Combined Polar Codes and 2-12 QAM over AWGN and Rayleigh Fading Channels

This article investigates a communication system using polar codes combined with a 2-12 QAM modulation scheme over channels disturbed by additive white Gaussian noise (AWGN) as well as over channels disturbed by Rayleigh fading in addition to AWGN. The 2-12 QAM modulation is compatible with legacy 16 QAM still widely used, and when combined with appropriate error correcting codes produces results that approach the Shannon limit


Introduction
Quadrature amplitude modulation (QAM) has been included in current wireless communication standards due to its high bandwidth efficiency [1].The 16 QAM constellation with Gray code mapping, performs well and in combination with LDPC codes or turbo codes it can approach the Shannon limit [2], [3].High power linear amplifiers are required at the transmitter side for some practical applications employing QAM modulation [4].However, due to non-linear distortions caused by the output of power amplifiers, the Shannon limit is far from being reached [5].Various linearization techniques [6], [7] have been introduced to reduce the non-linear effect, but changing hardware makes implementation difficult due to non-compatibility with systems already in use [8].
The recently proposed multidimensional 2-12 QAM modulation [8] presents relevant energy gains using just software mapping changes in the existing 16 QAM modulation.The 2-12 QAM modulation scheme uses two 16 QAM constellations, and eliminates the four highest energy symbols in each 16 QAM constellation to reduce the peak-to-average power ratio (PAPR).The 2-12 QAM maps 7 bits into a 4-dimensional symbol.
In a wireless mobile communication system, the transmitted signals suffer alterations due to different deleterious effects caused by the channel.As a consequence such signals might reach the receiver through multiple paths with different phases and amplitudes [9] thus causing signal fading.Signal fading can be characterized by a random process governing the magnitude of the transmitted signal [10] reaching the receiver.If there is no line of sight signal reception then the envelope of the received signal can be modeled by a Rayleigh probability distribution function [11].
To minimize the effect of noise in transmissions, error correcting codes are used to add redundant bits to the sequence to be sent in order to improve reliability [12].
The error correction capability of polar codes allowed them to be selected by the 3rd Generation Partnership Project (3GPP) as the code for the control channel in the 5th generation (5G) of mobile communication systems [13], [14].Polar codes present low complexity encoding and decoding algorithms [15].
In usual practice, the block length of polar code is a power of 2, therefore not being immediately compatible for combining with some modulation schemes, such as 2-12 QAM, for example.In this case, a kernel matrix of adequate size combined with a standard kernel matrix is recommended [16].Another alternative is the Arikan punctured kernel matrix [17], however both solutions require a high computational complexity.Another solution, which consists of appending a suffix of zeros to each codeword, can be employed in order to minimize computational complexity [18].
A polar code is designed to operate efficiently in a specific channel, and thus it is a challenge to use polar codes in a practical situation where the communication channel changes with time, as occurs in the presence of fading during transmission in free space.The problem presented by the use of polar codes over fading channels is addressed in [19], considering an additional constraint on the permissible average and peak power.Their model reduces the fading channel to a model consisting of a cascade of an AWGN channel followed by an erasure channel.In this article we apply polar codes to a fading channel without providing any further mitigation in order to cope with fading conditions.Our goal is to analyze the performance of polar codes combined with 2-12 QAM modulation in AWGN channels and Rayleigh fading channels disturbed by AWGN, when compared to legacy 16 QAM modulation under similar noise conditions, and discuss the results obtained by computer simulation.
The remaining sections of this paper are organized as follows.In Section 3, the 2-12 QAM modulation is described.A review of polar codes is presented in Section 4. The system model is described in Section 5.The simulated results are presented in Section 6.Finally, the conclusions are presented in Section 7.

Methods/Experimental
The studies that take place on this manuscript require data obtained from computational simulations with the goal of properly describe the BER performance of polar codes combined with 2-12 QAM modulation in AWGN channels and Rayleigh fading channels disturbed by AWGN, when compared to legacy 16 QAM modulation under similar noise conditions.The data acquired through simulations are them compared to other set of data that can be found on [8] for example.The authors would like to take the opportunity to state that there was no need to collect data from human beings given the purely computational nature of our study.

