Derivation Notes of Application of Ring Theory in TDI algorithm

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From Wei-Liang Qian
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The Time Delay Interferometry (TDI) laser-noise cancelation scheme involves the use of linear combinations of existing measurements to cancel out unknown noise terms. Since both the measurements and unknown noises can be expressed in terms of polynomials, the mathematics is readily chosen to be the (computational) algebraic geometry. This is because polynomials furnish an Abelian ring. To the best of our knowledge, the present status of the application is rather primitive.

The discussions follow largely the those presented in the review paper by Tinto and Dhurandhar. First, the measurements which contain the noise terms are called data streams. A simple but typical example is given as follows

<math>s_1=\mathcal{D}_3 p_2-p_1</math>
<math>s_{1'}=\mathcal{D}_2 p_3-p_1</math>,

together with all possible permutations <math>1\to 2\to 3\to 1</math>. In a slightly more general form, we have

<math>s_{i}=\mathcal{D}_{i-1} p_{i+1}-p_i</math>
<math>s_{i'}=\mathcal{D}_{i+1} p_{i-1}-p_i</math>.

where all the quantities are polynomials. Physically, <math>s_i</math> are defined as data streams, which correspond to the superposition of various phase signals <math>p_i</math> emitted at spacecraft <math>i</math>, inevitably affected by the time delays during their transmission, the latter are operations denoted by <math>\mathcal{D}_i</math>, with respect to the pathway associated with the arm <math>(i-1)\to(i+1)</math>.

One looks for the linear combination of <math>s_i</math> and <math>s_{i'}</math> that vanishes, namely,

<math>\sum_{i=1}^3 q_i s_{i}+q_{i'}s_{i'}=0</math>.

Since we are intended to cancel out any unknown but given form of noise embedded in <math>p_i</math>, the coefficients of all such terms must vanish. This gives rise to three equations. By some simple algebra, one may Gaussian eliminate two out of six coefficients of <math>q_i</math> and result in the following equation

<math>(1-\mathcal{D}_{1}\mathcal{D}_{2}\mathcal{D}_{3})q_3+(\mathcal{D}_{1}\mathcal{D}_{3})q_{1'}+\mathcal{D}_{1}(1-\mathcal{D}_{3}^2)q_{2'}+(1-\mathcal{D}_{1}^2)q_{3'}=0</math>.

The general solution of the above equation can be readily transformed into a mathematical problem using the module of syzygies. Although it might be rather straightforward for someone with an appropriate mathematical background, we briefly discuss the connection. First, apparently, the l.h.s. of the above equation furnishes a homomorphism, a map <math>\mathcal{R}^4\to\mathcal{R}</math>, from the four polynomials

<math>(q_3, q_{1'}, q_{2'}, q_{3'})</math>

to one. In fact, it defines an ideal in terms of four generators, namely,

<math>\mathcal{F}=\{f_1, f_2, f_3, f_4\}=\{(1-\mathcal{D}_{1}\mathcal{D}_{2}\mathcal{D}_{3}), (\mathcal{D}_{1}\mathcal{D}_{3}), \mathcal{D}_{1}(1-\mathcal{D}_{3}^2), (1-\mathcal{D}_{1}^2)\}</math>.

The solution of the above equation corresponds to the kernel of the homomorphism, known as the first module of syzygies. Physically, it exhausts all possible combinations that the laser noise will be entirely eliminated.

Dhurandhar et al. was the first who employed computational algebraic geometry to solve the above physical problem. The generator of the kernel can be encountered using the Groeber basis. In accordance with Becker and Weispfenning, we briefly reiterate what was described in the original paper and the review. First, one derives the generating set of the Groebner basis of the ideal. In practice, this can be readily done by using the Mathematica command

GroebnerBasis[].

For the present example, the Groebner basis <math>\mathcal{G}</math> is found to be

<math>\mathcal{G}=\{g_1, g_2, g_3\}=\{\mathcal{D}_{3}^2-1,\mathcal{D}_{2}^2-1,\mathcal{D}_{1}-\mathcal{D}_{2}\mathcal{D}_{3}\}</math>.

Then one proceeds to evaluate the generating set of the first module of syzygies, by manipulating the Groebner basis. It is straightforward to show that the transformation matrices <math>C</math> and <math>D</math> between <math>\mathcal{F}</math> and <math>\mathcal{G}</math>, namely

<math>f_i=d_{ij}g_j</math>,
<math>g_j=c_{ji}f_i</math>,

where <math>d_{ij}</math> and <math>c_{ji}</math> are the elements of the transformation matrices with <math>i=1, 2, 3, 4</math> and <math>j=1, 2, 3</math>. To be specific, they readily furnish four generators by each row of the four-by-four matrix

<math>A= I - D C</math>.

This is because

<math>(\delta_{ik}-d_{ij}c_{jk})f_k=0</math>.

Moreover, there are another three generators by computing the S-polynomials of the Groebner basis <math>\mathcal{G}</math>. Between the three elements, they give rise to three S-polynomials, denoted by the three rows of a matrix <math>B^*</math>. Since they are linear combinations of the elements of the Groebner basis, it is not difficult to show that they also furnish the generators of the module. It is noted that the above seven generators are not linearly independent, further refinement results in four generators, different options are usually denoted in the literature by either

<math>X^{(1)}, X^{(2)}, X^{(3)}, X^{(4)}</math>

or

<math>\alpha, \beta, \gamma, \zeta</math>.

Now, the first one of our questions is, once the above generators are obtained, what is the general procedure to decompose any element of the first module of syzygies in terms of these generators? One might try to do the long division, but what is the appropriate order (since, besides the apparent definition, we did not follow the mathematical details which lead to the Groebner basis.)

<math></math>