Update Content - 2020-12-11

2020-12-11 16:00:37 +01:00
parent 86eb3b2a13
commit 8f368b7515
11 changed files with 325 additions and 211 deletions
--- a/content/zettels/analog_to_digital_converters.md
+++ b/content/zettels/analog_to_digital_converters.md
@@ -23,9 +23,9 @@ Let's suppose that the ADC is ideal and the only noise comes from the quantizati
 Interestingly, the noise amplitude is uniformly distributed.

 The quantization noise can take a value between \\(\pm q/2\\), and the probability density function is constant in this range (i.e., it’s a uniform distribution).
-Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#org5848c2b)).
+Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#orgf547b74)).

-<a id="org5848c2b"></a>
+<a id="orgf547b74"></a>

 {{< figure src="/ox-hugo/probability_density_function_adc.png" caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}}

@@ -48,7 +48,7 @@ Thus, the two-sided PSD (from \\(\frac{-f\_s}{2}\\) to \\(\frac{f\_s}{2}\\)), we
  \int\_{-f\_s/2}^{f\_s/2} \Gamma(f) d f = f\_s \Gamma = \frac{q^2}{12}
 \end{equation}

-<div class="bred">
+<div class="important">
  <div></div>

 Finally, the Power Spectral Density of the quantization noise of an ADC is equal to:
@@ -62,7 +62,7 @@ Finally, the Power Spectral Density of the quantization noise of an ADC is equal

 </div>

-<div class="bgreen">
+<div class="exampl">
  <div></div>

 Let's take a 18bits ADC with a range of +/-10V and a sample frequency of 10kHz.
--- a/content/zettels/norms.md
+++ b/content/zettels/norms.md
@@ -4,23 +4,21 @@ author = ["Thomas Dehaeze"]
 draft = false
 +++

-Backlinks:
-
-   [Multivariable Control]({{< relref "multivariable_control" >}})
-
 Tags
 :

+\\[ \SI{1}{\meter\per\second} \\]
+
 Resources:

-   ([Skogestad and Postlethwaite 2007](#org44811fa))
-   ([Toivonen 2002](#orgfbd38d8))
-   ([Zhang 2011](#orgc3b14cc))
+-   ([Skogestad and Postlethwaite 2007](#org4fdbcff))
+-   ([Toivonen 2002](#org4782daf))
+-   ([Zhang 2011](#org9b9c22a))


 ## Definition {#definition}

-<div class="bblue">
+<div class="definition">
  <div></div>

 A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real number, denoted \\(\\|e\\|\\), that satisfies the following properties:
@@ -47,7 +45,7 @@ A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real nu

 ## Matrix Norms {#matrix-norms}

-<div class="bgreen">
+<div class="definition">
  <div></div>

 A norm on a matrix \\(\\|A\\|\\) is a matrix norm if, in addition to the four norm properties, it also satisfies the multiplicative property:
@@ -141,7 +139,7 @@ We now consider which system norms result from the definition of input classes a

 ### \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty--norm}

-<div class="bgreen">
+<div class="exampl">
  <div></div>

 Consider a proper linear stable system \\(G(s)\\).
@@ -159,7 +157,7 @@ In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as fo

 ### \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2--norm}

-<div class="bgreen">
+<div class="exampl">
  <div></div>

 Consider a strictly proper system \\(G(s)\\).
@@ -178,17 +176,17 @@ In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as fo

 The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}).

-As explained in ([Monkhorst 2004](#orgc4a9d92)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
+As explained in ([Monkhorst 2004](#orgb605c51)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:

 > The squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.


 ## Bibliography {#bibliography}

-<a id="orgc4a9d92"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
+<a id="orgb605c51"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.

-<a id="org44811fa"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
+<a id="org4fdbcff"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.

-<a id="orgfbd38d8"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
+<a id="org4782daf"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.

-<a id="orgc3b14cc"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
+<a id="org9b9c22a"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
--- a/content/zettels/sensor_noise_estimation.md
+++ b/content/zettels/sensor_noise_estimation.md
@@ -0,0 +1,120 @@
+++
+title = "Sensor Noise Estimation"
+author = ["Thomas Dehaeze"]
+draft = false
+++
+
+Tags
+:
+
+
+## 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>.
--- a/content/zettels/signal_to_noise_ratio.md
+++ b/content/zettels/signal_to_noise_ratio.md
@@ -10,7 +10,7 @@ Tags

 ## SNR to Noise PSD {#snr-to-noise-psd}

-From ([Jabben 2007](#org4650879)) (Section 3.3.2):
+From ([Jabben 2007](#orgf2f4e47)) (Section 3.3.2):

 > Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
 > In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
@@ -22,7 +22,7 @@ From ([Jabben 2007](#org4650879)) (Section 3.3.2):
 > \\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
 > with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.

-<div class="bgreen">
+<div class="exampl">
  <div></div>

 Let's take an example.
@@ -49,7 +49,7 @@ where \\(S\_{snr}\\) is the SNR in dB and \\(S\_\text{rms}\\) is the RMS value o
 If the full range is \\(\Delta V\\), then:
 \\[ S\_\text{rms} = \frac{\Delta V/2}{\sqrt{2}} \\]

-<div class="bgreen">
+<div class="exampl">
  <div></div>

 As an example, let's take a voltage amplifier with a full range of \\(\Delta V = 20V\\) and a SNR of 85dB.
@@ -66,7 +66,7 @@ The RMS value of the noise is then:
 If the wanted full range and RMS value of the noise are defined, the required SNR can be computed from:
 \\[ S\_{snr} = 20 \log \frac{\text{Signal, rms}}{\text{Noise, rms}} \\]

-<div class="bgreen">
+<div class="exampl">
  <div></div>

 Let's say the wanted noise is \\(1 mV, \text{rms}\\) for a full range of \\(20 V\\), the corresponding SNR is:
@@ -78,13 +78,13 @@ Let's say the wanted noise is \\(1 mV, \text{rms}\\) for a full range of \\(20 V

 ## Noise Density to RMS noise {#noise-density-to-rms-noise}

-From ([Fleming 2010](#orgf1518db)):
+From ([Fleming 2010](#orgf17a758)):
 \\[ \text{RMS noise} = \sqrt{2 \times \text{bandwidth}} \times \text{noise density} \\]

 If the noise is normally distributed, the RMS value is also the standard deviation \\(\sigma\\).
 The peak to peak amplitude is then approximately \\(6 \sigma\\).

-<div class="bgreen">
+<div class="exampl">
  <div></div>

 -   noise density = \\(20 pm/\sqrt{Hz}\\)
@@ -98,6 +98,6 @@ The peak-to-peak noise will be approximately \\(6 \sigma = 1.7 nm\\)

 ## Bibliography {#bibliography}

-<a id="orgf1518db"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
+<a id="orgf17a758"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.

-<a id="org4650879"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.
+<a id="orgf2f4e47"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.