# Parameter Optimization for an Accurate Swept-Sine Identification Procedure of Nonlinear Systems
## 2 Materials and Methods
### Discussion on the Pulse-Compression Identification Procedure
#### 2.2.1 Limitations in the Identification Procedure
Let us analyze, in detail, the procedure described in Section 2.1 and reformulate it in the frequency domain. The procedure makes no reference to limitations in the bandwidth of each of the signals involved, implicitly considering them to have infinite bandwidth (or, in the case of sampled signals, bandwidth up to the Nyquist frequency). This assumption is not fulfilled in practice. Let the nonlinear system be modeled exactly by an Hammerstein model of order OrdMax. If a sinusoidal signal \(s(t)=Cos(\phi(t))=Cos(2\pi\,\mathrm{f}_{0}t+\varphi_{0})\) is used as the input, the output from the model can be represented by the expression:  
\[\mathrm{u}(t)=\sum_{k=0}^{Ord_{Max}}H_{k}(\omega)\,\left[Cos(2\pi\,\mathrm{f} _{0}t+\varphi_{0})\right]^{k} \tag{8}\]  
where \(H_{k}(\omega)\) is the transfer function on the kth branch of the model, and, for generality, the zero frequency component (\(f_{MIN}=0\)) is considered. Each of the powers of the cosine function can be represented as a sum of its harmonics. The amplitude of these harmonics,some of which have zero amplitude, are given by the \([\mathrm{A}^{-1}]\) matrix, and the inverse of the matrix of the coefficients of the Chebyshev polynomials, given here in the OrdMax = 4 case:  
\[ \begin{bmatrix}\mathrm{Cos}[\omega_{0}\mathrm{t}]^{0}=1\\ \mathrm{Cos}[\omega_{0}\mathrm{t}]^{1}\\ \mathrm{Cos}[\omega_{0}\mathrm{t}]^{2}\\ \mathrm{Cos}[\omega_{0}\mathrm{t}]^{3}\\ \mathrm{Cos}[\omega_{0}\mathrm{t}]^{4}\end{bmatrix} = \begin{bmatrix}\mathrm{A }^{-1}\end{bmatrix} \cdot \begin{bmatrix}\mathrm{Cos}[0\;\omega_{0}\mathrm{t}]=1 \\ \mathrm{Cos}\;[1\omega_{0}\mathrm{t}]\\ \mathrm{Cos}\;[2\omega_{0}\mathrm{t}]\\ \mathrm{Cos}[3\omega_{0}\mathrm{t}]\\ \mathrm{Cos}[4\omega_{0}\mathrm{t}]\end{bmatrix} = \begin{bmatrix}1&0&0&0&0\\ 0&1&0&0&0\\ \frac{1}{2}&0&\frac{1}{2}&0&0\\ 0&\frac{3}{4}&0&\frac{1}{4}&0\\ \frac{3}{8}&0&\frac{1}{2}&0&\frac{1}{8}\end{bmatrix} \cdot \begin{bmatrix} \mathrm{Cos}[1\omega_{0}\mathrm{t}]\\ \mathrm{Cos}[2\omega_{0}\mathrm{t}]\\ \mathrm{Cos}[3\omega_{0}\mathrm{t}]\\ \mathrm{Cos}[4\omega_{0}\mathrm{t}]\end{bmatrix}  \tag{9}\]  
This equivalence makes it possible to give an alternative representation of the output signal \(\mathbf{u}(t)\), again of the additive type, in which the sum is extended to harmonics rather than to powers of the input signal. Expanding each power by means of the \([\mathrm{A}^{-1}]\) matrix and grouping by harmonics rather than by powers gives:  
\[\mathbf{u}(t)=\sum_{i=0}^{Ord_{Max}}\lvert G_{i}(i\;\omega_{0})\rvert\;\text{Cos }(i\;2\pi\;\mathrm{f}_{0}t+i\;\varphi_{0}+\varphi_{i}(i\;\omega_{0})) \tag{10}\]  
Figure 5: The signal at different stages of processing obtained starting from the data measured in the laboratory with the measuring bench described in Figure 4: (**a**) Signal received by the Rx probe; (**b**) output of the matched filter; (**c**,**d**) magnifications of the fundamental frequency and \(2^{\circ}\) harmonic peaks. The functions \(G_{i}(\omega)\), whose argument is \(\varphi_{i}(\omega)\), are obtained from the functions \(H_{k}(\omega)\) through the relationship that, again in the OrdMax = 4 case, is:  
\[ \begin{bmatrix}\mathrm{G}_{1}(\omega)\\ \mathrm{G}_{2}(\omega)\\ \mathrm{G}_{3}(\omega)\\ \mathrm{G}_{4}(\omega)\end{bmatrix} = \begin{bmatrix}\mathrm{A}_{\mathrm{C}}^{- 1}\end{bmatrix} ^{T}\cdot \begin{bmatrix}\mathrm{H}_{1}(\omega)\\ \mathrm{H}_{2}(\omega)\\ 0\ \frac{1}{2}\ 0\ \frac{1}{2}\\ 0\ 0\ \frac{1}{4}\ 0\\ 0\ 0\ 0\ \frac{1}{8}\end{bmatrix} \cdot \begin{bmatrix}\mathrm{H}_{1}(\omega)\\ \mathrm{H}_{2}(\omega)\\ \mathrm{H}_{3}(\omega)\\ \mathrm{H}_{4}(\omega)\end{bmatrix}  \tag{11}\]  
where superscript T denotes transposed matrix. In this case, the zero frequency component was not considered. The resulting matrix \([\mathrm{A}^{-1}]\), deprived of the first row and of the first column, is denoted \([\mathrm{A}_{\mathrm{C}}^{-1}]\) as in [19].