# Aliased laser noise and TDI coupling with LISA On table

Léon Vidal, Hubert Halloin, Nam Dam Quang, Pierre Prat, Antoine Petiteau

## ► To cite this version:

Léon Vidal, Hubert Halloin, Nam Dam Quang, Pierre Prat, Antoine Petiteau. Aliased laser noise and TDI coupling with LISA On table. Classical and Quantum Gravity, 2025, 42 (21), pp.215018. 10.1088/1361-6382/ae134d hal-05349662

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Submitted on 23 Jan 2026

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# Aliased laser noise and TDI coupling with LISA On table

Léon Vidal¹,*, Hubert Halloin¹, Nam Dam Quang¹,², Pierre Prat¹ and Antoine Petiteau³

¹ Université Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France

² Normandie Univ, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France

³ IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

E-mail: leon.vidal@apc.in2p3.fr

Received 25 July 2025; revised 6 October 2025

Accepted for publication 14 October 2025

Published 30 October 2025

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

Various noise sources will impact the laser interferometer space antenna (LISA) interferometric measurements, particularly laser frequency noise, which will dominate by several orders of magnitude. To achieve the required sensitivity for gravitational wave detection, noise reduction methods, such as the time-delay interferometry (TDI) algorithm, must be employed and tested using both computer and instrumental simulations. For this purpose, the electro-optical simulator LISA On table has been used to generate realistic LISA signals at the APC laboratory. This experiment also allows the testing of a LISA phasemeter (PM) prototype. In this paper, we experimentally demonstrate the coupling between the TDI algorithm and the PM anti-aliasing filters, which leads to the appearance of aliasing and non-stationary noise at high frequencies. We also propose an analytical model that allows us to estimate the impact of this coupling on our data and to constrain the design of the cascaded integrator-comb filter for a simplified LISA constellation. In addition, we explore the periodicity and impact of this non-stationarity in the case of a simplified LISA-like constellation with linearly evolving arm length.

Keywords: gravitational waves, instrumentation, LISA, interferometry, aliasing, time-delay interferometry, LISA on table

<figure><img src="image_3.png" /></figure>

Original Content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote>* Author to whom any correspondence should be addressed.</footnote>
# 1. Introduction

Laser interferometer space antenna (LISA) is an ESA space mission [1] aiming at measuring gravitational waves in the mihertz band. The instrument consists of 3 spacecraft (S/C) that follow Earth on its heliocentric orbit. The LISA orbital parameters are optimized so that the constellation forms a quasi-equilateral triangle with arm lengths $2.5 \times 10^6$ km long. These intrasatellite distances can vary in the order of percent over a few months with maximum relative speeds of $\pm 8 \text{ m s}^{-1}$ between S/C.

The S/C maintains laser links among them to form three giant interferometers [10] and three different interferometric measurements are performed in order to measure the distance fluctuations between distant free-falling test masses (TMs): scientific, test-mass and reference measurements. Each satellite is equipped with 2 moving optical sub-assembly (MOSA), which are at the core of the LISA instrument. Each MOSA is composed of several subsystems as the gravitational reference sensors with its TM, a Telescope in order to exchange laser beams with other satellites of the constellation, and an optical bench (OB) associated with its laser source and photodiodes (PDs).

The baseline laser system for LISA is based on a Nd:YAG (neodymium-doped yttrium aluminum garnet) non-planar ring oscillator (NPRO) operating at 1064 nm, used as the master oscillator in a master-oscillator-power-amplifier (MOPA) configuration [16]. Each satellite is also equipped with a Phasemeter (PM) [8] to digitize, filter, and down-sample the analog radio frequency (RF) signals coming from various PDs.

# 2. LISA on table (LOT)

The LOT experiment is an electro-optical simulator developed at the APC (AstroParticule et Cosmologie) laboratory [9, 12, 13] with the support of CNES. While the optical stage has been disassembled for practical reasons, the current electronic configuration still generates LISA-like data to test laser noise reduction in a simplified transponder-mode constellation [12, 23] (with the option to reassemble the optical stage for future work). Previous experimental demonstrations of LISA-like interferometry have validated key technologies such as PM linearity, clock synchronization, and laser-noise suppression using time-delay interferometry (TDI), including implementations with time-varying delays in hardware-in-the-loop configurations [14, 26, 27]. However, to our knowledge, none of these experiments have specifically investigated the coupling mechanisms that can lead to non-stationary laser-noise residuals in the presence of dynamic arm-length evolution. The present study experimentally validates this coupling mechanism for a transponder configuration and analytically extends the model to predict the residuals expected for evolving arm lengths in LISA.

## 2.1. *Assumptions on the simulator*

We describe the main assumptions made on the constellation, the consequences on the definition of the interferometric signals, and the formulation of Time Delay Interferometry (TDI) [22].

The current LOT interferometers allow one to simulate a simplified LISA constellation with only two arms: a *local* spacecraft (S/C) exchanges laser links with two *distant* ones. The constellation has only one clock on board, and therefore the differential clock noise [27] is not simulated in the current version of the interferometer.
In a S/C, both MOSAs exchange their laser signals through an optical fiber in order to perform the *TM* and *reference* measurements. This will induce a fiber noise in addition to laser phase noises. If the former is the same regardless of the direction of propagation in the fiber, or 'reciprocal', it can be removed in the first steps of the TDI algorithm [17]. In the LOT, we assume that it is reciprocal and, therefore, we do not simulate it.

One of the roles of the GRS is to shield its TM, which is used as an inertial reference for the GW measurements. Additionally, it performs a capacitive readout of the TM's position. This measurement is combined with the *TM* interferometric measurement to drive the drag-free attitude control system (DFACS), which adjusts the thrust of the micro-thrusters attached to the satellite and thus follows the TM in its movement along a geodesic. These systems were successfully validated by the LISA Pathfinder Mission [2]. We considered them *perfect* in our analysis, hence no environmental noise or residual acceleration noise is simulated: we assume in this article that S/C are in free fall and follow perfectly the TMs.

The secondary noise sources such as shot noise and electronic noise are quantified to be at least eight orders of magnitude below the laser frequency noise [7]. Their contribution in the final measurement noise is included in the LISA noise allocation. Although they are not simulated in this study, we compare their allocated levels with our results in section 4.5. The current configurations of the LOT simulate a single laser source for both arms of the interferometer, unlike the separate lasers used by each MOSA in LISA. The only noise source simulated in this study is the laser frequency noise, and only *inter-satellite* interferometers are considered.

After stabilization with a reference cavity, the requirement on the laser frequency noise level for LISA [16] is

$$S_{\nu}(f) \le 30 \frac{\text{Hz}}{\sqrt{\text{Hz}}} \sqrt{1 + \left(\frac{2 \text{mHz}}{f}\right)^4},$$

which can be approximated as white noise in the higher part of the LISA band [15]. Depending on the context:

* In sections 4.1–4.3, the noise was implemented as low-pass-filtered white noise, in order to minimize aliasing at the data generation frequency. Its level was deliberately set to $\sqrt{2} \times 100 \text{Hz} \sqrt{\text{Hz}}^{-1}$, providing an eight-order margin over the instrumental noise, consistent with the TDI efficiency requirement.

* In the numerical simulations of sections 4.4–4.6, the nominal $30 \text{Hz} \sqrt{\text{Hz}}^{-1}$ level was used, without low-pass filtering.

### 2.1.1. TDI variables.
The LOT simulates a constellation formed by a Michelson interferometer with two unequal and dynamic arms. We assume perfect phase locking of the distant lasers to the *master* laser and that the distant satellites act as perfect mirrors. This configuration is known as a *transponder*. The expression of a *inter-satellite* measurement $\xi_{ij}$ is simplified as the interference between a local signal and its delayed copy after a round trip through a LISA arm (between local S/C$_i$ and distant S/C$_j$). If we ignore the effect of the PM measurement on our signals, the TDI variables $\eta_{ij}$ given in equation (1), usually constructed by combining different interferometric measurements onboard in LISA [17], reduced in the LOT to simple *inter-satellite* measurements. In this study, they all share a common term of laser phase noise $p(t)$.
<figure><img src="image_4.png" /></figure>

(a) Schematic of the LISA On Table equivalent constellation in transponder mode.

(b) Diagram of the LISA On Table experiment.

<caption>**Figure 1.** Overview of the LISA On table setup.</caption>

We fix $i=1$ and $j=2, 3$, as shown in figure 1(a). The propagation time of the beam between two S/C is represented by the delay operator $\mathbf{D}_{ij}$, whose properties are given in section 3.1.1,

$$ \eta_{12}(t) \approx \xi_{12} = (\mathbf{D}_{12}\mathbf{D}_{21}-1)p(t) \\ \eta_{13}(t) \approx \xi_{13} = (\mathbf{D}_{13}\mathbf{D}_{31}-1)p(t). \qquad (1) $$

In the following sections of this article, the TDI variables $\eta_{12}$ and $\eta_{13}$ are replaced by other quantities that account for the effects of filtering and decimation in the PM, although they still serve as the inputs to the TDI algorithm.

## 2.2. Experimental setup

The experimental setup, represented in figure 1(b), consists on a data generation system which generates analog signals of our choice. These signals interfere in an electronic interferometer, producing beatnotes in the MHz range. These signals are then measured with a PM and transferred to the computer to apply the post-processing algorithm to suppress the laser frequency noise, i.e. TDI.

### 2.2.1. Data generation system.
The data generation system consists of a control-command unit that drives direct digital synthesizers (DDSs). These synthesizers generate radio-frequency sinusoidal signals $s(t)$ with amplitude $A(t)$, frequency $f(t)$ and initial phase $\phi_0(t)$. In the LOT, the frequency and phase resolutions of these signals are 3.6 µHz and 3.83 × 10² µrad, respectively. The time variable $t$ can also be modulated and delayed to simulate light propagation in the LISA arms. Each parameter can be defined as a combination of functions: constant, linear, sinusoidal, or white noise convolved with a combination of arbitrary pole-zero filter cells,

$$ s(t) = A(t) \cdot \sin(2\pi f(t) t + \phi_0(t)). \qquad (2) $$

We use a Labview program as an interface to choose the parameters of our simulation and the nature of the signals that we want to generate. A program in C generates binary data frames and establishes the link between the computer and the DDS through a PXIe-8361NI communication card. The frames are transmitted to the DDS at a frequency of $f_{frame} \approx 2.37$ kHz in
<figure><img src="image_5.png" /><figcaption>Figure 2. Schematic of LOT electronic interferometer.</figcaption></figure>

order to generate corresponding sinusoidal radio frequency (RF) signals to simulate the LISA lasers.

### 2.2.2. Electronical interferometer.
In the following study, we keep the amplitude and the initial phase constant. Only the frequency and delays varies over time to simulate the laser frequency noise. The main goal of the electronic interferometer, shown in figure 2, is to produce beat notes in the MHz range and simulate the science measurements $\xi_{12}$ and $\xi_{13}$ previously defined. Our DDS sources generate distant signals $s_{D1}(t)$ and $s_{D1'}(t)$, delayed with respect to a local signal $s_L(t)$. We write their expression as

$$ 
\begin{aligned}
s_L(t) &= A_0 \cdot \sin(2\pi f_L(t) t + \phi_0) \\
s_{D1}(t) &= A_1 \cdot \sin(2\pi f_{D1}(t) t + \phi_1) \\
s_{D1'}(t) &= A_{1'} \cdot \sin(2\pi f_{D1'}(t) t + \phi_{1'})
\end{aligned} 
\quad (3) $$

where we generally define $f(t) = f^0 + n(t)$ as the sum of a constant carrier frequency $f^0$ with a frequency noise term $n(t)$. The carrier frequencies for the three DDS channels are set to $f^0_L = 14.0001$ MHz, $f^0_{D1} = 25.5001$ MHz and $f^0_{D1'} = 25.7001$ MHz.

