- Research Article
- Open Access
TOA Estimator for UWB Backscattering RFID System with Clutter Suppression Capability
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 753129 (2010)
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.
Backscattering radio frequency identification (RFID) is a type of RFID technology employing tags that do not generate their own signals but reflect the received signals back to the readers. It is widely used in asset tracking, inventory management, health care monitoring, and other fields . Nowadays many RFID applications such as context-aware healthcare require accurate location information with extended operating range. The conventional backscattering RFID system using continuous wave (CW), however, cannot fulfill this requirement due to its poor distance estimation accuracy and limited operating range. Recently, ultra wideband (UWB) signal emerges as a viable solution for the new generation of backscattering RFID system. UWB is defined by Federal Communications Commission (FCC) as signals having a fractional bandwidth larger than 20% or absolute bandwidth of more than 500 MHz . Such enormous bandwidth brings many advantages such as higher ranging accuracy, lower probability of interception, and more resistance to multipath fading as compared to CW signal . Its capability in achieving accurate range and location estimation has been proven analytically and in experiments [4–7]. UWB signal has been applied to backscattering RFID system with localization functionality in [8, 9]. In , the concept of UWB pseudorandom backscattering tag is introduced. The tag receives signal only in certain time interval according to a pseudorandom time hopping sequence, slightly delays it and then reflects it back to the reader. In , a UWB backscattering RFID tag that is able to apply various modulation schemes is proposed and its potential operating range/data rate tradeoff in the presence of strong clutter is investigated.
Ranging for UWB backscattering RFID system requires estimation of time of arrival (TOA) of the tag response signal at the reader. However, recovering TOA information in environments with dense scatterers is challenging due to the undesired background clutter caused by scatterers other than RFID tags . Clutter paths arriving earlier than the direct path of the desired tag response signal may cause intolerable false alarm rate whereas those overlapping with the direct path may distort its pulse shape. As a result, the estimation accuracy can be severely degraded. To clean the received signal contaminated by clutter before applying any TOA estimation algorithm, the empty-room or frame-to-frame techniques used in the radar community may be applied . The empty-room technique subtracts the channel response measured in the absence of the target from any received signal comprising of both target response and clutter. Unfortunately, this technique is sensitive to environment changes since any change renders the previously measured empty-room response inappropriate for further use. For the frame-to-frame technique, each measurement consists of two received signals captured at different timings. The technique subtracts one signal from the other to eliminate the clutter that is assumed to be static in each measurement but may change from measurement to measurement. This technique fails if the target does not move or moves little between received timings of the two signals in the same measurement. Besides the empty-room and frame-to-frame techniques, it is also possible to mitigate clutter by proper selection of the modulation sequence used by the tags. In , it is shown that if the tag's modulation sequence for antipodal 2-PAM signaling fulfills certain criteria, by averaging over multiple symbols, the received signal is immune to clutter. The study, however, is carried out in the context of data communication under the assumption that perfect pulse synchronization to the first arriving path is achieved, that is, TOA information is known.
There are many existing TOA estimators proposed for UWB system. The channel estimators derived in [12, 13] are able to estimate the delays of the paths and hence can be implicitly used as TOA estimators. A generalized maximum likelihood (GML) estimator is proposed in , which performs the channel estimation in a predefined time interval prior to the largest sample and takes the timing of the first channel tap crossing a preset threshold as TOA. The subspace-based TOA estimators are pursued in [15, 16]. Frequency-domain super-resolution TOA estimation with diversity techniques is studied for indoor localization applications in . The computational complexity of the above estimators is high due to their requirements of either estimation of large number of multipath, or eigen value decomposition of matrices with large dimension. Recent works of TOA estimation focus on the development of low-complexity algorithms. A few suboptimal TOA estimation algorithms with reduced computational complexity are introduced in [18, 19]. The cyclostationarity nature of UWB signal is exploited to develop non-data-aided (NDA) TOA estimators [20, 21]. A novel "timing with dirty template" (TDT) synchronization criterion is established in  based on which both data-aided (DA) and NDA estimators are derived. As shown in , it is also possible to transform timing estimation into a maximum likelihood (ML) amplitude estimation problem and derive a closed form solution for the frame level timing offset. A few threshold-crossing TOA estimators applicable for energy detection (ED) receiver are developed in [24–26]. In , a normalized threshold selection adapted to the strength of the received channel profile is proposed. In , a threshold selection method based on Kurtosis value is developed which is proven to be robust to channel condition variation. In , the threshold is set as a function of propagation delay and simulation results show that considerable performance improvement can be achieved over conventional methods. A TOA estimator allowing long energy integration duration is derived based on unknown pulse shape and GML criterion . Based on least square (LS) criterion, TOA estimators for UWB system are derived in [28, 29]. To speed up the estimation process, different two-stage estimators are proposed where the first stage estimates a coarse TOA and the second stage refines the result [30, 31]. An excellent literature review of UWB TOA estimation is presented in . All these TOA estimators, however, are derived in the context of conventional UWB system which involves one-way channel propagation and the signal modulations are done