Bandlimited Field Sampling Using Mobile Sensors in the Absence of Location Information

# Bandlimited Field Sampling Using Mobile Sensors in the Absence of Location Information

## Abstract

Sampling of physical fields with mobile sensor is an emerging area. In this context, this work introduces and proposes solutions to a fundamental question: can a spatial field be estimated from samples taken at unknown sampling locations?

Unknown sampling location, sample quantization, unknown bandwidth of the field, and presence of measurement-noise present difficulties in the process of field estimation. In this work, except for quantization, the other three issues will be tackled together in a mobile-sampling framework. Spatially bandlimited fields are considered. It is assumed that measurement-noise affected field samples are collected on spatial locations obtained from an unknown renewal process. That is, the samples are obtained on locations obtained from a renewal process, but the sampling locations and the renewal process distribution are unknown. In this unknown sampling location setup, it is shown that the mean-squared error in field estimation decreases as where is the average number of samples collected by the mobile sensor. The average number of samples collected is determined by the inter-sample spacing distribution in the renewal process. An algorithm to ascertain spatial field’s bandwidth is detailed, which works with high probability as the average number of samples increases. This algorithm works in the same setup, i.e., in the presence of measurement-noise and unknown sampling locations.

Additive white noise, nonuniform sampling, signal reconstruction, signal sampling, wireless sensor networks

## I Introduction

Consider a mobile sensor which has to acquire a spatially smooth field by moving along a path or spatial trajectory [1, 2]. If the sensor is equipped with precise location information, high-precision quantizers, and negligible measurement-noise, then the field reconstruction process reduces to classical (noiseless) sampling and interpolation problems (see [3, 4, 5]). With precise location information, spatial field reconstruction or estimation has been addressed in the presence of quantization as well as measurement-noise (for example, see [6, 7, 8, 9, 10, 11]). In the context of spatial sampling with a mobile-sensor, a more challenging setup is when the location of samples collected is not known. Unknown sampling locations is a fundamentally new topic in spatial field sampling and will be the central theme of this work.

The motivation for mobile-sampling without the knowledge of locations is elucidated first. In practice, a device such as GPS (global positioning system) or other elaborate distributed localization mechanisms can be used to localize a sensor [12]. If a mobile-sensor’s path and its velocity are known, and if the mobile-sensor has an accurate clock, then the sampling locations can be calculated from sample timestamps [1]. However, all these elaborate mechanisms will add to the cost of mobile sensors, and increase its recording overhead. It would be desirable to get rid of the timestamps, the GPS, the distributed mechanisms for localization and the knowledge of velocity, and still reconstruct the spatial field to a desired accuracy. This is the core motivation behind the paper.

Spatially smooth, temporally fixed, and finite-support fields will be considered in this work. The smoothness of spatial field will be modeled by bandlimitedness. It will be assumed that the mobile sensor samples the field at locations obtained by an unknown renewal process. By unknown renewal process, we mean that the probability distribution of the inter-sample locations and even the locations at which field samples are obtained are not known. It will also be assumed that the field samples are affected by additive and independent noise with zero mean and finite variance. Except for independence, zero-mean, and finite variance, it is assumed that the noise distribution is also not known. In such a challenging setup, sampling rate (corresponding to oversampling) will be used in this work to decrease expected mean-squared error in field-reconstruction. In other words, the mobile sensor will collect a large number of readings on locations determined by an unknown renewal process and estimates will be developed in this work to drive down the expected mean-squared error in field-reconstruction with sampling rate.

The field sampling setup with a mobile sensor is illustrated in Fig. 1. To keep the analysis tractable, the spatial field is assumed to be one-dimensional in space and temporally fixed in this first exposition.1 The mobile sensor collects the spatial field’s values at unknown locations , , , derived from a renewal process with unknown inter-sample distribution [13].

The spatial field measurements are affected by additive and independent noise process , which has an unknown distribution with zero-mean and finite variance. In this challenging setup, the goal is to estimate the field from the readings collected by the mobile sensor.

