Multiuser Interference Mitigation in Non-coherent UWB Ranging via Nonlinear Filtering

—Ranging with energy detectors enables low-cost implementation. However, any interference can be quite detrimental to range accuracy. We develop a method that performs non-linear ﬁltering on the received signal energy to mitigate multiuser interference (MUI); and we test it over time-hopping and direct sequence impulse radio ultrawideband signals. Simulations conducted over IEEE 802.15.4a residential line of sight ultrawide-band multipath channels indicate that non-linear ﬁltering helps sustain range estimation accuracy in the presence of strong MUI.


I. INTRODUCTION
In time-of-arrival (ToA) based ranging, the range accuracy depends heavily on how well the ToA of a signal is estimated.Identifying multipath components and finding the leading path is crucial to decrease ranging errors.With its fractional bandwidth of 20%, or at least 500MHz bandwidth, an ultrawideband (UWB) signal provides high time resolution measured in nanoseconds; and it helps to separate individual multipath components better than narrowband signals [1].
In UWB ranging, tracking of the leading edge is challenging due to a vast number of multipaths and the fact that the line-of-sight (LoS) path may not have the highest amplitude.Traditionally, UWB approaches based on coherent reception require many rake fingers in order to combine energy from the received signal [2].However, there is a strong desire to drive down UWB radio cost.This has lead to an increased interest in alternative receiver techniques for UWB that do not require the hardware complexity of coherent rake receptions.
One intuitive approach has been a trade-off between high performance coherent receivers and low-complexity noncoherent receivers [3].However, one of the major drawbacks of a non-coherent receiver is its performance in the presence of multiuser interference (MUI).In a multiuser network, signals from multiple devices may interfere with a desired signal and deteriorate the range error drastically.This is due to the fact that interference suppression techniques such as CDMA are not readily applicable to simple non-coherent receivers.Typically, processing gain is obtained by coherently combining received signal energy according to transmitted time-hopping or DS patterns [4].However, in coherent energy combining, even a small amount of interference energy may be construed as a leading edge.Therefore, prior to coherent energycombining, it is prudent to remove as much MUI energy as possible.
In this paper, our scope is to make ranging via non-coherent radios resilient to MUI.We focus on simple energy detectors, and propose a MUI mitigation technique for time-hopping impulse-radio (TH-IR) [5] and direct sequence impulse radio (DS-IR) UWB systems to sustain sub-meter range accuracy when MUI is present.
The remainder of this paper is organized as follows.In Section II, the literature on UWB ranging is reviewed.In Section III, the TH-IR and DS-IR UWB signal models are given and then the proposed receiver architecture is described.In Section IV, MUI mitigation via non-linear energy filtering is explained.Section V is allocated to the discussion of simulation results.Finally, the paper is concluded in Section VI with a summary of our future work.

II. TOA BASED UWB RANGING
Acquisition of a signal can be achieved by locking onto the strongest multipath component, which would result in a coarse ToA estimate [6]- [11].However, precise ToA estimation requires identification of the leading path, which may not be the strongest.In [12], a generalized maximum likelihood (GML) approach is proposed to estimate the leading path by testing the paths prior to the strongest.A stopping rule is determined based on the statistics of the amplitude ratio and the delay between the strongest and the leading paths.However, very high sampling rates on the order of Nyquist rate is required.In [13], the authors relax the sampling rate requirements and propose a simpler threshold-based detection technique.In [14], the problem is approached as a break-point estimation for signal presence, where temporal correlation arising from the transmitted pulse is used to accurately partition the received signal.
Acquisition and ToA estimation can generally be achieved by using various transceiver types; e.g.matched filters (or stored-reference receivers), transmitted reference receivers, and energy detectors (ED) [6], [15].The use of energy detectors for synchronization and ToA estimation in UWB systems has been investigated in [15], [16], [17].ED receivers using threshold-based ToA estimation techniques are discussed in [18], [19], [20], a multi-scale product approach that improves the ranging accuracy was investigated in [21], and likelihood based techniques are proposed in [15].A twostep hybrid ToA estimation via ED and matched filters are also studied in [22], [23], where the energy-detection step provides a coarse ToA estimate, and the matched-filtering step refines the estimate.In [24], a matched-filter receiver's ability to differentiate between the desired user signal and interference for TH-IR UWB during synchronization is analyzed.
Our literature survey indicates that the ToA estimation problem for IR-UWB has been analyzed without consideration of MUI.Note that even though MUI mitigation is investigated extensively for IR-UWB systems for symbol detection [25]- [28], there is no reference that addresses interference mitigation for ToA estimation with non-coherent UWB radios to the best of our knowledge.This work is intended to fill this gap.