2-12 QAM
Fig. 1 ilustrates a scheme used for data transmission in several communication systems employing a rate 3/4 error correcting code, a 16 QAM modulator and a power amplifier (PA).The error correcting code is often referred as a forward error correcting (FEC) code [12].The system shown in Fig. 1 has a significant drawback as the power amplifier operates with an average power well below its maximum power [20], in order to avoid non-linear distortions.Aiming to operate effectively a power back-up is required from the power amplifier, with sufficient power to feed the 16 QAM modulation when highest energy symbols occur.This leads to a significant reduction in the overall system efficiency [21].Several techniques have been introduced in order to reduce this undesirable effect, such as non-linearity correction and reduction of PAPR, among others [6], [7], [22], [23], however significant changes are required to overcome this negative effect and, because of that, questions about compatibility with current systems are raised.In a 16 QAM system [24], a signal s i (t), 0 < t ≤ T , received in a time slot T is represented as where 1 ≤ i ≤ 16, A x and B y are the amplitudes corresponding to axes cos(2πf c t) and sin(2πf c t), respectively, and f c denotes the carrier frequency.The corresponding 16 QAM constellation is illustrated in Fig. 2, and this format consists of 16 2dimensional vectors, denoted by where i ∈ {1, . . ., 16}, y=1 and the distance between adjacent symbols is equal to 2d.The parameter d is related to the signal energy [24].
In order to use the 2-12 QAM modulation the transmission system of Fig. 1 should be replaced by the transmission system illustrated in Fig. 3, consisting of a rate 6/7 FEC code, a 2-12 QAM modulator and a PA.The 2-12 QAM modulation is based on the legacy 16 QAM modulation with mapping changes and offers energy gain and compatibility with current systems [8].
The 2-12 QAM is a multidimensional modulation obtained by eliminating the four highest energy symbols from two 2-dimensional 16 QAM constellations similar to the one depicted in Fig. 2. The eliminated symbols are those indicated by s 1 , s 4 , s 13 and s 16 , situated in the regions marked with ✕ in Fig. 4. As a result we have two constellations with twelve symbols each (2-12 QAM) to be mapped into a 4-dimensional signal.
The 16 QAM modulation combined with a rate 3/4 FEC has spectral efficiency of 3 bits/symbol [25].In order to maintain the same spectral efficiency in a 2-12 QAM  system, a rate 6/7 FEC and a new mapping is employed, by which a 7 bit block is associated to 2 twelve-ary symbols as illustrated in Fig. 3, by the block labeled as 7B -2TW mapping.One advantage of this format is the possibility of using a 16 QAM modulator/demodulator hardware and obtain 2-12 QAM modulation through a software update [8].As explained in [8], a 2-12 QAM signal s ij (t), 0 < t ≤ 2T , received in time slot 2T is represented as where the indices i and j are each associated with a distinct constellation in Fig. 4, i ∈ { 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15}, j ∈ {2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15}.A x and B y are amplitudes allowed in the axes cos(2πf c t) and sin(2πf c t), respectively, for one constellation, and C s and D z are amplitudes allowed in the axes cos(2πf c t) and sin(2πf c t), respectively, in the other constellation in Fig. 4, and f c denotes the carrier frequency.
The corresponding probability density function is given by where r 1 , r 2 , r 3 and r 4 are the components of the received vector r after transmission over an AWGN channel, given by where n 1 , n 2 , n 3 and n 4 are the additive Gaussian noise components with variance σ 2 = N 0 /2 and mean µ = 0.

Polar Codes
Polar codes, proposed by Arikan [15], can achieve the symmetric capacity of binary discrete memoryless channels (B-DMC) through channel polarization.Channel polarization refers to the operation that allows synthesizing N independent copies of a B-DMC W , from a set of N binary inputs {W . Following [15] a generic B-DMC, denoted as W : X → Y, is considered with a binary input alphabet X , i.e., X = {0, 1}, output alphabet Y, and transition probabilities W (y|x), x ∈ X , y ∈ Y.The goal of this technique is to select the most suitable channels for transmission of K information bits, with N encoded bits, and the code rate R is defined as R = K/N .The remaining N − K positions in a codeword are filled with a fixed value each, known by the decoder in advance, which are called frozen bits [15].The choice of the most suitable channels is made by the use of the Bhattacharyya parameter, defined as There are two phases to this operation, namely a channel combining phase and a channel splitting phase.The first phase consists of combining N copies of W to produce the channel W N : X N → Y N in a recursive manner, where N = 2 n , n ≥ 1.In its first level, two copies of W 1 are combined to form the channel W 2 : X 2 → Y 2 with the transition probabilities where u 1 ,u 2 are the inputs to channel W 2 and y 1 , y 2 are its outputs.The next phase, after having synthesized the channel vector N and u i as its input.The corresponding transition probabilities are defined by Given the bit sequence u , the codeword x is generated by the following operation where G N is the generator matrix of order N for the encoder [14], given by where B N is a bit-reversal permutation, F ⊗n is the n-th Kronecker power of F [15], and The decoding process takes place with the use of the Successive Cancelation (SC) algorithm, also proposed in [15].In order to estimate the decoded message, an SC decoder calculates the probability of the value of a certain bit by making use of values found on the encoder structure.In case of decoding a frozen bit, the value 0 is automatically assigned by the decoder to that frozen bit.

