# Analysis of Multi-Level Simultaneous Driving Technique for Capacitive Touch Sensors
## 3 Multi-Level Sequences for Capacitive Sensors
\begin{array}{rcl}r_{ij}&=&h_{1j}\mathcal{N}(\mu_{1j},z_{1j}^{2})+h_{2j} \mathcal{N}(\mu_{2j},z_{2j}^{2})+\cdots+h_{nj}\mathcal{N}(\mu_{nj},z_{nj}^{2}) \\ &=&\mathcal{N}(h_{1j}\mu_{1j},h_{1j}^{2}z_{1j}^{2})+\mathcal{N}(h_{2j}\mu_{2j}, h_{2j}^{2}z_{2j}^{2})+\cdots+\mathcal{N}(h_{nj}\mu_{nj},h_{nj}^{2}z_{nj}^{2}) \\ &=&\mathcal{N}\left(h_{1j}\mu_{1j}+h_{2j}\mu_{2j}+\cdots h_{nj}\mu_{nj},h_{1j}^ {2}z_{1j}^{2}+h_{2j}^{2}z_{2j}^{2}+\cdots h_{nj}^{2}z_{nj}^{2}\right)\\ &=&\mathcal{N}\left(\alpha V_{TX}\cdot n,z_{1j}^{2}+z_{2j}^{2}+\cdots+z_{nj}^{2 }\right)\end{array}  \tag{14}\]  
Therefore, the SNR for a bipolar \(H\) matrix can be defined as:  
\[ \begin{array}{rcl}\text{SNR}_{BH}(r_{ij})=\alpha V_{TX}\cdot\frac{n}{\sqrt{z _{1j}^{2}+z_{2j}^{2}+\cdots+z_{nj}^{2}}}.\end{array}  \tag{15}\]  
when \(z_{ij}\simeq z_{x}\) from the assumption, Equation (15) can be further simplified as,  
\[ \begin{array}{rcl}\text{SNR}_{BH}(r_{ij})\simeq\alpha V_{TX}\cdot\frac{ \sqrt{n}}{z_{x}}.\end{array}  \tag{16}\]  
When considering the SNR for the touch event, \(r_{ij}\) should be replaced by \(|\Delta r_{ij}|\) in (13) and (16). \(\alpha\) decreases by the increase of the input impedance of CA due to a touch event. A typical definition for SNR contains \(|\Delta r_{ij}|\) instead of \(r_{ij}\)[1; 27]. However, in this work, both \(\text{SNR}(|\Delta r_{ij}|)\) and \(\text{SNR}(r_{ij})\) definitions are employed. For equivalence to the existing definition [1; 27], the numerator in (12) and (15) corresponds to the mean value of the signal samples, and its additive noise \(z_{x}\) can be replaced by the standard deviation of them. Consequently, the acquired signal for one sequence in the \(M\)-\(H\) matrix is proportional to \(m\), which is identical to the sum of the TX levels in a column of \(\mathbf{H}\). For regular and Bush-type \(B\)-\(H\) matrices with \(n=m^{2}\), whose column sums and row sums are all equal to \(m\)[28], it can be concluded that \(\text{SNR}_{MH}(r_{ij})\simeq\text{SNR}_{BH}(r_{ij})\), according to (13) and (16). These special type \(H\) matrices are rare, and their orders are constrained as \(4l^{2}\), where \(l\) is a positive integer. For an irregular \(B\)-\(H\) matrix with \(n<m^{2}\), \(\text{SNR}_{MH}(r_{ij})>\text{SNR}_{BH}(r_{ij})\), where \(z_{ij}=z_{x}\). Figure 4 provides an example of \(H\) with \(N_{chip}=10\) in the form of an \(M\)-\(H\) matrix and the one corresponding output of the receiver with \(n=3\) and \(m=2\). The maximum column sum is identical to that of a typical Walsh-Hadamard (\(W\)-\(H\)) matrix with \(n=2\). Both the column sum and the row sum are equal to \(m\) in such \(M\)-\(H\) matrices. When a touch event decreases the capacitance across TX2 and RX1, the first and third receiving signals for the second TX sequence [6.67, \(-3.33\), 6.67] are decreased. Conversely, the second signal is increased due to the negative sign of the second code. The orders of \(M\)-\(H\) matrices can be constructed so that a given time slot for sensing can be fully utilized. To maximize the capacitive signals, the length in multiples of the code and chip sequence should match the required time slot [1]. This period is usually constrained by the scan rate or a reporting rate corresponding to the response time of the sensor system. Because typical \(H\) matrices exist in 2 and \(4l\)-length squares, it is difficult to only adjust the multiples of the columns in the \(H\) matrix to increase the signal gain. For example, \(H\) matrices with orders of 2 and \(4l\) do not fit time slots 3, 5, 6, 7, 9, and 10. Conversely, the suitability of \(M\)-\(H\) matrices with odd and even orders for the increased number of columns can be adjusted by removing unnecessary rows or by combining two or more \(M\)-\(H\) matrices. Multi-level \(\mathbf{H}_{n}\) matrices of order \(n\) can be expanded using the Kronecker product \(\otimes\). Let \(\mathbf{H}_{A}\) and \(\mathbf{H}_{B}\) be the \(M\)-\(H\) matrices satisfying (6)-(8). From \(\mathbf{H}_{C}=\mathbf{H}_{A}\otimes\mathbf{H}_{B}\), \(m_{C}=m_{A}\cdot m_{B}\) is derived.