POLAR CODED SIGNALING OVER A PARALLEL MULTICHANNEL/MULTICARRIER CHANNEL WITH RAYLEIGH FADING

NOTE: This section is Under-Edit if necessary: Construction began on October 22, 2025 and was finished on October 22, 2025.

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

by Darrell A. Nolta
October 22, 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', 'Polar Binary Codes & Successive Cancellation Decoding BFSK Signaling over a Rayleigh Fading Channel with Diversity', and 'Polar Binary Codes & Successive Cancellation: M-ary Signaling over a Parallel MultiChannel/MultiCarrier Channel' to verify the existence of Channel Polarization in simulated Memoryless, Memory, and Parallel MultiChannel/MultiCarrier 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 the Polar Codes feature.

So the obvious question is: Can the Channel Polarization phenomenon occur in Parallel MultiChannel/MultiCarriers Channels (PMC) with Memory such as an AWGN Memoryless (ML) PMC or a Discrete-Time (DT) FFT-based Discrete MultiTone (DMT) Modulation PMC 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 (P_b) Performance Minimization where the minimum BER value is striking less than the corresponding UnCoded system's BER value at an E_b/N₀ value] in DT FFT-based DMT Modulation PMCs with 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 PMC implementation is DT 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 (UPO: Universal Partial Order/Universal Reliability Sequence) Bit-Coordinate Channel-to-Frozen Bit assignment; and 2) using the 5G NR standard that specifies the 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 been revised to support this study of Polar Code Encoders, Parallel MultiChannel/MultiCarrier Channels [AWGN ML, CrossTalk (XTALK) PMC, and DT 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 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 or 'frozen' 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 * N_g} 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 {l_i} is the Group's set of the Number of Channel Input Bits.

As reported in this website paper titled, 'Polar Binary Codes & Successive Cancellation: M-ary Signaling over a Parallel MultiChannel/MultiCarrier Channel', Channel Polarization can occur in a Polar Coded MultiChannel/MultiCarrier Channel system where Information carrying bits ('free') are transmitted through Bit-Coordinate channels that are reliable (noiseless), the UPO chosen Bit-Coordinate channels.

So can Channel Polarization occur in a Polar Coded M-ary Signaling over a Parallel MultiChannel/MultiCarrier Channel with Rayleigh Fading?

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 a FFT-based DMT Modulation Parallel MultiCarrier Channel model consists of the following characteristics:

1) the Distinct N : {G * N_g}, 7-MC Group & {l_i} sets that describe 28 subchannels (or subcarriers) PMC Modulation and Demodulation subsystem is {4 * 32} &{l_i} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM} where PI/2 BPSK is Orthogonal BPSK.

2) 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) E_b/N₀^(k) values can become very large. To simplify this matter, for each P_b simulation, all the Signaling Schemes' E_b/N₀^(k) values are specified so that they are all equal. Thus, a plot's E_b/N₀ value is defined as

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

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.

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth.

This study involves Raleigh Fading model component of the T1 V2's PMC Fading model that is described below:

A Polar Coded (PC) AWGN Memoryless Parallel MultiChannel or DMT Modulation Parallel MultiCarrier Channel can be selected in T1V2 to experience Rayleigh or Rician Fading impairment. These multichannels support Coherent Signaling only (prior to a Fade event) and possess UnRestricted Bandwidth. This fading can exist in the parallel multichannels in one of two possible forms: 1) Fading exists in the SET of ALL SUBCHANNELS [AWGN Memoryless (ML) PMC] or SUBCARRIERS (DMT Modulation PMC); or 2) Fading exists in a SUBSET of SUBCHANNELS (AWGN Memoryless PMC) or SUBCARRIERS (DMT Modulation PMC). And the demodulation of the subchannel or subcarrier output of these parallel multichannels with fading can be performed without or with perfect Channel Side Information (CSI).

