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POLAR CODED SIGNALING over a MEMORY CHANNEL

NOTE: This section is Under-Edit if necessary: Construction began on September 18, 2025 and was finished on .

POLAR BINARY CODES & SUCCESSIVE CANCELLATION DECODING: BFSK Signaling over a Rayleigh Fading Channel with Diversity 

by Darrell A. Nolta
September 18, 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 paper titled 'Polar Binary Codes & Successive Cancellation Decoding: BPSK Signaling over a Coherent Memoryless Channel' to verify the existence of Channel Polarization in Memoryless Channels with Additive White Gaussian Noise (AWGN).

Please consult this paper 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 Non-Memoryless Channels such as in a Rayleigh Fading Memory Channel?

Key to understanding Channel Polarization is that for this 'Channel Polarization Occurrence in a Memory 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, Memory Channels [Burst, Fading, and InterSymbol Interference (ISI) 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 in Memory Channels. According to Arikan's theory of Channel Polarization, Channel Polarization should only occur in Memoryless Channels. Given the importance of Fading in Wireless Communication systems, we will study the use of the Rayleigh Fading Memory Channel model with and without Diversity as the signaling channel model along with the Alternative Polar Encoder model and the Successive Cancellation Algorithm Decoding model.

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

This Polar Coded Signaling over a Rayleigh Fading model consists of the following characteristics:

1) the Binary Frequency Shift Keying (BFSK) modulation scheme model is chosen for the T1 V2 Polar Coded Signaling over a Fading Channel model since M-ary FSK is Channel phase-independent, i.e., does not require Channel State Information (CSI).

2) the Rayleigh Fading (Frequency-NonSelective) Memory Channel is based on the -5.25 dB Normalized Energy Gain model; &

3) the Diversity model is based on Time Diversity (L, Number of Replicas of a Transmitted Channel Symbol) is equal to 3 and Time Diversity Symbol Replicas Spacing is equal to 4.

The simulated Bit Error Rate (BER) (Pb) performance of each Comm system was obtained and compared to the BER performance of a UnCoded BFSK Signaling over a Rayleigh Fading Channel with AWGN for each Information Bit Signal-to-Noise Ratio (Eb/N0) via a set of simulated Pb vs Eb/N0 graphs.

The five simulated BER vs. Eb/N0 graphs Figures 1 through 5 are shown below. Each figure of Figures 1 - 5 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 Figure 1, one can clearly see that Channel Polarization does not exist for the Use of the Non-UPO Bit-Coordinate Channel-to-Frozen Bit assignment in the N = 128, K = 64 Polar Coded Signaling over a Coherent Memoryless or Rayleigh Fading Channel. The Non-UPO BER curves (MLC & Rayleigh Fading channel) do not drop below their corresponding UnCoded versions and these BER curves do not drop below Pb value of 10-2.

In Figure 2, 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 Signaling over a Coherent Memoryless or Rayleigh Fading Channel. We observe the existence of the 'waterfall' portion of the BER curve for the Coherent Memoryless Channel as well as for the Rayleigh Fading Channel. 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.

In Figure 3, one can see the effect due to use of Diversity in N = 128, K = 64 Signaling over a Rayleigh Fading Channel. Without the use of Diversity, the BER performance for the Polar Coded Non-UPO cases does not drop below the UnCoded BFSK Signaling cases. But when Diversity is used, the BER performance for the Polar Coded Non-UPO cases drops below the UnCoded BFSK Signaling cases. Is this observed result do to Channel Polarization or Randomization of Channel Output due to the use of Diversity?

In Figure 4, one can clearly observe the effect of Channel Polarization in the case of N = 128, K = 64 Polar Coded Signaling over a Rayleigh Fading Channel when the UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard is used. The BER curve 's 'waterfall' is seen as well as its high SNR portion of BER curve is below it corresponding UnCoded BER curve.

In Figure 5, one can clearly observe the occurrence of the Channel Polarization phenomenon in Polar Coded Signaling over a Rayleigh Fading Channel when the UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard is used. One can clearly see that the use of Diversity enhances Channel Polarization, i.e., a deeper BER curve 'waterfall'. And, Diversity can compensate for the lack of Channel Polarization in the case of using Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment'

So in conclusion, we have shown the occurrence of Channel Polarization in N = 128, K = 64 Polar Coded Signaling over a Memory Channel (Rayleigh Fading Channel) as shown by the simulated BER results obtained from the use of T1 Version 2 LDPCC PC Revision. Further, we have shown that the use of Diversity can clearly improve the BER performance of Polar Coded Signaling over a Rayeigh Fading Channel. The BER results shown in Figure 5 clearly support these conclusions. This result has 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 or Memory 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 of 5G NR Polar Coded (N = 128, K = 64, Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BFSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN or a Rayleigh Fading Channel with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,000 Information (Info) Bits for UnCoded & 5G NR Polar Coded BFSK Signaling over a Coherent Vector MLC & a Vector Rayleigh Fading Memory Channel, 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);

