Shaped On-Off Keying Using Polar Codes

TL;DR

This paper introduces a polar code-based joint distribution matching and error correction scheme for power-efficient on-off keying, achieving near-optimal signaling and significant performance gains over uniform transmission.

Contribution

It proposes a novel polar coding scheme that jointly performs distribution matching and error correction for OOK, surpassing previous methods that used separate or time-sharing approaches.

Findings

Achieves 1.8 dB gain over uniform transmission at 0.25 bits/channel use

Uses a blocklength of 65,536 bits for numerical simulations

Enables asymptotically optimal signaling with polar codes

Abstract

The probabilistic shaping scheme from Honda and Yamamoto (2013) for polar codes is used to enable power-efficient signaling for on-off keying (OOK). As OOK has a non-symmetric optimal input distribution, shaping approaches that are based on the concatenation of a distribution matcher followed by systematic encoding do not result in optimal signaling. Instead, these approaches represent a time sharing scheme where only a fraction of the codeword symbols is shaped. The proposed scheme uses a polar code for joint distribution matching and forward error correction which enables asymptotically optimal signaling. Numerical simulations show a gain of 1.8 dB compared to uniform transmission at a spectral efficiency of 0.25 bits/channel use for a blocklength of 65,536 bits.

Equations

$$Y=aX+N$$

$$P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(1}\right)=p,\quad P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(0}\right)=1-p.$$

$$p^{*}(\gamma)=\operatorname*missing{argmax}_{p}\quad\mathop{}\!{\textnormal{I}}\mathopen{}\mathclose{{}\left(X\mathchar 59\relax Y}\right)\quad\text{s.t.}\quad pa^{2}=\gamma.$$

$$\mathbf{x}={\mathbf{uG}}_n,\text{with }{\mathbf{G}}_n={\mathbf{G}}_2^{\otimes n}\text{ and }{\mathbf{G}}_2=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right].$$

$$\mathcal{I}$$

$$\mathcal{F}$$

$$\displaystyle\lim_{N\to\infty}\,\,\frac{1}{N}\mathopen{}\mathclose{{}\left|\mathcal{I}}\right|$$

$$\displaystyle\lim_{N\to\infty}\,\,\frac{1}{N}\mathopen{}\mathclose{{}\left|\mathcal{F}}\right|$$

$$\mathop{}\!{\textnormal{I}}\mathopen{}\mathclose{{}\left(X\mathchar 59\relax Y}\right)=\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X}\right)-\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X\>\!\middle|\>\!Y}\right)=1-\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X\>\!\middle|\>\!Y}\right).$$

$$\mathcal{U}$$

$$\mathcal{D}$$

$$\displaystyle\lim_{N\to\infty}\,\,\frac{1}{N}\mathopen{}\mathclose{{}\left|\mathcal{U}}\right|$$

$$\displaystyle\lim_{N\to\infty}\,\,\frac{1}{N}\mathopen{}\mathclose{{}\left|\mathcal{D}}\right|$$

$$\lim_{N\to\infty}\,\,\frac{1}{N}\mathopen{}\mathclose{{}\left|\mathcal{U}\cap\mathcal{I}}\right|=\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X}\right)-\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X\>\!\middle|\>\!Y}\right)=\mathop{}\!{\textnormal{I}}\mathopen{}\mathclose{{}\left(X\mathchar 59\relax Y}\right)$$

$$P_{\mathopen{}\mathclose{{}\left.\bm{U}}\right.}\mathopen{}\mathclose{{}\left(\bm{u}}\right)=\prod_{i=1}^{N}P_{\mathopen{}\mathclose{{}\left.U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right.}\mathopen{}\mathclose{{}\left(u_{i}\>\!\middle|\>\!\bm{u}_{1}^{i-1}}\right).$$

$$u_{i}=\begin{cases}0&\text{with probability }P_{\mathopen{}\mathclose{{}\left.U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right.}\mathopen{}\mathclose{{}\left(0\>\!\middle|\>\!\bm{u}_{1}^{i-1}}\right),\\ 1&\text{with probability }P_{\mathopen{}\mathclose{{}\left.U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right.}\mathopen{}\mathclose{{}\left(1\>\!\middle|\>\!\bm{u}_{1}^{i-1}}\right).\end{cases}$$