In order to build the interference of the local signal with the distant ones, $s_L(t)$ is split and mixed with $s_{D1}(t)$ and $s_{D1'}(t)$. For each mixer, the output signal can be seen as the sum of two signals having for phase the sum and the difference of the phases of the mixer input signals. We then applied a low-pass filter with a cut-off frequency of 23 MHz in order to keep only the low-frequency beatnote. The expression of the resulting signals for the two interferometer arms can be written as

$$ 
\begin{aligned}
U_1^{-}(t) &= \frac{A_0 A_1}{2} \cdot \cos(2\pi (f_L(t) - f_{D1}(t)) t + \Delta\phi_1) \\
U_{1'}^{-}(t) &= \frac{A_0 A_{1'}}{2} \cdot \cos(2\pi (f_L(t) - f_{D1'}(t)) t + \Delta\phi_{1'}
\end{aligned} 
\quad (4) $$

with $\Delta\phi_1 = \phi_0 - \phi_1$ and $\Delta\phi_{1'} = \phi_0 - \phi_{1'}$.

Finally, these signals are coupled with a pilot tone to suppress the jitter noise of the PM analog-to-digital converters (ADCs) [17].
## 2.3. PM and acquisition

To acquire interferometric signals, the LOT is equipped with a prototype PM developed by the AEI [8, 25] with 8 inputs for signal measurements. A common 10 MHz clock is used and multiplied by four to drive the ADCs. Such a high sampling rate is required to correctly capture the MHz-range heterodyne beatnotes, but the resulting raw data stream largely exceeds what is relevant for simulating the mHz-Hz frequency band of LISA. For this reason, the PM implements a cascaded integrator-comb (CIC) filter for decimation, reducing the output rate while preserving the relevant frequency content, consistent with the CIC-based downsampling stage foreseen in the LISA PM design.

The signal phase variations are then measured with a digital phase-locked loop (DPLL), and stored in a frequency register. According to equations (4), the beat notes $f_{12}(t)$ and $f_{13}(t)$ of the interference signals entering channels 1 and 2 of the PM have the following expressions:

$$ 
\begin{aligned}
f_{12}(t) &= f_{D1}(t) - f_L(t) \\
f_{13}(t) &= f_{D1'}(t) - f_L(t).
\end{aligned}
\qquad (5) $$

## 2.4. Performances and intrinsic noise

The LOT is sensitive to environmental noise and also limited by a set of instrumental noises that depend on its components and chosen architecture. These disturbances define the intrinsic noise of the LOT. To distinguish between an instrumental limit and the performance limit of the TDI algorithm, we estimate this noise floor prior to running the experiment by excluding laser frequency noise over a period of 30 h. Our instrument sensitivity is restricted by the phase noise of the DDS, and we identify this noise as scintillation noise, with power decreasing as $f^{-1}$ (pink noise) between $10^{-4}$ Hz and 1 Hz. For every simulations presented in section 4, we plot the intrinsic noise in terms of amplitude spectral density (ASD), in units of Hz/√Hz, which can be approximated as :

$$ ASD_{\text{intr},f} = 3.10^{-6} \frac{\text{Hz}}{\sqrt{\text{Hz}}} \cdot f^{1/2}. \qquad (6) $$

# 3. Model formulation

## 3.1. Operators properties

### 3.1.1. Delay operators.
The constellation is composed of three satellites denoted SC$_i$ with $i \in 1, 2, 3$. By convention in figure 3, we indicate a MOSA$_{ij}$ according to the SC $i$ to which it belongs and with which distant satellite $j$ it shares an arm of the constellation. For example, MOSA$_{12}$ belongs to SC 1 and points its telescope towards SC 2. We denote $D_{ij}$ the delay operator associated with the laser beam propagation time from the satellite $j$ to a distant satellite $i$. The properties of the delay operators will be detailed in the following subsections for different constellation dynamics. The TDI algorithm applies these delay operators to interferometric signals to form equal virtual photon paths for laser noise suppression. In reality, a delayed value of this signal will be obtained by Lagrangian interpolation of the original signal at a shifted time $t - \tau$. However, here we do not take into account the interpolation or ranging errors [20]. The delays with LOT or TDI are assumed equivalent and we place ourselves in the continuous
<figure><img src="image_6.png" /><figcaption>Figure 3. Convention for the different MOSAs and delays in the LISA constellation.</figcaption></figure>

case. In these conditions, delaying a temporal signal $s(t)$ is defined by the application of the delay operator $D$, described in details in [21], such that:

$$D_{ji}s(t) = s(t - \tau_{ji}) = s(t - L_{ji}) \qquad (7)$$

where we define $\tau_{ji}$ as the propagation time between $S/C_i$ and $S/C_j$, and $L_{ji}$ as the associated arm length. Depending on the context, the latter will be used either as a propagation time or as the corresponding proper distance. We set the speed of light in vacuum to $c = 1$. The properties of this delay operator depend on the model we assume for the arms of LISA [17].

* For a static constellation, the propagation time $\tau_{ji}$ can then be written as a constant function of the arm length $L_{ji}$:

$$D_{ji}s(t) = s(t - L_{ji}) \qquad (8)$$

with $L_{ji}$ a constant arm length. The application of nested delay is obtained naturally:

$$D_{ijk}s(t) = s(t - L_{ij} - L_{jk}). \qquad (9)$$

One can verify that this result does not depend on which order we apply the successive operators. If the Sagnac effect due to rotation of the constellation is not modeled: $L_{ij} = L_{ji}$. In the case of a round-trip, one can combine this last arm length terms in a unique term $2L_{ij}$ as:

$$D_{iji}s(t) = s(t - 2L_{ij}). \qquad (10)$$

* For a constellation with moving $S/C$, the propagation time $\tau_{ji}$ is a time-dependent function of the arm length $L_{ji}(t)$,

$$D_{ji}s(t) = s(t - L_{ji}(t)). \qquad (11)$$

In this case, the successive application of delays operators is obtained by:

$$D_{ijk}s(t) = D_{ijs}(t - L_{jk}(t)) \\ = s((t - L_{ij}(t)) - L_{jk}(t - L_{ij}(t)))$$
where the time dependency of the operator implies their non-commutativity : $[D_{ij}, D_{jk}] s(t) = D_{ij}D_{jk}s(t) - D_{jk}D_{ij}s(t) \neq 0$. In the specific case of a round-trip delay $D_{ij}$ without the Sagnac effect, the commutativity of this operator is preserved.

In the following, we will not consider the Sagnac effect and use round-trip delay to simulate the LOT transponder constellation.

### 3.1.2. TDI for LOT data.
There are several generations of the TDI algorithm that differ depending on the dynamics of the constellation being considered. Here, we present the combinations $X_1$ and $X_2$ [21] that are used with the LOT data. The first generation of TDI is suitable for non-rotating constellations with fixed arms ($L_{ij} = L_{ji} = \text{constant}$). Since arm lengths are constant, the delay operators do not depend on time. Here we consider the Michelson combinations [3, 5, 17] in order to reduce laser frequency noise. If S/C 1 serves as the base for the interferometer, the combination can be written as:

$$X_1 = (\eta_{13} + D_{13}\eta_{31}) + D_{131}(\eta_{12} + D_{12}\eta_{21}) - [(\eta_{12} + D_{12}\eta_{21}) + D_{121}(\eta_{13} + D_{13}\eta_{31})].$$

It is possible to obtain the combinations $Y_1$ and $Z_1$ for the other satellites by cyclic permutation of the indices. It is noted that for the LOT (transponder) simulations, $\eta_{21} = \eta_{31} = 0$. Then, one can simplify the combination as follows (up to a sign):

$$X_1 \approx (1 - D_{121}) \eta_{13} - (1 - D_{131}) \eta_{12}. \qquad (12)$$

The LISA constellation will see the arm lengths change over time. The second generation of TDI is adapted for a constellation with varying arm lengths ($\dot{L} \neq 0$) and uses time-dependent delay operators. The second-generation Michelson combination $X_2$ cancels the laser noise to the first order of $\dot{L}$:

$$X_2 = (D_{13121} - 1)([\eta_{12} + D_{12}\eta_{21}] + D_{121}[\eta_{13} + D_{13}\eta_{31}]) \\ - (D_{12131} - 1)([\eta_{13} + D_{13}\eta_{31}] + D_{131}[\eta_{12} + D_{12}\eta_{21}])$$

with $Y_2$ and $Z_2$ obtained by cyclic permutation.

It is noted that, as with TDI 1 in the case of the LOT, only the terms of $\eta_{12}$ and $\eta_{13}$ are non-zero. Therefore, $X_2$ adapted to the LOT data is written:

$$X_2 \approx (D_{13121} - 1)(\eta_{12} + D_{121}\eta_{13}) - (D_{12131} - 1)(\eta_{13} + D_{131}\eta_{12}). \qquad (13)$$

### 3.1.3. CIC filters : cascaded integrator comb.
CIC filters are commonly used in digital signal processing systems, where the aim is to increase or decrease the data sampling frequency. They are generally composed of a succession of N moving averages (integrator) and N comb filters, with N is the order of the CIC filter (see figure 4). When the integrator stage comes before the comb stage, the filter functions as a decimator, reducing the sampling rate. Conversely, if the comb stage comes before the integrator, the filter acts as an interpolator, increasing the sampling rate. In the LOT, a CIC filter is used in the PM for the decimation of the data and allows to pass from an acquisition frequency $f_{ADC} = 40$ MHz, corresponding to the ADC sampling frequency, to a lower output sampling frequency $f_s = \frac{40 \cdot 10^6}{2^{20}}$ Hz. To avoid any aliasing with respect to Nyquist frequency $f_N = f_s/2$, higher frequencies are first filtered
<figure><img src="image_7.png" /><figcaption>Figure 4. Schematic for a N order CIC (downsampling).</figcaption></figure>

by the integrator before decimation. The transfer function $H_C(f)$ of a CIC filter can be written as:

$$H_C(f) = \left( \frac{\sin\left(RM\frac{\pi f}{f_{\text{ADC}}}\right)}{RM\sin\left(\frac{\pi f}{f_{\text{ADC}}}\right)} \right)^N e^{-jN(RM-1)\frac{\pi f}{f_{\text{ADC}}}} \quad (14)$$

where $N$ is the number of moving averages which defines the order of the filter, $R$ is the decimation or interpolation factor and $M$ is the differential delay between samples in the comb stage. The CIC filter of the PM used in LOT is characterized by the parameters $N = 2, R = 2^{20}$, and $M = 1$.

The downsampling process introduces aliasing of frequencies above the Nyquist frequency. Consequently, the amplitude of the aliased signal residuals depends on the filter parameters, with its behavior primarily governed by the filter order $N$.

A similar decimation stage is also required in the LISA PMs [26]. Although the heterodyne beatnotes are sampled at 80 MHz, the scientific frequency band of interest only extends from 0.1 mHz to 1 Hz. Transmitting and processing the full high-rate data stream would be neither feasible nor necessary. For this reason, the PM design employs a CIC stage as the first decimator before further filtering and data reduction, and to provide reduced-rate data streams to onboard subsystems [6, 27]. The implementation in LOT therefore models this step of the LISA data chain, enabling us to investigate residual noise effects associated with filtering and downsampling.

### 3.14. Spectrum of aliased signal.
We model decimation of the signal in the CIC filter as a sampling at the frequency $f_s$ of the filtered signal. The spectrum of a sampled signal $S_s(f)$ can be written as:

$$S_s(f) = \sum_{k=-\infty}^{+\infty} S(f - k f_s)$$

with $f \in [-f_s/2 : f_s/2]$.

