List of Figures

	Figure 1: Light from a laser is split into two beams, each injected into an arm formed by pairs of free-falling mirrors. Since the length of the two arms, $L1$ and $L2$ , are different, now the light beams from the two arms are not recombined at one photo detector. Instead each is separately made to interfere with the light that is injected into the arms. Two distinct photo detectors are now used, and phase (or frequency) fluctuations are then monitored and recorded there.
	Figure 2: Schematic diagram for $X$ , showing that it is a synthesized zero-area Sagnac interferometer. The optical path begins at an “x” and the measurement is made at an “o”.
	Figure 3: Schematic LISA configuration. The spacecraft are labeled 1, 2, and 3. The optical paths are denoted by $Li$ , $L ′i$ where the index $i$ corresponds to the opposite spacecraft. The unit vectors $ˆni$ point between pairs of spacecraft, with the orientation indicated.
	Figure 4: Schematic diagram of proof-masses-plus-optical-benches for a LISA spacecraft. The left-hand bench reads out the phase signals $s 1$ and $τ 1$ . The right-hand bench analogously reads out $′ s1$ and $′ τ1$ . The random displacements of the two proof masses and two optical benches are indicated (lower case $⃗δi,⃗δ′i$ for the proof masses, upper case $⃗Δi,Δ ′i$ for the optical benches).
	Figure 5: Schematic diagram of the unequal-arm Michelson interferometer. The beam shown corresponds to the term $2 2 (𝒟 2 − 1)(𝒟1 − 1)ϕ(t)$ in $X (t)$ which is first sent around arm 1 followed by arm 2. The second beam (not shown) is first sent around arm 2 and then through arm 1. The difference in these two beams constitutes $X (t)$ .
	Figure 6: Schematic diagrams of the unequal-arm Michelson, Monitor, Beacon, and Relay combinations. These TDI combinations rely only on four of the six one-way Doppler measurements, as illustrated here.
	Figure 7: The LISA Michelson sensitivity curve (SNR = 5) and the sensitivity curve for the optimal combination of the data, both as a function of Fourier frequency. The integration time is equal to one year, and LISA is assumed to have a nominal armlength $L$ = 16.67 s.
	Figure 8: The optimal SNR divided by the SNR of a single Michelson interferometer, as a function of the Fourier frequency $f$ . The sensitivity gain in the low-frequency band is equal to $√ -- 2$ , while it can get larger than 2 at selected frequencies in the high-frequency region of the accessible band. The integration time has been assumed to be one year, and the proof mass and optical path noise spectra are the nominal ones. See the main body of the paper for a quantitative discussion of this point.
	Figure 9: The SNRs of the three combinations $(A, E, T)$ and their sum as a function of the Fourier frequency $f$ . The SNRs of $A$ and $E$ are equal over the entire frequency band. The SNR of $T$ is significantly smaller than the other two in the low part of the frequency band, while is comparable to (and at times larger than) the SNR of the other two in the high-frequency region. See text for a complete discussion.
	Figure 10: Apparent position of the source in the sky as seen from LISA frame for $∘ ∘ (𝜃B = 90 ,ϕB = 0 )$ . The track of the source for a period of one year is shown on the unit sphere in the LISA reference frame.
	Figure 11: Sensitivity curves for the observables: Michelson, $max [X, Y,Z ]$ , $⃗v +$ , and network for the source direction ( $∘ 𝜃B = 90$ , $∘ ϕB = 0$ ).
	Figure 12: Ratios of the sensitivities of the observables network, $⃗v+,×$ with $max [X, Y,Z ]$ for the source direction $𝜃B = 90∘$ , $ϕB = 0∘$ .

	Abstract
1	Introduction
2	Physical and Historical Motivations of TDI
3	Time-Delay Interferometry
4	Algebraic Approach for Canceling Laser and Optical Bench Noises
	4.1	Cancellation of laser phase noise
	4.2	Cancellation of laser phase noise in the unequal-arm interferometer
	4.3	The module of syzygies
	4.4	Gröbner basis
	4.5	Generating set for the module of syzygies
	4.6	Canceling optical bench motion noise
	4.7	Physical interpretation of the TDI combinations
5	Time-Delay Interferometry with Moving Spacecraft
	5.1	The unequal-arm Michelson
	5.2	The Sagnac combinations
	5.3	Algebraic approach to second-generation TDI
	5.4	Solutions with one arm nonfunctional
6	Optimal LISA Sensitivity
	6.1	General application
	6.2	Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane
7	Experimental Aspects of TDI
	7.1	Time-delays accuracies
	7.2	Clocks synchronization
	7.3	Clocks timing jitter
	7.4	Sampling reconstruction algorithm
	7.5	Data digitization and bit-accuracy requirement
8	Concluding Remarks
	Acknowledgement
A	Generators of the Module of Syzygies
B	Conversion between Generating Sets
	References
	Footnotes
	Updates
	Figures