42 J.A. Bossert et al. Fig. 5.4 The mean and lowest percentage error data suggest a linear relationship between perturbation and error, while the highest percentage error data are still generally under 14 % for all simulations. This indicated that CS is not bit sensitive and capable of handling environmental variations The expected outcomes of the Yall1 algorithm are either a pass or a fail, but occasionally the Yall1 algorithm fails to process the compressed sample in any meaningful way, and a null recovery (a reconstructed signal of zero) is returned [12]. To quantify the rate at which the Yall1 algorithm returns a null recovery the null rate for all simulations and experimental tests were recorded and analyzed. Throughout all simulations, the null recovery rate was found to be 30 %, with little variation due to different perturbations in resistance values as described below. The sensitivity of GO and rGO to moisture, temperature, and solvents are one of the primary reasons it was chosen as a material to create the tamper evident seal [7, 9, 10]. However, for the seal to be usable without returning false-positives, the Yall1 algorithm needs to be able to reconstruct the original signal with low error, even with changes in environmental conditions. Under the assumption that changes in temperature, humidity, and of trace gasses and salts in the air affect all resistors on the seal equally, simulations were run where all resistances changed by an equivalent percentage. In total, 135 measurement matrices were tested with 57 perturbations ranging from 10 % to 10 % being applied to all resistors equally. The results are presented in Fig. 5.4, and suggest that there is a linear trend between the size of the perturbation and the error introduced into the reconstruction. This indicates that CS is robust with regards to homogeneous changes in resistance, and will not trigger false positives for minor environmental variations. The effects of measurement error were analyzed by applying a different, stochastic, perturbation to each resistor to emulate the random error introduced into the system by the DAC and the ADC. Again, 135 matrices were analyzed, each being perturbed 57 times. Each perturbation altered every element of the measurement matrix by a percentage randomly chosen from a normal distribution centered at zero, with a standard deviation of 2 %. This value for the standard deviation was chosen to fit well with the error observed in a closed DAC to ADC loop. The closed DAC to ADC loop connects the input of the ADC directly to the output of the DAC, allowing the creation of a mapping between the DAC output and ADC input. The results indicate that for the vast majority of the measurement matrices analyzed, small random perturbations have very little effect on the reconstruction error. Under the assumption that mechanical failure would introduce a large change in mechanical and electrical properties to one or more resistors [11], we simulated mechanical damage by introducing a large perturbation on a single resistor at a time. Eighty-five matrices were analyzed, and each of the eight resistors were perturbed by 75%, 50%,0%,50%,75%, and finally by setting the resistance equal to infinity. The results, presented in Fig. 5.5, show that for small changes, CS is relatively robust, and the reconstruction error increases slightly. However, for large changes such as ˙75 % the reconstruction error becomes much larger. Finally, by setting a resistance equal to infinity and opening the circuit, the reconstruction error is often above 100 %, which correctly indicates a seriously damaged seal. To ensure a correct identification of a damaged seal, only measurement matrices that exclude the possibility of false negatives (i.e. reconstruction error below 100 % when the seal is damaged) must be used to effectively run the seal.
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