# Inductance Estimation Based on Wavelet-GMDH for Sensorless Control of PMSM
## 3 Wavelet-GMDH Based Inductance Estimation Method  
### Signal Denoising via Discrete Wavelet Transform  
The wavelet transform was initially introduced by [PERSON] and [PERSON] [24] and was primarily employed for the evaluation of seismic data. Wavelets offer advantages over traditional Fourier methods when analyzing signals containing discontinuities and sharp spikes. Most data analysis applications utilize the continuous-time wavelet transform (CWT), providing an affine-invariant time-frequency representation. However, the discrete wavelet transform (DWT) is the most well-known version due to its outstanding signal compaction properties across various real-world signal classes, along with its computational efficiency. Additionally, the implementation of the DWT is straightforward as it relies on perfect reconstruction filter banks, as well as upsampling and downsampling techniques. The DWT translates continuous-time functions into a set of numbers [25]. \[c_{j,k}=\left\langle x(t),\varphi_{j,k}(t)\right\rangle=\int x(t)\varphi_{j,k}(t )dt \tag{5}\]  
\[d_{j,k}=\left\langle x(t),\psi_{j,k}(t)\right\rangle=\int x(t)\psi_{j,k}(t)dt, \tag{6}\]  
where \(x(t)\) is a real function, \(\varphi_{j,k}(t)\) is the scaling function, and \(\psi_{j,k}(t)\) is the wavelet function. In the DWT, the coefficients \(c_{j,k}\) are referred to as the smooth or approximation coefficients, while the coefficients \(d_{j,k}\) are termed the detail or wavelet coefficients. The inverse DWT is given by  
\[x(t)=\sum_{k}c_{j,k}\varphi_{j,k}(t)+\sum_{j=j}^{\infty}\sum_{k}d_{j,k}\psi_{j,k}(t), \tag{7}\]  
where \(J\) is the starting index, typically equal to zero. The DWT decomposes a given signal \(x(t)\) into its constituent components \(c_{j,k}\) and \(d_{j,k}\). The inverse DWT reconstructs the signal \(x(t)\) from its constituent components \(c_{j,k}\) and \(\hat{d}_{j,k}\). Indeed, the wavelet transform has demonstrated its efficiency in noise removal tasks. It offers an effective method to eliminate noisy components from received signals. The DWT typically proceeds in three phases; in the first phase, the input signal function is decomposed into piecewise functions, each composed of a wavelet and a coefficient. These wavelets are orthogonal to one another and collectively form an orthonormal basis. Furthermore, small coefficients typically represent noise, while large coefficients correspond to significant features within the received signal. Hence, in the second phase, the DWT approach filters these coefficients to retain the most significant ones. This phase is often referred to as thresholding, as any coefficient with a value below a defined threshold is discarded. Finally, the signal is reconstructed by computing the inverse DWT using the filtered coefficients. There has been extensive research into noise removal in signals using wavelet transform. A significant contribution is the work of [PERSON] and [PERSON] [25; 26], which focuses on the thresholding of wavelet coefficients and subsequent reconstruction. The model of a noisy signal \(r(n)\) may be written as  
\[r(n)=s(n)+e(n), \tag{8}\]  
where \(s(n)\) represents the original signal and \(e(n)\) represents the noise. A wavelet transform of the above equation yields the relation:  
\[W_{r}=W_{s}+W_{e}, \tag{9}\]  
where the vector \(W_{r}\) corresponds to the wavelet coefficients of the noisy signal, \(W_{s}\) contains the wavelet coefficients of the original signal, and \(W_{e}\) represents the noise wavelet coefficients. After applying a threshold \(\tau\), the modified wavelet coefficients \(W_{\tau\tau}\) of the degraded signal can be obtained. The inverse wavelet transform of \(W_{\tau\tau}\) yields the restored signal. Choosing the value of the threshold is a fundamental problem to prevent oversmoothing or undersmoothing. The most well-known threshold estimation methods include the SURE threshold method and the universal threshold method.