Phase readout, frequency distribution, and clock error modelling

We show in Fig. 5 an overview of the LISA phase readout chain, adapted from [15] (where technical details on the phase readout and frequency distribution system can be found). The optical beatnotes are converted to electrical signals by photoreceivers, which are then digitized by an ADC. The phase of these digital signals are then tracked by DPLLs.

The phasemeter is driven by an \(80\,\mathrm{MHz}\) clock signal. Inside the phasemeter, the ADC samples the electrical beatnotes at the same rate, with an additional timing jitter intrinsic to the ADC. This ADC jitter results in a phase error in the measured beatnotes, which is expected above the requirements. To correct for the ADC jitter, an additional periodic pilot tone signal at \(75\,\mathrm{MHz}\) is derived from the on-board clock and superimposed on each electrical signal fed to the phasemeter. The pilot tone phase is tracked alongside the main beatnotes in dedicated DPLL channels. By comparing the measured pilot tone phase against its nominal \(75\,\mathrm{MHz}\) value, the ADC jitter can be corrected in the main beatnotes, such that the pilot tone becomes the effective reference clock signal for the phase measurements.

Phase readout chain

Fig. 5 Overview of the phase readout chain of LISA, adapted from [15]. The photoreceiver converts the optical signal to electric signals, which are then digitized by the ADC. The intrinsic timing jitter of the ADC is corrected by a \(75\,\mathrm{MHz}\) pilot tone superimposed to the photoreceiver signal. The phasemeter DPLLs then tracks the beatnote phases.

Readout noise

We directly simulate the optical beatnote frequencies. Our simulated electrical signals are therefore the same quantities, with the addition of a readout noise term \(N^\text{ro}(\tau)\). This readout noise accounts for both shot noise and any errors due to front-end electronic in the photoreceivers.

Phasemeter and pilot tone

Because the photoreceiver signals are already simulated as discrete beatnote frequency samples, we do not directly simulate the digitization process of the ADC nor the phase tracking by the DPLLs.

Furthermore, we do not simulate the pilot tone correction, but assume that it perfectly removes the ADC jitter. We do account for timing errors in the pilot tone itself, which are also expected above the requirements. These pilot tone errors will be corrected using the sidebands introduced in Laser beam model. Refer to the subsequent sections for how we model the pilot tone and sideband signals.

Frequency distribution and clock signals

Frequency distribution scheme

Frequency distribution

Fig. 6 Overview over the USO frequency distribution on one optical bench, based on [15]. Not depicted is a possible electrical comparison between the \(2.4\,\mathrm{GHz}\) and \(2.401\,\mathrm{GHz}\) signals. Note that the phasemeter clock is not performance critical, and could be synthesized from either the \(2.4\,\mathrm{GHz}\) signal or directly from the USO, indicated by two possible connections in dotted lines.

Most subsystems on board LISA are driven by timing signals derived from the USO. In our simulation model, we focus on processes for which timing is performance critical, which are summarized in Fig. 6.

Following the current mission design, each LISA spacecraft uses one dedicated clock (realized by an USO), from which all timing signals are derived. As described above, the timing reference for all phasemeter measurements is the pilot tone, which is derived from the USO by first up-converting its nominal frequency[1] to \(2.4\,\mathrm{GHz}\), and then converting that signal to the desired \(\nu_\text{PT} = 75\,\mathrm{MHz}\) using frequency dividers. This conversion chain allows for a very stable phase relationship between the electrical pilot tone and the \(2.4\,\mathrm{GHz}\) optical sideband [15], which are used in postprocessing to reduce the timing errors of the pilot tone itself [2, 16, 17, 18, 19, 20].

The \(2.401\,\mathrm{GHz}\) sidebands used on right MOSAs, on the other hand, are less stable with respect to the pilot tone. This is acceptable, as additional clock noise in this signal can be corrected for using the sideband beatnotes in the REF [2, 20].

Lastly, any errors in the \(80\,\mathrm{MHz}\) phasemeter clock are also corrected by the pilot tone correction, such that it is not performance critical and could either be directly synthesized from the USO or from the \(2.4\,\mathrm{GHz}\) signal. This choice is currently irrelevant for our simulation since we directly simulate the pilot tone as reference clock for all measurements.

