## Appendix C Review on the Frequency-Domain-Based Bit Flipping Control Strategy
This further implies that  
\[\frac{d_{i}\overline{x}_{i}(0)}{\overline{b}_{i}}+\left.U(z)\right|_{z=d_{i}} =\left.\hat{S}(z)\right|_{z=d_{i}}\]  
for \(i=0,\cdots,P-1\). Define \(Y_{d_{i}}\) as the set of complex numbers where the elements inside \(Y_{d_{i}}\) can be represented as the binary combinations of the elements in the sequences \(\left\{d_{i}^{-k}:k\geq 0\right\}\). That is, \(\exists c_{k}\in\{1,-1\}\ \forall k\geq 0\) such that \(\sum\limits_{k\geq 0}c_{k}d_{i}^{-k}\in Y_{d_{i}}\). If \(\frac{d_{i}\overline{x}_{i}(0)}{\overline{b}_{i}}+\left.U(z)\right|_{z=d_{i}} \in Y_{d_{i}}\) for \(i=0,\cdots,P-1\), then there exists a binary sequence \(c_{k}\in\{1,-1\}\ \forall k\geq 0\) such that the single bit interpolative sigma delta modulator is internally stable with the z-transform of the quantizer output equal to \(\hat{S}(z)\). The above result characterizes a condition for stabilizing the single-bit high-order interpolative sigma delta modulator via flipping some values of the quantizer output. However, it can be seen from Section 2 that \(Y_{d_{i}}\) does not include all complex numbers. This implies that if \(\frac{d_{i}\overline{x}_{i}(0)}{\overline{b}_{i}}+\left.U(z)\right|_{z=d_{i}} \in C\backslash Y_{d_{i}}\), then there does not exist any binary sequence such that the sigma delta modulator can be stabilized. On the other hand, if \(\frac{d_{i}\overline{x}_{i}(0)}{\overline{b}_{i}}+\left.U(z)\right|_{z=d_{i}} \in Y_{d_{i}}\), then there exists a binary sequence such that the sigma delta modulator can be stabilized. For this case, it is required to find this binary sequence. In particular, an integer programming approach is employed to address this issue. The objective function of the optimization problem is to minimize the error energy between \(\frac{d_{i}\overline{x}_{i}(0)}{\overline{b}_{i}}+\left.U(z)\right|_{z=d_{i}}\) and \(\left.S(z)\right|_{z=d_{i}}\) subject to the binary constraint imposed to the quantizer output. That is, the optimization problem is formulated as follows:  
**Problem (P)**  
\[\min_{\mathbf{x}}\sum_{\forall i\in\{0,-P-1\}}\left|\psi_{i}^{T}\mathbf{x}^{ \prime}-\frac{d_{i}\overline{x}_{i}(0)}{\overline{b}_{i}}-\left.U(z)\right|_{ z=d_{i}}\right|^{2},\]  
subject to \(\mathbf{x}^{\prime}_{k}\in\{1,-1\}\ \forall k\geq 0\). Here, \(\mathbf{x}^{\prime}_{k}\) is the \(k^{\text{th}}\) element of \(\mathbf{x}^{{}^{\prime}}\) and \(d_{i}^{-k}\) is the \(k^{\text{th}}\) element of \(\psi_{i}\). If there exists a set of binary sequences such that the single-bit high-order interpolative sigma delta modulator is internally stable, then the objective functional value of Problem (**P**) for these binary sequences in this set will be exactly equal to zero. In this case, \(\mathbf{x}^{\prime}\) is the global minimum of Problem (**P**). Hence, by finding a solution of Problem (**P**), we can find a stable binary sequence stabilizing the sigma delta modulator. However, it is difficult to find a solution of Problem (**P**). This is because it is an infinite dimensional optimization problem. To address this issue, the quantizer output bit streams are divided into blocks with the length of each block is equal to \(M\). Denote the vector of the binary sequence in the \(m^{\text{th}}\) block as \(\mathbf{x}^{*}_{m}\) for \(m\geq 0\). Define a vector \(\boldsymbol{\psi}^{*}_{i,m}\) such that the \(k^{\text{th}}\) element of \(\boldsymbol{\psi}^{*}_{i,m}\) is equal to \(d_{i}^{-mM-k}\) for \(k=0,\cdots,M-1\).