-
-
- 1. Optimal Sensor Fusion for noise characteristics +
- 1. Optimal Sensor Fusion for noise characteristics
-
-
- 1.1. Architecture -
- 1.2. Noise of the sensors -
- 1.3. H-Two Synthesis -
- 1.4. Analysis -
- 1.5. H-Infinity Synthesis - +
- 2. Robustness to sensor dynamics uncertainty + -
- 2. Robustness to sensor dynamics uncertainty +
- 3. Equivalent Super Sensor
-
-
- 2.1. Unknown sensor dynamics dynamics -
- 2.2. Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor -
- 2.3. First Basic Example with gain mismatch -
- 2.4. TODO More Complete example with model uncertainty +
- 3.1. Sensor Fusion Architecture +
- 3.2. Equivalent Configuration +
- 3.3. Model the uncertainty of the super sensor +
- 3.4. Model the noise of the super sensor +
- 3.5. First guess
- - 3. Equivalent Super Sensor +
- 4. Optimal And Robust Sensor Fusion in Practice
-
-
- 3.1. Sensor Fusion Architecture -
- 3.2. Equivalent Configuration -
- 3.3. Model the uncertainty of the super sensor -
- 3.4. Model the noise of the super sensor -
- 3.5. First guess +
- 4.1. Measurement of the noise characteristics of the sensors + -
- 4. Optimal And Robust Sensor Fusion in Practice
-
-
-
- 4.1. Measurement of the noise characteristics of the sensors
-
-
-
- 4.1.1. Huddle Test -
- 4.1.2. Weights that represents the noises' PSD -
- 4.1.3. Comparison of the noises' PSD -
- 4.1.4. Computation of the coherence, power spectral density and cross spectral density of signals +
- 4.2. Estimate the dynamic uncertainty of the sensors +
- 4.3. Optimal and Robust synthesis of the complementary filters
- - 4.2. Estimate the dynamic uncertainty of the sensors -
- 4.3. Optimal and Robust synthesis of the complementary filters +
- 5. Methods of complementary filter synthesis
+
-
+
- 5.1. Complementary filters using analytical formula + -
- 5. Methods of complementary filter synthesis +
- 5.2. H-Infinity synthesis of complementary filters
-
-
- 5.1. Complementary filters using analytical formula - -
- 5.2. H-Infinity synthesis of complementary filters +
- 5.3. Feedback Control Architecture to generate Complementary Filters -
- 5.3. Feedback Control Architecture to generate Complementary Filters +
- 5.4. Analytical Formula found in the literature -
- 5.4. Analytical Formula found in the literature - - -
- 5.5. Comparison of the different methods of synthesis +
- 5.5. Comparison of the different methods of synthesis
-
-
- in section 1, the optimal design of the complementary filters in order to obtain the lowest resulting "super sensor" noise is studied +
- in section 1, the optimal design of the complementary filters in order to obtain the lowest resulting "super sensor" noise is studied
When blending two sensors using complementary filters with unknown dynamics, phase lag may be introduced that renders the close-loop system unstable.
-
-
- in section 2, the blending robustness to sensor dynamic uncertainty is studied. +
- in section 2, the blending robustness to sensor dynamic uncertainty is studied.
Then, three design methods for generating two complementary filters are proposed:
-
-
- in section 5.1, analytical formulas are proposed -
- in section 5.2, the \(\mathcal{H}_\infty\) synthesis is used -
- in section 5.3, the classical feedback architecture is used -
- in section 5.4, analytical formulas found in the literature are listed +
- in section 5.1, analytical formulas are proposed +
- in section 5.2, the \(\mathcal{H}_\infty\) synthesis is used +
- in section 5.3, the classical feedback architecture is used +
- in section 5.4, analytical formulas found in the literature are listed
-1 Optimal Sensor Fusion for noise characteristics
++1 Optimal Sensor Fusion for noise characteristics
The idea is to combine sensors that works in different frequency range using complementary filters. @@ -423,102 +419,122 @@ All the files (data and Matlab scripts) are accessible -
1.1 Architecture
++1.1 Architecture
-Let's consider the sensor fusion architecture shown on figure 1 where two sensors 1 and 2 are measuring the same quantity \(x\) with different noise characteristics determined by \(W_1\) and \(W_2\). +Let's consider the sensor fusion architecture shown on figure 1 where two sensors (sensor 1 and sensor 2) are measuring the same quantity \(x\) with different noise characteristics determined by \(N_1(s)\) and \(N_2(s)\).
