Difference between revisions of "HFI time response model"

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== Detection chain ==
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== LFER4 model ==
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<span style="color:red">(Francesco Piacentini)</span>
 
  
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This entire section has been moved to [[HFI_detection_chain | this page]].
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Here we describe the "Low frequency excess response" 4 model.
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+
If we write the input signal (power) on a bolometer as
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===Data compression===
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<math>\label{bol_in}
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s_0(t)=e^{i\omega t}</math>,
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The output of the readout electronics unit (REU) consists of one
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the bolometer physical impedance can be written as
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value for each of the 72 science channels for each half-period of
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<math>\label{bol_out} s(t)=e^{i\omega t}F(\omega)</math>,
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modulation. This number, <math>S_{REU}</math> , is the sum
+
where <math>\omega</math> is the angular frequency of the signal and <math>F(\omega)</math> is the complex intrinsic bolometer transfer function. For HFI the bolometer transfer function is modelled as the sum of four single pole low-pass filters: <math>\label{bol_tf}
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of the 40 16-bit ADC signal values obtained within the given
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F(\omega) = \sum_{i=0,4} \frac{a_i}{1 + i\omega\tau_i}</math>.
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half-period. The data processor unit (DPU) performs a lossy
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The modulation of the signal is performed with a square wave, written here as a composition of sine waves of decreasing amplitude: <math>\label{sigmod}
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quantization of <math>S_{REU}</math>.
 
−
 
 
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We define a compression slice of 254 <math>S_{REU}</math> values, corresponding
 
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to about 1.4 s of observation for each detector and to a
 
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strip of sky about 8 degrees long. The mean <math>\langle S_{REU} \rangle</math> of the data within
 
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each compression slice is computed, and data are demodulated
 
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using this mean:
 
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<math>S_{demod,i} = (S_{REU,i} − \langle S_{REU} \rangle ) ∗ (−1)^{i}</math>
 
−
 
 
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where <math>1 < i < 254</math> is the running index within the compression slice.
 
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The mean <math>\langle S_{demod} \rangle</math> of the demodulated data <math>S_{demod,i}</math>
 
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is computed and subtracted, and the resulting data slice is quantized
 
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according to a step size Q that is fixed per detector:
 
−
 
 
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<math> S_{DPU,i} = \mbox{round} \left[( S_{demod,i} − \langle S_{demod} \rangle) /Q \right ] </math>
 
−
 
 
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This is the lossy part of the algorithm: the required compression
 
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factor, obtained through the tuning of the quantization step Q,
 
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adds a measure of noise to the data. Assuming Gaussian white
 
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noise with standard deviation <math>\sigma</math>, a quantization setting of
 
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<math>\sigma</math>/Q = 2 adds 1% to the noise (Pajot et al. 2010; Pratt 1978).
 
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The value of <math>\sigma</math> was determined at the end of the CPV phase after
 
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subtraction of the signal from the timeline.
 
−
 
 
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The two means
 
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<math>\langle S_{REU} \rangle</math>
 
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and
 
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<math>\langle S_{demod} \rangle</math>
 
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are computed as
 
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32-bit words and sent through the telemetry, together with the
 
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<math>S_{DPU,i}</math> values.
 
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Variable-length encoding of the <math>S_{DPU,i}</math> values is
 
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performed on board, and the inverse decoding is applied on
 
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ground.
 
−
 
 
−
 
 
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Optimal use of the bandpass available for the downlink (75 kb/s 
 
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average for HFI science) was obtained initially by using a value
 
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of Q = <math>\sigma</math>/2.5 for all bolometer signals.
 
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After 12 December 2009, only for the 857 GHz detectors, the
 
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value was reset to Q = <math>\sigma</math>/2.0 to avoid data loss
 
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due to exceeding the limit of downlink rate.
 
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===Time response===
 
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−
 
 
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The HFI bolometers and readout electronics have a finite response time to changes in incident optical power.  The bolometers are thermal detectors of radiation whose response time is determined by the thermal circuit defined by the heat capacity of the detector and thermal conductivity.
 
−
 
 
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Due to Planck's nearly constant scan rate, the time response is degenerate with the optical beam.  However, because of the long time scale effects present in the time response, the  time response is deconvolved from the data in the processing of the HFI data (see [[TOI processing|TOI processing]]).
 
−
 
 
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The time response of the HFI bolometers and readout electronics is modeled as a Fourier domain transfer function (called the LFER4 model) consisting of the product of an bolometer thermal response <math>F(\omega)</math> and an electronics response <math>H'(\omega)</math>.  
 
−
 
 
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<math>\label{LFER4def}TF^{LFER4}(\omega) = F(\omega) H'(\omega)</math>
 
−
 
 
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=== LFER4 model ===
 
−
 
 
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If we write the input signal (power) on a bolometer as <math>\label{bol_in}
 
−
s_0(t)=e^{i\omega t}
 
−
</math> the bolometer physical impedance can be written as: <math>\label{bol_out}
 
−
s(t)=e^{i\omega t}F(\omega)
 
−
</math> where <math>\omega</math> is the angular frequency of the signal and <math>F(\omega)</math> is the complex intrinsic bolometer transfer function. For HFI the bolometer transfer function is modelled as the sum of 4 single pole low pass filters: <math>\label{bol_tf}
 
−
F(\omega) = \sum_{i=0,4} \frac{a_i}{1 + i\omega\tau_i}
 
−
</math> The modulation of the signal is done with a square wave, written here as a composition of sine waves of decreasing amplitude: <math>\label{sigmod}
 
 
s'(t)=e^{i\omega t}F(\omega)\sum_{k=0}^{\infty} \frac{e^{i\omega_r(2k+1)t}-e^{-i\omega_r(2k+1)t}}{2i(2k+1)}
 
s'(t)=e^{i\omega t}F(\omega)\sum_{k=0}^{\infty} \frac{e^{i\omega_r(2k+1)t}-e^{-i\omega_r(2k+1)t}}{2i(2k+1)}
−
</math> where we have used the Euler relation <math>\sin x=(e^{ix}-e^{-ix})/2i</math> and <math>\omega_r</math> is the angular frequency of the square wave. The modulation frequency is <math>f_{mod} = \omega_r/2\pi</math> and was set to <math>f_{mod} = 90.18759 </math>Hz in flight. This signal is then filtered by the complex electronic transfer function <math>H(\omega)</math>. Setting: <math>\omega_k^+=\omega+(2k+1)\omega_r</math> <math>\omega_k^-=\omega-(2k+1)\omega_r</math> we have: <math>\label{sigele}
+
</math>, where we have used the Euler relation <math>\sin x=(e^{ix}-e^{-ix})/2i</math> and <math>\omega_r</math> is the angular frequency of the square wave. The modulation frequency is <math>f_{\rm mod} = \omega_r/2\pi</math> and was set to <math>f_{\rm mod} = 90.18759 </math>Hz in flight. This signal is then filtered by the complex electronic transfer function <math>H(\omega)</math>. Setting <math>\omega_k^+=\omega+(2k+1)\omega_r</math> <math>\omega_k^-=\omega-(2k+1)\omega_r</math> we have <math>\label{sigele}
 
\Sigma(t)=\sum_{k=0}^\infty\frac{F(\omega)}{2i(2k+1)}\left[H(\omega_k^+)e^{i\omega_k^+t}-H(\omega_k^-)e^{i\omega_k^-t}\right]
 
