Discrete fourier transformbased TOA estimation in UWB systems
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Abstract
In this paper, we propose two time of arrival estimators for ultra wideband signals based on the phase difference between the discrete Fourier transforms (DFT) of the transmitted and received signals. The first estimator is based on the slope of the unwrapped phase and the second one on the absolute unwrapped phase. We derive the statistics of the unwrapped phase. We show that slopebased estimation almost achieves asymptotically the baseband CramerRao lower bound (CRLB), while the absolutephasebased estimator achieves asymptotically the passband CRLB. We compare the proposed estimators to the timedomain maximum likelihood estimator (MLE). We show that the MLE achieves the CRLB faster than the DFTbased estimator, while the DFTbased estimator outperforms the MLE for low signal to noise ratios. We describe also how to use the proposed estimators in multipath UWB channels.
Keywords
Mean Square Error Probability Density Function Discrete Fourier Transform Maximum Likelihood Estimator Additive White Gaussian NoiseI. Introduction
UWB has received increasing attention for many applications like positioning since the FCC (Federal Communications Commission) allowed in 2002 the unlicensed use of the spectrum between 3.1 and 10.6 GHz [1].
Thanks to their ultra wideband (UWB) larger than 500 MHz, UWB signals can be used for highly accurate positioning using the time of arrival (TOA) technique. Many TOA estimators have been proposed in the literature, especially for impulse radio UWB (IRUWB) signals. Most proposed estimators like the maximum likelihood estimator (MLE), the energybased estimators, the autocorrelationbased estimators, the thresholdbased estimators, and others are based on the time domain [2, 3, 4, 5, 6, 7, 8, 9, 10]. The drawback of timedomain estimators is that their precision is limited by the sampling frequency being used, and complex interpolation is required in order to improve the performance. Some other estimators for either electromagnetic or acoustic signals are using the discrete Fourier transform (DFT) of the received signal [11, 12, 13, 14, 15, 16, 17].
In this paper, we propose two estimators for the TOA based on the phase of the DFT of the received signal. The first estimator is based on the slope of the phase and the second one, on the absolute phase. For both estimators, we first compute local estimates at the different frequency components, and then we combine them in order to find the global estimates.
The main three contributions of this work are that:

we show that using the DFT, we can achieve asymptotically the CRLBs (CramerRao lower bound) using very simple estimators requiring only few samples and a sampling rate equal to the signal bandwidth. In our approach, the sampling period is much larger than the achieved accuracy, while in timedomainbased estimation, the sampling period must be smaller than the required accuracy. Another advantage of DFTbased estimation is that we do not need to identify the main lobe of the autocorrelation of the used pulses like in timedomain estimation.

we show that the MLE achieves the CRLB faster than the DFTbased estimator, while the DFTbased estimator outperforms the MLE for low SNRs.

we compute the statistics of the unwrapped phase of a noisy signal.
The main difference between this work and the previous works using the DFT approach is that in the previous works, the TOA is not estimated based on the phase of the DFT [13, 14], or the problem of phase ambiguity is not investigated (by assuming the maximum time delay smaller than the period of the highest frequency component) [15, 16], or the problem of phase ambiguity is solved using other approaches (Chinese remainder theorem [11, 12] or recursive correction of the TOA estimate [17]). The proposed estimators can be used for IRUWB signals as well as for multicarrier UWB (MCUWB) signals. Note that the main goal of this paper is to give the main ideas about DFTbased TOA estimation. Many improvements can be introduced in order to make the proposed estimators achieve performance closer to the CRLBs.
The paper is organized as follows. In Section II, we describe the system model. In Section III, we consider the MLE of the local phase and compute the statistics of the unwrapped phase. In Section IV, we derive the local slopebased and absolutephasebased TOA estimators. In Section V, we derive the global slopebased and absolutephasebased TOA estimators. In Section VI, we show how multipath UWB channel can be handled.
II. System model
We consider a transmitter and a receiver communicating through an additive white Gaussian noise (AWGN) channel.
where z[m] denotes the sample of the signal z(t) at t = mT_{ s }(T_{ s }= 1/B is the sampling period). (n[m]) is a white Gaussian sequence (i.e., the samples n[m] are independent and identically distributed (iid)). The variance of n[m] is given by ${\sigma}_{n}^{2}=2{N}_{0}B$ where 2N_{0} is the onesided power spectral density of the AWGN.
where ${\sigma}^{2}={\sigma}_{N}^{2}/2=M{N}_{0}B$ is the variance of ${x}_{{N}_{k}}$ and ${y}_{{N}_{k}}$.
