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Schematic representation of the covariance matrix, C , for the simple LISA model. Blocks of increasing grey rep- resent values of 2 σ p 2 + σ n 2 , σ p 2 , 0 and − σ p 2 .
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Data from the Laser Interferometer Space Antenna (LISA) is expected to be dominated by frequency noise from its lasers. However, the noise from any one laser appears more than once in the data and there are combinations of the data that are insensitive to this noise. These combinations, called time delay interferometry (TDI) variables, have receive...
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... is shown schematically in Fig. 3. It is now an exercise in linear algebra to compute the eigenvectors and eigenvalues of C. We use MAPLE [10] to do this calculation for us. There are 22 distinct eigen- values, the smallest of which is independent of the laser frequency noise. This minimal ...
Citations
... The first-generation TDI combinations can cancel out the laser frequency noise in a static unequal-arm configuration, and the second-generation TDI further cancels the frequency noise in time-dependent arm interferometry. The first-generation TDI configurations were applied for space-based GW detectors such as LISA, Taiji and TianQin in [19,[22][23][24][25][26][27][28][29][30][31][32][33], and the applications of second-generation TDI configurations were discussed in [20,21,[34][35][36][37][38][39][40]. For more discussion on TDI algorithm and its application to space-based GW detectors, please see [41][42][43] and references therein. ...
The accurate sky localization of gravitational wave (GW) sources is an important scientific goal for space-based GW detectors. Due to the effects of gravity on three spacecrafts, it is hard to maintain the equality of the arm length, so the time-delay interferometry (TDI) method is needed to cancel out the laser frequency noise for space-based GW detectors. By considering the first-generation TDI combination, we employ the Fisher information matrix to study the accuracy of sky localizations for future space-based GW detectors and their combined network. The main difference between future space-based GW detectors includes the time-changing orientation of the detector plane, the arm length, the orbital period of spacecrafts and the noise curve. We study the effects of these factors on the accuracy of source localization at different frequencies. We find that the amplitude modulation caused by the rotation of the detector plane can help LISA and Taiji not only to improve the accuracy of source localization but also to enlarge the sky coverage at frequencies below 1 mHz. As the frequency of monochromatic GWs increases, the Doppler modulation becomes dominate and the equatorial pattern appears in the sky map. The effect of arm length on the angular resolution mainly comes from the noise curve and it is almost the same for both heliocentric and geocentric constellations. The orbital period of the spacecrafts has little effect on the angular resolutions. The improvement on the angular resolutions by the network of combined detectors is small compared with a single detector and the angular resolutions are almost the same with and without the TDI combination.
... We do so by also using only analytic techniques. Recently matrix methods have also been employed, which lead to TDI observables albeit numerically [12,[17][18][19]. Although the TDI combinations we will derive in this article can be re-casted in matrix form, we will not do that here. ...
Time-Delay Interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the heterodyne Doppler measurements made by unequal-arm space-based gravitational wave interferometers. The space of all TDI combinations was first derived under the simplifying assumption of a stationary array, for which the three time-delay operators commute. In this model, any element of the TDI space can be written as a linear combination of four TDI variables, the generators of the "first-generation" TDI space. To adequately suppress the laser phase fluctuations in a realistic array configuration, the rotation of the array and the time-dependence of the six inter-spacecraft light-travel-times have to be accounted for. In the case of the Laser Interferometer Space Antenna (LISA), a joint ESA-NASA mission characterized by slowly time varying arm-lengths, it has been possible to identify data combinations that, to first order in the inter-spacecraft velocities, either exactly cancel or suppress the laser phase fluctuations below the level identified by the noise sources intrinsic to the heterodyne measurements (the so called "secondary" noises). Here we reanalyze the problem of exactly canceling the residual laser noise terms linear in the inter-spacecraft velocities. We find that the procedure for obtaining elements of the $2^{\rm nd}$-generation TDI space can be generalized in an iterative way. This allows us to "lift-up" the generators of the $1^{\rm st}$-generation TDI space and construct elements of the higher order TDI space.
... A new approach to the laser frequency noise problem has gained interest in the last two years, which formulates how the noise enters phase measurements with a design matrix, and interprets TDI as the solution to a linear algebra problem. Romano and Woan [11] took a first step in that direction (further explored in [12]) by showing * quentin.baghi@cea.fr that we can derive TDI variables from the eigenvectors of the laser noise covariance matrix, using a simple toy model in the time domain. More recently, we formalized this idea in the frequency domain, an approach that we named principal component interferometry (PCI), for which we provided first evidences for its suitability to parameter inference [13]. ...
... (17) to compute the q×q covarianceΣ en (f ) of the q lowest-variance aPCI variables. We assume that all single-link noises are uncorrelated and have the same PSD, set by Eq. (11). Hence, their covarianceΣ y (f ) is diagonal. ...
... In this section we write down the analytical expressions for the acceleration and OMS noises PSDs forming the secondary noises in Eq. (11). The acceleration noise PSD is ...