\Sigma(t)=\sum_{k=0}^\infty\frac{F(\omega)}{2i(2k+1)}\left[H(\omega_k^+)e^{i\omega_k^+t}-H(\omega_k^-)e^{i\omega_k^-t}\right]
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</math> This signal is then sampled at high frequency (<math>2 f_{mod} NS</math>). <math>NS</math> is one of the parameters of the HFI electronics and corresponds to the number of high frequency samples in each modulation semi-period. In order to obtain an output signal sampled every <math>\pi/\omega_r</math> seconds, we must integrate on a semiperiod, as done in the HFI readout. To also include a time shift <math>\Delta t</math>, the integral is calculated between <math>n\pi/\omega_r+\Delta t</math> and <math>(n+1)\pi/\omega_r+\Delta t</math> (with <math>T=2 \pi/\omega_r</math> period of the modulation). The time shift <math>\Delta t</math> is encoded in the HFI electronics by the parameter <math>S_{phase}</math>, with the relation <math>\Delta t = S_{phase}/NS/f_{mod} </math>.
+
</math>. This signal is then sampled at high frequency, (<math>2 f_{\rm mod} N_{\rm S}</math>). Here <math>N_{\rm S}</math> is one of the parameters of the HFI electronics and corresponds to the number of high frequency samples in each modulation semi-period. In order to obtain an output signal sampled every <math>\pi/\omega_r</math> seconds, we must integrate on a semiperiod, as done in the HFI readout. To also include a time shift <math>\Delta t</math>, the integral is calculated between <math>n\pi/\omega_r+\Delta t</math> and <math>(n+1)\pi/\omega_r+\Delta t</math> (with <math>T=2 \pi/\omega_r</math> period of the modulation). The time shift <math>\Delta t</math> is encoded in the HFI electronics by the parameter <math>S_{\rm phase}</math>, with the relation <math>\Delta t = S_{\rm phase}/N_{\rm S}/f_{\rm mod} </math>.
  
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After integration, the <math>n</math>-sample of a bolometer can be written as <math>\label{eqn:output}
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After integration, the <i>n</i>-sample of a bolometer can be written as <math>\label{eqn:output}
 
Y(t_n) = (-1)^n F(\omega) H'(\omega) e^{i t_n \omega}
 
Y(t_n) = (-1)^n F(\omega) H'(\omega) e^{i t_n \omega}
−
</math> where <math>\label{tfele}
+
</math>, where <math>\label{tfele}
 
H'(\omega) = \frac 12 \sum_{k=0}^\infty
 
H'(\omega) = \frac 12 \sum_{k=0}^\infty
 
e^{-i(\frac{\pi\omega}{2\omega_r}+\omega\Delta t)} \Bigg[
 
e^{-i(\frac{\pi\omega}{2\omega_r}+\omega\Delta t)} \Bigg[
Line 90: Line 23:
 
  \left(1-e^{\frac{i\omega_k^+\pi}{\omega_r}}\right)  
 
  \left(1-e^{\frac{i\omega_k^+\pi}{\omega_r}}\right)  
 
\\ - \frac{H(\omega_k^-)e^{i\omega_k^- \Delta t}}{(2k+1)\omega_k^-}  \left(1-e^{\frac{i\omega_k^-\pi}{\omega_r}}\right)
 
\\ - \frac{H(\omega_k^-)e^{i\omega_k^- \Delta t}}{(2k+1)\omega_k^-}  \left(1-e^{\frac{i\omega_k^-\pi}{\omega_r}}\right)
−
\Bigg]
+
\Bigg].
−
</math>
 
−
 
 
−
The output signal in equation eqn:output can be demodulated (thus removing the <math>(-1)^n</math>) and compared to the input signal in equation bol_in. The overall transfer function is composed of the bolometer transfer function and the effective electronics transfer function, <math>H'(\omega)</math>: <math>TF(\omega) = F(\omega) H'(\omega)
 
 
</math>
 
</math>
  
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The shape of <math>H(\omega)</math> is obtained combining low and high-pass filters with Sallen Key topologies (with their respective time constants) and accounting also for the stray capacitance low pass filter given by the bolometer impedance combined with the stray capacitance of the cables. The sequence of filters that define the electronic band-pass function <math>H(\omega) = h_0*h_1*h_2*h_3*h_4*h_{5}</math> are listed in table table:readout_electronics_filters.
+
The output signal in equation eqn:output can be demodulated (thus removing the (-1)<sup><i>n</i></sup>) and compared to the input signal in equation bol_in. The overall transfer function is composed of the bolometer transfer function and the effective electronics transfer function, <math>H'(\omega)</math>: <math>TF(\omega) = F(\omega) H'(\omega)</math>.
−
 
 
−
=== Parameters of LFER4 model ===
 
  
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The LFER4 model has are a total of 10 parameters(<math>A_1</math>,<math>A_2</math>,<math>A_3</math>,<math>A_4</math>,<math>\tau_1</math>,<math>\tau_2</math>,<math>\tau_3</math>,<math>\tau_4</math>,<math>S_{phase}</math>,<math>\tau_{stray}</math>) 9 of which are independent, for each bolometer. The free parameters of the LFER4 model are determined using in-flight data in the following ways:
+
The shape of <math>H(\omega)</math> is obtained combining low- and high-pass filters with Sallen-Key topologies (with their respective time constants) and accounting also for the stray capacitance low-pass filter given by the bolometer impedance combined with the stray capacitance of the cables. The sequence of filters that define the electronic band-pass function <math>H(\omega) = h_0*h_1*h_2*h_3*h_4*h_{5}</math> are listed in the following table.
  
−
* <math>S_{phase}</math> is fixed at the value of the REU setting.
+
<center>
−
* <math>\tau_{stray}</math> is measured during the QEC test during CPV.
+
{| class="wikitable"  align="center" style="text-align:left" border="1" cellpadding="5" cellspacing="0"
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* <math>A_1</math>, <math>\tau_1</math>, <math>A_2</math>, <math>\tau_2</math> are fit forcing the compactness of the scanning beam.
+
|+ '''HFI electronics filter sequence.'''  Here we define <math>s = i \omega</math>.
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* <math>A_3</math>, <math>\tau_3</math>, <math>A_{4}</math> <math>\tau_4</math> are fit by forcing agreement of survey 2 and survey 1 maps.
+
|- bgcolor="ffdead"
−
* The overall normalization of the LFER4 model is forced to be 1.0 at the signal frequency of the dipole. 
+
! Filter || Parameters || Function
−
 
 
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The details of determining the model parameters are given in (reference P03c paper) and the best-fit parameters listed here in table table:LFER4pars.
 
