NN-MI: Neural Network Achievable Information Rate Computation for Channels with Memory

This repository contains the program code for the paper "Neural Network-Based Successive Interference Cancellation for Non-Linear Bandlimited Channels", which was accepted to the IEEE Transactions on Communications on 11 August, 2024. A preprint of the accepted journal version is available. A shorter conference version is also available.

The code computes achievable information rates under successive interference cancellation (SIC). At each SIC stage a recurrent NN estimates (conditional) a-posteriori probabilities.

We provide three example applications.

Tip

The NN parameters have been chosen to work over a wide range of SNRs. Fine-tuning the parameters based on model memory, constellation size or SNR can lead to better performance and faster training.

Citation

The software is provided under the open-source MIT license. If you use the software in your academic work, please cite the accompanying document as follows:

D. Plabst, T. Prinz, F. Diedolo, T. Wiegart, G. Böcherer, N. Hanik and G. Kramer, "Neural Network-Based Successive Interference Cancellation for Non-Linear Bandlimited Channels," IEEE Trans. Commun., to appear. Available: https://arxiv.org/abs/2401.09217

The corresponding BibTeX entry is: nnmi/cite.bib.

Example 1: Fiber-Channel with Square-Law Detector

Consider short-range optical communication with a square-law detector (SLD) at the receiver, i.e., a single photodiode performs the optical-to-electrical conversion¹². The standard single-mode fiber (SSMF) between transmitter and receiver causes chromatic dispersion, leading to intersymbol interference (ISI). We consider thermal noise from photodetection. The model is

$$ Y(t) = h(t) * (\left|X(t)\right|^2 + N(t)) $$

with the

• baseband signal $X(t) = \sum_\kappa X_\kappa g(t-\kappa T)$ and symbol period $T$
• u.i.i.d. discrete channel inputs $X_\kappa$ from the constellation $\mathcal{A}$
• filter $g(t)$ that combines transmit pulseshaping (DAC) and linear fiber effects
• filter $h(t)$ that models linear effects at the receiver
• real white Gaussian noise $N(t)$.

The SLD doubles the signal bandwidth. We use a low-pass filter $h(t)$ with twice the bandwidth of $X(t)$. We oversample $Y(t)$ to obtain sufficient statistics. The samples are passed to the NN-SIC receiver. Due to the receiver squaring operation $|\cdot|^2$, symbol detection becomes a phase retrieval problem.

We compare two setups.

• ex1a_nnmi.py computes information rates for back-to-back configuration (without fiber).
• ex1b_nnmi.py computes information rates for 30 km of SSMF.

The fiber is operated at the C band carrier 1550 nm; see [Tab. II]. The DAC is operated at a symbol rate of 35 GBd and performs sinc pulseshaping. We use 4-ASK modulation with constellation $\mathcal{A} = \left\lbrace\pm 1, \pm 3\right\rbrace$. The example program considers NN-SIC with four stages and plots stage rates for $s=1,\ldots,4$ and the average rate across all stages (red). The noise variance is set to $\sigma^2 = 1$ and we vary the average transmit power $P_\text{tx}$. Hence SNR $= P_\text{tx}$.

Example 2: Linear Baseband Communication with AWGN

Consider linear baseband communication with AWGN. The model is

$$ Y(t) = h(t) * \left(h_\text{ch}(t) * X(t) + N(t)\right) $$

with the

• baseband signal $X(t) = \sum_\kappa X_\kappa g(t-\kappa T)$ and symbol period $T$
• u.i.i.d. discrete channel inputs $X_\kappa$ from the constellation $\mathcal{A}$
• RRC filter $g(t)$ with roll-off factor $\alpha$ as the DAC response
• frequency-selective channel $h_\text{ch}(t)$
• RRC receive matched-filter $h(t)$
• circularly-symmetric (c.s.) complex white Gaussian noise $N(t)$.

Example 2a: No ISI

Consider a flat channel $h_\text{ch}(t)$ that passes $X(t)$ without distortion. The combination of transmit filter and matched-filter is a Nyquist filter and sampling at symbol rate provides sufficient statistics. The equivalent discrete system is memoryless with c.s. complex AWGN:

$$ Y = X + N. $$

Separate detection and decoding (SDD) $S=1$ and joint detection and decoding (JDD) are therefore equally good.

Running ex2a_nnmi.py uses NN-SIC to compute achievable information rates. The plotted rates are the same as the single-letter mutual information in Fig. 1³. We also plot the capacity of the memoryless AWGN channel, achieved by Gaussian signalling. For real modulation, the SNR definition takes only the real component of the noise into account.

Example 2b: ISI

Consider the model⁴:

$$ Y[\kappa] = (X * h)[\kappa] + N[\kappa] $$

with the

• discrete channel $(h_\kappa)_{\kappa=-3}^{3} = (0.19, 0.35, 0.46, 0.5, 0.46, 0.35, 0.19)$
• u.i.i.d. discrete channel inputs $X_\kappa$ from the BPSK constellation $\mathcal{A} = \lbrace\pm 1\rbrace$
• real AWGN $N_\kappa$.

