Schematic of the measurement setup, including the resonator and network analyzer, with the corresponding signal and noise inputs marked. At node 1: is the input signal from the network analyzer, and are the noises associated with the input signal and the ambience (e.g., thermomechanical noise), respectively. At node 2: is the resonator output signal and is the intermediate stage noise. The measured output signal is .
(a) The magnitude of the measured signal amplitude compared to the contributing SHO transfer function , the initial least-square curve fit , and root-mean-square noise , (b) illustration of the proposed iterative procedure for step-by-step elimination of the noise effect, with intermediate signals and of the iteration step are marked. In both graphs the signal amplitude is shown in linear scale. In the presented data, the SNR at the resonance frequency, i.e., , is 3.0; also, the extracted -factor based on is 1074, while the actual resonance -factor, which is extracted based on , is 1280.
Flowchart of proposed iterative curve fitting method.
Measured resonance characteristics (magnitude) for different signal powers. The corresponding measurements are performed with minimum sequential time delay to prevent possible variations in the noise power density. (The presented magnitudes are in decibel, with the noise power set as reference.)
Implementation of the proposed iterative fitting method on an actual measured signal . Because of magnitude fluctuations, the partial noise power is calculated by considering the average of squares of the first 10 data points of . In this graph, the SNR is 3.5, and the magnitude is in linear scale.
-factor as a function of the SNR, extracted from the measurement data by using the two conventional methods, i.e., single least-square curve fit (square) and the 3 dB bandwidth methods (circle), and also the proposed iterative fitting method (diamond). No meaningful -factor data were obtained with the 3 dB bandwidth method for SNR smaller than 2, and with the single fit method for SNR smaller than 1.
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