Unknown location information on samples, (low-precision) quantization, knowledge of the spatial field’s bandwidth, and presence of additive measurement-noise are four perils in the process of field estimation (or reconstruction). In this work, leaving quantization issues aside, the other three ailments will be addressed together in a mobile-sampling framework. At first, a reconstruction method is discussed where the bandwidth of the signal is known. Next, an algorithm will be outlined in this work which handles an unknown (but finite) bandwidth signal. The main results shown in this work are as follows:

1. For the field sampling setup illustrated in Fig. 1 and with some regularity conditions on the inter-sample location distribution, the expected mean-squared error in field estimation is upper bounded as , where is the sampling density (or is the expected number of samples realized by the renewal process in the interval of sampling). This result holds when zero-mean finite-variance additive noise is present during sampling, and when the sampling locations are obtained at unknown locations generated from an unknown renewal process.

2. To address unknown bandwidth, an algorithm will be presented to ascertain the correct bandwidth of the spatial field (with high probability) in the mobile-sensor sampling paradigm. This algorithm works in the presence of measurement-noise and when field samples are obtained at unknown locations generated from an unknown renewal process. The proposed algorithm requires the knowledge of mean and variance of squared-noise.2

Prior art: The topic of unknown but uniformly distributed sampling locations was recently introduced in the context of spatial sampling in a finite interval [14]. A detailed differentiation with this earlier work is needed, since the setup and results appear to be similar in nature. The estimates used are also the same in the two works (see (7)). If sensors are uniformly distributed in an interval, as in [14], then their ordered (noise-affected) readings can be used to obtain a bandlimited field estimate with mean-squared error of . The derivation of this result utilizes established facts on the mean-squared deviations of ordered uniformly distributed random variables [14, 15]. It is also known that ordered uniformly distributed random variables can be understood as realizations of a Poisson renewal process [13]. In contrast, in this current work the renewal process distribution is assumed to be unknown. This setup is much more challenging than the previous work, since the distribution properties are assumed to be unknown in the current work. As a result, the mean-squared error analysis is derived from first principles. The final result on mean-squared error analysis results in an upper bound of , which is the same as in prior work in spite of unknown distribution properties of the renewal process. One important aspect of result in this work is that both the estimate and the mean-squared error result are universal in nature (see (7) and Theorem III.1). In summary, the sampling location’s distribution is assumed to be unknown, and is the major difference with past work; and, the problem addressed in the current work is more difficult!

Sampling and reconstruction of discrete-time bandlimited signals from samples taken at unknown locations was first studied by Marziliano and Vetterli [16]. This problem is addressed completely in a discrete-time setup and solutions are combinatorial in nature. A recovery algorithm for bandlimited signals from a finite number of ordered nonuniform samples at unknown sampling locations has been proposed by Browning [17]. This algorithm works in a deterministic setup. Estimation of periodic bandlimited signals, where samples are obtained at unknown locations obtained by a random perturbation of equi-spaced deterministic grid, has been studied by Nordio et al. [18]. More generally, the topic of sampling with jitter on the sampling locations [3],[19, Chap. 3.8] is well known in the literature.

This work is different from previous literature in the following non-trivial aspects—(i) the sampling locations are generated according to an unknown renewal process, where even the distribution defining the renewal process is unknown; and, (ii) the field is affected by zero-mean additive independent noise with finite variance, where the distribution of noise is also not known.3

Notation: Spatial fields which are temporally fixed will be denoted by and its variants. The -norm of a field will be denoted by . The spatial derivative of will be denoted by . The number of (random) samples will be denoted by , while will denote the expected value of . Expectation will be denoted by . The set of integers, real numbers, and complex numbers will be denoted by , , and , respectively. Finally, .

Organization: Spatial field’s model, the reconstruction distortion criterion, the sampling model using a mobile sensor, and the noise model on measurement (or field) is explained in Section II. The estimation of spatial field with known bandwidth and samples taken on an unknown renewal process is addressed in Section III. An algorithm for spatial field estimation with unknown bandwidth and samples taken on an unknown renewal process addressed in Section IV. Measurement-noise is considered both in Section III and Section IV. Simulation results are presented in Section V. Finally, conclusions will be presented in Section VI.