$$u_{i}=\begin{cases}0&\text{if }P_{\mathopen{}\mathclose{{}\left.U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right.}\mathopen{}\mathclose{{}\left(0\>\!\middle|\>\!\bm{u}_{1}^{i-1}}\right)\geq P_{\mathopen{}\mathclose{{}\left.U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right.}\mathopen{}\mathclose{{}\left(1\>\!\middle|\>\!\bm{u}_{1}^{i-1}}\right),\\ 1&\text{else}.\end{cases}$$

$$\displaystyle R_{\text{loss}}=\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X}\right)-\frac{\mathopen{}\mathclose{{}\left|\mathcal{U}}\right|}{N}$$

$$\displaystyle R_{\text{loss}}=\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X}\right)-\mathopen{}\mathclose{{}\left\lfloor\log_{2}\binom{N}{P_{X}(0)\cdot N}}\right\rfloor/N.$$

$$\frac{1}{N}\sum _{i=1}^N(aE[X_i]{)}^2=\gamma .$$

Full text

I Introduction

Power efficient signaling requires a non-uniform input distribution for many channels. Combining the optimal input distribution with FEC (FEC) is not straightforward: conventional schemes (e.g., [1, Sec. 6.2], [2]) place the shaping operation after FEC encoding so that it needs to be reversed before (or performed jointly with) the FEC decoding. This is prone to error propagation and synchronization issues [3].

In [4], the authors build on the reverse concatenation principle [5] (the shaping operation is performed before the FEC encoding) and introduce the concept of sparse-dense transmission. The term “sparse-dense” reflects the composition of a FEC codeword with a sparse (ones and zeros are not equally distributed) and dense part (zeros and ones are approximately uniformly distributed). The sparse part is realized with mapping techniques (e.g., look-up tables) and its distribution is maintained by systematic encoding.

In general, any communication scheme using this approach operates in a TS (TS) fashion as only a fraction of the codeword symbols is shaped. The explicit integration of a variable-to-fixed length DM (DM) in a sparse-dense setup is done for the first time in [6, Sec. 7.3]. The suboptimality of TS can be circumvented by the approach of [6, Sec. 7.4] which uses a chaining construction to concatenate subsequent FEC frames. However, this is of limited practical use because of error propagation and increased latency. In [7], the authors use sparse-dense transmission with a fixed-to-fixed length CCDM (CCDM) and LDPC (LDPC) codes for power efficient signaling with OOK (OOK). Herein, gains of about $1\text{\,}\mathrm{d}\mathrm{B}$ are observed for transmission at a spectral efficiency of $0.25\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}~{}\mathrm{u}\mathrm{s}\mathrm{e}~{}\mathrm{(}\mathrm{b}\mathrm{p}\mathrm{c}\mathrm{u}\mathrm{)}$.

Recently, PAS (PAS) was proposed [8], which exploits the symmetry of the optimal input distribution for the AWGN (AWGN) channel with a bipolar modulation format (e.g., ASK) such that the suboptimality of a sparse-dense scheme can be circumvented. For sign-symmetric input distributions, e.g., Gaussian or Gaussian like distributions, PAS factors the input distribution into amplitude and sign parts that are stochastically independent. Using systematic encoding, the non-uniform distribution on the amplitudes is preserved, while the parity bits are mapped to the sign. In [9], syndrome shaping is introduced, an architecture which extends PAS to arbitrary input distributions and codes with systematic encoding. However, current implementations support matching rates close to one only.

Non-coherent modulation schemes such as OOK generally do not have a symmetric input distribution such that PAS can not be used and schemes like [7] still exhibit a gap to capacity. In this work, we analyze a PS (PS) approach for OOK that uses a method by Honda and Yamamoto [10, 11] where polar codes [12, 13] perform joint distribution matching and FEC. This idea was also applied in [14] with the intention to avoid an additional DM [15] and to use a single component for distribution matching and FEC. We apply this principle to OOK and show gains of $1.8\text{\,}\mathrm{d}\mathrm{B}$ over uniform signaling at a spectral efficiency of $0.25\text{\,}\mathrm{b}\mathrm{p}\mathrm{c}\mathrm{u}$. The proposed scheme outperforms sparse-dense signaling [7] with CCDM.