III. Statistics of the unwrapped MLE of the phase
In this section, we consider the MLE of the phase and compute the statistics of its unwrapped version.
It is obvious that the time delay can be estimated from an estimation of (5) as either: (i) the phase to angular frequency ratio or (ii) the slope of the phase with respect to the angular frequency. For both approaches, the estimated phase must be continuous. With the former it must also be around the true value, while with th e latter a constant offset along the frequency axis is accepted. As in practice the phase is computed modulo 2π (wrapped phase), an unwrapped version of it is needed in order to rebuild the continuous phase.
where the nonambiguity condition (2π Δfτ < π) must be respected. Unwrap procedure described above is well known and "unwrap" MATLAB function can be used to perform unwrapping.
As in practice the true value of the phase is unknown we can start the unwrap procedure from an arbitrary k_{0} by taking ${\stackrel{\u0303}{\phi}}_{{k}_{0}}={\widehat{\phi}}_{{k}_{0}}$, then running the unwrap procedure for k_{0} +1,..., M/2  1 and k_{0}  1,..., M/2. It is obvious that the unwrapped phase may have an offset (almost constant ∀k) with respect to the true phase dependent on the offset at the starting point $\left(2\pi \left({f}_{{k}_{0}}+{f}_{c}\right)\tau {\widehat{\phi}}_{{k}_{0}}\right)$.
where {·}* denotes the complex conjugate. The estimates ${\widehat{\phi}}_{k}$ at different frequencies k are independent because the noise samples N_{k} are independent.
where $\text{erfc}\left(z\right)=\left(2/\sqrt{\pi}\right){\int}_{z}^{+\infty}{\text{e}}^{{\xi}^{2}}d\xi $ denotes the complementary error function, and the superscript ^{ wr }the wrapped phase. ${T}_{{\widehat{\phi}}_{k}}^{wr}\left({\widehat{\phi}}_{k}\right)$ is 2π periodic and can be defined on any interval $\left({I}_{{c}_{k}}=\left[{c}_{k}\pi ,{c}_{k}+\pi \right]\right)$ of width 2π. ${\int}_{{I}_{{c}_{k}}}{T}_{{\widehat{\phi}}_{k}}^{wr}\left({\widehat{\phi}}_{k}\right)d{\widehat{\phi}}_{k}=1\forall {c}_{k}$. It is shown in [20] that the distribution of the wrapped phase can be approximated by a normal distribution if the local SNR ν_{ k }is sufficiently high, and by a uniform distribution if ν_{k} is very low.
Let us now compute the PDF of the unwrapped MLE ${\widehat{\phi}}_{k}$ of the phase. Assume that we start the unwrap procedure from k = 0 (so, we have ${\stackrel{\u0303}{\phi}}_{0}={\widehat{\phi}}_{0}$). Let ${T}_{{\stackrel{\u0303}{\phi}}_{k}}\left({\stackrel{\u0303}{\phi}}_{k}\right)$ be the marginal PDF of ${\widehat{\phi}}_{k}$. Below, we will show that ${T}_{{\stackrel{\u0303}{\phi}}_{k}}\left({\stackrel{\u0303}{\phi}}_{k}\right)$ can be computed recursively for k = 1,..., M/2  1 and k =  1,..., M/2 starting from ${\widehat{\phi}}_{0}$.
However, some errors multiple of  2π can be introduced during the unwrap procedure as can be seen in Figure 1b, c for two other realizations of the of the wrapped phase ${\widehat{\phi}}_{k}$. This happens when the unwrap procedure should add a multiple of 2π to the next phase (for instance at k =  3 in Figure 1b), but does not do it because the absolute difference between the neighboring noisy phases is less than $\pi \left({\widehat{\phi}}_{3}{\widehat{\phi}}_{4}\le \pi \right)$. Every time this phenomenon happens, an additional error of 2π will be introduced.
Note that errors multiple of 2π can also be introduced. This happens when the unwrap procedure should not add a multiple of 2π to the next phase, but does it because the absolute difference between the neighboring noisy phases is greater than π. These errors occur rarely if the slope of the true phase is positive.