For Rayleigh or Rician Fading, the first form [Fading exists in the SET of ALL SUBCHANNELS (AWGN ML PMC) or SUBCARRIERS (DMT Modulation PMC)], the Fading normalized 'Energy Gain' Distribution (Distrb) Polar Coded PMC) Groups Type can be one of two possible subtypes:

1) Rayleigh/Rician Fading 'Energy Gain': Constant Distribution over all of the PMC Groups, or

2) Rayleigh/Rician Fading 'Energy Gain': Gaussian Distribution over all of the PMC Groups.

For each PMC Group, the Fading 'Energy Gain' Distrb Polar Coded Parallel Multiple Channel SubChannel Type can be one of two possible subtypes:

1) Constant Distribution over all of the SubChannels of a PMC Group; or 2) Gaussian Distribution over all of the SubChannels of a PMC Group.

Note that a PMC Group is identified by its Group Index Number (No.) and a PMC SubChannel of a Group is identified by its Group and SubChannel Index Numbers. The PMC Group or SubChannel indices are Zero-based numbers.

The User-specified Rayleigh or Rician Fading normalized 'Energy Gain' (dB) is specified at the Group level as the Baseband Fading Average (Avg) Received SNR to Transmitted SNR (Es/N0) Ratio dB as a constant value or Gaussian Group Mean value.

Note for Rician Fading, the Fading Component's Mean (gamma) as a Fraction of its Standard Deviation is also specified by the User.

For the Gaussian Distribution over all of the PMC Groups, the Group Gaussian Mean can be chosen to be one of three possible Group No. Mean types depending on the even/odd characteristic of the Number of PMC Groups:

1) Smallest Group No. (0); 2) Largest Group No.; or 3) Middle Group No..

The Group Mean Number type 1 or 2 is possible for the Even Number case and type 1, 2, or 3 is possible for the Odd Number case.

The User will specify the Fading PMC Groups 'Energy Gain' Gaussian Distribution Group Standard Deviation that is dependent on the number of Groups.

For the Gaussian Distribution over all of the PMC Group SubChannels, the Group SubChannel Gaussian Mean can be chosen to be one of three possible types depending on the even/odd characteristic of the Number of Subchannels of a PMC Group:

1) Smallest SubChannel No. (0); 2) Largest SubChannel No.; or 3) Middle SubChannel No..

The Group SubChannel Mean Number type 1 or 2 is possible for the Even Number case and type 1, 2, or 3 is possible for the Odd Number case.

Note all of the PMC Groups contain the same number of SubChannels.

The User will specify the Fading PMC Group Subchannel 'Energy Gain' Gaussian Distribution Group Subchannel Standard Deviation that is dependent on the number of Group SubChannels.

For Rayleigh or Rician Fading, the second form [Fading exists in a SUBSET of SUBCHANNELS (AWGN ML PMC) or SUBCARRIERS (DMT Modulation PMC)] allows the User to specify a randomly determined subset of SubChannels of a Group for all Groups that will experience Fading. The indices of these Fading Group SubChannels are selected via a Randomly Uniform Generation process. The User will specify the Number of Group SubChannels that are subjected to the Fading.

Note the Number of Fading SubChannels can be selected to be one to the total number of SubChannels for a Group.

The User will specify the Random Number Generation Seed (a Positive No.). Next, the User will decide the Number of Random Numbers to be generated for the Fading SubChannel Distribution prior to its generation. This number choice can be Default No. (100 Thousand) or User Specified. Then, the User will decide on the Maximum Number of Random Numbers that are necessary for the Number of Fading Group SubChannels Indices to be generated as specified by the User. This number can be Default No. (100 Thousand) or User Specified.

After the Distribution of Fading Group SubChannels has been selected, the User will then specify the normalized Baseband 'Energy Gain' [Avg Received SNR to Transmitted SNR (Es/N0) Ratio, in dB due to Fading which will be the same (constant) for all of the randomly selected Rayleigh or Rician Fading SubChannels for a Group.

Note for Rician Fading, the Fading Component's Mean (gamma) as a Fraction of its Standard Deviation is also specified by the User.

