a polar codeword; and   transmitting the polar codeword to a receiving wireless communication device over a wireless air interface.   
     
     
         2 . The method of  claim 1 , wherein calculating the respective cumulative sum for each synthetic channel of the set of synthetic channels further comprises:
 for a synthetic channel of the set of synthetic channels, calculating the cumulative sum of a binary representation of a position of the synthetic channel within the set of synthetic channels.   
     
     
         3 . The method of  claim 2 , wherein the cumulative sum comprises a vector having a number of vector components equal to a corresponding number of binary components of the binary representation of the position. 
     
     
         4 . The method of  claim 3 , wherein calculating the cumulative sum for the synthetic channel of the set of synthetic channels further comprises:
 adding a binary component of the binary representation of the position of the synthetic channel to a corresponding immediately prior vector component of the vector to produce a next vector component in the vector for the synthetic channel.   
     
     
         5 . The method of  claim 2 , wherein sorting the set of synthetic channels further comprises:
 comparing the cumulative sums of a first synthetic channel of the set of synthetic channels and a second synthetic channel of the set of synthetic channels, wherein the second synthetic channel has a later position in the set of synthetic channels than the first synthetic channel; and   upgrading the second synthetic channel with respect to the first synthetic channels or the first synthetic channel with respect to the second synthetic channel in the order of synthetic channels based on the comparing of the cumulative sums.   
     
     
         6 . The method of  claim 5 , further comprising:
 when the cumulative sum of the second synthetic channel is component-wise greater than the cumulative sum of the first synthetic channel, upgrading the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels.   
     
     
         7 . The method of  claim 5 , wherein comparing the cumulative sums of the first synthetic channel and the second synthetic channel further comprises:
 comparing corresponding components in the cumulative sums of first synthetic channel and the second synthetic channel.   
     
     
         8 . The method of  claim 7 , wherein comparing the cumulative sums of the first synthetic channel and the second synthetic channel further comprises:
 calculating a respective difference between each set of corresponding components in the cumulative sums of the second synthetic channel and the first synthetic channel.   
     
     
         9 . The method of  claim 8 , further comprising:
 when a sum of the respective differences is greater than zero, upgrading the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels.   
     
     
         10 . The method of  claim 9 , wherein calculating the respective difference between each set of corresponding components in the cumulative sums of the second synthetic channel and the first synthetic channel further comprises:
 calculating a first difference between a first component of the cumulative sum of the second synthetic channel and a corresponding first component of the cumulative sum of the first synthetic channel.   
     
     
         11 . The method of  claim 8 , further comprising:
 when a sum of the respective differences is equal to zero, comparing a last component of the cumulative sum of the first synthetic channel to a corresponding last component of the cumulative sum of the second synthetic channel;   if the last component of the cumulative sum of the first synthetic channel is greater than the corresponding last component of the cumulative sum of the second synthetic channel, upgrading the first synthetic channel with respect to the second synthetic channel in the order of synthetic channels;   if the last component of the cumulative sum of the first synthetic channel is less than the corresponding last component of the cumulative sum of the second synthetic channel, upgrading the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels; and   if the last component of the cumulative sum of the first synthetic channel is equal to the corresponding last component of the cumulative sum of the second synthetic channel, comparing a next-to-last component of the cumulative sum of the first synthetic channel to a next-to-last component of the cumulative sum of the second synthetic channel.   
     
     
         12 . The method of  claim 8 , further comprising:
 when a sum of the respective differences is equal to zero, comparing a first component of the cumulative sum of the first synthetic channel to a corresponding first component of the cumulative sum of the second synthetic channel;   if the first component of the cumulative sum of the first synthetic channel is greater than the corresponding first component of the cumulative sum of the second synthetic channel, upgrading the first synthetic channel with respect to the second synthetic channel in the order of synthetic channels;   if the first component of the cumulative sum of the first synthetic channel is less than the corresponding first component of the cumulative sum of the second synthetic channel, upgrading the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels; and   if the first component of the cumulative sum of the first synthetic channel is equal to the corresponding first component of the cumulative sum of the second synthetic channel, comparing a second component of the cumulative sum of the first synthetic channel to a second component of the cumulative sum of the second synthetic channel.   
     