## Ii Field model, distortion, sampling process by mobile sensor, and measurement-noise model

The models used for theoretical analysis presented in Section III and Section IV will be discussed in this section. Field models are discussed first.

### Ii-a Field model

It will be assumed that the field of interest is one dimensional, temporally fixed, and spatially bandlimited. Let be the field, where is the spatial dimension. For reasons stated in Sec. I, it will be assumed that the temporal variation of is negligible, i.e., during the sampling process. Temporally fixed assumption on the spatial field is suitable when the speed of the mobile sensor is much higher than the temporal rate of change of the field (also see [1] for discussions). Temporally fixed fields are more tractable to analysis, and this assumption will also help in understanding the effect of location-unawareness on the field sampling process in isolation.

Without loss of generality, it is assumed that , is periodic and bandlimited with period equal to , and has a bandwidth , where is a known positive integer. All these assumptions imply that has a Fourier series in the interval with bounded Fourier series coefficients

 g(x) =b∑k=−ba[k]exp(j2πkx) (1) with a[k] =∫10g(x)exp(−j2πkx)dx. (2)

Based on Bernstein inequality [20], it follows that

 |g′(x)|≤2bπ∥g∥∞≤2bπ. (3)

### Ii-B Distortion criterion

For a first exposition, a mean-squared error will be used as the distortion metric. If is any estimate of the field , then the distortion is defined as

 D :=E[∫10∣∣ˆG(x)−g(x)∣∣2dx] =b∑k=−bE[|A[k]−a[k]|2] (4)

where is the Fourier series representation of .

### Ii-C Renewal process based sampling model

It will be assumed that be a renewal process which are the separations between the sampling locations of a mobile sensor.4 The mobile sensor traveling in the interval obtains samples at the locations , , , , where is the random number of samples from the renewal process that fall in the interval . The random number of samples that will be obtained in satisfies a stopping rule

 X1+X2+…XM≤1 and X1+X2+…XM+1>1

and, therefore, is a well defined (measurable) random variable [21]. Let have the common distribution . For analysis purposes, it will be assumed that

 0

where is a finite constant independent of . In other words, , or is a bounded random variable. The random variable is the (random but large) number of readings made to determine the field by a mobile sensor, which is unaware of its sampling locations. The main help in field reconstruction is from the parameter . It will be shown that as the parameter becomes large, there exists an estimate of the field which converges to the true field with respect to the distortion in (4). Our estimate will work without the knowledge of (exact values of) , and more ambitiously the distribution of will also be unknown. It is assumed however that the distribution of has a non-zero support only on .

### Ii-D Measurement-noise model

It will be assumed that the field samples are affected by additive measurement-noise; that is, is sampled, where is additive and independent. The measurement-noise process is independent, which means for any positive integer , are i.i.d. for distinct . Observe that is not a continuous-time white noise process, but its sampled version will be a discrete-time white noise process. It is also assumed that is independent of the renewal-process which generates the sampling locations.

## Iii Field estimation from samples obtained on an unknown renewal process

In this section, the Fourier series of the periodic bandlimited field will be estimated by observing measurement-noise affected samples obtained on an unknown renewal process. As outlined in Section I and Section II, the term unknown renewal process refers to unknown sampling locations and unknown inter-sample spacing distribution.

Recall that the sampling locations are defined as

 S1 =X1, S2 =X1+X2=S1+X2, and SM =X1+…+XM=SM−1+XM. (6)

The sampling locations are unknown, as well as with unknown distribution. Noise-affected field values are available for the estimation of the field . The key to our estimation procedure will be the field’s Fourier series coefficient estimate defined by

 ˆA\footnotesize gen[k] :=1MM∑i=1{g(Si)+W(Si)}exp(−j2πkiM) (7)

where . This estimate works without the knowledge of since the noise-affected field values are the samples recorded by the mobile sensor in our model. This formula is a Riemann-sum like approximation to the Fourier series integral formula in (2) with two assumptions: (i) the sample locations given by are “near” the grid points ; and (ii) the measurement-noise part in (7) averages out to “near-zero”. The effect of these assumptions is analyzed; and, in the estimation of Fourier series coefficients from (in the mean-squared sense) the following theorem is noted.

