Time-delay interferometry infinity for tilt-to-length noise estimation in LISA

The Laser Interferometer Space Antenna (LISA) mission is a space-borne observatory designed to detect and characterize gravitational wave sources inaccessible to ground-based detectors. The mission relies on laser interferometry to measure changes in space-time. In this context, non-avoidable noise sources within the LISA system, including tilt-to-length (TTL) coupling, reduce the detector’s resolution and complicate achieving the mission’s goals unless appropriate mitigation strategies are implemented. This paper applies time-delay interferometry infinity (TDI- ∞ ) for TTL noise estimation in LISA and assesses its suitability from the perspective of system identification and calibration for the first time. The recently published TDI- ∞ concept indicates a different frequency response within the LISA measurement band compared to the standard methodology of TDI second-generation. Our main contribution is demonstrating the advantages of the TDI- ∞ algorithm for TTL noise calibration in space-borne interferometer constellations. Specifically, we show that this algorithm improves the estimation performance of TTL noise due to its frequency behavior while requiring less computation time. The reduction in computational effort afforded by the TDI- ∞ algorithm could accelerate the availability of calibrated TDI data for astrophysical analysis if required by the LISA science community. The improvement in estimation performance underscores the concept’s potential to enhance the detector’s sensitivity further.

The Laser Interferometer Space Antenna (LISA) mission is a space-borne observatory designed to detect and characterize gravitational wave sources inaccessible to ground-based detectors. The mission relies on laser interferometry to measure changes in space-time. In this context, non-avoidable noise sources within the LISA system, including tilt-to-length (TTL) coupling, reduce the detector's resolution and complicate achieving the mission's goals unless appropriate mitigation strategies are implemented. This paper applies time-delay interferometry infinity (TDI-∞) for TTL noise estimation in LISA and assesses its suitability from the perspective of system identification and calibration for the first time. The recently published TDI-∞ concept indicates a different frequency response within the LISA measurement band compared to the standard methodology of TDI second-generation. Our main contribution is demonstrating the advantages of the TDI-∞ algorithm for TTL noise calibration in space-borne interferometer constellations. Specifically, we show that this algorithm improves the estimation performance of TTL noise due to its frequency behavior while requiring less computation time. The reduction in computational effort afforded by the TDI-∞ algorithm could accelerate the availability of calibrated TDI data for astrophysical analysis if required * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Introduction
The section provides an introduction to the Laser Interferometer Space Antenna (LISA) constellation as a means to facilitate the understanding of derivations that follow in later sections. Then, the concept of time-delay interferometry (TDI) and the relevance of tilt-to-length (TTL) coupling noise calibration are presented. Figure 1 depicts the LISA constellation and the reference systems used in the paper with corresponding index notations. The triangular constellation moves in an Earth-like orbit around the Sun with a nominal distance between remote spacecraft (SC) of 2.5 million kilometers. Each SC carries two Moving Optical Subassemblies (MOSA) and each MOSA hosts an optical bench (OB), a telescope, and a free-falling test mass (TM) [1]. An MOSA can be rotated perpendicular to the constellation plane about its pivot axis. The TMs are to be shielded from any noise, such as solar radiation pressure and should ideally be affected only by incident gravitational waves. The goal of LISA is to resolve distance changes between TMs of remote SC related to ripples in space time [2]. Therefore, various interferometric measurements are performed onboard, which are combined in postprocessing to obtain signals accessible for astrophysical purposes. In this context, the calculation of TDI observables is required, the concept of which is introduced in the next section.