We recently introduced the basic concepts of an approach to filtering strongly laser-noise dominated space-based gravitational-wave data, like LISA's phase comparison data streams, which does not rely on independent knowledge of a temporal delays pattern in the dominant noise that generates the data. Instead, our automated Principal Component Interferometry (aPCI) approach only assumes that one can produce some linear combinations of the temporally nearby regularly spaced phase measurements, which cancel the laser noise. Then we let the data reveal those combinations, thus providing us with a set of laser-noise-free data channels. Our basic approach relied on the simplifying additional assumption that laser-noise-cancelling data combinations or the filters which lead to the laser-noise-free data streams are time-independent. In LISA, however, these filters will vary as the constellation armlengths evolve. Here, we discuss a generalization of the basic aPCI concept compatible with data dominated by a still unmodeled but slowly varying dominant noise covariance. We find that despite its independence on any model, the aPCI processing successfully mitigates laser frequency noise below the other noise sources level, and that its sensitivity to gravitational waves is the same as the state-of-the-art second-generation time-delay interferometry, up to a 2% error.
... Romano and Woan first came up with the idea of using matrices for TDI by employing the method of principal component analysis [16]. This idea was further investigated by Leighton in a Ph.D. thesis [17]. ...
Time-delay interferometry (TDI) is a data processing technique that cancels the large laser phase fluctuations affecting the one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. By taking finite linear combinations of properly time-shifted Doppler measurements, laser phase fluctuations can be removed at any time t and gravitational wave signals can be studied at the requisite level of sensitivity. In the past, other approaches to this problem have been proposed. Recently, matrix-based approaches have been put forward; two such approaches are by Vallisneri et al. and Tinto et al. In this paper, we establish a close relationship between these approaches. In fact, we show that the matrices involved in defining the operators in the two approaches exhibit an isomorphism, and therefore, in both approaches one is dealing with matrix representations of the time-delay operators.
... In fairly recent years a novel approach was adopted by Vallisneri et al. [15] in which the problem has been formulated in terms of matrices (see also [16] where principal component analysis has been employed). In this approach, the data are discretized and a design matrix representing the delays is defined. ...
Time-Delay Interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. By taking finite linear combinations of properly time-shifted Doppler measurements, laser phase fluctuations are removed at any time t and gravitational wave signals can be studied at a requisite level of sensitivity. In the past, other approaches to this problem have been proposed. Recently, matrix based approaches have been put forward; two such approaches are by Vallisneri et al. and Tinto, Dhurandhar and Joshi. In this paper we establish a close relationship between these approaches. In fact we show that the matrices involved in defining the operators in the two approaches exhibit an isomorphism and therefore in both approaches one is dealing with matrix representations of the time-delay operators.
... An alternative and a totally different approach has been adopted in (Romano and Woan 2006) from the point of view of Bayesian statistical inference. The covariance matrix of the six elementary data streams sampled at integer multiples of the timedelay is first formed. ...
... Both are numerically based and can in principle identify TDI combinations capable of canceling the laser noises in the presence of arbitrarily large inter-spacecraft velocities. The one proposed in Vallisneri et al. (2020) in particular follows from a Bayesian formulation of TDI (Romano and Woan 2006) in which the laser noises are characterized by having variances much larger than those associated with all other noises. The method relies on using the entire temporal duration of the one-way Doppler measurements and by relating them to the laser noises through an identified linear relationship. ...
Equal-arm detectors of gravitational radiation allow phase measurements many orders of magnitude below the intrinsic phase stability of the laser injecting light into their arms. This is because the noise in the laser light is common to both arms, experiencing exactly the same delay, and thus cancels when it is differenced at the photo detector. In this situation, much lower level secondary noises then set the overall performance. If, however, the two arms have different lengths (as will necessarily be the case with space-borne interferometers), the laser noise experiences different delays in the two arms and will hence not directly cancel at the photo detector. To solve this problem, a technique involving heterodyne interferometry with unequal arm lengths and independent phase-difference readouts has been proposed. It relies on properly time-shifting and linearly combining independent Doppler measurements, and for this reason it has been called time-delay interferometry (TDI). This article provides an overview of the theory, mathematical foundations, and experimental aspects associated with the implementation of TDI. Although emphasis on the application of TDI to the Laser Interferometer Space Antenna mission appears throughout this article, TDI can be incorporated into the design of any future space-based mission aiming to search for gravitational waves via interferometric measurements. We have purposely left out all theoretical aspects that data analysts will need to account for when analyzing the TDI data combinations.
... Romano and Woan [13] showed that we could think of TDI combinations as low-variance principal components of the laser noise covariance matrix. This idea relied on a matrix representation of the interferometric measurements and was also explored in Ref. [14]. ...
... As a result, decomposing the data with PCA and keeping only the lowest variance components is equivalent to restricting the likelihood to its most significant terms (i.e., the terms that are the most relevant for GW parameters inference). The data-driven PCA process outlined here is similar to the TDI analysis, which has already been shown to be equivalent to an approximation of the likelihood [13,15,16]. Depending on the choice of q, Eq. (18) can be considered as a family of approximations for Eq. ...
