Closed-Form Relations for Resonance Detection Error Using Statistical Analysis of Amplitude Noise

The optimization of resonance-tracking sensors relies critically on proper estimation of resonance detection error. We study this error in two common resonance detection algorithms: the absolute minimum method and the linear regression method. Closed-form relations for the error originating from additive noise are presented. The formulation accommodates the majority of lineshapes of practical interest and a wide variety of noise statistics. Lorentzian and Fano line shapes are studied here with further detail for their practical importance. It is discussed that while the performance of the absolute minimum method depends on the tail of the noise probability distribution function, for the linear regression method the total noise power is the determining characteristic of the noise. This fact is explained for the specific case of quantization noise. The results of this study enable a quantitative comparison of the performance of resonance-based sensors, which is a center piece in the optimization of their limit of detection.