We distinguish in the sum the terms of positive and negative frequencies by redefining its bounds:

$$S_s(f) = S(f) + \sum_{k=1}^{+\infty} S(f - k f_s) + S(f + k f_s).$$
<figure><img src="image_8.png" /><figcaption>Figure 5. Decimation of a signal spectrum $S(f)$ at frequency $f_s$.</figcaption></figure>

As we are dealing here only with real signals, we can use $S(-f) = S^*(f)$, which implies on this last equation :

$$S_s(f) = S(f) + \sum_{k=1}^{+\infty} S^*(kf_s - f) + S(kf_s + f). \qquad (15)$$

To simplify notations we introduce $S_{k_+}$ and $S_{k_-}$ as :

$$\begin{aligned} S_{k_+} &= S(kf_s + f) \\ S_{k_-} &= S(kf_s - f). \end{aligned} \qquad (16)$$

At the end, the decimation operator, represented in figure 5, can be written as :

$$S_s(f) = S(f) + \sum_{k=1}^{+\infty} S_{k-}^* + S_{k+}. \qquad (17)$$

We notice that we lose the multiplicative properties of the transfer functions. If we consider that our signal before decimation is the product between an input signal $e(f)$ and a transfer function $H(f)$ such that: $S(f) = H(f) \cdot e(f)$, the decimation of $S(f)$ is not the product of the decimations of $e(f)$ and $H(f)$ :

$$\begin{aligned} S_s(f) &= (H(f) \cdot e(f))_s \\ &\neq H_s(f) \cdot e_s(f). \end{aligned}$$

In the following, we apply equation (17) as a decimation operator.

## 3.2. Non commutativity between delay operator and CIC filtering

The commutativity of delay and decimation operators is tested as described in figure 6. We generate a signal of spectrum $S(f)$ and successively apply decimation and delay operators, as defined previously. We model our PM as the application of a low-pass filter with transfer function $H_c$ followed by a decimator and determine the global transfer function implemented in figure 6. Even if the delay operator is done by Lagrangian interpolation in LOT [11], we assume here that they are perfect delays and take the form of a simple phase term in Fourier space. For a constant delay due to a round-trip propagation in the arm $ij$ ($\tau_{iji} = L_{iji} = \tau$), the delay operator applied to a signal $s(t)$ can be written as

$$D_{\tau} s(t) = s(t - L_{iji}) = s(t - \tau).$$

Its Fourier transform is written

$$\tilde{D}_{\tau} s(t) = S(f) e^{-i2\pi f \tau}. \qquad (18)$$

Using equation (18) for constant delays in Fourier space, we can write the spectrum of a filtered, decimated and then delayed signal $T(f, \tau)$ in the first arm:
<figure><img src="image_9.png" /><figcaption>**Figure 6.** This schematic represents the experimental setup to estimate the residual aliased laser noise in a simple configuration. The PM measurement is modeled with a succession of two operators (filtering and sampling). The output *R* is the difference between a delayed signal (of spectrum *S*(*f*)) measured and a measured signal that is then delayed.</figcaption></figure>

$$ 
\begin{aligned}
T(f, \tau) &= \tilde{D}_{\tau} (H_C \cdot S)_s (f) \\
&= H_C(f) S(f) e^{-j2\pi f \tau} + \sum_{k=1}^{+\infty} (H_{C,k_-}^* S_{k_-}^* + H_{C,k_+} S_{k_+}) e^{-j2\pi f \tau}.
\end{aligned} 
\quad (19) $$

In the other arm, we delay the data with the LOT before measuring it with the PM. We reverse the order of application of the operators and transfer functions to obtain the expression of a delayed, filtered and decimated signal *K*(*f*, *τ*):

$$ 
\begin{aligned}
K(f, \tau) &= (H_C \cdot \tilde{D}_{\tau} S)_s (f) \\
&= H_C(f) S(f) e^{-j2\pi f \tau} + \sum_{k=1}^{+\infty} (H_{C,k_-}^* S_{k_-}^* e^{j2\pi (kf_s-f)\tau} + H_{C,k_+} S_{k_+} e^{-j2\pi (kf_s+f)\tau}) .
\end{aligned} 
\quad (20) $$

We define the difference between the two paths as $R(f, \tau) = T(f, \tau) - K(f, \tau)$ and can write its expression using equations (19) and (20):

$$ R(f, \tau) = e^{-j2\pi f \tau} \sum_{k=1}^{+\infty} [I_{k_-}^*(\tau) + I_{k_+}(\tau)] \quad (21) $$

where:

$$ 
\begin{aligned}
I_{k_-}(\tau) &= H_{C,k_-} S_{k_-} \cdot (1 - e^{-j2\pi (kf_s)\tau}) \\
I_{k_+}(\tau) &= H_{C,k_+} S_{k_+} \cdot (1 - e^{-j2\pi (kf_s)\tau}) .
\end{aligned} 
$$

The power spectral density $PSD_R(f)$ of these residuals is computed in appendix B. In the LOT, our input signal is a Gaussian white noise, of constant ASD *B*, filtered with a low-pass filter, of transfer function $H_A(f)$. In the following, we group the CIC filter transfer function with this low pass filter in a unique filter term $H(f) = H_A(f) \cdot H_C(f)$, and the $PSD_R(f)$ can be written as:

$$ PSD_R(f) = 4B^2 \sum_{k=1}^{+\infty} \sin^2(\pi k f_s \tau) \cdot (|H_{k-}|^2 + |H_{k+}|^2). \quad (22) $$
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Illustration of TDI variables processed by the phasemeter (PM).</figcaption></figure>

Notice that this residual is suppressed when $\tau$ is an integral multiple of the sampling rate:
$$ \tau = \frac{n}{f_s} $$

$$ \sin^2 \left( \pi k f_s \frac{n}{f_s} \right) = 0 \quad \Rightarrow \quad \mathcal{PSD}_R \left( f, \frac{n}{f_s} \right) = 0. $$

This result shows the non-commutativity between CIC filtering and delay operators in presence of aliasing, as was experimentally demonstrated in [23] and in [20] where the analytic formulation and numerical simulations are compared.

As TDI consists of applying delays to decimated delayed signals, one can expect to have this effect in LISA, depending on the parameters of the decimation filters as well as the orbits.

### 3.3. Fixed unequal arms constellation

In order to compute this effect for an unequal and static arms constellation, we first model a simple TDI variable as a combination of a signal and its delayed copy. As shown in section 2, this variable is expressed in LOT as an inter-satellite signal in a transponder mode configuration.

In figure 7(a), we first interfere a *local* signal of the spectrum $S(f)$ with a *distant* identical signal, delayed by the round-trip propagation between SC 1 and 2. For this, we apply successive delay operator and note that $L_{12} = L_{21}$

$$ \begin{aligned} D_{121} s(t) &= s(t - L_{21} - L_{12}) \\ &= s(t - \tau_{121}) \end{aligned} $$

with $\tau_{121} = 2L_{12}$. Then we write the interferometric signal as

$$ \xi_{12}(t) = s(t - \tau_{121}) - s(t) \qquad (23) $$

and its associated Fourier transform

$$ \tilde{\xi}_{12}(f) = (e^{-j2\pi f\tau_{121}} - 1)S(f). \qquad (24) $$

Then we apply the CIC filtering transfer function (equation (14)) and decimation operator (equation (17)) to equation (24) in order to model the PM acquisition. The resulting variable $L(f, \tau_{121})$, which is the TDI variable that we consider in this static case, can be written as:
$$ 
\begin{aligned}
L(f, \tau_{121}) &= (H_C \cdot \tilde{\xi}_{12})_s(f) \\
&= H_C(f) \tilde{\xi}_{12}(f) + \sum_{k=1}^{+\infty} (H_{C,k_-}^* \tilde{\xi}_{12,k_-}^* + H_{C,k_+} \tilde{\xi}_{12,k_+}).
\end{aligned} 
\quad (25) $$

This expression is deduced for the other arm 13 of the LOT, $L(f, \tau_{131})$, by replacing $\tilde{\xi}_{12}(f)$ and $\tau_{121}$ by $\tilde{\xi}_{13}(f)$ and $\tau_{131}$ respectively. We construct a simplified Michelson combination $\tilde{X}_1$ in the case of a transponder mode constellation:

$$ \tilde{X}_1(f) = (1 - \tilde{D}_{\tau_{131}}) L(f, \tau_{121}) - (1 - \tilde{D}_{\tau_{121}}) L(f, \tau_{131}) \quad (26) $$

and model this combination applied to our two simplified variables coming from the PM in figure 7(b).

The resulting expression can be written as a function of the average of the two arm length $\tau = \frac{\tau_{121} + \tau_{131}}{2}$ and their difference $\Delta\tau = \tau_{121} - \tau_{131}$ which are contained in the term $E(kf_s, \tau, \Delta\tau)$:

$$ X_1(f, \tau, \Delta\tau) = \sum_{k=1}^{+\infty} H_{C,k_-}^* S_{k_-}^* \cdot E(kf_s, \tau, \Delta\tau) + H_{C,k_+} S_{k_+} \cdot E(-kf_s, \tau, \Delta\tau) $$

with $E(kf_s, \tau, \Delta\tau)$ given in equation (27):

$$ E(kf_s, \tau, \Delta\tau) = 2je^{-j2\pi f\tau} \times (e^{j2\pi kf_s\tau} [\sin(\pi(kf_s - f)\Delta\tau) - \sin(\pi kf_s\Delta\tau)] + \sin(\pi f\Delta\tau)). \quad (27) $$

In the following, we do not explicitly include the dependency on $\tau$ and $\Delta\tau$ in the expression for $E$. As in equation (22), the $\mathcal{PSD}_{X_1}$ associated with laser noise residuals for a fixed transponder mode constellation is expressed as a function of various filter transfer functions combined in the term $H(f)$. If the input signal $S(f)$ is white noise, its power spectral density can be replaced by a constant, such that $\mathcal{PSD}_S(f) = B^2$:

$$ \mathcal{PSD}_{X_1}(f) = B^2 \cdot \sum_{k=1}^{+\infty} (|H_{k_-} E(kf_s)|^2 + |H_{k_+} E(-kf_s)|^2). \quad (28) $$

We can demonstrate the cancellation of this effect when $\tau_{121}$ and $\tau_{131}$ are both multiples of the sampling period $1/f_s$ (see appendix C):

$$ 
\begin{aligned}
E(kf_s) &= E(-kf_s) = 0 \\
\Rightarrow \mathcal{PSD}_{X_1}(f) &= 0.
\end{aligned} 
$$

### 3.4. Flexing unequal arms constellation

We apply the same method used in the previous paragraph, but this time considering arm lengths that vary linearly with time. The difference from the fixed-arm model arises from the time dependency of the delay operators. To compute a round-trip delay for our signals, we successively apply a delay operator in the time-dependent case, as defined in equation (12). This round-trip operator can be expressed as:

$$ D_{ijis}(t) = s((t - L_{ij}(t)) - L_{ji}(t - L_{ij}(t))). \quad (29) $$
We consider a linearly time-dependent arm-length $L_{ij}(t)$ as :

$$L_{ij}(t) = L_{ij,0} + \dot{L}_{ij} \cdot t$$

with $L_{ij,0}$ the arm size at the origin between satellites i and j, used here as a duration, and $\dot{L}_{ij}$ the relative velocity between these two satellites.