−
 
 
−
 
 
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===HFI electronics filter sequence ===
 
−
 
 
−
{| class="wikitable" style="text-align: left; border-collapse: collapse; border-width: 1px; border-style: solid; border-color: #000"
 
−
|+HFI electronics filter sequence. We define $s = i \omega$
 
 
|-
 
|-
−
! style="border-style: solid; border-width: 1px"|Filter
+
|0. Stray capacitance low-pass filter || <math>\tau_{\rm stray}= R_{\rm bolo} C_{\rm stray}</math> || <math>h_0 = \frac{1}{1.0+\tau_{\rm stray}*s}</math>
−
! style="border-style: solid; border-width: 1px"|Parameters
 
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! style="border-style: solid; border-width: 1px"|Function
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|0. Stray capacitance low pass filter
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|1. Low-pass filter || <math>R_1=1</math>k<math>\Omega</math> <br /> <math>C_1=100</math>nF || <math>h_1 = \frac{2.0+R_1*C_1*s}{2.0*(1.0+R_1*C_1*s)}</math>
−
|style="border-style: solid; border-width: 1px"|<math>\tau_{stray}= R_{bolo} C_{stray}</math>
 
−
|style="border-style: solid; border-width: 1px"|<math>h_0 = \frac{1}{1.0+\tau_{stray}*s}</math>
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|1. Low pass filter
+
|2. Sallen-Key high-pass filter || <math>R_2=51</math>k<math>\Omega</math><br /> <math>C_2=1\mu</math>F || <math>h_2= \frac{(R_2*C_2*s)^2}{(1.0+R_2*C_2*s)^2}</math>3
−
|style="border-style: solid; border-width: 1px"|<math>R_1=1</math>k<math>\Omega</math> <br /> <math>C_1=100</math>nF
 
−
|style="border-style: solid; border-width: 1px"|<math>h_1 = \frac{2.0+R_1*C_1*s}{2.0*(1.0+R_1*C_1*s)}</math>
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|2. Sallen Key high pass filter
+
|3. Sign reverse with gain || || <math>h_3=-5.1</math>
−
|style="border-style: solid; border-width: 1px"|<math>R_2=51</math>k<math>\Omega</math><br /> <math>C_2=1\mu</math>
 
−
|style="border-style: solid; border-width: 1px"|<math>h_2= \frac{(R_2*C_2*s)^2}{(1.0+R_2*C_2*s)^2}</math>3
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|3. Sign reverse with gain
+
|4. Single pole low-pass filter with gain || <math>R_4=10</math>k<math>\Omega</math><br /> <math>C_4=10</math>nF || <math>h_4= \frac{1.5}{1.0+R_4*C_4*s}</math>
−
|style="border-style: solid; border-width: 1px"|
 
−
|style="border-style: solid; border-width: 1px"|<math>h_3=-5.1</math>
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|4. Single pole low pass filter with gain
+
|5. Single pole high-pass filter coupled to a Sallen-Key low-pass filter || <math>R_9=18.7</math>k<math>\Omega</math><br /><math>R_{12}=37.4</math>k<math>\Omega</math><br /> <math>C=10.0</math>nF<br /><math>R_{78}=510</math>k<math>\Omega</math><br /> <math>C_{18}=1.0\mu</math>F<br /><math>K_3 = R_9^2*R_{78}*R_{12}^2*C^2*C_{18}</math><br /> <math>K_2 = R_9*R_{12}^2*R_{78}*C^2+R_{9}^2*R_{12}^2*C^2+R_9*R_{12}^2*R_{78}*C_{18}*C</math><br /> <math>K_1 =R_9*R_{12}^2*C+R_{12}*R_{78}*R_9*C_{18}</math> || <math>h_{5} = \frac{2.0*R_{12}*R_9*R_{78}*C_{18}*s}{s^3*K_3 +  
−
|style="border-style: solid; border-width: 1px"|<math>R_4=10</math>k<math>\Omega</math><br /> <math>C_4=10</math>nF
 
−
|style="border-style: solid; border-width: 1px"|<math>h_4= \frac{1.5}{1.0+R_4*C_4*s}</math>
 
−
|-
 
−
|style="border-style: solid; border-width: 1px"|5. Single pole high pass filter coupled to a Sallen Key low pass filter
 
−
|style="border-style: solid; border-width: 1px"|<math>R_9=18.7</math>k<math>\Omega</math><br /><math>R_{12}=37.4</math>k<math>\Omega</math><br /> <math>C=10.0</math>nF<br /><math>R_{78}=510</math>k<math>\Omega</math><br /> <math>C_{18}=1.0\mu</math>F<br /><math>K_3 = R_9^2*R_{78}*R_{12}^2*C^2*C_{18}</math><br /> <math>K_2 = R_9*R_{12}^2*R_{78}*C^2+R_{9}^2*R_{12}^2*C^2+R_9*R_{12}^2*R_{78}*C_{18}*C</math><br /> <math>K_1 =R_9*R_{12}^2*C+R_{12}*R_{78}*R_9*C_{18}</math>
 
−
|style="border-style: solid; border-width: 1px"|<math>h_{5} = \frac{2.0*R_{12}*R_9*R_{78}*C_{18}*s}{s^3*K_3 +  
 
 
       s^2*K_2+  
 
       s^2*K_2+  
 
       s*K_1 + R_{12}*R_9 } </math>
 
       s*K_1 + R_{12}*R_9 } </math>
  
 
|}
 
|}
 +
</center>
  
 +
== Parameters of LFER4 model ==
  
 +
The LFER4 model has are a total of 10 parameters for each bolometer (<math>A_1</math>,<math>A_2</math>,<math>A_3</math>,<math>A_4</math>,<math>\tau_1</math>,<math>\tau_2</math>,<math>\tau_3</math>,<math>\tau_4</math>,<math>S_{\rm phase}</math>,<math>\tau_{\rm stray}</math>) nine of which are independent. The free parameters of the LFER4 model are determined using in-flight data in the following ways:
  
−
{| class="wikitable" style="text-align: left; align: center; border-collapse: collapse; border-width: 1px; border-style: solid; border-color: #000"
+
* <math>S_{\rm phase}</math> is fixed at the value of the REU setting;
−
|+ Parameters for LFER4 model.
+
* <math>\tau_{\rm stray}</math> is measured during the QEC test of the CPV phase;
−
|-
+
* <math>A_1</math>, <math>\tau_1</math>, <math>A_2</math>, <math>\tau_2</math> are fit by forcing the compactness of the scanning beam;
−
|style="border-style: solid; border-width: 1px"|Bolometer
+
* <math>A_3</math>, <math>\tau_3</math>, <math>A_{4}</math> <math>\tau_4</math> are fit by forcing agreement of Survey 2 and Survey 1 maps;
−
|style="border-style: solid; border-width: 1px"|<math>A_1</math>
+
* the overall normalization of the LFER4 model is forced to be 1.0 at the signal frequency of the dipole. 
−
|style="border-style: solid; border-width: 1px"|<math>\tau_1</math> (s)
+
 
−
|style="border-style: solid; border-width: 1px"|<math>A_2</math>
+
The details of determining the model parameters are given in (reference P03c paper) and the best-fit parameters are listed below.
−
|style="border-style: solid; border-width: 1px"|<math>\tau_2</math> (s)
+
 
