An Innovative Finite Impulse Response Filter Design Using a Combination of L1/L2 Regularization to Improve Sparsity and Smoothness
Mohamed H. M. Nerma, Abdelrahman Osman Elfaki, Anas Bushnag, Mohammed Hassan Alnemari · Electronics · 2025
This paper presents an innovative method for designing finite impulse response (FIR) filters. The method optimizes the desired frequency response attributes while simultaneously increasing the sparsity of the filter coefficients. Traditional FIR filter design techniques, such as the window method (FirW) and the Parks–McClellan (FirPM) algorithm, excel in meeting precise frequency-domain requirements but often result in dense impulse responses. In scenarios with limited resources, a sparse filter, which has numerous zero or nearly zero coefficients, has advantages such as decreased computational complexity, lower power consumption, and simplified hardware integration. The proposed (L1/L2 regularization) approach defines filter design as an iterative optimization challenge that decreases a composite objective function. This function combines an error term based on the L2-norm to measure deviation from the target frequency response and an L1-norm-based regularization term to encourage coefficient sparsity. By adjusting the regularization parameter λ, users can balance performance in the frequency domain with the level of impulse response sparsity. Extensive simulations reveal that compared with filters designed using FirW and FirPM, this method produces filters with competitive frequency characteristics while achieving significantly higher sparsity. This finding highlights its considerable potential for effective hardware and software implementation. The proposed FIR filter design method presents a compelling alternative to conventional paradigms, particularly for applications where implementation efficiency is a critical design constraint.