II Preliminaries

II-A Notation

Random variables are denoted by uppercase letters, e.g., $X$, while realizations or deterministic variables are denoted by lowercase letters, e.g., $x$. Vectors are denoted by a bold font, e.g., $\bm{x}$ for deterministic vectors and $\bm{X}$ for random vectors. Bold capital letters are also used for deterministic matrices. We write $\bm{x}_{i}^{j}=[x_{i},\dots,x_{j}]$. The notation \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X}\right) denotes the entropy of the random variable $X$ in bits. Similarly, \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X\>\!\middle|\>\!Y}\right) is the conditional entropy of $X$ given $Y$. The MI (MI) of $X$ and $Y$ is denoted by \mathop{}\!{\textnormal{I}}\mathopen{}\mathclose{{}\left(X\mathchar 59\relax Y}\right).

II-B System Model

Consider an AWGN channel

[TABLE]

where $X\in\{0,1\}$, $a$, $N$, and $Y$ denote the transmit signal, symbol amplitude, additive white Gaussian noise, and received signal respectively. With OOK modulation, $X$ is distributed according to

[TABLE]

The additive noise is assumed to have zero mean and unit variance. The SNR (SNR) is $\gamma=pa^{2}$ and an achievable rate is \mathop{}\!{\textnormal{I}}\mathopen{}\mathclose{{}\left(X\mathchar 59\relax Y}\right). Fig. 1 depicts \mathop{}\!{\textnormal{I}}\mathopen{}\mathclose{{}\left(X\mathchar 59\relax Y}\right) versus the SNR for two different choices of P_{\mathopen{}\mathclose{{}\left.X}\right.}: the blue curve is for uniform P_{\mathopen{}\mathclose{{}\left.X}\right.} (i.e., $p=0.5$) and the red curve is for a P_{\mathopen{}\mathclose{{}\left.X}\right.} that is optimized for each SNR, i.e.,

[TABLE]

There is a significant gain in power efficiency for non-uniform input symbols, e.g., for a rate of $0.25\text{\,}\mathrm{b}\mathrm{p}\mathrm{c}\mathrm{u}$ the optimal input distribution gains approximately $2\text{\,}\mathrm{d}\mathrm{B}$ over uniform inputs.

II-C Polar Codes

Polar codes [12, 13] are linear block codes with block length $N=2^{n}$ for $n\in\mathbb{N}$ and dimension $K$. The codeword $\bm{x}\in\mathbb{F}_{2}^{N}$ is generated from the input $\bm{u}\in\mathbb{F}_{2}^{N}$ by using

[TABLE]

$\bm{G}_{2}^{\otimes n}$ denotes the $n$-th Kronecker power of $\bm{G}_{2}$. The codeword is transmitted over a memoryless channel P_{\mathopen{}\mathclose{{}\left.Y\>\!\middle|\>\!X}\right.}. The received signals are collected in the vector $\bm{y}\in\mathbb{R}^{N}$. The bits of $\bm{u}$ asymptotically polarize into two sets [13]:

[TABLE]

for $\delta>0$. For finite $N$ we have a vanishing fraction of bits with \delta<\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right)<1-\delta. With SC (SC) decoding, bit $i$ in $\bm{u}$ is reliable if $i\in\mathcal{I}$. Otherwise, the bit $i$ is unreliable. The unreliable bits are frozen, i.e., they are set to a fixed value that is known both at the encoder and the decoder. The reliable bits are used for information transmission.

Arıkan [13] showed that for a B-DMC (B-DMC), we asymptotically have

[TABLE]

For symmetric channels (i.e., \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(Y\>\!\middle|\>\!X}\right)=\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(Y\>\!\middle|\>\!X=x}\right),\forall x), the capacity achieving distribution $P_{X}$ is uniform, and we have

[TABLE]

Thus polar codes achieve the symmetric capacity of B-DMCs.

III Polar Codes with Non-Uniformly Distributed Codewords

III-A Polarization for Non-Uniformly Distributed Codewords

Suppose that we want to create a codeword $\bm{x}$, where the codeword symbols have a non-uniform distribution. Honda and Yamamoto [10] showed that this is possible using a non-linear coding scheme based on polar codes. With the constraint on the distribution of the codewords, the bit positions in $\bm{u}$ do not only polarize asymptotically into $\mathcal{I}$ and $\mathcal{F}$, but also into

[TABLE]

The $i$-th bit position of $\bm{u}$ can be used for uniform data if $i\in\mathcal{U}$. If, however, $i\in\mathcal{D}$, then the value of $u_{i}$ is (almost) deterministic given the previous values $\bm{u}_{1}^{i-1}$. Thus the bit positions $i\in\mathcal{D}$ can not be used for data transmission, but are frozen to a value that depends (non-linearly) on the previous input. We describe the encoding procedure in Sec. III-B.

