Encoding-decoding convolutional neural networks (CNNs) play a central role in data-driven noise reduction and can be found within numerous deep learning algorithms. However, the development of these CNN architectures is often done in an ad hoc fashion and theoretical underpinnings for important design choices are generally lacking. Up to now, there have been different existing relevant works that have striven to explain the internal operation of these CNNs. Still, these ideas are either scattered and/or may require significant expertise to be accessible for a bigger audience.

Designing filters with perfect frequency responses (i.e., flat passbands, sharp transition bands, highly suppressed stopbands, and linear phase responses) is always the ultimate goal of any digital signal processing (DSP) practitioner. High-order finite impulse response (FIR) filters may meet these requirements when we put no constraint on implementation complexity. In contrast to FIR filters, infinite impulse response (IIR) filters, owing to their recursive structures, provide an efficient way for high-performance filtering at reduced complexity.

Analyzing the magnitude response of a finite-length sequence is a ubiquitous task in signal processing. However, the discrete Fourier transform (DFT) provides only discrete sampling points of the response characteristic. This work introduces bounds on the magnitude response, which can be efficiently computed without additional zero padding. The proposed bounds can be used for more informative visualization and inform whether additional frequency resolution or zero padding is required.

A computational experiment is deemed reproducible if the same data and methods are available to replicate quantitative results by any independent researcher, anywhere and at any time, granted that they have the required computing power. Such computational reproducibility is a growing challenge that has been extensively studied among computational researchers as well as within the signal processing and machine learning research community.

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. 

Discrete-time rational transfer functions are often converted to parallel second-order sections due to better numerical performance compared to direct form infinite impulse response (IIR) implementations. This is usually done by performing partial fraction expansion over the original transfer function.