WO2014120178A1

WO2014120178A1 - Data interpolation and resampling

Info

Publication number: WO2014120178A1
Application number: PCT/US2013/024023
Authority: WO
Inventors: Pavel Kornilovich; Raul Hernan Etkin; Farzad PARVARESH
Original assignee: Hewlett-Packard Development Company, L.P.
Priority date: 2013-01-31
Filing date: 2013-01-31
Publication date: 2014-08-07

Abstract

According to an example, a method for data interpolation and resampling may include receiving timestamps t _s[m] for a sensor disposed in a network, where m is an interval for the received timestamps t _s[m], receiving values of a signal x[m] corresponding to the timestamps t _s[m] from the sensor, and determining, by a processor, values of the signal x[m] at a new set of time values t _d[m] based on the timestamps t _s[m] and the values of the signal x[m] corresponding to the timestamps t _s[m]. The values of the signal x[m] at the new set of time values t _d[m] may be determined by using a frequency domain approach that uses a phase shift to compensate for a time offset between the new set of time values t _d[m] and the timestamps t _s[m].

Description

DATA INTERPOLATION AND RESAMPLING

BACKGROUND

[0001] Sensing applications typically include analysis of data from a plurality of sensors across a network. For example, an application related to seismic oil exploration may survey an area using accelerometers (i.e., sensor nodes) located in a regular or irregular grid. For such an application, seismic waves may be induced, for example, by explosives, vehicles, etc., and the reflections of the seismic waves may be measured by the accelerometers. For the various sensors used with such sensing applications, the sensed data may need to be synchronized across the network.

BRIEF DESCRIPTION OF DRAWINGS

[0002] Features of the present disclosure are illustrated by way of example and not limited in the following figure(s), in which like numerals indicate like elements, in which:

[0003] Figure 1 illustrates an architecture of a data interpolation and resampling apparatus, according to an example of the present disclosure;

[0004] Figure 2 illustrates a method for data interpolation and resampling, according to an example of the present disclosure;

[0005] Figure 3 illustrates further details of the method for data interpolation and resampling, according to an example of the present disclosure; and

[0006] Figure 4 illustrates a computer system, according to an example of the present disclosure.

DETAILED DESCRIPTION

[0007] For simplicity and illustrative purposes, the present disclosure is described by referring mainly to examples. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. It will be readily apparent however, that the present disclosure may be practiced without limitation to these specific details. In other instances, some methods and structures have not been described in detail so as not to unnecessarily obscure the present disclosure.

[0008] Throughout the present disclosure, the terms "a" and "an" are intended to denote at least one of a particular element. As used herein, the term "includes" means includes but not limited to, the term "including" means including but not limited to. The term "based on" means based at least in part on.

[0009] Sensing applications typically need sensed data from various sensors to be synchronized across a network. For example, a sensing application may include a plurality of sensors disposed across a network, with each sensor including a free running sampling clock (i.e. not synchronized) on the sensor nodes in the network. For example, sensors may use sampling clocks that are based on crystal oscillators. While such crystal oscillators may have acceptable short term frequency stability, such oscillators may eventually drift with respect to one another over extended periods of time. As a result, the sampling clocks from the different sensors may need to be synchronized. Data from such sensors may be tagged with appropriate timestamps, and synchronized in post processing. For example, a central timing source (e.g., a global positioning system (GPS)) may be used to distribute a timing reference throughout the network. The sampled data recorded by the sensors may be aligned according to this timing reference of the central timing source.

[0010] A data interpolation and resampling apparatus and a method for data interpolation and resampling are disclosed herein for post processing alignment of the sampled data recorded on the sensors by interpolation and resampling. The apparatus and method disclosed herein may use timestamps and sampled (i.e., discrete time) data to obtain an approximation to a continuous time signal, which can be then be resampled at any desired sampling times. For example, the apparatus and method disclosed herein may use the same set of sampling times when resampling data of all the sensors to obtain synchronized data that is consistent across the network. The apparatus and method disclosed herein may use a frequency domain approach to efficiently synchronize the data of multiple sensors to make the data consistent with respect to the timing reference of the central timing source.