We proceed to a Taylor expansion up to the first order of $\dot{L}_{ij}$ and we note the successive application of two delay operators such that

$$D_{ij} s(t) \approx s(t(1 - 2\dot{L}_{ij}) + L_{ij,0} \cdot \dot{L}_{ji} - 2L_{ij,0}) \quad (30)$$

and its associated Fourier transform, computed in the appendix A

$$\tilde{D}_{ij} s(t) \approx k_{ij}(f) \cdot S(f) + h_{ij}(f) \cdot \frac{dS}{df}(f) \quad (31)$$

with $k_{ij}(f) = (1 + 2\dot{L}_{ij} - 3j2\pi fL_{ij,0}\dot{L}_{ij})e^{-j2\pi f2L_{ij,0}}$ and $h_{ij}(f) = 2f\dot{L}_{ij} \cdot e^{-j2\pi f2L_{ij,0}}$.

Referring to figure 7(a), we define our inter-satellite measurement in the Fourier space $\tilde{\xi}_{ij}(f)$ for a flexing arm as

$$\tilde{\xi}_{ij}(f) = g_{ij}(f) \cdot S(f) + h_{ij}(f) \cdot \frac{dS}{df}(f) \quad (32)$$

with $g_{ij}(f) = k_{ij}(f) - 1$.

Let us define $M_{ij}(f)$ as the spectrum of $\xi_{ij}$ after filtering (equation (14)) and decimation (equation (17)) by a CIC filter. We can write the result for the arm-length between S/C 1 and S/C 2 :

$$\begin{aligned} M_{12}(f) &= (H_C(f) \cdot \tilde{\xi}_{12}(f))_s \\ &= H_C(f) \left( g_{12}(f) S(f) + h_{12}(f) \frac{dS(f)}{df} \right) + \sum_{k=1}^{+\infty} \left[ H_{C,k-}^* \left( g_{12,k-}^* S_{k-}^* + h_{12,k-}^* \frac{dS_{k-}^*}{df} \right) \right. \\ & \qquad \left. + H_{C,k+} \left( g_{12,k+} S_{k+} + h_{12,k+} \frac{dS_{k+}}{df} \right) \right], \end{aligned} \quad (33)$$

where the expression for the other arm $M_{13}$ is obtained by substituting in $M_{12}$ index 2 by 3. This last equation shows a dependency on the input noise spectrum derivative $\frac{dS(f)}{df}$. We aim to simplify the following developments by neglecting these terms (in our case, derivatives of filtered Gaussian white noise).

Since $S(f)$ is the Fourier transform of a stochastic input noise, we cannot describe a specific realization analytically. Instead, we consider its statistical properties via the ASD, defined as:

$$\text{ASD}_S(f) = \sqrt{\langle |S(f)|^2 \rangle}. \quad (34)$$

In the case of filtered white Gaussian noise, the ASD typically follows a power-law behavior:

$$\text{ASD}_S(f) \propto f^{\alpha}, \quad (34)$$

where $\alpha$ depends on the filter characteristics. The relative contribution of the derivative term in (32) can thus be estimated as:

$$\left| \frac{h_{ij}(f)}{g_{ij}(f)} \cdot \frac{1}{S(f)} \cdot \frac{dS}{df} \right| \quad (35)$$
Assuming a power-law spectrum $S(f) \sim f^{\alpha}$, we obtain:

$$ \frac{1}{S(f)} \cdot \frac{dS}{df} \sim \frac{\alpha}{f}, \qquad (36) $$

and therefore the condition for neglecting the derivative term becomes:

$$ \left| \frac{h_{ij}(f)}{g_{ij}(f)} \cdot \frac{\alpha}{f} \right| \ll 1. \qquad (37) $$

This local condition on the slope of the spectrum is satisfied at sufficiently high frequencies or when $\alpha$ is moderate and the ratio $h_{ij}(f)/g_{ij}(f)$ remains small. In our case, we assume Gaussian white noise filtered by an order-12 filter, leading to $\alpha = -12$ beyond the cutoff. By choosing its cut-off frequency such that the injected noise behaves approximately as white noise within the LOT's bandwidth, this condition is well satisfied and the derivative term in (32) can be neglected. We plot in figure E1 this condition for typical values of velocities $\dot{L} \approx 10 \text{ m s}^{-1}$ and sizes of arms $L = 2.5 \cdot 10^9 \text{ m}$.

With this approximation, equation (32) reduces to:

$$ \tilde{\xi}_{ij}(f) \approx g_{ij}(f) \cdot S(f) \qquad (38) $$

As a consequence, equation (33) is simplified as:

$$ M_{12}(f) = H_C(f) g_{12}(f) S(f) + \sum_{k=1}^{+\infty} \left[ H_{C,k_{-}}^{*} g_{12,k_{-}}^{*} S_{k_{-}}^{*} + H_{C,k_{+}} g_{12,k_{+}} S_{k_{+}} \right] \qquad (39) $$

Rewriting in Fourier space the expression of TDI 2 given by equation (13), we obtain the combination $X_2$ now taking into account the aliasing effect such that:

$$ X_{2.0}(f) = (1 - \tilde{D}_{12131}) [M_{13}(f) + \tilde{D}_{131} M_{12}(f)] \\ - (1 - \tilde{D}_{13121}) [M_{12}(f) + \tilde{D}_{121} M_{13}(f)] \qquad (40) $$

One can show that, if we restrict our analysis to first order in $\dot{L}$, equation (31) can be applied as a round trip delay operator and its successive application can be achieved directly in frequency domain. This is convenient for defining more complex delays, such as the one present in the $X_2$ combination. We present an alternative method in appendix A, which consists of deriving the $X_2$ combination in the time domain and then doing Fourier transform to obtain the corresponding PSD. The results are identical for both approaches up to the first order in $\dot{L}$.

We assume that the noise can be considered quasi-stationary on LISA data timescale. Considering only white input noise $S(f)$, $PSD_S(f) = B^2$ is a constant and we can include aliasing into $X_2$ transfer function so that:

$$ \mathcal{P}SD_{X_{2.0}}(f) = B^2 |Q_{2.0}(f)|^2 \qquad (41) $$

where $Q_{2.0}$ is obtained from equations (39) and (40) with $S(f) = 1$.

### 3.5. Application of the analytical model to LISA's TDI and CIC filter coupling

In this section, we apply the analytical model developed in the previous section to estimate the impact of the coupling between CIC filters and TDI in the context of the LISA instrument.

Similarly to the LOT, the LISA PM measures the phase fluctuations of various signals using a DPLL. This loop generates a local copy of the signals and outputs data at a sampling rate of 80 MHz. To make these signals usable for the DFACS and for downlink transmission to
<figure><img src="image_11.png" /><figcaption>Figure 8. Application of two CIC Filters on a measurement of signal $\tilde{\xi}_{ij}(f)$.</figcaption></figure>

Earth, the data must be decimated to 16 Hz and 4 Hz, respectively. That decimation is achieved through a cascade of two CIC filters, and three finite impulse response (FIR) filters to prevent the folding of laser noise into LISA's bandwidth [25]. We assume that the first CIC filter decimates the data by a factor of $R_1 = 2^3$, reducing the frequency from $f_{in,1} = 80$ MHz to $f_{out,1} = 10$ MHz. This is followed by a second CIC filter, which further reduces the frequency from $f_{in,2} = 10$ MHz to $f_{out,2} = 640$ Hz with a decimation factor of $R_2 = 5^6$. The application of LISA's CIC filters on a measurement with spectrum $\tilde{\xi}_{ij}(f)$ is illustrated in figure 8.

The spectrum of a filtered measurement at the output of the two CIC filters $U_{ij}(f)$, taking into account residual aliasing, is defined in equation (42), combining equation (17) twice:

$$U_{ij}(f) = \left( H_{C,2}(f) \cdot \left( H_{C,1}(f) \cdot \tilde{\xi}_{ij}(f) \right)_{f_{out,1}} \right)_{f_{out,2}}. \quad (42)$$

We adapted our Python code used to compute the aliased noise spectrum of the LOT data for LISA's filtering scheme. However, we applied it to the simplified transponder architecture as defined in the LOT experiment. For a static constellation, the input signal $\tilde{\xi}_{ij}$ is described by expression (24), while for a flexing constellation, equation (32) shall be used. Since additional filtering does not change previously aliased noise, FIR filters are not considered here.

# 4. Results and discussion

In this section, we present the result of the impact of non-commutativity of CIC filter and delay operator in the data. Then, the TDI of the residuals laser noise and aliasing of the LOT data will be shown to demonstrate the efficiency of TDI to suppress the laser noise, for both first and second generations. We will also compare the PSD of the experimental data to the analytical models that we introduced in the section 3.4. Based on the study of LOT data, we apply the same method to the LISA design in order to discuss the aliased noise with different constellation configurations. Finally, we do a study on the dependence of the CIC and TDI coupling with the varying arm length, which reveals a non-stationary source of noise in the LISA data. The data and analysis codes used in this section can be found here [24].

## 4.1. CIC filtering and delay coupling

In the simulations conducted with the LOT, the laser frequency noise is modeled as a Gaussian white-noise process with a one-sided ASD of $\sqrt{2} \times 100$ Hz/$\sqrt{Hz}$, which drives an autoregressive moving-average (ARMA) filter. The ARMA process directly generates low-pass-filtered noise sequences, preventing aliasing above the data frame generation frequency $f_{frame} \approx 2.37$ kHz. Its effective transfer function $H_A(f)$ depends on the filter order $n$ and cutoff frequency $f_c$:

$$H_A(f) = \left( \frac{1}{1 + if/f_c} \right)^n, \quad (43)$$
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Results for the configuration shown in figure 6. The dark blue curve shows $T(f, \tau)$, the spectrum of a low-pass-filtered noise sequence generated with the LOT, measured with the phasemeter and then delayed. The residual $R(f, \tau) = T(f, \tau) - K(f, \tau)$ (light blue, experimental; dashed red, analytical) corresponds to the laser frequency noise residuals due to the non-commutativity of delay and filtering/decimation operations, with $K(f, \tau)$ the same noise sequence delayed first and then measured with the phasemeter. The low-pass filter has order 8 with a cutoff frequency of 15 Hz.</figcaption></figure>

with $f_c \ll f_{frame}$. Note that although this low-pass filtering does not reproduce the flat high-frequency spectrum expected for the LISA laser, its cutoff frequency (15 Hz in section 4.1, 150 Hz in sections 4.2 and 4.3) lies well above the LISA band, so that the noise remains effectively white in-band; the filtering only reduces the level of aliased noise, which is properly accounted for in our analytical model.

In order to verify our analytic model described in 3.2, we employ the simplified setup shown in figure 6 using two DDS for signal and another one to generate the pilot tone. A first DDS channel generates a signal $s_1(t)$ with frequency $f_1(t) = f_{car,1} + (n_\nu(t) * h_A)(t)$, where $f_{car,1} = 45$ MHz is a constant carrier frequency. The term $n_\nu(t)$ represents a white Gaussian noise process used to simulate laser frequency noise. This noise is passed through a digital filter with impulse response $h_A$. The second DDS directly generates a delayed version of $s_1 : s_2(t) = s_1(t - \tau)$, with $\tau = 10$ s. The delay is obtained by using a Lagrangian interpolation of order 31.

By subtracting the two series and plotting its ASD in figure 9, we obtain the light blue curve, which is consistent with the analytic equation (22) represented by the red curve. In fact, we observe a residual laser noise with a slope proportional to $f^2$, characteristic of our CIC filter order $N = 2$. As first discussed in [27] and later in [23], this result shows that delaying a decimated signal and decimating a delayed signal are not equivalent operations due to the presence of aliasing. TDI involves applying delays to propagating and decimated signals. Therefore, we can expect residual CIC aliasing to occur in the LISA data, depending on the parameters of the decimation filters and the orbits.
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Coupling between TDI 1.0 and the CIC filter for a static constellation. The TDI variable $L_{12}$ (blue) is a combination of frequency laser noise terms, with a noise level of $100 * \sqrt{2}$ Hz/$\sqrt{Hz}$. The low pass filter (ARMA) applied on laser noise is of order 12 and has a cut off frequency of 150 Hz. We show the comparison of residuals after applying TDI on the LOT data (light blue) and the analytical formula of X1 (red), with the LOT intrinsic noise (black). The latter is computed by applying the same analysis method to noiseless simulated signals (see section 2.4).</figcaption></figure>

## 4.2. TDI 1.0 laser noise residuals and aliasing

We compare here the predictions of our analytical model, using equation (28) defined in section 3.3, and the data generated with the LOT, following the equivalent configuration shown in figures 7(a) and (b). The laser noise residuals obtained are due to the coupling between CIC filtering and TDI 1.0 in a static transponder configuration, where the round-trip time of flight is chosen as $\tau_{121} = 16.6$ s and $\tau_{131} = 16.76$ s for arms 12 and 13, respectively. The ASDs of these residuals are compared in units of Hz/$\sqrt{Hz}$ in figure 10.