Concept of TDI
The LISA orbits have been carefully selected to guarantee the constellation's stability throughout its planned lifetime. To comply with the bandwidth limitation of the interferometric beat note detection system, the relative velocity between opposing SC must not exceed 10 m s −1 during the ten years mission duration [3]. Although the relative SC velocities are already limited, LISA's unequal interferometer arm lengths pose a significant challenge to detecting gravitational waves. This is because the relative long arm phase measurements are highly disturbed by the influence of laser phase noise then. The frequency instability of a laser source is unproblematic for equal-arm interferometers. Here, laser frequency noise fed into two equal-length interferometer arms at one point in time returns from these interferometer arms to the detector simultaneously, thus suppressing itself in relative phase measurements [4]. The suppression does not occur automatically for LISA. Instead, suppression must be achieved in postprocessing applying the TDI algorithm [5]. The basic idea of TDI is to record the noise of a laser source twice using independent detectors on different SC. The measurements are then appropriately shifted in time relative to each other and linearly combined so that laser phase noise is suppressed in the linear combination of the measurements. The linear combinations are called TDI observables or TDI variables. Since TDI does not suppress the science signal, the output of the algorithm is suitable for gravitational wave analyses.
TDI-unsuppressed noise sources deteriorate the science performance and impede the realization of the mission objectives. This includes TTL noise, which, unfortunately, appears in the TDI output as a dominant noise contributor at a few Millihertz depending on the SC jitter spectrum and TTL coupling coefficients, for example. TTL coupling is not directly measurable and, therefore, considered to be estimated via the TDI observables. Subsequently, the estimates are used to correct the TDI output.
Before discussing the effect of TTL coupling and its estimation in more detail, the secondgeneration TDI algorithm will be presented. This algorithm is standard in current LISA data processing research. Several generations of TDI have evolved over time. First-generation TDI suppresses laser phase noise in the case of stationarity only, i.e. a constellation without temporal variations of unequal arm lengths. TDI generation '1.5' enables suppression for a rotating constellation with time-invariant arm lengths, and TDI second generation suppresses laser phase noise if the arm lengths evolve linearly with time [6].
The time-delay interferometry infinity (TDI-∞) [7] algorithm focused on in this paper describes the highest of all possible TDI generations and suppresses laser phase noise for any relative dynamics of the LISA SC. Since the TTL estimation performance of a TDI-∞ based estimator will be compared to the second-generation TDI reference at the end of the paper, analytical expressions of a specific set of second-generation TDI variable will be presented as well as the mathematical modeling of LISA's measurement setup, which is required for both concepts.
1.2.1. Phasemeter modeling equations. Three interferometric measurements are recorded per OB. For a detailed explanation of LISA's optical design, please refer to the corresponding literature [6,8,9]. First, a long arm measurement is performed at the carrier frequency (s c ij (t) in the following) and at the sideband frequency (s sb ij (t) in the following). The laser beam of the local laser of OB OBij is brought to interference with the incoming laser beam of the remote OB OBji with {i, j} ∈ 1, 2, 3 and i ̸ = j. The occurrence of laser phase noise p ij (t) of OBij and p ji (t) of OBji in the long arm measurement performed at OBij is described by (1) and (2). The term h ij (t) covers the gravitational wave signal, n TTL,c ij (t) and n TTL,sb ij (t) represent long arm TTL noise, while n c ij (t) and n sb ij (t) denote remaining noise sources, for example, OB and TM displacement noise as well as clock noise.
Laser phase noise from the remote OB OBji is detected on the OB OBij with a delay defined by the light travel time t j.i from OBji of SCj to OBij of SCi. The long arm measurement on OB OBij at timestep t is, therefore, determined by p ji emitted at t − t j.i , i.e. p ji (t − t j.i ). This is equivalent to the notation p ji;j.i (t), where the time delay is covered by the post-semicolon index. Hence, the delay notation i.j refers to the time delay between the emission of a signal at one SC, denoted by the index i, and its detection at another, denoted by the index j. Figure 1 illustrates the nomenclature by highlighting the SC indices and the inter-SC delay names defined by its two neighboring SC. The notation will also be relevant for TTL modeling in section 1.3. The second measurement is that of the TM interferometer. The laser of an OB is directed to the local TM acting as a mirror, reflected and interfered at the detector with the laser of the neighboring OB of the same SC. For i + j ⩽ 3 the measurement of the TM interferometer of the OB OBij is given by (3a), and for i + j > 3 by (3b), respectively.
The delayed arrival of laser noise from the neighboring OB of the same SC is negligible. Consequently, no delay occurs in the modeling of the local TM interferometer measurement. OB displacement noise, TM displacement noise, and secondary noise sources affecting the measurement are covered by n ϵ ij (t). While the sum of a long arm measurement and two measurements of the TM interferometers already provides the distance between remote TMs, one needs a third measurement per OB to cancel common mode phase fluctuations in the TM interferometer. The required measurement is the one of the reference interferometer. The reference measurement of OBij is described by (4a) (for i + j ⩽ 3) and (4b) (for i + j > 3). Laser phase noise appears analogously to (3). However, n τ ij (t) differs from n ϵ ij (t) because the reference interferometer is not affected by OB displacement or TM displacement noise. The measurement provides a reference about the relative phase noise of the laser sources involved and is required by TDI as additional information. Detailed information can be found in [6] (4b) Figure 2 shows a block diagram that illustrates the noise reduction pipeline for generating LISA's science signals, which will be detailed in the further course. References are intended to increase comprehensibility and facilitate the classification of subsequent text passages.