[0011] According to an example, the data interpolation and resampling apparatus may include a memory storing machine readable instructions to receive timestamps for a sensor disposed in a network, and receive values of a signal corresponding to the timestamps from the sensor. The data interpolation and resampling apparatus may further include the machine readable instructions to determine values of the signal at a new set of time values. The values of the signal at the new set of time values may be determined based on the timestamps and the values of the signal corresponding to the timestamps by using a frequency domain approach that uses a phase shift to compensate for a time offset between the new set of time values and the timestamps. The frequency domain approach may further determine a time-domain stretch or contraction needed to compensate for a difference in actual and desired sampling frequencies. The data interpolation and resampling apparatus may further include a processor to implement the machine readable instructions.

[0012] The data interpolation and resampling apparatus and the method for data interpolation and resampling provide for efficient processing of large amounts of data with limited computational resources. In addition, the apparatus and method disclosed herein provide for combining of the interpolation and resampling operations with other data processing operations, such as filtering or correlation in the frequency domain, without the need to perform additional forward and inverse transforms. The apparatus and method disclosed herein may also be applied to seismic surveys that do not use synthetically generated stimuli, and allow for frequency corrections with fine granularity.

[0013] Figure 1 illustrates an architecture of a data interpolation and resampling apparatus 100, according to an example. Referring to Figure 1 , the apparatus 100 is depicted as including a signal interpolation and resampling module 101 to receive a vector of timestamps and a value of a band-limited signal at the timestamps from sensors 102, which may be denoted as sensors 1-N. The signal interpolation and resampling module 101 may determine values of the signal at a new set of time values. The values of the signal at the new set of time values may be output as an interpolated and resampled signal 103.

[0014] The modules and other components of the apparatus 100 that perform various other functions in the apparatus 100, may comprise machine readable instructions stored on a non-transitory computer readable medium. In addition, or alternatively, the modules and other components of the apparatus 100 may comprise hardware or a combination of machine readable instructions and hardware.

[0015] As described herein, data from the sensors 102 may be tagged with appropriate timestamps, and synchronized in post processing. For example, each of the sensors 102 may receive a signal from a central timing source (e.g., a GPS) and include a local timing source (e.g., a quartz clock). The GPS signal may be used to distribute a timing reference throughout the network including the sensors 102. The sampled data recorded by the sensors 102 may be aligned according to this timing reference of the central timing source. For example, by comparing the local quartz signal (which may slowly vary) with the reference GPS signal, a post processing determination can be made on the deviation of the local quartz signal from the reference GPS signal. For example, the timestamps may be generated at predetermined intervals (e.g., 2.00000 milliseconds apart) that may differ from the interval (e.g., 1.99999 milliseconds apart) of the sampled signal for a sensor 102, or different intervals (e.g., 2.00001 milliseconds apart, etc.) for the sampled signals for different sensors 102.

[0016] The signal interpolation and resampling module 101 may receive a vector of timestamps t_s[m] and a value of a band-limited signal x[m] at the timestamps from each of the sensors 102. The signal interpolation and resampling module 101 may use the timestamps t_s[m] and the value of the band-limited signal x[m] to determine values of the signal at a new set of time values, which may be denoted as tj[m]. For example, signal interpolation and resampling module 101 may use timestamps t_s[m] (e.g., 2 milliseconds apart) and the value of the band- limited signal x[m] to determine values of the signal at a new set of time values (e.g., 1 millisecond apart, 3 milliseconds apart, etc.). The sensor 102 may have fewer or more timestamps compared to the number of data samples. For the case where the sensor 102 has fewer timestamps compared to the number of data samples, the additional timestamps, so that a timestamp is provided for each data sample, may be derived mathematically using the available timestamps. For example, linear interpolation may be used between the available timestamps to derive any missing timestamps. For the case where the sensor 102 has more timestamps compared to the number of data samples, the additional data samples, so that data sample is provided for each timestamp, may be derived mathematically using the available data samples. The band-limited signal x[m] may be considered a short sequence corresponding, for example, to a single seismic shot. The band-limited signal x[m] may include a length N, and may be indexed in an interval m = -Ni , . . . , Nz, where Λ/ι and N₂ may be denoted as follows:

