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# 物理代写|传感器代写Sensor代考|PHYS213 Deviations from Ideality: Errors

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## 物理代写|传感器代写Sensor代考|Deviations from Ideality: Errors

The deterministic model is advantageous when dealing with transduction chains, gains, and ranges. However, for a more significant degree of knowledge, this is not sufficient: real sensors (as anything of the real world) are affected by unpredictable errors/fluctuations arising from the physical nature of the environment and the interface.
We could roughly classify errors in two main categories:

Random errors: Unpredictable deviations of the output due to stochastic temporal variations. For stationary input and constant conditions of the sensing system, repeated readouts provide different results at any sample. They are realizations of random physical processes and are also referred to as noise. ${ }^8$

Systematic errors: Constant deviations of the sensor response from the ideal characteristic. For stationary input and constant conditions of the sensing system, repeated readouts provide the same error. This is given, for example, by nonlinear deviations of the real characteristic from the ideal one or by the output variation induced by influence parameters. Even if the error is fixed for repeated conditions, it should be pointed out that it has a degree of unpredictability so that stochastic models should describe it.

mode) in a real sensing system, it will be mapped into the output with some error added: $\Delta y_S+\Delta y_E$, as shown in Fig. 2.25A. In this case, the error could be precisely known and, by acquiring a large amount of data, we can characterize the error model using, for example, the associated probability distributions.

Conversely, if we know the error model and only observe the readout value (operating or prediction mode), we do not know the single error (outcome) entity even if we have characterized the error model. Therefore, since the errors have e stochastic behavior, the input could no longer be determined but estimated or predicted, as shown in Fig. 2.25B, under some degree of uncertainty.

## 物理代写|传感器代写Sensor代考|The Input−Output Duality of a Single Error

Following the previous discussion, the output variation $\Delta y$ of the readout could be dependent on both a signal $\Delta y_S$ or error $\Delta y_E$ (either systematic or random) of the system
$$\Delta y=\Delta y_S+\Delta y_E .$$
Owing to the linear behavior of the system, we can write
$$\Delta y=S \cdot \Delta x=S \cdot\left(\Delta x_S+\Delta x_E\right)$$
so that
$$\Delta x=\Delta x_S+\Delta x_E=\frac{\Delta y_S}{S}+\frac{\Delta y_E}{S},$$

where $\Delta x_E$ is referred to as the input-referred error. Therefore, the error could be modeled as something that is summed up to the input signal to give the same deviation of the output. We will see that if the error is ascribed to noise, it is referred to as input-referred noise (IRN) or equivalent input noise (EIN) or referred to input (RTI) noise.

We can graphically represent this relationship as in Fig. 2.26A, where we can see that the error $\Delta y_E$ (either systematic or random) of a real system is summed to the output signal $\Delta y_s$. Now, we can model the error as a contribution added to the output of an ideal system, as shown in Fig. 2.26B, which is also called output-referred error. Like the signal, we can model the same error by an additional fictitious input-referred contribution $\Delta x_E$ summed to the input of an ideal system in order to get the same output result. In other terms, we can map the output error into an input-referred error by dividing it by the gain of the system. Similarly, a source of errors placed at the input could be mapped to the output by multiplying it to the gain the system.

## 物理代写|传感器代写Sensor代考|The Input-Output Duality of a Single Error

$$\Delta y=\Delta y_S+\Delta y_E .$$

$$\Delta y=S \cdot \Delta x=S \cdot\left(\Delta x_S+\Delta x_E\right)$$

$$\Delta x=\Delta x_S+\Delta x_E=\frac{\Delta y_S}{S}+\frac{\Delta y_E}{S},$$

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