Clock-signal model

We model the pilot tone signal as a periodic signal of the form

\[V_\text{PT}(\tau) = \cos(2\pi \nu_\text{PT}[\tau + q_1(\tau)]).\]

Here, \(q_1(\tau)\) describes the timing deviations of the pilot tone generated on spacecraft 1 with respect to the TPS \(\tau_1\), expressed in the latter. Note that the dominant noise source in the pilot tone generation is the USO itself [15], such that we assume the statistical properties of the pilot tone noise to be identical to those of the USO noise.

We further decompose \(q_1(\tau)\) using two time series,

(11)\[q_1(\tau) = q_1^o(\tau) + q_1^\epsilon(\tau),\]

to model large deterministic effects (such as clock frequency offsets and drifts) and small in-band stochastic fluctuations. As before, we do not simulate the \(75\,\mathrm{MHz}\) signal itself, but only \(q_1^o(\tau)\) and \(q_1^\epsilon(\tau)\) (or rather \(\dot q_1^o(\tau)\) and \(\dot q_1^\epsilon(\tau)\) as the pilot tones fractional frequency fluctuations).

The clock signal is used to create the sidebands, as described in Local beams. The total phase of the sideband modulation signals is modeled as

\[\nu^m_{12} \cdot (\tau + q_1(\tau) + M_{12}(\tau)).\]

Here, \(\nu_{12}^m\) is the constant nominal frequency [2] of the modulating signal on optical bench 12. Imperfections in the frequency conversion between the pilot tone and the sidebands are modeled by an additional modulation noise term \(M_{12}(\tau)\).

Timer model

In order to model timestamping and pseudoranging (see PRN Ranging), we not only need the frequency fluctuations of the local clock, but also the time shown by each spacecraft timer. These times must be tracked down to at least nanosecond-precision while reaching values of around \(10^8\,\mathrm{s}\) at the end of the 10 years of extended mission. The use of double-precision floating-point numbers is not compatible with such a dynamic range. Therefore, we simulate offsets of that timer relative to the associated TPS \(\delta\hat\tau_1^{\tau_1}(\tau) \equiv \delta\hat\tau_1(\tau)\), called timer deviations, which evolve slowly with time. The total clock time [3] \(\hat\tau_1^{\tau_1}(\tau)\) as a function of the TPS can then be computed by

(12)\[\hat\tau_1^{\tau_1}(\tau) = \tau + \delta\hat\tau_1(\tau).\]

Timer deviations are closely related to the clock timing jitter,

\[\delta\hat\tau_1(\tau) = q_1(\tau) + \delta\hat\tau_{1,0}.\]

In this equation, \(\delta\hat\tau_{1,0}\) accounts for the fact that we don’t know the true time \(\tau_{1,0}\) at which we turn on the timer, i.e., we can’t directly relate the initial phase of the clock signal \(q_1(\tau_{1,0})\) to any external time frame.

Signal sampling

Signal sampling in terms of phase

The photoreceiver signals recorded, say, on spacecraft 1, are generated according to the TPS \(\tau_1\). The measurements that are eventually telemetered, however, are recorded and timestamped with clock time \(\hat{\tau}_1\). As a consequence, we need to resample the photoreceiver signals from the TPS to the clock time frame.

If a photoreceiver signal \(\Phi_\text{PD}\) is expressed in terms of phase, this can be achieved following

\[\Phi_\text{PD}^{\hat\tau_1}(\tau) = \Phi_\text{PD}^{\tau_1}(\tau_1^{\hat\tau_1}(\tau)).\]

Therefore, we need to compute the TPS \(\tau_1^{\hat\tau_1}(\tau)\) given a given clock time \(\tau\). This quantity can be computed by applying the inverse transformation and therefore writing

\[\begin{split}\tau &\equiv \hat\tau_1^{\tau_1}(\tau_1^{\hat\tau_1}(\tau)) \\ &= \tau_1^{\hat\tau_1}(\tau) + \delta\hat\tau_1(\tau_1^{\hat\tau_1}(\tau)).\end{split}\]

Rearranging, we get

\[\tau_1^{\hat\tau_1}(\tau) = \tau - \delta\hat\tau_1(\tau_1^{{\hat\tau_1}}(\tau)).\]

