# Analysis of Multi-Level Simultaneous Driving Technique for Capacitive Touch Sensors
## 3 Multi-Level Sequences for Capacitive Sensors  
Consider the \(M\)-\(H\) matrix applicable to \(\mathbf{H}^{*}\) in (1). A circulant matrix can be used to form \(M\)-\(H\) matrices [6]. Suppose that the first row of the \(M\)-\(H\) matrix is \((abb\cdots b)\), where \(a\) and \(b\) are real numbers. The remaining rows are represented in the form of \((bab\cdots b)\), \((bba\cdots b)\), \(\cdots\), and \((bbb\cdots a)\), as shown in Figure 3. The code values \(a\) and \(b\) in \(\mathbf{H}^{*}\) define the relative TX driving voltage levels in a sequence, and the column sum of \(\mathbf{H}^{*}\) represents the maximum input range of the receiver, as well as the peak swing levels in the corresponding TXs-to-RX paths of a touch screen. We formulate two properties for \(M\)-\(H\) matrices as follows:  
\[\text{Orthogonality}: 2 ab+(n-2)b^{2}=0 \tag{6}\] \[\text{Max. column sum}: m=a+(n-1)b<\tfrac{M}{aV_{TX}} \tag{7}\]  
where \(n\) is the order of the \(M\)-\(H\) matrix. \(M\) is the maximum capacity for capacitive signals limited by the sensing circuit. This implies that the effective signal is only accepted in [-\(M\), +\(M\)] V. In real application, the inequality in (7) can be satisfied by controlling \(|Z_{f}|\) to adjust \(\alpha\). Using (6) and (7), the following is obtained:  
\[a = m\left(2/n-1\right) \tag{8}\] \[b = 2m/n\]  
which are functions of \(m\) for the given \(n\). In (1), the multiplication of \(H^{\top}\) at receiver \(j\) does not increase the signal gain against the noise component while \(\mathbf{X}_{j}=\alpha V_{TX}\mathbf{H}\) determines the capacitive signal levels. It is assumed that \(c_{j}\) consists of the same constant \(c\). If every element \(x_{ij}\in\mathbf{X}_{j}\) is replaced with the Gaussian random variables \(\mathcal{N}(\mu_{x},z_{x}^{2})\) with a mean \(\mu_{x}\) and the variance \(z_{x}^{2}\), Equation (1) can be rewritten as:  
\[\mathbf{r}_{j}=\left[\mathcal{N}(\mu_{1j},z_{1j}^{2}),\mathcal{N}(\mu_{2j},z_{2j} ^{2}),\cdots,\mathcal{N}(\mu_{ij},z_{ij}^{2}),\cdots,\mathcal{N}(\mu_{nj},z_{ nj}^{2})\right]\cdot\mathbf{H}^{\top}. \tag{9}\]  
Figure 3: Circulant matrix for constructing multi-level \(H\) matrices. Figure 2: Signal processing flow and possible noise components. Then, the received signal \(r_{ij}\) for the \(i\)-th driver and \(j\)-th receiver can be decoded as:  
\[r_{ij}=\left[\mathcal{N}(\mu_{1j},z_{1j}^{2}),\mathcal{N}(\mu_{2j},z_{2j}^{2}), \cdots,\mathcal{N}(\mu_{ij},z_{ij}^{2}),\cdots,\mathcal{N}(\mu_{nj},z_{nj}^{2}) \right]\cdot\mathbf{h}_{i}^{\top} \tag{10}\]  
where \(\mathbf{h}_{i}\) denotes the \(i\)-th row vector in \(\mathbf{H}\). In regular \(H\)-like \(M\)-\(H\) matrices, all the elements in \(\mathbf{X}\) are identical. If \(r_{ij}\) is assumed to be \(r_{x}\), Equation (10) can be expanded by using the same random variable \(N(\mu_{x},z_{x}^{2})\):  
\[ \begin{array}{rcl}r_{ij}&=&a\mathcal{N}\left(\mu_{x},z_{x}^{2}\right)+b \mathcal{N}\left(\mu_{x},z_{x}^{2}\right)+\cdots+b\mathcal{N}\left(\mu_{x},z_ {x}^{2}\right)\\ &=&\mathcal{N}\left(a\mu_{x},a^{2}z_{x}^{2}\right)+\mathcal{N}\left(b\mu_{x},b ^{2}z_{x}^{2}\right)+\cdots+\mathcal{N}\left(b\mu_{x},b^{2}z_{x}^{2}\right) \\ &=&\mathcal{N}\left(\alpha V_{TX}\left\{a^{2}+(n-1)b^{2}\right\},a^{2}z_{x}^{ 2}+(n-1)b^{2}z_{x}^{2}\right)\end{array}  \tag{11}\]  
where \(\mu_{x}=m\cdot\alpha V_{TX}=\left\{a+(n-1)b\right\}\cdot\alpha V_{TX}\). The property is applied for the sum of the independent random variables [26] to (11). Then, the SNR for \(r_{ij}\) can be defined as:  
\[ \begin{array}{rcl}\text{SNR}\left(r_{ij}\right)&=&\alpha V_{TX}\cdot\frac{a ^{2}+(n-1)b^{2}}{\sqrt{a^{2}z_{x}^{2}+(n-1)b^{2}z_{x}^{2}}}\\ &=&\alpha V_{TX}\cdot\frac{\sqrt{a^{2}+(n-1)b^{2}}}{z_{x}}.\end{array}  \tag{12}\]  
By substituting (8) into (12), the following is obtained:  
\[ \begin{array}{rcl}\text{SNR}_{MH}(r_{ij})=\alpha V_{TX}\cdot\frac{m}{z_{x}}.\end{array}  \tag{13}\]  
Similarly, according to (10), for the bipolar \(H\) matrix where \(|h_{ij}|=1\),  
\[