Ni = (N - l)/2] Equation (1)

A¾ = l(N - 1)/2J Equation (2)

For Equations (1) and (2), L J and H respectively denote the floor and ceiling functions. During the time of the seismic shot, the sampling clock frequency f_s(t) may be considered to be constant, and the desired sampling clock frequency f_d(t) may also be considered to be constant. The constant sampling clock frequency f_s(t) and desired sampling clock frequency f_d(t) may be modeled such that: t_s [m] = 77? / _s + ( s Equation (3) t_d[m] = m/f_d + ο,ι Equation (4)

Equations (3) and (4) may be considered true for all m corresponding to the seismic shot. Further, for Equations (3) and (4), f_s and f_d may be respectively denoted as the actual and desired sampling frequencies during the shot, and Φ₃ and _ά may be denoted as time offsets.

[0017] The actual sampling frequency f_s may be computed using the

timestamps, for example, as follows: fs = U - i)/(t_s[j] - t_s[i]) _{Equatjon (5)}

Using Equation (5), the actual sampling frequency f_s may be computed for some /≠ j. According to another example, the timestamp values may be fitted over an interval with a line using, for example, a least squares approach to obtain an estimate of the sampling frequency using the reciprocal of the slope of the line.

The desired sampling time sequence {*rf[^m3}>» may be considered to be a design parameter, and thus, f_d may also be considered to be a design parameter. The actual and desired sampling frequencies f_s and f_d, respectively, may differ by a few parts per million, and <p_s and φ_£ may differ by a few milliseconds, or a fraction of a millisecond.

[0018] The time offset correction may be computed as follows:

11 7 ,

Φά - Φβ = td[m] - t₈ [n] + - - - -

Jd Equation (6)

Based on the Equations (1) - (6), the signal interpolation and resampling module 101 may interpolate and resample a given data sequence at desired sampling times using a frequency domain approach. The interpolation and resampling operations for the signal interpolation and resampling module 101 may be expressed as follows:

Js m=— j _vVj

Equation (7) x[m] ∑ X[k] exp ( j^'¾m ) for m =—Λ^Γχ, . . . , JV₂

^Λ

^{j 7} Equation (8)

Equation (7) may be considered analogous to a discrete Fourier transform (DFT), where the factor fd/f_* introduces the time-domain stretch or contraction needed to compensate for the difference in sampling frequencies. Further, the term Ψ<ι - _s in the exponent of Equation (7) may introduce a phase shift that compensates for the needed time offset.

[0019] The desired resampled sequence for Equation (8) may be output as the interpolated and resampled signal 103. Equation (8) may be considered to be an inverse discrete Fourier transform (IDFT), which implies that the sequence {^χίΜ} for Equation (7) is the DFT of the desired resampled sequence {·¾[?»]} for Equation (8). Therefore, additional processes, such as filtering or time-domain correlation (with, for example, the vibroseis sweep signal which sweeps through a frequency range) may be efficiently implemented in the frequency domain by multiplying i^xM} for Equation (7) with the proper frequency domain signal (e.g., filter frequency response, or conjugate of the vibroseis sweep signal spectrum) without the need of an additional DFT or fast Fourier transform (FFT) step.