We can solve this implicit equation for \(\tau_1^{\hat\tau_1}(\tau)\) iteratively, by computing

\[ \begin{align}\begin{aligned}\delta\hat\tau_1^{(0)}(\tau) &= \delta\hat\tau_1(\tau),\\\delta\hat\tau_1^{(n+1)}(\tau) &= \delta\hat\tau_1(\tau - \delta\hat\tau_1^{(n)}(\tau)),\end{aligned}\end{align} \]

such that

\[\lim_{n \rightarrow \infty} \delta\hat\tau_1^{(n)}(\tau) = \delta\hat\tau_1(\tau_1^{{\hat\tau_1}}(\tau)).\]

Since the timer deviations are evolving slowly, the iteration converges quickly. In our simulations, we stop after two iterations, such that

\[\tau_1^{\hat\tau_1}(\tau) \approx \tau - \delta\hat\tau_1^{(2)}(\tau).\]

We can then plug the previous equation in the earlier expression to write all frame-independent measurements as a functions of the correct recording times \(\Phi_\text{PD}^{\hat\tau_1}(\tau)\), given the same quantities expressed in the TPS. We find

\[\Phi_\text{PD}^{\hat\tau_1}(\tau) \approx \Phi_\text{PD}^{\tau_1}(\tau - \delta\hat\tau_1^{(2)}(\tau)).\]

This operation can be implemented with time-varying fractional delay filter (interpolation).

We introduce the timestamping operator \(\mathcal{T}_1\), which shifts a signal \(s(\tau)\) from the TPS to the clock time of spacecraft 1. Formally, its action is given by

\[\mathcal{T}_1 s(\tau) = s(\tau - \delta\hat\tau_1^{(2)}(\tau)).\]

Using this shorthand notation, the previous equation now reads

\[\Phi_\text{PD}^{\hat\tau_1}(\tau) = \Phi_\text{PD}^{\tau_1}(\tau_1^{\hat\tau_1}(\tau)) \approx \mathcal{T}_1 \Phi_\text{PD}^{\tau_1}(\tau).\]

Note that this is only valid for measurements expressed in phase, as frequencies are not frame-independent quantities.

Sampling errors in terms of frequency

The effect of sampling can also be expressed in terms of total frequency, where it manifests itself as a Doppler-like frequency shift.

In the following paragraph, we compute frequencies by taking the derivative of phase with respect to the clock time, since this is the time reference that the phasemeter will use to measure the signal frequency. From the previous equation, and denoting function composition as \((\Phi^{\tau_1}_\text{PD} \circ \tau_1^{\hat\tau_1})(\tau) = \Phi^{\tau_1}_\text{PD}(\tau_1^{\hat\tau_1}(\tau))\), we have

\[\nu_\text{PD}^{\hat\tau_1}(\tau) = \frac{d}{d\tau} \Phi_\text{PD}^{\hat\tau_1}(\tau) = \frac{d}{d\tau} (\Phi^{\tau_1}_\text{PD} \circ \tau_1^{\hat\tau_1})(\tau).\]

Using the chain rule,

\[\nu_\text{PD}^{\hat\tau_1}(\tau) = \nu_\text{PD}^{\tau_1}(\tau_1^{\hat\tau_1}(\tau)) \times \frac{d\tau_1^{\hat\tau_1}}{d\tau}(\tau).\]

To compute the derivative of \(\tau_1^{\hat\tau_1}(\tau)\), let us differentiate the defining implicit equation,

\[\frac{d\tau_1^{\hat\tau_1}}{d\tau}(\tau) = 1 - \frac{d}{d\tau}(\delta\hat\tau_1 \circ \tau_1^{\hat\tau_1})(\tau) = 1 - \frac{d\delta\hat\tau_1}{d\tau}(\tau_1^{\hat\tau_1}(\tau)) \times \frac{d\tau_1^{\hat\tau_1}}{d\tau}(\tau).\]