at the transmitter. Therefore, the channel response to a single transmitted UWB pulse only contains the information of a single data bit. The UWB backscattering RFID system, however, presents a different propagation scenario. It involves roundtrip channel propagation and the signal modulation is performed at the tag which lies in the middle of the roundtrip channel. If the clock of the backscattering tag does not synchronize with the clock of the reader, such modulation incurs mismatch, that is, the front part of a channel response received at the tag may be modulated by one bit while the tail is modulated by the next data bit. Hence the received signal model of UWB backscattering system is significantly different from that of conventional UWB system. Furthermore, the aforementioned TOA estimators do not include clutter in their signal models during the derivation processes and hence they lack of proven clutter suppression capability. Consequently, those TOA estimators cannot be directly applied to UWB backscattering RFID s
According to the above discussion, the peculiar transmission and modulation characteristics of UWB backscattering system together with the clutter suppression requirement call for a dedicated treatment of the derivation for TOA estimator. With these requirements in mind, we derive a novel NDA LS TOA estimator for the UWB backscattering RFID system with antipodal 2-PAM based on the tag structure proposed in . The proposed estimator is able to recover the TOA information even when the clocks of readers and tags are asynchronous. By examining the properties of the derived estimator, we find that it has inherent immunity to clutter for arbitrary data sequence which is highly desirable for backscattering RFID system. Unlike the aforementioned empty-room and frame-to-frame techniques, such immunity holds regardless of environment changes and does not matter if the tags move or keep stationary. Simulation results indicate that the estimator is robust in undersampling operation.
The rest of the paper is organized as follows. Section 2 describes the system model and signal model. Section 3 derives the LS estimator whose immunity to clutter is discussed in Section 4. Simulation results are presented in Section 5. Finally, a conclusion is drawn in Section 6.
2. System Model and Signal Model
Here, we derive a NDA LS TOA estimator for the UWB backscattering RFID system with the tag proposed in , assuming that the clocks of the reader and tag are asynchronous. Tags under consideration are implemented with antipodal 2-PAM modulators. To initialize the TOA estimation, a reader transmits UWB pulses to a targeted tag and estimates the TOA of the roundtrip channel response. As depicted in Figure 1, the returned roundtrip tag response (including direct transmission and channel echoes) is contaminated by the clutter reflected directly by surrounding scatterers without passing through the tag . With the TOA metric recovered from the received signal, the range between the reader and the tag can be calculated. The range measurements from different readers can be combined to estimate the location of the desired tag. Without loss of generality, we will focus on the TOA estimation between one reader and one tag.
As illustrated in Figure 1, the tag under consideration consists of an antenna, an antipodal 2-PAM modulator and a bipolar sequence generator with output being 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 . The structural mode scattering is solely determined by the physical properties of the antenna and is independent of the load condition of the antenna. Thus the signal caused by structure mode scattering cannot be modulated by the 2-PAM modulator and hence does not carry any data information in the tag. In contrast, the signal incurred by the antenna mode scattering is received by the antenna and modulated with a bipolar sequence generated by spreading a ranging sequence 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.
To initialize the TOA estimation process, the reader repetitively sends UWB pulses starting at timing with respect to its own clock. Every symbol consists of frames with one pulse per frame. The transmitted signal is given by
where is the transmitted UWB pulse and is the frame duration.
After propagating through the downlink channel (from reader to tag), the transmitted signal with its channel echoes arrives at the tag. Let be the number of paths in the downlink channel. The downlink channel response to is modeled as
where 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
Let be a rectangular window with unit amplitude in and zero elsewhere. As illustrated in Figure 2(b), the modulator function of the tag is given by
where 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
where has time support on . The output of the modulator is expressed as
Substituting (3) and (5) into (6) yields
where , and . The physical explanations of and are given as follows. As shown in Figures 2(b)–2(d), due to the asynchronism between the clocks of the reader and the tag, for , 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 .
Let 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
where 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 1, besides the returned tag response, the reader also inevitably receives clutter. The clutter has consistent waveform over different frames since it is not modulated by the tag. The clutter channel response to a single transmitted UWB pulse is
where 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.
Let us define the true TOA 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
where 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).
In the above derivation, we have assumed that the uplink channel has the same true maximum one-way channel delay spread 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.
3. LS TOA Estimation
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]:
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 . 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.
4. Immunity of the Estimator to Clutter
Any TOA estimator for UWB backscattering RFID system should posses clutter suppression capability, especially in an environment with dense scatterers. In , for data communication applications, clutter is suppressed by choosing a data sequence with zero mean (or quasi-zero mean). Intuitively we would have expected that similar sequence selection should also be imposed to ensure that the estimator derived in Section 3 is immune to clutter. However, we are going to show that the derived estimator is indeed inherently immune to clutter for arbitrary data sequence.
