## 2 Wavelet Transform Algorithm Transmittance
### Algorithm Transmittance Impact on Error Signals  
Assume that the signal \(x(t)\), processed by the analyzed WT algorithm, can be described in an ideal case as follows:  
\[\hat{x}(i)=\sum_{j=0}^{\infty}E_{x,\omega}\Big{(}\omega_{j}\Big{)}\sin\!\left( \omega_{j}!T_{s}+\varphi_{x,\omega}\Big{(}\omega_{j}\Big{)}\right),\]  
where \(T_{s}=\frac{1}{f_{s}}\) is the sampling period corresponding to sampling frequency \(f_{s}\), \(E_{x,\omega}(\omega)\) is the amplitude, and \(\varphi_{x,\omega}(\omega)\) is the phase shift of the signal harmonic of pulsation \(\omega\). In the real case, the quantity \(x(i)\) can be described as follows:  
\[\hat{x}(i)=\hat{x}(i)+e_{x,r}(i)+\sum_{j=0}^{\infty}E_{x,\omega}\Big{(}\omega_ {j}\Big{)}\sin\!\left(iT_{s}\omega_{j}+\varphi_{x,\omega}\Big{(}\omega_{j} \Big{)}\right), \tag{3}\]  
where \(e_{x,r}(i)\) is random error (non-deterministic), \(E_{x,\varepsilon}(\omega)\) is the amplitude, and \(\varphi_{x,\varepsilon}(\omega)\) is the phase shift of the deterministic error. According to Equation (3), it is possible to define static error signal \(e_{x,s}(i)\) (for which subsequent values do not change for a single WT algorithm run) as follows:  
\[e_{x,s}(i)=E_{x,\varepsilon}(0)\sin\!\left(\varphi_{x,\varepsilon}(0)\right),\]where in case of \(\varphi_{x,e}(0)=\pi\) rad, the equation in question can be written as \(e_{x,s}(i)=E_{x,e}(0)\). In case of harmonics with non-zero pulsation, the dynamic error signal \(e_{x,d}(i)\) can be defined as follows:  
\[e_{x,d}(i)=\sum_{j=1}^{\infty}E_{x,e}\Big{(}\omega_{j}\Big{)}\sin\biggl{(}iT_{s }\omega_{j}+\varphi_{x,e}\Big{(}\omega_{j}\Big{)}\biggr{)}.\]  
The random error signal \(e_{x,r}(i)\) cannot be described in a deterministic form, but statistical parameters such as variance, expected value, and the shape of the probability density function can describe its properties. According to [5; 23; 24], a single row of the transmission matrix, as indicated in Equation (1), and the related transmittance described in Equation (2), can be analyzed in a manner analogous to a finite impulse response (FIR) filter [25]. For each output quantity, it is possible to analyze how the algorithm introduces error signals present at its input to this quantity. It should also be noted that in the case of the discussed family of algorithms, the transfer function associated with the selected quantity is linear and time invariant. Therefore, the deterministic error component of the input quantity, as described in Equation (3), is transferred to the \(i\)-th output of the WT algorithm according to the following relationship:  
\[e_{X,s,i}\big{(}j\big{)}=K_{i}(0)E_{x,e}(0)\sin\bigl{(}\varphi_{x,e}(0)+ \varphi_{i}(0)\bigr{)},\]  
The discussed transmittance also affects the variance of the analyzed error signals. In the case of both deterministic and random signals, the following relationship can be written [23; 26]:  
\[\sigma_{X,i}^{2}(\omega)=K_{i}^{2}(\omega)\sigma_{x}^{2}(\omega)=\sigma_{x}^{ 2}(\omega)\biggl{|}H_{i}\Bigl{(}\mathrm{e}^{\mathrm{j}\omega T_{s}}\Bigr{)} \biggr{|}^{2}, \tag{4}\]  
where for analyzed error signal \(\sigma_{x}^{2}\) is the variance in the WT algorithm input, and \(\sigma_{X,i}^{2}\) is the variance in the \(i\)-th WT algorithm output. Given the above equation, the single harmonic variance of the dynamic error signal \(e_{X,d,i}(j)\) in the algorithm output can be determined according to the relationship [24]  
\[\sigma_{X,d,i}^{2}(\omega)=\frac{1}{2}E_{x,e}^{2}(\omega)\biggl{|}H_{i}\Bigl{(} \mathrm{e}^{\mathrm{j}\omega T_{s}}\Bigr{)}\biggr{|}^{2}. \tag{5}\]  
In the case of static error signals, the presented equation simplifies (for \(\omega=0\) rad/s, it becomes \(\mathrm{e}^{\mathrm{j}\omega T_{s}}=1\)), so it can be written as follows:  
\[\sigma_{X,s,i}^{2}=\bigl{|}H_{i}(1)\bigr{|}^{2}\sigma_{x,s}^{2}=\sigma_{x,s}^ {2}\left(\sum_{j=0}^{N-1}a_{i,j}\right)^{2}.\]  
In the case of random error signals, it can be observed that the algorithm in question processes \(N\) subsequent realizations of these signals.