On Probability Shaping for 5G MIMO Wireless Channel with Realistic LDPC Codes

The physical meaning of the problem (2) is to minimize the constellation energy at a fixed constellation entropy. The entropy $H(X)$ means the amount of information transmitted by the constellation, and the energy $\operatorname{\mathbb{E}}[|X|^{2}]$ means the power the transmitter has to expend in transmitting the data.

There is no analytical expression for the problem (2), and so the constellation points are assumed to have a Maxwell-Boltzmann distribution (3) since it is close to the optimal distribution [11] and maximizes the entropy of the constellation with a constraint on its energy:

Parameter $\mu=\mu(\hat{p}_{1},\dots\hat{p}_{m})$ — is a scaling of constellation points:

where $p_{i}$ is the uniform distribution, $\hat{p}_{i}$ is the optimal distribution, and $x_{i}$ is the complex coordinates of the points.

For the practical finite block-length codes, it is required to implement the Enumerative Sphere Shaping (ESS) method [7].

The energy minimization process is fairly straightforward. We take the constellation points with the smallest absolute value with a higher probability, and the points with the largest absolute value with a lower probability. Thus, we are more interested in constellation points that have more zeros than ones at the amplitude bit positions in the binary representation, and then it is sufficient to maximize the probability of zero at the amplitude bit positions. We also assume that the sign bits are uniformly distributed, i.e. the probability of zero and one of the first two bits in the binary representation of each constellation point is equal to $\frac{1}{2}$.

As noted above, amplitude bit one corresponds to more distant points from the origin, and amplitude bit zero corresponds to closer points. Thus, we can assume that the energy of a sequence of $n$ amplitude bits consisting of $k$ ones and $n-k$ zeros, is equal to

It can be seen that the nearest points to the origin have the lowest energy.

For a given number of input amplitude bits $k$ and block length $n$, the most efficient way to change probabilities is to map all possible $2^{k}$ realisations to the $2^{k}$ sequences of $n$ amplitude bits with minimal energy. After that, we can calculate the probability of one $p_{a}(1)$ and probability of zero $p_{a}(0)$ in a set of blocks of length $n$.

Now if we know the distribution of the amplitude bits, then we can find the probability of the constellation points. Each constellation point contains two sign bits and two amplitude bits, so the probability of a point is equal to

where $k$ is the number of zero amplitude bits in the bit representation of the constellation point.

After we have changed the distribution of constellation points, we can calculate the scaling parameter $\mu$ as the ratio of the initial energy to the received energy:

where $\widehat{X}$ is a new random variable with distribution 5. Finally, we shift points of the constellation by multiplying them by the parameter $\mu$, thereby reducing the probability of error.

In this subsection we describe the model provided in Fig. 4. Initially, the input is $k$ informational bits with a uniform distribution. These bits are divided into two groups, one of which will be the amplitude bits, and the other group will be part of the sign bits. Amplitude bits are transmitted through the shaper block, which works according to the algorithm described above. The shaper output is a block of a different length, in which the amplitude bits are already distributed according to the algorithm. We will denote the number of bits in the first group by $k_{sh}$, the number of bits in the second group by $k_{sign}$ and the number of bits at the shaper output by $n_{sh}$.

After that, $k_{sign}$ bits and $n_{sh}$ amplitudes bits are concatenated and encoded using the LDPC procedure. The LDPC procedure, in turn, generates additional $\Delta n$ check bits, which are also considered to be uniformly distributed. We will denote the number of bits at the encoder output as $n_{FEC}=k_{sign}+n_{sh}+\Delta n$. Note that the $k_{sign}+\Delta n$ bits are sign bits, which have a uniform distribution, while the amplitude bits $n_{sh}$ are distributed according to the algorithm. The number of sign $k_{sign}$ and error correction bits $\Delta n$ is equal to the number of shaper output bits, i.e. $n_{sh}=k_{sign}+\Delta n$.

For this procedure, coderate $R\in(0,1]$ is fixed, while shaper input size $k_{sh}$ and shaper output $n_{sh}$ vary. The values of $k_{sign}$ and $n_{FEC}$ can be calculated using the code rate formulas.

Extra bits are now shared between Shaper with rate $R_{sh}=\frac{k_{sh}}{n_{sh}}$ and Encoder with rate $R_{\textit{FEC}}=\frac{n_{sh}+k_{sign}}{n_{\textit{FEC}}}$, afterall $R=\frac{1}{2}(R_{sh}+2R_{\textit{FEC}}-1)$.

The problem is to find the optimal proportion between $R_{sh}$ and $R_{\textit{FEC}}$.

In Tabs. 1, 2 examples of generated probabilities for QAM-16 and QAM-64 (Fig. 1) using the ESS method [7] are given. Note that for QAM-64 and above the probabilities of zeros and ones depend on each other, so joint probabilities need to be determined.

For mapping purposes, we form a special PS matrix (Fig. 5) with uniform sign bits and non-uniform amplitude bits, following the data flow scheme (Fig. 4). We generate $k_{sign}$ sign bits with equal probability of zeros and ones $p_{s}=\frac{1}{2}$, and $n_{sh}$ amplitude bits with unequal probability of zeros and ones: $p_{a}\neq\frac{1}{2}$ such that $p_{a}(0)>p_{a}(1)$. The probability of amplitude bits can be determined by the proper values of $k_{sh}$ and $n_{sh}$ using the ESS method [7].