−
|style="border-style: solid; border-width: 1px"|<math>A_3</math>
+
<center>
−
|style="border-style: solid; border-width: 1px"|<math>\tau_3</math> (s)
+
{| class="wikitable"  align="center" style="text-align:center" border="1" cellpadding="5" cellspacing="0"  width=500px
−
|style="border-style: solid; border-width: 1px"|<math>A_4</math>
+
|+ '''LFER4 model parameters'''
−
|style="border-style: solid; border-width: 1px"|<math>\tau_4</math> (s)
+
|- bgcolor="ffdead"
−
|style="border-style: solid; border-width: 1px"|<math>\tau_{stray}</math> (s)
+
! Bolometer || <math>A_1</math> || <math>\tau_1</math> (s) || <math>A_2</math> || <math>\tau_2</math> (s) || <math>A_3</math> || <math>\tau_3</math> (s) || <math>A_4</math> || <math>\tau_4</math> (s) || <math>\tau_{\rm stray}</math> (s) || <math>S_{\rm phase}</math> (s)
−
|style="border-style: solid; border-width: 1px"|<math>S_{phase}</math> (s)
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|100-1a
+
| 100-1a|| 0.392|| 0.01|| 0.534|| 0.0209|| 0.0656|| 0.0513|| 0.00833|| 0.572|| 0.00159|| 0.00139
−
|style="border-style: solid; border-width: 1px"|0.392
 
−
|style="border-style: solid; border-width: 1px"|0.01
 
−
|style="border-style: solid; border-width: 1px"|0.534
 
−
|style="border-style: solid; border-width: 1px"|0.0209
 
−
|style="border-style: solid; border-width: 1px"|0.0656
 
−
|style="border-style: solid; border-width: 1px"|0.0513
 
−
|style="border-style: solid; border-width: 1px"|0.00833
 
−
|style="border-style: solid; border-width: 1px"|0.572
 
−
|style="border-style: solid; border-width: 1px"|0.00159
 
−
|style="border-style: solid; border-width: 1px"|0.00139
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|100-1b
+
| 100-1b|| 0.484|| 0.0103|| 0.463|| 0.0192|| 0.0451|| 0.0714|| 0.00808|| 0.594|| 0.00149|| 0.00139
−
|style="border-style: solid; border-width: 1px"|0.484
 
−
|style="border-style: solid; border-width: 1px"|0.0103
 
−
|style="border-style: solid; border-width: 1px"|0.463
 
−
|style="border-style: solid; border-width: 1px"|0.0192
 
−
|style="border-style: solid; border-width: 1px"|0.0451
 
−
|style="border-style: solid; border-width: 1px"|0.0714
 
−
|style="border-style: solid; border-width: 1px"|0.00808
 
−
|style="border-style: solid; border-width: 1px"|0.594
 
−
|style="border-style: solid; border-width: 1px"|0.00149
 
−
|style="border-style: solid; border-width: 1px"|0.00139
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|100-2a
+
| 100-2a|| 0.474|| 0.00684|| 0.421|| 0.0136|| 0.0942|| 0.0376|| 0.0106|| 0.346|| 0.00132|| 0.00125
−
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|-
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|-
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|style="border-style: solid; border-width: 1px"|100-3b
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|-
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|style="border-style: solid; border-width: 1px"|100-4a
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|100-4b
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−
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|-
 
|-
−
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−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|143-1b
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−
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|143-2a
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−
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−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|143-2b
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|143-3a
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|143-3b
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|-
 
|-
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|-
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|143-5
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|143-6
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|143-7
+
| 143-7|| 0.414|| 0.00543|| 0.562|| 0.00543|| 0.0185|| 0.0449|| 0.00545|| 0.314|| 0.00186|| 0.00139
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−
|style="border-style: solid; border-width: 1px"|0.00139
 
 
|-
 
|-
−
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−
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|217-5b
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|-
 
|-
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|-
 
|-
−
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|-
 
|-
−
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|-
 
|-
−
|style="border-style: solid; border-width: 1px"|217-7b
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|-
 
|-
−
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−
|style="border-style: solid; border-width: 1px"|0.0128
 
−
|style="border-style: solid; border-width: 1px"|0.382
 
−
|style="border-style: solid; border-width: 1px"|0.00179
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|217-8b
+
| 217-8b|| 0.512|| 0.00703|| 0.41|| 0.00703|| 0.0639|| 0.0272|| 0.0139|| 0.232|| 0.00173|| 0.00125
−
|style="border-style: solid; border-width: 1px"|0.512
 
−
|style="border-style: solid; border-width: 1px"|0.00703
 
−
|style="border-style: solid; border-width: 1px"|0.41
 
−
|style="border-style: solid; border-width: 1px"|0.00703
 
−
|style="border-style: solid; border-width: 1px"|0.0639
 
−
|style="border-style: solid; border-width: 1px"|0.0272
 
−
|style="border-style: solid; border-width: 1px"|0.0139
 
−
|style="border-style: solid; border-width: 1px"|0.232
 
−
|style="border-style: solid; border-width: 1px"|0.00173
 
−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|217-1
+
| 217-1|| 0.0136|| 0.00346|| 0.956|| 0.00346|| 0.0271|| 0.0233|| 0.00359|| 1.98|| 0.00159|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.0136
 
−
|style="border-style: solid; border-width: 1px"|0.00346
 
−
|style="border-style: solid; border-width: 1px"|0.956
 
−
|style="border-style: solid; border-width: 1px"|0.00346
 
−
|style="border-style: solid; border-width: 1px"|0.0271
 
−
|style="border-style: solid; border-width: 1px"|0.0233
 
−
|style="border-style: solid; border-width: 1px"|0.00359
 
−
|style="border-style: solid; border-width: 1px"|1.98
 
−
|style="border-style: solid; border-width: 1px"|0.00159
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|217-2
+
| 217-2|| 0.978|| 0.00352|| 0.014|| 0.0261|| 0.00614|| 0.042|| 0.00194|| 0.686|| 0.0016|| 0.00125
−
|style="border-style: solid; border-width: 1px"|0.978
 
−
|style="border-style: solid; border-width: 1px"|0.00352
 
−
|style="border-style: solid; border-width: 1px"|0.014
 
−
|style="border-style: solid; border-width: 1px"|0.0261
 
−
|style="border-style: solid; border-width: 1px"|0.00614
 
−
|style="border-style: solid; border-width: 1px"|0.042
 
−
|style="border-style: solid; border-width: 1px"|0.00194
 
−
|style="border-style: solid; border-width: 1px"|0.686
 
−
|style="border-style: solid; border-width: 1px"|0.0016
 
−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|217-3
+
| 217-3|| 0.932|| 0.00355|| 0.0336|| 0.00355|| 0.0292|| 0.0324|| 0.00491|| 0.279|| 0.00174|| 0.00125
−
|style="border-style: solid; border-width: 1px"|0.932
 
−
|style="border-style: solid; border-width: 1px"|0.00355
 
−
|style="border-style: solid; border-width: 1px"|0.0336
 
−
|style="border-style: solid; border-width: 1px"|0.00355
 
−
|style="border-style: solid; border-width: 1px"|0.0292
 
−
|style="border-style: solid; border-width: 1px"|0.0324
 
−
|style="border-style: solid; border-width: 1px"|0.00491
 
−
|style="border-style: solid; border-width: 1px"|0.279
 
−
|style="border-style: solid; border-width: 1px"|0.00174
 
−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|217-4
+
| 217-4|| 0.658|| 0.00135|| 0.32|| 0.00555|| 0.0174|| 0.0268|| 0.00424|| 0.473|| 0.00171|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.658
 
