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content/zettels/sensor_noise_estimation.md
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title = "Sensor Noise Estimation"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
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@@ -12,18 +12,17 @@ Tags
Measuring the noise level of inertial sensors is not easy as the seismic motion is usually much larger than the sensor's noise level.
A technique to estimate the sensor noise in such case is proposed in ([Barzilai, VanZandt, and Kenny 1998](#org65ed433)) and well explained in ([Poel 2010](#org02bd600)) (Section 6.1.3).
A technique to estimate the sensor noise in such case is proposed in (<a href="#citeproc_bib_item_1">Barzilai, VanZandt, and Kenny 1998</a>) and well explained in (<a href="#citeproc_bib_item_2">Van der Poel 2010</a>) (Section 6.1.3).
The idea is to mount two inertial sensors closely together such that they should measure the same quantity.
This is represented in Figure [1](#orgbc58a8d) where two identical sensors are measuring the same motion \\(x(t)\\).
This is represented in Figure [1](#figure--fig:huddle-test-setup) where two identical sensors are measuring the same motion \\(x(t)\\).
<a id="orgbc58a8d"></a>
<a id="figure--fig:huddle-test-setup"></a>
{{< figure src="/ox-hugo/huddle_test_setup.png" caption="Figure 1: Schematic representation of the setup for measuring the noise of inertial sensors." >}}
{{< figure src="/ox-hugo/huddle_test_setup.png" caption="<span class=\"figure-number\">Figure 1: </span>Schematic representation of the setup for measuring the noise of inertial sensors." >}}
<div class="definition">
<div></div>
Few quantities that will be used to estimate the sensor noise are now defined.
This include the **Coherence**, the **Power Spectral Density** (PSD) and the **Cross Spectral Density** (CSD).
@@ -35,7 +34,7 @@ where \\(|P\_{x}(\omega)|\\) is the output PSD of signal \\(x(t)\\) and \\(|C\_{
The PSD and CSD are defined as follow:
\begin{align}
|P\_x(\omega)| &= \frac{2}{n\_d T} \sum^{n\_d}\_{n=1} \left| x\_k(\omega, T) \right|^2 \\\\\\
|P\_x(\omega)| &= \frac{2}{n\_d T} \sum^{n\_d}\_{n=1} \left| x\_k(\omega, T) \right|^2 \\\\
|C\_{xy}(\omega)| &= \frac{2}{n\_d T} \sum^{n\_d}\_{n=1} [ x\_k^\*(\omega, T) ] [ y\_k(\omega, T) ]
\end{align}
@@ -76,11 +75,11 @@ Now suppose that:
- sensor noises are modelled as input noises \\(n\_1(t)\\) and \\(n\_2(s)\\)
- sensor noises are uncorrelated and each are uncorrelated with \\(x(t)\\)
Then, the system can be represented by the block diagram in Figure [2](#org1dabfe7), and we can write:
Then, the system can be represented by the block diagram in Figure [2](#figure--fig:huddle-test-block-diagram), and we can write:
\begin{align}
P\_{y\_1y\_1}(\omega) &= |H\_1(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_1}(\omega) ) \\\\\\
P\_{y\_2y\_2}(\omega) &= |H\_2(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_2}(\omega) ) \\\\\\
P\_{y\_1y\_1}(\omega) &= |H\_1(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_1}(\omega) ) \\\\
P\_{y\_2y\_2}(\omega) &= |H\_2(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_2}(\omega) ) \\\\
C\_{y\_1y\_2}(j\omega) &= H\_2^H(j\omega) H\_1(j\omega) P\_{x}(\omega)
\end{align}
@@ -90,21 +89,20 @@ And the CSD between \\(y\_1(t)\\) and \\(y\_2(t)\\) is:
\gamma^2\_{y\_1y\_2}(\omega) = \frac{|C\_{y\_1y\_2}(j\omega)|^2}{P\_{y\_1}(\omega) P\_{y\_2}(\omega)}
\end{equation}
<a id="org1dabfe7"></a>
<a id="figure--fig:huddle-test-block-diagram"></a>
{{< figure src="/ox-hugo/huddle_test_block_diagram.png" caption="Figure 2: Huddle test block diagram" >}}
{{< figure src="/ox-hugo/huddle_test_block_diagram.png" caption="<span class=\"figure-number\">Figure 2: </span>Huddle test block diagram" >}}
Rearranging the equations, we obtain the PSD of \\(n\_1(t)\\) and \\(n\_2(t)\\):
\begin{align}
P\_{n1}(\omega) = \frac{P\_{y\_1}(\omega)}{|H\_1(j\omega)|^2} \left( 1 - \gamma\_{y\_1y\_2}(\omega) \frac{|H\_1(j\omega)|}{|H\_2(j\omega)|} \sqrt{\frac{P\_{y\_2}(\omega)}{P\_{y\_1}(\omega)}} \right) \\\\\\
P\_{n1}(\omega) = \frac{P\_{y\_1}(\omega)}{|H\_1(j\omega)|^2} \left( 1 - \gamma\_{y\_1y\_2}(\omega) \frac{|H\_1(j\omega)|}{|H\_2(j\omega)|} \sqrt{\frac{P\_{y\_2}(\omega)}{P\_{y\_1}(\omega)}} \right) \\\\
P\_{n2}(\omega) = \frac{P\_{y\_2}(\omega)}{|H\_2(j\omega)|^2} \left( 1 - \gamma\_{y\_1y\_2}(\omega) \frac{|H\_2(j\omega)|}{|H\_1(j\omega)|} \sqrt{\frac{P\_{y\_1}(\omega)}{P\_{y\_2}(\omega)}} \right)
\end{align}
If we assume the two sensor dynamics to be the same \\(H\_1(s) \approx H\_2(s)\\) and the PSD of \\(n\_1(t)\\) and \\(n\_2(t)\\) to be the same (\\(P\_{n\_1}(\omega) \approx P\_{n\_2}(\omega)\\)) which is most of the time the case when using two identical sensors, we obtain this approximate equation:
<div class="important">
<div></div>
\begin{equation}
|P\_{n\_1}(\omega)| \approx \frac{P\_{y\_1}}{|H\_1(j\omega)|^2} \big( 1 - \gamma\_{y\_1y\_2}(\omega) \big)
@@ -113,9 +111,9 @@ If we assume the two sensor dynamics to be the same \\(H\_1(s) \approx H\_2(s)\\
</div>
## Bibliography {#bibliography}
<a id="org65ed433"></a>Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” _Review of Scientific Instruments_ 69 (7):2767–72. <https://doi.org/10.1063/1.1149013>.
<a id="org02bd600"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. <https://doi.org/10.3990/1.9789036530163>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” <i>Review of Scientific Instruments</i> 69 (7): 2767–72. doi:<a href="https://doi.org/10.1063/1.1149013">10.1063/1.1149013</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. doi:<a href="https://doi.org/10.3990/1.9789036530163">10.3990/1.9789036530163</a>.</div>
</div>