## 3 Performance Evaluation  
Since the control objective is to guarantee the stability of the sigma delta modulator and the stability only depends on the input signal for a given sigma delta modulator and the control strategy, the input dynamical range is employed as the criterion for the comparison. As discussed in Section 1 that the clipping method [7] is the most common control method, the clipping method [7] is compared. More precisely, the clipping rule is to reset the state vector \(\mathbf{x}(k)\) to the zero vector when the absolute values of the outputs of the loop filter are greater than or equal to one. In this paper, a single bit fifth order interpolative sigma delta modulator realized using the state space representation with the following state space matrices is considered:  
\[\mathbf{A}=\left[ \begin{array}{cccc}0.9990&1.0000&0.0000&0.0000\\ -0.0020&0.9990&0.0530&0.1764&0.5164\\ 0&0&0.9997&1.0000&-0.0000\\ 0&0&-0.0007&0.9997&0.3567\\ 0&0&0&0&1.0000\\ \end{array} \right],\]  
\[\mathbf{B}=\left[ \begin{array}{cccc}0&0&0&0&1\\ \end{array} \right]^{T}\]  
and  
\[\mathbf{C}=\left[ \begin{array}{cccc}0.0307&0.4153&0.0828&0.2754&0.8063\\ \end{array} \right]\]  
This sigma delta modulator is chosen for the comparison because it is widely used in the audio industry [7]. It is worth noting that the eigenvalues of \(\mathbf{A}\) are \(0.9990+0.0445j\), \(0.9990-0.0445j\), \(0.9997+0.0264j\), \(0.9990-0.0264j\) and \(1\). As all the eigenvalues are on the unit circle, this loop filter is not bounded input bounded output stable. For the ease of the implementation, assume that the initial state vector is zero. Since this loop filter contains a DC pole, the step input is applied to this sigma delta modulator for illustration purposes. This is because this input will result to the occurrence of the resonance. Therefore, control is required for stabilizing the sigma delta modulator. Here, three-step input signals with different step sizes are illustrated. For the first case, an input step size equal to \(0.24\) is applied to the sigma delta modulator. That is, \(u(k)=0.24\) for \(k\geq 0\). Figure 1(a),b show the outputs of the loop filters when our control strategy and the clipping method [7] are applied, respectively. Figure 1(c),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 1(a) that the output of the loop filter under our proposed control strategy is bounded. This implies that there exists a binary sequence in \(Y_{d_{i}}\) or the neural network finds a solution of Problem (P) such that the objective functional value of Problem (P) is exactly equal to zero. Besides, it can be seen from Figure 1(c) that the output of the loop filter is a wide band signal. Its frequency content is very rich. This means that the sigma delta modulator exhibits the chaotic behavior. On the other hand, it can be seen from Figure 2b that the output of the loop filter under the clipping strategy [7] is periodic. This is because the state vectors are reset to the same state every time when the reset action takes place. Besides, it can be seen from Figure 2d that the output of the loop filter only consists of the DC frequency and the Nyquist sampling frequency. This implies that the sigma delta modulator exhibits the limit cycle behavior. For the audio application, this frequency component with the high amplitude refers to the annoying audio tune, which should be avoided. Figure 2: Cont. In order to evaluate the performance on the analog to digital conversion of the sigma delta modulator under various control strategies, the peak signal to noise ratio is employed as the performance index. Since the passband of the loop filter is \(\left(-\frac{\pi}{64},\frac{\pi}{64}\right)\), the ideal lowpass filter with the passband equal to \(\left(-\frac{\pi}{64},\frac{\pi}{64}\right)\) is applied to the quantizer output. Here, the ideal lowpass filtering is implemented using the discrete Fourier transform approach.