In [10], it was additionally shown that

[TABLE]

Therefore, the fraction of bits that can be used for uniform data is asymptotically \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X}\right). Fig. 2 shows a graphical representation of the input $\bm{U}$. The fraction of bits that can be transmitted reliably (i.e., where \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right)\approx 0) is (asymptotically) 1-\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X\>\!\middle|\>\!Y}\right) and the fraction of bits that can be used for uniform data such that a shaped codeword can be obtained is (asymptotically) \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X}\right). Conditioning does not increase entropy and thus \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right)\leq\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right). It follows that for a bit at position $i$ with \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right)\approx 1, we also have \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right)\approx 1, i.e., $i\in\mathcal{F}$ implies $i\in\mathcal{U}$ and thus $\mathcal{F}\subset\mathcal{U}$.

A bit $i$ in $\bm{u}$ can be used for information if it is reliable (i.e., $i\in\mathcal{I}$) and if uniform data is allowed at this position (i.e., $i\in\mathcal{U}$). The set of bits that can be used for information transmission is thus $\mathcal{U}\cap\mathcal{I}=\mathcal{U}\setminus\mathcal{F}$ with

[TABLE]

and the scheme can achieve capacity on asymmetric B-DMCs [10].

In the procedure of encoding, decoding, and code construction we will handle three different types of bit positions in the input $\bm{u}$:

• •

If $i\in\mathcal{U}\cap\mathcal{I}$, bit position $i$ will be used for uniform data.

• •

If $i\in\mathcal{F}$, bit position $i$ will be frozen to a value known to the encoder and the decoder.

• •

If $i\in\mathcal{D}$, bit position $i$ will be set to a value depending on the previous input $\bm{u}_{1}^{i-1}$ during encoding. The value is not known to the decoder.

III-B Encoding

The requirement on P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(x}\right) induces a constraint on the joint distribution P_{\mathopen{}\mathclose{{}\left.\bm{U}}\right.}\mathopen{}\mathclose{{}\left(\bm{u}}\right). The task of the encoder is to generate a $\bm{u}$ that contains data and fulfills this constraint. The codeword $\bm{x}$ is generated from $\bm{u}$ as in (4). Honda and Yamamoto [10] observed that P_{\mathopen{}\mathclose{{}\left.\bm{U}}\right.}\mathopen{}\mathclose{{}\left(\bm{u}}\right) can be calculated efficiently using a polar decoder. Using the chain rule, P_{\mathopen{}\mathclose{{}\left.\bm{U}}\right.}\mathopen{}\mathclose{{}\left(\bm{u}}\right) can be decomposed as

[TABLE]

When a SC polar decoder is initialized just with information on the distribution of $\bm{x}$, i.e., with a LLR (LLR) L=\log(P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(0}\right)/P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(1}\right)), it outputs for bit position $i$ the probability P_{\mathopen{}\mathclose{{}\left.U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right.}\mathopen{}\mathclose{{}\left(u_{i}\>\!\middle|\>\!\bm{u}_{1}^{i-1}}\right) given a realization of $\bm{u}_{1}^{i-1}$.

Honda and Yamamoto [10] thus proposed an encoding scheme that successively encodes bit by bit as follows: If $i\in\mathcal{U}\cap\mathcal{I}$, then $u_{i}$ is used for (uniform) data. If $i\in\mathcal{F}$, $u_{i}$ is chosen from a uniform distribution and the value is assumed to be known at the decoder as well (the value can be chosen once and kept constant for every block). Otherwise (i.e., if $i\in\mathcal{D}$), $u_{i}$ is set according to

[TABLE]

This method is called randomized rounding rule in [11].

A simplified approach is an encoding rule called the argmax rule in [11]. Here, for the values of $u_{i}$ with $i\in\mathcal{D}$, one chooses

[TABLE]

The randomized coding rule yields provable capacity achieving results, whereas the argmax rule yields better finite length results [11, 16] and does not need any randomness for encoding or decoding.

III-C List Encoding

During successive encoding a hard decision for the bits $u_{i}$ with $i\in\mathcal{D}$ must be done using (16) or (17). This hard decision may not be ideal especially for bit positions where \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right) is not polarized perfectly. One could thus follow the idea of [17] and use a SCL (SCL) decoder for encoding that branches a list when a hard decision is done. This idea was also applied in [14]. The list can be pruned with the usual metric used for SCL decoding. At the end, the SCL encoder outputs a list of valid codewords, i.e., all codewords contain the encoded data. We choose the codeword that has an empirical distribution closest to the target distribution.