-\(n_1\) and \(n_2\) are white noise (constant power spectral density over all frequencies). +\(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise:
+\begin{equation} +\label{org31f32a3} + \Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_1}(\omega) = 1 +\end{equation} ---
++
Figure 1: Fusion of two sensors
-We consider that the two sensor dynamics \(G_1\) and \(G_2\) are ideal (\(G_1 = G_2 = 1\)). We obtain the architecture of figure 2. +We consider that the two sensor dynamics \(G_1(s)\) and \(G_2(s)\) are ideal: +
+\begin{equation} +\label{org10f3718} + G_1(s) = G_2(s) = 1 +\end{equation} + ++We obtain the architecture of figure 2.
--
++
Figure 2: Fusion of two sensors with ideal dynamics
-\(H_1\) and \(H_2\) are complementary filters (\(H_1 + H_2 = 1\)). The goal is to design \(H_1\) and \(H_2\) such that the effect of the noise sources \(n_1\) and \(n_2\) has the smallest possible effect on the estimation \(\hat{x}\). +\(H_1(s)\) and \(H_2(s)\) are complementary filters: +
+\begin{equation} +\label{orgf0f9605} + H_1(s) + H_2(s) = 1 +\end{equation} + ++The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the estimation \(\hat{x}\).
We have that the Power Spectral Density (PSD) of \(\hat{x}\) is: -\[ \Gamma_{\hat{x}} = |H_1 W_1|^2 \Gamma_{n_1} + |H_2 W_2|^2 \Gamma_{n_2} \] +\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \]
And the goal is the minimize the Root Mean Square (RMS) value of \(\hat{x}\): -\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Gamma_{\hat{x}}(\omega) d\omega} \] -
- --As \(n_1\) and \(n_2\) are white noise: \(\Gamma_{n_1} = \Gamma_{n_2} = 1\) and we have: -\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 W_1|^2(\omega) + |H_2 W_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2 \] -
- --Thus, the goal is to design \(H_1\) and \(H_2\) such that \(H_1 + H_2 = 1\) and such that \(\left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2\) is minimized. -
- --For that, we will use the \(\mathcal{H}_2\) Synthesis.
+\begin{equation} +\label{orge1ffe08} + \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega} +\end{equation}-1.2 Noise of the sensors
++1.2 Noise of the sensors
--Let's define the noise characteristics of the two sensors by choosing \(W_1\) and \(W_2\): +Let's define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\):
-
-
- Sensor 1 characterized by \(W_1\) has low noise at low frequency (for instance a geophone) -
- Sensor 2 characterized by \(W_2\) has low noise at high frequency (for instance an accelerometer) +
- Sensor 1 characterized by \(N_1(s)\) has low noise at low frequency (for instance a geophone) +
- Sensor 2 characterized by \(N_2(s)\) has low noise at high frequency (for instance an accelerometer)
-omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4; -W1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000); +N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100); omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8; -W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; +N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
-1.3 H-Two Synthesis
++1.3 H-Two Synthesis
--We use the generalized plant architecture shown on figure 4. +As \(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise: \(\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1\) and we have: +\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \] +Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized. +
+ ++For that, we use the \(\mathcal{H}_2\) Synthesis. +
+ ++We use the generalized plant architecture shown on figure 4.
-+-
Figure 4: \(\mathcal{H}_2\) Synthesis - Generalized plant used for the optimal generation of complementary filters
@@ -526,9 +542,9 @@ We use the generalized plant architecture shown on figureThe transfer function from \([n_1, n_2]\) to \(\hat{x}\) is: -\[ \begin{bmatrix} W_1 H_1 \\ W_2 (1 - H_1) \end{bmatrix} \] +\[ \begin{bmatrix} N_1 H_1 \\ N_2 (1 - H_1) \end{bmatrix} \] If we define \(H_2 = 1 - H_1\), we obtain: -\[ \begin{bmatrix} W_1 H_1 \\ W_2 H_2 \end{bmatrix} \] +\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \]
@@ -536,11 +552,11 @@ Thus, if we minimize the \(\mathcal{H}_2\) norm of this transfer function, we mi
-We define the generalized plant \(P\) on matlab as shown on figure 4. +We define the generalized plant \(P\) on matlab as shown on figure 4.