Rayleigh Fading (Frequency-NonSelective): -5.25 dB Normalized Energy Gain;

Maximum Likelihood (ML) or ML Energy Detector Demodulation for UnCoded BFSK Signaling over Coherent MLC or Rayleigh Fading Channel, respectively;

ML Demodulation for 5G NR Polar Coded BFSK Signaling over a Coherent Memoryless Channel;

Ml Energy Detector Demodulation for 5G NR Polar Coded BFSK Signaling over a Rayleigh Fading Channel; &

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



Figure 2. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 128, K = 64, UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BFSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN or a Rayleigh Fading Channel with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,000 Information (Info) Bits for UnCoded & 5G NR Polar Coded BFSK Signaling over a Coherent Vector MLC & a Vector Rayleigh Fading Memory Channel, respectively;

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;

Rayleigh Fading (Frequency-NonSelective): -5.25 dB Normalized Energy Gain;

Maximum Likelihood (ML) or ML Energy Detector Demodulation for UnCoded BFSK Signaling over Coherent MLC or Rayleigh Fading Channel, respectively;

ML Demodulation for 5G NR Polar Coded BFSK Signaling over a Coherent Memoryless Channel;

ML Energy Detector Demodulation for 5G NR Polar Coded BFSK Signaling over a Rayleigh Fading Channel; &

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



Figure 3. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 128, K = 64, Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BFSK Signaling over a Rayleigh Fading Channel with AWGN with or without Diversity:

Equal probable I.I.D. Source for 10 Million and 1,000,000 Information (Info) Bits for UnCoded & 5G NR Polar Coded BFSK Signaling over a Vector Rayleigh Fading Memory Channel, respectively;

Equal probable I.I.D. Source for 1,000,192 Info Bits for 5G NR Polar Coded BFSK Signaling over a Vector Rayleigh Fading Memory Channel with Diversity;

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);

Rayleigh Fading (Frequency-NonSelective): -5.25 dB Normalized Energy Gain;

Time Diversity (L, Number of Replicas of a Transmitted Channel Symbol) = 3 & Symbol Replicas Spacing = 4;

ML Energy Detector Demodulation for UnCoded BFSK Signaling over Rayleigh Fading Channel;

ML Energy Detector Demodulation for 5G NR Polar Coded BFSK Signaling over a Rayleigh Fading Channel; &

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



Figure 4. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 128, K = 64, UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BFSK Signaling over a Rayleigh Fading Channel with AWGN with or without Diversity:

Equal probable I.I.D. Source for 10 Million and 1,000,000 Information (Info) Bits for UnCoded & 5G NR Polar Coded BFSK Signaling over a Vector Rayleigh Fading Memory Channel, respectively;

Equal probable I.I.D. Source for 1,000,192 Info Bits for 5G NR Polar Coded BFSK Signaling over a Vector Rayleigh Fading Memory Channel with Diversity;

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;

Rayleigh Fading (Frequency-NonSelective): -5.25 dB Normalized Energy Gain;

Time Diversity (L, Number of Replicas of a Transmitted Channel Symbol) = 3 & Symbol Replicas Spacing = 4;

ML Energy Detector Demodulation for UnCoded BFSK Signaling over Rayleigh Fading Channel;

ML Energy Detector Demodulation for 5G NR Polar Coded BFSK Signaling over a Rayleigh Fading Channel; &

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



Figure 5. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 128, K = 64, Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment or UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BFSK Signaling over a Rayleigh Fading Channel with AWGN with or without Diversity:

Equal probable I.I.D. Source for 10 Million and 1,000,000 Information (Info) Bits for UnCoded & 5G NR Polar Coded BFSK Signaling over a Vector Rayleigh Fading Memory Channel, respectively;

Equal probable I.I.D. Source for 1,000,192 Info Bits for 5G NR Polar Coded BFSK Signaling over a Vector Rayleigh Fading Memory Channel with Diversity;

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

Rayleigh Fading (Frequency-NonSelective): -5.25 dB Normalized Energy Gain;

Time Diversity (L, Number of Replicas of a Transmitted Channel Symbol) = 3 & Symbol Replicas Spacing = 4;

ML Energy Detector Demodulation for UnCoded BFSK Signaling over Rayleigh Fading Channel and for 5G NR Polar Coded BFSK Signaling over a Rayleigh Fading Channel with or without Diversity; &

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