My three years of service as the editor-in-chief (EIC) of Signal Processing Magazine ( SPM ) are now coming to a close. During the past three years, many of us were deeply affected by serious political, social, and environmental events such as the war in Ukraine; protests for freedom in Iran; coups d’état in Africa; the COVID-19 pandemic; seisms in Turkey, Syria, and Morocco; huge floods in Libya and India; gigantic fires in North America and Southern Europe; and an avalanche of stones in the Alps, to name a few. In such a context, I believe that the IEEE slogan, “Advancing Technology for Humanity,” is incredibly relevant and timely.

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.

My end of term as IEEE Signal Processing Society (SPS) president is fast approaching. It has been an incredible experience that has provided me with so many opportunities to engage with our members around the globe, forge relationships with other IEEE Societies, and meet a diverse range of people that I hope will become active members of our Society in the future. It has been a great privilege to be at the helm of a Society that garners such a high level of worldwide respect and recognition.

The objectives of IEEE Signal Processing Magazine ( SPM ) are to propose, for any IEEE Signal Processing Society (SPS) member and beyond, a wide range of tutorial articles on both methods and applications in signal and image processing. The articles are divided into different categories: feature articles, column and forum articles, and articles in special issues, the specificities of which are detailed on the SPM webpage “Information for Authors - SPM”.

Deep learning (DL) has been wildly successful in practice, and most of the state-of-the-art machine learning methods are based on neural networks (NNs). Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep NNs (DNNs). In this article, we present a relatively new mathematical framework that provides the beginning of a deeper understanding of DL. This framework precisely characterizes the functional properties of NNs that are trained to fit to data. The key mathematical tools that support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory, which are all techniques deeply rooted in signal processing.