−
|style="border-style: solid; border-width: 1px"|0.00135
 
−
|style="border-style: solid; border-width: 1px"|0.32
 
−
|style="border-style: solid; border-width: 1px"|0.00555
 
−
|style="border-style: solid; border-width: 1px"|0.0174
 
−
|style="border-style: solid; border-width: 1px"|0.0268
 
−
|style="border-style: solid; border-width: 1px"|0.00424
 
−
|style="border-style: solid; border-width: 1px"|0.473
 
−
|style="border-style: solid; border-width: 1px"|0.00171
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-3a
+
| 353-3a|| 0.554|| 0.00704|| 0.36|| 0.00704|| 0.0699|| 0.0305|| 0.0163|| 0.344|| 0.0017|| 0.00125
−
|style="border-style: solid; border-width: 1px"|0.554
 
−
|style="border-style: solid; border-width: 1px"|0.00704
 
−
|style="border-style: solid; border-width: 1px"|0.36
 
−
|style="border-style: solid; border-width: 1px"|0.00704
 
−
|style="border-style: solid; border-width: 1px"|0.0699
 
−
|style="border-style: solid; border-width: 1px"|0.0305
 
−
|style="border-style: solid; border-width: 1px"|0.0163
 
−
|style="border-style: solid; border-width: 1px"|0.344
 
−
|style="border-style: solid; border-width: 1px"|0.0017
 
−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-3b
+
| 353-3b|| 0.219|| 0.00268|| 0.671|| 0.00695|| 0.0977|| 0.0238|| 0.0119|| 0.289|| 0.00157|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.219
 
−
|style="border-style: solid; border-width: 1px"|0.00268
 
−
|style="border-style: solid; border-width: 1px"|0.671
 
−
|style="border-style: solid; border-width: 1px"|0.00695
 
−
|style="border-style: solid; border-width: 1px"|0.0977
 
−
|style="border-style: solid; border-width: 1px"|0.0238
 
−
|style="border-style: solid; border-width: 1px"|0.0119
 
−
|style="border-style: solid; border-width: 1px"|0.289
 
−
|style="border-style: solid; border-width: 1px"|0.00157
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-4a
+
| 353-4a|| 0.768|| 0.00473|| 0.198|| 0.00993|| 0.0283|| 0.0505|| 0.00628|| 0.536|| 0.00181|| 0.00125
−
|style="border-style: solid; border-width: 1px"|0.768
 
−
|style="border-style: solid; border-width: 1px"|0.00473
 
−
|style="border-style: solid; border-width: 1px"|0.198
 
−
|style="border-style: solid; border-width: 1px"|0.00993
 
−
|style="border-style: solid; border-width: 1px"|0.0283
 
−
|style="border-style: solid; border-width: 1px"|0.0505
 
−
|style="border-style: solid; border-width: 1px"|0.00628
 
−
|style="border-style: solid; border-width: 1px"|0.536
 
−
|style="border-style: solid; border-width: 1px"|0.00181
 
−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-4b
+
| 353-4b|| 0.684|| 0.00454|| 0.224|| 0.0108|| 0.0774|| 0.08|| 0.0149|| 0.267|| 0.00166|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.684
 
−
|style="border-style: solid; border-width: 1px"|0.00454
 
−
|style="border-style: solid; border-width: 1px"|0.224
 
−
|style="border-style: solid; border-width: 1px"|0.0108
 
−
|style="border-style: solid; border-width: 1px"|0.0774
 
−
|style="border-style: solid; border-width: 1px"|0.08
 
−
|style="border-style: solid; border-width: 1px"|0.0149
 
−
|style="border-style: solid; border-width: 1px"|0.267
 
−
|style="border-style: solid; border-width: 1px"|0.00166
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-5a
+
| 353-5a|| 0.767|| 0.00596|| 0.159|| 0.0124|| 0.0628|| 0.0303|| 0.0109|| 0.357|| 0.00156|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.767
 
−
|style="border-style: solid; border-width: 1px"|0.00596
 
−
|style="border-style: solid; border-width: 1px"|0.159
 
−
|style="border-style: solid; border-width: 1px"|0.0124
 
−
|style="border-style: solid; border-width: 1px"|0.0628
 
−
|style="border-style: solid; border-width: 1px"|0.0303
 
−
|style="border-style: solid; border-width: 1px"|0.0109
 
−
|style="border-style: solid; border-width: 1px"|0.357
 
−
|style="border-style: solid; border-width: 1px"|0.00156
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-5b
+
| 353-5b|| 0.832|| 0.00619|| 0.126|| 0.0111|| 0.0324|| 0.035|| 0.0096|| 0.397|| 0.00166|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.832
 
−
|style="border-style: solid; border-width: 1px"|0.00619
 
−
|style="border-style: solid; border-width: 1px"|0.126
 
−
|style="border-style: solid; border-width: 1px"|0.0111
 
−
|style="border-style: solid; border-width: 1px"|0.0324
 
−
|style="border-style: solid; border-width: 1px"|0.035
 
−
|style="border-style: solid; border-width: 1px"|0.0096
 
−
|style="border-style: solid; border-width: 1px"|0.397
 
−
|style="border-style: solid; border-width: 1px"|0.00166
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-6a
+
| 353-6a|| 0.0487|| 0.00176|| 0.855|| 0.006|| 0.0856|| 0.0216|| 0.0105|| 0.222|| 0.00199|| 0.00125
−
|style="border-style: solid; border-width: 1px"|0.0487
 
−
|style="border-style: solid; border-width: 1px"|0.00176
 
−
|style="border-style: solid; border-width: 1px"|0.855
 
−
|style="border-style: solid; border-width: 1px"|0.006
 
−
|style="border-style: solid; border-width: 1px"|0.0856
 
−
|style="border-style: solid; border-width: 1px"|0.0216
 
−
|style="border-style: solid; border-width: 1px"|0.0105
 
−
|style="border-style: solid; border-width: 1px"|0.222
 
−
|style="border-style: solid; border-width: 1px"|0.00199
 
−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-6b
+
| 353-6b|| 0.829|| 0.00561|| 0.127|| 0.00561|| 0.0373|| 0.0252|| 0.00696|| 0.36|| 0.00228|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.829
 
−
|style="border-style: solid; border-width: 1px"|0.00561
 
−
|style="border-style: solid; border-width: 1px"|0.127
 
−
|style="border-style: solid; border-width: 1px"|0.00561
 
−
|style="border-style: solid; border-width: 1px"|0.0373
 
−
|style="border-style: solid; border-width: 1px"|0.0252
 
−
|style="border-style: solid; border-width: 1px"|0.00696
 
−
|style="border-style: solid; border-width: 1px"|0.36
 
−
|style="border-style: solid; border-width: 1px"|0.00228
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-1
+
| 353-1|| 0.41|| 0.000743|| 0.502|| 0.00422|| 0.0811|| 0.0177|| 0.0063|| 0.329|| 0.00132|| 0.00097
−
|style="border-style: solid; border-width: 1px"|0.41
 
−
|style="border-style: solid; border-width: 1px"|0.000743
 
−
|style="border-style: solid; border-width: 1px"|0.502
 
−
|style="border-style: solid; border-width: 1px"|0.00422
 
−
|style="border-style: solid; border-width: 1px"|0.0811
 
−
|style="border-style: solid; border-width: 1px"|0.0177
 
−
|style="border-style: solid; border-width: 1px"|0.0063
 
−
|style="border-style: solid; border-width: 1px"|0.329
 
−
|style="border-style: solid; border-width: 1px"|0.00132
 
−
|style="border-style: solid; border-width: 1px"|0.00097
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-2
+
| 353-2|| 0.747|| 0.00309|| 0.225|| 0.00726|| 0.0252|| 0.0447|| 0.00267|| 0.513|| 0.00154|| 0.00097
−
|style="border-style: solid; border-width: 1px"|0.747
 