System Model
The block diagram of the proposed system is shown in Fig. 5.The idea is to investigate the performance of the proposed system to transmit information, when polar codes are combined with 2-12 QAM modulation over AWGN channels and Rayleigh fading channels affected by AWGN.Initially, an information source generates a sequence u of equiprobable bits to be encoded as indicated in (10).Then the generated codeword is split into 7 bit blocks to be mapped into 2-12-ary symbols according to the 2-12 QAM scheme descried earlier.Because the block length of the polar code used in this article is a power of 2, which is not an integer multiple of 7, at least in principle, it is not compatible with the 2-12 QAM modulation.In order to circumvent this difficulty a possible approach is to develop a scheme that combines the standard polarization kernel with an elaborate 7 × 7 design kernel [15], [16].In order to reduce the complexity of code construction and decoding algorithm in this article an uncoded suffix of zeros corresponding to s = 7 − (N mod 7) is added to each codeword [18].Then this modified codeword x ′ , with suffix s, has length N s given by After being mapped according to the 2-12 QAM scheme, the modulated sequence is transmitted through AWGN channels and Rayleigh fading channels disturbed by AWGN, with transmission rate R = K/N s .The channel output is denoted by the vector r k in ( 14), 1 ≤ k ≤ N s /7.
where n 1k , n 2k , n 3k and n 4k are the additive Gaussian noise components with variance σ 2 = N 0 /2 and mean µ = 0, A xk , B yk , C sk and D zk are the set of true hypothesis, α is the fading amplitude of a channel obeying the Rayleigh distribution with probability density function given by [26] considering a normalized fading, that is, E[α 2 ] = 1, where E[.] denotes the expected value operator [27].
After transmission over AWGN channels or Rayleigh fading channels disturbed by AWGN, and assuming equiprobable bits, P (b = 0) = P (b = 1) = 1/2, the Log Likelihood Ratio (λ t ) for bit x t is calculated according to [28] as follows where 1 ≤ t ≤ N s .The decision rule employed is b t = 0 if λ ≥ 0 and b t = 1 otherwise, and the probability p(r k |α k , x k = b) can be written as (17).The last s bits provided by the 2-12 QAM demodulator are punctured and a sequence of length N is decoded through the conventional SC algorithm.
6 Results and Discussion Fig. 6 shows performance curves expressed by bit error rate (BER) versus E b /N 0 for a system using uncoded 2-12 QAM with average energy E s ≈ 7.33 dB and variance σ 2 ≈ 0.52/10 (E b /N0) .For comparison purposes, a system using uncoded 16 QAM modulation format with Gray code mapping, with average energy E s = 10 ) is also shown in Fig. 6.Observing the curves in Fig. 6, we notice that the system employing uncoded 2-12 QAM modulation presents an energy gain of approximately 3.1 dB for BER = 10 −4 when compared to the system that uses uncoded 16 QAM modulation, both systems operating over AWGN channels.
We consider next a coded communication system as indicated in Fig. 3 fed with an information sequence u of length K = 880 as the input to the encoder, using a polar code of rate R = 6/7 and block length 1024.A codeword x of length N = 1024 has a suffix of 5 bits added in order to be compatible with 2-12 QAM, thus resulting in a lengthened codeword x ′ of length N s = 1029.The modulated sequence passes through an AWGN channel with variance σ 2 ≈ 0.61/10 (E b /N0) .At the receiver, after calculation of the LLRs and puncturing the suffix, the SC algorithm is employed.The corresponding performance curves, expressed by BER versus E b /N 0 , are also shown in Fig. 6.The system using 2-12 QAM with a polar code, when compared to uncoded 2-12 QAM presents energy gains of about 1.5 dB and 2.3 dB for BER = 10 −3 and 10 −4 , respectively, with a block length N = 1024.
For comparison purposes, a coded system as ilustrated in Fig. 1, with 16 QAM modulation is considered which employs Gray code mapping and a polar code with block length N = 1024 and rate R = 3/4 in AWGN channel with variance σ 2 ≈ 1.67/10 (E b /N0) .The corresponding BER versus E b /N 0 curve is shown in Fig. 6.The system employing 2-12 QAM modulation and a polar code with block length N = 1024 presents a gain of approximately 1.3 dB for BER = 10 −3 .Fig. 6 also shows performance curves when a polar code of block length N = 512 is employed.When considering 2-12 QAM modulation, the suffix to be added has length s = 6 bits and the length of codeword x ′ is N s = 518.Considering the same spectral efficiency, performance curves for the system using 2-12 QAM with code rate R = 6/7 presents an energy gain of approximately 0.8 dB for BER = 10 −3 when compared to a system using 16 QAM with code rate R = 3/4, and block length N = 512.The 2-12 QAM combined with a polar code with block length N = 1024 presents an energy gain of about 0.6 dB for BER = 10 −4 when compared to the system that uses 2-12 QAM and a polar code with block length N = 512.The system using uncoded 2-12 QAM has an energy gain of about 1.2 dB for BER = 10 −2 when compared with the 16 QAM combined and a polar code with block length N = 1024.Fig. 7 shows performance curves for combined polar codes and 2-12 QAM system, indicating an energy gain of approximately 2 dB for BER = 10 −2 when compared to the system using 16 QAM modulation format with Gray code mapping, both over uncoded AWGN channels and Rayleigh fading channels disturbed by AWGN.Fig. 7 illustrates the performance of a system using polar codes of rate R = 6/7, block length N = 1024 with an appended suffix of s = 5 bits to achieve compatibility with the 2-12 QAM scheme, and considering AWGN channels and Rayleigh fading channels disturbed by AWGN, when the SC decoding algorithm is used.This system presents an energy gain of about 2 dB for BER = 10 −3 over the 16 QAM modulation with Gray code mapping with polar code of rate R = 3/4 and block length N = 1024 also over AWGN channels and Rayleigh fading channels disturbed by AWGN.
The system performance when 2-12 QAM and a polar code with block length N = 1024 are employed, shows an energy gain of about 5.5 dB for BER = 10 −2 when compared to uncoded 2-12 QAM, both over AWGN channels and Rayleigh fading channels disturbed by AWGN.The performance of 16 QAM combined with a polar code of block length N = 1024 over AWGN channels offers an energy gain of about 6 dB for BER = 10 −2 in comparison to uncoded 16 QAM over the same channels conditions.For Rayleigh fading channels disturbed by AWGN this system offers an energy gain of about 6 dB for BER = 10 −2 in comparison to uncoded 16 QAM over the same channels conditions.Fig. 7 also shows the performance of systems using 2-12 QAM and 16 QAM both combined with a polar code of block length N = 512 and over AWGN channels and Rayleigh fading channels disturbed by AWGN.

