Onboard processing

In this section, we describe the processing steps the readout signals undergo on board the spacecraft, and in particular the filtering and downsampling steps. The sampling rates used in our simulation are shown schematically in Fig. 7. We then give the expressions of the main measurement signals, which are the main outputs of the simulation.

Fig. 7 Overview of the real signal sampling rates, from continuous optical and electrical signals, to the cascaded downsampled signals down to the telemetered 4 Hz data. We also indicate the sampling rates used in the simulation (in grey boxes) to represent these signals; continuous and high-frequency signals are represented by discrete 16 Hz simulated quantities, while the telemetry data is simulated at their true 4 Hz rate.

Filtering and downsampling

Physics sampling rate

As described in Phase readout, frequency distribution, and clock error modelling and following, the current mission design, the onboard phasemeters track the phase (or, equivalently, the instantaneous frequency) of sampled and digitized versions of the MHz beatnotes using DPLLs running at 80 MHz.

For performance reasons, we cannot simulate continuous analog signals nor DPLL signals at their real sampling rate. Instead, we use a discretized representation and rely on high-level models to capture the most significant effects. In our simulations, continuous quantities, as well as photoreceiver signals and beatnote measurements, are simulated at the physics rate

\[f_s^{\mathrm{phy}} = 16\,\mathrm{Hz}.\]

Note that this physics rate matches the penultimate downsampling step of the real onboard decimation chain (described below), which is used by the DFACS.

Antialiasing filters

The current mission design suggests that the raw 80 MHz phasemeter beatnote signals are then filtered and downsampled to various lower sampling rates, and ultimately to the final measurement rate of 4 Hz. This last measurement sampling rate is in line with the mission instrument design, and compatible with the limited telemetry budget and the required bandwidth for on-ground processing. The 4 Hz data are then telemetered down to Earth.

High-order digital low-pass FIR filters, as well as cascading filters, are expected to be used to prevent noise aliasing in the frequency band relevant for LISA data analysis, between 1e-4 Hz and 1 Hz [21]. These filters must strongly attenuate the signals above the Nyquist frequency, while maintaining a gain close to unity and low phase distortion below 1 Hz. Their precise implementation is still under development.

In the simulation, we use a single digital symmetrical FIR filter to go from \(f_s^{\mathrm{phy}}\) to the final measurement sampling rate of

\[f_s^{\mathrm{meas}} = 4\,\mathrm{Hz}.\]

Decimation

Once the beatnote frequency measurements are filtered, we use a four-fold decimation (we select 1 sample out of 4) to produce the final 4 Hz telemetry data. They are the main output of the simulation.

Analytically, we model the filtering and downsampling step with the continuous, linear filter operator \(\mathcal{F}\), which is applied to the beatnote frequency measurements.

Telemetered beatnote measurements

We summarize here the downsampled, filtered beatnote measurements output by the phasemeter, i.e., the interspacecraft, test-mass, reference carrier and sideband beatnote frequencies. They are ultimately telemetered down to Earth[1].

Beatnote measurement notation

For these beatnote measurements, we introduce a clear notation that uses the name of the associated interferometer and its index, complemented by the type of beam (carrier or sideband). The real phasemeter will only produce the total frequency or the total phase of the signal. For our studies, however, it is often useful to also have access to the underlying offsets and fluctuations in two separate variables, which is why we give here the signals in this form. The simulation will provide an additional output for the total frequency, given as the sum of the two components.

For readability’s sake, we drop all time arguments. We use delay operators to account for time shifts that appear when propagating signals. We denote \(\mathcal{D}_{12}\) the delay operator associated with the PPR \(d^o_{12}(\tau)\) defined in Distant beams, such that for any signal \(s(\tau)\),

(13)\[\mathcal{D}_{12} s(\tau) = s(\tau - d^o_{12}(\tau))\]

Furthermore, we introduce the Doppler-delay operator, which is defined as

\[\dot{\mathcal{D}}_{12} s(\tau) = (1 - \dot d^o_{12}(\tau)) s(\tau - d^o_{12}(\tau))\]

We also make use of the timestamping operators \(\mathcal{T}_i\) introduced in Signal sampling, and the downsampling and filtering operator \(\mathcal{F}\).

