Tutorial Review of N-Path Filters and Their Time-Domain Interpretation

Reconfigurable radio-frequency (RF) front ends employ N-path filters to achieve digitally tunable frequency selectivity, high linearity, and low static power. However, their linear periodically time-varying (LPTV) operation complicates analysis because an input tone is translated to multiple output harmonics. This tutorial review synthesizes the principal methods for analyzing N-path filters, comparing continuous-time window function analysis, discrete-time ordinary differential equation (ODE) modeling, and adjoint network methods. We evaluate and compare their underlying assumptions, outputs, and computational burdens. Additionally, we present an educational time-domain interpretation based on orthogonal sine/cosine excitation. This viewpoint connects capacitor averaging and path-to-path phase cancellation with harmonic transfer functions (HTFs). Rather than replacing rigorous HTF formulations, this interpretation provides a physically intuitive explanation for the fundamental coefficient (Formula presented.) and the gain-null condition at (Formula presented.). The numerical integration of the switched-RC equations serves as a consistency check. For a four-path example with (Formula presented.), the numerical values of (Formula presented.) and (Formula presented.) differ from the intuitive limits by less than (Formula presented.) dB. The residual responses at (Formula presented.) and (Formula presented.) are (Formula presented.) dB and (Formula presented.) dB, respectively. Finally, we extend the orthogonal-excitation relationship to extract higher-order HTFs. This tutorial synthesis clarifies how these established analytical methods relate and guides selection for specific applications.