TDI second generation.
TDI second generation aims at eliminating laser phase noise p ij by linearly combining time-shifted interferometer measurements of the constellation. The suppression is exact for linear evolving arm lengths. Otherwise, residual laser noise remains, which is shown to have an uncritical magnitude in the TDI output [10]. The suppression is done step by step, applying the full-removal algorithm proposed in [6]. First, OB displacement noise, clock noise, and three of the six laser phase noise terms (p 13 , p 21 , p 32 ) are canceled by a dedicated algorithm (not presented here) executed prior to TDI second generation. The six output quantities (one per OB) are based on (1)-(4) and denoted by γ ij in the paper. They are, in addition to non-suppressed noise sources and the gravitational wave signal, affected by the remaining three laser phase noise terms p 12 , p 23 , and p 31 . Setting non-suppressed noise sources and the gravitational wave signal to zero, the γ ij variables scale with the different laser phase noises, as [6]: Note that the referenced literature denotes the variables termed γ ij in (5) as η ij . The change in name was made here to avoid confusion with the η ij degrees of freedom of the linear TTL model presented later. TDI eliminates the remaining three laser noise terms in the final step of the full removal algorithm. Note that there are several combination possibilities of γ ij , which lead to a suppression of the mentioned noise terms [11]. In (6) the TDI Michelson variable is given, which requires measurements from four of the six OBs of the constellation [12], i.e. OB12, OB21, OB13, and OB31.
The γ ij variables have to be delayed multiple times to achieve laser noise suppression in X. For readability, multiple delays are abbreviated by letters, the meaning of which can be found in table 1 of the appendix. Note that X is centered around SC1 synthesizing a virtual Michelson interferometer [13]. This can be understood from figure 3. The figure labels the measurements required for constructing TDI Michelson X, Y, and Z as well as their assignment to the individual LISA SC in terms of color. The TDI second-generation Michelson variables Y and Z are centered around SC2 and SC3, respectively. They are generated when cyclic permuting indices in (6).

