Abstract
Time of arrival (TOA) estimation in multipath dense environment for UWB backscattering radio frequency identification (RFID) system is challenging due to the presence of strong clutter. In addition, the backscattering RFID system has peculiar signal transmission and modulation characteristics, which are considerably different from conventional communication and localization systems. The existing TOA estimators proposed for conventional UWB systems are inappropriate for the backscattering RFID system since they lack the required clutter suppression capability and do not account for the peculiar characteristics of backscattering system. In this paper, we derive a nondata-aided (NDA) least square (LS) TOA estimator for UWB backscattering RFID system. We show that the proposed estimator is inherently immune to clutter and is robust in under-sampling operation. The effects of various parameter settings on the TOA estimation accuracy are also studied via simulations.
or 
. The load condition of the tag antenna is determined by the 2-PAM modulator which is controlled by the sequence generator. The 2-PAM modulator retains the polarity of its input signal if +1 is generated by the sequence generator, and it inverts the polarity if 
1 is generated. The tag antenna acts like a scatterer and its scattering mechanism can be classified as structural mode and antenna mode [
with a pseudonoise (PN) code 
which has period 
, that is, 
for all 
. The PN code 
is unique for each tag and is used for multiuser interference suppression or spectrum smoothing. The ranging sequence 
is periodic with period 
. In the following discussion, the tag response refers to the signal caused by the antenna mode scattering while the unmodulated signal caused by the structural mode scattering is treated as part of the clutter.
with respect to its own clock. Every symbol consists of 
frames with one pulse per frame. The transmitted signal is given by
is the transmitted UWB pulse and 
is the frame duration.
be the number of paths in the downlink channel. The downlink channel response to 
is modeled as
is the pulse of the l th path, 
is the associated relative path delay, 
and 
are the propagation delays of the l th path and the direct path, respectively. 
has a support region on 
and 
is the true maximum delay spread of the one-way channel. Suppose that 
is the processing delay from the tag's receiving antenna to the modulator. The noise-free signal fed to the modulator of the tag is expressed as
be a rectangular window with unit amplitude in 
and zero elsewhere. As illustrated in Figure 
and 
are integers denoting the ranging symbol offset and PN code offset between the reader and the tag, 
is the timing offset between the clocks of the reader and the tag. Let 
be the modular operation
and 
be the integer floor operation. Equivalently, (4) may be also written as
has time support on 
. The output of the modulator is expressed as
, 
and 
. The physical explanations of 
and 
are given as follows. As shown in Figures 
, the channel response 
may be modulated by two consecutive data bits if the transition edge between the two data bits splits the channel response into two parts. Here 
accounts for the front part of the channel response whereas 
represents the tail. For 
, we have 
since the support region of 
does not overlap with the nonzero region of 
. Similarly, for 
, we have 
.
be the delay from the output of the modulator to the transmitting antenna of the tag. The modulated tag response 
is delayed by 
seconds and retransmitted back to the reader via the uplink channel (from tag to reader). Assuming that the uplink channel has 
paths, its responses to 
and 
are given by
and 
are the responses of the l th path to 
and 
, respectively, 
and 
are the relative propagation delays associated with 
and 
, 
is the propagation delay of the direct path in the uplink channel, and 
and 
are the propagation delays of the l th path. As illustrated in Figure 
is
is the response of the l th path in the clutter channel, 
is the propagation delay of that path, and 
is the number of clutter paths. 
has time support on 
, where 
is the maximum clutter spread.
as the arrival timing of the returned direct path in response to the pulse transmitted at 
. By this definition, 
is equal to the roundtrip propagation delay plus the processing delay of the tag, that is, 
, where 
is the total processing delay of the tag. For ranging applications, 
may be known by the reader via precalibration or communications between the tag and the reader so that eventually it can be calibrated out of estimated 
. Assume that 
follows a uniform distribution, that is, 
, where 
is the maximum TOA. After passing through a zonal bandpass filter with bandwidth 
, the overall signal received by the reader can be expressed as
accounts for thermal noise and multiple access interference, the double-sided power spectral density of 
of 
, 
is the overall clutter waveform with 
defined in (9), 
is the uplink channel response to 
and can also be interpreted as the overall roundtrip tag response to the UWB pulse transmitted at 
, 
and 
have been defined in (8).
as the downlink channel. We further assume that 
and 
so that the interframe interference (IFI) is avoided. As implied by (10), the tag response energy may vary for different frames since the data bits modulating 
and 
may be different. It is useful to define an averaged symbol energy to noise ratio 
, where 
is the energy per symbol of the tag response averaged over all possible data bits, that is, 
and 
is the expectation operator. Another parameter of interest is the signal-to-clutter ratio 
, where 
and 
is the energy per symbol of the clutter.
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.
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
and 
are 
column vectors containing samples of clutter and noise, respectively, 
is the sample vector of tag response and is given by
and 
are data bits modulating the partial tag responses 
and 
, respectively.
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:
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.
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 
[
.
. This can be readily shown by differentiating (14) twice with respect to 
, which yields
, 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
is an 
vector containing the samples averaged over 
frames, 
, 
, 
and 
are defined as
is a constant and is irrelevant to 
. Using (18), Equation (
can be developed as
is a constant irrelevant to 
and 
.
into (13) and dropping the factor 
which is irrelevant to the decision-making process, we reach
, 
, 
, 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.
, the intermediate function 
expressed in (23) also fulfills the two special properties. First, Equation (
and 
.