This model often describes magnetic recording channels. The program ex2b_nnmi.py uses NN-SIC to calculate the achievable rates. We compare the results with the exact information rates calculated with the forward-backward algorithm (FBA)⁴. The NN-SDD rates are within 0.1 dB of the FBA reference and NN-SIC with $S=4$ stages is very close to the JDD rate. The JDD rate can be approached by increasing the number of SIC steps.

Example 3: Baseband Communication with Nonlinear Power Amplifier

Consider baseband communication with AWGN

$$ Y(t) = h(t) * \big(f(X(t)) + N(t)\big) $$

where the real baseband signal $X(t)$ passes through a nonlinear power amplifier (PA)

$$ f(x) = \sqrt{P_\mathrm{max}} \cdot \tanh\left( \frac{x}{\sqrt{P_\mathrm{max}}} \right) $$

with peak output power $P_\mathrm{max}$. The parameters are the

• baseband signal $X(t) = \sum_\kappa X_\kappa g(t-\kappa T)$ and symbol period $T$
• u.i.i.d. discrete channel inputs $X_\kappa$ from the constellation $\mathcal{A}$
• RRC filter $g(t)$ with roll-off factor $\alpha$ that models the DAC
• receive filter $h(t)$
• real-valued white Gaussian noise $N(t)$.

We compare two setups.

• ex3a_nnmi.py performs receiver matched-filtering, symbol-rate sampling and SDD.
• ex3b_nnmi.py uses a brickwall receive filter $h(t)$ with cutoff frequency $2/T$ and performs 4-fold oversampling with SIC.

We consider 4-ASK modulation, set the noise variance $\sigma^2 = 1$ and vary the average transmit power $P_\text{tx}$ before the PA. We define SNR = $P_\text{tx}$. Achievable information rates are plotted without PA constraints (blue) and 7 dBW transmit peak power. For higher transmit powers, the PA significantly broadens the transmit signal spectrum and oversampling with SIC increases information rates.

Usage

One may execute the examples providing command line parameters:

python exN_nnmi.py -m 4-ASK -S 4

where one must replace N by [1a,1b,2a,3a,3b] to choose the example of interest. The prompt example computes achievable information rates for 4-ASK modulation and $S=4$ SIC stages, and saves the result as a CSV file. The CSV file lists the rates pre stage and the average rate under SIC.

Further options can be found by executing:

python exN_nnmi.py --help

which outputs:

usage: exN_nnmi.py [-h] [--stages STAGES] [--mod_format MOD_FORMAT] [--indiv_stage INDIV_STAGE] [--device {cpu,cuda:0,cuda:1}]

NN-MI: Neural Network Achievable Information Rate Computation for Channels with Memory

options:
-h, --help                                            show this help message and exit
--stages STAGES, -S STAGES                            number of successive interference cancellation stages
--mod_format MOD_FORMAT, -m MOD_FORMAT                M-ASK, M-PAM, M-SQAM (star-QAM), M-QAM (square) modulation with order M
--indiv_stage INDIV_STAGE, -s INDIV_STAGE             simulation of a single individual stage
--device {cpu,cuda:0,cuda:1}, -d {cpu,cuda:0,cuda:1}  run code on cpu, cuda:0 or cuda:1

Software Requirements

The code runs under Python >= 3.9.6 and dependencies nnmi/requirements.txt:

asciichartpy==1.5.25
matplotlib==3.7.2
numpy==1.25.2
scikit_commpy==0.8.0
scipy==1.11.4
tabulate==0.9.0
torch==2.0.1

Footnotes

1. D. Plabst, Tobias Prinz, Thomas Wiegart, Talha Rahman, Nebojša Stojanović, Stefano Calabrò, Norbert Hanik and Gerhard Kramer, "Achievable Rates for Short-Reach Fiber-Optic Channels With Direct Detection," in J. Lightw. Technol., vol. 40, no. 12, pp. 3602-3613, 15 June15, 2022, doi: 10.1109/JLT.2022.3149574. [Xplore] ↩

2. T. Prinz, D. Plabst, T. Wiegart, S. Calabrò, N. Hanik and G. Kramer, "Successive Interference Cancellation for Bandlimited Channels with Direct Detection," in IEEE Trans. Commun., vol. 72, no. 3, pp. 1330-1340, March 2024, doi: 10.1109/TCOMM.2023.3337254. [Xplore] ↩

3. G. D. Forney and G. Ungerboeck, "Modulation and coding for linear Gaussian channels," in IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2384-2415, Oct. 1998, doi: 10.1109/18.720542. [Xplore] ↩

4. D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic and W. Zeng, "Simulation-Based Computation of Information Rates for Channels With Memory," in IEEE Trans. Inf. Theory, vol. 52, no. 8, pp. 3498-3508, Aug. 2006, doi: 10.1109/TIT.2006.878110. [Xplore] ↩ ↩²