We will also use a shorthand notation for the beatnote frequency offsets in the TPS, which we define by

(14)\[ \begin{align}\begin{aligned}a^\mathrm{c}_{12} &\equiv \nu_{\mathrm{sci}_{12},\mathrm{c}}^o = \dot{\mathcal{D}}_{12} O_{21} - \nu_0 \dot d^o_{12} - O_{12}\\a_{12}^\mathrm{sb} &\equiv \nu_{\mathrm{sci}_{12},\mathrm{sb}}^o = a^\mathrm{c}_{12} + \dot{\mathcal{D}}_{12} \left[ \nu_{21}^m (1 + \dot q_2^o) \right] - \nu_{12}^m (1 + \dot q_1^o)\\b^\mathrm{c}_{12} &\equiv \nu_{\mathrm{ref}_{12},\mathrm{c}}^o = O_{13} - O_{12}\\b_{12}^\mathrm{sb} &\equiv \nu_{\mathrm{ref}_{12},\mathrm{sb}}^o = b^\mathrm{c}_{12} + (\nu_{13}^m - \nu_{12}^m) (1 + \dot q_1^o)\end{aligned}\end{align} \]

In addition, most of the laser-related terms \(p_{12}\), \(O_{12}\) will be determined by the laser locking scheme, as described in Laser locking and frequency planning.

Interspacecraft interferometer beatnote frequencies

The carrier-carrier beatnote frequency measurement in the SCIs contains the delayed distant and local laser frequency fluctuations \(\dot{p}_{21}\) and \(\dot{p}_{12}\), as well as the delayed distant and local optical-bench path length noises appearing as Doppler shifts \(\dot{N}^\mathrm{ob}_{\mathrm{sci}_{12} \leftarrow 21}\) and \(\dot{N}^\mathrm{ob}_{\mathrm{sci}_{12} \leftarrow 12}\). The effect of the gravitational-wave signal \(\dot{H}_{12}\) also appears as an extra Doppler shift on the distant beam, alongside TTL effects. Lastly, the readout \(\dot{N}^\mathrm{ro}_{\mathrm{sci}_{12},\mathrm{c}}\) and clock noise \(a^\mathrm{c}_{12} \dot q_1^\epsilon/(1 + q_1^o)\) terms are added.

(15)\[ \begin{align}\begin{aligned}\mathrm{sci}_{12,\mathrm{c}}^o &= \mathcal{F} \mathcal{T}_1 a^\mathrm{c}_{12}\\\mathrm{sci}_{12,\mathrm{c}}^\epsilon &= \mathcal{F} \mathcal{T}_1 \Big\{ \dot{\mathcal{D}}_{12} \dot{p}_{21} + (\nu_0 + \mathcal{D}_{12} O_{21}) (\dot H_{12} - \dot{\mathrm{TTL}}_{12})\\&\quad + \frac{\nu_0}{c} \dot{N}^\mathrm{ob}_{\mathrm{sci}_{12} \leftarrow 21} - \left( \dot{p}_{12} + \frac{\nu_0}{c} \dot{N}^\mathrm{ob}_{\mathrm{sci}_{12} \leftarrow 12} \right) + \dot{N}^\mathrm{ro}_{\mathrm{sci}_{12},\mathrm{c}} - \frac{a^\mathrm{c}_{12} \dot{q}_1^\epsilon}{1 + \dot{q}_1^o} \Big\}\\\mathrm{sci}_{12,\mathrm{c}} &= \mathrm{sci}_{12,\mathrm{c}}^o + \mathrm{sci}_{12,\mathrm{c}}^\epsilon\end{aligned}\end{align} \]