###### Theorem III.1

Let be as defined in (7). Let be i.i.d. positive (inter-sample spacing) random variables such that , and distribution of has support in . Let be a measurement-noise process independent of with zero-mean and finite variance. Then

 E[∣∣ˆA\footnotesize gen[k]−a[k]∣∣2]≤Cn (8)

where is a constant that is independent of , and depends on renewal process parameter and the signal bandwidth parameter . Correspondingly the distortion in (4) is bounded as or .

Proof: To maintain the flow of the results, the key ideas and inequalities in the proof will be proved in this section while the technically detailed (mundane) statistical calculations will be presented in Appendix A and Appendix B.

The estimate in (7) has been designed with the assumption that has been observed at , has been observed at , and has been observed at for various values of . This is our key statistical approximation. The mean-squared error in making this approximation will be analyzed next. The estimate consists of two conceptual parts

 ˆA\footnotesize gen[k] =1MM∑i=1g(Si)exp(−j2πkiM)ˆA[k]+ 1MM∑i=1W(Si)exp(−j2πkiM)W\footnotesize avg[k] (9)

where the first and second terms correspond to the signal and the measurement-noise part, respectively. These terms will be analyzed separately. Let

 AR[k]:=1MM∑i=1g(i/M)exp(−j2πki/M) (10)

be the -point Riemann approximation of in (2). The approximation is random due to presence of . Since , therefore

 E[∣∣ˆA\footnotesize gen[k]−a[k]∣∣2]=2E[∣∣ˆA[k] −a[k]∣∣2]+ 2E[|W\footnotesize avg[k]|2]. (11)

Next, from the triangle inequality [22],

 ∣∣ˆA[k]−a[k]∣∣≤|ˆA[k]−AR[k]|+|AR[k]−a[k]| (12)

and , it follows that

 E[∣∣ˆA[k]−a[k]∣∣2]≤2E[|ˆA[k]− AR[k]|2]+ 2E[|AR[k]−a[k]|2] (13)

The term will be bounded using the smoothness properties of and the mean-value theorem; and, the mean-squared value of will be upper-bounded using eaxchangeability of conditioned on the stopping time  [21]. The next parts are devoted to these analyses. First, is considered. Note that

 |ˆA[k] −AR[k]| =∣∣∣1MM∑i=1g(Si)exp(−j2πkiM)− 1MM∑i=1g(iM)exp(−j2πkiM)∣∣∣ (14) ≤1MM∑i=1∣∣∣g(Si)−g(iM)∣∣∣ (15)

where the last step using the triangle inequality [22]. Since , for any real numbers and any integer , so

 |ˆA[k] −AR[k]|2 ≤1M2MM∑i=1∣∣∣g(Si)−g(iM)∣∣∣2 (16) ≤∥g′∥2∞1MM∑i=1∣∣∣Si−iM∣∣∣2, (17)

where the last step uses for any . Taking expectations in (17),

 E[|ˆA[k] −AR[k]|2] ≤∥g′∥2∞E[1MM∑i=1∣∣∣Si−iM∣∣∣2]. (18)

From Appendix A, it is noted that

 E[1MM∑i=1∣∣∣Si−iM∣∣∣2]≤C1n (19)

for any and as becomes large, and the constant depends on and is independent of .

For , it is shown in Appendix B that

 |a[k]−AR[k]| ≤C2M (20) or E[|a[k]−AR[k]|2] ≤E(C22M2)≤C2λ2n2. (21)

The constant depends on the field’s bandwidth parameter and is independent of .

