SYSTEMS AND METHODS FOR USE IN RECONSTRUCTION OF ANALOG SIGNALS

Abstract

Some embodiments relates to the technique, including systems and methods, for use in analog-to-digital conversion (ADC). A sampling system is presented for sampling an input analog signal including a train of pulses of a predetermined shape and allowing a recovery of the degrees of freedom of the signal. The sampling system includes: a kernel for selectively passing components of an input signal, and an integrate and fire time encoding machine (IF-TEM), wherein the kernel has a size of kernel support set being in a predetermined relation with a number of degrees of freedom in the finite rate of innovation signal.

Claims

1. A signal processing system for processing an input analog signal, x(t), comprising a train of L pulses of a predetermined shape, the system comprising a sampling system comprising: a kernel characterized by predetermined size of a kernel support set, , and configured to receive the input analog signal and generate a kernel-filtered signal, y(t); a sampler configured as an integrate and fire time encoding machine, IF-TEM, being parametrized by one or more predetermined characteristic parameters comprising one or more positive real numbers, said sampler being operable to receive the kernel-filtered signal and produce sampling data indicative of a series of N time-encodings, t.sub.n, of the kernel-filtered signal forming discrete time representation of said analog signal; wherein said size of the kernel support set is in a predetermined relation with a number F of degrees of freedom in the input signal defined by characteristic parameters of the input signal.

2. (canceled)

3. The system of claim 1, characterized by at least one of the following: said characteristic parameters defining the degrees of freedom comprise amplitude and time delays of said train of L pulses forming the input signal, and said characteristic parameters defining the degrees of freedom comprise amplitudes and time parameters of symmetric and anti-symmetric parts of the pulses.

4. (canceled)

5. The system of claim 1, wherein said kernel is configured with a minimal transmission coefficient for zero frequency component of the input signal as compared to transmission coefficients for other frequency components to thereby substantially suppress transmission of the zero frequency component of the input signal, thereby enabling noise-resilient reconstruction of the input analog signal.

6. (canceled)

7. The system of claim 1, wherein the IF-TEM comprises: an integrator and a discharge circuit, a comparator, and a switch positioned to receive a signal transmitted from the comparator for initiating reset of the integrator and for switching on the discharge circuit, wherein the discharge circuit comprises a capacitor configured to operate in its linear zone to provide rapid and complete discharge of the integrator.

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13. The system according to claim 1, further comprising a reconstruction system configured to receive data indicative of the time-encodings of the kernel-filtered signal and process said data indicative of the time-encodings of the kernel-filtered signal, said processing comprising: creating data indicative of vector representation, y, of the kernel-filtered signal by utilizing data indicative of the one or more characteristic parameters of the IF-TEM, and by utilizing data indicative of the kernel support set, to define a linear relation between said data indicative of vector representation, y, of the kernel-filtered FRI signal and data indicative of a vector representation, {circumflex over (x)}, of Fourier series coefficients of the signal, thereby enabling reconstruction of the input analog signal, x(t).

14. The system according to claim 13, wherein the reconstruction system comprises: an analyzer configured and operable to analyze the data indicative of the time-encodings of the kernel-filtered signal and utilize the data indicative of the characteristic parameters of the IF-TEM to generate data indicative of vector representation, y, of the kernel-filtered signal; a processor configured and operable to utilize said data indicative of the time-encodings of the kernel-filtered signal and said data indicative of the kernel support set and define a linear relation between the data indicative of the vector representation, y, of the kernel-filtered signal and the vector representation, {circumflex over (x)}, of the Fourier series coefficients of the input signal; and a signal reconstructor processor configured and operable to utilize said linear relation to determine, from said data indicative of the vector representation, y, the vector representation, {circumflex over (x)}, of the Fourier series coefficients of the input signal, thereby enabling reconstruction of the analog signal x(t).

15. The system according to claim 14, wherein said signal reconstructor processor comprises an extractor utility configured and operable to process the vector representation, {circumflex over (x)}, of the Fourier series coefficients and extract parameters of the L pulses forming the input analog signal.

16. The system according to claim 13, wherein said linear relation comprises a characteristic matrix having pseudoinverse representation thereof and being configured for describing a relation between said data indicative of vector representation, y, of the kernel-filtered FRI signal and data indicative of a vector representation, {circumflex over (x)}, of Fourier series coefficients of the signal.

17. The system according to claim 16, characterized by at least one of the following: said processor comprises a matrix creator utility configured and operable to process the data indicative of the time encodings of the kernel-filtered signal utilizing said data indicative of the kernel support set to create said characteristic matrix and is configured and operable to utilize said characteristic matrix to process said data indicative of the vector representation, y, of the kernel-filtered signal by applying thereto the pseudoinverse representation of said characteristic matrix to obtain the vector representation, {circumflex over (x)}, of the Fourier series coefficients of the input signal, and said kernel is configured with a minimal transmission coefficient for zero frequency component of the input signal as compared to transmission coefficients for other frequency components to thereby substantially suppress transmission of the zero frequency component of the input signal, said kernel support set includes integers symmetric around zero: ={K, . . . , 1,1, . . . , K}; said reconstruction system being configured and operable to create said characteristic matrix being a Vandermonde type invertible matrix having linearly independent columns, said characteristics matrix describing the relation between partial sums vector z of said vector representation y, of the kernel-filtered FRI signal and a vector {circumflex over (z)} associated with said vector representation {circumflex over (x)}, of the Fourier series coefficients of the FRI signal.

18. The system according to claim 13, characterized by at least one of the following: said kernel support set , includes integers symmetric around zero: ={K, . . . , 1, 0, 1, . . . , K}; said kernel is configured with a minimal transmission coefficient for zero frequency component of the input signal as compared to transmission coefficients for other frequency components to thereby substantially suppress transmission of the zero frequency component of the input signal, said kernal support set includes integers symmetric around zero: ={K, . . . , 1,1, . . . , K}; the reconstruction system is configured and operable to determine the parameters of the input signal with a reconstruction error not exceeding 25 dB, for the input signal being sampled by the sampling system at asynchronous sampling rates of at least 10 times lower than the Nyquist rate.

19. (canceled)

20. The system according to claim 16, wherein said kernel is configured with a minimal transmission coefficient for zero frequency component of the input signal as compared to transmission coefficients for other frequency components to thereby substantially suppress transmission of the zero frequency component of the input signal, said kernel support set includes integers symmetric around zero: ={K, . . . , 1,1, . . . , K}; said reconstruction system being configured and operable to create said characteristic matrix being a Vandermonde type invertible matrix having linearly independent columns, said characteristic matrix describing the relation between partial sums vector z of said vector representation y, of the kernel-filtered FRI signal and a vector {circumflex over (z)} associated with said vector representation {circumflex over (x)}, of the Fourier series coefficients of the FRI signal, the reconstruction system being configured and operable to carry out the following: utilize said vector representation, y, of the kernel-filtered FRI signal to generate the partial sums vector {circumflex over (z)}: utilize the partial sums vector z and said characteristic matrix to determine the vector {circumflex over (z)} being in a predetermined relation with the vector representation, {circumflex over (x)}, of the Fourier series coefficients of the analog signal; and determine the vector representation, {circumflex over (x)}, from said vector z by selecting predetermined elements of the vector {circumflex over (z)}.