III. RANGING SIGNAL WAVEFORMS AND RECEIVER FRONT-END
In [19], four different waveforms were compared from the ranging perspective; In this work, we adopt two of them: DS-IR and TH-IR (see Fig. 1).These are currently under consideration for standardization in the IEEE 802.15.4aTask Group.
Each IEEE 802.15.4a packet contains a preamble that consists of multiple repetition of a base symbol waveform; and the preamble is used for acquisition/syncronization and ranging.We adopt the IEEE 802.15.4a terminology and use the following notations in the sequel: E (k) s denotes the symbol energy from the kth user, N sym is the number of symbol repetition within the preamble, ω is the transmitted pulse shape with unit energy, T sym is the symbol duration, T p is the pulse duration, k is the TOA of the kth user's signal and η is the zero-mean AWGN with variance σ 2 n = N0 2 .L k denotes the total number of multipath components for the kth user, γ l,k and τ l,k represents the amplitude and delay of the lth multipath component for the kth user respectively, and N s is the total number of pulses per symbol.
A receiver can process the preamble by either template matching (coherent) or energy detection (ED).Even though coherent ranging is superior, the ED receiver has its own advantages such as simplicity, operability at sub-Nyquist sampling rates (which determines the range resolution) and low cost.They are also more resilient to pulse shape distortion.
The ED receiver we study in this paper is illustrated in Fig. 2. It first feeds the received signal (after a band pass filter) into a square-law device, integrates its output, and then samples periodically.We denote these generated energy samples as z[n], and the sampling interval and the number of samples per symbol as t s and n b = Tsym t s , respectively.The z[n] are then regrouped into a 2D matrix.
Once a matrix is formed, it is passed through a non-linear filter to enhance desired signal energy parts and remove the MUI.Afterwards, the matrix is converted back to 1D time series to locate the leading edge, by means of adaptive searchback and threshold techniques.In what follows, we present signal models for DS-IR and TH-IR systems.

A. DS-IR
In DS-IR, a symbol interval is divided into two halves.A group of closely spaced pulses called burst is transmitted either in the first or the second half in a pseudo-random pattern.With such an orthogonal burst positioning, ranging can be performed in the presence of multiple simultaneously operating devices.The received DS-IR symbol waveform from user k can be written as where d ≥ T p .The polarities of the pulses in a burst are used to convey data for coherent reception.Therefore, the spacing between the pulses enables coherent receivers to demodulate the data.
If there are K simultaneously transmitting users, the received signal would be where b λ,k ∈ {0, 1} is the λth symbol of kth user, T ppm is the modulation index (i.e.delay) for pulse-burst position modulation (PPM).Note that varying T ppm would change the inter-burst interval.Hence, multiple orthogonal waveforms can be generated, and each can be assigned to users of different networks.
The ED output samples at the desired receiver with the DS-IR waveforms is where n = 1, 2, ..., N b , and Fig. 2. Illustration of the energy imaging ranging receiver while processing ED outputs.

B. TH-IR
In TH-IR, a symbol is divided into virtual time intervals T f called frames, which is further decomposed into smaller time slots T (th) c called chips.A single pulse is transmitted in each frame on a chip location specified by a user specific pseudorandom time-hopping code.The received TH-IR signal from user k is where c j,k and d j,k are the TH codes and polarity scrambling codes of user k, respectively.If K users are transmitting N sym symbols simultaneously, each with a unique TH code, the received signal by the desired user becomes The collected energy samples at the ED receiver would be

C. Conventional Energy Combining (Conv)
A conventional receiver coherently combines the energies over N sym symbols to improve the signal-to-noise ratio (SNR) using the bit sequence of the desired user in the DS-IR case 1 , and over N sym × N s pulse positions using the TH sequences of the desired user in the TH-IR case.Then, a searchback algorithm is applied to locate the leading signal energy.
In this paper, we adopt the searchback scheme presented in [19].With the assumption that the receiver is perfectly synchronized to the strongest energy sample, the algorithm tries to identify the leading edge by searching the samples backward within a predetermined window starting from the strongest sample.In non-LoS environments, the strongest path may arrive as much as 60ns after the first path [29].At 4ns sampling period, this would correspond to 15 samples.Therefore, in the searchback algorithm (see Fig. 3), it would be sufficient to have W = 15.
Each sample within the searchback window is compared to a threshold.Even if it is smaller than the threshold, the algorithm does not terminate; and it allows up to w cls consecutive noiseonly samples.This is because of the fact that clustering of the multipath components yields noise-only regions between the clusters.The threshold ξ that corresponds to a fixed P f a is given by 2 where µ ed and σ ed are the mean and the variance of noise-only samples.The optimal threshold is a function of w cls .

