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POLAR CODED SIGNALING OVER A PARALLEL MULTICHANNEL/MULTICARRIER CHANNEL

NOTE: This section is Under-Edit if necessary: Construction began on October 17, 2025 and was finished on October 17, 2025.

POLAR BINARY CODES & SUCCESSIVE CANCELLATION DECODING: M-ary Signaling over a Parallel MultiChannel/MultiCarrier Channel 

by Darrell A. Nolta
October 16, 2025

The AdvDCSMT1DCSS (T1) Professional (T1 Version 2) 5G NR LDPCC PC Revision system tool has been used to create a set of Polar Codes (N, K) Encoders and associated Successive Cancellation Decoders and to investigate the phenomenon of Channel Polarization that was discovered by Erdal Arikan as described in his 2009 published paper (titled: 'Channel Polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels') .

T1 V2 LDPCC PC Revision has been used as reported on this website by the papers titled 'Polar Binary Codes & Successive Cancellation Decoding: BPSK Signaling over a Coherent Memoryless Channel' and 'Polar Binary Codes & Successive Cancellation Decoding BFSK Signaling over a Rayleigh Fading Channel with Diversity' to verify the existence of Channel Polarization in simulated Memoryless and Memory Channels with Additive White Gaussian Noise (AWGN).

Please consult these papers on this website to learn about the attributes of T1 V2 and Polar Coding and Decoding. Also, consult the T1 V2 5G NR LDPCC PC Revision 'Key Capabilities and Features Guide' on this website for the description of T1 V2 5G NR LDPCC PC Revision 1 and Polar Codes feature.

So the obvious question is: Can the Channel Polarization phenomenon occur in Parallel MultiChannel/MultiCarriers Channels (PMC) without or with Memory such as an AWGN Memoryless (ML) PMC or a Discrete-Time (DT) FFT-based Discrete MultiTone (DMT) Modulation PMC without or with Rayleigh or Rician Fading.?

This paper will report results of the use of T1 V2 to investigate the behavior (physical manifestation) of Channel Polarization [Polar Coding system's Bit Error Rate (BER) or Bit Error Probability (Pb) Minimization where the minimum BER value is striking less than the corresponding UnCoded system's BER value at a Eb/N0 value] in AWGN ML PMCs without Fading and FFT-based DMT Modulation PMCs without Fading component of a simulated Polar Coding & Successive Cancellation Decoding system.

An important note to be recognized is that the DMT Modulation PMC is also known as Orthogonal frequency-division Multiplexing (OFDM) PMC. T1 V2's OFDM implementation is FFT-Based.

Key to understanding Channel Polarization is that for this 'Channel Polarization Occurrence in a Parallel MultiChannel/MultiCarrier Channel' study, we have two major cases: 1) using Consecutive Non-UPO Bit-Coordinate Channel-to-Frozen Bit assignment; and 2) using the 5G NR standard that specifies the UPO UPO Bit-Coordinate Channel-to-Frozen Bit assignment.

Remember that we want to minimize the Bit Error Rate (BER) of the transmission of Information Bits through a Noisy Channel. Thus, we assign the Least Reliable Bit-Coordinate Channels to the set of Frozen Bits and assign the Most Reliable Bit-Coordinate Channels Bits to the set of Information/Data Bits.

The AdvDCSMT1DCSS (T1) Professional (T1 Version 2) 5G NR LDPCC PC Revision system tool has now been revised to support this study of Polar Code Encoders, Parallel MultiChannel/MultiCarrier Channels [AWGN ML, CrossTalk (XTALK) PMC, and FFT-based DMT Modulation PMC Channels] and Successive Cancellation Decoding.

Specifically, this paper is focused on the use of T1 V2 5G NR LDPCC PC Revision to investigate whether or not the phenomenon of Channel Polarization can occur when transmission occurs in separate Parallel MultiChannel SubChannels or in separate MultiCarrier Channel SubCarriers/SubChannels. In the two previous studies, a single Memoryless or Memory Channel was used multiple times to transmit the Polar Encoder output of N bits where N is the CodeWord Blocklength.

According to Arikan's theory of Channel Polarization, Channel Polarization should only occur in Memoryless Channels. In this theory, polarized channels (Bit-Coordinate channels) are created out of N independent copies of a binary-input discrete memoryless channel by an n-fold Kronecker product of a basic polarization kernel (G2) transformation. The input to each Bit-Coordinate channel is a bit taken from an encoding vector (known as a 'free' bit) and the output of this channel is a CodeWord bit.