     
         13 . The method of  claim 5 , further comprising:
 when the cumulative sum of the second synthetic channel is component-wise less than the cumulative sum of the first synthetic channel, determining a sequence of the first synthetic channel and the second synthetic channel in the order of synthetic channels by comparing the cumulative sums of each of the first synthetic channel and the second synthetic channel to the cumulative sum of at least one additional synthetic channel.   
     
     
         14 . An apparatus configured for polar coding, the apparatus comprising:
 a transceiver;   a memory; and   a processor communicatively coupled to the transceiver and the memory, the processor configured to:
 calculate a respective cumulative sum for each synthetic channel of a set of synthetic channels; 
 sort the set of synthetic channels in order of reliability based on the respective cumulative sums to produce an order of synthetic channels; 
 identify K best synthetic channels of the set of synthetic channels in accordance with the order of synthetic channels; 
 place information bits within information bit locations of an information block corresponding to the K best synthetic channels; 
 place frozen bits within frozen bit locations of the information block corresponding to N−K synthetic channels; 
 polar code the information block to produce a polar codeword; and 
 transmit the polar codeword to a receiving wireless communication device over a wireless air interface via the transceiver. 
   
     
     
         15 . The apparatus of  claim 14 , wherein the processor is further configured to:
 for a synthetic channel of the set of synthetic channels, calculate the cumulative sum of a binary representation of a position of the synthetic channel within the set of synthetic channels.   
     
     
         16 . The apparatus of  claim 15 , wherein the cumulative sum comprises a vector having a number of vector components equal to a corresponding number of binary components of the binary representation of the position. 
     
     
         17 . The apparatus of  claim 16 , wherein the processor is further configured to:
 add a binary component of the binary representation of the position of the synthetic channel to a corresponding immediately prior vector component of the vector to produce a next vector component in the vector for the synthetic channel.   
     
     
         18 . The apparatus of  claim 15 , wherein the processor is further configured to:
 compare the cumulative sums of a first synthetic channel of the set of synthetic channels and a second synthetic channel of the set of synthetic channels, wherein the second synthetic channel has a later position in the set of synthetic channels than the first synthetic channel; and   upgrade the second synthetic channel with respect to the first synthetic channels or the first synthetic channel with respect to the second synthetic channel in the order of synthetic channels based on the comparing of the cumulative sums.   
     
     
         19 . The apparatus of  claim 18 , wherein the processor is further configured to:
 when the cumulative sum of the second synthetic channel is component-wise greater than the cumulative sum of the first synthetic channel, upgrade the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels.   
     
     
         20 . The apparatus of  claim 18 , wherein the processor is further configured to:
 compare corresponding components in the cumulative sums of first synthetic channel and the second synthetic channel.   
     
     
         21 . The apparatus of  claim 20 , wherein the processor is further configured to:
 calculate a respective difference between each set of corresponding components in the cumulative sums of the second synthetic channel and the first synthetic channel.   
     
     
         22 . The apparatus of  claim 21 , wherein the processor is further configured to:
 when a sum of the respective differences is greater than zero, upgrade the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels.   
     
     
         23 . The apparatus of  claim 22 , wherein the processor is further configured to:
 calculate a first difference between a first component of the cumulative sum of the second synthetic channel and a corresponding first component of the cumulative sum of the first synthetic channel.   
     
     
         24 . The apparatus of  claim 21 , wherein the processor is further configured to:
 when a sum of the respective differences is equal to zero, compare a last component of the cumulative sum of the first synthetic channel to a corresponding last component of the cumulative sum of the second synthetic channel;   if the last component of the cumulative sum of the first synthetic channel is greater than the corresponding last component of the cumulative sum of the second synthetic channel, upgrade the first synthetic channel with respect to the second synthetic channel in the order of synthetic channels;   if the last component of the cumulative sum of the first synthetic channel is less than the corresponding last component of the cumulative sum of the second synthetic channel, upgrade the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels; and   if the last component of the cumulative sum of the first synthetic channel is equal to the corresponding last component of the cumulative sum of the second synthetic channel, compare a next-to-last component of the cumulative sum of the first synthetic channel to a next-to-last component of the cumulative sum of the second synthetic channel.   
     