−
|style="border-style: solid; border-width: 1px"|0.00309
 
−
|style="border-style: solid; border-width: 1px"|0.225
 
−
|style="border-style: solid; border-width: 1px"|0.00726
 
−
|style="border-style: solid; border-width: 1px"|0.0252
 
−
|style="border-style: solid; border-width: 1px"|0.0447
 
−
|style="border-style: solid; border-width: 1px"|0.00267
 
−
|style="border-style: solid; border-width: 1px"|0.513
 
−
|style="border-style: solid; border-width: 1px"|0.00154
 
−
|style="border-style: solid; border-width: 1px"|0.00097
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-7
+
| 353-7|| 0.448|| 0.0009|| 0.537|| 0.0041|| 0.0122|| 0.0273|| 0.00346|| 0.433|| 0.00178|| 0.00125
−
|style="border-style: solid; border-width: 1px"|0.448
 
−
|style="border-style: solid; border-width: 1px"|0.0009
 
−
|style="border-style: solid; border-width: 1px"|0.537
 
−
|style="border-style: solid; border-width: 1px"|0.0041
 
−
|style="border-style: solid; border-width: 1px"|0.0122
 
−
|style="border-style: solid; border-width: 1px"|0.0273
 
−
|style="border-style: solid; border-width: 1px"|0.00346
 
−
|style="border-style: solid; border-width: 1px"|0.433
 
−
|style="border-style: solid; border-width: 1px"|0.00178
 
−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|353-8
+
| 353-8|| 0.718|| 0.00223|| 0.261|| 0.00608|| 0.0165|| 0.038|| 0.00408|| 0.268|| 0.00177|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.718
 
−
|style="border-style: solid; border-width: 1px"|0.00223
 
−
|style="border-style: solid; border-width: 1px"|0.261
 
−
|style="border-style: solid; border-width: 1px"|0.00608
 
−
|style="border-style: solid; border-width: 1px"|0.0165
 
−
|style="border-style: solid; border-width: 1px"|0.038
 
−
|style="border-style: solid; border-width: 1px"|0.00408
 
−
|style="border-style: solid; border-width: 1px"|0.268
 
−
|style="border-style: solid; border-width: 1px"|0.00177
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|545-1
+
| 545-1|| 0.991|| 0.00293|| 0.00743|| 0.026|| 0.00139|| 2.6||   0||   0|| 0.00216|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.991
 
−
|style="border-style: solid; border-width: 1px"|0.00293
 
−
|style="border-style: solid; border-width: 1px"|0.00743
 
−
|style="border-style: solid; border-width: 1px"|0.026
 
−
|style="border-style: solid; border-width: 1px"|0.00139
 
−
|style="border-style: solid; border-width: 1px"| 2.6
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|0.00216
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|545-2
+
| 545-2|| 0.985|| 0.00277|| 0.0128|| 0.024|| 0.00246|| 2.8||   0||   0|| 0.00187|| 0.00097
−
|style="border-style: solid; border-width: 1px"|0.985
 
−
|style="border-style: solid; border-width: 1px"|0.00277
 
−
|style="border-style: solid; border-width: 1px"|0.0128
 
−
|style="border-style: solid; border-width: 1px"|0.024
 
−
|style="border-style: solid; border-width: 1px"|0.00246
 
−
|style="border-style: solid; border-width: 1px"| 2.8
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|0.00187
 
−
|style="border-style: solid; border-width: 1px"|0.00097
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|545-4
+
| 545-4|| 0.972|| 0.003|| 0.0277|| 0.025|| 0.000777|| 2.5||   0||   0|| 0.00222|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.972
 
−
|style="border-style: solid; border-width: 1px"|0.003
 
−
|style="border-style: solid; border-width: 1px"|0.0277
 
−
|style="border-style: solid; border-width: 1px"|0.025
 
−
|style="border-style: solid; border-width: 1px"|0.000777
 
−
|style="border-style: solid; border-width: 1px"| 2.5
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|0.00222
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|857-1
+
| 857-1|| 0.974|| 0.00338|| 0.0229|| 0.025|| 0.00349|| 2.2||   0||   0|| 0.00176|| 0.00111
−
|style="border-style: solid; border-width: 1px"|0.974
 
−
|style="border-style: solid; border-width: 1px"|0.00338
 
−
|style="border-style: solid; border-width: 1px"|0.0229
 
−
|style="border-style: solid; border-width: 1px"|0.025
 
−
|style="border-style: solid; border-width: 1px"|0.00349
 
−
|style="border-style: solid; border-width: 1px"| 2.2
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|0.00176
 
−
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|857-2
+
| 857-2|| 0.84|| 0.00148|| 0.158|| 0.00656|| 0.00249|| 3.2||   0||   0|| 0.0022|| 0.00125
−
|style="border-style: solid; border-width: 1px"|0.84
 
−
|style="border-style: solid; border-width: 1px"|0.00148
 
−
|style="border-style: solid; border-width: 1px"|0.158
 
−
|style="border-style: solid; border-width: 1px"|0.00656
 
−
|style="border-style: solid; border-width: 1px"|0.00249
 
−
|style="border-style: solid; border-width: 1px"| 3.2
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|   0
 
−
|style="border-style: solid; border-width: 1px"|0.0022
 
−
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|857-3
+
| 857-3|| 0.36|| 4.22e-05|| 0.627|| 0.0024|| 0.0111|| 0.017|| 0.002|| 1.9|| 0.00152|| 0.00126
−
|style="border-style: solid; border-width: 1px"|0.36
 
−
|style="border-style: solid; border-width: 1px"|4.22e-05
 
−
|style="border-style: solid; border-width: 1px"|0.627
 
−
|style="border-style: solid; border-width: 1px"|0.0024
 
−
|style="border-style: solid; border-width: 1px"|0.0111
 
−
|style="border-style: solid; border-width: 1px"|0.017
 
−
|style="border-style: solid; border-width: 1px"|0.002
 
−
|style="border-style: solid; border-width: 1px"| 1.9
 
−
|style="border-style: solid; border-width: 1px"|0.00152
 
−
|style="border-style: solid; border-width: 1px"|0.00126
 
 
|-
 
|-
−
|style="border-style: solid; border-width: 1px"|857-4
+
| 857-4|| 0.278|| 0.0004|| 0.719|| 0.00392|| 0.00162|| 0.09|| 0.00152|| 0.8|| 0.00149|| 0.000558
−
|style="border-style: solid; border-width: 1px"|0.278
 
−
|style="border-style: solid; border-width: 1px"|0.0004
 
−
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|style="border-style: solid; border-width: 1px"|0.00392
 
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|style="border-style: solid; border-width: 1px"|0.00162
 
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|style="border-style: solid; border-width: 1px"|0.09
 
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|style="border-style: solid; border-width: 1px"|0.00152
 
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|style="border-style: solid; border-width: 1px"| 0.8
 