Finally, after the PS matrix is constructed, the mapping to the QAM is performed. With mapping procedure, bits are converted to the constellation points (or symbols) using the mapping table, which gives a specific coordinate on the complex plane for each unique sequence of bits. Notice that, given bit probabilities, there is a one-to-one correspondence to symbol probabilities.

After mapping procedure is done the symbols go through the MIMO channel, demodulation and decoding, probability deshaping and BLER calculation, which are described in Sec. 2 and Fig. 6. The demodulation and deshaping methods are the same procedures described earlier and are performed in reverse order. The decoding procedure is a complex process, which uses loopy belief propagation [14] to iteratively recover the correct bits (LLRs).

To consistent all the shapes $k_{sh}$, $k_{sign}$ and $\Delta n$, and form the PS matrix (Fig. 5) we solve the system of integer equations (7) finding LCM. Hereafter, the values of $N_{fr}^{sh}$ and $N_{fr}^{\textit{FEC}}$ define the multiplicative constants balancing these equations. The values of $N_{A}$ and $N_{S}$ define the number of amplitude and sign bits in the code block. It is implicitly assumed that everywhere in Figs. (4) (5) the values are $k_{sh}:=N_{fr}^{sh}k_{sh}$, $k_{sign}:=N_{fr}^{sh}k_{sign}$, $\Delta n:=N_{fr}^{\textit{FEC}}\Delta n$, $n_{\textit{FEC}}:=N_{fr}^{\textit{FEC}}n_{\textit{FEC}}$:

where the value of $C$ defines the constellation system, i.e. $C=1$ — QAM-16, $C=2$ — QAM-64, $C=3$ — QAM-256 and so on.

The energy per bit to noise ratio $E_{b}/N_{0}$ is a normalized SNR measure, also known as SNR per bit. The $E_{b}/N_{0}$ measure can be used to express the relationship between signal power and noise power.

The energy per bit measure $E_{b}$ is the energy we use to transmit one bit of information with the total power $P$ and the LDPC coderate $R$:

The noise measure $N_{0}$ is the noise variance per real and imaginary parts:

Thus, $E_{b}/N_{0}$ can be expressed in terms of SNR:

In decibel, $E_{b}/N_{0}$ is

We use the value of $E_{b}/N_{0}$ in the Monte Carlo experiments. For a given value of $E_{b}/N_{0}$ with the power $P$ and coderate $R$ the variable noise power $\sigma^{2}$ disturbs the symbols transmitted over the channel (1).

This study considers OFDM MIMO with a base station and a user equipped with multiple cross-polarised antennas. We provide simulations in Sionna [8] on the OFDM channel using 5G LDPC codes. The architecture of the system consists of LDPC, Bit Interleaver, Resource Grid Mapper, LS Channel Estimator, Nearest Neighbor Demapper, LMMSE Equalizer [13], OFDM Modulator and presented in Fig. 6. Optimization variables are constellation type, a bit order, coderate, BLER, SNR, code block sizes and 5G model (LOS D, NLOS A).

The system uses soft estimates of LLRs for the decoder. Channel model is chosen to be OFDM 5G 2.6 GHz with delay spread of 40ns. The block size is 1536 with $10^{5}$ number of Monte-Carlo trials in the simulations and so in total $1.536\cdot 10^{8}$ bits were processed for each point of $E_{b}/N_{0}$. For all simulation, 20 iterations of LDPC have been used. The code source is random binary tensors. The system use 3GPP wireless both Line of Sight (LOS) and Non Line of Sight (NLOS) channel models D and A. The model is simulated in real time domain considering inter-symbol (IS) and inter-carrier (IC) interferences. In Tab. 3 we provide simulation parameters for Sionna.

In Figs. 7 and 8, we provide an experiment for both QAM-16 LOS Model D and NLOS Model A probability shaped (PS) constellations. We present experiments Coded BLER (see Fig. 6) with Gaussian transmit signal with QAM16 Baseline and amplitude PS with different shaping parameters, where coderate is $r$, block size is $n$ and parameters of PS are $n_{sh}$ and $k_{sh}$.

There is a local optimum for $k_{sh}=192$ PS QAM-16 in both LOS and NLOS models. It is noteworthy that the optimal parameter $k_{sh}$ is the same for the different LOS and NLOS models, which tells us that the chosen parameters are stable. Note that with wrong parameter settings, e.g. a strong shaping factor $k_{sh}=128$, the quality of the PS method is worse than that of baseline QAM.

In Fig. 9 present experiments with AWGN channel and LDPC, which show a higher gain than for the OFDM channel model.

In Tab. 4, we provide gains in dB of the proposed PS method at 10% BLER. The PS method achieves 0.56 dB gain in Model A NLOS Uplink compared to the baseline. From the massive experiments, we conclude that the proposed PS method constellations superior the baseline QAM for real 5G Wireless System using FEC LDPC.

The model codes and related experiments can be found in the repository¹¹1https://github.com/eugenbobrov/On-Probabilistic-QAM-Shaping-for-5G-MIMO-Wireless-Channel-with-Realistic-LDPC-Codes.