Finally, the mean-squared value of has to be characterized. For this part, note that

 E(|W\footnotesize avg[k]|2) =E∣∣ ∣∣{1MM∑i=1W(Si)exp(−j2πkiM)}∣∣ ∣∣2 =E[1M2M∑i,l=1W(Si)W(Sl)× exp(−j2πkiM)exp(j2πklM)] (22) \lx@stackrel(a)=E[1M2M∑i=1|W(Si)|2] (23) =E[σ2M]≤σ2λn. (24)

The equality in follows since the noise process is independent, are distinct, and (which depends on sampling process) is independent of the measurement-noise process .

Putting together results from (18), (19), (21), and (24) in (11),

 E[ |ˆA\footnotesize gen[k]−a[k]|2] (25) ≤4∥g′∥2∞C1n+4C22λ2n2+2σ2λn (26) ≤(4(2b+1)2π2C1+4C22λ2n+2σ2λ)1n (27) ≤Cn (28)

for some which does not depend on . Observe that the constant becomes larger with larger bandwidth (), larger noise variance (), and a larger spread of renewal distribution (). A larger value of , indicates that the (unknown) sampling locations are more spread-out around their mean value of . This results in a worse proportionality constant. The main result of the theorem is complete.

The following remarks are useful in the context of above result.

###### Remark 1

The estimate works on the principle that is nearly equal to , and statistical averaging (with large ) is expected to help in convergence of towards . The exact rate of convergence is shown to be . The simulations presented in Section IV illustrate a mean-squared error proportional to . This suggests that the upper-bound is tight, at least for the estimate presented in (9).

###### Remark 2

It was assumed that the distribution of has a support in , where is a finite constant. This assumption simplifies the analysis but omits distribution with infinite support such as the exponential distribution. In the special case when is exponentially distributed, conditioned on , the random variables will correspond to ordered random variables. In case if is exponentially distributed, the mean-squared error between and can be shown to be by using existing results in the literature [14].

###### Remark 3

From the proof in Appendix A, the decay in distortion will hold if decreases proportionally to and is proportional to . The latter condition can be established easily by Wald’s identity [21]. The former condition will need some sophisticated statistical analysis with stopping-times and has been left as a future work. At a high level, can be expected to decrease as since almost-surely as the sampling rate increases. The assumption that makes the mean-squared error analysis a little convenient.

###### Remark 4

Renewal process with small mean (of ) result in a ‘pontogram’, which is connected to the Brownian Bridge [23]. In the spatial-sampling context, if is the number of samples taken up till location (with in this work’s notation), then will be a generalized pontogram as a function of . Then, it is known that the worst deviation of the Pontogram from a Brownian bridge is negligible (with high probability) as increases [23, Theorem 2.1]. This indicates that, in the limit of large, the mobile sensor will be sampling the spatial field on a Brownian bridge! The properties of a bandlimited field being observed on a Brownian bridge (at unknown points) is an interesting topic of study for future research.

Like in a Brownian bridge, this result also suggests that the variance of sample-locations in the middle is larger than those at the edges of the sampling interval. In the future, it would be interesting to design estimates of which utilize this property.

###### Remark 5

An inspection of proof of Theorem III.1 suggests that its result will hold if is a field in any set of bounded dynamic-range fields having an orthonormal basis, and having smoothness properties such as finite derivative over the entire class. Bandlimitedness would translate to having finite degrees of freedom (or finite number of non-zero coordinates in the orthonormal basis). Orthonormal basis would imply that the degrees of freedom can be obtained using a suitable inner product, which can be approximated using a Riemann sum (see (2) and (9)). Smoothness properties of the set of fields will enable counterparts of (3) required for approximation analysis (see (17)). This generalization, we believe, is analogous to the Fourier series development followed in this work. This generalization is not established in the current work due to space constraints, and more importantly for simplicity of exposition.

## Iv Bandwidth determination using field samples obtained on an unknown renewal process

In some applications, the bandwidth or the essential bandwidth of the spatial field may not be known [24]. This can be because the essential bandwidth of spatial fields change with time [25], or because the field being observed is not characterized for bandwidth, or because the sampling path is not a straight line.5 Under some technical assumptions, an algorithm is outlined to find the bandwidth of the field in an asymptotic setting where the sampling rate increases asymptotically.