These results suggest the validity of equation (28). The aliased noise dominates the noise residuals over the frequency band and, similarly to the previous subsection, we observe a slope proportional to $f^2$, characteristic of the CIC filter of order $N = 2$.

## 4.3. TDI 2.0 laser noise residuals and aliasing

We present here the predictions obtained using equation (41) and compare them with the data acquired from the LOT during the TDI 2.0 tests, described in section 3.4, as shown in figure 11.

This simulated constellation features arm lengths that vary linearly with time as $L_{ij}(t) = L_{ij,0} + \dot{L}_{ij} \times t$. We initially set arm lengths of about 2.5 millions km for arm 12 and add 25000 km for arm 13. This corresponds to delays at the PM starting time of $L_{12,0} = 8.339$ 16s and $L_{13,0} = 8.4225$ s, and to velocities (or delay drifts) of $\dot{L}_{12} = 5$ ms$^{-1}$ and $\dot{L}_{13} = 10$ ms$^{-1}$ for arms 12 and 13, respectively. We derive the laser noise residuals resulting from the coupling between the CIC filters and TDI 2.0 in a flexing transponder configuration. The modeled and
<figure><img src="image_14.png" /><figcaption>**Figure 11.** Coupling between TDI 2.0 and the CIC filter for a flexing constellation. The low pass filter (ARMA) applied on laser noise is of order 12 and has a cut off frequency of 150 Hz. The TDI variable $M_{12}$ (dark blue), referring to equation (39), is a combination of frequency laser noise terms, with a noise amplitude of $100 * \sqrt{2} \text{ Hz}/\sqrt{\text{Hz}}$, measured with the PM. We show the comparison of residuals after applying TDI on the LOT data (light blue) and the analytical formula of X2 (red), with the LOT intrinsic noise (black).</figcaption></figure>

experimental residuals are compared in units of $Hz \sqrt{Hz}^{-1}$, along with the intrinsic noise of the LOT, as shown in figure 11.

The residuals follow the expected $f^2$ trend, consistent with a CIC filter of order $N=2$. As in the static-arm case, the experimental residuals agree well with our model.

4.4. *Estimation of aliased noise for a fixed-arm constellation after TDI 1.0*

In the following calculations, we assume for the laser frequency noise a white noise with a one-sided spectral amplitude density of $30 \frac{\text{Hz}}{\sqrt{\text{Hz}}}$, which is the expected frequency noise level for the LISA laser [15, 16]. Using the analytical model describing the coupling between CIC filters and the TDI 1.0 (equation (28)), and adapting it to LISA's decimation scheme 8, we estimate the folded noise residual within LISA's bandwidth. The folded laser noise is then compared to the noise allocation for an LISA-like PM (appendix D.1), multiplied by TDI 1.0 transfer function [18].

The blue curve on figure 12 represents the laser noise, first filtered and decimated by the initial CIC₁ filter of order $N_1 = 2$, then filtered and decimated by the second CIC₂ filter of order $N_2 = 2$, and finally processed using TDI 1.0. The Nyquist frequency of the first filter, CIC₁, is $f_{N,1} = 5$ MHz, which is more than four orders of magnitude higher ($R_2 = 5^6$) than that of the second filter, CIC₂, with $f_{N,2} = 320$ Hz. Consequently, the aliased noise from the first filter is negligible within the second filter bandpass. In fact, the trend of the aliased noise after CIC of order N follows $f^N$, which implies that for an input noise amplitude of 30 Hz $\sqrt{Hz}^{-1}$, the level of aliased noise due to CIC₁ at $f_{nyquist,2} = f_{out,2}/2$ is approximately $\frac{30}{R_2^2} \approx 1.2 \cdot 10^{-7} \text{ Hz} \sqrt{Hz}^{-1}$. Thus, the spectral of the laser noise residues is dominated by the order of the second CIC filter.
<figure><img src="image_15.png" /><figcaption>**Figure 12.** Coupling between the two LISA CIC filters and TDI 1.0. The round-trip time for the arms is constant and chosen as integer and half-integer multiples of the sampling period of the second CIC filter (this configuration maximizes the aliasing effect): $L_{121} = 10240/f_{out,2} = 16$ s and $L_{131} = 10448.5/f_{out,2} \approx 16.325$ s, with $f_{out,2} = 640$ Hz.</figcaption></figure>

The noise level intersects LISA's PM noise allocation around 6 mHz and reaches more than 7 times the noise allocation at $f = 1$ Hz. The same procedure is repeated for a configuration with $N_2 = 3$, resulting in the red curve. In this case, the aliased noise level from $CIC_2$ remains lower than the PM noise allocation in the whole LISA band and reaches 1.12% of the PM noise allocation at $f = 1$ Hz.

4.5. *Estimation of aliased noise for a flexing-arm constellation after TDI 2.0*

We use the analytical model of the coupling between the CIC filters and the TDI 2.0 algorithm from equation (41). By simplifying the LISA decimation scheme shown in figure 8 to include only $CIC_2$, the computation time is significantly reduced. In fact, applying a second decimation operator increases quadratically the computational load. Figure 13 compares the laser noise residuals of our model with the noise allocation of a LISA PM multiplied by the TDI 2.0 transfer function [18]. The blue curve represents the laser noise filtered and aliased by the second CIC filter, $CIC_2$, of order $N_2 = 2$, processed using TDI 2.0. It can be observed that, in this case, the resulting noise level exceeds the LISA PM noise allocation (black curve) between 6 mHz and 1 Hz. By increasing the filter order to $N_2 = 3$, the aliased noise level becomes lower than the allocation. This result suggests that a filter order $N_2 > 2$ is required to avoid excessive laser noise aliasing within the LISA frequency band.

It is also observed that the shape of the aliased noise curve for $N_2 = 3$ changes around 0.3 Hz. This behaviour does not appear in the case of coupling with TDI 1.0 for fixed arms, suggesting that it depends on the drift terms $\dot{L}_{12}$ and $\dot{L}_{13}$. Although we attribute this residual to flexing filtering [5], we leave a quantitative analysis for future studies.

For reference, the secondary noise sources in LISA-shot noise, electronic noise, and other metrology noises-are expected to contribute at the level of $\sim 4 \times 10^{-8} \text{Hz} \sqrt{\text{Hz}}^{-1}$ at
<figure><img src="image_16.png" /><figcaption>**Figure 13.** Coupling between the two CIC filters in LISA and TDI 2.0. The arm lengths vary linearly with time. Their initial lengths are set as $L_{12,0} = \frac{10240}{2f_{\text{out},2}} = 8$ s and $\dot{L}_{12} = \frac{5 \text{ m} \cdot \text{s}^{-1}}{c}$. For arm 13, the initial length is $L_{13,0} = \frac{10448.5}{2f_{\text{out},2}} = 8.1625$ s and $\dot{L}_{13} = \frac{10 \text{ m} \cdot \text{s}^{-1}}{c}$.</figcaption></figure>

10 mHz [7]. The laser frequency noise, initially $30\,\text{Hz}^{-1}$, is expected to be reduced by TDI to $\sim 2 \times 10^{-9}\,\text{Hz}^{-1}$. When including the CIC-TDI coupling, we estimate maximum residuals of order $4 \times 10^{-8}\,\text{Hz}^{-1}$ for a CIC filter of order 2 in both fixed and flexing-arm configurations, decreasing below $6 \times 10^{-13}\,\text{Hz}^{-1}$ and $10^{-10}\,\text{Hz}^{-1}$ for order 3 (see figure 13). At 10 mHz, the $N_2 = 2$ coupling is therefore comparable to the secondary-noise level; for $f < 10$ mHz the secondary noises dominate, whereas for $f > 10$ mHz the $f^{N_2}$ scaling implies that the $N_2 = 2$ coupling likely exceeds the metrology contribution. For $N_2 = 3$, the coupling remains below the secondary-noise floor over the LISA band.

## 4.6. Study of the non-stationarity of CICxTDI coupling

We aim here to study the dependence of the aliased noise amplitude from the final CIC filter in LISA to the arm lengths of the constellation. It can be demonstrated that the aliasing effect cancels out for a fixed-arm configuration using TDI 1.0 provided that the roundtrip delays ($L_{121}$ and $L_{131}$) are integer multiples of the output sampling period (see appendix C).

We then consider a set of constellations with arm 12 fixed and arm 13 going to $[21/f_{\text{out},2} : 21.5/f_{\text{out},2}]$. For each of them, we apply TDI 2.0. In the following we assume that the data duration on which TDI is applied is short enough to approximate the constellation as having fixed arms during that period. Our analysis is restricted to the higher range of the LISA frequency band ($10^{-2}\,\text{Hz}$ to $1\,\text{Hz}$), as aliased noise has the greatest impact at these frequencies.

We calculate that, at a maximum arm-length drift rate of $10\,\text{ms}^{-1}$, the propagation delay would reach half-integer multiples of $1/f_{\text{out},2}$-the points where the coupling effect cancels-only after a timescale about 23 times longer than our acquisition duration (1000 s).
<figure><img src="image_17.png" /><figcaption>**Figure 14.** ASD of the simulated aliased laser noise, initially set to (30 Hz $\sqrt{Hz}^{-1}$), for different values of arm 13's length, after applying TDI 2.0 and using a filter with parameters $N=2$, $RM = 5^6$, $f_{in} = 1 \times 10^7$ Hz, and $f_{out,2} = 640$ Hz. Here, we set $L_{12} = 5120/f_{out,2} = 8$ s and vary $L_{13}$ from $5141/f_{out,2}$ s to $5141.25/f_{out,2}$ s. The curve corresponding to the minimum residual ($L_{13} = 5141/f_{out,2}$ s) is not displayed, as it falls outside the plotting range and would compromise the readability of the other curves. The maximum residual is reached for $L_{13} - L_{12} = 21.25/f_{out,2}$ (red curve).</figcaption></figure>

The residuals of TDI 2.0 in the upper part of LISA's band (10⁻² Hz to 1 Hz) for different values of arm 13's length are plotted in figure 14.

As expected, the aliasing effect disappears for $L_{13} - L_{12} = 21/f_{out,2}$ because it corresponds to round-trip arm lengths being integers of $1/f_{out,2}$.

We also note that a maximum of noise is reached when $L_{13} - L_{12} = 21.25/f_{out,2}$, i.e. a differential roundtrip propagation time being half an integer of the CIC₂ sampling period. To illustrate the temporal variation of this non-stationarity, we plot the value of the ASD near a maximum ($f = 0.98$ Hz) for 2 filter order ($N_2 = 2$ and $N_2 = 3$) and $L_{13} - L_{12}$ varying from 21 to 21.5 times the filter sampling period. The results are shown in figures 15 and 16.