In the cases of Polar Coded FFT-based DMT Modulation Parallel MultiChannel/MultiCarrier (MC) Channel with Fading, Cyclic Prefix (CP) Append option can be chosen. The CP type is DMT Symbol-based. And its Cyclic Prefix Length is User-Specified with its maximum length equal to the number of IFFT samples for the particular FFT-based DMT Modulation MC Channel. Further, the FFT-based DMT Modulation MC Channel with Fading is simulated using the Linear Convolution or Cyclic (Circular) Convolution method (User-Specified).

This T1 V2 simulated FFT-based DMT Modulation PMC Rayleigh Fading model was chosen to have the following characteristics:

1) Rayleigh Fading exists in the SET of ALL SUBCARRIERS/SUBCHANNELS;

2) Rayleigh/Rician Fading 'Energy Gain': Constant Distribution over all of the PMC Groups;

3) For each PMC Group, the Fading 'Energy Gain' Distribution Polar Coded Parallel Multiple Channel SubChannel Type is chosen to have Constant Distribution over all of the SubChannels of a PMC Group;

4) The User-specified Rayleigh Fading normalized 'Energy Gain' (dB) is specified at the Group level as the Baseband Fading Average (Avg) Received SNR to Transmitted SNR (Es/N0) Ratio dB as a constant value of -5.25 dB; &

5) A Polar Coded DMT Symbol-based Cyclic Prefix (CP) Append model using on of two possible CP lengths were used: 16 & 64 IDFT Samples. The CP Lengths 16 or 64 is 1/4 of the number or equal to the number of IFFT samples DMT Symbol, respectively.

The demodulation of the subchannel or subcarrier output of this Parallel DMT Modulation MultiChannel/MultiCarrier Channel with Rayleigh Fading is performed with perfect Channel Side Information (CSI).

The Bit Error Rate (BER) (P_b) performance of each simulated Polar Coded 7-MC per group DT FFT-based DMT Modulation PMC (28 SubCarriers/SubChannels, CP of 16 or 64 IFFT samples) with Raleigh Fading Communications (Comm) system was obtained and compared to the BER performance of a UnCoded M-ary Signaling over a 7-MC DT FFT-based DMT Modulation PMC Communications (Comm) system for each Information Bit Signal-to-Noise Ratio (E_b/N₀) via a set of simulated P_b vs E_b/N₀ graphs.

The five simulated BER vs. E_b/N₀ graphs Figures 1 through 5 are shown below. Each figure of Figures 1 - 5 clearly shows the effect of choosing the appropriate Cyclic Prefix Length (L), i.e., the number of IFFT samples used for the DMT Symbol-based Cyclic Prefix and 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 ('free' bits).

In Figures 1, we see the severe deterioration of BER that is caused by Rayleigh Fading and Cyclic Prefix (CP) Length (L) being too small (almost flat curves & P_b values > 10^-1). Channel Polarization is not observed in this graph: Non-UPO Bit-Coordinate Channel Assignment is used [the Polar Code BER curve (CP L = 64) does not drop below the UnCoded BER curve (CP L = 16) .

In Figures 2, we see the severe deterioration of BER that is caused by Rayleigh Fading and Cyclic Prefix (CP) Length (L) being too small (almost flat curves & P_b values > 10^-1). . Channel Polarization is clearly observed in this graph: UPO Bit-Coordinate Channel Assignment is used and CP Length is equal to the number of IFFT samples [the Polar Code BER curve (CP L = 64) drops below the UnCoded BER curve (CP L = 16) and whose values are striking smaller than the UnCoded BER curve [CP L = 16 that is equal to the number of IFFT samples] values at the high SNR.portion of the curve.

In Figures 3, we see the severe deterioration of BER that is caused by Rayleigh Fading and CP L = 16 being too small (almost flat curves & P_b values > 10^-1). Clearly, Channel Polarization is not observed in this graph even though UPO Bit-Coordinate Channel Assignment is used.

In Figures 4, we clearly observe Channel Polarization in this graph. UPO Bit-Coordinate Channel Assignment is used and the CP L = 64 which is equal to the number of IFFT samples for the DMT Symbol.