Consider a spatial field which has a finite but unknown bandwidth parameter . From (9) and the result in Theorem III.1, for will converge (in the mean-squared sense, and therefore in probability) to zero. This observation can be used to design a reconstruction algorithm for a spatially bandlimited process with unknown bandwidth. The following assumptions are made for this section:

1. The spatial field has a finite but unknown bandwidth parameter (see (2)).

2. All the non-zero Fourier series coefficients are larger than in magnitude, where is a constant.

3. The measurement-noise is zero-mean, its second moment (variance) is known, and its fourth moment is finite (i.e., ).

A non-zero constant (in the second assumption above) is needed to ascertain the bandwidth, since any statistical estimate (such as ) will be negligible outside the bandwidth but not exactly zero. A threshold parameter ensures that the sampling rate can be made large enough to get rid of negligible but otherwise non-zero Fourier series coefficients.6

For any , an estimate for can be obtained from in (9). The tricky part is determination of , that is, when to stop the Fourier series coefficient estimation! In other words, a stopping condition is needed. The next paragraph summarizes this stopping condition and then an algorithm for bandwidth determination is presented, with a sketch of its technical correctness.

Consider a simplified problem, where a bandlimited but unknown bandwidth signal is available. It is known that the bandwidth of is finite, but its value is not known. To reconstruct the field, the bandwidth of is required. Let be the Fourier series of . The Fourier series coefficients can be sequentially computed. The main issue is when the Fourier series coefficients computation should be stopped? To this end, note that since is available so is its energy . By Parseval’s theorem, it is known that

 ∫10|h(x)|2dx=∞∑k=−∞|c[k]|2. (29)

So, can be computed till the energy in the collected coefficients matches with that of . Since the energy of is finite, and is finite by assumption, so this process will end in number of steps. An adaptation of this idea will be used in the stochastic sampling setup with a mobile sensor. Since accurate approximations of the field’s energy and Fourier series coefficients are available only for , so an approximate adaptation of this algorithm is needed to address finite but large values of .

An estimate of spatial field energy (see (29)) is needed since the field is not available in entirety but only through noise-affected samples at unknown locations. The spatial field’s energy estimate is defined as

 Eg:=1MM∑i=1[g(Si)+W(Si)]2−σ2, (30)

where is the noise variance and is assumed to be known. The intuition in the above estimate is that and are independent (and hence uncorrelated), and will average near . The analysis in Appendix C shows that the mean-squared value of is bounded as

 ≤~Cn (31)

where is some constant independent of and depends on only through . Therefore converges to the field energy in the mean-squared sense (and hence in probability).

As sampling rate becomes larger, the empirical energy in (30) converges in mean-squared sense to the true energy of . An estimate for Fourier series coefficients has been presented in (9), which converges in mean-squared sense as sampling rate becomes large. If this estimate is below the threshold by some margin, the Fourier series coefficient can be set to zero by a thresholding operation. For asymptotic , this process will result in (mean-squared) correct Fourier series coefficients. Similarly, as noted earlier, each converges in mean-squared sense to the correct . This motivates the following estimation algorithm, if each Fourier series coefficient is more than in magnitude, where is a positive parameter.

1. Calculate an estimate for the spatial field’s energy as in (30).

3. Calculate the Fourier series coefficient and as in (9). If the estimates are more than in magnitude, retain them. Otherwise, set and as zero. It will be shown shortly that non-zero coefficients succeed while zero coefficients fail in this test with high probability, as increases asymptotically.

4. Increase by . Repeat the process in previous step till

 −Δ22≤B∑k=−B∣∣ˆA% \footnotesize gen[k]∣∣2−Eg≤Δ22. (32)

It will be shown shortly that this test will be met only by the correct bandwidth with high probability, as increases asymptotically.

Claim in Item 3) above follows by Chebychev inequality [21]. If , it is noted that

 P[|ˆA\footnotesize gen[k]−a[k]|<1/3√n] ≤E∣∣ˆA\footnotesize gen[k]−a[k]∣∣2(1/3√n)2 (33) ≤Cn1/3 (34)

which means that with high probability. Since by assumption, so

 Δ−13√n≤|a[k]|−|a[k]−ˆA%\footnotesizegen[k]|≤|ˆA\footnotesize gen[k]| (35)

with high probability. This establishes (the obvious) that will have a magnitude greater than with high probability if has a magnitude greater than .