In the case of M-ary Signaling over a Parallel MultiChannel or MultiCarrier Channel, one or more Polar CodeWord bits are mapped to a subchannel or subcarrier, respectively.

As a reminder, a Polar Coded Parallel MultiChannel (MC) is partitioned into G parallel subchannel groups where a subchannel group consists of K parallel subchannels. The set {G * Ng} represents the possible partitions of the Polar Code's blocklengh (N). This approach is used for the Polar Code & Signaling over a PMC application because N can be very large and the process of Codebits to Channel Input Bits assignment can quickly become unmanageable. Note {li} is the Group's set of the Number of Channel Input Bits.

So it is an important question to ask if the Channel Polarization can occur in a Polar Coded MultiChannel/MultiCarrier Channel system where Information carrying bits are transmitted through Bit-Coordinate channels that are reliable (noiseless).

This study involves the Polar Code Encoder model as specified by the 5G NR standard (3GPP set of TS 38.212 Version 16.2.0 Release 16 Standard). For use in T1 V2, it is described as the Alternative or Input G2 Kernels (IG2K) Polar Code Encoder model. This model is not the Arikan Polar Code Encoder Model. The specific Polar Code used for this study are N = 128, K = 64.

The matching Polar Code Decoder is defined as the Alternative or Output G2 Kernels (OG2K) Polar Code Decoder model.

This Polar Coded Signaling over an AWGN ML PMC or FFT-based DMT Modulation MultiCarrier Channel model consists of the following characteristics:

1) the Distinct 7-MC's {G * Ng} & {li} sets that describe 28 subchannels (or subcarriers) PMC Modulation and Demodulation subsystem is {4 * 32} &{li} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM}; or

2) the Distinct 4-MC's {G * Ng} & {li} sets that describe 64 subchannels (or subcarriers) PMC Modulation and Demodulation subsystem is {16 * 8} &{li} = {1,1,2,4} <=>{BPSK,PI/2 BPSK,QPSK,16-QAM};

where PI/2 BPSK is Orthogonal BPSK.

3) Since a PMC simulation consists of a number of Signaling Schemes (Distinct) the possible choices for the set of Signal Scheme's Signal-to-Noise Ratio (SNR) Eb/N0(k) values can become very large. To simplify this matter, for each Pb simulation, all the Signaling Schemes' Eb/N0(k) values are specified so that they are all equal. Thus, a plot's Eb/N0 value is defined as

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K Signaling Schemes.

4) Each Polar Coded Discrete-Time Waveform (DTW) AWGN PMC subchannel consists of half-cosine orthonormal baseband shaping pulse, 8 symbols per symbol period, and half-cosine matched filter demodulator front-end.

These DTW subchannels possess a NonDistorting, UnRestricted Bandwidth;

5) 28 Discrete-Time (DT) FFT-Based DMT Modulation SubCarriers/SubChannels & 64 IFFT (Inverse Discrete Fourier Transform) Samples per Frame where the MultiChannel Channel is Transmitted over a Single Channel or

64 Discrete-Time (DT) FFT-Based DMT Modulation SubCarriers/SubChannels & 16 IFFT Samples per Frame where the MultiChannel Channel is Transmitted over a Single Channel.

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth;

The simulated Bit Error Rate (BER) (Pb) performance of each 7-MC DTW AWGN ML (or DT FFT-based DMT Modulation) PMC Comm system was obtained and compared to the BER performance of a UnCoded M-ary Signaling over a 7-MC DTW AWGN ML (or FFT-based DMT Modulation) PMC Communications (Comm) system for each Information Bit Signal-to-Noise Ratio (Eb/N0) via a set of simulated Pb vs Eb/N0 graphs,

The simulated Bit Error Rate (BER) (Pb) performance of each 4-MC DTW AWGN ML (or DT FFT-based DMT Modulation) PMC Comm system was obtained and compared to the BER performance of a UnCoded M-ary Signaling over a 4-MC DTW AWGN ML (FFT-based DMT Modulation) PMC Comm system for each Information Bit Signal-to-Noise Ratio (Eb/N0) via a set of simulated Pb vs Eb/N0 graphs.

The eight simulated BER vs. Eb/N0 graphs Figures 1 through 8 are shown below. Each figure of Figures 1 - 6 clearly shows the effect of using or not using the 5G NR UPO method of choosing the set of Bit-Coordinate Channels to transmit the Information Bits.

