A Trick for Designing Composite Filters With Sharp Transition Bands and Highly Suppressed Stopbands

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 [1], [2], [3]. 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 [1]

where ω∗ is the notch frequency. The frequency response HN(ω) is obtained by replacing z by ω through the relationship z=ejω; α and ρ control the frequency response of the notch filter. For stability, we have to set 0≤ρ<1 and α≥0. An example of the frequency magnitude response for the case α=0 is plotted in Figure 1. The frequency magnitude response becomes sharper when α increases [1]. We can see that the frequency magnitude responses also change with respect to the choice of ρ. The choices of α and ρ thus provide degrees of freedom in shaping the transition band of a base filter.