As already mentioned, the presence of these secondary lobes is due to errors multiple of ±2π introduced by the unwrap procedure. The main lobe becomes weaker and secondary lobes stronger as the frequency increases which means that we have more chance that such an error occurs. This is due to the fact that the unwrapping is performed recursively for increasing frequencies (see Figure 1ac), so the ±2π errors accumulate over the course of the procedure. If we increase the number of samples or decrease the global SNR, we will obtain more secondary lobes at · · ·,  4π, 2π, 2π, 4π, · · · from the main lobe. Errors multiple of  2π (resp. 2π) are more frequent if the slope of the true phase is positive (resp. negative). Obviously, the unwrapped phase is biased, and both the bias and the variance increase with the frequency due to the accumulation of ±2π errors.
We can see in Figure 3a that the simulated variance and MSE closely follow the theoretical ones, which validates our theoretical approach. However, variance and MSE are not following the CRLB, and they increase with the frequency due to the errors multiple of ±2π which are introduced by the unwrap procedure.
In Figure 3b where the local SNRs are sufficiently high (ν_{k} = 28 dB, ∀k), we can see that the derived and simulated variance and MSE are very close to the CRLB. In fact for high SNRs, the wrapped phases are unwrapped correctly because the errors multiple of ±2π become very rare.
IV. Slopebased and absolutephasebased local TOA estimators
In the last section, we have studied the unwrapped MLE ${\stackrel{\u0303}{\phi}}_{k}$ of the phase φ_{ k }. In this section, we propose two local TOA estimators based on ${\stackrel{\u0303}{\phi}}_{k}$.
where "round" denotes the "round to nearest integer" function, and ${\stackrel{\u0303}{\tau}}^{bb}$ the global slopebased estimator. ${\stackrel{\u0303}{\tau}}^{bb}$ is given in Section V as a linear combination of ${\stackrel{\u0303}{\tau}}_{k}^{bb}$.
As for sufficiently high SNRs, the unwrapped phase becomes unbiased and its variance converges to its CRLB (1/ν_{ k }), we can deduce from (20) and (21) (resp. (17) and (22)) that the local passband (resp. baseband) TOA estimator becomes also unbiased and achieves the local passband CRLB (resp. the sum of the local baseband CRLB of f_{0} and f_{ k }).
V. Slopebased and absolutephasebased global TOA estimators
In this section, we derive the global TOA estimators based on the local TOA estimators studied in section IV.
The global baseband (resp. passband) TOA estimator ${\stackrel{\u0303}{\tau}}^{bb}$ (resp. ${\stackrel{\u0303}{\tau}}^{pb}$) is defined as the minimumvariance unbiased linear combination of the local estimators ${\stackrel{\u0303}{\tau}}_{k}^{bb},k=M/2,...,M/21$ (resp ${\stackrel{\u0303}{\tau}}_{k}^{pb}$).
As the covariances and variances of the local estimators ${\stackrel{\u0303}{\tau}}_{k}^{bb}$ and ${\stackrel{\u0303}{\tau}}_{k}^{pb}$) are unknown, we compute the global estimators from (24) and (25) by assuming that ${\stackrel{\u0303}{\phi}}_{k}$ achieves the CRLB ${c}^{{\phi}_{k}}$, and substituting ${\sigma}_{{\stackrel{\u0303}{\phi}}_{k}}^{2}$ by $1/{\rho}_{{S}_{k}}^{2}$ (proportional to ${c}^{{\phi}_{k}}$).
where ν is the global SNR given in (9) and ${\beta}_{s}^{2}={\sum}_{k}4{\pi}^{2}{\rho}_{{S}_{k}}^{2}{f}_{k}^{2}/{\sum}_{k}{\rho}_{{S}_{k}}^{2}$ the discrete mean quadratic bandwidth of s[k].
where s_{ pb }(t) and r_{ pb }(t) denote the real passband transmitted and received signals and ⊗ the convolution operator.
We can see that the global baseband estimator almost achieves asymptotically the baseband CRLB. We can also see that both the MLE and the global passband estimator achieve asymptotically the passband CRLB. However, ${\stackrel{\u0303}{\tau}}^{ml}$ achieves c^{ pb }faster than ${\stackrel{\u0303}{\tau}}^{pb}$. Many improvements can be introduced to our estimators in order to make them achieve the CRLBs faster. Hereafter, we will describe briefly one more baseband estimator and one more passband estimator.