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[[Category:HFI design, qualification and performance|012]]

Latest revision as of 09:19, 10 December 2014

LFER4 model[edit]

Here we describe the "Low frequency excess response" 4 model. If we write the input signal (power) on a bolometer as [math]\label{bol_in} s_0(t)=e^{i\omega t}[/math], the bolometer physical impedance can be written as [math]\label{bol_out} s(t)=e^{i\omega t}F(\omega)[/math], where [math]\omega[/math] is the angular frequency of the signal and [math]F(\omega)[/math] is the complex intrinsic bolometer transfer function. For HFI the bolometer transfer function is modelled as the sum of four single pole low-pass filters: [math]\label{bol_tf} F(\omega) = \sum_{i=0,4} \frac{a_i}{1 + i\omega\tau_i}[/math]. The modulation of the signal is performed with a square wave, written here as a composition of sine waves of decreasing amplitude: [math]\label{sigmod} s'(t)=e^{i\omega t}F(\omega)\sum_{k=0}^{\infty} \frac{e^{i\omega_r(2k+1)t}-e^{-i\omega_r(2k+1)t}}{2i(2k+1)} [/math], where we have used the Euler relation [math]\sin x=(e^{ix}-e^{-ix})/2i[/math] and [math]\omega_r[/math] is the angular frequency of the square wave. The modulation frequency is [math]f_{\rm mod} = \omega_r/2\pi[/math] and was set to [math]f_{\rm mod} = 90.18759 [/math]Hz in flight. This signal is then filtered by the complex electronic transfer function [math]H(\omega)[/math]. Setting [math]\omega_k^+=\omega+(2k+1)\omega_r[/math] [math]\omega_k^-=\omega-(2k+1)\omega_r[/math] we have [math]\label{sigele} \Sigma(t)=\sum_{k=0}^\infty\frac{F(\omega)}{2i(2k+1)}\left[H(\omega_k^+)e^{i\omega_k^+t}-H(\omega_k^-)e^{i\omega_k^-t}\right] [/math]. This signal is then sampled at high frequency, ([math]2 f_{\rm mod} N_{\rm S}[/math]). Here [math]N_{\rm S}[/math] is one of the parameters of the HFI electronics and corresponds to the number of high frequency samples in each modulation semi-period. In order to obtain an output signal sampled every [math]\pi/\omega_r[/math] seconds, we must integrate on a semiperiod, as done in the HFI readout. To also include a time shift [math]\Delta t[/math], the integral is calculated between [math]n\pi/\omega_r+\Delta t[/math] and [math](n+1)\pi/\omega_r+\Delta t[/math] (with [math]T=2 \pi/\omega_r[/math] period of the modulation). The time shift [math]\Delta t[/math] is encoded in the HFI electronics by the parameter [math]S_{\rm phase}[/math], with the relation [math]\Delta t = S_{\rm phase}/N_{\rm S}/f_{\rm mod} [/math].

After integration, the n-sample of a bolometer can be written as [math]\label{eqn:output} Y(t_n) = (-1)^n F(\omega) H'(\omega) e^{i t_n \omega} [/math], where [math]\label{tfele} H'(\omega) = \frac 12 \sum_{k=0}^\infty e^{-i(\frac{\pi\omega}{2\omega_r}+\omega\Delta t)} \Bigg[ \frac{H(\omega_k^+)e^{i\omega_k^+ \Delta t}}{(2k+1)\omega_k^+} \left(1-e^{\frac{i\omega_k^+\pi}{\omega_r}}\right) \\ - \frac{H(\omega_k^-)e^{i\omega_k^- \Delta t}}{(2k+1)\omega_k^-} \left(1-e^{\frac{i\omega_k^-\pi}{\omega_r}}\right) \Bigg]. [/math]

The output signal in equation eqn:output can be demodulated (thus removing the (-1)n) and compared to the input signal in equation bol_in. The overall transfer function is composed of the bolometer transfer function and the effective electronics transfer function, [math]H'(\omega)[/math]: [math]TF(\omega) = F(\omega) H'(\omega)[/math].

The shape of [math]H(\omega)[/math] is obtained combining low- and high-pass filters with Sallen-Key topologies (with their respective time constants) and accounting also for the stray capacitance low-pass filter given by the bolometer impedance combined with the stray capacitance of the cables. The sequence of filters that define the electronic band-pass function [math]H(\omega) = h_0*h_1*h_2*h_3*h_4*h_{5}[/math] are listed in the following table.

HFI electronics filter sequence. Here we define [math]s = i \omega[/math].
Filter Parameters Function
0. Stray capacitance low-pass filter [math]\tau_{\rm stray}= R_{\rm bolo} C_{\rm stray}[/math] [math]h_0 = \frac{1}{1.0+\tau_{\rm stray}*s}[/math]
1. Low-pass filter [math]R_1=1[/math]k[math]\Omega[/math]
[math]C_1=100[/math]nF		 [math]h_1 = \frac{2.0+R_1*C_1*s}{2.0*(1.0+R_1*C_1*s)}[/math]
2. Sallen-Key high-pass filter [math]R_2=51[/math]k[math]\Omega[/math]
[math]C_2=1\mu[/math]F		 [math]h_2= \frac{(R_2*C_2*s)^2}{(1.0+R_2*C_2*s)^2}[/math]3
3. Sign reverse with gain [math]h_3=-5.1[/math]
4. Single pole low-pass filter with gain [math]R_4=10[/math]k[math]\Omega[/math]
[math]C_4=10[/math]nF		 [math]h_4= \frac{1.5}{1.0+R_4*C_4*s}[/math]
5. Single pole high-pass filter coupled to a Sallen-Key low-pass filter [math]R_9=18.7[/math]k[math]\Omega[/math]
[math]R_{12}=37.4[/math]k[math]\Omega[/math]
[math]C=10.0[/math]nF
[math]R_{78}=510[/math]k[math]\Omega[/math]
[math]C_{18}=1.0\mu[/math]F
[math]K_3 = R_9^2*R_{78}*R_{12}^2*C^2*C_{18}[/math]
[math]K_2 = R_9*R_{12}^2*R_{78}*C^2+R_{9}^2*R_{12}^2*C^2+R_9*R_{12}^2*R_{78}*C_{18}*C[/math]
[math]K_1 =R_9*R_{12}^2*C+R_{12}*R_{78}*R_9*C_{18}[/math]		 [math]h_{5} = \frac{2.0*R_{12}*R_9*R_{78}*C_{18}*s}{s^3*K_3 + s^2*K_2+ s*K_1 + R_{12}*R_9 } [/math]

Parameters of LFER4 model[edit]

The LFER4 model has are a total of 10 parameters for each bolometer ([math]A_1[/math],[math]A_2[/math],[math]A_3[/math],[math]A_4[/math],[math]\tau_1[/math],[math]\tau_2[/math],[math]\tau_3[/math],[math]\tau_4[/math],[math]S_{\rm phase}[/math],[math]\tau_{\rm stray}[/math]) nine of which are independent. The free parameters of the LFER4 model are determined using in-flight data in the following ways:

  • [math]S_{\rm phase}[/math] is fixed at the value of the REU setting;
  • [math]\tau_{\rm stray}[/math] is measured during the QEC test of the CPV phase;
  • [math]A_1[/math], [math]\tau_1[/math], [math]A_2[/math], [math]\tau_2[/math] are fit by forcing the compactness of the scanning beam;
  • [math]A_3[/math], [math]\tau_3[/math], [math]A_{4}[/math] [math]\tau_4[/math] are fit by forcing agreement of Survey 2 and Survey 1 maps;
  • the overall normalization of the LFER4 model is forced to be 1.0 at the signal frequency of the dipole.