By similar argument, if , then

 P[|ˆA\footnotesize gen[k]−a[k]|<Δ−13√n] ≤E∣∣ˆA\footnotesize gen[k]−a[k]∣∣2(Δ−n1/3)2 ≤Cn(Δ−n1/3)2 (36)

which converges to zero with increasing . So, if , then is smaller than with high probability.

Claim in Item 4) follows by Chebychev inequality and finiteness of . From (31) and Chebychev inequality, it follows that

 (37)

or with high probability or is close to . The following inequalities are noted, each of which holds with high probability. From (34) and (36),

 a[k]≠0, ||ˆA\footnotesize gen[k]|2−|a[k]|2|≤O(1/3√n) (38) a[k]=0, |ˆA\footnotesize gen[k]|=0 (39)

where the second equality is achieved by thresholding the near-zero against . That is, each estimated (for ) is equal to zero or at a maximum distance of to a non-zero . So the maximum difference between coefficient energies is

 B∑k=−B|ˆA\footnotesize gen[k]|2−B∑k=−B|a[k]|2=O(13√n). (40)

By Parseval’s relation,

 B∑k=−B|a[k]|2=∫10g2(x)dx (41)

where the maximum number of nonzero is , since by assumption. Finally, the maximum difference between energy estimate of the field and the actual energy is

 ∣∣∣Eg−∫10g2(x)dx∣∣∣≤13√n (42)

From (40), (41), and (42), it follows that

 Eg−B∑k=−B|ˆA\footnotesize gen[k]|2=O(1/3√n) (43)

only for the correct value of bandwidth . Therefore, the stopping condition in (32) will be met with high probability.

###### Remark 6

It must be noted that the above algorithm works for asymptotic and convergence rate guarantees are not given in this first exposition. It is possible to select some other function of instead of in Item 3) and some other threshold than , and optimize the probability of error for a finite (but large) . This analysis is left for future work, but simulations based on the above algorithm will be presented.

## V Simulation results

Some simulation results are presented in Fig. 2. In these simulations, additive measurement-noise was generated using Uniform random variables. The random sampling locations were obtained using a renewal process with uniform inter-sample spacing distribution. The knowledge of this distribution, as explained in (9) and Section III, was not used in the field reconstruction. The field was generated in different ways for the first and second plots, and the third plot. This is explained next.

In the first and second plots, which evaluates the performance of field estimate in Section III for field with known bandwidth, it is assumed that there is a field with Fourier series coefficients given by

 a[0]=0.2445,a[1]=−0.0357+j0.0478 a[2]=0.0978+j0.0729,a[3]=−0.1796−j0.0756 a[k]=¯a[−k], and a[k]=0 for |k|>3. (44)

These Fourier series coefficients were obtained by independent trials of Uniform random variables (to obtain the real and imaginary parts). Conjugate symmetry ensures that the field is real-valued. Finally, the field was scaled to limit its dynamic range within .

For the third plot, which evaluates the bandwidth estimation algorithm of Section IV, it is assumed that there is a field with Fourier series coefficients given by

 a[0] =0.1,a[1]=−0.1,a[12]=0.1, a[k] =¯a[−k], and a[k]=0 otherwise. (45)

In these simulations is set as Larger values of are more desirable for the algorithm proposed in this section. Coefficient threshold check tests whether the non-zero coefficients in the Fourier series of are estimated as more than and the zero coefficients are (see Item 3) in the algorithm above). The stopping rule in (32) requires that the estimated field energy is within of the original. This requirement is very stringent and as a result, the stopping rule condition is violated (never or wrongly met) for smaller values of . For , the threshold is positive only when . For this reason, simulation begins from in the third plot.