III-D Decoding

The decoder estimates $\bm{u}$ from the noisy channel observations $\bm{y}$. The estimates are stored in a vector $\hat{\bm{u}}$. Decoding is performed with a SC or SCL decoder [17].

To show the capacity achieving property it is assumed in [10] that the decoder has knowledge about the values of $u_{i}$ with $i\in\mathcal{D}$. This knowledge can be obtained by running a SC decoder initialized with the LLR L=\log(P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(0}\right)/P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(1}\right)) that mimics the encoder and successively calculates the probabilities P_{\mathopen{}\mathclose{{}\left.U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right.}\mathopen{}\mathclose{{}\left(u_{i}\>\!\middle|\>\!\hat{\bm{u}}_{1}^{i-1}}\right) (it is assumed that previous bits have been decoded correctly, i.e., $\bm{\hat{u}}_{1}^{i-1}=\bm{u}_{1}^{i-1}$). The random choices of (16) can be recovered by using a pseudo-random number generator at the encoder and the decoder that is initialized with the same seed.

Simplifications are possible: As \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right)\leq\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right), it follows that if \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right) is close to zero (i.e., if $i\in\mathcal{D}$), then also \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right) is close to zero (i.e., $i\in\mathcal{I}$) and the bits at position $i$ with $i\in\mathcal{D}$ can be estimated reliably without running a second decoder. Thus, a practical SC or SCL decoder implementation works as follows: if $i\in\mathcal{F}$, then $\hat{u}_{i}$ is set to the known frozen value. Otherwise (i.e., if $i\in\mathcal{U}\cap\mathcal{I}$ or if $i\in\mathcal{D}$), $\hat{u}_{i}$ is decoded regularly. This idea is also used in [14], and it keeps the complexity at the receiver almost identical to a receiver for uniformly distributed codewords.

III-E Code Construction

Code construction consists of finding the four sets $\mathcal{I}$, $\mathcal{F}$, $\mathcal{U}$, and $\mathcal{D}$. For finite length simulations, we slightly deviate from the definitions in (5) and (10) and pursue the following strategy: we choose the sets $\mathcal{D}$ and $\mathcal{F}$. Then, $\mathcal{U}$ and $\mathcal{I}$ are given by $\mathcal{U}=\{1,\dots,N\}\setminus\mathcal{D}$ and $\mathcal{I}=\{1,\dots,N\}\setminus\mathcal{F}$, respectively. To choose $\mathcal{D}$ and $\mathcal{F}$, we first estimate an ordering of \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right) and \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right), respectively. Second, for a fixed transmission rate $R$, we find a tradeoff between the size of $\mathcal{D}$ and $\mathcal{F}$ such that $|\mathcal{D}|+|\mathcal{F}|=N(1-R)$.

We use a Monte Carlo approach to estimate the ordering of \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right) and \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right) as described in the following. We remark that one can extend the Tal-Vardy construction [18] by using the method of [19] to estimate the entropy values with less computational effort. To estimate \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right) with the Monte Carlo approach, a SC decoder is initialized with the LLR \log(P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(0}\right)/P_{\mathopen{}\mathclose{{}\left.X}\right.}\mathopen{}\mathclose{{}\left(1}\right)). When choosing the inputs $u_{i}$ successively using the randomized rounding rule, the SC decoder successively outputs P_{\mathopen{}\mathclose{{}\left.U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right.}\mathopen{}\mathclose{{}\left(u_{i}\>\!\middle|\>\!\bm{u}_{1}^{i-1}}\right). Sampling over many frames, one can estimate \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right). Furthermore, a transmission over the channel with the randomly generated data is simulated and a SC decoder is applied. If the decoder produces a wrong decision for bit $i$, the error counter for this bit position is increased by one and the error is corrected. After many trials, the error counter for each bit position gives a reliability order for the bit positions and — as the entropy \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right) is a monotone function of the error rate of bit position $i$ — an order for the entropies \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right).

We now choose the $D$ bits with lowest \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1}}\right) to form the set $\mathcal{D}$ and the $N(1-R)-D$ bits with highest \mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(U_{i}\>\!\middle|\>\!\bm{U}_{1}^{i-1},\bm{Y}}\right) to form the set $\mathcal{F}$. The optimal $D$ can be found by numerical simulations. Numerical results show that one has to choose a $D$ that is only slightly higher than the asymptotic limit N(1-\mathop{}\!{\textnormal{H}}\mathopen{}\mathclose{{}\left(X}\right)) for good results. Depending on the choice of $D$, there is a slight mismatch between the target distribution and the empirical distribution in $\bm{x}$. This stems from the finite length rate loss of the DM process, which is discussed in Sec. IV-A.