For a constellation with one fixed arm (an integer multiple of $1/f_{out,2}$) and the other growing at a constant velocity of $v = 10 \text{ m} \cdot \text{s}^{-1}$, we estimate that the regions where this coupling cancels would be crossed with a period of: $T_{\text{coupling}} = \frac{c}{2f_{\text{out},2} \cdot v} \approx 2.34 \cdot 10^4 \text{ s}$ minimum, while the maximum can be computed as a function of the relative speeds in LISA arms given by the orbits. This corresponds to a frequency of $f_{\text{coupling}} = \frac{1}{T_{\text{coupling}}} \approx 4.27 \cdot 10^{-5} \text{ Hz}$.

In agreement with our assumptions about the constellation, this noise could manifest itself as a fluctuation in the background noise of LISA with a minimum period of approximately 6.5 h and an ASD of 0.71 nm $\sqrt{Hz}^{-1}$ for a final CIC filter of order $N_2 = 2$ at approximately 1 Hz. For $N_2 = 3$, the ASD decreases to 1.08 pm $\sqrt{Hz}^{-1}$ at approximately 1 Hz, in both cases following a dependency of $f^{N_2}$. These curves confirm that a final CIC filter of order 3 is required to guarantee a residual PM noise compatible with the requirement at any time.
<figure><img src="image_18.png" /><figcaption>Figure 15. ASD of the simulated aliased noise at 0.98 Hz after applying TDI 2.0, for different values of arm 13's length. We set $L_{12} = 5120/f_{out,2} = 8$ s and vary $L_{13} - L_{12}$ from 21 to 21.5 integer multiples of the sampling period of LISA's final CIC filter (order 2).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 16. ASD of the aliased noise at 0.98 Hz after applying TDI 2.0, for different values of arm 13's length. The same arm model as in figure 15 is used.</figcaption></figure>

Realistic LISA orbits induce periodic and non linear arm length fluctuations [7]. To illustrate their effect, we consider a simplified orbital model in which the arm lengths evolve as

$$L_{ij}(t) = L_0 + \Delta L \sin(\omega t + \phi_{ij}), \quad (44)$$
with $\Delta L/L_0 = 1\%$, $\omega = 2\pi/(1 \text{ year})$, and a phase difference of $\phi_{13} - \phi_{12} = 2\pi/3$. In this configuration, the relative velocity between arms does not exceed $\approx 8.6 \text{ m s}^{-1}$, leading to a modulation in the amplitude of the non-stationary period that reaches a maximum value of $\approx T_{\text{coupling}} \times 10/8.6$. The corresponding modulation frequency, below $\sim 3.7 \times 10^{-5} \text{ Hz}$, lies well within the acceleration-noise-dominated regime of LISA measurements, while above 1 mHz the coupling can be considered quasi-static. Therefore, we estimate that realistic orbital variations are expected to mainly affect the periodicity of the non-stationary residuals rather than their amplitude, without significantly altering the conclusions of this study.

Sources within the LISA band should be largely unaffected by the aliasing residuals, provided that a CIC filter of order at least three is implemented. In this configuration, the residual laser frequency noise represents only a small fraction of the secondary-noise level, resulting in a negligible variation in the signal-to-noise ratio for typical sources such as massive black-hole binaries or galactic binaries [7]. However, non-stationary aliasing could become more critical for searches targeting sources with intrinsically low signal-to-noise ratios, for which a quantitative impact assessment will require dedicated future studies.

## 5. Summary

Our study investigates the coupling between CIC filtering and TDI in the context of LISA-like laser frequency noise suppression. Using the LOT simulator, we analyze the impact of non-commutativity between delay and decimation operations on laser noise residuals. Our simulations confirm that the aliasing effects introduced by the CIC filtering of order $N$ produce residual noise with a characteristic $f^N$ slope, which must be mitigated to meet LISA's phase measurement requirements.

We validate an analytical model predicting laser noise aliasing in TDI 1.0 and TDI 2.0 configurations. For a static transponder setup, the model accurately reproduces the measured noise residuals. In a dynamic constellation with flexing arms, we identify a dependence of aliased noise to arm-length evolution, with periodic variations linked to the output sampling period of the final CIC filter.

Our results indicate that for LISA, a CIC filter order of at least $N_2 = 3$ is required to maintain aliasing below the allocated PM noise level. Finally, we analyze the nonstationary behavior of the CIC-TDI coupling, showing that aliasing effects cyclically cancel out when differential round trip delays align with integer multiples of the final CIC sampling period.

## Data availability statement

The data that support the findings of this study are openly available at the following URL/DOI: https://gitlab.in2p3.fr/leonvidal/LISA_ON_table.

## Acknowledgments

We thank CNES and Université Paris Cité for supporting this research. We also wish to thank the referees for their insightful comments, which helped to improve and clarify this manuscript. We are grateful to the AEI for providing the phasemeter used in this study. The author also acknowledges the MOWGLI team for providing a supportive environment that made it possible to complete this work.
Appendix A. TDI X2 in Fourier space

We want to determine the expression (30) in the Fourier space. To do this, we rewrite the latter as follows:

$$D_{ij} s(t) \approx s(t(1 - 2\dot{L}_{ij}) + L_{ij,0} \cdot \dot{L}_{ji} - 2L_{ij,0}) = s(\alpha t + \beta) \quad (A.1)$$

with $\alpha = 1 - 2\dot{L}_{ij}$, and $\beta = L_{ij,0} \cdot \dot{L}_{ji} - 2L_{ij,0}$

The Fourier transform of a function that depends linearly on time, such as $x(\alpha t + \beta)$, can be written as:

$$ \begin{aligned} \mathcal{F}\mathcal{T}(x(\alpha t + \beta))(f) &= \int_{-\infty}^{\infty} x(\alpha t + \beta) e^{-j2\pi ft} dt \\ &= \int_{-\infty}^{\infty} x(\tau) e^{-j2\pi f \frac{\tau - \beta}{\alpha} \frac{d\tau}{|\alpha|}} \\ &= \frac{e^{j2\pi f \beta / \alpha}}{|\alpha|} \mathcal{F}\mathcal{T}(x(t)) \left(\frac{f}{\alpha}\right). \end{aligned} $$

By substituting the values of a and b into equation (A.2), we obtain the Fourier transform of expression (30).

$$\tilde{D}_{ij} s(t) (f) = \frac{1}{|1 - 2\dot{L}_{ij}|} e^{-j2\pi f \frac{(2L_{ij,0} - \dot{L}_{ij}L_{ij,0})}{1 - 2\dot{L}_{ij}}} S\left(\frac{f}{1 - 2\dot{L}_{ij}}\right). \quad (A.2)$$

Since the relative velocity of our satellites is much smaller than c, this implies that $2\dot{L} \ll 1$. We can then simplify our expression using a first-order Taylor expansion:

$$\Rightarrow S\left(\frac{f}{1 - 2\dot{L}_{ij}}\right) \approx S(f) + 2f\dot{L}_{ij}\frac{dS(f)}{df} \quad (A.3)$$

and also for the exponential term:

$$ \begin{aligned} \Rightarrow e^{-j2\pi f \frac{(2L_{ij,0} - \dot{L}_{ij}L_{ij,0})}{1 - 2\dot{L}_{ij}}} &\approx e^{-j4\pi f L_{ij,0}} + \dot{L}_{ij} \\ &\quad \cdot \left[ e^{-j2\pi f \frac{(2-\dot{L}_{ij})L_{ij,0}}{1-2\dot{L}_{ij}}} (-j2\pi f L_{ij,0}) \cdot \frac{d}{d\dot{L}_{ij}} \left( \frac{2-\dot{L}_{ij}}{1-2\dot{L}_{ij}} \right) \right]_{\dot{L}_{ij}=0} \\ &= (1 - 3j2\pi f L_{ij,0} \dot{L}_{ij}) e^{-j4\pi f L_{ij,0}}. \end{aligned} $$

With these simplifications and by substituting expressions (A.3) and (A.4) into (A.2), we obtain a round-trip delay operator at first order in $\dot{L}_{ij}$ in Fourier space:

$$\tilde{D}_{ij} s(t) \approx k_{ij}(f) \cdot S(f) + h_{ij}(f) \cdot \frac{dS}{df}(f) \quad (A.4)$$

with

$$ \begin{aligned} k_{ij}(f) &= (1 + 2\dot{L}_{ij} - 3j2\pi f L_{ij,0} \dot{L}_{ij}) e^{-j4\pi f L_{ij,0}}, \\ h_{ij}(f) &= 2f\dot{L}_{ij} \cdot e^{-j4\pi f L_{ij,0}} \end{aligned} $$

In a more general way [4], one can use the expression for nested delays applied to a time-dependent signal $x(t)$:

$$y(t) = D_{i_1 i_2 \dots i_n} x(t) = x \left( S_n t - \sum_{k=1}^{n} \frac{S_k}{S_k} L_{ik} \right) \quad (A.5)$$
where

$$S_k = \prod_{p=1}^{k} (1 - \dot{L}_{ip}) \quad \text{and} \quad S_0 = 1 \qquad (A.6)$$

and if we stop at first order on $\dot{L}$.

$$S_k \approx 1 - \sum_{p=1}^{k} \dot{L}_{ip}. \qquad (A.7)$$

Substituting this in (A.5), we get a general formulation :

$$\begin{aligned} y(t) &= x \left[ \left( 1 - \sum_{p=1}^{n} \dot{L}_{ip} \right) t - \sum_{k=1}^{n-1} \left( 1 - \sum_{p=k+1}^{n} \dot{L}_{ip} \right) L_{ik} - L_{in} \right] \\ &= x \left[ \left( 1 - \sum_{p=1}^{n} \dot{L}_{ip} \right) t + \sum_{k=1}^{n-1} \sum_{p=k+1}^{n} \dot{L}_{ip} \cdot L_{ik} - \sum_{k=1}^{n} L_{ik} \right]. \end{aligned} \qquad (A.8)$$

In the following, we use these expressions to get the expression of our delays present in the combination X2 in time domain. Then, we pass in frequency domain and get the expression of the respective PSD.

In the LOT we do some approximation on the arms length and inter-satellite velocities as :

$$\begin{aligned} L_{12} &= L_{21}, L_{13} = L_{31} \\ \dot{L}_{12} &= \dot{L}_{21}, \dot{L}_{13} = \dot{L}_{31}. \end{aligned} \qquad (A.9)$$

Let us define a signal successively delayed by a round trip in the two arms $a(t) = D_{12131}x(t)$:

$$\begin{aligned} a(t) &= x \left[ (1 - 2\dot{L}_{12} - 2\dot{L}_{13})t + (\dot{L}_{12} + 2\dot{L}_{13})L_{12} + 2\dot{L}_{13} \cdot L_{21} + \dot{L}_{13} \cdot L_{31} - 2L_{12} - 2L_{13} \right] \\ &= x \left[ (1 - 2\dot{L}_{12} - 2\dot{L}_{13})t + (\dot{L}_{12} + 4\dot{L}_{13})L_{12} + \dot{L}_{13}L_{13} - 2(L_{12} + L_{13}) \right]. \end{aligned} \qquad (A.10)$$

Similarly for $b(t) = D_{13121}x(t)$

$$b(t) = x \left[ (1 - 2\dot{L}_{13} - \dot{L}_{12})t + (\dot{L}_{13} + 4\dot{L}_{12})L_{13} + \dot{L}_{12}L_{12} - 2(L_{12} + L_{13}) \right]. \qquad (A.11)$$

Adding an extra round-trip delay gives the bigger terms present in the $X_2$ combination $c(t) = D_{1213131}x(t)$

$$\begin{aligned} c(t) &= x \left[ (1 - 2\dot{L}_{12} - 2\dot{L}_{13} - 2\dot{L}_{14})t + (\dot{L}_{12} + 4\dot{L}_{13} + 4\dot{L}_{14})L_{12} + (\dot{L}_{13} + 4\dot{L}_{14})L_{13} \right. \\ & \qquad \left. + \dot{L}_{13}L_{14} - 2L_{12} - 4L_{14} \right] \\ &= x \left[ (1 - 2\dot{L}_{12} - 4\dot{L}_{13})t + \dot{L}_{12}L_{12} + 6\dot{L}_{13}L_{13} + 8\dot{L}_{13}L_{12} - 2L_{12} - 4L_{13} \right]. \end{aligned} \qquad (A.12)$$