[0020] Direct computation of Equations (7) and (8) may have an 0(N²) computational complexity. However, the operations for Equations (7) and (8) may be efficiently implemented, for example, by using the Chirp z-Transform (CZT) and FFT techniques, which provide an 0(N log/V) computational complexity implementation for Equations (7) and (8). [0021] In order to efficiently implement Equations (7) and (8) to respectively compute {^xi^k)} and {x[m]}, Equation (7) may be rewritten as follows:

Equation (9)

Equation (10)

=Φ[λ] - X[k] Equation (11)

For Equation (11), the term Φ[&] introduces the phase shift necessary to compensate for the offset (φ_ύ - ₃) . As a result, X[k] coincides with i^xl^k]} when (Φ_ά— φ,) = 0. Because of the factor * in the exponent inside the sum in Equation (10), the FFT may not be used to directly compute X[k\. However, X[k] may be computed using the CZT as follows:

Equation (12)

Equation (13) Equation (14)

Where: y[m] = ι exp -j-rr - and = exp j— -

Equation (15)

[0022] The convolution ? [Α- - m] may be implemented using the

FFT. By noting that in the previous sum, the index of the z sequence ranges from -A/i - N₂ to Λ/ι + N2, the sequence length may be at least 2Ni + 2N₂ + 1. For integer constants i > + N₂ and M₂≥ Λ/ι + Λ/₂, Mi and M₂ may be chosen such that i + M₂ + 1 is a power of 2, to simplify the implementation of the FFT. Further y[^m] and ^:t^mi may be defined as follows:

Equation (16)

:[m] -J i - iV₂≤ m≤ i + iV₂

m 0 -Mi≤ m <— Λ^*ι - i¥₂

0 Ni + N₂ < m M₂ Equation (17)

Further, the corresponding DFTs are as follows:

Equation (18)

Equation (19)

The DFTs for Equations (18) and (19) may be specified for k = -M_1: . . . , M₂, which may be computed efficiently using the FFT. Further, by setting ^WW ⁼ -H^ fe] f_or k = -M . . . , M₂, the convolution may be computed as follows:

M .

2 Α'?η

w\m\ W[k} e p [ j-

+ M₂ + 1

Equation (20)

∑ y[m]z[k - in) = w[k] , for /,· = -Λ^ N₂

Equation (21)

For Equations (20) and (21), the sequence w may be efficiently computed using the inverse fast Fourier transform (IFFT).

[0023] As a result, ^ M may be determined as follows:

Equation (22)

For Equation (22), since the FFT and IFFT has computational complexity

0((Mi + M₂ + 1) log(Aii + M₂ + 1)) = 0(N log N), _and the phase shifts in Equations (11) and (14) have computational complexity O(N), the computation of has complexity 0(N logN). Moreover, since Equation (8) can be implemented with an IFFT with complexity 0(N logN), the frequency domain resampling approach of the signal interpolation and resampling module 101 has complexity O(N logN).

[0024] The signal interpolation and resampling module 101 , which uses FFT and IFFT processes, may benefit from operating over sequences whose lengths are powers of 2, or powers that can be factorized with small factors. In such cases, the sequences of length N = 2 may be used for some integer K. Further, zeros may be appended to the original sequence {x[m] in order to satisfy the length requirements, which can be removed after resampling.

[0025] The processes implemented by the signal interpolation and resampling module 101 may be combined with other processing steps in the frequency domain. For example, multiplication in the frequency domain corresponds to circular convolution in the time domain. When convolution is needed, the signal and filter sequences may be appropriately appended with zeros. For the sequence x[m] of length n, x[m] may be indexed in the range m = -Hi , . . . , n₂, where "i ⁼ Γ(« - i)/²l and 2 ^{^} L( ^_ !)/²! Further the sequence y[m] may be indexed in the range ^m— ^~h , and I may be defined such that I = + h + 1.

[0026] The operation of filtering the sequence x with a filter with impulse response y may be implemented by first appending / zeros at the beginning and end of the sequence x to obtain a new sequence χ of length n + 2l. Next, rii + I— h zeros may be appended at the beginning of the sequence y, and 7i2 + I— zeros may be appended at the end of the sequence y to obtain a new sequence y . Additional zeros may be added to both sequences for a power of 2 length.

