## 2 Theoretical Model of a System in the Presence of Gaussian Noise
### N-Correlator Receiver  
In this case, the source can generate \(K\)-bit symbols [1]. Thus, the number of required sequences will be \(N=2^{K}\). The modulation and demodulation will be the same as in the case of a single-correlator receiver. However, the correlation for each sequence must be performed in its own receiver correlator. Therefore, at the receiver, a bank of \(N\) correlators needs to be implemented, as shown in Figure 1.  
The receiver for binary symbols transmission is analyzed in previous section. In that case it was sufficient to have the first correlator in the receiver and the symbol to sequence conversion is performed by a multiplier as shown separately in Figure 1. For that case the random variable \(w_{1}\) at the output of the first correlator is calculated in Equation (7) and the related mean, variance and probability of error in Equations (9), (12) and (16). If the source generates \(N\) symbols, which correspond to combinations of message bits, the receiver will use all \(N\) correlators in Figure 1. The decision circuit now is a comparator of correlator outputs. The order of these outputs corresponds to the order of the spreading sequences that could be sent at the transmitter side. The decision circuit decides according to this decision rule: the sequence/symbol received corresponds to the maximum output value of the decision circuit. This, if the maximum is at the output three the decision sequence will say the third symbol was sent. Suppose that the first sequence is sent. Following the procedure of a single-correlator receiver modeling, the output of the \(n\)-th correlator can be expressed as  
\[w_{n}=\sum_{i=1}^{2\beta}z_{i}c_{in}=\sqrt{E_{c}}\sum_{i=1}^{2\beta}c_{i1}c_{ in}+\sqrt{E_{N}}\sum_{i=1}^{2\beta}n_{i}c_{in}. \tag{18}\]  
The first term is inter-sequence interference and the second term is noise term. The value \(w_{n}\) is a random sample of a random variable defined for the \(n\)-th correlator output. Due to the central limit theorem (CLT) this random variable can be approximated by the Gaussian random variable with zero mean  
\[\eta_{wn}=\sqrt{E_{c}}\sum_{i=1}^{2\beta}E\{c_{i1}c_{in}\}+\sqrt{E_{N}}\sum_{ i=1}^{2\beta}E\{n_{i}c_{in}\}=0. \tag{19}\]  
and variance  
\[\sigma_{wn}^{2}=2\beta E_{c}E\{c_{i1}^{2}\}E\{c_{in}^{2}\}+2\beta E_{N}\sigma _{ni}^{2}E\{c_{in}^{2}\}. \tag{20}\]  
In the case of binary chip transmission this variance is  
\[\sigma_{wn}^{2}=2\beta(E_{c}+E_{N}).\]  
The density function of this random variable can be approximated by Gaussian density expressed as  
\[f_{wn}(w_{n})=\frac{1}{\sqrt{2\pi\sigma_{wn}^{2}}}e^{\left(-w_{n}^{2}/2\sigma _{wn}^{2}\right)}. \tag{21}\]  
Suppose the threshold value inside the decision circuit is \(w_{1}=w\). The probability of correct decision is equal to the conditional probability of that all outputs are less than this threshold value for the given value \(w_{1}=w\), _i.e._,  
\[\mathrm{P}(c|w_{1}=w)=\mathrm{P}(w_{2}<w,...,w_{N}<w|w_{1}=w)=\prod_{n=2}^{N} \mathrm{P}(w_{n}<w|w_{1}=w) \tag{22}\]  
Inserting Equation (21), and having in mind statistical independence between variables \(w_{n}\), this expression becomes  
\[\prod_{n=2}^{N}\int_{-\infty}^{w}\frac{1}{\sqrt{2\pi\sigma_{wn}^{2}}}e^{ \left(-w_{n}^{2}/2\sigma_{wn}^{2}\right)}dw_{n}=[1-\int_{w}^{\infty}\frac{1}{ \sqrt{2\pi\sigma_{wn}^{2}}}e^{\left(-w_{n}^{2}/2\sigma_{wn}^{2}\right)}dw_{n} ]^{N-1} \tag{23}\]  
This expression can be simplified, by applying the error complementary function and binomial theorem, and expressed in this form  
Figure 1.0  
This expression gives one for \(i=0\). If \((-1)\) is taken out of the brackets of _erfc_, the expression becomes \[ \begin{split}\mathrm{P}(c|w_{1}=w)=1+\sum_{i=1}^{N-1}\binom{N-1}{i} \left(-1\right)^{i}[\frac{1}{2}erfc\frac{w_{1}}{\sqrt{2\sigma_{wn}^{2}}}]^{i}= \\ 1-\sum_{i=1}^{N-1}\binom{N-1}{i}\left(-1\right)^{i+1}[\frac{1}{2 }erfc\frac{w_{1}}{\sqrt{2\sigma_{wn}^{2}}}]^{i}\end{split}