21. (canceled)

22. The system of claim 17, wherein said kernel is configured with a minimal transmission coefficient for zero frequency component of the input signal as compared to transmission coefficients for other frequency components to thereby substantially suppress transmission of the zero frequency component of the input signal, said kernel support set includes integers symmetric around zero, ={K, . . . , 1,1, . . . , K}; said reconstruction system being configured and operable to create said characteristic matrix being a Vandermonde type invertible matrix having linearly independent columns, said characteristic matrix describing the relation between partial sums vector z of said vector representation y, of the kernel-filtered FRI signal and a vector {circumflex over (z)} associated with said vector representation {circumflex over (x)}, of the Fourier series coefficients of the FRI signal, wherein each successive component of the vector of partial sums, being a successive partial sum, is determined as a sum of a preceding partial sum and a linear transform, y.sub.n, of a respective difference between two consecutive time encodings t.sub.n+1 and t.sub.n defined as: y.sub.n=b(t.sub.n+1t.sub.n)+.

23. The system of claim 15, wherein said extractor utility is configured and operable to apply spectral analysis to the vector representation, {circumflex over (x)}, of the Fourier series coefficients to thereby extract the parameters of the L pulses of the input signal.

24. (canceled)

25. The system of claim 1, characterized by at least one of the following: the kernel support set, is configured with a number K of non-zero frequency components in the input signal, said number K being defined to allow determination of a minimum number of Fourier series coefficients (FSC) of the input signal allowing a unique reconstruction of the input signal; the kernel comprises any one of the following, a sinc function kernel, a sum-of-sincs (SoS) kernel, a sum-of-modulated spline kernel, a polynomial-reproducing kernel, or an exponential-reproducing kernel; the kernel is compactly supported.

26. The system of claim 25, wherein the kernel support set, , is configured with a number K of non-zero frequency components in the input signal, said number K being defined to allow determination of a minimum number of Fourier series coefficients (FSC) of the input signal allowing a unique reconstruction of the input signal, said kernel being configured with a minimal transmission coefficient for zero frequency component of the input signal as compared to transmission coefficients for other frequency components to thereby substantially suppress transmission of the zero frequency component of the input signal, said support set of indices being chosen to exclude zero and include integers symmetric around zero: $=\left\{-K,.Math.,-1,1,.Math.,K\right\}.$

27. (canceled)

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31. The system of claim 13, wherein said kernel support set, , includes integers symmetric around zero: ={K, . . . , 1, 0, 1, . . . , K}, the kernel being configured with the support set satisfying a condition that ||2 L, where 2 L is the number F of the degrees of freedom in the input signal having said L pulses of the predetermined shape defining L amplitudes and L time delays characterizing the input signal.

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39. The system of claim 1, wherein said input signal has a predetermined time characteristic T; the kernel is configured with the support set satisfying a condition that ||L; said L pulses are of the predetermined shape such that the input signal is characterized by L amplitudes and L time delays of said L pulses defining the number F=2 L of degrees of freedom in the input signal, and the time delays are arranged in accordance with a predetermined time grid; and the IF-TEM is configured to sample a minimum value of the number N of the time-encodings, N2K+2, within the predetermined time characteristic T of the input signal, wherein K satisfies a condition that KL, to thereby enable reconstruction of the L amplitudes and the L time delays of the L pulses, and wherein the time characteristic T is one of the following: a period T of a periodic input signal in the form of said train of L pulses; or a time interval T defined by a finite time interval [0, T) of time delays of a finite train of the L pulses of a nonperiodic input signal.

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44. An analog to digital converter (ADC) comprising the signal processing system of claim 1.

45. A signal reconstruction system for reconstructing an analog signal, x(t), in the form of a train of L pulses of a predetermined shape, from data indicative of discrete time representation of said analog signal generated by the signal processing system ofan claim 1, the signal reconstruction system being configured and operable to carry out the following: utilizing input data comprising data indicative of the kernel support set used in generation of said kernel-filtered signal and data indicative of the characteristic parameters of the IF-TEM, to analyze said series of N time-encodings and create a linear relation between data indicative of vector representation, y, of the kernel-filtered signal and data indicative of a vector representation, {circumflex over (x)}, of Fourier series coefficients of the signal, thereby enabling reconstruction of the input analog signal, x(t).

46. A signal reconstruction system for reconstructing an analog signal, x(t), formed by a train of L pulses of a predetermined shape, the system being configured and operable to carry out the following: receiving data indicative of discrete time representation of said analog signal being indicative of a series of N time-encodings, t.sub.n, obtained by an integrate and fire time encoding mechanism (IF-TEM), applied to a kernel-filtered signal, y(t); utilizing input data comprising data indicative of a kernel support set used in generation of said kernel-filtered signal and data indicative of one or more characteristic parameters of the IF-TEM, to analyze said data indicative of the series of N time-encodings and create a linear relation between data indicative of vector representation, y, of the kernel-filtered signal and data indicative of a vector representation, {circumflex over (x)}, of Fourier series coefficients of the signal, thereby enabling reconstruction of the input signal, x(t).

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Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0052] The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

[0053] In order to better understand the subject matter that is disclosed herein and to exemplify how it may be carried out in practice, embodiments will now be described, by way of non-limiting examples only, with reference to the accompanying drawings:

[0054] FIG. 1 schematically illustrates a known in the art kernel-based sampling framework, where a signal x(t) is first filtered by a sampling kernel s(t) and then instantaneous uniform samples are measured at a sub-Nyquist rate;

[0055] FIG. 2A schematically illustrates a signal processing system according to some embodiments of the present disclosure;

[0056] FIG. 2B exemplifies an IF-TEM with a spike trigger reset where the IF-TEM is parametrized by one or more of the following positive real numbers: bias, b, scaling factor x, and reset-associated threshold ;

[0057] FIG. 3A illustrates, by way of block diagram, the configuration and operation of a signal reconstruction system according to the present disclosure;

[0058] FIGS. 3B and 3C illustrate, by way of block diagrams, two specific not limiting examples of the configuration and operation of the signal reconstruction system of FIG. 3A;

[0059] FIG. 4 shows an example of IF-TEM hardware sampling including the IF-TEM input signal y(t) (blue), the integrator output (green), and the IF-TEM output time instances (red);

[0060] FIGS. 5A to 5F shows sampling and reconstruction of a stream of Dirac impulses using TEM by applying the SoS kernel, according to the present disclosure, wherein both and are chosen uniformly at random over (0,1); FIGS. 5A to 5C show the input signal and its reconstruction for L=3 (FIG. 5A), L=5, (FIG. 5B), and L=10 (FIG. 5C), respectively. FIGS. 5D to 5F show the filtered signal y(t) and the time instants t.sub.n for L=3 (FIG. 5D), L=5, (FIG. 5E), and L=10 (FIG. 5F), respectively;

[0061] FIGS. 6A and 6B show sampling and reconstruction of stream of pulses using TEM by applying the SoS kernel, according to the present disclosure, wherein FIG. 6A shows the input signal and its reconstruction for L=3, and FIG. 6B shows the filtered signal y(t) and the time instants t.sub.n for L=3;

[0062] FIGS. 7A and 7B show examples of (t) and r(t) when the matrix A has large condition number, according to the present disclosure, wherein in FIG. 7A condition numbers of A is 30 and of B is 3, and in FIG. 7B condition numbers of A is 3000 and of B is 5.