[0027] The resampled spectrum of A- may be computed using Equation (7). The frequency response of the filter y may be obtained by computing the DFT (FFT) of y over the range ~ⁿi— · · · , ?½ ^~M. The corresponding spectra may be given by X and f . The filtered sequence in the time domain may be obtained by multiplying the spectrum of the resampled signal with the frequency response of the filter, and computing the inverse DFT as follows: m, for m -r i 11-2

n + 21

J . ~ ·^> · · · '^■" " Equation (23)

The correlation operation with a (real) signal {y[m]}^ll may be performed by first appending zeros to the sequences, and obtaining the sequences ^χ and y over the range— 7?. χ - / n + 1. Additional zeros may be added to both sequences for a power of 2 length. When both x and y need to be resampled, Equation (7) may be used to compute the resampled spectra, for example, with different parameters, to obtain X and Ϋ. For correlation, X may be multiplied with Ϋ*, where '*' denotes the complex conjugate. The correlated signal in the time domain may be obtained by computing the inverse DFT as follows:

^H = ^ ^ ∑ Μ «Φ

^" Α·=-,»,-⁽ ^ r¾) ^{for m} =→i - i "i + ⁱ

Equation (24)

For Equations (23) and (24), the inverse DFT may be computed using the IFFT.

[0028] Figures 2 and 3 respectively illustrate flowcharts of methods 200 and 300 for data interpolation and resampling, corresponding to the example of the data interpolation and resampling apparatus 100 whose construction is described in detail above. The methods 200 and 300 may be implemented on the data interpolation and resampling apparatus 100 with reference to Figure 1 by way of example and not limitation. The methods 200 and 300 may be practiced in other apparatus.

[0029] Referring to Figure 2, for the method 200, at block 201 , timestamps for a sensor disposed in a network may be received. For example, referring to Figure 1 , the signal interpolation and resampling module 101 may receive the timestamps t_s[m] for a sensor 102 disposed in a network. The network may also include a plurality of sensors disposed across the network.

[0030] At block 202, values of a signal corresponding to the timestamps may be received from the sensor. For example, referring to Figure 1 , the signal interpolation and resampling module 101 may receive values of the signal x[m] corresponding to the timestamps from the sensor 102.

[0031] At block 203, values of the signal at a new set of time values may be determined based on the timestamps and the values of the signal corresponding to the timestamps by using a frequency domain approach that uses a phase shift to compensate for a time offset between the new set of time values and the timestamps. For example, referring to Figure 1 , the signal interpolation and resampling module 101 may determine values of the signal at a new set of time values td[m] based on the timestamps t_s[m] and the values of the signal corresponding to the timestamps t_s[m] by using a frequency domain approach (e.g., see Equations (7) and (8)) that uses a phase shift to compensate for a time offset between the new set of time values tdm] and the timestamps t_s[m]. Using the frequency domain approach may further include determining a time-domain stretch or contraction needed to compensate for a difference in actual and desired sampling frequencies. For example, Equation (7) may be considered analogous to a DFT, where the factors f<i/fs introduce the time-domain stretch or contraction needed to compensate for the difference in sampling frequencies. Using the frequency domain approach may further include determining an exponent of the phase shift between the new set of time values and the timestamps. For example, the term '·^''</ - Φ.<< in the exponent of Equation (7) may introduce a phase shift that compensates for the needed time offset.

[0032] Referring to Figure 3, for the method 300, at block 301 , the signal interpolation and resampling module 101 may receive the timestamps t_s[m] for a sensor 102 disposed in a network.

[0033] At block 302, the signal interpolation and resampling module 101 may receive values of the signal x[m] corresponding to the timestamps t_s[m] from the sensor 102.

