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.