, 
, 
, 
, 
. 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
and 
into (23) and simplifying, we have the following expression for 
depending only on 
:
. To cover this exceptional situation, the following analysis is carried out. The equation 
can be factorized as
, 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
into (31) yields
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
are the final estimates of 
. And the TOA is estimated as
. 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.
is computed based on sample set 
which is a function of the true tag response 
, clutter 
, and noise 
. Recall that 
is computed using (27) for 
and using (33) for 
. To explicitly reveal the possible effects of the clutter term 
on 
given in (27), the signal model 
given in (11) is substituted into (27), which yields
and 
and 
are 
vectors with their k th element given by 
,
have been completely cancelled out during the derivation process, leaving no clutter term in the final expression. This finding suggests that clutter does not have any effect on the value of decision function. Therefore, the decision function (27) is immune to the clutter.
, for all 
which leads to equations 
and 
as discussed in Section 3. With these equations and (
is a 
vector with its 
th element given by
in (38) implies that the decision function (33) is also irrelevant to the clutter.
which is the same setting as used in [
and 
of total channel energy of CM1 and CM2, respectively. Since no UWB channel model for clutter has been reported in the literature, the impulse response of the clutter channel used in simulations is generated as follows. Let 
be the impulse response of the clutter. 
is generated by
is the convolution operator, and 
is a one-way channel impulse response generated independently and randomly from the CM1 and CM2 channel models used for the tag response generation, 
is a Dirac delta function and 
is the propagation delay of the first path in the clutter. Assume that 
follows a uniform distribution, that is, 
. The length of clutter responses are truncated beyond 
which is slightly longer than the true maximum roundtrip delay spread of tag responses 
.
. Without loss of generality, we set the tag processing delay to zero in the simulation, that is, 
. The rest of system settings are 
, 
. For each channel realization, the sequence 
and 
are randomly generated. The TOA estimation error for the 
th channel realization is 
and the mean absolute error (MAE) defined by 
is used as performance criterion where 
is the total number of channel realizations and is set to 1000 in the simulation. The symbols adopted in the simulations are summarized in Table 
, 
, and 
. The SCR is set to 
representing the extreme environment where the clutter overwhelms the signal. It is found that regardless of the variation in channel condition and the number of observed symbols, the estimator is immune to the clutter. Figure 
on the performance with 
, 
, and 
. As 
increases from 
to 
, the variation of MAE is insignificant which suggests that the impact of 
on the estimation accuracy is trivial. Moreover, according to (34), the computational complexity of the proposed estimator increases as 
increases. Hence 
should be minimized for TOA estimation purpose during system design phase.
is closely related to the performance of TOA estimation since it directly determines 
which is included in the discrete signal model used to derive LS estimator. Figure 
for different SNR values in CM1 channels which have RMS delay spread of around 
. The system settings are 
and 
. There is an optimal setting of 
for every SNR curve. This optimal setting ranges from 
to 
as SNR varies from 
to 
and it increases as SNR increases. The MAE is much more sensitive to the setting of 
for signal with high SNR. 
and 
. The system settings are 
and 
. 
is the Nyquist rate while 
is the sampling rate equal to the inverse of pulse width. As expected, the MAE increases as the sampling rate decreases. The MAE performance difference between different sampling rates is reduced as SNR decreases. At moderate SNR level, for instance, 
, the MAE difference between 
and 
is only 
for 
and 
for 
. Therefore, we may conclude that the estimator is robust over undersampling operation.
to obtain energy samples and then searches the direct path sample within a time period prior to the strongest sample. Note that 
is the time resolution for this estimator. The search back window is denoted as 
and the number of energy samples within the windows are 
. The first sample crossing a predefined normalized threshold 
is detected as the direct path sample. The energy samples for this estimator can be expressed as
is defined by
and 
are minimum and maximum samples among 
. The TOA is estimated as
. This estimator is referred to as ED estimator with normalized threshold hereafter. 
) ED estimator with normalized threshold (solid line); (
) estimator derived based on GML criterion (dash line); (
) the estimator proposed in this paper (dash-dot line). The settings are E
s
/N
0
= 35 dB, N
s
= 8, N
f
= 4, 
= 1 ns, 
= 1 ns, and 
= 1ns, 
= 40 ns.
comparable to the delay spread of propagation channel. Unlike the estimator in [
which determines the time resolution. The samples are
, 
, and 
so that the time resolution of the three estimators, that is, the ED estimator with normalized threshold, the GML estimator, and the estimator presented in this paper, are the same. The rest of parameter settings are 
, 
, and 
. Figure 
and 
, it can be shown that
. Letting 
and using 
again, it is straightforward to show that 
. Substituting 
into (A.1) yields 
, where 
. Denoting 
and noting that 
for 
, it is straightforward to derive that 
with which 
becomes
, we can also show that 
. Therefore, we can make the denotation 
where 
is a constant.
, (21) may be rewritten as
, we have 
and 
. Therefore, (A.3) may be rewritten as
as a constant and letting 
,
, (A.4) becomes
, it can be shown that
, for all 
, the first equation (
is always true and the equality holds if and only if 
for all 
[
, for all 
and 
is a constant. Note that 
is equivalent to 
since 
and 
can only be 
or 
. In conclusion, when 
, for all 
or 
, for all 
are fulfilled, at least one of (29) and (30) holds and the expression (27) becomes invalid.