In this section, a LS TOA estimator is derived based on which is the presumed maximum delay spread of the one-way channel. Generally, is not equal to its true value . As shown in Section 5, the discrepancy between and does affect the estimation accuracy and the optimum value of may be determined via simulation.

The received wave form expressed in (10) is sampled at frequency with corresponding period , where is the total number of samples per frame. is observed over time duration and is sampled at timings with , , and . Here, is the number of observed symbols and is assumed to be integer multiples of the ranging sequence's period , that is, and is an integer. The reason for considering multiple symbols for TOA estimation is to provide processing gain to suppress noise. The channels for both the tag response and clutter are assumed to be static during the sampling duration. The sample vector in the *n* th frame of the *m* th symbol interval is denoted as . Let and . Let and be the sample vectors of and . Define two column vectors and , where represents a column vector with all elements being zeros. Both and contain elements each. may be modeled as

where and are column vectors containing samples of clutter and noise, respectively, is the sample vector of tag response and is given by

Here, and are data bits modulating the partial tag responses and , respectively.

To proceed, let be the candidate values of parameters among which is the parameter of interest whereas the rest are nuisance parameters. are chosen to minimize the following nonlinear LS function:

where the norm operator computes the Euclidean distance of vector . The search for global minimum of a general multidimensional nonlinear function usually involves numerical searching using the genetic algorithms, grid searchers or other computational intensive algorithms. Fortunately, the nonlinear cost function (13) has some special properties that allow us to first reduce the variables set to be optimized from to , which drastically reduces the computational complexity. This reduction procedure can lead to the global minimum of the cost function. The simplified cost function with the reduced variable set is then minimized by searching over all possible discrete values of the variables to reach the global minimum. The details of minimization procedure are given as follows.

The cost function has two special properties. The first property is that it is a general convex quadratic function of . By substituting (12) into (13), it can be readily shown that the nonlinear function can be transformed into the following general convex quadratic form of [34, 35]:

where .

The second property is that the cost function is twice continuously differentiable with respect to . This can be readily shown by differentiating (14) twice with respect to , which yields

The treatments for such special unconstrained convex optimization problem can be found in many books such as [34]. For such special convex function of , the partial differentiation equation with respect to the variable leads to the global minimum of the function. For this reason, we can substitute the solution of partial differential equation into (13) to eliminate . Solving for yields the intermediate expression

where is an vector containing the samples averaged over frames, , , and are defined as

As shown in Appendix A

where is a constant and is irrelevant to . Using (18), Equation (16) is rewritten as

Before proceeding to the next step, let us first define a few terms that will be used in the later discussions

It is shown in Appendix A that can be developed as

where is a constant irrelevant to and .

With equations (19)–(22), by substituting into (13) and dropping the factor which is irrelevant to the decision-making process, we reach

where , , , and and are vectors with their respective *k* th elements being , , and , for all . Note that , and only depend on the observed samples, the ranging and the PN codes.

Similar to the original cost function, the intermediate function expressed in (23) also fulfills the two special properties. First, Equation (23) may be transformed into convex quadratic forms of and .

where , , , , . Since , we have . Therefore, and . Second, it can be readily shown that (24) is twice differentiable about and , respectively. According to the previous discussions, we may conclude that the solutions of the two partial differentiation equations and lead to the global minimum of . Solving the two differentiation equations for and gives

Solving (25) gives the following intermediate expressions for tag response:

With (26), substituting and into (23) and simplifying, we have the following expression for depending only on :

Note that (27) is invalid when . To cover this exceptional situation, the following analysis is carried out. The equation can be factorized as

which leads to

In Appendix B, we show that at least one of (29) and (30) holds when the conditions , for all or , for all is met. Next, we will give some intuition for these two conditions. The condition , for all is met when every bit in the sequence has the same polarity, that is, , for all or , for all . In this case the tag response will retain polarity over different frames, appearing as "unmodulated" signal like clutter. Therefore, there is no way the tag response can be distinguished from the clutter, which is undesired. In the following discussion, we assume that such undesired sequence is deliberately discarded in the system design so that applies. The second condition , for all is fulfilled when consists of alternative and , that is, it is a sequence of. With such sequence, we have and . Together with (23), it is straightforward to show that

Solving (25) gives

With (32), substituting into (31) yields

According to the above discussion, we can conclude our final estimator: for leading to , the decision function is directly computed based on (33); else is computed using (27). Subsequently, the candidate values minimizing the value of the decision function is adopted as final estimates

where are the final estimates of . And the TOA is estimated as

Equation (34) indicates that the final solution involves a minimum search procedure over a three-dimensional space span by variables . The complexity of this searching procedure is proportional to the number of possible discrete values of , , and , that is, proportional to the maximum number of samples prior to the TOA sample , the number of symbols , and the number of frames per symbol . Reducing or can lower the computational complexity. As a tradeoff, the TOA estimation accuracy will decrease accordingly. However, the simulation results in Section 5 will reveal that the TOA performance is very robust to the reduction in and is reduced by only about 3 dB for halving . Reducing also reduces the computational complexity which causes insignificant variation of TOA estimation as shown in the simulation results in Section 5. Therefore, our scheme does not require large and should be minimized for TOA estimation during system design phase. This minimum value of should be determined by other aspects of system design such as spectrum smoothing or the number of users in the system, which is out of the scope of this paper. Hence, by carefully setting , and , satisfactory performance can be achieved with reasonable complexity.