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Designing a perfect filter (i.e., flat passband, sharp transition band, and highly suppressed stopband) is always the goal of digital signal processing practitioners. This goal is reachable if we make no consideration of implementation complexity. In other words, the challenge of designing a high-performance filter is to leverage the distortion tradeoff in the passband, transition band, and stopband. In this article, we introduce a trick for designing a composite filter by cascading the base filter with a notch filter for sharpening its transition band. The stopband of the base filter is then further shaped by complementary comb filters (CCFs), which are simple in complexity. Performing a delicate calculation of all these component filters, the passband of the composite filter is flat, and a near-perfect filter is obtained with reasonable complexity.
The purpose of a notch filter is to remove narrow-band interference and to preserve the intended signal. These filters have wide applications for removing narrow-band interference in communication, image processing, audio processing, and biomedical engineering , , . Ideally, a notch filter has a 3-dB notch bandwidth equal to zero and a unit gain in the passband. The design challenge of notch filters is to approach these design targets. The novelty of this article lies in how to utilize the imperfection of a notch filter (i.e., a nonzero notch band) to sharpen the transition band of a base filter. Then, the stopband of the cascaded filter is easily further suppressed by CCFs, again through the cascade technique. We first introduce the design of a low-pass filter (LPF) and then extend its applicability to filtering high-pass and bandpass signals.
The transfer function of a feasible notch filter suitable for designing a composite filter is