[0034] At block 303, in order to determine the desired resampled sequence { [m]} for Equation (8), the desired sampling frequency and the desired phase 4>_d may be received to determine the new set of time values tdm] by Equation (4). The actual sampling frequency f_s may be determined using Equation (5). Further, the time offset correction may be determined using Equation (6). In order to determine the sequence {^xl^k } for Equation (7), block 304 may be used as a function call to determine the sequence ]}.

[0035] At block 304, values for the band-limited signal x[m], the desired sampling frequency , the actual sampling frequency f_s, the desired phase <p_d , and the actual phase ¾. may be received. At block 304, y\^m\ and ^z\^m\ may be determined using Equation (15). The convolution - m)

may be implemented using the FFT Further v[^m] and ⁷! may be determined using Equations (16) and (17). The corresponding DFTs and Z[k) may be determined using Equations (18) and (19). The DFTs for Equations (18) and (19) may be specified for k = -Mi_, . . . , M₂, which may be determined efficiently using the FFT. Further, by setting M ⁼ ^Ι ^^'] for k = -M^ . . . , M₂, the convolution may be determined using Equation (20). The sequence w may be efficiently determined using the IFFT using Equation (20). As a result, _mav e_e determined using Equation (22). The ^Λ i'^vJ determined using Equation (22) may be returned to block 303, which then determines the desired resampled sequence {r[?n]} for Equation (8). The desired resampled sequence {.%?]} for Equation (8) may be output as the interpolated and resampled signal 103. Equation (8) may be considered to be an IDFT, which implies that the sequence {^χ[ ]} for Equation (7) is the DFT of the desired resampled sequence { «]} for Equation (8).

[0036] Figure 4 shows a computer system 400 that may be used with the embodiments described herein. The computer system 400 may represent a generic platform that may include components that may be in a server or another computer system. The computer system 400 may be used as a platform for the apparatus 100. The computer system 400 may execute, by a processor or other hardware processing circuit, the methods, functions and other processes described herein. These methods, functions and other processes may be embodied as machine readable instructions stored on computer readable medium, which may be non-transitory, such as, for example, hardware storage devices (e.g., RAM (random access memory), ROM (read only memory), EPROM (erasable, programmable ROM), EEPROM (electrically erasable, programmable ROM), hard drives, and flash memory).

[0037] The computer system 400 may include a processor 402 that may implement or execute machine readable instructions performing some or all of the methods, functions and other processes described herein. Commands and data from the processor 402 may be communicated over a communication bus 404. The computer system 400 may also include a main memory 406, such as, for example, a random access memory (RAM), where the machine readable instructions and data for the processor 402 may reside during runtime, and a secondary data storage 408, which may be non-volatile and stores machine readable instructions and data. The memory and data storage may be examples of computer readable mediums. The memory 406 may include a data interpolation and resampling module 420 including machine readable instructions residing in the memory 406 during runtime and executed by the processor 402. The data interpolation and resampling 420 may include the modules of the apparatus 100 shown in Figure 1.

[0038] The computer system 400 may include an I/O device 410, such as, for example, a keyboard, a mouse, a display, etc. The computer system 400 may include a network interface 412 for connecting to a network. Other known electronic components may be added or substituted in the computer system 400.

[0039] While the embodiments have been described with reference to examples, various modifications to the described embodiments may be made without departing from the scope of the claimed embodiments.

Claims

What is claimed is:

1. A method for data interpolation and resampling, the method comprising: receiving timestamps t_s[m] for a sensor disposed in a network, where m is an interval for the received timestamps t_s[m]; receiving values of a signal x[m] corresponding to the timestamps t_s[m] from the sensor; and determining, by a processor, values of the signal x[m] at a new set of time values tdm] based on the timestamps t_s[m] and the values of the signal x[m] corresponding to the timestamps t_s[m] by: using a frequency domain approach that uses a phase shift φ_ά - #_s to compensate for a time offset between the new set of time values tdm] and the timestamps t_s[m], where <p_d is a desired sampling phase, and <p_s is an actual sampling phase, and applying a time-domain stretch or contraction needed to compensate for a difference in an actual sampling frequency f_s and a desired sampling frequency f_d.

