signals, correctly rescaled, to obtain
νPDτi,e(τ) 1+q˙oi(τ)− ν
τi,o
PD(τ)q˙ei(τ)
[1+q˙oi(τ)]2 . (6.31)
Note that all quantities are still evaluated atτ.
Lastly, these quantities are resampled at τ−δτˆi2(τ)using Lagrange interpola-tion to retrieve eqs. (6.28) and (6.29).
For clarity’s sake, we drop the time arguments in our signals, such that they are inherently given in the corresponding spacecraft THE. We introduce the first-order timestamping operator Ti, which shifts a signals(τ)from the TPS to the THE of spacecrafti. Formally, its action is given by
Tis(τ) =s(τ−δτˆi2(τ)). (6.32) Its pendant for frequency includes the rescaling by 1+qoi, and is defined as
T˙is(τ) =Ti
s(τ) 1+q˙oi(τ)
. (6.33)
The photodiode signals read, in the THE, νPDτˆi,o ≈T˙iντPDi,o and νPDτˆi,e(τ)≈T˙i
νPDτi,e− ν
τi,o PDq˙ei 1+q˙oi
, (6.34)
where we suppressed the explicit time argument.
6.3 pseudo-ranging
As sketched in section3.2, LISA noise reduction algorithms, such as TDI, require knowledge of the propagation time of the laser beams between the spacecraft, orranging. Technically, this is realized in LISA by generating a PRN code representing the time shown by the local clock on the sending spacecraft, which is modulated and transmitted via the outgoing laser beam. This code is then recovered from the phase measurement of that beam, and correlated with a local copy of the code generated according to the receiving spacecrafts clock. We call the result the measured pseudo-range (MPR). See section3.6.4 (and references there) for more details on the technical implementation.
In this section, we first derive the equations for the MPRs in terms of clock time differences, c.f. section6.3.1. Since we only simulate timer deviations, we reformulate these equations in section6.3.2so they are directly applicable to our simulation.
Pseudo-ranging simulation is performed at fsphy, while the MPRs are ulti-mately downsampled to a lower rate fsmeas, alongside the other measure-ments.
78 clock and timing distribution model
6.3.1 Pseudo-ranging as clock time difference
Since the PRN code is modulated onto the laser beam, it suffers the same delays when propagating between the spacecraft3.
As described in section5.2.3, we do not model the actual phase modulation of the PRN. Instead, we directly compute the difference Rij between times shown by local clocks (i.e. THE times).
Following the conventions used in section5.5, we consider in the following paragraphs a beam received by optical bench ijat the receiver TPSτ, which was emitted from optical bench ji at emitter TPSτji(τ) = τ−dij(τ). Here, the PPRdij(τ)contains the photon time of flight, as well as the conversion between time coordinates.
Conceptually, the MPR measures the difference between the THE ˆτiτi(τ) shown by the local clock of the receiving spacecraft at the event of reception of the beam, and the THE ˆτjτj(τ−dij(τ)) shown by the local clock of the sending spacecraft at the event of emission of the beam. Thus, we can model the MPR as the difference
Rij(τ) =τˆi(τ)−τˆj(τ−dij(τ)) +NijR(τ), (6.35) whereNijR(τ)is a ranging noise term modelling imperfections in the overall correlation scheme. See appendix D.7for its model in the simulation.
6.3.2 Pseudo-ranging in terms of timer deviations
As explained in section6.1.3, we do not simulate the total THE ˆτi(τ)for each spacecraft, but only deviationsδτˆi(τ)from the TPS,
τˆi(τ) =τ+δτˆi(τ) and ˆτj(τ) =τ+δτˆj(τ). (6.36) Inserting these definitions into eq. (6.35) yields
Rij(τ) =δτˆi(τ)−δτˆj τ−dij(τ)−dij(τ)+NijR(τ). (6.37) Let us defineδτˆisci←j(τ),
δτˆisci←j(τ)≈δτˆj(τ−dij(τ))−dij(τ), (6.38) the clock time of the sending spacecraft propagated to the photodiode of the distant inter-spacecraft interferometer.
We can then express the MPR as the simple difference
Rij(τ)≈δτˆi(τ)−δτˆiscij←j(τ) +NijR(τ). (6.39) This is the measurement we generate in the simulation, where we make the additional assumption that dij ≡ doij for this measurement. This is valid,
3 We assume here that the vacuum between the satellites is sufficiently good that we can neglect (or compensate) any dispersion effects.