IV. ENERGY MATRIX FORMATION
SNR is one of the parameters that range estimation accuracy heavily depends on.Although the SNR can be improved via processing gain by coherently combining received signal energy samples [22], Fig. 4 illustrates how poor the ranging performance would be after coherent energy combining in the presence of MUI.In the given TH-IR example, the symbol consists of four frames, and signal energy is integrated and sampled at a period such that it produces four samples in each frame and total 16 samples per symbol.The TH code of the desired signal is {0, 4, 4, 3}, and that of the interference is {0, 4, 5, 4}.Coherent combining requires energy samples z[n] of the received signal to be combined in accordance with the matched TH code.In this example, it would produce the combined energy values where 0 ≤ n ≤ 3, assuming that TOA ambiguity is as much as the frame duration.If there is no interference, E [1] = 4A and E[n] = 0 for n = 1 and the TOA index is 1.In the presence of interference, the time of arrival information is very likely to be impacted, and it is easy to see in the example that TOA index becomes 0 because E[0] = 2A (see Fig. 4d).
We have now illustrated that signal design itself and coherent energy combining is not sufficient to deal with detrimental impact of interference.A solution simply lies in looking at the collected energy samples from a different view: twodimensional energy matrix.Let us create a so-called energy matrix Z of size M × N , where M is the number of frames processed and N the number of energy samples collected from each frame.Referring to the previous example, the size of Z would be 4 × 4, it would be populated as Filling out each column of Z with samples grouped according to the received signal's TH pattern would form vertical lines, whenever signal energy is present in all of those samples (Fig. 4e).The detection of the left most vertical line would give the time index of the first arriving signal energy.If the interference follows a different TH pattern, intuitively the energy matrix of the interference would not form a vertical line (Fig. 4f).The CONV does not account for the MUI and it directly aggregates the energy samples.This would be equivalent to summing the rows of Z along each column, yielding an energy vector.Note that the column sum of the matrix in Fig. 4e generates the energy vector in Fig. 4c, and that of (e) + (f ) results in Fig. 4d.
Applying conventional leading edge detection techniques on the energy vector in Fig. 4d would cause erroneous ranging due to interference.It is clear from the illustrations that the energy matrix would provide an insight into the presence and whereabout of interference energy; and non-linear filters can be applied onto the matrix to mitigate this interference.The following subsections explain how to form an energy matrix from DS-IR and TH-IR waveforms.

A. Energy Matrix of DS-IR
Let λ denote the row index (which is also the symbol index), and κ denote the column index of the matrix.Then, the samples in (3) can be used to populate the matrix as follows where 1 ≤ λ ≤ N sym and 1 ≤ κ ≤ n b A typical energy matrix of a DS-IR signal after passing through an IEEE 802.15.4aCM1 channel is given in Fig. 5 while the E b /N 0 is 16dB for the desired received signal and 10dB for the interference.Clearly, the desired signal forms a vertical line indicating multi-path components, whereas the interference pattern is intermittent.
Self interference may be present too in the energy matrix.This happens when only some of the samples of a column do overlap with the energy from bursts.
The energy vector z(ds) that the Conv receiver generates would be equivalent to the column-sum of Z (ds) .
where 1 N sym is a column vector of all ones.

B. Energy Matrix of TH-IR
In TH-IR, energy samples given in eq.6 are grouped together according to the transmitted TH code, and samples of the same group are used to populate a column of the energy matrix Z (th) .As a result, there would be N s × N sym rows.
where λ(j) = N s (λ − 1) + j, and j ∈ {1, 2, ..., N s }.Note that we assume T c to be an integer multiple of t s in order to be able to collect the energies over integer number of pulses.
A typical energy matrix of a TH-IR signal after passing through an IEEE 802.15.4aCM1 channel is given in Fig. 6.The E b /N 0 is 16dB for the desired received signal and 10dB for the interference.Note that MUI and self-interference would cause short discrete lines.The actual ToA corresponds to the most left continuous vertical line in Z (th) .
A cause of the self-interference is the imperfect autocorrelation of the TH codes.Note that the energy samples of a column are grouped according to the desired user's TH code.It is possible that only some of the grouped samples would contain energy from the received signal due to a partial overlap with the signal's TH pattern.Especially, if the uncertainty region for the ToA is larger than T f , the energy collection process would cause more self-interference.Non-linear filters would not be able to distinguish self-interference from MUI.
Furthermore, to suppress noise N img matrices can be superposed, relying on the assumption that the statistics of interference and noise are stationary.The Conv would columnsum Z (th) and would perform edge detection on z(th) .
V. NON-LINEAR MATRIX FILTERING In this section, we consider two non-linear filters for interference mitigation: minimum filter and median filter.In the following discussion, without losing generality, we drop the superscript of the energy matrix for DS-IR and TH-IR, and refer to it as Z.