IV Numerical Results

IV-A Finite Length Rate Loss Evaluation

The rate loss [8, Sec. V-B] is an important metric to analyze the finite length performance of a DM scheme. Assume an output blocklength of $N$ bits. The rate loss is then defined as

[TABLE]

for the polar DM. For the CCDM, we have

[TABLE]

We numerically characterize $R_{\text{loss}}$ for different DM architectures in Fig. 3. For this, we fix a desired DM rate of $0.5\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{/}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}~{}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{b}\mathrm{o}\mathrm{l}$ and evaluate (18) for different blocklengths $N$. We observe that CCDM is superior to the polar DM for all considered lengths. Its superior performance for long blocklengths is to be expected from previous results [20], which showed the optimality of CCDM for fixed-to-fixed matching and $N\to\infty$. We remark that the polar DM rate loss can be decreased if the list encoding of Sec. III-C is used, see Fig. 3.

IV-B Coded Results

We evaluate the performance of the presented transmission scheme. For fixed $D$ we estimate the empirical codeword distribution and scale the amplitude $a$ so that we transmit at the target SNR, i.e., we choose $a$ such that

[TABLE]

Fig. 4 shows a numerical example for $N=$65,536$$ and transmission rate $R=0.25$. At this rate gains up to $2\text{\,}\mathrm{d}\mathrm{B}$ can be expected from Fig. 1. With SC decoding, the shaped polar code ( ) gains about $1.8\text{\,}\mathrm{d}\mathrm{B}$ at a FER (FER) of $10^{-3}$ compared to the polar code with uniform codewords ( ). With SCL encoding and decoding (both with list size $32$) and an outer CRC, the shaped polar code ( ) gains around $1.8\text{\,}\mathrm{d}\mathrm{B}$ compared to the uniform reference ( ). The performance of the polar code at this blocklength is limited by the relatively small list size. Increasing the list size can further improve the performance, e.g., when choosing $L=256$ ( ) the performance improves by $0.2\text{\,}\mathrm{d}\mathrm{B}$ compared to $L=32$. We also include the performance of LDPC codes with blocklength $64,800$ bits using the time-sharing based PS scheme from [7]. The LDPC code with uniform signaling ( ) is taken from the DVB-S2 standard [21]. The difference between the TS1 ( ) and TS2 ( ) code is that TS2 uses different signaling amplitudes on the systematic and parity parts. Both codes have been optimized individually for the respective scenario. A CCDM [15] is used in both cases as a DM.

In Fig. 5, we depict the performance for a scenario with $R=2/3$, where gains up to $0.9\text{\,}\mathrm{d}\mathrm{B}$ can be expected. The polar codes ( : shaped, : uniform) have a blocklength of $1024\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}$, while the reference LDPC codes from the Wimax standard [22] have a blocklength of $1056\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}$ and code rates of 2/3 ( : uniform) and 3/4 ( : shaped). We depict a curve for TS1 only as it turns out (both by achievable rate analysis and finite length simulations) that the gain of TS2 over TS1 vanishes with increasing rate [7]. As expected from previous works [23], polar codes with SCL show an excellent performance for short to medium blocks. In all LDPC cases, two hundred belief propagation iterations are performed. We also include two finite length random coding union (RCU) bounds based on saddlepoint approximations of the RCU bound [24]. At a FER of ${10}^{-3}$, we operate within $0.6\text{\,}\mathrm{d}\mathrm{B}$ of these bounds.

V Conclusion

We applied the shaping scheme by Honda and Yamamoto for polar codes [10] to OOK transmission. Compared to previous approaches, the proposed scheme is asymptotically optimal and shows superior performance for finite length. Especially for low transmission rates, the performance is substantially better than a TS based LDPC implementation. Future work may also compare shaped OOK to pulse position modulation based schemes such as [25] with a multilevel coding/multistage decoding architecture.

Acknowledgements

The authors would like to thank Ido Tal and Boaz Shuval for motivating this study, as well as Gerhard Kramer for helpful comments and discussions. The authors would also like to thank the anonymous reviewers who provided valuable input and ideas for improvement.