Similarly for $d(t) = D_{1312121}x(t)$

$$d(t) = x \left[ (1 - 2\dot{L}_{13} - 4\dot{L}_{12})t + \dot{L}_{13}L_{13} + 6\dot{L}_{12}L_{12} + 8\dot{L}_{12}L_{13} - 2L_{13} - 4L_{12} \right]. \qquad (A.13)$$

As we have done earlier for the round-trip delay, we want to get these expressions in the Fourier domain. Let us start with the Fourier transform of $\tilde{A}(f) = \mathcal{F}\mathcal{T}\{a(t)\}$:

$$\tilde{A}(f) = \mathcal{F}\mathcal{T}\{x \left[ (1 - 2\dot{L}_{12} - 2\dot{L}_{13})t + (\dot{L}_{12} + 4\dot{L}_{13})L_{12} + \dot{L}_{13}L_{13} - 2(L_{12} + L_{13}) \right]\} \qquad (A.14)$$
One can use equation (A.2) and get :

$$ \tilde{A}(f) = \frac{1}{|1 - 2\dot{L}_{12} - 2\dot{L}_{13}|} \exp \left[ j\omega \frac{(\dot{L}_{12} + 4\dot{L}_{13}) L_{12} + \dot{L}_{13} L_{13} - 2(L_{12} + L_{13})}{1 - 2\dot{L}_{12} - 2\dot{L}_{13}} \right] \times \tilde{X} \left( \frac{f}{1 - 2\dot{L}_{12} - 2\dot{L}_{13}} \right). \quad (A.15) $$

Because we assume $\dot{L}_{12}, \dot{L}_{13} \ll 1$

$$ \begin{aligned} \tilde{A}(f) & \approx \frac{1}{(1 - 2\dot{L}_{12} - 2\dot{L}_{13})} (1 + j\omega [L_{12} - 4(L_{12} + L_{13})]) \dot{L}_{12} + j\omega [4L_{12} + L_{13} - 4(L_{12} + L_{13})] \dot{L}_{13} \\ & \times \exp[-2j\omega(L_{12} + L_{13})] \times [\tilde{X}(f) + 2(\dot{L}_{12} + \dot{L}_{13})f \frac{d\tilde{X}(f)}{df}] \\ & \approx [1 + 2(\dot{L}_{12} + \dot{L}_{13})] [1 - 3j\omega(L_{12}\dot{L}_{12} + L_{13}\dot{L}_{13}) - 4j\omega L_{13}\dot{L}_{12}] \\ & \times \exp[-2j\omega(L_{12} + L_{13})] [\tilde{X}(f) + 2(\dot{L}_{12} + \dot{L}_{13})f \frac{d\tilde{X}(f)}{df}] \\ & \approx [1 + 2(\dot{L}_{12} + \dot{L}_{13}) - 3j\omega(L_{12}\dot{L}_{12} + L_{13}\dot{L}_{13}) - 4j\omega L_{13}\dot{L}_{12}] \exp[-2j\omega(L_{12} + L_{13})] \tilde{X}(f) \\ & + 2(\dot{L}_{12} + \dot{L}_{13}) \exp[-2j\omega(L_{12} + L_{13})] f \frac{d\tilde{X}(f)}{df}. \end{aligned} \quad (A.16) $$

Similarly, we obtain the expression for $\tilde{B}(f) = \mathcal{FT}\{b(t)\}$, exchanging the index 12 and 13 in the expression of $\tilde{A}(f)$:

$$ \tilde{B}(f) \approx [1 + 2(\dot{L}_{12} + \dot{L}_{13}) - 3j\omega(L_{12}\dot{L}_{12} + L_{13}\dot{L}_{13}) - 4j\omega L_{12}\dot{L}_{13}] \exp[-j\omega 2(L_{12} + L_{13})] \tilde{X}(f) \\ + 2(\dot{L}_{12} + \dot{L}_{13}) \exp[-j\omega 2(L_{12} + L_{13})] f \frac{d\tilde{X}(f)}{df}. \quad (A.17) $$

We now calculate the expression for $\tilde{C}(f) = \mathcal{FT}\{c(t)\}$:

$$ \begin{aligned} c(t) &= x [(1 - 2\dot{L}_{12} - 4\dot{L}_{13})t + \dot{L}_{12}L_{12} + 6\dot{L}_{13}L_{13} + 8\dot{L}_{13}L_{12} - 2L_{12} - 4L_{13}] \\ &= x(\alpha t + \beta) \end{aligned} \quad (A.18) $$

where this time $\alpha = (1 - 2\dot{L}_{12} - 4\dot{L}_{13})$

and $\beta = \dot{L}_{12}L_{12} + 6\dot{L}_{13}L_{13} + 8\dot{L}_{13}L_{12} - 2L_{12} - 4L_{13}$.

Using equation (A.2), we get:

$$ \begin{aligned} \tilde{C}(f) & \approx (1 + 2\dot{L}_{12} + 4\dot{L}_{13}) [1 + jw(L_{12} - 2(2L_{12} + 4L_{13}))\dot{L}_{12} \\ & \quad + j\omega[6L_{13} + 8L_{12} - 4(2L_{12} + 4L_{13})]\dot{L}_{13}] \\ & \times \exp[-jw(2L_{12} + 4L_{13})] [\tilde{X}(f) + 2f(\dot{L}_{12} + 2\dot{L}_{13})\frac{\partial\tilde{X}(f)}{\partial f}] \\ & \approx (1 + 2\dot{L}_{12} + 4\dot{L}_{13}) [1 - 3jwL_{12}\dot{L}_{12} - 10j\omega L_{13}\dot{L}_{13} - 8j\omega L_{13}\dot{L}_{12}] \\ & \quad \times \exp[-jw(2L_{12} + 4L_{13})] \tilde{X}(f) \\ & \quad + 2f(\dot{L}_{12} + 2\dot{L}_{13}) \exp[-jw(2L_{12} + 4L_{13})] \frac{\partial\tilde{X}(f)}{\partial f} \end{aligned} $$
$$ \approx [1 + 2\dot{L}_{12} + 4\dot{L}_{13} - 3jwL_{12}\dot{L}_{12} - 10j\omega L_{13}\dot{L}_{13} - 8j\omega L_{13}\dot{L}_{12}] \\ \times \exp[-jw(2L_{12} + 4L_{13})]\tilde{X}(f) \\ + 2f(\dot{L}_{12} + 2\dot{L}_{13})\exp[-jw(2L_{12} + 4L_{13})]\frac{\partial\tilde{X}(f)}{\partial f}. \qquad (A.19) $$

The equivalent expression for $\tilde{D}(f) = \mathcal{F}\mathcal{T}\{d(t)\}$ is obtained by exchanging the indices 12 and 13 in the expression of $\tilde{C}(f)$:

$$ \tilde{D}(f) \approx [1 + 2\dot{L}_{13} + 4\dot{L}_{12} - 3jwL_{13}\dot{L}_{13} - 10j\omega L_{12}\dot{L}_{12} - 8j\omega L_{12}\dot{L}_{13}] \\ \times \exp[-jw(2L_{13} + 4L_{12})]\tilde{X}(f) \\ + 2f(\dot{L}_{13} + 2\dot{L}_{12})\exp[-jw(2L_{13} + 4L_{12})]\frac{\partial\tilde{X}(f)}{\partial f}. \qquad (A.20) $$

Apply these equations to (40):

$$ \begin{aligned} X_{2,0}(f) &= (-1 + \tilde{D}_{131} + \tilde{D}_{13121} - \tilde{D}_{1213131}) \tilde{M}_{12}(f) \\ &\quad + (1 - \tilde{D}_{121} - \tilde{D}_{12131} + \tilde{D}_{1312121}) \tilde{M}_{13}(f) \\ &\approx -\tilde{M}_{12}(f) \\ &\quad + (1 + 2\dot{L}_{13} - 3j\omega L_{13}\dot{L}_{13}) e^{-2j\omega L_{13}} \tilde{M}_{12}(f) \\ &\quad + 2f\dot{L}_{13} \cdot e^{-2j\omega L_{13}} \frac{d\tilde{M}_{12}(f)}{df} \\ &\quad + [1 + 2(\dot{L}_{12} + \dot{L}_{13}) - 3j\omega(L_{12}\dot{L}_{12} + L_{13}\dot{L}_{13}) - 4j\omega L_{12}\dot{L}_{13}] \\ &\quad \times \exp[-2j\omega(L_{12} + L_{13})] \tilde{M}_{12}(f) \\ &\quad + 2f(\dot{L}_{12} + \dot{L}_{13}) \exp[-2j\omega(L_{12} + L_{13})] \frac{d\tilde{M}_{12}(f)}{df} \\ &\quad - [1 + 2\dot{L}_{12} + 4\dot{L}_{13} - 3jwL_{12}\dot{L}_{12} - 10j\omega L_{13}\dot{L}_{13} - 8j\omega L_{13}\dot{L}_{12}] \\ &\quad \times \exp[-2jw(L_{12} + 2L_{13})] \tilde{M}_{12}(f) \\ &\quad - 2f(\dot{L}_{12} + 2\dot{L}_{13}) \exp[-2jw(L_{12} + 2L_{13})] \frac{d\tilde{M}_{12}(f)}{df} \\ &\quad + \tilde{M}_{13}(f) - (1 + 2\dot{L}_{12} - 3j\omega L_{12}\dot{L}_{12}) e^{-2j\omega L_{12}} \tilde{M}_{13}(f) \\ &\quad - 2f\dot{L}_{12} \cdot e^{-2j\omega L_{12}} \frac{d\tilde{M}_{13}(f)}{df} \\ &\quad - [1 + 2(\dot{L}_{13} + \dot{L}_{12}) - 3j\omega(L_{13}\dot{L}_{13} + L_{12}\dot{L}_{12}) - 4j\omega L_{13}\dot{L}_{12}] \\ &\quad \times \exp[-2j\omega(L_{12} + L_{13})] \tilde{M}_{13}(f) \\ &\quad - 2f(\dot{L}_{12} + \dot{L}_{13}) \exp[-2j\omega(L_{12} + L_{13})] \frac{d\tilde{M}_{13}(f)}{df} \\ &\quad + [1 + 2\dot{L}_{13} + 4\dot{L}_{12} - 3jwL_{13}\dot{L}_{13} - 10j\omega L_{12}\dot{L}_{12} - 8j\omega L_{12}\dot{L}_{13}] \\ &\quad \times \exp[-2jw(L_{13} + 2L_{12})] \tilde{M}_{13}(f) \\ &\quad + 2f(\dot{L}_{13} + 2\dot{L}_{12}) \exp[-2jw(L_{13} + 2L_{12})] \frac{d\tilde{M}_{13}(f)}{df}. \qquad (A.21) \end{aligned} $$

Substituting (39) into the above equation, we can obtain the final result.
It is possible to show that successive application of the round-trip delay operator in the frequency domain, introduced in equation (A.4), gives the same result as the general approach operated first in the time domain (equation (A.8)) and then transformed in the frequency domain up to the first order in $\dot{L}$.