A. Minimum Filter -Min
To remove outliers, which are most likely due to interference, in Z, we apply length-W minimum filter along each column.The minimum filter replaces the center sample with the minimum of the samples within the filter window.Then, the elements of the new energy matrix Z (min) would be where λ ∈ {1, 2, ..., N sym − W + 1} for DS-IR and λ ∈ {1, 2, ..., N s N sym − W + 1} for TH-IR.Once the interference is removed, Z (min) is converted to a vector by the column-sum operation, z(ds,min) = 1 Nsym−W +1 Z (ds,min) , ( 14) where Z (ds,min) indicates Min filtered matrix for the DS-IR and Z (th,min) for the TH-IR.Note that while it significantly removes the interference, The Min filter may also degrade the desired signal.

B. Median Filter -Median
One well-known non-linear filter is the median filter.Median filters are special cases of stack filters and they have been widely used in digital image and signal processing [30], [31] to remove singularities caused by noise.A median filter replaces the center value in a given data set with the median of the set.A longer filter length makes output noise more colored and becomes less effective to mitigate interference, because any unsuppressed interference energy may propagate onto its neighboring samples.We use length-3 median filter in the simulations and refer to it as Median.One way to prevent output noise from being colored is to apply the median-filter in non-overlapping windows.In appendix A, we quantify the impact of non-overlapping median filtering on detection performance of DC signals in white Gaussian noise to provide some insight into more complex detection problems.In ( 16), Z (med) is the energy matrix at the output of the median filter.
After converting Z (med) into an energy vector, we have The leading edge search is performed on z(ds,med) for DS-IR waveforms and on z(th,med) for TH-IR waveforms.Note that both minimum and median filtering add to the low complexity of an energy-detection receiver.Assume that z[n] are provided by a 16-bit ADC.Then, the memory requirement for storing Z of size M × N would be 2MN Bytes.It is known that sorting W numerals has an inherent computational complexity of O(W logW ).Thus, the overall complexity of applying Median or Min would be M (N −W +1)O(W logW ).

VI. SIMULATION RESULTS
The DS-IR and TH-IR signals are transmitted over IEEE 802.15.4aCM1 (residential line-of-sight) channels.For performance comparison, we use mean absolute error (MAE) of ToA estimations over 1000 realizations.A DS-IR or TH-IR symbol waveforms of length 512ns are considered; the other simulation settings are as follows: T sym = 512ns, T ppm = 256ns, T f = 128ns, T p = 4ns, w cls = 2, T c = 4ns for TH-IR and 6ns for DS-IR, and the integration interval is 4ns.Energy images are obtained using 80 symbols (therefore yielding 80 rows for DS-IR, and 320 rows for TH-IR), and the images are further assumed to be averaged over 250 realizations 3 .For TH-IR, the time hopping sequence for the desired user is c j,1 = [1, 1, 4, 2], and for the interfering user c j,2 = [1, 4, 2, 1], where there are T f /T c = 64 chip positions per frame 4 .
We compare the ranging accuracy of the searchback algorithm described in Fig. 3 under different interference levels.Let E (1) and E (2) denote the symbol energies received from the desired user and the interfering user, respectively (we also use E b for desired user's bit energy).Then, we simulate the interference levels where E (2) /N 0 ∈ {−∞, 0, 5, 10}dB.Energy matrices are constructed, and MAEs before (Conv) and after non-linear filtering (Min, Median) are obtained for all cases using a non-linear filter window length of 3.

A. DS-IR
The MAE results in Fig. 8a) show that in the absence of MUI, the Conv and Median outperforms Min by achieving MAE of as low as 2ns at E b /N 0 values less than 14dB.Intuitively, this is because of the fact that when noise is the dominant term, the Min penalizes the signal.
However, at higher E b /N 0 , the MAE of the Min is better than those of both Conv and Median, because at high E b /N 0 , self interference becomes the dominant factor, and (for certain channel realizations) the multipath components from a previous symbol may extend into the searchback window and still degrade the ranging accuracy of Conv and Median (see Fig. 5).Minimum filter remains effective to mitigate self interference at high SNRs.