As the unwrapped phase errors described above generate large negative slopes, and as the time delay can be assumed positive by putting the reference pulse at the beginning of the observation period, we can mitigate these errors by keeping only the positive values of ${\stackrel{\u0303}{\tau}}_{k}^{sp}$. Let ${\underset{}{\stackrel{\u0303}{\tau}}}_{+}^{sp}$ be the vector containing the positive values of ${\stackrel{\u0303}{\tau}}_{k}^{sp}$ and ${\underset{}{\underset{}{\Gamma}}}_{{\underset{}{\stackrel{\u0303}{\tau}}}_{+}^{sp}}$ its covariance matrix.
Now, instead of unwrapping the phase recursively, we unwrap each ${\stackrel{\u0303}{\phi}}_{k}$ (wrapped phase) with respect to $2\pi \left({f}_{k}+{f}_{c}\right){\stackrel{\u0303}{\tau}}_{+}^{sp}$ in order to get ${\stackrel{\u0303}{\phi}}_{k}^{sp}$ (new unwrapped phase located around the true phase). The new global baseband (resp. passband) estimator ${\stackrel{\u0303}{\tau}}_{+}^{bb}$ (resp. ${\stackrel{\u0303}{\tau}}_{+}^{pb}$) is obtained as before from (16) and (24) (resp. (19) and (25)), but after substituting ${\stackrel{\u0303}{\phi}}_{k}$ by ${\stackrel{\u0303}{\phi}}_{k}^{sp}$ (resp. ${\stackrel{\u0303}{\phi}}_{k}+\stackrel{\u0303}{\Delta}\phi $ by ${\stackrel{\u0303}{\phi}}_{k}^{sp}$) in (16) (resp. (19)).
The MSEs of ${\stackrel{\u0303}{\tau}}_{+}^{bb}$ and ${\stackrel{\u0303}{\tau}}_{+}^{pb}$ obtained by simulation are shown in Figure 5. We can see that ${\stackrel{\u0303}{\tau}}_{+}^{bb}$ and ${\stackrel{\u0303}{\tau}}_{+}^{pb}$ achieve c^{ bb }and c^{ pb }faster than ${\stackrel{\u0303}{\tau}}^{bb}$ and ${\stackrel{\u0303}{\tau}}^{pb}$, respectively. Still, the MLE achieves c^{ bb }and c^{ pb }faster than ${\stackrel{\u0303}{\tau}}^{bb}$ and ${\stackrel{\u0303}{\tau}}^{pb}$. However, for small SNRs (ρ < 15 dB), the new passband estimator outperforms the MLE.
Fianlly, the main advantage of the MLE is that it achieves the CRLB faster, while the main two advantages of the new estimator are that: i) it requires a sampling rate and a number of samples much smaller than those required by the MLE and that ii) it outperforms the MLE for low SNRs.
VI. TOA estimation in multipath channels
where we have considered the modulus in order to get only one peak per MPC (the used baseband pulse must have only one lobe). The coarse estimates of τ^{(l)}can be obtained as locations of the peaks of ${\Gamma}_{{r}_{MP,}s}\left(t\right)$ crossing a given threshold. Once the coarse estimates are obtained, we can apply our DFTbased estimators by taking a window around each MPC slightly larger than the pulse width. The final estimates of τ^{(l)}are expected to have the same characteristics shown throughout this paper if the MPCs are not overlapping.
VII. Conclusion
Two TOA estimators are proposed based on the absolute phase and the slope of the unwrapped phase of the DFT of the received signal. The slopebased TOA estimation is used as a coarse estimation in order to rebuild the absolute unwrapped phase and to compute the absolutephasebased estimator. The statistics of the unwrapped phase are computed. It has been shown that the slopebased estimator almost achieves asymptotically the baseband CRLB, while the absolutephasebased estimator achieves asymptotically the passband CRLB. The proposed estimators are compared to the timedomain MLE estimator. It has been shown that the MLE achieves the CRLB faster than the DFTbased estimator, while the DFTbased estimator outperforms the MLE for low SNRs. It has also been also described how the proposed estimators can be used in multipath UWB channels. The main theoretical results are validated by simulation.
Notes
Acknowledgements
The authors would like to thank the FP7 NEWCOM++, the DGTRE COSMOS, and the RADIANT projects for the financial support and the scientific inspiration. They also would like to thank Sinan Gezici and Davide Dardari for the useful discussions with them.
Supplementary material
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