# Iterative Receiver Design for the Estimation of Gaussian Samples in Impulsive Noise +
## 2 System Model
### Markov Middleton Class A Noise  
In this model, the number \(i\) of interferers, at time \(k\), can exceed one and can be virtually any integer number, despite that a maximum number \(M\) of interferers is usually considered for practical reasons. The Gaussian interferers are all independent of each other and are assumed to have the same power, \(\sigma_{I}^{2}\). As in the Bernoulli-Gaussian and in the Markov-Gaussian models of Section 2.1, the interferers are superimposed on an independent background noise sequence \(\{n_{k}^{G}\}\) with variance \(\sigma_{G}^{2}\), so that the overall noise power is \(\sigma_{i}^{2}=\sigma_{G}^{2}+i\sigma_{I}^{2}\), whenever the channel is in state \(i\), i.e., \(i\) interferers are active. The pdf of noise samples is thus:  
\[p(n_{k})=\sum_{i=0}^{\infty}\frac{P_{i}}{\sqrt{2\pi\sigma_{i}^{2}}}\text{exp} \Bigg{\{}-\frac{n_{k}^{2}}{2\sigma_{i}^{2}}\Bigg{\}}. \tag{7}\]  
In the Middleton class A model, the channel state is assumed to follow a Poisson distribution [4], so that in (7), \(P_{i}=e^{-A}A^{i}(i!)^{-1}\), where \(A\) is called the impulsive index and can be interpreted as the average number of active interferers per time unit. When the channel state accounts for \(i\) interferers, we can thus express the total noise power as:  
\[\sigma_{i}^{2}=\bigg{(}1+\frac{i}{A\Gamma}\bigg{)}\sigma_{G}^{2} \tag{8}\]  
Figure 1: State diagram of the Markov-Gaussian noise.  
in which a new parameter \(\Gamma=\sigma_{G}^{2}(A\sigma_{I}^{2})^{-1}\) represents the ratio between the power of background noise and the average power of interferers. The average power of Markov Middleton class A noise is thus:  
\[\sigma^{2}=E\Big{[}\sigma_{i}^{2}\Big{]}=\sum_{i=0}^{\infty}\sigma_{i}^{2}P_{i }=\bigg{(}1+\frac{1}{\Gamma}\bigg{)}\sigma_{G}^{2} \tag{9}\]  
In order to avoid the practical complications of an extremely large (and unlikely) number of interferers, by exploiting the rapidly decreasing values of the Poisson distribution, an approximation of the Middleton class \(A\) noise is usually introduced by truncating its pdf to a maximum value for the channel state \(i\leq M\), as follows:  
\[p(n_{k})=\sum_{i=0}^{M-1}\frac{P_{i}^{\prime}}{\sqrt{2\pi\sigma_{i}^{2}}}\text {exp}\Bigg{\{}-\frac{n_{k}^{2}}{2\sigma_{i}^{2}}\Bigg{\}}\ \,\ \ \ P_{i}^{\prime}=\frac{P_{i}}{\sum_{j=0}^{M-1}P_{j}}\ \ \ . \tag{10}\]  
In order to account for memory in the Middleton class A model, a hidden Markov model (HMM) is assumed to govern the underlying transitions among channel states, so that successive noise states are correlated [6]. The corresponding transition probability matrix is thus:  
\[\mathbf{P}=\left[\begin{array}{cccc}x+(1-x)P_{0}^{\prime}&(1-x)P_{1}^{\prime}& \cdots&(1-x)P_{M-1}^{\prime}\\ (1-x)P_{0}^{\prime}&x+(1-x)P_{1}^{\prime}&\cdots&(1-x)P_{M-1}^{\prime}\\ \vdots&\vdots&\vdots&\vdots\\ (1-x)P_{0}^{\prime}&(1-x)P_{1}^{\prime}&\cdots&x+(1-x)P_{M-1}^{\prime}\\ \end{array}\right]\ \ \, \tag{11}\]  
in which each row sums to one and \(x\) is the correlation parameter. Referring to the state diagram in Figure 2, the channel can remain in the present state \(i_{k}\in\{0,1,\cdots,M-1\}\) with probability \(x\) or otherwise switch to one of the states (including \(i_{k}\)), with a (pruned, due to (11)) Poisson distribution. The average duration of an impulsive event with \(m\) interferers, i.e., the average permanence time within a state \(m\), can be computed as:  
\[T_{m}=\frac{1}{(1-x)(1-P_{m}^{\prime})}\ \ . \tag{12}\]  
The Markov-Middleton class A model reduces to the simpler memoryless Middleton class A model in the case \(x=0\). At the same time, the Markov-Gaussian model is an instance of the Markov-Middleton class A model, for \(M=2\). Despite the latter being a more general model that includes the earlier, we illustrate and analyze both, in line with the existing literature that has developed along the two tracks.  
In any case, the expression of noise samples to be considered in the observation model (1) is the one in (4).  
Figure 2: State diagram of the Markov–Middleton class \(A\) noise.