[0063] FIG. 8 shows the average condition number of matrices A and B as a function of L, according to the present disclosure;

[0064] FIG. 9 shows a performance comparison of Algorithm 1 and Algorithm 2 in the presence of continuous-time white Gaussian noise with zero mean and variance 0.001, according to the present disclosure;

[0065] FIG. 10 shows a comparison of Algorithm 1 and Algorithm 2 with noisy input signal;

[0066] FIGS. 11A to 11D show a comparison of with zero (Algorithm 1, FIG. 11A), without zero (Algorithm 2, FIG. 11B), and estimation of signal parameters via rotational invariance technique (ESPRIT, FIG. 11C) approaches for off-grid time delays with perturbation in the time encodings, and FIG. 11D shows one-dimensional MSE plots for K=2 L+1 and K=5 L;

[0067] FIG. 12 shows a comparison of resolutions of the with zero (Algorithm 1) and without zero (Algorithm 2) approaches when the difference between the time delays, .sub.2-.sub.1, is varied and the time encodings are perturbed with =0.008 (low noise) and =0.014 (high noise);

[0068] FIG. 13 shows a comparison of firing rates, FR-A and FR-B denoting mean firing rates of with zero and without zero approaches, respectively. Maximum and minimum rates are denoted by maxFR and minFR, respectively; the rate of innovation (Rol) is shown with dashed line;

[0069] FIG. 14 is an illustration of intersection of a trigonometric polynomial (t) and a straight line r(t);

[0070] FIGS. 15 to 17 schematically present examples of IF-TEMs according to the present disclosure;

[0071] FIG. 18; schematically illustrates an ADC according to the present disclosure;

[0072] FIGS. 19A and 19B present an example of a perfect reconstruction of an FRI signal from IF-TEM measurements using sampling kernel without zero and partial summation, wherein FIG. 19A shows the input signal and its reconstruction for L=5, and FIG. 19B shows the filtered signal y(t) and the time instants t.sub.n;

[0073] FIG. 20 presents, according to a present disclosure, an average condition number of matrices B and C as a function of the number of FRI pulses L.

[0074] FIGS. 21A and 21B present, according to a present disclosure, a comparison of approaches in Algorithm 2 and Algorithm 3 for off-grid time delays with perturbation in the time encodings;

[0075] FIG. 22 presents, according to a present disclosure, a 1 MHz filter Bode plot, the filter being configured to remove the zeroth frequency, wherein the magnitude (in blue) and phase (on red) are plotted on a logarithmic frequency scale;

[0076] FIG. 23 presents, according to a present disclosure, the FRI-TEM hardware prototype containing a signal generator, a sampling kernel and an IF-TEM sampler;

[0077] FIG. 24 presents, according to a present disclosure, a block diagram of the analog board;

[0078] FIG. 25 presents an IF-TEM hardware board;

[0079] FIG. 26 schematically shows an example of an integrator circuit wherein the hardware implementation includes an operational amplifier, a resistor R and a capacitor C;

[0080] FIGS. 27A and 27B present, according to a present disclosure, an input signal x(t) (green), BPF output y(t) (yellow), and the IF-TEM output resulting in 19 samples (blue) (FIG. 27A) and sampling and reconstruction using IF-TEM hardware: the input signal x(t) (blue) and its reconstruction (red) (FIG. 27B);

[0081] FIGS. 28A and 28B present, according to a present disclosure an input analog signal x(t) (green), BPF output y(t) (yellow), and the IF-TEM output resulting in 19 samples (blue) (FIG. 28A) and sampling and reconstruction using IF-TEM hardware: the input signal x(t) (blue) and its reconstruction (red) (FIG. 28B);

[0082] FIGS. 29A and 29B present, according to a present disclosure, an input analog signal x(t) (yellow), BPF output y(t) (green), and the IF-TEM output resulting in 21 samples (blue) (FIG. 29A) and sampling and reconstruction using IF-TEM hardware: the input signal x(t) (blue) and its reconstruction (red) (FIG. 29B).

[0083] FIGS. 30A and 30B present, according to a present disclosure, an input analog signal x(t) (green), BPF output y(t) (yellow), and the IF-TEM output resulting in 22 samples (blue) (FIG. 30A) and sampling and reconstruction using IF-TEM hardware: the input signal x(t) (blue) and its reconstruction (red) (FIG. 30B);

[0084] FIG. 31 presents, according to a present disclosure, a comparison between the reconstruction using the hardware measurements and the simulation;

[0085] FIG. 32 presents, according to a present disclosure, a schematic illustration of a sampling and reconstruction setup for sampling and reconstruction of continuous-time ECG signals;

[0086] FIG. 33 presents, according to a present disclosure, an example of reconstruction results of a single ECG pulse from [6];

[0087] FIGS. 34A and 34B present, according to a present disclosure, IF-TEM sampling and reconstruction example, wherein FIG. 34A shows the sampling mechanism of a filtered ECG pulse by IF-TEM; and FIG. 34B shows reconstructed R waves of the ECG signal;

[0088] FIGS. 35A to 35C present an example of HR monitoring performance for SNR=2 [dB] using various techniques: VPW-FRI from [4](FIG. 35A), IF-TEM from [5](FIG. 35B), and ECG-TEM (FIG. 35C) according to the present disclosure;

[0089] FIGS. 36A to 36D present evaluation of HRM performance of various reconstruction techniques vs. SNR using various statistical metrics: HR RMSE (FIG. 36A), HR MAE (FIG. 36B), HR PCC (FIG. 36C), and HR Success Rate (FIG. 36D).

DETAILED DESCRIPTION OF THE ASPECTS AND EMBODIMENTS

[0090] FIG. 1 illustrates the known in the art FRI sampling scheme in which the signal is first filtered by a sampling kernel to remove a redundancy, and then instantaneous samples are measured at a sub-Nyquist rate.

[0091] As described above, the technique of the present disclosure in one of its aspects provides a novel approach for sampling an input analog signal formed by a train of pulses of a predetermined shape. This approach is aimed at reducing the sampling rate (to the minimal rate enabling appropriate signal recovery) as well as reducing the power consumption. and also allowing robust signal reconstruction at noisy conditions during the sampling. In another aspect, the present disclosure provides a novel approach for the signal reconstructions technique, which enables robust reconstruction even for signals sampled under noisy conditions and provides high-quality (perfect) reconstruction while eliminating need for any simulation procedure.

[0092] Reference is made to FIG. 2A schematically illustrating a signal processing system 100 of the present disclosure. The system 100 includes a sampling system 102 configured and operable for processing an input signal, x(t), being an analog signal (e.g., FRI signal or VPW signal) comprising a train of L pulses of a predetermined shape (e.g., fixed shape or variable shape).

[0093] As also shown in the figure, the sampling system 102 may be directly connected (via wires or wireless signal transmission using any known suitable communication technique and communication protocols) to a signal reconstruction system 180. Alternatively, or additionally, communication of the output of the sampling system to the reconstruction system may be via a remote storage device 104 which is connectable/accessible by the sampling system 102 and reconstruction system 180. Generally, the sampling and reconstruction systems 102 and 180 may be integral within a signal processing system 100.

[0094] The sampling system 102 receives an input analog signal x(t), applies the sampling procedure thereto and outputs data indicative of time-encodings data to of the sampled input signal (i.e., discrete time representation of the input signal). The data indicative of time-encodings data t.sub.z (e.g., the time-encodings themselves or differences between each two consecutive time-encodings) is properly stored in a memory 150 of the system 102 and/or in the storage device 104, and this data is further processed by the reconstruction system 180 to restore the input signal.

