Scaling-Free Transformations for Stability-Guaranteed Variable Notch-Frequency Filters

Digital notch filters can be employed in removing a single or multiple unwanted frequency components in digital signals. For example, the signal components to be removed include the interference signals from power lines. Such unwanted frequencies are called notch frequencies, and notch filters play the central role in removing the unwanted signal components. Ideally, the magnitude response of a notch filter at notch-frequency (NF) should be zero. To implement a notch filter with a recursive structure, it is mandatory to guarantee its stability. Otherwise, the notch filter becomes useless. A useful technique is presented in this paper for guaranteeing the stability of a recursive variable NF filter. By employing this stability-guarantee technique, a notch filter can be designed with a recursive structure, and its stability is absolutely ensured. This stability-guarantee technique is derived from the well-known stability-triangle condition, and it utilizes parameter transformations. After showing the stability-guarantee technique, we apply it to the design of a digital notch filter whose NF is variable. This illustrative example not only verifies the guaranteed stability, but also demonstrates the high design accuracy.