B. TH-IR
In general, the TH-IR waveform yields higher MAEs when compared to the DS-IR for the simulated set of parameters.This can be explained by higher self interference from autocorrelation sidelobes of TH-IR waveforms; even though TH sequences with a large zero correlation zones are used in our simulations, for the channels with large maximum excess delays, the performance is degraded.In the DS-IR case, the Min effectively suppressed self interference even at high E b /N 0 .
An interesting observation with TH-IR waveforms is that there exists an optimum E b /N 0 and the MAE starts increasing beyond this value even if there is no MUI.This is due to the fact that increasing the E b /N 0 also increases the energy of autocorrelation sidelobes; since threshold is set based only on the noise level, stronger self interference starts degrading the performance after the optimum SNR level 5 .
The presence of interference at levels of E (2) /N 0 = 0dB or higher drastically hits the performance of Conv and Median and as a result their MAE never goes below 6ns; whereas the MAE of Min remains the same as no interference case when E (2) /N 0 = {0, 5}dB.Even when E (2) /N 0 = 10dB, the MAE floor of the Min approaches 5ns at very high SNR (E (1) /N 0 = 18dB).
These results suggest that a better search-back and thresholding techniques need to be developed for the TH-IR case for more accurate ranging performance.Also, the energy matrix with minimum filtering proves to be effective to deal with interference in TH-IR case too.

VII. CONCLUSION
In this paper, we introduce a ranging method that uses a matrix of received energy samples from a square-law device, and applies non-linear filtering on the matrix to remove the outliers caused by interference.The recommended non-linear filter, based on the simulation results, is the minimum filter.After the non-linear filtering, energy values along each column of the matrix are aggregated.Hence, the two dimensional data are converted into an energy vector.Then, a search-back algorithm is run on the energy vector to locate the leading signal energy.
The effectiveness of this approach is proven by simulations conducted using IEEE 802.15.4a channel models.Non-linear filtering changes noise and signal characteristics.Due to space limitations, the impact of non-linear filtering on the receiver detection performance will be studied in a different article.
This study reveals the following: • Ranging is quite sensitive to interference, since the leading edge sample may be very weak compared to interference samples.• A single interference energy sample may prolong the searchback process, and increase ranging error.• In addition to multiuser interference, the searchback algorithm has to deal with self interference.All in all, the paper presents a framework and provides practical algorithms to deal with multiuser interference in TOA estimation via non-coherent ultrawideband systems.Our future work includes development of adaptive algorithms (e.g.minimum and median filters with adaptive window size) for enhanced ranging accuracy under varying levels of interference, and quantification of the impact of non-linear filtering on detection performance.zero mean and variance σ 2 .Assume that there are N i.i.d.observations of the test data z[n].When there is no signal, the data set belongs to a noise only hypothesis H 0 , and when signal is present it belongs to hypothesis H 1 .
The probability of detection, P D , with the Neyman-Pearson detector for this problem is given in [32] as Note that After length-W median filtering with nonoverlapping windows, the new observation set would have only N/W samples and the noise variance would be scaled by f (W ), where f () indicates the function of.Since the input distribution is Gaussian, the output would approximate to Gaussian with the same mean, but lower variance [33].
Theoretically, the output density of the length 3 median filter is where Q z is the complementary cumulative distribution function and p z (y) is the density of the input data.Our numerical analysis indicates that f (3) = 0.44, and it provides a tight approximation to (21).The Kolmogorov-Smirnow test to compare the approximated density function and ( 21) results in the significance level of 0.1%.Then, in consideration of the approximation, the probability of detection P m D after median filtering becomes Here the problem of detecting a DC level in Gaussian noise is addressed for its simplicity, and in Fig. 7 it is shown that median filtering in non-overlapping windows would lower the probability of detection.If the length-3 median filter is applied with two-sample overlapping windows, the output noise would be a colored Gaussian, but the size of the observation set would remain as N.It may be possible to observe an increase in detection performance.Quantification of the impacts of median filtering with overlapping windows on the detection performance of non-coherent receivers will be studied in detail in another study.
(ds) j,k ∈ {±1} are the binary sequences for the kth user, and T (ds) c is the chip duration (pulse repetition interval) such that T (ds) c

Fig. 3 .
Fig. 3. Pseudocode for the adaptive searchback algorithm to locate the leading signal energy

Fig. 4 .
Fig. 4. Illustration of coherent energy combining in 1D a) Energy samples from TH-IR desired user, b) Energy samples from TH-IR interference c) Coherent combining of energy samples without interference d) Coherent combining of energy samples with interference e) Energy image of the desired signal, Z f) Energy image of the interference

Fig. 7 .
Fig.7.Degradation in probability of detection after length-3 median filtering for "DC level detection in Gaussian noise" problem.Note that W = 3, f (3) = 0.44.Note that median filtering with non-overlapping windows degrades detection performance.