[0095] The sampling system 102 includes a kernel (filter) 120 (represented by a function g(t)) characterized by a predetermined size of a kernel support set, , and configured to receive the analog input signal x(t) and generate a kernel-filtered signal, y(t); and a sampler 140 configured as an integrate and fire time encoding machine, IF-TEM, which receives the kernel-filtered signal, y(t), and produces sampling data indicative of a series of time-encodings, {t.sub.n}, being discrete time representation of the analog signal x(t). The IF-TEM is parametrized by predetermined characteristic parameter(s) comprising one or more, e.g., at least three, positive real numbers. According to the present disclosure, the kernel is configured such that a size of the kernel support set is in a predetermined relation with a number F of degrees of freedom in the input signal.

[0096] The input analog signal is typically in the form of a series of L pulses with a known/predetermined shape and can thus be represented by a sum of up to L such pulses, presenting a parametric signal, uniquely defined by a known/predetermined number F of degrees of freedom which in turn is defined by characteristic parameters of the signal, e.g., amplitude and time delay of pulses; or amplitude and time parameters of symmetric and anti-symmetric parts of the pulses.

[0097] In some embodiments, the kernel 120 is further configured with a minimal transmission coefficient for zero frequency component of an input signal as compared to transmission coefficients for other frequency components to thereby substantially suppress transmission of the zero-frequency component of the input signal. This feature of zero suppression enables noise-resilient reconstruction of the input analog signal. This is also specifically shown in FIG. 2A exemplifying kernel function 160 being a band pass filter (BPF) function configured to suppress the zero-frequency component of the input signal.

[0098] As will be described further below, the reconstruction system 180 receives the data indicative of the time-encodings data t.sub.n, as well as kernel-related data KD and IF-TEM related data TD, and properly processes and analyses the time-encoded data utilizing the kernel- and TEM-related data to reconstruct the input signal x(t).

[0099] FIG. 2B shows an embodiment of the present disclosure where the IF-TEM 140 is employing an alternative sampling mechanism which does not require a global clock. Here, the samples are irregularly spaced threshold-based samples. In sampling with a TEM, an analog signal is represented by a set of time instants at which a specific phenomenon is observed; for example, the filtered input signal, or its function crosses a certain threshold. In the embodiment of FIG. 2B the TEM is parametrized by one or more or in some embodiments at least three positive real numbers comprising: a bias b satisfying a condition c<b< where c is a maximum real value of the input signal, a scaling factor , and a threshold . In particular, in the IF-TEM embodiment described in the disclosure, the signal input into the IF-TEM is biased with b to make it positive.

[0100] Reference is made to FIG. 3A showing a block diagram of the configuration and operation of the signal reconstruction system 180 according to the present disclosure. The system 180 includes an analyzer 182, processor 183 and signal reconstructor processor 188. The analyzer 182 is configured as a time-encoding analyzer, which analyzes the data indicative of the time-encodings {t.sub.n}(i.e., the time-encodings themselves and/or differences between two consecutive time-encodings) to generate data indicative of the vector representation, y, of the kernel-filtered signal of the kernel-filtered signal. This analyses utilize the data indicative of the characteristic parameters of the IF-TEM, TD, as will be described further below. The processor 183 is configured and operable to utilize the data indicative of the time-encodings {t.sub.n} and also the data indicative of the kernel support set, KD, and define a linear relation between the data indicative of the vector representation, y, of the kernel-filtered signal and vector representation, {circumflex over (x)}, of the Fourier series coefficients of the input signal x(t). The signal reconstructor 188 is configured and operable to utilize the linear relation to determine the vector representation, {circumflex over (x)}, of the Fourier series coefficients of the input signal from said data indicative of the vector representation, y, and by this reconstruct the analog signal x(t).

[0101] FIGS. 3A and 3B illustrate two examples of the configuration and operation of the above-described signal reconstruction system. The systems of FIGS. 3A and 3B are denoted 180A and 180B, respectively. To facilitate understanding, the same reference numbers are used to identify functionally similar components of the systems shown in all the examples.

[0102] Each of the reconstruction system 180A, 180B is configured to receive data indicative of the time-encodings t.sub.n of the kernel-filtered signal y(t) and process this data by utilizing the kernel-related data (i.e., data indicative of the kernel support set, ,) and data indicative of the characteristic parameter(s) of the IF-TEM (e.g., one or more of or three positive real numbers), to create the linear relation between y and {circumflex over (x)}, and by this reconstruct the analog signal x(t). In these specific examples, the linear relation is in the form of a characteristic matrix M having pseudoinverse representation thereof and describing a relation between the data indicative of vector representation, y, of the kernel-filtered signal y(t) and data indicative of the vector representation, {circumflex over (x)}, of Fourier series coefficients (FSCs) of the analog signal x(t). This enables reconstruction of the input analog signal, x(t), e.g. using spectral analysis.

[0103] As shown in FIG. 3B, the reconstruction system 180A includes analyzer 182; processor 183 including a matrix creator utility 184 and FCSs vector determinator 186; and signal reconstructor 188 including a pulse parameters' extractor utility 189. The analyzer 182 is configured and operable as described above to analyze the data indicative of the time-encodings t.sub.n of the kernel-filtered signal y(t) and utilize data indicative of the characteristic parameter(s) of the IF-TEM, TD, to generate data indicative of vector representation, y, of the kernel-filtered FRI signal. The matrix creator utility 184 is configured and operable to process the data indicative of the time encodings data t.sub.n of the kernel-filtered signal utilizing the data indicative of the kernel support set , KD, to create the characteristic matrix M, e.g. matrix A or B having left-inverse matrices A.sup. or B.sup. as will be described further below. The processor 186 is configured and operable to utilize the so-created characteristic matrix to process the data indicative of the vector representation, y, of the kernel-filtered FRI signal by applying to this vector y pseudoinverse representation (e.g., left-inverse) of the characteristic matrix, A.sup. or B.sup. to obtain the vector representation, {circumflex over (x)}, of the FSCs of the input signal. The extractor utility 189 is configured and operable to process the vector representation, {circumflex over (x)}, of the Fourier series coefficients and extract parameters of the L pulses forming the input signal x(t).

[0104] More specifically, as shown in FIG. 3B, the reconstruction system 180A receives the sampling data comprising the series of time-encodings, {t.sub.n}, of the kernel-filtered signal y(t) being discrete time representation of the analog signal x(t). The time encoding analyzer 182 calculates the vector representation y of the kernel-filtered signal {y.sub.n, n} defined by

[00002]$y_n\stackrel{}{=}{}_{t_n}^{t_{n+1}}y\left(s\right)\mathrm{ds}=-b\left(t_{n+1}-t_n\right)+.$

Here, t.sub.n and t.sub.n+1 are respectively n-th and (n+1)-th time encodings, and b, and are characteristic parameters of the IF-TEM corresponding to, respectively, bias b (c<b<, where c is a maximum real value of the input signal), scaling factor , and threshold .

[0105] It should be noted that although in the description presented here the characteristic parameters of the IF-TEM are represented by three positive real numbers including bias b (c<b<, where c is a maximum real value of the input signal), scaling factor , and threshold , the principles of the technique of the present disclosure is not limited to this specific example. For example, if the signal being reconstructed is a positive signal, there is no need to use a bias b parameter for reconstruction.

[0106] It can be shown that each y.sub.n represents an average value of y(t) (in the following referred to as y(t) for clarity of presentation) in the interval between two consecutive times t.sub.n and t.sub.n+1. The matrix creator 184 utilizes the knowledge about the frequencies transmitted by the chosen sampling kernel (i.e., the kernel support set data contained in KD) to analyze the received data indicative of the time encodings and define a characteristic matrix M having inverse representation thereof M.sup.+ (e.g., matrix A having left-inverse representation), connecting/describing a relation between data indicative of the vector representation, y, of the kernel-filtered signal y(t) and data indicative of vector representation, {circumflex over (x)}, of FSCs of the analog signal. This matrix depends on the specific kernel and kernel support set as will be described in detail further below.