TTL coupling noise
TTL coupling describes the impact of angular SC jitter and jitter of their optomechanical subassemblies on longitudinal interferometer measurements. A variety of causes produces TTL noise, including misalignment of optical components and wavefront errors of the transmitted laser beams. A detailed explanation of the physical background of this complex effect can be found in [14][15][16]. An illustrative example of geometrical TTL coupling is shown in figure 4.
Here, the combination of a laser beam offset d and an interferometer mirror tilt β results in a TTL-induced path length L. Assuming small angles and linearity, the one-way path length change is given by the beam offset and tilt product d · β. In reality, non-geometrical effects will also contribute to the overall TTL noise level. The authors of [17] provide a detailed description of the various TTL mechanisms. In this paper, TTL coupling noise of the long arm interferometer is considered. Two long arm TTL categories are distinguished per OB OBij, i.e. TTL noise related to the receiving laser beam (RX TTL) and TTL noise related to the transmitted laser beam (TX TTL). The path length changes due to RX and TX TTL are denoted by n RX ij (t) and n TX ij (t) in subsequent equations. TX TTL includes all sources of cross-coupling in the transmitting path of the laser beam, from TX laser injection on the local OB to far-field reception by the telescope on the remote SC. TX TTL, which is caused by the MOSA of OB OBij, i.e, n TX ij (t), is measured on the OB OBji. This is contrary to RX TTL. RX TTL includes all sources of cross-coupling in the path of the receiving laser beam, from the wavefront of the RX beam received by the entrance pupil of the telescope to the detector in the long arm interferometer. Therefore, RX TTL, which is caused by the MOSA of OBij, i.e. n RX ij (t), is also measured on the OB OBij. For both categories, the TTL contribution is modeled by the product of a TTL coupling factor and the corresponding rotation. The linear TTL coupling model is shown by (7) and (8) assuming small angles and linearity. A TTL reference frame Ω Cij , is defined for each laser beam of an SC, the origin of which is body-fixed in the center of the respective TM, see figure 1. The x Cij -axis aligns with the sensitive axis of the respective MOSA in reverse direction of the incoming laser beam. The z Cij -axis points perpendicular in the direction of the solar arrays, and y Cij completes the right-hand system. The angles of the TTL reference frame are denoted by {θ, η, ϕ}. TTL arises for rotations around y Cij and z Cij . It is worth noting that θ is of no interest as any rotations around x Cij -the axis defined by the incoming laser beam-cause negligible TTL noise. The rotational angles {η, ϕ } contain SC and MOSA jitter contributions. In [18], mathematical transformations between {η, ϕ} and SC and MOSA degrees of freedom are derived. The relations are of interest when designing optimal on-orbit excitation strategies for dedicated TTL calibration campaigns. They are not required when applying TDI-∞ for TTL noise estimation based on residual pointing jitter Equations (7) and (8) can be used to describe TTL noise in (1) and (2). The TTL contributions covered by n TTL,c ij (t) and n TTL,sb ij (t) are given by: TX TTL measured on the OB OBij and caused by OBji occurs time-delayed analogously to laser phase noise explained previously. The factor 2π/λ ji (t) converts the TTL contribution from meters to radians considering the wavelength λ ji (t) of the laser hosted by OB OBji. The phasemeter model of (1)-(4) as well as (7)-(9) will be required for estimating the coupling factors via TDI second generation and TDI-∞, in particular. TTL coupling noise calibration with TDI second generation is covered in [18][19][20][21][22], for example. It should be noted that TTL coupling in LISA's TM interferometers is not calibrated here. This is mainly because the arm lengths of these interferometers are relatively small and do not extend over millions of kilometers as they do in the long-arm measurements for the associated TX and RX TTL contributions. In this context, TTL coupling in the TM interferometers is treated as unsuppressed noise in the TDI output, which degrades the estimation performance of both TX and RX TTL coupling factors.

Extension of TDI infinity
The TDI-∞ concept is first introduced by [7]. TDI-∞ observables are not explicitly specified, as in second generation TDI Michelson, but via an implicit matrix-vector equation to be evaluated numerically. In addition to suppressing laser phase noise for any relative dynamics of the SC, the authors of [7] point out several other advantages of TDI-∞ over previous TDI generations that simplify data processing for LISA. For instance, there is no need to select a specific set of analytical TDI observables or model their power-spectral densities. Additionally, TDI-∞ uses simpler phase measurements, which enables direct computation of gravitational wave theoretical templates, facilitating comparison with observational data. Another advantage of TDI-∞ is its ability to handle measurement gaps automatically. This feature is handy when dealing with data from a multi-SC constellation such as LISA. TDI-∞ can handle the shift between different independent combinations when the number of available laser links changes, further simplifying the data processing. Finally, the relationship between TDI-∞ Figure 5. TDI-∞ Toy Model. The toy model, as proposed in [7], is used to convey the fundamental idea of the TDI-∞ concept. and the computation of matrix null-space bases opens up new possibilities for sophisticated algorithms, including parallelized or streaming variants suited to GPUs. These developments could lead to even more efficient data processing in the future. Section 4 will show that the frequency response of TDI-∞ differs from previous TDI generations holding a further benefit in the context of TTL factor estimation. Particularly, the different frequency response of TDI-∞ produces more accurate estimation results compared to TDI second generation considering equal inputs for both algorithms. This is crucial for estimating the TTL coupling factors of the LISA constellation and motivates the analysis at hand. Figure 5 depicts the toy model on the basis of which the TDI-∞ approach is derived in [7]. A laser source feeds two interferometer arms of different lengths corresponding to LISA's long arm interferometers. The round-trip light travel times are denoted by t 1.1 and t 2.2 . The measurements of the detectors are described by (10) and (11). The terms h 1 (t) and h 2 (t) cover the gravitational wave signal; n TTL 1 (t) and n TTL 2 (t) TTL noise, and n 1 (t) and n 2 (t) noise sources analogously to the corresponding expressions of (1)-(4)