In Figures 1, 2, and 3 one can clearly see that Channel Polarization effect does exist for the Use of the 5G NR standard's UPO Bit-Coordinate Channel-to-Frozen Bit assignment in the N = 128, K = 64 Polar Coded (PC) Signaling over a Coherent M-ary Signaling Parallel MultiChannel /MultiCarrier Channel (7-MC or a multiple of 7-MC DTW AWGN ML & DT FFT-based DMT Modulation). We observe the existence of the 'waterfall' portion of the BER curve for the UPO & Coherent Parallel MultiChannel cases. And, the high SNR portion of the Coded BER curve drops below its corresponding UnCoded BER curve such that the BER for the Polar Coded Signaling curve goes to zero before the corresponding BER for the UnCoded Signaling curves goes to zero. The PC Non-UPO BER curve never drops below the UnCoded BER curve in each figure.

In Figure 3, one can see that the BER curves for AWGN ML PMC and FFT-based DMT Modulation cases appear to overlap except at the high SNR portion of the curves for the Non-UPO, UnCoded, and UPO subsets of BER curves.

In Figures 4, 5, and 6 one can clearly see that Channel Polarization effect does exist for the Use of the 5G NR standard's UPO Bit-Coordinate Channel-to-Frozen Bit assignment in the N = 128, K = 64 Polar Coded (PC) Signaling over a Coherent M-ary Signaling Parallel MultiChannel /MultiCarrier Channel (4-MC or a multiple of 4-MC DTW AWGN ML & DT FFT-based DMT Modulation). We observe the existence of the 'waterfall' portion of the BER curve for the UPO & Coherent Parallel MultiChannel cases. And, the high SNR portion of the Coded BER curve drops below its corresponding UnCoded BER curve such that the BER for the Polar Coded Signaling curve goes to zero before the corresponding BER for the UnCoded Signaling curves goes to zero. The PC Non-UPO BER curve never drops below the UnCoded BER curve in each figure.

In Figure 6, one can see that the BER curves for AWGN ML PMC and FFT-based DMT Modulation cases appear to overlap except at the high SNR portion of the curves for the Non-UPO, UnCoded, and UPO subsets of BER curves.

In Figure 7, one can see that the BER curves for AWGN ML PMC and FFT-based DMT Modulation cases appear to overlap except at the high SNR portion of the curves for the Non-UPO subset of BER curves. The 28 or 64 SubChannels AWGN ML PMC and the 28 or 64 SubCarriers/SubChannels FFT-based DMT Modulation PMC appear to overlap each other except at the high SNR portion.

In Figure 8, one can see that the BER curves for AWGN ML PMC and FFT-based DMT Modulation cases appear to overlap except at the high SNR portion of the curves for the UPO subset of BER curves. The 28 or 64 SubChannels AWGN ML PMC and the 28 or 64 SubCarriers/SubChannels FFT-based DMT Modulation PMC appear to overlap each other except at the high SNR portion.

The BER results shown in Figure 7 and 8 may indicate that the T1 V2 simulated Pb are reliable as shown in Figures 1 - 6.

So in conclusion, we have shown the occurrence of Channel Polarization in N = 128, K = 64 Polar Coded Signaling over a DTW AWGN ML PMC and DT FFT-based DMT Modulation PMC as shown by the simulated BER results obtained from the use of T1 Version 2 LDPCC PC Revision. The BER results shown in Figure 1 - 6 clearly support these conclusions.

These results have extreme importance: it may indicate that the Erdal Arikan's theory of Channel Polarization has broader applications in Digital Communications systems. This is true because the Arikan's theory provides a deterministic construction method of Channel Encoders and Channel Decoders that satisfies the Shannon's Noisy Channel Coding Theorem.

T1 Professional (T1 V2) 5G NR LDPCC PC Revision now offers the 5G NR Polar Codes in addition to 5G NR LDPC along with the Gallager, Array, Repeat-Accumulate (RA), and Permutation and Quasi-Cyclic Protograph-Based) LDPC codes construction. This T1 V2 revision supports Gallager, Array, RA, Protograph-Based, and 5G NR LDPC Channel Coding for Signaling over a Memoryless, Memory, or Parallel Multichannel. The Layered Sum-Product Algorithm (SPA) and the OMS Check Message scheme is supported by this T1 V2 revision addition to the Flooding SPA and the Theoretical Check Message scheme for 5G NR Decoding. And, this T1 V2 revision supports the Quantization of SPA Channel Decoder Messages for 5G NR Coded Signaling over a MLC.