Note that the final decision function 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
where and and are vectors with their k th element given by ,
By observing (36) and (37), it is found that all the terms with symbolic term 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.
Next, to prove that (33) is immune to the clutter, we recall that (33) is used for , for all which leads to equations and as discussed in Section 3. With these equations and (11), (33) is written as
where is a vector with its th element given by
The absence of clutter term in (38) implies that the decision function (33) is also irrelevant to the clutter.
Consequently, based on the above discussions, it can be concluded that the proposed estimator is immune to the clutter. This conclusion is also supported by the simulation results presented in the next section.
5. Simulation Results
For the tag response, the impulse responses of both downlink and uplink channels are generated from the channel models CM1 for residential LOS environment and CM2 for residential NLOS environment as described in IEEE802.15.4a standard . The lengths of the generated one-way channel impulse responses are truncated beyond 60 ns, that is, which is the same setting as used in  and can capture 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
where 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 .
The transmitted UWB pulse is shaped as the second derivative of Gaussian pulse with width of . 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 1.
Figure 3compares the MAEs of the LS TOA estimator in the scenarios with clutter and without clutter. The system settings are , , 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 3 indicates that the MAE of CM1 is generally lower than that of CM2 especially in the high SNR region. The reason is that the direct path is statistically stronger in CM1 than in CM2 due to the obstacles in CM2 environment that may severely attenuate the direct path. It can also be observed that using more symbols improves the performance of the estimator. In both CM1 and CM2, for a large range of SNR, doubling the number of sampled symbols results in 3 dB less SNR to achieve a given MAE.
Figure 4 investigates the effects of the number of frames per symbol 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.
Setting of the presumed maximum one-way channel delay spread 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 5 presents the results of MAE versus 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.
Figure 6 shows the MAE of the estimator operating with various sampling rates for the presumed delay spread of 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.
In Figure 7, we compare the performance of the estimator proposed in this paper with two estimators proposed in [24, 27]. The two estimations are described as follows. The TOA estimator developed in  passes the received waveform to a square-law device, integrates the output successively with time interval 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
The normalized threshold is defined by
where and are minimum and maximum samples among . The TOA is estimated as
where . This estimator is referred to as ED estimator with normalized threshold hereafter.
The estimator presented in  is derived based on GML criterion with the assumption that the shape of the received waveform is unknown. The estimator performs energy integrations successively with long time duration comparable to the delay spread of propagation channel. Unlike the estimator in , the time intervals of the integrations are allowed to overlap with each other if multiple integrators are implemented in parallel. The starting timings of two adjacent integrations are separated by a fixed delay which determines the time resolution. The samples are
The arrival timing of the maximum sample is taken as TOA
Among the two estimators, the ED estimator with normalized threshold is more robust to the channel and noise variation since its adaptive threshold setting while the estimator derived based on GML criterion can potentially achieve higher resolution in practical system since it decouples the time resolution from the length of integration interval and the short interval is difficult to be implemented due to receiver hardware limitation .
To perform a fair comparison, we set , , 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 7 indicates that the estimator presented in this paper is superior to the two estimators presented in [24, 27]. The ED estimator with normalized threshold in  and the GML estimator in  have comparable accuracy and the performances of both of them degrade rapidly as SCR decreases. On the contrary, the accuracy of the estimator proposed in this paper remains constant as the SCR varies.
A novel NDA LS TOA estimator is proposed as a solution to overcome the undesired clutter signal for TOA estimation problem in UWB backscattering RFID system. Both theoretical study and simulation results indicate that the estimator is inherently immune to the clutter signal regardless of SNR variation. Simulation results also show that the performance of the estimator depends on the number of sampled symbols as well as the presumed channel delay spread setting. Also the study shows that the estimator is robust over undersampling operation.
To prove (18), using conditions and , it can be shown that
where . 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
Analogous to the proof for , we can also show that . Therefore, we can make the denotation where is a constant.
To prove (22), letting , (21) may be rewritten as
Noting that for , we have and . Therefore, (A.3) may be rewritten as
Finally, denoting as a constant and letting ,, (A.4) becomes
Using (22) and noting that , it can be shown that
Equation (B.1) implies that only when , for all , the first equation (29) holds.
With (17) and (22), (B.1) may be derived as
To get more insight into the second equation, we invoke the Cauchy-Schwarz inequality which states that is always true and the equality holds if and only if for all . Applying the inequality to the right side of (B.2) yields
and the equality holds if , 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.
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This paper was supported by the ASTAR SERC Project under Grant no. 052-121-0086.
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Xu, C., Law, C.L. TOA Estimator for UWB Backscattering RFID System with Clutter Suppression Capability. J Wireless Com Network 2010, 753129 (2010). https://doi.org/10.1155/2010/753129
- Mean Absolute Error
- Delay Spread
- Uplink Channel
- Generalize Maximum Likelihood
- Channel Delay Spread