[0107] The processor 186 defines the FSCs vector as {circumflex over (x)} and utilizes the known properties of the sampling kernel allowing to write the relation between the IF-TEM measurements and the FSCs in a matrix form, e.g., y=A{circumflex over (x)}, and operates to determine the vector representation, {circumflex over (x)}, of the FSCs by applying the inverse representation (e.g., left-inverse) of the matrix, e.g., A.sup., to the vector representation, y, of the kernel-filtered signal. Finally, the extractor utility 189 extracts the unknown parameters (e.g., L amplitudes and L delays) of the input signal from the vector representation, {circumflex over (x)}, of FSCs, thereby recovering the analog signal x(t). It is noted that several known techniques may be used for recovering the signal from its FSCs, spectral analysis method being one non-limiting example.

[0108] The reconstruction system 180B exemplified in FIG. 3C is configured generally similar to the above-described reconstruction system 180A, namely includes analyzer 182, matrix creator utility 184; processor 186; and signal reconstructor 188 including a pulse parameters' extractor utility 189. In the system 180B, the processor 183 also includes a partial sum generator 190.

[0109] The reconstruction system 180B receives the sampling data comprising the data indicative of the series of time-encodings, {t.sub.n}, of the kernel-filtered signal y(t) being discrete time representation of the analog signal x(t). The time encoding analyzer 182 analyzes the data indicative of the time-encodings t.sub.n of the kernel-filtered signal y(t), and utilizes data indicative of the characteristic parameter(s) of the IF-TEM, TD, to generate data indicative of the vector representation y of the kernel-filtered FRI signal {y.sub.n, n} defined by

[00003]$y_n\stackrel{}{=}{}_{t_n}^{t_{n+1}}y\left(s\right)\mathrm{ds}=-b\left(t_{n+1}-t_n\right)+.$

As noted above, each y.sub.n represents an average value of y(t) in the interval between two consecutive times t.sub.n and t.sub.n+1.

[0110] This vector representation y is processed by the partial sum generator 190 to determine a vector of partial sums z: {z.sub.n, n} defined by:

[00004]$z_n={.Math.}_{i=1}^{n-1}{y}_i.$

[0111] The matrix creator 184 utilizes the knowledge about the frequencies transmitted by the chosen sampling kernel (i.e., the kernel support set, KD,) to analyze the received data indicative of the time encodings t.sub.n of the kernel-filtered signal and create the characteristic matrix M configured as described above. In this example, such matrix is denoted C and is configured as a Vandermonde type matrix having linearly independent columns, as will be described further below. This invertible matrix C describes the relation between partial sums vector z (constituting data indicative of vector representation y of the kernel-filtered signal) and a vector {circumflex over (z)} (constituting data indicative of vector representation, {circumflex over (x)}, of FSCs of the input signal x(t)), i.e., {circumflex over (z)}=C.sup.z.

[0112] The processor 186 operates to utilize the matrix inverse Ct to determine the vector {circumflex over (z)} and then determine the vector representation {circumflex over (x)} of FSCs as

[00005]$\hat{x}={\left[\hat{z}\left[-K\right],.Math.,\hat{z}\left[-1\right],\hat{z}\left[1\right],.Math.,\hat{z}\left[K\right]\right]}^T{}^{2K}.$

[0113] The extractor utility 189 extracts the unknown parameters (e.g., L amplitudes and L delays) of the input signal from the vector representation, {circumflex over (x)}, of FSCs, thereby recovering the analog signal x(t). As noted above, several suitable known techniques may be used for recovering the analog signal from its FSCs, spectral analysis method being one non-limiting example.

[0114] FIG. 4 illustrates IF-TEM sampling using the system 102 of the present disclosure: the IF-TEM input signal y(t) generated by the kernel 120 (i.e., acted on by the kernel function g(t)) is denoted 140A and is drawn in blue, the integrator output denoted 140B is drawn in green, and the IF-TEM output time instances, {t.sub.n}, calculated from recorded differences between consecutive events (time-encodings) when the threshold is met, are denoted 140C and are pointed with red. The number of time instants (time-encodings) per unit time, N/T, also called a firing rate (T being a predetermined time characteristic of the input signals, as described below), is proportional to the local frequency content of such an IF-TEM input signal. For example, there are more firings in a time interval where the signal amplitude (after biasing) varies rapidly compared to time intervals with relatively slow variations.

[0115] To this end, the kernel 120 has a size of the kernel support set in a predetermined relation with the number F of degrees of freedom in the input signal defined by characteristic parameters of the input signal, and preferably has a minimal transmission coefficient for zero frequency component of the input signal as compared to transmission coefficients for other frequency components to thereby substantially suppress transmission of the zero frequency component while allowing passage of a number K of non-zero frequency components in the input signal. The number K is defined to allow determination of a minimum number of Fourier-series coefficients (FSC) of the input signal allowing a unique reconstruction of the input signal.

[0116] The size of the kernel support and transmission coefficient frequency dependence of the kernel 120 are arranged to provide an alias cancellation for the input signal. Examples of such kernels may be found in [2] and they can also be called annihilating kernels.

[0117] In general, the kernel 120 may comprise any one of the following: a sinc function kernel, a sum-of-sincs (SoS) kernel, a sum-of-modulated spline kernel, a polynomial-reproducing kernel, or an exponential-reproducing kernel. The exponential reproducing kernel may be of maximum-order and minimum support or satisfying at least one generalized Strang-Fix condition.

[0118] A crucial component of a sampling architecture is the sampling kernel. Generally, sampling kernels with compact support are preferable from a hardware implementation perspective. In some embodiments of the present disclosure, the kernel has a support set of indices chosen to exclude zero and include integers symmetric around zero:

[00006]$=\left\{-K,.Math.,-1,1,.Math.,K\right\}.$

[0119] In some embodiments, the kernel is configured with the support set of indices satisfying a condition that ||2 L, where 2 L is a number of degrees of freedom in the input signal having L pulses of a predetermined shape defining L amplitudes and L time delays characterizing the input signal.

[0120] The input signal has a predetermined time characteristic T which may describe the period of a periodic input signal, the predetermined shape of the input signal defining the rate of innovation to be 2 L/T.

[0121] A periodic signal is assumed during the detailed description of the methods of the present disclosure, however, the same methods described in the disclosure, may be applied to nonperiodic signals which may be defined as a finite train of pulses having time delays restricted to lie in a finite time interval [0, T) and the nonperiodic signal having a finite support. It will be shown further below that all theorems regarding sampling and reconstruction of T-periodic signals are equally applicable to the nonperiodic case as well.

[0122] In the non-limiting example of L pulses of a predetermined shape defining L amplitudes and L time delays characterizing the input signal, the TEM 140 may be configured to sample a minimum number N, N2K+2, of the time-encodings {t.sub.n} within a period T of the input signal, wherein K satisfies a condition that K2 L, to thereby enable reconstruction of the L amplitudes and the L time delays of the L pulses.

[0123] The three positive real numbers parametrizing the TEM, {b, , }, are chosen such that

[00007]$\frac{b-c}{}\frac{2K+2}{T},$

where

[00008]$\frac{2K+2}{T}$

is a minimal required sampling rate of the TEM for a robust recovery of the input signal in presence of noise. This minimal required sampling rate of the TEM being more than twice the rate of innovation, 2 L/T.