This T1 V2 revision supports Gallager, Array, RA, Protograph-Based, 5G NR LDPC and 5G NR Polar Channel Coding for Signaling over a Memoryless, Memory Channel, or Parallel MultiChannel/MultiCarrier Channel.

In conclusion, the User via T1 V2 5G NR LDPCC PC Revision can get experience with the Generation of 5G NR, and Gallager, Array, Repeat-Accumulate, Protograph-based (Permutation and Quasi-Cyclic) LDPC codes and the Sum-Product Algorithm as applied to Iterative Decoding in simulated Digital Communication Systems for Spacecraft and Mobile Communications and Digital Storage Systems LDPC Coding applications.

The User via T1 V2 5G NR LDPCC PC Revision can get experience with the use of Polar Codes and associated SC/SCL Decoders to achieve Channel Polarization and apply it to complex Digital Communication Systems for Spacecraft and Mobile Communications and Digital Storage Systems Polar Coding applications.



Figure 1. Bit Error Probability for Successive Cancellation Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO and UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Coherent Discrete-Time Waveform (DTW) AWGN ML Parallel MultiChannel (PMC) with AWGN:

Equal probable I.I.D. Source for 1 Million Information (Info) Bits for UnCoded & 1 Million Info Bits for 5G NR Polar Coded M-ary Signaling over a Coherent 7-MC (7 SubChannels MultiChannel) DTW AWGN PMC & 28 SubChannels DTW AWGN PMC, respectively;

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive) & UPO Bit-Coordinate Channel-to-Frozen Bit Assignment;

The Distinct 7-MC Group Signaling Schemes consist of {li} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K = 7 Signaling Schemes.

28 DTW subchannels: Each Polar Coded DTW AWGN PMC subchannel consists of half-cosine orthonormal baseband shaping pulse, 8 symbols per symbol period, and half-cosine matched filter demodulator front-end;

These DTW subchannels possess a NonDistorting, UnRestricted Bandwidth;

Successive Cancellation (SC) Decoding Algorithm is implemented by the T1V2 's Alternative Decoder using UnQuantized Messages.



Figure 2. Bit Error Probability for Successive Cancellation Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO and UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Coherent FFT-based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with AWGN:

Equal probable I.I.D. Source for 1 Million Information (Info) Bits for UnCoded & 1 Million Info Bits for 5G NR Polar Coded M-ary Signaling over a Coherent 7-MC (7 SubCarriers/SubChannel MultiChannel) & 28 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive) & UPO Bit-Coordinate Channel-to-Frozen Bit Assignment;

The Distinct 7-MC Group Signaling Schemes consist of {li} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K = 7 Signaling Schemes.

MultiChannel Channel Transmitted over a Single Channel

28 Discrete-Time (DT) FFT-Based DMT Modulation SubCarriers/SubChannels & 64 IFFT Samples per Frame;

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth and No Cyclic Prefix Append;

Successive Cancellation (SC) Decoding Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.



Figure 3. Bit Error Probability for Successive Cancellation Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO and UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Coherent DTW AWGN ML PMC or Coherent FFT-based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with AWGN:

Equal probable I.I.D. Source for 1 Million Information (Info) Bits for UnCoded & 1 Million Info Bits for 5G NR Polar Coded M-ary Signaling over a Coherent 7-MC (7 Subchannels MultiChannel) & 28 SubChannels AWGN ML PMC or 7-MC (7 SubCarriers/SubChannels MultiChannel) & 28 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive) & UPO Bit-Coordinate Channel-to-Frozen Bit Assignment;

The Distinct 7-MC Group Signaling Schemes consist of {li} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K = 7 Signaling Schemes.

Successive Cancellation (SC) Decoding Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages



Figure 4. Bit Error Probability for Successive Cancellation Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO and UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Coherent Discrete-Time Waveform (DTW) AWGN ML Parallel MultiChannel (PMC) with AWGN:

Equal probable I.I.D. Source for 1 Million Information (Info) Bits for UnCoded & 1 Million Info Bits for 5G NR Polar Coded M-ary Signaling over a Coherent 4-MC (4 SubChannels MultiChannel) DTW AWGN PMC & 64 SubChannels DTW AWGN PMC, respectively;

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive) & UPO Bit-Coordinate Channel-to-Frozen Bit Assignment;

The Distinct 4-MC Group Signaling Schemes consist of {li} = {1,1,2,4} <=> {BPSK,PI/2 BPSK,QPSK,16-QAM};

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K = 4 Signaling Schemes.