-@@ -553,83 +569,69 @@ And we do the \(\mathcal{H}_2\) synthesis using theP = [0 W2 1; - W1 -W2 0]; +
P = [0 N2 1; + N1 -N2 0];
h2syncommand.-What is minimized is
- -norm([W1*H1,W2*H2], 2). --Finally, we define \(H_2 = 1 - H_1\). +Finally, we define \(H_2(s) = 1 - H_1(s)\).
-H2 = 1 - H1;
-1.4 Analysis
--The complementary filters obtained are shown on figure 5. The PSD of the 6. -Finally, the RMS value of \(\hat{x}\) is shown on table 1. -The optimal sensor fusion has permitted to reduced the RMS value of the estimation error by a factor 8 compare to when using only one sensor. +The complementary filters obtained are shown on figure 5. +
+ ++The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. 6. +
+ ++The Cumulative Power Spectrum (CPS) is shown on Fig. 7. +
+ ++The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors.
-+ +-+-PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2; +PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2; +PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2; +
++ +-
Figure 6: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal (png, pdf)
-
+Table 1: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters ++-CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1); +CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2); +CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2); +
+- - -- - - - - -rms value -- - -Sensor 1 -1.1e-02 -- - -Sensor 2 -1.3e-03 -- - -Optimal Sensor Fusion -1.5e-04 --1.5 H-Infinity Synthesis
----1.5.1 First idea
-+-+1.4 H-Infinity Synthesis - method A
+Another objective that we may have is that the noise of the super sensor \(n_{SS}\) is following the minimum of the noise of the two sensors \(n_1\) and \(n_2\): \[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \] @@ -639,41 +641,40 @@ Another objective that we may have is that the noise of the super sensor \(n_{SS In order to obtain that ideal case, we need that the complementary filters be designed such that:
\begin{align*} - & H_1(j\omega) = 1 \text{ and } H_2(j\omega) = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\ - & H_1(j\omega) = 0 \text{ and } H_2(j\omega) = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega) + & |H_1(j\omega)| = 1 \text{ and } |H_2(j\omega)| = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\ + & |H_1(j\omega)| = 0 \text{ and } |H_2(j\omega)| = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega) \end{align*}-Which is indeed impossible in practice. +Which is indeed impossible in practice.
-We could try to use filters with the highest possible order to approximate that. -However, we may have robustness and implementation problems. +We could try to approach that with the \(\mathcal{H}_\infty\) synthesis by using high order filters. +
+ ++As shown on Fig. 3, the frequency where the two sensors have the same noise level is around 9Hz. +We will thus choose weighting functions such that the merging frequency is around 9Hz. +
+ ++The weighting functions used as well as the obtained complementary filters are shown in Fig. 8.
---omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4; -N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000); - -omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8; -N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; -
-n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2; -W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n; +W1a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n; n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2; -W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n; +W2a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
-@@ -681,12 +682,12 @@ W2 = ( -P = [W1 -W1; - 0 W2; - 1 0]; +
P = [W1a -W1a; + 0 W2a; + 1 0];
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +
[H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +[H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); Resetting value of Gamma min based on D_11, D_12, D_21 terms Test bounds: 0.1000 < gamma <= 10500.0000 @@ -722,115 +723,92 @@ Test bounds: 0.1000 < gamma <= 10500.0000
--H1 = 1 - H2; +
H1a = 1 - H2a;
+ -
+- - -+ +We then compute the Power Spectral Density as well as the Cumulative Power Spectrum. +
-- - - - -rms value -- - -Sensor 1 -1.1e-02 -- - -Sensor 2 -1.3e-03 -- - -Optimal Sensor Fusion -3.2e-04 -+PSD_Ha = abs(squeeze(freqresp(N1*H1a, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2a, freqs, 'Hz'))).^2; +CPS_Ha = 1/pi*cumtrapz(2*pi*freqs, PSD_Ha); +
+-1.5.2 Second idea
-++1.5 H-Infinity Synthesis - method B
+We have that: -\[ \Phi_{n_{ss}} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} \] +\[ \Phi_{\hat{x}}(\omega) = \left|H_1(j\omega) N_1(j\omega)\right|^2 + \left|H_2(j\omega) N_2(j\omega)\right|^2 \]
-We model the Power Spectral Densities of the two sensors by weighting functions \(N_1(s)\) and \(N_2(s)\). -\[ \Phi_{n_{ss}} = \left|H_1 N_1\right|^2 + \left|H_2\right N_2|^2 \] +Then, at frequencies where \(|H_1(j\omega)| < |H_2(j\omega)|\) we would like that \(|N_1(j\omega)| = 1\) and \(|N_2(j\omega)| = 0\) as we discussed before. +Then \(|H_1 N_1|^2 + |H_2 N_2|^2 = |N_1|^2\).