## Appendix B. Compute PSD of $R(f, \tau)$

The difference $R(f, \tau)$ between the two arms of figure 6 is obtained as:

$$ 
\begin{aligned}
R(f, \tau) &= T(f, \tau) - K(f, \tau) \\
&= \tilde{D}_{\tau} (H_C \cdot S)_s (f) - (H_C \cdot \tilde{D}_{\tau} S)_s (f) \\
&= H_C(f) S(f) e^{-j2\pi f \tau} - H_C(f) S(f) e^{-j2\pi f \tau} + \left[ \sum_{k=1}^{+\infty} H_{C,k_{-}}^{*} S_{k_{-}}^{*} + H_{C,k_{+}} S_{k_{+}} \right] e^{-j2\pi f \tau} \\
&\quad - \left[ \sum_{k=1}^{+\infty} H_{C,k_{-}}^{*} S_{k_{-}}^{*} e^{j2\pi (kf_{s}-f)\tau} + H_{C,k_{+}} S_{k_{+}} e^{-j2\pi (kf_{s}+f)\tau} \right] \\
&= \sum_{k=1}^{+\infty} \left[ H_{C,k_{-}}^{*} S_{k_{-}}^{*} \left( e^{-j2\pi f \tau} - e^{j2\pi (kf_{s}-f)\tau} \right) + H_{C,k_{+}} S_{k_{+}} \left( e^{-j2\pi f \tau} - e^{-j2\pi (kf_{s}+f)\tau} \right) \right] \\
&= e^{-j2\pi f \tau} \sum_{k=1}^{+\infty} \left[ H_{C,k_{-}}^{*} S_{k_{-}}^{*} \left( 1 - e^{j2\pi (kf_{s})\tau} \right) + H_{C,k_{+}} S_{k_{+}} \left( 1 - e^{-j2\pi (kf_{s})\tau} \right) \right].
\end{aligned}
$$

For clarity we define the function $I$ as:

$$ 
\begin{aligned}
I(kf_{s}+f, \tau) &= H_{C,k_{+}} S_{k_{+}} \left(1 - e^{-j2\pi (kf_{s})\tau}\right) \\
I(kf_{s}-f, \tau) &= H_{C,k_{-}} S_{k_{-}} \left(1 - e^{-j2\pi (kf_{s})\tau}\right).
\end{aligned}
$$

By injecting this term into equation (B.1), we express $R(f, \tau)$ more simply:

$$ R(f, \tau) = e^{-j2\pi f \tau} \sum_{k=1}^{+\infty} [I^*(kf_s - f, \tau) + I(kf_s + f, \tau)]. \quad (B.1) $$

To compute the power spectral density of $R(f, \tau)$, we start computing its squared modulus:

$$ 
\begin{aligned}
|R(f, \tau)|^2 &= \sum_{k_1=1}^{+\infty} \sum_{k_2=1}^{+\infty} [I^*(k_1 f_s - f, \tau) + I(k_1 f_s + f, \tau)] \cdot [I(k_2 f_s - f, \tau) + I^*(k_2 f_s + f, \tau)] \\
&= \sum_{k_1=1}^{+\infty} \sum_{k_2=1}^{+\infty} I^*(k_1 f_s - f, \tau) \cdot I(k_2 f_s - f, \tau) + I(k_1 f_s + f, \tau) \cdot I^*(k_2 f_s + f, \tau) \\
&\quad + I(k_1 f_s + f, \tau) \cdot I(k_2 f_s - f, \tau) + I^*(k_1 f_s - f, \tau) \cdot I^*(k_2 f_s + f, \tau).
\end{aligned}
$$

The Nyquist theorem guarantees $f \in ]0 : f_s/2[$:

$$ \Rightarrow k_1 f_s - f \neq k_2 f_s + f \quad \forall k_1, k_2 \geq 1. \quad (B.2) $$
For the LOT case, the input signal is a white Gaussian noise $N(f)$ filtered by an ARMA filter with transfer function $H_A(f)$:

$$S(f) = H_A(f) \cdot N(f).$$

Using the properties of white noise, we can write that:

$$\Rightarrow \langle N(f) N^*(f') \rangle = \delta_{f-f'} \langle |N(f)|^2 \rangle$$

$$\Rightarrow \langle S(f) S^*(f') \rangle = \delta_{f-f'} \langle |S(f)|^2 \rangle.$$

Equation (B.2) is then simplified as:

$$|R(f, \tau)|^2 = \sum_{k=1}^{+\infty} [I^*(kf_s - f, \tau) \cdot I(kf_s - f, \tau)] + [I(kf_s + f, \tau) \cdot I^*(kf_s + f, \tau)] \\ = \sum_{k=1}^{+\infty} |I(kf_s - f, \tau)|^2 + |I(kf_s + f, \tau)|^2.$$

We introduce the transfer function of the ARMA filter $H_A$ used in the LOT to filter the simulated white noise. Then, we combine the filters transfer functions to use a unique filter term $H(f) = H_A(f) \cdot H_C(f)$. By expanding the expression of $I$ and writing the spectrum of our white noise $N(f)$, we get from the previous equation:

$$|R(f, \tau)|^2 = \sum_{k=1}^{+\infty} \left[ |N_{k_{-}}|^2 |H_{k_{-}}|^2 \cdot \left| (1 - e^{-j2\pi(kf_s)\tau}) \right|^2 + |N_{k_{+}}|^2 |H_{k_{+}}|^2 \cdot \left| (1 - e^{-j2\pi(kf_s)\tau}) \right|^2 \right].$$

Using the Wiener-Khintchine theorem and assuming that the PSD of our white noise $PSD_N$ is constant $B^2$ for any frequency $f$: $PSD_N(kf_s + f) = PSD_N(kf_s - f) = PSD_N(f) = B^2$, we express the PSD of $R(f, \tau)$ as $PSD_R(f)$:

$$PSD_R(f) = 4B^2 \sum_{k=1}^{+\infty} (\sin^2(\pi k f_s \tau)) \cdot (|H_{k_{-}}|^2 + |H_{k_{+}}|^2).$$

## Appendix C. Effect cancellation for a fixed arm transponder mode constellation

We show here that the residual laser noise cancels out when $\tau_{121}$ and $\tau_{131}$ are multiples of the sampling period $1/f_s$ and distinguish two cases:

* $\tau_{121}$ and $\tau_{131}$ are even or odd multiples of $1/f_s$. This implies that:

$$\Delta\tau = \tau_{121} - \tau_{131} = \frac{2p}{f_s} \text{ is an even multiple of } 1/f_s$$

$$\tau = \frac{\tau_{121} + \tau_{131}}{2} = \frac{n}{f_s} \text{ is a multiple of } 1/f_s.$$
In this case we compute $E(kf_s)$ in the equation (27) term by term:

$$e^{j2\pi kf_s\tau} = e^{j2\pi k \cdot n} = \mathbf{1}, \forall k$$

$$\sin(\pi(kf_s - f)\Delta\tau) = \sin\left(\pi(kf_s - f)\frac{2p}{f_s}\right) = -\sin\left(2\pi p \frac{f}{f_s}\right), \forall k$$

$$\sin(\pi kf_s \Delta\tau) = \sin(2\pi k p) = 0$$

$$\sin(\pi f \Delta\tau) = \sin\left(\pi \frac{f}{f_s} 2p\right).$$

Substituting these expressions into $E(kf_s)$, we get:

$$E(kf_s) = 2je^{-j2\pi f\tau}\left(\left[-\sin\left(2\pi p \frac{f}{f_s}\right) - 0 \cdot e^{-j2\pi f\tau}\right] + \sin\left(\pi \frac{f}{f_s} 2p\right)\right) = 0.$$

This result is true whatever the summation index k so $\tilde{X}_1(f, \tau, \Delta\tau) = 0$

* $\tau_{121}$ and $\tau_{131}$ are of opposite parity. This implies that:

$$\Delta\tau = \tau_{121} - \tau_{131} = \frac{2p+1}{f_s} \text{ is an odd multiple of } 1/f_s$$

$$\tau = \frac{\tau_{121} + \tau_{131}}{2} = \frac{2n+1}{2f_s} \text{ is a half multiple of } 1/f_s$$

We calculate $E(kf_s)$ this time in the second case:

$$e^{j2\pi kf_s\tau} = e^{j2\pi kf_s \frac{2n+1}{2f_s}} = \begin{cases} 1 & \text{if } k \text{ even} \\ -1 & \text{if } k \text{ odd} \end{cases}$$

$$\sin(\pi(kf_s - f)\Delta\tau) = \sin\left(\pi(kf_s - f)\frac{2p+1}{f_s}\right)$$

$$= \begin{cases} -\sin\left(\pi(2p+1)\frac{f}{f_s}\right) & \text{if } k \text{ even} \\ \sin\left(\pi(2p+1)\frac{f}{f_s}\right) & \text{if } k \text{ odd} \end{cases}$$

$$\sin(\pi kf_s \Delta\tau) = \sin\left(\pi kf_s \frac{2p+1}{f_s}\right)$$

$$= \sin(2\pi k (2p+1)) = 0$$

$$\sin(\pi k f \Delta\tau) = \sin\left(\pi f \frac{2p+1}{f_s}\right)$$

$$= \sin\left(\pi (2p+1) \frac{f}{f_s}\right).$$
Substituting these expressions into $E(kf_s)$, we obtain:

$$ 
\begin{aligned}
E(kf_s) &= 2je^{-j2\pi f\tau} \left[ \pm 1 \left[ \mp \sin \left( \pi (2p+1) \frac{f}{f_s} \right) - \right. \right. \\
&\qquad \left. \left. + \sin \left( \pi (2p+1) \frac{f}{f_s} \right) \right] \right] \\
&= 0.
\end{aligned}
$$

This result is also true whatever the summation index k so $\tilde{X}_1(f, \tau, \Delta\tau) = 0$.

For both possible cases, the terms $E(kf_s) = E(-kf_s) = 0$, so the combination cancels perfectly:

$$ \Rightarrow \tilde{X}_1(f, \tau, \Delta\tau) = 0 \qquad (C.1) $$

when $\tau_{121}$ and $\tau_{131}$ are multiples of $1/f_s$.

## Appendix D. Useful equations

### D.1. Phasemeter noise allocation

We approximate the noise amplitude spectral density allocated to a single link of a LISA phase-meter (at 1 pm $\sqrt{Hz}$) [19] with this expression:

$$ ASD_{PM,f} = 5.9 \cdot 10^{-6} \cdot \left( 1 + \left( \frac{f}{3 \times 10^{-3}} \right)^{-4} \right)^{0.5} \cdot \frac{f}{2\pi} \qquad (D.1) $$

where we expressed it in units of $\frac{Hz}{\sqrt{Hz}}$.

### D.2. CIC filter derivative

$$ \frac{dH_C}{df}(f) = \left( N \cdot \left( \frac{\sin(RM\pi f/f_s)}{RM\sin(\pi f/f_s)} \right)^{N-1} \cdot \frac{\pi}{f_s} \cdot \left( \frac{\cos(RM\pi f/f_s)}{\sin(\pi f/f_s)} - \frac{\sin(RM\pi f/f_s)\cos(\pi f/f_s)}{RM(\sin(\pi f/f_s))^2} \right) \right) \cdot \exp\left(-iN\frac{(RM-1)\pi f}{f_s}\right) - iN\frac{(RM-1)\pi}{f_s} \cdot H_C(f) $$
## Appendix E. Figures

<figure><img src="image_20.png" /><figcaption>**Figure E1.** Condition on the nature of the noise spectrum we consider for the validity of our approximation on equation (38).</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure E2.** Transfer function of the CIC filtering implemented in the LOT.</figcaption></figure>

## ORCID iDs

Léon Vidal ID 0000-0002-7492-2126

Hubert Halloin ID 0000-0002-9250-4622

Nam Dam Quang ID 0000-0003-3125-6909

Pierre Prat ID 0000-0002-2129-236X

Antoine Petiteau ID 0000-0002-7371-9695
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