Numerical formulation of TDI infinity observables
The authors of [7] convert (10) and (11) into a combined matrix-vector equation considering n distinct time steps: with: The design matrix M is to be filled so that (10) and (11) requiring TM = 0. The row vectors of the TDI matrix T ∈ R n×2n are in the null space of M T . Therefore, T is determined by the null space of M T . It is easy to see that o suppresses laser phase noise for the applied toy model when TM = 0 holds.

Extension of time delay interferometry infinity for the LISA constellation
Thanks to the structured derivation of TDI-∞ in [7], its application to the LISA constellation is straightforward. Analogous to TDI second generation, it is convenient to use the six γ ij variables as input for the approach. Thus, only three laser noise terms have to be suppressed by TDI-∞, namely p 12 , p 23 , and p 31 . The measurement vector s takes the following form: The laser noise vector p that considers p 12 , p 23 , and p 31 at all observation times is given by: Equation (12) can be used equally with the proposed extension to describe the impact of laser noise p, the gravitational wave signal, and other noise sources n on s of (20). The design matrix M ∈ R 6n×3n is to be filled so that the relations in (5) are met and the form of n TTL is derived in the next section. The TDI-∞ observable that considers the realistic LISA setup is given by (19) taking into account (20) and (22) this time. Equation (24) illustrates the shape of M for the simplified case where light travel times are integer multiples of the sample cadence ∆T. In this case, no interpolation between samples is necessary, and M only contains 0,−1, and +1 entries. The light travel times in the example of (24) are t 2.
In section 4 a spectral analysis is presented, proving that suppression of laser noise works with TDI-∞ under realistic LISA conditions. Note that the various TDI concepts can only suppress laser noise if it is measured on different detectors. Therefore, an initialization phase occurs for second-generation TDI at the beginning of restarting measurement campaigns, the length of which depends, for example, on the light travel times between the SC. The incomplete measurements can be excluded for TDI-∞ by truncating the design matrix M. Then, the TDI-∞ matrix T continues to identify laser noise-free combinations, as addressed in [23]. The results shown in section 4 confirm the research of [7] and the essentially equivalent formulation to TDI-∞ presented in [24] . . .

Calibration of TTL noise
The calibration of the disturbing TTL noise is done in two steps. First, the TTL coupling factors have to be estimated. This requires the TDI variables, the measured SC and MOSA attitudes as well as the arm lengths of the constellation, as will be seen later. Then the product of coupling factors and attitude measurements is formed and subtracted from the TDI variables.

TTL coupling noise in time delay interferometry infinity
To set up the TDI-∞ based TTL estimator, the occurrence of the coupling factors in the γ ij variables must first be known. For this reason, (7) and (8) need to be inserted into the phasemeter model of (1) and (2). Then, the first steps of the full-removal algorithm of [6] need to be applied again (not presented here). This results in (5) It can be concluded with the help of (7) and (8) that each γ ij variable is characterized by four different TTL coupling factors (two RX and two TX coupling factors). Hence, twenty-four coupling factors C RX/TX ij,η/ϕ need to be estimated for the entire LISA constellation. To describe the impact of TTL noise on o the TTL contribution n TTL of (12) is written in matrix-vector form: with c ∈ R 24×1 and A γ ∈ R 6n×24 . The vector c contains the twenty-four coupling factors C RX/TX ij,η/ϕ and A γ is filled such that (25) is met regarding the TTL contribution. Note that A γ contains η ij (t) and ϕ ij (t) measurements as well as their delayed versions simultaneously. This is different to the approach followed in (12), where M only contained weighting factors defined by the time-variant arm lengths of the constellation. Merging both, attitude measurements and their delayed versions into one matrix, i.e. A γ , allows to factor out the TTL coupling parameters. Thus, the estimation task reduces to the determination of c.
The impact of TTL noise on the TDI-∞ observable o, i.e. o TTL , follows, when inserting (26) into (19) setting laser phase noise, gravitational wave signals and other noise contributions to zero:

The TTL coupling estimators
The selection of an appropriate estimation method depends on the context in which the estimator is to be used. Important information on this can be found in [25][26][27][28]. The problem of TTL noise estimation can be assigned to the general case of determining a periodically constant state vector. Hence, a standard least-squares estimator is used to determine the coupling factors based on the TDI-∞ observable. The goal is to minimize the squared difference between o TTL derived in the previous section and o affected additionally by unsuppressed noise sources, e.g. TM displacement noise and interferometer readout errors, according to n in (12) min The solution of (28) is denoted c o and given by: The performance of the TDI-∞ based estimator is compared in the next section with an analog least squares estimator using the TDI Michelson variables X, Y, Z of (6). This estimator is derived in [18] and solves the following system by linear regression: with: A X,Y,Z ∈ R 3n×24 and y = ( containing the noisy TDI Michelson variables. The matrix A X,Y,Z is defined analogously to A γ mapping the occurrence of TTL in the three TDI Michelson variables X, Y, Z, instead of the occurrence in the six y ij variables. The solution of (30) is termed c X,Y,Z . It is given by: Note that, while considering colored noise for the simulation campaign of section 4, the whitening technique of input signals followed by the authors of [21] will not be applied. TTL estimation through linear regression will be performed without this additional data preprocessing step for both TDI concepts. To enhance the estimation accuracy of TTL factors, whitening and other filtering techniques can be employed. However, research on optimal TDI pre-processing of LISA's raw measurements is beyond the scope of this article.

