## 3 Performance Evaluation
To further demonstrate the effects of the input step size on the stability of the sigma delta modulator, Figure 5a,b show the maximum absolute value of the output of the loop filter against the input step size under our proposed control strategy and the clipping control strategy [7], respectively. For our proposed control strategy, it can be seen from Figure 5a that the input step sizes being smaller than or equal to 0.58 will yield the maximum absolute values of the outputs of the loop filter being equal to nonzero. This means that the sigma delta modulator can be stabilized. However, the modulus of \(U(z)\big{|}_{z=d_{i}}\) increases as the input step size increases. Eventually, \(\frac{d\widetilde{x_{i}}(0)}{\widetilde{b_{i}}}+\left.U(z)\right|_{z=d_{i}} \
ot\in Y_{d_{i}}\). In this case, the sigma delta modulator cannot be stabilized. Therefore, it can be seen from Figure 5a that the maximum absolute values of the outputs of the loop filter are equal to zero when the input step sizes are larger than 0.58. On the other hand, for the clipping control method [7], it can be seen from Figure 5b that the input step sizes being smaller than or equal to 0.24 will yield the maximum absolute values of the outputs of the loop filter being equal to nonzero. However, the maximum absolute values of the outputs of the loop filter are equal to zero when the input step sizes are larger than 0.58. This also demonstrates that our proposed control method yields a wider stability region in terms of the input dynamical range. Figure 4: (**a**) Output of the loop filter when \(u(k)=0.58\) for \(k\geq 0\) under our proposed control strategy. (**b**) Output of the loop filter when \(u(k)=0.58\) for \(k\geq 0\) under the clipping strategy [7]. (**c**) Magnitude response of the output of the loop filter when \(u(k)=0.58\) for \(k\geq 0\) under our proposed control strategy. (**d**) Magnitude response of the output of the loop filter when \(u(k)=0.58\) for \(k\geq 0\) under the clipping strategy [7]. Figure 6: **(a)** Output of the loop filter when \(u(k)=0.24\sin(0.0142\pi k)\) for \(k\geq 0\) under our proposed control strategy. **(b)** Output of the loop filter when \(u(k)=0.24\sin(0.0142\pi k)\) for \(k\geq 0\) under the clipping control strategy [7]. **(c)** Magnitude response of the output of the loop filter when \(u(k)=0.24\sin(0.0142\pi k)\) for \(k\geq 0\) under our proposed control strategy. **(d)** Magnitude response of the output of the loop filter when \(u(k)=0.24\sin(0.0142\pi k)\) for \(k\geq 0\) under the clipping control strategy [7]. Figure 7: Cont. To demonstrate the application value of our proposed method, the sigma delta modulator is applied to an electromyogram [1]. Figure 8 shows the electromyogram. Figure 9a,b show the outputs of the loop filters under our control strategy and the clipping method [7], respectively. Figure 9c,d show the magnitude responses of the outputs of the loop filters under our control strategy and the clipping method [7], respectively. It can be seen from Figure 9a,c that the sigma delta modulator exhibits similar behavior to before for our proposed control strategy. This demonstrates the generality of our proposed method and the possibility of applying our proposed method to some practical signals. Figure 8: Electromyogram. Figure 7: (**a**) Output of the loop filter when \(u(k)=0.24\sin(0.0084\pi k)\) for \(k\geq 0\) under our proposed control strategy. (**b**) Output of the loop filter when \(u(k)=0.24\sin(0.0084\pi k)\) for \(k\geq 0\) under the clipping control strategy [7]. (**c**) Magnitude response of the output of the loop filter when \(u(k)=0.24\sin(0.0084\pi k)\) for \(k\geq 0\) under our proposed control strategy. (**d**) Magnitude response of the output of the loop filter when \(u(k)=0.24\sin(0.0084\pi k)\) for \(k\geq 0\) under the clipping control strategy [7]. Figure 9: _Cont._