[0124] As mentioned above, the reconstruction system 180 is configured to receive and process the series of time-encodings, {t.sub.n}, to reconstruct the input signal. The inventors show that with a proper choice of the sampling kernel, e.g., a sum-of-sincs kernel, the FSCs of the input signal can be uniquely identified from the IF-TEM sampling data comprising the time encodings {t.sub.n}. It should be noted that filter 120 may be understood as spreading the information of the FRI signal (since the purpose of the sampling kernel is to distribute the FRI signal information in such a way that the parameter-estimation block, i.e., the reconstruction system 180 would be enabled to estimate the time delays and amplitudes using a finite number of low-rate samples).

[0125] The authors in [1] aimed at investigating a supposition that by using sampling kernels that have a frequency-domain alias cancellation properties, FRI signals with 2 L degrees of freedom can be recovered from (2 L+2) IF-TEM firing instants. The influence of noise was not addressed. Also, perfect recovery is not established: rather, the reconstruction algorithm is based on an assumption that certain invertibility conditions for a matrix used for reconstruction are guaranteed; and in simulations the invertibility conditions were satisfied for a large number of experiments.

[0126] The inventors of the present disclosure have considered sampling analog signals using an IF-TEM, and more generally, TEM. They have found theoretical guarantees for the recovery (so-called perfect recovery, i.e., recovery in the absence of noise) of an input analog signal formed by a train of pulses of the predetermined shape, i.e., the lowest possible sampling rate, for the cases where the signal time delays are either on or off grid. That is, the inventors have designed a new sampling technique that can be used to ensure recovery in the noise-free setting, but that is also more robust in the presence of noise than some known techniques: it results in a lower error in the presence of noise for the same number of measurements. This allows for more easily producing, with lower costs, of more energy-efficient ADCs for signals.

[0127] Specifically, the reconstruction system 180 is further configured to perform consecutive partial summations of discrete representations of the input signal derived from the series of time-encodings and predetermined positive real number(s) parametrizing the IF-TEM comprising: a bias b satisfying a condition c<b< where c is a maximum real value of the input signal and/or a scaling factor , and/or a threshold ; and obtaining a vector of partial sums, to thereby provide noise-resilient reconstruction of the input signal.

[0128] Each successive component of the vector of partial sums, being a successive partial sum, is determined as a sum of a preceding partial sum and a linear transform, y.sub.n, of a respective difference between two consecutive time encodings t.sub.n+1 and t.sub.n defined as: y.sub.n=b(t.sub.n+1t.sub.n)+.

[0129] As already noted above, the present disclosure provides methods for robust reconstruction of analog signals also in the presence of noise. Excluding the zero frequency from the sampling kernel was found by the inventors to enable robust recovery of signal in the presence of perturbations in the time encodings. However, the reconstruction method mentioned above requires the condition that K2 L resulting in an increased firing rate.

[0130] An alternative approach to reduce the firing rate is to assume that the time delays are arranged in accordance with a predetermined time grid. In this case the requirement on the kernel reduces to KL and further below the inventors provide the details of reconstructing the analog signal from the L values of FSCs. Thus, the kernel may be configured with the support set of indices satisfying a condition that ||L. In this case, the three positive real numbers parametrizing the IF-TEM, {b, K, (}, are chosen such that

[00009]$\frac{b-c}{}\frac{2K+2}{T}$

where

[00010]$\frac{2K+2}{T}$

is the minimal required sampling rate of the TEM for a robust recovery of the input signal in presence of noise, the minimal required sampling rate of the TEM being higher than once and lower than twice the rate of innovation.

[0131] In another embodiment of the present disclosure, the kernel 120 is configured with a support set of indices satisfying a condition that ||4 L+1, where 4 L is the number of degrees of freedom in the input signal having L ECG pulses, the pulses being variable-pulse-width (VPW) ECG pulses of a predetermined shape.

[0132] The ECG signal is an example of a nonperiodic signal in the form of a finite train of pulses having time delays restricted to lie in a finite time interval and the nonperiodic signal having a finite support, the finite time interval being [0, T). Thus, the rate of innovation of an ECG signal is defined as 4 L/T in the present disclosure.

[0133] Thus, the IF-TEM 140 of FIG. 3A is configured to sample a minimum number N, N8 L+2, of the time-encodings within a time interval T of the input signal, to thereby enable to reconstruct 4 L parameters of the L VPW ECG pulses. Specifically, the three positive real numbers parametrizing the TEM, {b, , }, are chosen such that

[00011]$\frac{b-c}{}\frac{8L+2}{T}$

where

[00012]$\frac{8L+2}{T}$

is the minimal required sampling rate of the TEM for a robust recovery of the ECG signal.

[0134] Further, the inventors have built and tested hardware prototypes of respective samplers and ADCs.

[0135] Furthermore, the inventors have found a reconstruction technique which is more robust with respect to the hardware noise.

[0136] The usability of the specific variations of the sampling system 100 is supported by the theoretical guarantees, as developed by the inventors of the present disclosure, for perfect reconstruction of analog signals with arbitrary, but known pulse shapes. Also, the inventors have designed a sampling kernel and IF-TEM sampler with improved noise robustness. Specifically, since analog signals with F degrees of freedom can be perfectly reconstructed from F consecutive Fourier series coefficients (FSCs), the inventors have chosen sampling kernels with the alias-cancellation condition to annihilate the undesirable FSCs. The filtered signal, with fewer FSCs, is applied to an IF-TEM, and firing instants are measured. The inventors have found and shown that F+2 firing instants are sufficient to uniquely determine F FSCs from which the original signals can be recovered. Furthermore, the inventors have established conditions on the IF-TEM parameters that ensure that the minimum firing rate is achieved.

[0137] However, the inventors have then advanced much further. While the above sampling approach leads to perfect reconstruction of the signal in the absence of noise, the reconstruction can be highly sensitive to noise as the inventors have found via simulations. To address this issue, the inventors have designed a further modified sampling and reconstruction mechanism. In particular, the inventors have found that the zeroth Fourier coefficient of the filtered signal results in an unstable inverse matrix while computing the FSCs from the time instants in the presence of noise. To improve noise robustness, the inventors have modified the sampling kernel by removing the zero-frequency component. For this modified method, the inventors have found that 2F+2 time instants are sufficient for perfect recovery when the time-delays of the signal are off-grid, whereas F+2 firings are sufficient when the delays are on grid.

[0138] Through simulations, the inventors have shown that for the same number of firing rates (beyond F+2 firings), the mean squared error in the estimation of the on-grid time delays in the proposed approach is 2-6 dB lower compared to the one in [1]. In the case of off-grid time delays, the inventors have considered perturbations in the time encoding. The inventors have shown that their approach may bring more than 3-10 dB gain in terms of error compared to the method in [1] for the same number of measurements. In addition, the inventors have shown that their approach has better resolution ability in the presence of noise. Specifically, when the signal pulses are close to each other, their method is able to distinguish them.