In Figures 5, we clearly observe effect of using the appropriate CP L (to compensate for the Rayleigh Fading): too small of CP L allows severe BER deterioration due to Rayleigh Fading. Once the CP L is set to a appropriate length (set to DMT Symbol Length) and UPO Bit-Coordinate Channel Assignment is selected, the UPO BER curve clearly exhibit the Channel Polarization phenomenon.

In conclusion, this paper's P_b results for the T1 V2 simulated Polar Coded M-ary Signaling over a DT FFT-based DMT Parallel MultiCarrier Channel & Successive Decoding Comm system demonstrate the effect of Rayleigh Fading and the use of a compensating scheme (Cyclic Prefix added to the DMT Symbol). The orthogonality of the subcarriers is affected by the distortion of the transmitting channel by a Raleigh fade. The Rayleigh Fading (multipath) induces InterSymbol Interference (ISI) in the SubCarriers/SubChannels that leads to Unequal E_b/N₀ values across the subcarriers/subchannels. The samples of Cyclic Prefix serve as a Guard Band of the DMT Symbol that will result in a reduction of the ISI.

If the Rayleigh Fading process is compensated for by the use of a Cyclic Prefix with length equal to DMT Symbol length, Channel Polarization occurs for the N = 128 K = 64 Polar Coded (using the UPO Bit-Coordinate Channel Assignment method) M-ary Signaling over a FFT-based DMT Modulation PMC, and SC Decoding (using Perfect Channel Side Information Demodulation) Comm system. The behavior of the BER curves in Figure 2, 4, and 5 clearly supports these conclusions.

Does the Channel Polarization phenomenon occurs in this above described Polar Coding Comm system subjected to Rayleigh Fading for larger Polar Code Blocklength N (256, 512, or 1024) and corresponding larger Cyclic Prefix Lengths?

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 (SC) 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 Discrete-Time (DT) FFT-based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with Rayleigh Fading & 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 7-MC (7 SubCarriers/SubChannel MultiChannel) & 28 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Polar 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 & UnCoded 7-MC Signaling Schemes consist of {l_i} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

For each simulated P_b value, E_b/N₀ = E_b/N₀⁽¹⁾ = E_b/N₀⁽²⁾ =…= E_b/N₀^(K), for 1 through K = 7 Signaling Schemes;

MultiChannel Channel Transmitted over a Single Channel: 7 DT UnCoded FFT-Based DMT Modulation SubCarriers/SubChannels & 16 IFFT Samples per Frame; & 28 DT FFT-Based DMT Modulation SubCarriers/SubChannels & 64 IFFT Samples per Frame;

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth;

Frequency-Selective Rayleigh Fading can affect all subcarriers: -5.25 dB Normalized Energy Gain for a subcarrier with fading;

Cyclic Prefix (CP) Append Length (L); UnCoded Signaling: 4 & 16 Samples & Polar Coded Signaling: 16 & 64 Samples; &

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



Figure 2. Bit Error Probability for Successive Cancellation (SC) Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Discrete-Time (DT) FFT-based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with Rayleigh Fading & 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 7-MC (7 SubCarriers/SubChannel MultiChannel) & 28 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Polar 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;

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

For each simulated P_b value, E_b/N₀ = E_b/N₀⁽¹⁾ = E_b/N₀⁽²⁾ =…= E_b/N₀^(K), for 1 through K = 7 Signaling Schemes;

MultiChannel Channel Transmitted over a Single Channel: 7 DT UnCoded FFT-Based DMT Modulation SubCarriers/SubChannels & 16 IFFT Samples per Frame; & 28 DT FFT-Based DMT Modulation SubCarriers/SubChannels & 64 IFFT Samples per Frame;

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth;

Frequency-Selective Rayleigh Fading can affect all subcarriers: -5.25 dB Normalized Energy Gain for a subcarrier with fading;

Cyclic Prefix (CP) Append Length (L); UnCoded Signaling: 4 & 16 Samples & Polar Coded Signaling:: 16 & 64 Samples; &