In the first plot, the convergence of random realizations of to can be observed with increasing . The graph for is near identical to the true field and cannot be seen in the graph. The mean-squared error, averaged over random trials, decreases as as illustrated in the second (log-log) plot. This is in consonance with our results in Theorem III.1. Finally, in the third plot, it is observed that the stopping rule check and coefficient threshold check are met successfully if exceeds . These numerical values will change depending on the value of .

## Vi Concluding remarks

This work introduced the estimation of bandlimited spatial fields from noise-affected samples taken at unknown sampling locations, where the locations are generated from a renewal process with unknown distribution. Sampling rate was used to combat against additive measurement-noise as well as unknown sampling locations. A spatial field estimate, which converges to the true spatial field in the mean-squared sense at the rate , was presented and its analysis was the first main result of the work. A spatial field bandwidth determination algorithm from field samples collected at unknown sampling locations, which works with probability one as sampling rate increases asymptotically, was proposed and its correctness was the second main result of this work. Simulation results, consonant with the theoretical analysis, were also presented.

This work opens a flurry of interesting ideas related to reconstruction of spatial fields from samples collected at unknown sampling locations. How can the field estimates be derived (developed) when the renewal process distribution or the noise distribution is known? Is the distortion result developed in this paper optimal in certain circumstances? How will the distortion change in the presence of sample quantization? How should the field estimation change if a fraction of field samples are taken at known locations or without measurement-noise or both? What field estimation or reconstruction strategy should be used to tackle the sampling of non-bandlimited fields? In all these, and many more, questions we expect sampling rate to play a fundamental role in the obtained answers.

## Appendix A Mean-squared closeness of renewal-process sampling grid to a uniform grid

For analysis purposes, let

 RM=1−(X1+X2+…XM) (46)

be the remaining distance between the last location of sampling and the end of field support (or sampling vehicle’s terminal stop). Observe that is bounded since

 RM≤XM+1≤λn. (47)

First the average value of will be determined, conditioned on . Since by definition, so

 E(SM +RM|M=m)=1 (48) or mE(X1|M=m) +E(RM|M=m)=1 (49) i.e., E(X1|M=m) =1m−E(RM|M=m)m. (50)

Since so the second term is expected to be negligible with large sampling rate . That is, conditional average of is nearly conditioned on . In (49), conditioned on , the exchangeable nature of is used along with .

To determine the expectation of average mean-squared error between and equi-spaced grid, consider the conditional expectation of the following error-term:

 E[ (51) =E[1mm∑i=1{i∑l=1(Xl−1m)}2∣∣∣M=m] =E[1mm∑i=1i∑l=1i∑p=1(Xl−1m)(Xp−1m)∣∣∣M=m] \lx@stackrel(a)=E[1mm∑i=1i(X1−1m)2+ i(i−1)(X1−1m)(X2−1m)∣∣∣M=m] (52) (X1−1m)(X2−1m)∣∣∣M=m] (53) =(m+1)2am+m2−13bm (54)

where follows by exchangeability of conditioned on , and

 am :=E[(X1−1m)2∣∣∣M=m] and bm :=E[(X1−1m)(X2−1m)∣∣∣M=m] (55)

By definition conditioned on , so

 E[(Sm−1)2|M=m]=E[(Rm)2|M=m] (56)

and therefore,

 E⎡⎣{m∑i=1(Xi−1m)}2∣∣∣M=m⎤⎦−E[R2m∣∣M=m]=0

since . A rearrangement of the above equation results in,

 mam +m(m−1)bm=E[R2m∣∣M=m] or bm =1m(m−1)(−mam+E[R2m∣∣M=m]). (57)

With from (57), the conditional error term in (54) can be rewritten as

 E [1mm∑i=1{i∑l=1(Xl−1m)}2∣∣∣M=m] (58) =(m+1)2am+m2−13m(m−1)(−mam+E[R2m∣∣M=m]) =(m+1)6am+m+13mE[R2m∣∣M=m] ≤(m+1)6am+23λ2n2 (59)

where the last step is obtained since and by assumption on the inter-sample spacing distribution. So,

 E[ 1MM∑i=1(Si−iM)2] ≤16E((M+1)aM)+23λ2n