Simulation results
In this section, the suppression of laser phase noise and the performance of the TDI-∞ based estimator are analyzed for the LISA constellation. Figure 6 shows the suppression performance of TDI-∞ o and second-generation TDI Michelson X concerning laser phase noise. Incomplete measurements, as discussed in [23], are excluded to avoid their effect on the spectral illustration of the figure. The result of this simulation and all subsequent simulations applies to the case of a non-static LISA constellation with unequal interferometer arms and SC orbits, as modeled in [29]. A nominal SC-to-SC distance of 2.5 million kilometers is assumed. Doppler shifts are neglected similarly to the simulator presented in [6]. The Amplitude Spectral Density (ASD) considered for the laser noise source terms p ij of (1) to (4) is shown by the magenta curve. Residual laser phase noise in o, i.e. o res. pij , is shown in blue. Analog for X in green. TDI-∞ suppresses laser phase noise by the same order of
Coupling factors are randomly selected within ±2 mm rad −1 . The values of the longarm TTL factors are in the order of a few millimeters, taking into account the telescope magnification [15]. The level of colored noise, i.e. n of (12), and the estimation window length are chosen such that the maximum estimation errors do not exceed 1% of the maximum coupling factor. For this reason, the relative comparison of the estimation performances between the TDI-∞ and TDI second generation based estimators is of central interest considering identical simulation conditions. It is natural to ask whether other scenarios with different noise assumptions affect the comparison results. While this is generally the case regarding the absolute performance metrics, the relative difference between the TDI concepts' estimation results remains, especially for the most realistic noise assumptions currently implemented in the Airbus internal LISA simulator. Figure 7 compares the estimation performance resulting from (29) and (31) by plotting the differences between the estimated TTL coupling vectors and the truth c true . To obtain a statistically significant estimation result according to the European Cooperation for Space Standardization (ECSS) standard, 2000 individual simulation runs were performed with varying noise seeds. The estimation via TDI-∞ shows a better performance for the test case considering that the 99.73% value of the absolute estimation errors is about 20% smaller compared to TDI second generation. During the authors' analyses, it is observed that the individual TTL factors are not equally well estimated. This is partly due to the correlated MOSA jitter. Besides, the TTL transfer functions that pertain to the η-and ϕ-related TTL noise contribution in X exhibit differences due to the geometric configuration of the MOSA with a nominal inner angle of 60 • as depicted in figure 1. The transfer function is not presented here. Consequently, the contribution of ϕ-related TTL factors on X (and also o) is higher than that of η-related TTL factors. Hence, a higher error variance is obtained for η-related TTL factor estimation. This observation is not illustrated here. However, it is important to note that the required estimation window length is determined by those coupling factors that exhibit the lowest observability in the corresponding TDI observable. Further analyses are needed to verify if the statement holds when Doppler shifts are considered. This will be addressed in the future. Figure 8 explains why TTL noise estimation with TDI-∞ yields a better performance. The figure shows the transfer function of o and X for the isolated propagation of TTL, i.e. o TTL and X TTL , and noise sources that deteriorate the TTL estimation accuracy, i.e. o n and X n . In this example, white noise is assumed for n. Transfer functions can be calculated analytically for X by substituting the signal of interest into (6) and Fourier transforming the expression. This is done once for TTL noise in figure 8 with the assumptions described (dark green curve; derivation is not presented). The other curves represent LSD representations of time simulations. The characteristic high-pass transfer behavior of X has been explained in [18]. It differs fundamentally from TDI-∞ as can be seen from the blue curves. Due to the implicit formulation of o, an analytical transfer function cannot be easily given. In general, similar estimation behavior is expected because the ratio of TTL noise to other noise sources in o and the analog ratio in X are comparable. However, let us consider the flat response behavior of TDI-∞ and the inexistence of zeros in the TDI-∞ transfer function, which are characteristic for TDI second generation. Signal components at zero locations of second-generation TDI are available to the TTL estimator when applying the TDI-∞ concept. Hence, a slightly better estimation performance can be expected.
Future work will address the accuracy of TTL factor estimation with Michelson-like TDI-∞ variables; variables based on (19) that do not combine all γ ij variables simultaneously, but only those applied per Michelson variable of (6). Figure 9 shows a comparison of the computational effort for the two estimators. Note that it is not the evaluation of (29) and (31), which is time-consuming, but evaluating the matrices A γ and A X,Y,Z . The corresponding computing times are denoted by t A γ and t A X,Y,Z . For the chosen implementation, where each delayed signal is evaluated once per timestep, the calculation of A γ takes 60% less time on average. This is because A γ does not have to consider the complex structure of TTL in TDI second generation, as it is the case with A X,Y,Z , but only the structure of TTL in the simple γ ij variables of (25).
If the reader is interested in the absolute computing time, it is worth noting that a data stream of 600 s is generated per simulation run for figure 7. Due to the interpolation kernel order and the TDI initialization phase, which is described in [6], approximately 100 s must be removed, leaving 500 s of valid data for the estimation itself. The estimation with (31) requires approximately 500 s to generate an estimate based on 500 s of 5 Hz interferometric measurements, while the estimation with (29) requires an average of 200 s. Based on this, linear extrapolations can be performed to first approximate the computational effort regarding longer estimation windows. In general, it needs to be clarified with the scientific community if the computation time has practical relevance, considering that these calculations will be performed on-ground. The reduced computational effort would undoubtedly accelerate the availability of calibrated TDI combinations for astrophysical purposes.

Conclusion
The study presented the application of TDI-∞ for TTL noise estimation in the context of the proposed LISA mission. Therefore, the algorithm was extended to account for LISA's complex measurement setup, and a TDI-∞ based estimator was presented. Our analysis shows that the TDI-∞ possesses a frequency response significantly different from the TDI second generation. Due to the absence of zeros in the TDI-∞ transfer function, access to frequency intervals is provided, which are unavailable for TDI second generation during TTL noise estimation. This indicates the advantage of the TDI-∞ algorithm from the perspective of TTL calibration. The study compared the performance of the TDI-∞ based TTL estimator with an estimator that utilizes TDI second generation variables. The results demonstrate that the estimation error of the TDI-∞ based estimator is 20% lower in the chosen simulation scenario, with a 60% reduction in computation time. The findings suggest TDI-∞ be a promising methodology for LISA science data processing research with the potential of further improving the precision of space-borne gravitational wave detector measurements. It is worth noting that the paper also analyzed the suppression of dominant laser phase noise for a non-static constellation and realistic LISA orbits and found that the TDI-∞ approach shows no decrease in performance compared to TDI second generation. Future studies need to assess the impact of Doppler shifts.

Data availability statement
The data cannot be made publicly available upon publication because they contain commercially sensitive information. The data that support the findings of this study are available upon reasonable request from the authors.