2. The method of claim 1 , wherein the signal x[m] is band-limited.

3. The method of claim 1 , wherein the network includes a plurality of sensors disposed across the network.

4. The method of claim 1 , wherein applying the time-domain stretch contraction further comprises: using a factor fu/fs to introduce the time-domain stretch or contraction.

5. The method of claim 1 , wherein using the frequency domain approach further comprises: determining an exponent of the phase shift Φ_α - <¾. between the new set of time values t^m] and the timestamps t_s[m].

6. The method of claim 1 , wherein determining the values of the signal x[m] at the new set of time values t_d[m] includes an 0(N log/V) computational complexity, where N is a length of the signal x[m].

7. The method of claim 1 , wherein determining the values of the signal x[m] at the new set of time values tj[m] further comprises: using a Chirp z-Transform (CZT) technique to provide an 0(N log/V) computational complexity.

8. The method of claim 1 , further comprising: implementing at least one of filtering and time-domain correlation in a frequency domain without applying additional discrete Fourier transform (DFT) or fast Fourier transform (FFT) techniques for the implementation.

9. A data interpolation and resampling apparatus comprising: a memory storing machine readable instructions to: receive timestamps f_s[m] for a sensor disposed in a network, where m is an interval for the received timestamps t_s[m]; receive values of a signal x[m] corresponding to the timestamps t_s[m] from the sensor; and determine values of the signal x[m] at a new set of time values tdm] based on the timestamps t_s[m] and the values of the signal x[m] corresponding to the timestamps t_s[m] by using a frequency domain approach that uses a phase shift φ_ά - <p_£ to compensate for a time offset between the new set of time values ¾[m] and the timestamps t_s[m], where <p_d is a desired sampling phase, and <p_s is an actual sampling phase; and a processor to implement the machine readable instructions.

10. The apparatus of claim 9, wherein the network includes a plurality of sensors disposed across the network.

11. The apparatus of claim 9, wherein using the frequency domain approach further comprises: determining an exponent of the phase shift φ_ά - <p_g between the new set of time values t_d[m] and the timestamps t_s[m].

12. The apparatus of claim 9, wherein the machine readable instructions to determine values of the signal x[m] at the new set of time values tdm] further comprise: applying a time-domain stretch or contraction needed to compensate for a difference in an actual sampling frequency f_s and a desired sampling frequency f_d.

13. The apparatus of claim 12, wherein applying the time-domain stretch or contraction further comprises: using a factor f<i/.fs to introduce the time-domain stretch or contraction.

14. A non-transitory computer readable medium having stored thereon machine readable instructions for data interpolation and resampling, the machine readable instructions when executed cause a computer system to: receive timestamps t_s[m] for a sensor disposed in a network, where m is an interval for the received timestamps t_s[m]\ receive values of a signal x[m] corresponding to a plurality of the timestamps t_s[m] from the sensor; and determine, by a processor, values of the signal x[m] at a new set of time values t_<£m] based on the timestamps t_s[m] and the values of the signal x[m] corresponding to the plurality of the timestamps t_s[m] by: using a frequency domain approach that uses a phase shift Φ_ά - <p_s to compensate for a time offset between the new set of time values t_d[m] and the timestamps t_s[m], where φ_ό is a desired sampling phase, and <p_s is an actual sampling phase, and applying a time-domain stretch or contraction needed to compensate for a difference in an actual sampling frequency f_s and a desired sampling frequency f_d by using a factor fd/fs to introduce the time-domain stretch or contraction.

15. The non-transitory computer readable medium of claim 14, wherein using the frequency domain approach further comprises: determining an exponent of the phase shift φ_ά - φ₃ between the new set of time values tj[m] and the timestamps t_s[m].