Conclusions
This article presents performance results for a new scheme for data transmission over channels disturbed by AWGN or by Rayleigh fading plus AWGN, using 2-12 QAM modulation and polar codes.The main results are summarized as follows.
• AWGN channels, at BER = 10 For future work other decoding schemes such as Successive Cancelation List decoding can be used [29] as well as generalized fading channel distributions [30].

Figure 1
Figure 1 Block diagram of a transmitter for a 16 QAM system with a rate 3/4 FEC code

Figure 3
Figure 3 Block diagram of a transmitter for a 2-12 QAM system with a rate 6/7 FEC code

Figure 5
Figure 5 Block diagram of the proposed transmission system employing coded 2-12 QAM modulation

Figure 6
Figure 6 BER versus E b /N 0 performance over AWGN channels of uncoded 2-12 QAM and uncoded 16 QAM, and 2-12 QAM and 16 QAM combined with polar codes

Figure 7
Figure 7 BER versus E b /N 0 performance over Rayleigh fading channels disturbed by AWGN of uncoded 2-12 QAM and uncoded 16 QAM, and 2-12 QAM and 16 QAM combined with polar codes −4 a) Uncoded 2-12 QAM has a 3 dB gain in E b /N 0 over uncoded 16 QAM.b) Coded 16 QAM has a 4 dB gain in E b /N 0 over uncoded 16 QAM, for the same spectral efficiency, employing a polar code of block length 1024.c) Coded 2-12 QAM has a 1.5 dB gain in E b /N 0 over coded 16 QAM, for the same spectral efficiency, employing polar codes of block length 1024 in both cases.• Rayleigh fading with AWGN channels, at BER = 10 −3 a) Uncoded 2-12 QAM has an estimated gain of 2.5 dB in E b /N 0 over uncoded 16 QAM.b) Coded 16 QAM has an estimated gain of at least 10 dB in E b /N 0 over uncoded 16 QAM, for the same spectral efficiency, employing a polar code of block length 1024.c) Coded 2-12 QAM has a 2 dB gain in E b /N 0 over coded 16 QAM, for the same spectral efficiency, employing either polar codes of block length 1024 or 512.