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Markdown
121 lines
5.0 KiB
Markdown
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title = "Sensor Noise Estimation"
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author = ["Thomas Dehaeze"]
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draft = false
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## Estimation of the Noise of Inertial Sensors {#estimation-of-the-noise-of-inertial-sensors}
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Measuring the noise level of inertial sensors is not easy as the seismic motion is usually much larger than the sensor's noise level.
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A technique to estimate the sensor noise in such case is proposed in ([Barzilai, VanZandt, and Kenny 1998](#org7fe766e)) and well explained in ([Poel 2010](#org964c18e)) (Section 6.1.3).
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The idea is to mount two inertial sensors closely together such that they should measure the same quantity.
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This is represented in Figure [1](#org53e9426) where two identical sensors are measuring the same motion \\(x(t)\\).
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<a id="org53e9426"></a>
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{{< figure src="/ox-hugo/huddle_test_setup.png" caption="Figure 1: Schematic representation of the setup for measuring the noise of inertial sensors." >}}
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<div class="definition">
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<div></div>
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Few quantities that will be used to estimate the sensor noise are now defined.
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This include the **Coherence**, the **Power Spectral Density** (PSD) and the **Cross Spectral Density** (CSD).
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The coherence between signals \\(x\\) and \\(y\\) is defined as follow
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\\[ \gamma^2\_{xy}(\omega) = \frac{|C\_{xy}(\omega)|^2}{|P\_{x}(\omega)| |P\_{y}(\omega)|} \\]
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where \\(|P\_{x}(\omega)|\\) is the output PSD of signal \\(x(t)\\) and \\(|C\_{xy}(\omega)|\\) is the CSD of signals \\(x(t)\\) and \\(y(t)\\).
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The PSD and CSD are defined as follow:
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\begin{align}
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|P\_x(\omega)| &= \frac{2}{n\_d T} \sum^{n\_d}\_{n=1} \left| x\_k(\omega, T) \right|^2 \\\\\\
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|C\_{xy}(\omega)| &= \frac{2}{n\_d T} \sum^{n\_d}\_{n=1} [ x\_k^\*(\omega, T) ] [ y\_k(\omega, T) ]
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\end{align}
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where:
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- \\(n\_d\\) is the number for records averaged
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- \\(T\\) is the length of each record
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- \\(x\_k(\omega, T)\\) is the finite Fourier transform of the kth record
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- \\(x\_k^\*(\omega, T)\\) is its complex conjugate
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The Matlab function `mscohere` can be used to compute the coherence:
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```matlab
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%% Parameters
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Fs = 1e4; % Sampling Frequency [Hz]
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win = hanning(ceil(10*Fs)); % 10 seconds Hanning Windows
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%% Coherence between x and y
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[pxy, f] = mscohere(x, y, win, [], [], Fs); % Coherence, frequency vector in [Hz]
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```
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Alternatively, it can be manually computed using the `cpsd` and `pwelch` commands:
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```matlab
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%% Manual Computation of the Coherence
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[pxy, f] = cpsd(x, y, win, [], [], Fs); % Cross Spectral Density between x and y
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[pxx, ~] = pwelch(x, win, [], [], Fs); % Power Spectral Density of x
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[pyy, ~] = pwelch(y, win, [], [], Fs); % Power Spectral Density of y
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pxy_manual = abs(pxy).^2./abs(pxx)./abs(pyy);
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```
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</div>
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Now suppose that:
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- both sensors are modelled as LTI systems \\(H\_1(s)\\) and \\(H\_2(s)\\)
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- sensor noises are modelled as input noises \\(n\_1(t)\\) and \\(n\_2(s)\\)
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- sensor noises are uncorrelated and each are uncorrelated with \\(x(t)\\)
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Then, the system can be represented by the block diagram in Figure [2](#org0e1cf4a), and we can write:
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\begin{align}
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P\_{y\_1y\_1}(\omega) &= |H\_1(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_1}(\omega) ) \\\\\\
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P\_{y\_2y\_2}(\omega) &= |H\_2(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_2}(\omega) ) \\\\\\
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C\_{y\_1y\_2}(j\omega) &= H\_2^H(j\omega) H\_1(j\omega) P\_{x}(\omega)
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\end{align}
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And the CSD between \\(y\_1(t)\\) and \\(y\_2(t)\\) is:
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\begin{equation}
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\gamma^2\_{y\_1y\_2}(\omega) = \frac{|C\_{y\_1y\_2}(j\omega)|^2}{P\_{y\_1}(\omega) P\_{y\_2}(\omega)}
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\end{equation}
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<a id="org0e1cf4a"></a>
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{{< figure src="/ox-hugo/huddle_test_block_diagram.png" caption="Figure 2: Huddle test block diagram" >}}
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Rearranging the equations, we obtain the PSD of \\(n\_1(t)\\) and \\(n\_2(t)\\):
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\begin{align}
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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) \\\\\\
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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)
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\end{align}
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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:
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<div class="important">
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<div></div>
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\begin{equation}
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|P\_{n\_1}(\omega)| \approx \frac{P\_{y\_1}}{|H\_1(j\omega)|^2} \big( 1 - \gamma\_{y\_1y\_2}(\omega) \big)
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\end{equation}
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</div>
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## Bibliography {#bibliography}
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<a id="org7fe766e"></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>.
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<a id="org964c18e"></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>.
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