digital-brain/content/zettels/sensor_noise_estimation.md

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title = "Sensor Noise Estimation"
author = ["Thomas Dehaeze"]
draft = false
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## Estimation of the Noise of Inertial Sensors {#estimation-of-the-noise-of-inertial-sensors}

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).

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)\\).

<a id="org53e9426"></a>

{{< figure src="/ox-hugo/huddle_test_setup.png" caption="Figure 1: 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).

The coherence between signals \\(x\\) and \\(y\\) is defined as follow
\\[ \gamma^2\_{xy}(\omega) = \frac{|C\_{xy}(\omega)|^2}{|P\_{x}(\omega)| |P\_{y}(\omega)|} \\]
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)\\).

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 \\\\\\
  |C\_{xy}(\omega)| &= \frac{2}{n\_d T} \sum^{n\_d}\_{n=1} [ x\_k^\*(\omega, T) ] [ y\_k(\omega, T) ]
\end{align}

where:

-   \\(n\_d\\) is the number for records averaged
-   \\(T\\) is the length of each record
-   \\(x\_k(\omega, T)\\) is the finite Fourier transform of the kth record
-   \\(x\_k^\*(\omega, T)\\) is its complex conjugate

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

%% 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

pxy_manual = abs(pxy).^2./abs(pxx)./abs(pyy);
```

</div>

Now suppose that:

-   both sensors are modelled as LTI systems \\(H\_1(s)\\) and \\(H\_2(s)\\)
-   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:

\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) ) \\\\\\
  C\_{y\_1y\_2}(j\omega) &= H\_2^H(j\omega) H\_1(j\omega) P\_{x}(\omega)
\end{align}

And the CSD between \\(y\_1(t)\\) and \\(y\_2(t)\\) is:

\begin{equation}
  \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>

{{< figure src="/ox-hugo/huddle_test_block_diagram.png" caption="Figure 2: 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\_{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)
\end{equation}

</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="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>.