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Thomas Dehaeze
2021-05-02 20:37:00 +02:00
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content/zettels/sensor_noise_estimation.md
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@@ -12,13 +12,13 @@ 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](#org7fe766e)) and well explained in ([Poel 2010](#org964c18e)) (Section 6.1.3).
A technique to estimate the sensor noise in such case is proposed in ([Barzilai, VanZandt, and Kenny 1998](#org4702c9a)) and well explained in ([Poel 2010](#orgeaef46f)) (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](#org53e9426) where two identical sensors are measuring the same motion \\(x(t)\\).
This is represented in Figure [1](#org030f5c0) where two identical sensors are measuring the same motion \\(x(t)\\).
<a id="org53e9426"></a>
<a id="org030f5c0"></a>
{{< figure src="/ox-hugo/huddle_test_setup.png" caption="Figure 1: Schematic representation of the setup for measuring the noise of inertial sensors." >}}
@@ -49,23 +49,23 @@ where:
The Matlab function `mscohere` can be used to compute the coherence:
```matlab
%% Parameters
Fs = 1e4; % Sampling Frequency [Hz]
win = hanning(ceil(10*Fs)); % 10 seconds Hanning Windows
%% Parameters
Fs = 1e4; % Sampling Frequency [Hz]
win = hanning(ceil(10*Fs)); % 10 seconds Hanning Windows
%% Coherence between x and y
[pxy, f] = mscohere(x, y, win, [], [], Fs); % Coherence, frequency vector in [Hz]
%% Coherence between x and y
[pxy, f] = mscohere(x, y, win, [], [], Fs); % Coherence, frequency vector in [Hz]
```
Alternatively, it can be manually computed using the `cpsd` and `pwelch` commands:
```matlab
%% Manual Computation of the Coherence
[pxy, f] = cpsd(x, y, win, [], [], Fs); % Cross Spectral Density between x and y
[pxx, ~] = pwelch(x, win, [], [], Fs); % Power Spectral Density of x
[pyy, ~] = pwelch(y, win, [], [], Fs); % Power Spectral Density of y
%% Manual Computation of the Coherence
[pxy, f] = cpsd(x, y, win, [], [], Fs); % Cross Spectral Density between x and y
[pxx, ~] = pwelch(x, win, [], [], Fs); % Power Spectral Density of x
[pyy, ~] = pwelch(y, win, [], [], Fs); % Power Spectral Density of y
pxy_manual = abs(pxy).^2./abs(pxx)./abs(pyy);
pxy_manual = abs(pxy).^2./abs(pxx)./abs(pyy);
```
</div>
@@ -76,7 +76,7 @@ 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](#org0e1cf4a), and we can write:
Then, the system can be represented by the block diagram in Figure [2](#orgec7c79b), and we can write:
\begin{align}
P\_{y\_1y\_1}(\omega) &= |H\_1(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_1}(\omega) ) \\\\\\
@@ -90,7 +90,7 @@ 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="org0e1cf4a"></a>
<a id="orgec7c79b"></a>
{{< figure src="/ox-hugo/huddle_test_block_diagram.png" caption="Figure 2: Huddle test block diagram" >}}
@@ -113,8 +113,9 @@ If we assume the two sensor dynamics to be the same \\(H\_1(s) \approx H\_2(s)\\
</div>
## Bibliography {#bibliography}
<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>.
<a id="org4702c9a"></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="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>.
<a id="orgeaef46f"></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>.