The details of determining the model parameters are given in (reference P03c paper) and the best-fit parameters are listed below.

LFER4 model parameters
Bolometer [math]A_1[/math] [math]\tau_1[/math] (s) [math]A_2[/math] [math]\tau_2[/math] (s) [math]A_3[/math] [math]\tau_3[/math] (s) [math]A_4[/math] [math]\tau_4[/math] (s) [math]\tau_{\rm stray}[/math] (s) [math]S_{\rm phase}[/math] (s)
100-1a 0.392 0.01 0.534 0.0209 0.0656 0.0513 0.00833 0.572 0.00159 0.00139
100-1b 0.484 0.0103 0.463 0.0192 0.0451 0.0714 0.00808 0.594 0.00149 0.00139
100-2a 0.474 0.00684 0.421 0.0136 0.0942 0.0376 0.0106 0.346 0.00132 0.00125
100-2b 0.126 0.00584 0.717 0.0151 0.142 0.0351 0.0145 0.293 0.00138 0.00125
100-3a 0.744 0.00539 0.223 0.0147 0.0262 0.0586 0.00636 0.907 0.00142 0.00125
100-3b 0.608 0.00548 0.352 0.0155 0.0321 0.0636 0.00821 0.504 0.00166 0.00125
100-4a 0.411 0.0082 0.514 0.0178 0.0581 0.0579 0.0168 0.37 0.00125 0.00125
100-4b 0.687 0.0113 0.282 0.0243 0.0218 0.062 0.00875 0.431 0.00138 0.00139
143-1a 0.817 0.00447 0.144 0.0121 0.0293 0.0387 0.0101 0.472 0.00142 0.00125
143-1b 0.49 0.00472 0.333 0.0156 0.134 0.0481 0.0435 0.27 0.00149 0.00125
143-2a 0.909 0.0047 0.0763 0.017 0.00634 0.1 0.00871 0.363 0.00148 0.00125
143-2b 0.912 0.00524 0.0509 0.0167 0.0244 0.0265 0.0123 0.295 0.00146 0.00125
143-3a 0.681 0.00419 0.273 0.00956 0.0345 0.0348 0.0115 0.317 0.00145 0.00125
143-3b 0.82 0.00448 0.131 0.0132 0.0354 0.0351 0.0133 0.283 0.00161 0.000832
143-4a 0.914 0.00569 0.072 0.0189 0.00602 0.0482 0.00756 0.225 0.00159 0.00125
143-4b 0.428 0.00606 0.508 0.00606 0.0554 0.0227 0.00882 0.084 0.00182 0.00125
143-5 0.491 0.00664 0.397 0.00664 0.0962 0.0264 0.0156 0.336 0.00202 0.00139
143-6 0.518 0.00551 0.409 0.00551 0.0614 0.0266 0.0116 0.314 0.00153 0.00111
143-7 0.414 0.00543 0.562 0.00543 0.0185 0.0449 0.00545 0.314 0.00186 0.00139
217-5a 0.905 0.00669 0.0797 0.0216 0.00585 0.0658 0.00986 0.342 0.00157 0.00111
217-5b 0.925 0.00576 0.061 0.018 0.00513 0.0656 0.0094 0.287 0.00187 0.00125
217-6a 0.844 0.00645 0.0675 0.0197 0.0737 0.0316 0.0147 0.297 0.00154 0.00125
217-6b 0.284 0.00623 0.666 0.00623 0.0384 0.024 0.0117 0.15 0.00146 0.00111
217-7a 0.343 0.00548 0.574 0.00548 0.0717 0.023 0.0107 0.32 0.00152 0.00139
217-7b 0.846 0.00507 0.127 0.0144 0.0131 0.0479 0.0133 0.311 0.00151 0.00139
217-8a 0.496 0.00722 0.439 0.00722 0.0521 0.0325 0.0128 0.382 0.00179 0.00111
217-8b 0.512 0.00703 0.41 0.00703 0.0639 0.0272 0.0139 0.232 0.00173 0.00125
217-1 0.0136 0.00346 0.956 0.00346 0.0271 0.0233 0.00359 1.98 0.00159 0.00111
217-2 0.978 0.00352 0.014 0.0261 0.00614 0.042 0.00194 0.686 0.0016 0.00125
217-3 0.932 0.00355 0.0336 0.00355 0.0292 0.0324 0.00491 0.279 0.00174 0.00125
217-4 0.658 0.00135 0.32 0.00555 0.0174 0.0268 0.00424 0.473 0.00171 0.00111
353-3a 0.554 0.00704 0.36 0.00704 0.0699 0.0305 0.0163 0.344 0.0017 0.00125
353-3b 0.219 0.00268 0.671 0.00695 0.0977 0.0238 0.0119 0.289 0.00157 0.00111
353-4a 0.768 0.00473 0.198 0.00993 0.0283 0.0505 0.00628 0.536 0.00181 0.00125
353-4b 0.684 0.00454 0.224 0.0108 0.0774 0.08 0.0149 0.267 0.00166 0.00111
353-5a 0.767 0.00596 0.159 0.0124 0.0628 0.0303 0.0109 0.357 0.00156 0.00111
353-5b 0.832 0.00619 0.126 0.0111 0.0324 0.035 0.0096 0.397 0.00166 0.00111
353-6a 0.0487 0.00176 0.855 0.006 0.0856 0.0216 0.0105 0.222 0.00199 0.00125
353-6b 0.829 0.00561 0.127 0.00561 0.0373 0.0252 0.00696 0.36 0.00228 0.00111
353-1 0.41 0.000743 0.502 0.00422 0.0811 0.0177 0.0063 0.329 0.00132 0.00097
353-2 0.747 0.00309 0.225 0.00726 0.0252 0.0447 0.00267 0.513 0.00154 0.00097
353-7 0.448 0.0009 0.537 0.0041 0.0122 0.0273 0.00346 0.433 0.00178 0.00125
353-8 0.718 0.00223 0.261 0.00608 0.0165 0.038 0.00408 0.268 0.00177 0.00111
545-1 0.991 0.00293 0.00743 0.026 0.00139 2.6 0 0 0.00216 0.00111
545-2 0.985 0.00277 0.0128 0.024 0.00246 2.8 0 0 0.00187 0.00097
545-4 0.972 0.003 0.0277 0.025 0.000777 2.5 0 0 0.00222 0.00111
857-1 0.974 0.00338 0.0229 0.025 0.00349 2.2 0 0 0.00176 0.00111
857-2 0.84 0.00148 0.158 0.00656 0.00249 3.2 0 0 0.0022 0.00125
857-3 0.36 4.22e-05 0.627 0.0024 0.0111 0.017 0.002 1.9 0.00152 0.00126
857-4 0.278 0.0004 0.719 0.00392 0.00162 0.09 0.00152 0.8 0.00149 0.000558

(Planck) High Frequency Instrument

Readout Electronic Unit

Calibration and Performance Verification

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