A Unified Framework for Group Delay Flattening: Stable and Hardware-Efficient All-Pass Networks

This paper presents a unified framework for group delay (GD) equalization and phase distortion compensation using cascaded all-pass filter (APF) networks. The proposed approach addresses the critical limitation of finite impulse response (FIR) equalizers, which require prohibitively high orders for adequate wideband and multiband performance. We consider GD flattening as a stability-constrained optimization problem, adjusting APF poles to minimize root mean square and maximum GD deviations over prescribed frequency bands. A hybrid optimization strategy ensures robust convergence to high-quality solutions. The framework is extensively benchmarked against FIR and spline-based equalizers, demonstrating superior GD flatness and lower phase distortion at a fraction of the computational complexity. Results across single-band, multiband, and wideband scenarios show that the APF equalizers reduce GD error by orders of magnitude and remain robust under noise and coefficient quantization. Furthermore, we introduce a differentiable extension of the APF cascade, trainable with the Adam optimizer. This differentiable Adam-based extension demonstrates faster adaptations. At relaxed error levels, both classical re-optimization and Adam-based fine-tuning converge within a small number of iterations. Hardware feasibility is confirmed through FPGA synthesis and CMOS-level analysis, highlighting low latency and resource efficiency. The proposed solution establishes a practical, high-performance path for deployable phase equalization in modern communication, radar, and biomedical systems.