With a laser interferometric gravitational-wave detector in separate free flying spacecraft, the only way to achieve detection is to mitigate the dominant noise arising from the frequency fluctuations of the lasers via postprocessing. The noise can be effectively filtered out on the ground through a specific technique called time-delay interferometry (TDI), which relies on the measurements of time-delays between spacecraft and careful modeling of how laser noise enters the interferometric data. Recently, this technique has been recast into a matrix-based formalism by several authors, offering a different perspective on TDI, particularly by relating it to principal component analysis (PCA). In this work, we demonstrate that we can cancel laser frequency noise by directly applying PCA to a set of shifted data samples, without any prior knowledge of the relationship between single-link measurements and noise, nor time-delays. We show that this fully data-driven algorithm achieves a gravitational-wave sensitivity similar to classic TDI.
... We would like to briefly mention here another matrix based approach. Romano and Woan [8] have used Bayesian inference to set up a noise covariance matrix of the data streams. Then by performing a principal component analysis of the covariance matrix, they identify the principal components with large eigenvalues with the laser noise and so distinguish it from other ambient noises and signal which correspond to small eigenvalues. ...
... A Bayesian inference approach has been adopted by Romano and Woan [8]. They set up a noise covariance matrix of the data streams y j , y j and perform a principal component analysis. ...
Time-delay interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. By taking finite linear combinations of properly time-shifted Doppler measurements, laser phase fluctuations are removed at any time t and gravitational wave signals can be studied at a requisite level of precision. In this article we show the delay operators used in TDI can be represented as matrices acting on arrays associated with the laser noises and Doppler measurements. The matrix formulation is nothing but the group theoretic representation (ring homomorphism) of the earlier approach involving time-delay operators and so in principle is the same. It is shown that the homomorphism is valid generally and we cover all situations of interest. To understand the potential advantages the matrix representation brings, care must be taken by the data analyst to account for the light travel times when linearly relating the one-way Doppler measurements to the laser noises. This is especially important in view of the future gravitational wave projects envisaged. We show that the matrix formulation of TDI results in the cancellation of the laser noises at an arbitrary time t by only linearly combining a finite number of samples of the one-way Doppler data measured at and around time t.
... We would like to briefly mention here another matrix based approach. Romano and Woan [8] have used Bayesian inference to set up a noise covariance matrix of the data streams. Then by performing a principal component analysis of the covariance matrix, they identify the principal components with large eigenvalues with the laser noise and so distinguish it from other ambient noises and signal which correspond to small eigenvalues. ...
... A Bayesian inference approach has been adopted by Romano and Woan [8]. They set up a noise covariance matrix of the data streams y j , y j and perform a principal component analysis. ...
Time-Delay Interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. By taking finite linear combinations of properly time-shifted Doppler measurements, laser phase fluctuations are removed at any time $t$ and gravitational wave signals can be studied at a requisite level of precision. In this article we show the delay operators used in TDI can be represented as matrices acting on arrays associated with the laser noises and Doppler measurements. We show here that the matrix formulation is nothing but the group theoretic representation of the earlier approach involving time-delay operators and so in principle is the same. To understand the potential advantages the matrix representation brings, care must be taken by the data analyst to account for the light travel times when linearly relating the one-way Doppler measurements to the laser noises. This is especially important in view of the future gravitational wave projects envisaged. We show that the matrix formulation of TDI results in the cancellation of the laser noises at an arbitrary time $t$ by only linearly combining a finite number of samples of the one-way Doppler data measured at and around time $t$.
... It can also be formalized algebraically in terms of polynomial syzygies [17]. Last, it can be recast as an application of principal component analysis [18,19]-a formalism closely related to ours, and discussed further below. ...
... In Ref. [18], Romano and Woan identify the TDI observables with the small-eigenvalue eigenvectors of the y covariance matrix (N + MCM † in our notation), and emphasize that its singular value decomposition factorizes the y likelihood into a TDI term (a sufficient statistic for astrophysical inference), and a laser-dominated term (useful for laser-noise monitoring but not GW detection). They also recover the classical TDI expressions by analyzing the covariance matrix for integer-∆t laser delays. ...
The space-based gravitational-wave observatory LISA relies on a form of synthetic interferometry (time-delay interferometry, or TDI) where the otherwise overwhelming laser phase noise is canceled by linear combinations of appropriately delayed phase measurements. These observables grow in length and complexity as the realistic features of the LISA orbits are taken into account. In this paper we outline an implicit formulation of TDI where we write the LISA likelihood directly in terms of the basic phase measurements, and we marginalize over the laser phase noises in the limit of infinite laser-noise variance. Equivalently, we rely on TDI observables that are defined numerically (rather than algebraically) from a discrete-filter representation of the laser propagation delays. Our method generalizes to any time dependence of the armlengths; it simplifies the modeling of gravitational-wave signals; and it allows a straightforward treatment of data gaps and missing measurements.
