Highly accurate and real-time determination of resonant characteristics: complex linear regression of the transmission coefficient

For measuring high-frequency properties of materials by resonant techniques, one needs to obtain the resonant characteristics of the resonator, i.e., the resonant frequency f_R and the bandwidth of the resonance f_B. For precise measurements, accurate and real-time determination of the resonant characteristics is required. We present a complex linear regression method, which contains only the linear square-fit algorithm, and is applicable to the complex transmission coefficient data S₂₁(f) with arbitrary phase error. Numerical simulations and experimental data show very high accuracy and stability of this technique, when compared with the standard 3-dB method and the Lorentzian-fit method. The optimum frequency span of the measurement is found to be approximately 1.5f_B, where f_B is the bandwidth of the resonance.