# Polar Codes and a M-ary Modulation-Based OFDM-PLC System
## 3 System Model
### PLC Channel Noise Model  
The noise situation in actual PLC channels is extremely complex and is mainly divided into two parts: background noise and impulse noise. The background noise includes colored noise, narrowband noise, and periodic impulse noise that is asynchronous with the power frequency. It is usually more stable, has low power, and its intensity does not change dramatically over time. The impulse noise includes periodic impulse noise synchronized with the power frequency and burst impulse noise. It is highly random, with high impulse intensity that decays rapidly within a short period of time, and its power spectral density increases as the frequency decreases. The superposition of this noise results in severe interference to the PLC channel and poor reliability of data transmission. The Middleton class-A model, as a non-Gaussian noise model, is often used to characterize the noise characteristics of the PLC channel, but its simulated impulses are independent sampling points, which do not conform to the physical characteristics of impulse noise and cannot reproduce the randomness of impulse occurrence in real scenes. Therefore, this paper uses a multi-damping sinusoidal function superimposed to simulate impulse noise and the modeling of the single impulse noise waveform function is as follows:  
\[n_{k}(t)=E_{k}\cdot e^{\frac{-t}{\lambda}}\cdot sin[2\pi f(t)+\varphi]\cdot u(t) \tag{13}\]  
where \(E_{k}\) is the impulse amplitude; \(\lambda\) is the impulse time constant, which is about one-fifth of the impulse width \(t_{k,w}\); \(u(t)\) is the unit step function; \(\varphi\) is the initial phase, \(\varphi\in(0,2\pi)\); \(f\) is the pseudo frequency of the decaying sine function described by the Weibull distribution at frequencies between \(0\sim 30\) MHz. The probability density of the Weibull distribution is as follows:  
\[f(x)=abx^{b-1}e^{-ax^{b}} \tag{14}\]  
where \(a\) and \(b\) are constants. When \(f<3\) MHz, the optimal values of \(a\) and \(b\) are \(5\times 10^{-7}\) and \(6.07\), respectively. The optimal values of \(a\) and \(b\) are \(0.59\) and \(2.27\), respectively, when \(f>3\) MHz [24].  
To characterize the impulse amplitude and width, parameters \(\mu\) and \(p\) are introduced. \(\mu\) is the power ratio between impulse noise and background noise, \(\mu=\frac{\sigma_{1}^{2}}{\sigma_{B}^{2}}\), which gives the impulse amplitude as \(E_{k}=\sqrt{\frac{\sigma_{0}^{2}}{\mu}}\), the total noise power is \(\sigma^{2}=\sigma_{1}^{2}+\sigma_{B}^{2}\). \(p\) is the probability of impulse noise width occurring in a signal period \(t\). If \(b(n)\) satisfies theBernoulli distribution process, and \(p\{b(k)=1\}=p\), \((k=0,1,\ldots)\), then the impulse width is \(t_{k,w}=\sum_{l=0}^{N-1}b(i)\).  
Therefore, the \(k\) impulse noises occurring in a signal period \(t\) can be represented as:  
\[N(t)=\sum_{k}n_{k}(t-t_{k}) \tag{15}\]  
where \(t_{k}\) is the time at which the \(k\) impulses arrive, which needs to be determined by measuring the data, and can be represented by a random variable to reflect the randomness of impulse noise. The impulse parameter \(m\) is introduced to characterize the probability of impulse arrival time \(t_{k}\) within a signal period \(t\), \(m\in\left[10^{-3},10^{-1}\right]\). Then, the number \(k\) of impulse noise occurrences in one period can be represented as \(k=m*t\). Figure 3 shows the simulated PLC random impulse noise waveform generated when \(m=0.02\), and the inset shows the waveform of a single impulse noise generated by (13).