The sideband-sideband beatnote frequency measurement is similar with two main differences. First, the distant and local laser frequency fluctuations are affected by the coupling of the modulation frequency with clock jitter and modulation noise \(\nu_{21}^m (\dot{q}_2^\epsilon + \dot{M}_{21})\) and \(\nu_{12}^m (\dot{q}_1^\epsilon + \dot{M}_{12})\). Secondly, the distant sideband beatnote frequency offsets are shifted by the modulation frequency affected by out-of-band clock errors \(\nu^m_{21} (1 + \dot{q}^o_2)\). Overall, we get

(16)\[ \begin{align}\begin{aligned}\mathrm{sci}_{12,\mathrm{sb}}^o &= \mathcal{F} \mathcal{T}_1 a_{12}^\mathrm{sb}\\\mathrm{sci}_{12,\mathrm{sb}}^\epsilon &= \mathcal{F} \mathcal{T}_1 \Big\{ \dot{\mathcal{D}}_{12} \left( \dot{p}_{21} + \nu_{21}^m (\dot{q}_2^\epsilon + \dot{M}_{21}) \right) + \left( \nu_0 + \mathcal{D}_{12} [O_{21} + \nu^m_{21} (1 + \dot{q}^o_2)] \right) (\dot{H}_{12} - \dot{\mathrm{TTL}}_{12})\\&\quad + \frac{\nu_0}{c} \dot{N}^\mathrm{ob}_{\mathrm{sci}_{12} \leftarrow 21} - \left( \dot{p}_{12} + \nu_{12}^m (\dot{q}_1^\epsilon + \dot{M}_{12}) + \frac{\nu_0}{c} \dot{N}^\mathrm{ob}_{\mathrm{sci}_{12} \leftarrow 21} \right) + \dot{N}^\mathrm{ro}_{\mathrm{sci}_{12},\mathrm{sb}} - \frac{a_{12}^\mathrm{sb} \dot{q}_1^\epsilon}{1 + \dot{q}_1^o} \Big\}\\\mathrm{sci}_{12,\mathrm{sb}} &= \mathrm{sci}_{12,\mathrm{sb}}^o + \mathrm{sci}_{12,\mathrm{sb}}^\epsilon\end{aligned}\end{align} \]

Reference interferometer beatnote frequencies

The carrier-carrier beatnote frequency measurement in the REFs contains the frequency fluctuations of the adjacent and local laser beams \(\dot{p}_{13}\) and \(\dot{p}_{12}\), as well as the associated optical-bench path length noises \(\dot{N}^\mathrm{ob}_{\mathrm{ref}_{12} \leftarrow 13}\) and \(\dot{N}^\mathrm{ob}_{\mathrm{ref}_{12} \leftarrow 12}\). The adjacent beam that travels through the optical fiber picks up the backlink noise \(\dot{N}^\mathrm{bl}_{12}\). The readout \(\dot{N}^\mathrm{ro}_{\mathrm{ref}_{12},\mathrm{c}}\) and clock noise \(b^\mathrm{c}_{12} \dot{q}_1^\epsilon/(1 + q_1^o)\) terms are then added.

(17)\[ \begin{align}\begin{aligned}\mathrm{ref}_{12,\mathrm{c}}^o &= \mathcal{F} \mathcal{T}_1 b^\mathrm{c}_{12}\\\mathrm{ref}_{12,\mathrm{c}}^\epsilon &= \mathcal{F} \mathcal{T}_1 \Big\{ \dot{p}_{13} + \frac{\nu_0}{c} \left( \dot{N}^\mathrm{ob}_{\mathrm{ref}_{12} \leftarrow 13} + \dot{N}^\mathrm{bl}_{12} \right) - \left( \dot{p}_{12} + \frac{\nu_0}{c} \dot{N}^\mathrm{ob}_{\mathrm{ref}_{12} \leftarrow 12} \right) + \dot{N}^\mathrm{ro}_{\mathrm{ref}_{12},\mathrm{c}} - \frac{b^\mathrm{c}_{12} \dot{q}_1^\epsilon}{1 + \dot{q}_1^o} \Big\}\\\mathrm{ref}_{12,\mathrm{c}} &= \mathrm{ref}_{12,\mathrm{c}}^o + \mathrm{ref}_{12,\mathrm{c}}^\epsilon\end{aligned}\end{align} \]