64 DTW subchannels: Each Polar Coded DTW AWGN PMC subchannel consists of half-cosine orthonormal baseband shaping pulse, 8 symbols per symbol period, and half-cosine matched filter demodulator front-end;

These DTW subchannels possess a NonDistorting, UnRestricted Bandwidth;

Successive Cancellation (SC) Decoding Algorithm is implemented by the T1V2 's Alternative Decoder using UnQuantized Messages.



Figure 5. Bit Error Probability for Successive Cancellation Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO and UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Coherent FFT- based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with AWGN:

Equal probable I.I.D. Source for 1 Million Information (Info) Bits for UnCoded & 1 Million Info Bits for 5G NR Polar Coded M-ary Signaling over a Coherent 4-MC (4 SubCarriers/SubChannels MultiChannel) & 64 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive) & UPO Bit-Coordinate Channel-to-Frozen Bit Assignment;

The Distinct 4-MC Group Signaling Schemes consist of {li} = {1,1,2,4} <=> {BPSK,PI/2 BPSK,QPSK,16-QAM};

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K = 4 Signaling Schemes.

MultiChannel Channel Transmitted over a Single Channel

64 Discrete-Time (DT) FFT-Based DMT Modulation SubCarriers/SubChannels & 16 IFFT Samples per Frame;

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth and No Cyclic Prefix Append;

Successive Cancellation (SC) Decoding Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.



Figure 6. Bit Error Probability for Successive Cancellation Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO and UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Coherent DTW AWGN ML PMC or Coherent FFT-based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with AWGN:

Equal probable I.I.D. Source for 1 Million Information (Info) Bits for UnCoded & 1 Million Info Bits for 5G NR Polar Coded M-ary Signaling over a Coherent 4-MC (4 Subchannels MultiChannel) & 28 SubChannels AWGN ML PMC or 7-MC (7 SubCarriers/SubChannels MultiChannel) & 28 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive) & UPO Bit-Coordinate Channel-to-Frozen Bit Assignment;

The Distinct 4-MC Group Signaling Schemes consist of {li} = {1,1,2,4} <=> {BPSK,PI/2 BPSK,QPSK,16-QAM};

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K = 4 Signaling Schemes.

Successive Cancellation (SC) Decoding Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.



Figure 7. Bit Error Probability for Successive Cancellation Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Coherent 28 or 64 SubChannels AWGN ML PMC with AWGN; or 28 or 64 SubCarrierS/SubChannels FFT-based DMT Modulation PMC with AWGN:

Equal probable I.I.D. Source for 1 Million Info Bits for 5G NR Polar Coded M-ary Signaling over a Coherent 28 or 64 SubChannels AWGN ML PMC; or 28 or 64 SubCarrierS/SubChannels FFT-based DMT Modulation PMC

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive)

The Distinct 7-MC Group Signaling Schemes for 28 SubChannels/SubCarriers PMC consist of {li} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

The Distinct 4-MC Group Signaling Schemes for 64 SubChannels/SubCarriers PMC consist of {li} = {1,1,2,4} <=> {BPSK,PI/2 BPSK,QPSK,16-QAM};

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K = 7 or 4 Signaling Schemes.

Successive Cancellation (SC) Decoding Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.



Figure 8. Bit Error Probability for Successive Cancellation Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Coherent 28 or 64 SubChannels AWGN ML PMC with AWGN; or 28 or 64 SubCarrierS/SubChannels FFT-based DMT Modulation PMC with AWGN:

Equal probable I.I.D. Source for 1 Million Info Bits for 5G NR Polar Coded M-ary Signaling over a Coherent 28 or 64 SubChannels AWGN ML PMC; or 28 or 64 SubCarrierS/SubChannels FFT-based DMT Modulation PMC

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard;

The Distinct 7-MC Group Signaling Schemes for 28 SubChannels/SubCarriers PMC consist of {li} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

The Distinct 4-MC Group Signaling Schemes for 64 SubChannels/SubCarriers PMC consist of {li} = {1,1,2,4} <=> {BPSK,PI/2 BPSK,QPSK,16-QAM};

Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K) , for 1 through K = 7 or 4 Signaling Schemes.

Successive Cancellation (SC) Decoding Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.