## 2 Theoretical Aspects
### Noise Power on the Output \(y_{H}(t)\) and on Functions \(g(t)\) and \(h(t)\)
By following this reasoning, it is straightforward to evaluate the variance of noise sequences \(n_{h_{i}}(t)\) superimposed on each one of the \(h_{i}(t)\) kernels from the variance of the noise sequences \(n_{g_{i}}(t)\) added to the \(g_{i}(t)\): the noise sequences \(n_{g_{i}}(t)\) (uncorrelated, all with zero mean and variance \(\sigma^{2}g\)) are combined with each other by using the same coefficients as the \(g_{i}(t)\); in the linear combination, the noise amplitudes \(n_{g_{i}}(t)\) will be altered according to the coefficients \([R_{c}]^{-1}[A_{c}]^{T}\) seen above. The noise sequences \(n_{g_{i}}(t)\) are not correlated with each other, so the variances of noise sequences \(n_{h_{i}}(t)\), denoted as \(\sigma^{2}{}_{h_{i}}\), will then be obtained by combining the variances \(\sigma^{2}{}_{g}\) with coefficients equal to the square of those in the matrix \([R_{c}]^{-1}[A_{c}]^{T}\). We define the matrix \([V_{c}]=\left\{[R_{c}]^{-1}[A_{c}]^{T}\right\}^{2}\), where by \(\{[. ]\}^{2}\) we denote the matrix whose terms are the squares of the individual elements of the matrix \([.]\). If \(\left[\sigma^{2}{}_{h}\right]\) and \(\left[\sigma^{2}{}_{g}\right]\) are the vectors of \(\sigma^{2}{}_{h_{i}}\) and \(\sigma^{2}{}_{g}\), we have:  
\[\left[\sigma^{2}{}_{h}\right]=\left\{[R_{c}]^{-1}[A_{c}]^{T}\right\}^{2}.\left[ \sigma^{2}{}_{g}\right] \tag{10}\]  
In the example of model order \(N_{H}=4\), remembering that the variances \(\sigma^{2}{}_{g}\) are all equal to each other, the variances \(\sigma^{2}{}_{h_{i}}\) will be obtained via the linear combination:  
\[\left[ \begin{array}{c}\sigma^{2}{}_{h_{1}}\\ \sigma^{2}{}_{h_{2}}\\ \sigma^{2}{}_{h_{3}}\\ \sigma^{2}{}_{h_{4}}\end{array} \right]=\left[ \begin{array}{ccc}\left(\frac{ 1}{R}\right)^{2}&0&\left(\frac{-3}{R}\right)^{2}&0\\ 0&\left(\frac{2}{R^{2}}\right)^{2}&0&\left(\frac{-8}{R^{2}}\right)^{2}\\ 0&0&\left(\frac{4}{R^{2}}\right)^{2}&0\\ 0&0&0&\left(\frac{8}{R^{2}}\right)^{2}\end{array} \right].\left[ \begin{array}[] {c}\sigma^{2}{}_{g}\\ \sigma^{2}{}_{g}\\ \sigma^{2}{}_{g}\\ \sigma^{2}{}_{g}\end{array} \right] \tag{11}\]  
Frequency Band of Functions \(g(t)\) and \(h(t)\): Noise Reduction on \(h(t)\) by Lowpass Filtering  
Equation (10) expresses the connection between the variance \(\sigma^{2}{}_{g}\) of noise superimposed to each function \(g_{i}(t)\) and the variance \(\sigma^{2}{}_{h_{i}}\) of noise overlapping the functions \(h_{k}(t)\) in the case where the identification procedure is the one described in the previous sections. We already mentioned that, under the assumption of stationarity of the system under measurement [20], proposes to adopt repetitions of the input swept sine signal to reduce the amount of uncertainties arising from the presence of superimposed noise. In the present paper we propose a different way to achieve a mitigation of the effects of additive noise. The procedure we propose deserves some further consideration about the bandwidths of the different signals involved. Equation (10) connects the variances of noise processes superimposed on functions of different types, considered all full-band, i.e., with harmonic components that can reach the Nyquist frequency. In reality, the different signals whose variances are combined in (10) relate to different frequency bands: If the excitation signal \(x(t)\) has harmonic components between \(f_{MIN}\) and \(f_{MAX}\), the kernel \(h_{k}(t)\) present on the \(k-th\) branch of the Hammerstein model will be excited in the frequency band associated with the \(k-th\) power of that signal, that is, in the frequency range between \(f_{MIN}\) and \(k*f_{MAX}\), having excluded the DC component, if any; it is, therefore, sufficient to characterize the kernel \(h_{k}(t)\) up to frequency \(k*f_{MAX}\), since it will not be stressed for higher frequencies.