# Iterative Receiver Design for the Estimation of Gaussian Samples in Impulsive Noise +
## 5 Results  
We evaluated, by numerical simulation, the mean-squared error (MSE) versus the average signal-to-noise ratio (SNR) in a bursty impulsive noise scenario with a maximum of \(M-1\) interferers, i.e., with \(M\) noise states, where the zeroth state corresponds to background noise only. As discussed in Section 2, the peculiarity of this system is that the actual ratio of signal-to-noise power can dramatically change at every time epoch. Figure 4 shows the MSE for the estimated signals corrupted by impulsive noise with \(M=2,4,16\) states (Figure 4a-c), where the following system parameters were chosen for the simulation. The correlation parameter \(x\) was set to 0.9, meaning that the channel state is expected to switch to a new state (including the present one) with probability 0.1; adjacent noise samples are thus highly correlated, i.e., the noise is bursty. The impulsive index \(A\), i.e., the average number of active interferers, was chosen to be either 0.2 or one, which statistically corresponds to a \"hardly present interferer\" or to an \"almost constantly present\" one (at least). \(\Gamma\) is the ratio between the power of background noise and the average power of interferers and was set to 0.01, corresponding to a strongly impulsive noise where interference can boost noise power by a factor of \(\Gamma^{-1}=100\). The signal parameters, in the AR(1) model, were set as follows: \(a_{1}=0.9\) and \(\sigma_{s}^{2}=1\) (i.e., normalized signal power). We transmitted 100 frames of 1000 samples each (for a total number of \(10^{5}\) received samples). The curve labeled GAKS in Figure 4 shows the performance of a \"genie-aided Kalman smoother\" and refers to an ideal receiver with perfect channel state information, i.e., with exact knowledge of \(i_{k}\), hence of noise variance at every time epoch. In this case, the system reduces to the classical estimation of correlated Gaussian samples, for which the Kalman smoother is optimal; hence, the GAKS curve represents a lower bound for the performance of any non-ideal receiver. The curve labeled PIS in Figure 4 is obtained through the parallel iterative schedule algorithm discussed in Section 4.3. A comparison of Figure 4a-c reveals that the simulation results are very close to each other. A fine checking of the numerical values reveals, for example, that in the \(M=2\) case of Figure 4a, at \(\text{SNR}=-5\) dB, the MSE is \(-5.598\) dB for \(A=1\) and \(-8.763\) dB for \(A=0.2\). In the \(M=4\) case of Figure 4b, at the same SNR values, the MSE changes to \(-5.038\) and \(-8.573\), while in the \(M=16\) case of Figure 4c, the MSE values change to \(-5.103\) and \(-8.708\). This implies that the performance of the estimator is not considerably changed when we opt for noise models with more than \(M=2\) states, which corresponds to the Markov-Gaussian model of Section 2.1. The reason is that a larger maximum number of interferers implies a smaller amount of power per interferer, so that, for a given average SNR, it is the average number of active interferers \(A\), rather than its maximum value \(M-1\), that determines the performance. Results in Figure 4 confirm, as expected, that a larger \(A\) degrades the performance of the estimator. For this reason, in the following simulations, we limited our attention to the case \(M=4\) only. We considered the same noise parameters as in [6], i.e., a correlation parameter \(x=0.98\), corresponding to an impulsive noise with increased burstiness, while the values of \(A=0.2\) or \(0.8\) were slightly decreased (compared to Figure 4). The values \(\Gamma=0.01\) and \(0.001\), considered in Figure 5a,b, account for a strongly impulsive noise, where the impulsive events can increase the noise power up to \(1000\) times the background noise power. Figure 5 shows the performance of an estimator that implements the EP and TP algorithms described in Section 4.4 (curves labeled EP and TP) that pursue the approximation of Gaussian mixture messages.