# Analysis of Multi-Level Simultaneous Driving Technique for Capacitive Touch Sensors
## 4 Results and Discussion  
### Simulation Results with Constant \(z_{x}\)  
In this subsection, the conventional \(W\)-\(H\) matrix, the tuned \(H\) matrix, which is described in [1], and the proposed \(M\)-\(H\) matrices are compared when the signal capacity of the receiver is limited. To maximize the SNR in \(M\)-\(H\) and \(B\)-\(H\) matrices, we can scale \(m\) up to \(M/\alpha V_{TX}\), as defined in (7).  
Figure 6 summarizes the capacitive signal gain and sensing time for a sequence, sampled at the output of the decoder for the corresponding \(H\) matrix, where a code consists of 10 chips with a 200 kHz driving frequency. The transfer functions of the TXs-to-RX network and CA were extracted using the circuit design and measurement data for a bar-type touch panel. A total of 10,000 runs of a Gaussian random noise simulation was iterated to obtain the significant digits. Simulated results for SNRs could be verified by (13) and (16). In this simulation, the TX voltage levels of the \(M\)-\(H\) matrices and tuned \(H\) matrices were scaled according to the column sum of the \(W\)-\(H\) matrix for a given \(n\). The TX levels were adjusted so that the maximum column sum was equal to the same target \(m\) (\(=n\)) for all \(H\) matrices for a given number of multi-channels or codes. Kronecker products were used in extracting \(M\)-\(H\) matrices (\(n\) = 8, 9, 10, 12, 14, 15, 16) to reduce the maximum TX level. Even if there are no sensing time constraints, the \(M\)-\(H\) matrices yielded higher SNRs than the other \(H\) matrices, except in the nine-channel case. For the tuned \(H\) matrix with \(n\) = 9, the maximum column sum (\(m\)) can be decreased to 3 [1]. It shows \(n\) = \(m^{2}\), as in the \(M\)-\(H\) matrix. Since its code length (\(=12\)) is greater than 9, the resulting SNR can be higher than that of the \(M\)-\(H\) matrix. However, most irregular \(B\)-\(H\) matrices in \(n\)\(<\)\(m^{2}\) showed lower SNRs, as stated in Section 3. The conventional \(B\)-\(H\) sequences including the tuned \(H\) matrices, require two or 4\(l\) length of codes. The \(M\)-\(H\) matrices give the same code length for a given \(n\). As shown in Figure 6, this advantage results in a shorter sensing time than the \(B\)-\(H\) matrices. The current model assumes a fixed number of cycles (\(N_{chip}\times N_{code}\)) in an acquisition of capacitive signals. However, if the receiver circuit will be able to support the integration of the charges for every cycle, the SNRs of the \(M\)-\(H\) codes in Figure 6 will be more improved while their sensing times are increased to those of the tuned \(H\) matrices.  
Figure 5: Expansion of an \(M\)-\(H\) matrix using Kronecker product.  
When improving the SNR, it is inevitable that the total energy consumption for concurrent TX driving increases. This is not due to the type of code, but to the driving levels and the resulting SNR. The code value and \(V_{TX}\) determine the peak-to-peak swing level of the TX. Even if the polarity of the code sequence still affects the number of transitions during the driving, its power consumption is a function of squared code [1]. Let \(n_{MH}\) and \(n_{BH}\) be the orders of the \(M\)-\(H\) and \(B\)-\(H\) codes, respectively. If the target code length is defined as \(n_{MH}\), because of its universality for orders, the sums of squared codes for the \(B\)-\(H\) and \(M\)-\(H\) matrices can be defined as \(n_{MH}\cdot n_{BH}\) and \(n_{MH}\cdot m^{2}\), respectively. As stated in Section 3, \(n_{BH}\) determines the SNR of the \(B\)-\(H\) codes while \(m\) defines the SNR of the \(M\)-\(H\) codes. Thus, both energy consumption and the SNR for the \(M\)-\(H\) case will be higher than the ones of irregular \(B\)-\(H\) matrices where \(n_{BH}<m^{2}\). Otherwise, if \(n_{BH}=m^{2}\), SNRs and energy consumptions are nearly the same in the two cases.