## 4 BER Computational Strategies for Decision Unit--Background Tutorial
### Importance Sampling Scheme  
As earlier concluded, a small BER demands a large \(K\). From a DU point of view, this is considered a fatal limitation of MC implementation, specifically for spread spectrum (SS) systems [92] (such as CDMA systems) in which every transmitted bit must be modulated via the SS code with an abundance of bits.  
A modified MC method called the importance sampling (IS) method can be utilised to decrease BER simulation complexity for SS systems [93]. Further, ref. [94] introduces a method for estimating the bit error rate (BER) based on IS applied to trapping sets. Considering the IS approach, the noise source statistics in the system are biased so that bit errors occur with greater \(p_{e}\), thus minimising the needed execution time. For instance, for a BER equal to \(10^{-5}\), practically, we artificially degrade the performance of the channel, pushing the BER to \(10^{-2}\).  
To explain the IS approach, let \(g(\cdot)\) be the original noise probability density function (PDF) and let \(g^{*}(\cdot)\) be the rising noise PDF utilising an external noise source. Hence, the weighting coefficient can be expressed as follows:  
\[w(x)=\frac{g(x)}{g^{*}(x)} \tag{9a}\]For a simple threshold-dependent decision element, the expression that describes an error takes place as soon as there is a significant excursion of the threshold \(D_{th}\) as follows:  
\[a_{k}=0\;for\left\{\begin{array}{l}error\;count=1\;if\;x_{k}\;\geq D_{th}\\ error\;count=0\;otherwise\end{array}\right. \tag{9b}\]  
Then, \(p_{e}\) is given as follows:  
\[p_{e}=\int_{-\infty}^{\infty}I(x)\;g(x)dx \tag{9c}\]  
\(I(x)\) is an indicator function, which equals 1 when an error takes place; otherwise, it equals 0. Hence, we can express that with regard to the natural estimator of the expectation (i.e., sample mean) as follows:  
\[\hat{p_{e}}=\frac{1}{K}\sum_{k=1}^{K}I(x_{k}) \tag{9d}\]  
Hence, concerning the PDF of the noise (i.e., \(r_{th}(t)\) in Equation (4)) and using Equations (9c) and (9d), we obtain  
\[p_{e}=\int_{-\infty}^{\infty}I(x)\;\frac{g(x)}{g^{*}(x)}g^{*}(x)dx=\int_{- \infty}^{\infty}I^{*}(x)\;g^{*}(x)dx=E[I^{*}(x)] \tag{9e}\]  
Equation (9e) is not just a mathematical expression; it represents the noise processes statistics that influence, and the prediction is achieved with regard to \(g^{*}(\cdot)\). As in the preceding subsection, we may attain the estimator using the sample mean;  
\[\hat{p_{e}}=\frac{1}{K}\sum_{k=1}^{K}I^{*}(a_{k})\;=\frac{1}{K}\sum_{k=1}^{K}w (a_{k})I(a_{k}) \tag{9f}\]  
Regarding Equation (9d), in Equation (9f), the weight parameter \(w(x)\) needs to be evaluated at \(\ae_{k}\). This means reducing the \(\sigma_{e}\), which can be accomplished by establishing external noise of biassed density.  
IS-based BER estimation performance relies crucially on the biassing scheme \(w(x)\). An accurate estimate of the BER can be attained with a brief simulation run time if a good biassing scheme is configured for a specified receiver circuitry system. Contrarily, the BER estimate might even converge at a slower rate than the conventional MC simulation. This implies the IS technique must not be regarded as a generic approach for estimating every receiving system's bit error rate (BER).