Using the previous definition, we find \(\frac{d\delta\hat\tau_1}{d\tau}(\tau) = \dot q_1(\tau)\). Inserting this identity, we can rearrange the previous equation to get

\[\frac{d\tau_1^{\hat\tau_1}}{d\tau}(\tau) = \frac{1}{1 + \dot q_1(\tau_1^{\hat\tau_1}(\tau))},\]

which finally yields for the total frequency,

\[\nu_\text{PD}^{\hat\tau_1}(\tau) = \frac{\nu_\text{PD}^{\tau_1}(\tau_1^{\hat\tau_1}(\tau))}{1 + \dot q_1(\tau_1^{\hat\tau_1}(\tau))} \approx \mathcal{T}_1 \left[ \frac{ \nu_\text{PD}^{\tau_1}(\tau)}{1 + \dot{q}_1(\tau)} \right].\]

Sampling in two-variable decomposition

We now want to describe the effect of timing errors in the framework of two-variable decomposition. This will allow us to split the sampling errors derived previously into large deterministic offsets in the measurement timestamps, and small stochastic fluctuations that enter as an additional noise term. The latter represent what is often referred to as clock noise [10].

However, we want to make it clear once more that this decomposition is entirely artificial. Both slow drifts and in-band clock noise describe the same physical process, namely the instability of the USO, on different time scales.

The sampling process applies to the total phase of each photoreceiver signal, given by

\[\Phi_\text{PD}^{\hat\tau_1}(\tau) = \Phi_\text{PD}^{\tau_1}(\tau_1^{\hat{\tau}_1}(\tau)) = \phi^{o}_\text{PD}(\tau_1^{\hat{\tau}_1}(\tau)) + \phi^\epsilon_\text{PD}(\tau_1^{\hat{\tau}_1}(\tau)).\]

Since \(\phi^o_\text{PD}(\tau)\) is very quickly evolving, small (first-order) timing fluctuations in \(\tau_1^{\hat{\tau}_1}(\tau)\) must appear in the measurement described by \(\phi^\epsilon_\text{PD}(\tau)\). Thus, we must account for the cross coupling between \(\phi^o_\text{PD}(\tau)\) and \(\phi^\epsilon_\text{PD}(\tau)\), and we cannot simply time shift both components individually.

We can insert the previous definitions into the implicit equation to get

\[\tau_1^{\hat{\tau}_1}(\tau) = \tau - \delta\hat\tau_{1,0} - q_1^o(\tau_1^{\hat{\tau}_1}(\tau)) - q_1^\epsilon(\tau_1^{\hat{\tau}_1}(\tau)),\]

We model clock noise fractional frequency fluctuations \(\dot q_1^\epsilon\) as band-limited noise, such that they remain small, and we can expand the \(\Phi_\text{PD}\) term to first order in \(q_1^\epsilon\) and drop mixed terms with \(\phi^\epsilon_\text{PD}\),

\[\begin{split}\Phi_\text{PD}^{\hat\tau_1}(\tau) ={}& \phi^{o}_\text{PD}(\tau - \delta\hat\tau_{1,0} - q_1^o(\tau_1^{\hat{\tau}_1}(\tau))) + \phi^\epsilon_\text{PD}(\tau_1^{\hat{\tau}_1}(\tau)) \\ &- \nu^{o}_\text{PD}(\tau - \delta\hat\tau_{1,0} - q_1^o(\tau_1^{\hat{\tau}_1}(\tau))) q_1^\epsilon(\tau_1^{\hat{\tau}_1}(\tau)).\end{split}\]

Finally, we obtain the two variable-decomposition for the resampled photoreceiver phase.

\[ \begin{align}\begin{aligned}\phi^{\hat\tau_1,o}_\text{PD}(\tau) &\approx \phi^{o}_\text{PD}(\tau - \delta\hat\tau_{1,0} - \mathcal{T}_1 q_1^o(\tau)),\\\phi^{\hat\tau_1,\epsilon}_\text{PD}(\tau) &\approx \mathcal{T}_1 \phi^\epsilon_\text{PD}(\tau) - \nu^{o}_\text{PD}(\tau - \delta\hat\tau_{1,0} - \mathcal{T}_1 q_1^o(\tau)) \mathcal{T}_1 q_1^\epsilon(\tau).\end{aligned}\end{align} \]