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



Figure 3. Bit Error Probability for Successive Cancellation (SC) Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO & UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Discrete-Time (DT) FFT-based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with Rayleigh Fading & 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 7-MC (7 SubCarriers/SubChannel MultiChannel) & 28 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Polar 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 & UnCoded 7-MC Signaling Schemes consist of {l_i} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

For each simulated P_b value, E_b/N₀ = E_b/N₀⁽¹⁾ = E_b/N₀⁽²⁾ =…= E_b/N₀^(K), for 1 through K = 7 Signaling Schemes;

MultiChannel Channel Transmitted over a Single Channel: 7 DT UnCoded FFT-Based DMT Modulation SubCarriers/SubChannels & 16 IFFT Samples per Frame; & 28 DT FFT-Based DMT Modulation SubCarriers/SubChannels & 64 IFFT Samples per Frame;

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth;

Frequency-Selective Rayleigh Fading can affect all subcarriers: -5.25 dB Normalized Energy Gain for a subcarrier with fading;

Cyclic Prefix (CP) Append Length (L); UnCoded Signaling: 4 Samples & Polar Coded Signaling:: 16 Samples; &

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



Figure 4. Bit Error Probability for Successive Cancellation (SC) Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, Non-UPO & UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Discrete-Time (DT) FFT-based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with Rayleigh Fading & 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 7-MC (7 SubCarriers/SubChannel MultiChannel) & 28 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Polar 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 & UnCoded 7-MC Signaling Schemes consist of {l_i} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

For each simulated P_b value, E_b/N₀ = E_b/N₀⁽¹⁾ = E_b/N₀⁽²⁾ =…= E_b/N₀^(K), for 1 through K = 7 Signaling Schemes;

MultiChannel Channel Transmitted over a Single Channel: 7 DT UnCoded FFT-Based DMT Modulation SubCarriers/SubChannels & 16 IFFT Samples per Frame; & 28 DT FFT-Based DMT Modulation SubCarriers/SubChannels & 64 IFFT Samples per Frame;

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth;

Frequency-Selective Rayleigh Fading can affect all subcarriers: -5.25 dB Normalized Energy Gain for a subcarrier with fading;

Cyclic Prefix (CP) Append Length (L); UnCoded Signaling: 16 Samples & Polar Coded Signaling:: 64 Samples; &

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



Figure 5. Bit Error Probability for Successive Cancellation (SC) Algorithm Decoding of 5G NR Polar Coded (N = 128, K = 64, UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) M-ary Signaling over a Discrete-Time (DT) FFT-based Discrete MultiTone (DMT) Modulation Parallel MultiChannel (PMC) with Rayleigh Fading & 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 7-MC (7 SubCarriers/SubChannel MultiChannel) & 28 SubCarriers/SubChannels FFT-based DMT Modulation PMC, respectively;

5G NR Polar 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 & UnCoded 7-MC Signaling Schemes consist of {l_i} = {1,1,2,4,8,8,8} <=> {BPSK,PI/2 BPSK,QPSK 16-QAM,256-QAM,256-QAM,256-QAM};

For each simulated P_b value, E_b/N₀ = E_b/N₀⁽¹⁾ = E_b/N₀⁽²⁾ =…= E_b/N₀^(K), for 1 through K = 7 Signaling Schemes;

MultiChannel Channel Transmitted over a Single Channel: 7 DT UnCoded FFT-Based DMT Modulation SubCarriers/SubChannels & 16 IFFT Samples per Frame; & 28 DT FFT-Based DMT Modulation SubCarriers/SubChannels & 64 IFFT Samples per Frame;

These DT subchannels possess a NonDistorting, UnRestricted Bandwidth;

Frequency-Selective Rayleigh Fading can affect all subcarriers: -5.25 dB Normalized Energy Gain for a subcarrier with fading;

Cyclic Prefix (CP) Append Length (L); UnCoded Signaling: 4 & 16 Samples & Polar Coded Signaling:: 16 & 64 Samples; &

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