## 3 Performance Evaluation
That is, those discrete Fourier transform coefficients outside the frequency band \(\left(-\frac{\pi}{64},\frac{\pi}{64}\right)\) are set to zero. It is found that the signal to noise ratio of the signal based on our control strategy is 36.98 dB. On the other hand, the signal to noise ratio of the signal based on the clipping method [7] is \(-1.76\) dB. This is because the output of the loop filter only consists of the DC frequency and the Nyquist sampling frequency while there is no DC component in the input signal. Therefore, the signal to noise ratio of the signal is significantly worse. Now, consider another input step size. Here, the input step size is incremented by 0.01. More precisely, an input step size of 0.25 is applied. That is, \(u(k)=0.25\) for \(k\geq 0\). Figure 3a,b show the outputs of the loop filters when our control strategy and the clipping method [7] are applied, respectively. Figure 3c,d show the magnitude responses of the outputs of the loop filters when our control strategy and the clipping method [7] are applied, respectively. It can be seen from Figure 3a,c that the sigma delta modulator exhibits the similar behavior as before if our proposed control strategy is applied. On the other hand, it is worth noting that the increase in the input step size causes the absolute value of the output of the loop filter greater than or equal to one at every time instant if no control force is applied. Therefore, if the clipping control strategy [7] is applied, then the clipping action takes place at every time instant. Hence, it can be seen from Figure 3b,d that the output of the loop filter is always equal to zero if the clipping control strategy [7] is applied. In this case, the sigma delta modulator is not working properly. Figure 2: (**a**) Output of the loop filter when \(u(k)=0.24\) for \(k\geq 0\) under our proposed control strategy. (**b**) Output of the loop filter when \(u(k)=0.24\) for \(k\geq 0\) under the clipping strategy [7]. (**c**) Magnitude response of the output of the loop filter when \(u(k)=0.24\) for \(k\geq 0\) under our proposed control strategy. (**d**) Magnitude response of the output of the loop filter when \(u(k)=0.24\) for \(k\geq 0\) under the clipping strategy [7]. Figure 3: (**a**) Output of the loop filter when \(u(k)=0.25\) for \(k\geq 0\) under our proposed control strategy. (**b**) Output of the loop filter when \(u(k)=0.25\) for \(k\geq 0\) under the clipping strategy [7]. (**c**) Magnitude response of the output of the loop filter when \(u(k)=0.25\) for \(k\geq 0\) under our proposed control strategy. (**d**) Magnitude response of the output of the loop filter when \(u(k)=0.25\) for \(k\geq 0\) under the clipping strategy [7]. Likewise, it is found that the signal to noise ratio of the signal based on our control strategy is 35.37 dB. On the other hand, the signal to noise ratio of the signal based on the clipping method [7] is exactly 0 dB. Next, a very large input step, which is equal to 0.58, is applied. Figure 4a,b show the outputs of the loop filters when our control strategy and the clipping method [7] are applied, respectively. Figure 4c,d show the magnitude responses of the outputs of the loop filters when our control strategy and the clipping method [7] are applied, respectively. It can be seen from Figure 4a,c that the sigma delta modulator also exhibits the similar behavior as before if our proposed control strategy is applied. Similarly, it can be seen from Figure 4b,d that the output of the loop filter is always zero under the clipping control strategy [7]. From here, it can be seen that our proposed control method can achieve a larger stability region in terms of the input dynamical range. Figure 4: \(Cont.\)  
Likewise, it is found that the signal to noise ratio of the signal based on our control strategy is 32.52 dB. On the other hand, the signal to noise ratio of the signal based on the clipping method [7] is exactly 0 dB.