# Wavelet Decomposition Prediction for Digital Predistortion of Wideband Power Amplifiers
## 3 The Proposed WDP Architecture
Get \(\mathbf{S}\), \(\mathbf{S}_{l}\), \(S_{h}\) through preparation. Obtain DPD output: \(\mathbf{y}_{\text{DPD}}=f_{\text{DPD}}(\theta_{\text{DPD}};\mathbf{S},\mathbf{S}_{l},S_{h})\). ```

**Algorithm 1**_Cont_. ### PA Nonlinear Modeling by WDP

During the PA nonlinear modeling stage, we adopt a combination of WD and UGRNN methods, where WD serves as a signal analysis technique that extracts time-frequency features by progressively decomposing the time series into low-frequency and high-frequency subsequences. We define the input time series as \(\mathbf{x}=\{x_{1},\dots,x_{m},\dots,x_{M}\}\). The high-frequency and low-frequency subsequences generated at the \(i\)-th decomposition layer are denoted as \(\mathbf{x}^{h}(i)\) and \(\mathbf{x}^{l}(i)\), respectively. At the \((i+1)\)-th level, WD applies a low-pass filter, \(\mathbf{l}=\{l_{1},\dots,l_{k},\dots,l_{K}\}\), and a high-pass filter, \(\mathbf{h}=\{h_{1},\dots,h_{k},\dots,h_{K}\}\), where \(K\ll M\). The low-frequency components from the previous layer are then convolved as

\[a_{n}^{l}(i+1)=\sum_{k=1}^{K}x_{n+k-1}^{l}(i)\cdot l_{k}, \tag{7}\]

\[a_{n}^{h}(i+1)=\sum_{k=1}^{K}x_{n+k-1}^{l}(i)\cdot h_{k}, \tag{8}\]where \(x_{n}^{l}(i)\) represents the \(n\)-th element of the low-frequency component at the \(i\)-th decomposition layer. The low-frequency \(\mathbf{x}^{l}(i)\) and high-frequency \(x^{h}(i)\) components at the \(i\)-th layer are derived from the \(\frac{1}{2}\) down-sampling of the intermediate variable sequences \(\mathbf{a}^{l}(i)=\left\{a_{1}^{l}(i),a_{2}^{l}(i),\dots\right\}\) and \(\mathbf{a}^{h}(i)=\left\{a_{1}^{h}(i),a_{2}^{h}(i),\dots\right\}\). The method proposed in [29] can approximate the implementation of WD within a NN architecture. The WD model decomposes time series using the following two functions:

\[\mathbf{a}^{l}(i) =\mathbf{W}^{l}(i)\mathbf{x}^{l}(i-1)+\mathbf{b}^{l}(i), \tag{9}\] \[\mathbf{a}^{h}(i) =\mathbf{W}^{h}(i)\mathbf{x}^{h}(i-1)+\mathbf{b}^{h}(i), \tag{10}\]

where \(\mathbf{b}^{l}(i)\) and \(\mathbf{b}^{h}(i)\) are trainable bias vectors, and their initial values are all random values close to zero. It is not difficult to find that Equations (7)-(10) have the same form of WD function. \(x_{n}^{l}(i)\) and \(x_{n}^{h}(i)\) are the \(n\)-th elements of the low-frequency and high-frequency components in the \(i\)-th layer, respectively, which are down-sampled from the intermediate variables \(\mathbf{a}^{l}(i)\) and \(\mathbf{a}^{h}(i)\) using average pooling. To achieve the convolution defined in Equation (3), it is necessary to set the initial values of the weight matrices \(\mathbf{W}^{l}(i)\) and \(\mathbf{W}^{h}(i)\) as

\[\mathbf{W}^{l}(i)=\left[ \begin{array}{cccccccc}l_{1}&l_{2}&l_{3}&\dots&l_{K}& \varepsilon&\dots&\varepsilon\\ \varepsilon&l_{1}&l_{2}&\dots&l_{K-1}&l_{K}&\dots&\varepsilon\\ \vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\vdots\\ \varepsilon&\varepsilon&\varepsilon&\dots&\dots&\dots&1&l_{K}\\ \varepsilon&\varepsilon&\varepsilon&\dots&\dots&\dots&\varepsilon&l_{1}\\ \end{array} \right], \tag{11}\]

\[\mathbf{W}^{h}(i)=\left[ \begin{array}{cccccccc}h_{1}&h_{2}&h_{3}&\dots&h_{K}& \varepsilon&\dots&\varepsilon\\ \varepsilon&h_{1}&h_{2}&\dots&h_{K-1}&h_{K}&\dots&\varepsilon\\ \vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\vdots\\ \varepsilon&\varepsilon&\varepsilon&\dots&h_{1}&\dots&h_{K-1}&h_{K}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\vdots\\ \varepsilon&\varepsilon&\varepsilon&\dots&\dots&\dots&h_{1}&h_{2}\\ \varepsilon&\varepsilon&\varepsilon&\dots&\dots&\dots&\varepsilon&h_{1}\\ \end{array} \right], \tag{12}\]

where \(\mathbf{W}^{l}(i)\) and \(\mathbf{W}^{h}(i)\in\mathbb{R}^{P\times P}\), and \(P\) is the size of \(\mathbf{x}^{l}(i-1)\). The values in the weight matrix that do not belong to the filter coefficients are filled with a random value, \(\varepsilon\), where the random value \(\varepsilon\) needs to satisfy \(|\varepsilon|\ll|l|\), \(\forall l\in\mathbf{I}\), and \(|\varepsilon|\ll|h|\), \(\forall h\in\mathbf{h}\).