For frequency data, we start with the previous result, and decompose again clock noise \(\dot q_1\) into two variables, as explained above. We then expand it to first order in \(\dot q_1^\epsilon\) to get

\[\frac{1}{1 + \dot q_1^o(\tau_1^{\hat\tau_1}(\tau)) + \dot q_1^\epsilon(\tau_1^{\hat\tau_1}(\tau))} \approx \frac{1}{1 + \dot q_1^o(\tau_1^{\hat\tau_1}(\tau))} - \frac{\dot q_1^\epsilon(\tau_1^{\hat\tau_1}(\tau))}{[1 + \dot q_1^o(\tau_1^{\hat\tau_1}(\tau))]^2}.\]

So in total, we have

\[\nu_\text{PD}^{\hat\tau_1}(\tau) \approx \nu_\text{PD}^{\tau_1}(\tau_1^{\hat{\tau}_1}(\tau)) \times \left[\frac{1}{1 + \dot q_1^o(\tau_1^{\hat\tau_1}(\tau))} - \frac{\dot q_1^\epsilon(\tau_1^{\hat\tau_1}(\tau))}{[1 + \dot q_1^o(\tau_1^{\hat\tau_1}(\tau))]^2}\right].\]

We now expand \(\nu_\text{PD}^{\tau_1}(\tau) = \nu_\text{PD}^{\tau_1,o}(\tau) + \nu_\text{PD}^{\tau_1,\epsilon}(\tau)\) (and in the same way its counterpart \(\nu_\text{PD}^{\hat\tau_1}(\tau)\)), and neglect the small coupling of \(q_1^\epsilon(\tau)\) to the already small fluctuations \(\nu_\text{PD}^{\tau_1,\epsilon}(\tau)\). We collect the terms to express the photodiode signal offsets \(\nu_\text{PD}^{\hat\tau_1,o}(\tau)\) and fluctuations \(\nu_\text{PD}^{\hat\tau_1,\epsilon}(\tau)\) after shifting to the clock time, using the previous result,

\[ \begin{align}\begin{aligned}\nu^{\hat\tau_1,o}_\text{PD}(\tau) &\approx \frac{\mathcal{T}_1 \nu_\text{PD}^{\tau_1,o}(\tau)}{1 + \mathcal{T}_1 \dot q_1^o(\tau)},\\\nu^{\hat\tau_1,\epsilon}_\text{PD}(\tau) &\approx \frac{\mathcal{T}_1 \nu_\text{PD}^{\tau_1,\epsilon}(\tau)}{1 + \mathcal{T}_1 \dot q_1^o(\tau)} - \frac{\mathcal{T}_1\nu_\text{PD}^{\tau_1,o}(\tau) \mathcal{T}_1 \dot q_1^\epsilon(\tau)}{[1 + \mathcal{T}_1 \dot q_1^o(\tau)]^2}.\end{aligned}\end{align} \]

To simplify further our equations, we define the frequency timestamping operator, which includes the rescaling by \(1 + \dot{q}_1^o\). It is formally defined by its action on a signal \(s(\tau)\),

\[\mathcal{D}_1 s(\tau) = \mathcal{T}_1 \left[\frac{s(\tau)}{1 + \dot q_1^o(\tau)}\right] = \frac{\mathcal{T}_1 s(\tau)}{1 + \mathcal{T}_1 \dot q_1^o(\tau)},\]

Now, photoreceiver frequency signals in the clock time frame of spacecraft 1 read

\[ \begin{align}\begin{aligned}\nu^{\hat\tau_1,o}_\text{PD}(\tau) &\approx \mathcal{D}_1 \nu_\text{PD}^{\tau_1,o}(\tau),\\\nu^{\hat\tau_1,\epsilon}_\text{PD}(\tau) &\approx \mathcal{D}_1\left[ \nu_\text{PD}^{\tau_1,\epsilon}(\tau) - \frac{\nu_\text{PD}^{\tau_1,o}(\tau) \dot q_1^\epsilon(\tau)}{1 + \dot q_1^o(\tau)}\right].\end{aligned}\end{align} \]

[1]

No final decision has been made on the precise USO frequency that will be used for LISA.

[2]

By definition, these frequencies are at their nominal values. The real modulation signals will have a frequency offset due to the terms \(q_1\) and \(M_{12}\) in the total phase.

[3]

This timescale will be realized in practice by the so-called SCET, which is the only timescale directly available onboard the satellites.