## 1 Introduction  
A single-bit high-order interpolative sigma delta modulator [1; 2] consists of a negative feedback of a high order loop filter and a single bit quantizer [3]. If the loop filter is designed properly, then the magnitude of the noise transfer function can be very small at the signal band and the noise is mainly localized outside the signal band. On the other hand, the magnitude of the signal transfer function can also be very small outside the signal band and the signal is mainly localized in the signal band. As the signal and the noise are separated in different frequency bands, a very good analog to digital conversion performance can be achieved by applying a simple lowpass filtering at the quantizer output. This technique is known as the noise shaping technique [1; 4]. As an oversampling operation for an audio signal can be implemented using an existing hardware [5; 6], a single-bit high-order interpolative sigma delta modulator is widely employed for the analog to digital conversion in the audio devices [7; 8].  
However, the loop filter is required to be unstable (not bounded input bounded output stable) in order to achieve a high signal to noise ratio. Therefore, a single-bit high-order interpolative sigma delta modulator may suffer from the internal instability problem [9; 10]. That is, the state variables of the loop filter may diverge. To address this issue, control is required for the stabilization purpose [7; 11; 12; 13].  
The most common control method is clipping. Here, the values of the output of the loop filter are clipped to a certain value when they exceed a certain value [7]. However, this control method usually results to the occurrence of a limit cycle [14]. This is because it will reach the same set of state vectors every time when the control action takes place. Another common control method is to reset the state vector to a certain vector in its largest invariant set if the largest invariant set of the state vectors exists [11]. However, the determination of the largest invariant set of the state vectors requires a very high computational power [15]. Therefore, the implementation cost of this approach is very expensive.  
To address the drawbacks of the existing clipping control strategy, a sliding mode control-based method was proposed [12; 13]. However, the conventional sliding mode control technique usually requires an amplifier with the gain higher than the saturation level of the quantizer. Equivalently speaking, this flipping strategy is valid only for the input signal with a very small dynamic range.  
To address the drawbacks of the above control strategies, a frequency-domain-based bit flipping control strategy was proposed [1]. Although the theoretical analysis was discussed in detail in [1], the implementation of this control strategy and the detail performance evaluation have not performed. The main contribution of this paper is to propose an implementation to realize this control strategy such that the stability region in terms of the input dynamical range can increase significantly.  
The outline of this paper is as follows. In Section 2, the implementation of the frequency-domain-based bit flipping controller is proposed. In Section 3, the performance evaluations are presented. Finally, a conclusion is drawn in Section 4.