[0139] In the following, the above embodiments, examples and developments are further explained, based on some specific examples presenting at times further suitable, but optional features, as well as their combinations and sub-combinations, which all have to be considered as disclosed:

[0140] The inventors have considered, as an example, an IF-TEM with the refractory period approximated as zero as shown in FIG. 2B. The input to the IF-TEM is a bounded signal y(t), and the output is a series of firing or time instants. An IF-TEM is parametrized by positive real numbers b, , and and works as follows: A bias b is added to a c-bounded signal y(t) where |y(t)|c<b<, and the sum is integrated and scaled by . When the resulting signal reaches the threshold , the time difference with such previous event is recorded in a memory (a hardware component) of the sampler or the ADC, and hence instant t.sub.n is calculated, and the integrator is reset. The process is repeated for subsequent time instants, i.e., if a time instant t.sub.n was obtained, the next time instant t.sub.n+1 satisfies:

[00013]$\begin{array}{cc}\frac{1}{}{}_{t_n}^{t_{n+1}}\left(y\left(s\right)+b\right)\mathrm{ds}=.& \left(1\right)\end{array}$

The time encodings {t.sub.n, n} form a discrete representation y.sub.n of the analog kernel-filtered signal y(t) and the objective is to reconstruct y(t) from them and then reconstruct the analog input signal x(t) from its kernel-filtered analog signal y(t). Typically, reconstruction is performed by using an alternative set of discrete representations {y.sub.n, n} defined as

[00014]$\begin{array}{cc}{y}_n\stackrel{}{=}{}_{t_n}^{t_{n+1}}y\left(s\right)\mathrm{ds}=-b\left(t_{n+1}-t_n\right)+.& \left(2\right)\end{array}$

The measurements {y.sub.n, n} are derived from the time encodings {t.sub.n, n} and IF-TEM parameters {b, , }.

[0141] Although reconstruction methods vary for different classes of signals, for perfect recovery of any signal, the firing rate is required to satisfy a lower bound that depends on the degrees of freedom of the signal. The firing rate of an IF-TEM is bounded both from above and below, where the bounds are a function of the IF-TEM parameters and an upper-bound on the signal amplitude. Using Eq. (2) and the fact that |y(t)|c, it can be shown that for any two consecutive time instants:

[00015]$\begin{array}{cc}\frac{}{b+c}{t}_{n+1}-t_n\frac{}{b-c}.& \left(3\right)\end{array}$

[0142] The inequalities in Eq. (3) imply that in any arbitrary, non-zero, observation interval T.sub.obs, the maximum and minimum number of firings are

[00016]$\begin{array}{cc}{T}_{\mathrm{obs}}\frac{b+c}{}\mathrm{and}{T}_{\mathrm{obs}}\frac{b-c}{},& \left(4\right)\end{array}$

respectively. Thus, the firing rate of an IF-TEM F.sub.R with parameters b, , and is upper and lower bounded as

[00017]$\begin{array}{cc}\frac{b-c}{}{F}_R\frac{b+c}{}.& \left(5\right)\end{array}$

[0143] As an example, to demonstrate the techniques of the disclosure, the inventors recover a continuous-time FRI signal x(t) of the form of (Eq. (6) below, from the time instances {t.sub.n}.

[0144] It will be assumed in the following that the input signal is T-periodic, for simplicity of presentation only. However, the techniques of the disclosure are applicable to non-periodic FRI signals as well, as will be described further below. Thus, the inventors demonstrate their method with a T-periodic FRI signal of the form:

[00018]$\begin{array}{cc}x\left(t\right)={.Math.}_p{.Math.}_{=1}^L{a}_{}h\left(t-{}_{}-\mathrm{pT}\right),& \left(6\right)\end{array}$

where h(t) is a known, real-valued pulse and the amplitudes and delays {()|[0, T),

[00019]${a_{}(0,a_{\max }]\}}_{=1}^L$

are unknown parameters. This signal model is used in applications such as radar, ultrasound, and more. In these applications, h(t) denotes a known transmit pulse which is reflected from L targets. The reflected signal is modeled as x(t) where and denote the amplitude and time-delay corresponding to the -th target. It is noted that the signal form of Eq. (6) is a non-limiting example described here to exemplify the principles of the sampling and reconstruction techniques of the disclosure. Other signal forms will be considered further below.

[0145] In general, FRI signals have wide bandwidth due to short duration pulses h(t). However, by using the structure of the signal, FRI signals can be sampled at sub-Nyquist rates and with the knowledge of the pulse h(t) they can be reconstructed. The sampling is typically achieved by passing x(t) through a designed sampling kernel, e.g., s(t) and then measuring low-rate samples y(nT) of the filtered signal y(t) as shown in FIG. 1. The kernel is designed such that the FRI parameters

[00020]${\left\{a_{},{}_{}\right\}}_{=1}^L$

are computed accurately from the samples. In particular, it was shown that 2 L samples of y(t) in an interval of length T, that are measured either uniformly or non-uniformly, are sufficient to determine

[00021]${\left\{a_{},{}_{}\right\}}_{=1}^L$

uniquely. The reconstruction or determination of the parameters from the samples is achieved by applying spectral analysis methods such as the annihilating filter (in other words, alias-cancellation filter).

[0146] As discussed above, a conventional FRI sampling scheme, such as in FIG. 1, has a sampler which is controlled by a global clock that operates at the rate of innovation

[00022]$\frac{2L}{T}$

Hz. For a large L or a small T, the sampling rate increases, and the global clock requires high power. In this case, according to the present disclosure, an IF-TEM sampler is well suited as it does not require a global clock.

[0147] The inventors have considered the problem of perfect recovery of the FRI parameters

[00023]${\left\{a_{},{}_{}\right\}}_{=1}^L$

using an IF-TEM sampling scheme as, for example shown in FIG. 2A. Specifically, they have considered designing the sampling kernel g(t) and an IF-TEM such that the FRI parameters are uniquely determined from the time-encodings by keeping the firing rate close to the rate of innovation.

[0148] In the following it is shown that, according to the present disclosure, in the noiseless case, perfect recovery is guaranteed using as few as 2 L+2 firings in an interval of length T. Further below, an alternative approach that is more robust to noise is presented.

[0149] The inventors have used the fact that the FRI signal x(t) in Eq. (6) can be perfectly reconstructed from its 2 L FSCs. The inventors have derived conditions on the IF-TEM parameters and the sampling kernel g(t) such that 2 L FSCs of the input FRI signal can be uniquely recovered from the IF-TEM output.

[0150] The inventors have begun by explicitly relating the input signal x(t) of Eq. (6) to its FSCs (cf. Eq. (10)) following [2].

[0151] Since x(t) is a T-periodic signal it has a Fourier series representation

[00024]$\begin{array}{cc}x\left(t\right)={.Math.}_k\stackrel{}{x}\left[k\right]e^{\mathrm{jk}{}_{0}t},& \left(7\right)\end{array}$

where

[00025]${}_0=\frac{2}{T}.$

The Fourier-series coefficients {circumflex over (x)}[k] are given by

[00026]$\begin{array}{cc}\stackrel{}{x}\left[k\right]=\frac{1}{T}\hat{h}\left(k{}_0\right){.Math.}_{=1}^L{a}_{}{e}^{-\mathrm{jk}{}_0{}_{}},& \left(8\right)\end{array}$

where, () is the continuous-time Fourier transform of h(t) and the inventors have assumed that (k.sub.0)0 for k where is a given set of indices. Since x(t) is real-valued, its FSCs {circumflex over (x)}[k] are complex conjugate pairs, that is,

[00027]$\begin{array}{cc}{\stackrel{}{x}}^{*}\left[-k\right]=\stackrel{}{x}\left[k\right].& \left(9\right)\end{array}$