     
         25 . The apparatus of  claim 21 , wherein the processor is further configured to:
 when a sum of the respective differences is equal to zero, compare a first component of the cumulative sum of the first synthetic channel to a corresponding first component of the cumulative sum of the second synthetic channel;   if the first component of the cumulative sum of the first synthetic channel is greater than the corresponding first component of the cumulative sum of the second synthetic channel, upgrade the first synthetic channel with respect to the second synthetic channel in the order of synthetic channels;   if the first component of the cumulative sum of the first synthetic channel is less than the corresponding first component of the cumulative sum of the second synthetic channel, upgrade the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels; and   if the first component of the cumulative sum of the first synthetic channel is equal to the corresponding first component of the cumulative sum of the second synthetic channel, compare a second component of the cumulative sum of the first synthetic channel to a second component of the cumulative sum of the second synthetic channel.   
     
     
         26 . The apparatus of  claim 18 , wherein the processor is further configured to:
 when the cumulative sum of the second synthetic channel is component-wise less than the cumulative sum of the first synthetic channel, determine a sequence of the first synthetic channel and the second synthetic channel in the order of synthetic channels by comparing the cumulative sums of each of the first synthetic channel and the second synthetic channel to the cumulative sum of at least one additional synthetic channel.   
     
     
         27 . An apparatus within a wireless communication network, comprising:
 means for calculating a respective cumulative sum for each synthetic channel of a set of synthetic channels;   means for sorting the set of synthetic channels in order of reliability based on the respective cumulative sums to produce an order of synthetic channels;   means for identifying K best synthetic channels of the set of synthetic channels in accordance with the order of synthetic channels;   means for placing information bits within information bit locations of an information block corresponding to the K best synthetic channels;   means for placing frozen bits within frozen bit locations of the information block corresponding to N−K synthetic channels;   means for polar coding the information block to produce a polar codeword; and   means for transmitting the polar codeword to a receiving wireless communication device over a wireless air interface.   
     
     
         28 . The apparatus of  claim 27 , further comprising:
 means for calculating, for a synthetic channel of the set of synthetic channels, the cumulative sum of a binary representation of a position of the synthetic channel within the set of synthetic channels, wherein the cumulative sum comprises a vector having a number of vector components equal to a corresponding number of binary components of the binary representation of the position;   means for comparing the cumulative sums of a first synthetic channel of the set of synthetic channels and a second synthetic channel of the set of synthetic channels, wherein the second synthetic channel has a later position in the set of synthetic channels than the first synthetic channel; and   means for upgrading the second synthetic channel with respect to the first synthetic channel or the first synthetic channel with respect to the second synthetic channel in the order of synthetic channels based on the comparing of the cumulative sums.   
     
     
         29 . The apparatus of  claim 28 , further comprising:
 means for upgrading the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels when the cumulative sum of the second synthetic channel is component-wise greater than the cumulative sum of the first synthetic channel.   
     
     
         30 . The apparatus of  claim 28 , further comprising:
 means for calculating a respective difference between each set of corresponding components in the cumulative sums of the second synthetic channel and the first synthetic channel;   means for upgrading the second synthetic channel with respect to the first synthetic channels in the order of synthetic channels when a sum of the respective differences is greater than zero;   means for comparing a last component of the cumulative sum of the first synthetic channel to a corresponding last component of the cumulative sum of the second synthetic channel when the sum of the respective differences is equal to zero;   means for upgrading the first synthetic channel with respect to the second synthetic channel in the order of synthetic channels when the last component of the cumulative sum of the first synthetic channel is greater than the corresponding last component of the cumulative sum of the second synthetic channel;   means for upgrading the second synthetic channel with respect to the first synthetic channel in the order of synthetic channels when the last component of the cumulative sum of the first synthetic channel is less than the corresponding last component of the cumulative sum of the second synthetic channel; and   means for comparing a next-to-last component of the cumulative sum of the first synthetic channel to a next-to-last component of the cumulative sum of the second synthetic channel when the last component of the cumulative sum of the first synthetic channel is equal to the corresponding last component of the cumulative sum of the second synthetic channel.