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 H0(f) and the gain-null condition at fin=kNfs. The numerical integration of the switched-RC equations serves as a consistency check. For a four-path example with Γ=τ/(RC)=0.02, the numerical values of |H0(fs)| and |H0(2fs)| differ from the intuitive limits by less than 0.001 dB. The residual responses at 4fs and 8fs are -49.95 dB and -55.97 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.