-Then, at frequencies where \(|H_1| < |H_2|\) we would like that \(|N_1| = 1\) and \(|N_2| = 0\) as we discussed before. -Then \(|H_1 N_1|^2 + |H_2 N_2|^2 = |H_1 N_1|^2\). -
- --However, if we design \(H_2\) such that when \(|N_1| < |N_2|\), we have that: +We know that this is impossible in practice. A more realistic choice is to design \(H_2(s)\) such that when \(|N_2(j\omega)| > |N_1(j\omega)|\), we have that: \[ |H_2 N_2|^2 = \epsilon |H_1 N_1|^2 \] +
+ ++Which is equivalent to have (by supposing \(|H_1| \approx 1\)): +\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \] +
+ +And we have:
\begin{align*} - \Phi_{n_{ss}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\ - &= (1 + \epsilon)\left|H_1 N_1\right|^2 \\ - &\approx \left|H_1 N_1\right|^2 + \Phi_{\hat{x}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\ + &= (1 + \epsilon) \left| H_1 N_1 \right|^2 \\ + &\approx \left|N_1\right|^2 \end{align*}-We have then that: -\[ |H_2 N_2| = \sqrt{\epsilon} |H_1 N_1| \] -We suppose that \(|H_1| = 1\): -\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \] -We can use that for the maximum magnitude of \(|H_2|\) at frequencies where \(|N_1| < |N_2|\) -
- --Similarly, we have that the maximum magnitude of \(|H_1|\) at frequencies where \(|N_1| > |N_2|\) is: +Similarly, we design \(H_1(s)\) such that at frequencies where \(|N_1| > |N_2|\): \[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \]
-We could take \(\epsilon = 1\), then we increase the noise of the super sensor just by a factor \(\sqrt{2}\) over the all bandwidth from the idea case. +For instance, is we take \(\epsilon = 1\), then the PSD of \(\hat{x}\) is increased by just by a factor \(\sqrt{2}\) over the all frequencies from the idea case.
-This could be used as weighting functions for the \(\mathcal{H}_\infty\) synthesis of the complementary filters. +We use this as the weighting functions for the \(\mathcal{H}_\infty\) synthesis of the complementary filters. +
+ ++The weighting function and the obtained complementary filters are shown in Fig. 9.
-omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4; -N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000); +
epsilon = 2; -omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8; -N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; +W1b = 1/epsilon*N1/N2; +W2b = 1/epsilon*N2/N1; + +W1b = W1b/(1 + s/2/pi/1000); % this is added so that it is proper
-- -W1 = N1/N2; -W2 = N2/N1; - -W1 = W1/(1 + s/2/pi/1000); % this is added so that it is proper -
--@@ -838,49 +816,77 @@ W1 = W1/ -P = [W1 -W1; - 0 W2; - 1 0]; +
P = [W1b -W1b; + 0 W2b; + 1 0];
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +
[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -Test bounds: 0.0000 < gamma <= 2625.0000 +[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +Test bounds: 0.0000 < gamma <= 32.8125 gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f -2.625e+03 1.8e+01 -2.3e-12 6.3e+00 -1.7e-14 0.0000 p -1.313e+03 1.8e+01 -3.3e-12 6.3e+00 -5.9e-13 0.0000 p - 656.250 1.8e+01 -1.8e-12 6.3e+00 -1.2e-13 0.0000 p - 328.125 1.8e+01 -8.3e-13 6.3e+00 -7.5e-14 0.0000 p - 164.063 1.8e+01 -3.8e-13 6.3e+00 -6.1e-14 0.0000 p - 82.031 1.8e+01 -3.3e-22 6.3e+00 -5.4e-14 0.0000 p - 41.016 1.8e+01 -3.4e-12 6.3e+00 -1.4e-16 0.0000 p - 20.508 1.8e+01 -6.4e-14 6.3e+00 -4.4e-13 0.0000 p - 10.254 1.8e+01 -1.4e-13 6.3e+00 -4.9e-14 0.0000 p - 5.127 1.7e+01 -1.1e-12 6.3e+00 -1.6e-13 0.0000 p - 2.563 1.7e+01 -6.5e-13 6.3e+00 -7.3e-14 0.0000 p - 1.282 1.4e+01 -8.9e+04# 6.3e+00 -1.6e-14 0.0000 f - 1.923 1.6e+01 -4.6e+05# 6.3e+00 -2.1e-13 0.0000 f - 2.243 1.7e+01 -1.9e+06# 6.3e+00 -9.9e-14 0.0000 f - 2.403 1.7e+01 -3.7e-11 6.3e+00 -2.1e-12 0.0000 p - 2.323 1.7e+01 -4.6e+06# 6.3e+00 -5.8e-14 0.0000 f - 2.363 1.7e+01 -1.2e+07# 6.3e+00 -3.3e-14 0.0000 f - 2.383 1.7e+01 -6.8e+07# 6.3e+00 -1.2e-13 0.0000 f - 2.393 1.7e+01 -5.5e-11 6.3e+00 -1.8e-12 0.0000 p - 2.388 1.7e+01 1.3e-22 6.3e+00 -2.5e-15 0.0000 p - 2.386 1.7e+01 -1.5e+08# 6.3e+00 -1.4e-13 0.0000 f - 2.387 1.7e+01 -4.0e+08# 6.3e+00 -1.0e-13 0.0000 f - 2.388 1.7e+01 -2.1e+09# 6.3e+00 -4.2e-13 0.0000 f + 32.812 1.8e+01 3.4e-10 6.3e+00 -2.9e-13 0.0000 p + 16.406 1.8e+01 3.4e-10 6.3e+00 -1.2e-15 0.0000 p + 8.203 1.8e+01 3.3e-10 6.3e+00 -2.6e-13 0.0000 p + 4.102 1.8e+01 3.3e-10 6.3e+00 -2.1e-13 0.0000 p + 2.051 1.7e+01 3.4e-10 6.3e+00 -7.2e-16 0.0000 p + 1.025 1.6e+01 -1.3e+06# 6.3e+00 -8.3e-14 0.0000 f + 1.538 1.7e+01 3.4e-10 6.3e+00 -2.0e-13 0.0000 p + 1.282 1.7e+01 3.4e-10 6.3e+00 -7.9e-17 0.0000 p + 1.154 1.7e+01 3.6e-10 6.3e+00 -1.8e-13 0.0000 p + 1.089 1.7e+01 -3.4e+06# 6.3e+00 -1.7e-13 0.0000 f + 1.122 1.7e+01 -1.0e+07# 6.3e+00 -3.2e-13 0.0000 f + 1.138 1.7e+01 -1.3e+08# 6.3e+00 -1.8e-13 0.0000 f + 1.146 1.7e+01 3.2e-10 6.3e+00 -3.0e-13 0.0000 p + 1.142 1.7e+01 5.5e-10 6.3e+00 -2.8e-13 0.0000 p + 1.140 1.7e+01 -1.5e-10 6.3e+00 -2.3e-13 0.0000 p + 1.139 1.7e+01 -4.8e+08# 6.3e+00 -6.2e-14 0.0000 f + 1.139 1.7e+01 1.3e-09 6.3e+00 -8.9e-17 0.0000 p - Gamma value achieved: 2.3882 + Gamma value achieved: 1.1390
--H1 = 1 - H2; +
H1b = 1 - H2b;
- 4.1. Measurement of the noise characteristics of the sensors
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