6.3 pseudo-ranging 79 since the terms contained in deij only create timing jitters much less than a nanosecond (cf. section5.5).
Notice that in eq. (6.39), we compute the MPR as a function of the receiving TPSs, so that formallyRij = Rτiji. In reality, the MPR is measured according to the THE of the receiving spacecraft, Rτijˆi. Similarly to all other measurement, we simulate this by first generating Rτiji and then resampling the resulting time series to get Rτijˆi, as described in section 6.2.
O N B O A R D P R O C E S S I N G
7
The onboard phasemeters track the phase evolution (or, almost equivalently, the instantaneous frequencies) of sampled and digitized versions of the MHz beatnotes described in section5.6, using a digital phase-locked loop (DPLL) which runs at 80 MHz. The resultant phase is downsampled in multiple steps (cf. [36]) to the final measurement rate.
As described above, we do not simulate the phasemeter at this high sampling rate, but instead rely on high-level models to capture the most significant effects.
During the sampling process, clock imperfections couple into the measure-ments. We described this effect in chapter6, and modelled its impact on the phase of the digitized signal by shifting all measurements from the TPS to the THE. We will now assume that the DPLL is able to perfectly reconstruct the phase of this digitized signal.
Following [47], we assume that one of the last downsampling steps inside the phasemeter creates a timeseries at fsphy=16 Hz, which then gets further filtered and decimated to fsmeas =4 Hz.
We model this last step in the processing chain by including a digital finite impulse response (FIR) filter in our simulation, which we model in section7.1.
The resulting phasemeter signals are given in section7.2.
These phasemeter equations still contain unevaluated terms for the laser offsets and frequency fluctuations, which are determined by the laser locking schemes. We describe these locking conditions and the different locking schemes in section7.3.
Finally, we conclude by describing additional data streams available from contact to ground stations in section 7.4.
7.1 filtering and downsampling
High-order digital low-pass FIR filters are expected to be used to prevent noise aliasing in the frequency band relevant for LISA data analysis, between 10−4Hz and 1 Hz [56]. Therefore, they must strongly attenuate the signals
As explained below, the transition band can actually extend slightly above the Nyquist frequency, up to
fsmeas−1 Hz.
above the Nyquist frequency, while maintaining a high gain and low phase distortion below 1 Hz. In reality, this will be performed by cascading fil-ters to go all the way from 80 MHz to fsmeas = 4 Hz [36], and their precise implementation is under development.
81
82 onboard processing
Figure 7.1: Antialias-ing filter transfer func-tion magnitude. The transition band (grey) is chosen to avoid aliasing into the mea-surement band (blue).
0 1 2 3 4 5 6 7 8
10−16 10−12 10−8 10−4 100
Frequency [Hz]
Magnitude
Filter transfer function Aliased transfer function
In the simulation, we only use a single filtering and downsampling step to go from fsphyto the final measurement sampling rate of fsmeas=4 Hz.
We build a digital symmetrical FIR filter from a Kaiser windowing function, using the following parameters:
• the transition band extends from 1.1 Hz to 2.9 Hz,
• the minimum attenuation above 2.9 Hz is 240 dB.
Note that the filter transition band extends above the Nyquist frequency, such that there will be a significant amount of aliasing during downsampling, as depicted in fig.7.1. However, since aliasing happens by reflection across the Nyquist frequency, any noise in the band[fs/2, fs−1 Hz]will be aliased into the band[1 Hz,fs/2], such that it stays outside our measurement band of [10−4Hz, 1 Hz]. We will revisit how the aliased noise appears in our final signals in section12.3.3.
The filter coefficientsck are computed based on these parameters using the
kaiserord method of the scipy library [95]. The filter is then applied by computing then0thsample of the output seriesyfrom the input seriesxvia
y[n] =
∑
N k=0ckx[n−k], (7.1)
with Nas the order of the filter.
Analytically, we model the filter by applying a filter operatorFto the photo-diode signals. I.e., we simply write
y(t) =Fx(t), (7.2)
where we extend the definition in eq. (7.1) to also be applicable to a continous time function. Inspecting eq. (7.1),Fis a linear operator.
7.2 phasemeter signals 83