The expression for the sideband-sideband beatnote frequency measurement follows the same logic, with the adjacent and local laser frequency fluctuations affected by the in-band clock and modulation noises \(\nu_{13}^m (\dot{q}_1 + \dot{M}_{13})\) and \(\nu_{12}^m (\dot{q}_1 + \dot{M}_{12})\),

(18)\[ \begin{align}\begin{aligned}\mathrm{ref}_{12,\mathrm{sb}}^o &= \mathcal{F} \mathcal{T}_1 b^\mathrm{sb}_{12}\\\mathrm{ref}_{12,\mathrm{sb}}^\epsilon &= \mathcal{F} \mathcal{T}_1 \Big\{ \dot{p}_{13} + \nu_{13}^m (\dot{q}_1 + \dot{M}_{13}) + \frac{\nu_0}{c} \left( \dot{N}^\mathrm{ob}_{\mathrm{ref}_{12} \leftarrow 13} + \dot{N}^\mathrm{bl}_{12} \right)\\&\quad - \left( \dot{p}_{12} + \nu_{12}^m (\dot{q}_1 + \dot{M}_{12}) + \frac{\nu_0}{c} \dot{N}^\mathrm{ob}_{\mathrm{ref}_{12} \leftarrow 12} \right) + \dot{N}^\mathrm{ro}_{\mathrm{ref}_{12},\mathrm{sb}} - \frac{b^\mathrm{sb}_{12} \dot{q}_1^\epsilon}{1 + \dot{q}_1^o} \Big\}\\\mathrm{ref}_{12,\mathrm{sb}} &= \mathrm{ref}_{12,\mathrm{sb}}^o + \mathrm{ref}_{12,\mathrm{sb}}^\epsilon\end{aligned}\end{align} \]

Test-mass interferometer beatnote frequencies

The carrier-carrier beatnote frequency measurements in the TMI have the same form as for the REF, with the exception of the additional local test-mass noise term \(\dot{N}^\delta_{12}\),

(19)\[ \begin{align}\begin{aligned}\mathrm{tmi}_{12,\mathrm{c}}^o &= \mathcal{F} \mathcal{T}_1 b^\mathrm{c}_{12}\\\mathrm{tmi}_{12,\mathrm{c}}^\epsilon &= \mathcal{F} \mathcal{T}_1 \Big\{ \dot{p}_{13} + \frac{\nu_0}{c} \left( \dot{N}^\mathrm{ob}_{\mathrm{tmi}_{12} \leftarrow 13} + \dot{N}^\mathrm{bl}_{12} \right) - \left( \dot{p}_{12} + \frac{\nu_0}{c} \left( \dot{N}^\mathrm{ob}_{\mathrm{tmi}_{12} \leftarrow 12} + 2 \dot{N}^\delta_{12} \right) \right)\\&\quad + \dot{N}^\mathrm{ro}_{\mathrm{tmi}_{12},\mathrm{c}} - \frac{b^\mathrm{c}_{12} \dot{q}_1^\epsilon}{1 + \dot{q}_1^o} \Big\}\\\mathrm{tmi}_{12,\mathrm{c}} &= \mathrm{tmi}_{12,\mathrm{c}}^o + \mathrm{tmi}_{12,\mathrm{c}}^\epsilon\end{aligned}\end{align} \]

As mentioned previously, we do not model sideband-sideband beatnote measurements in the TMIs.

Warning

Due to gls{dfacs}, the test mass noise should actually not show up in the test mass ifo anymore, since the spacecraft follows the test mass (at least to a certain degree, at low frequencies).

[1]

As mentioned before, there are other data streams, such as the angular readouts provided by DWS, see Auxiliary measurements. The MPR measurements are described in PRN Ranging.