HFI time response model

LFER4 model (PR1 /2013 release)

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

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

PR2 (2015) and PR3 (2018) Model

For the data associated to the PR2 (2015) and PR3 (2018) releases, the bolometer time transfer function is described by the sum of five single-pole low-pass functions, each with a time constant τi and an associated amplitude ai. The parameters for the model are in the table below (from Planck-2015-A07).