The sequence

[00028]$\begin{array}{cc}\frac{\stackrel{}{x}\left[k\right]}{\stackrel{}{h}\left(k{}_0\right)}=\frac{1}{T}{.Math.}_{=1}^L{a}_{}{e}^{-\mathrm{jk}{}_0{}_{}},& \left(10\right)\end{array}$

consists of a sum of L complex exponentials. From the theory of high-resolution spectral estimation, it is well known that 2 L consecutive samples of

[00029]$\frac{\stackrel{}{x}\left[k\right]}{\stackrel{}{h}\left(k{}_0\right)}$

are sufficient to determine

[00030]${\left\{a_{},{}_{}\right\}}_{=1}^L.$

For example, one can apply a well-known annihilating filter method to compute

[00031]${\left\{a_{},{}_{}\right\}}_{=1}^L.$

In practice, the pulse h(t) has short-duration and wide bandwidth. Hence, there always exist 2 L or more non-vanishing Fourier samples (k.sub.0) that are computed a priori. To determine the FRI parameters, it is needed to compute 2 L consecutive values of {circumflex over (x)}[k]. The problem is then reduced to that of uniquely determining the desired number of FSCs from the signal measurements. Since x(t) typically consists of a large number of FSCs, a sampling kernel design which removes unnecessary FSCs and thus reduces the sampling rate is discussed in the following.

[0152] Since a minimum of 2 L FSCs are sufficient for uniquely recovering the FRI signal, the sampling kernel g(t) is designed to remove or annihilate any additional FSCs. The filtered signal y(t) is given by

[00032]$\begin{array}{cc}\begin{array}{c}y\left(t\right)=\left(x*g\right)\left(t\right)={}_{-}^{}x\left(\right)g\left(t-\right)d\\ ={.Math.}_k\stackrel{}{x}\left[k\right]{}_{-}^{}g\left(t-\right)e^{\mathrm{jk}{}_0}d\\ ={.Math.}_k\stackrel{}{x}\left[k\right]\stackrel{}{g}\left(k{}_o\right)e^{\mathrm{jk}{}_{0}t}.\end{array}& \left(11\right)\end{array}$

To restrict the summation to a finite number of terms and annihilate the unwanted FSCs the inventors have defined the filter to satisfy the following condition in the Fourier domain:

[00033]$\begin{array}{cc}\stackrel{}{g}\left(k{}_0\right)=\{\begin{array}{cc}1& \mathrm{if}k,\\ 0& \mathrm{otherwise},\end{array}& \left(12\right)\end{array}$

where is a set of integers such that |2 L.

[0153] One particular choice of the sampling kernel, in this example, is a sum-of-sincs (SoS) kernel [2] generated by

[00034]$\begin{array}{cc}\stackrel{}{g}\left(\right)={.Math.}_k\sin c\left(\frac{}{{}_0}-k\right),& \left(13\right)\end{array}$$\mathrm{and}$$\begin{array}{cc}g\left(t\right)=\{\begin{array}{cc}{.Math.}_k{e}^{\mathrm{jk}{}_{0}t},& t(-\frac{T}{2},\frac{T}{2}]\\ 0& \mathrm{elsewhere},\end{array}& \left(14\right)\end{array}$

The sampling kernel g(t) is designed to pass the coefficients {circumflex over (x)}[k], k while suppressing all other coefficients {circumflex over (x)}[k], k.Math.. Note that one can also apply a lowpass filter (e.g., substantially or approximately ideal) with an appropriate cutoff frequency to remove the FSCs. However, the impulse response of an ideal lowpass filter has infinite support, whereas the SoS kernel has compact support.

[0154] Using a SoS kernel, the filtered signal y(t) is

[00035]$\begin{array}{cc}y\left(t\right)={.Math.}_k\stackrel{}{x}\left[k\right]\stackrel{}{g}\left(k{}_0\right)e^{\mathrm{jk}{}_{0}t}={.Math.}_k\stackrel{}{x}\left[k\right]e^{\mathrm{jk}{}_{0}t}.& \left(15\right)\end{array}$

The filtered signal y(t) is sampled by an IF-TEM which requires its input to be real-valued and bounded. Since {circumflex over (x)}[k] are conjugate symmetric, to ensure that y(t) is real-valued, the support set is chosen to be symmetric around zero, that is, is, for example, given as

[00036]$\begin{array}{cc}=\left\{-K,.Math.,K\right\}& \left(16\right)\end{array}$

where KL to ensure that there are at-least 2 L FSCs of x(t) retained in y(t). From Eq. (6) and y(t)=(x*g)(t), it can be shown that

[00037]$\begin{array}{cc}c\stackrel{}{=}\underset{t}{\max }.Math.y\left(t\right).Math.{\mathrm{La}}_{\max }{.Math.\left(h*g\right).Math.}_{}& \left(17\right)\end{array}$$\begin{array}{cc}{\mathrm{La}}_{\max }{.Math.g.Math.}_{}{.Math.h.Math.}_1,& \left(18\right)\end{array}$

where Young's convolution inequality is used. Since |g(t)|| and is a finite set, g(t) is bounded. Hence y(t) is bounded provided that the maximal amplitude .sub.max< and the pulse h(t) is absolutely integrable. In the remaining sections, both these conditions are considered as held.

[0155] The IF-TEM input is the filtered signal y(t), which is the T-periodic signal defined in Eq. (15). The output of the IF-TEM is a set of time instants Given {t.sub.n} one can determine the measurements {y.sub.n} by using Eq. (2).

[0156] In order to express the relation between the measurements y.sub.n and the FSCs (that are to be determined) in a matrix form (defined by kernel and the time encodings), a matrix A is defined:

[00038]$\begin{array}{cc}A=\left[\begin{array}{ccccc}{e}^{-\mathrm{jK}{}_0{t}_2}-e^{-\mathrm{jK}{}_0{t}_1}& .Math.& t_2-t_1& .Math.& e^{\mathrm{jK}{}_0{t}_2}-e^{\mathrm{jK}{}_0{t}_1}\\ e^{-\mathrm{jK}{}_0{t}_2}-e^{-\mathrm{jK}{}_0{t}_2}& .Math.& t_3-t_2& .Math.& e^{\mathrm{jK}{}_0{t}_3}-e^{\mathrm{jK}{}_0{t}_2}\\ .Math.& & .Math.& & .Math.\\ e^{-\mathrm{jK}{}_0{t}_N}-e^{-\mathrm{jK}{}_0{t}_{N-1}}& .Math.& t_N-t_{N-1}& .Math.& e^{\mathrm{jK}{}_0{t}_N}-e^{\mathrm{jK}{}_0{t}_{N-1}}\end{array}\right]& \left(19\right)\end{array}$

The relation between the measurements y.sub.n and the desired FSCs is given by [1]:

[00039]$\begin{array}{cc}\begin{array}{c}{y}_n={}_{t_n}^{t_{n+1}}y\left(t\right)\mathrm{dt}\\ ={}_{t_n}^{t_{n+1}}{.Math.}_{k\left\{0\right\}}\stackrel{}{x}\left[k\right]e^{\mathrm{jk}{}_{0}t}\mathrm{dt}+{}_{t_n}^{t_{n+1}}\stackrel{}{x}\left[0\right]\mathrm{dt}\\ ={.Math.}_{k\left\{0\right\}}\stackrel{}{x}\left[k\right]\frac{\left(e^{\mathrm{jk}{}_0{t}_{n+1}}-e^{\mathrm{jk}{}_0{t}_n}\right)}{\mathrm{jk}{}_0}+\stackrel{}{x}\left[0\right]\left(t_{n+1}-t_n\right)\end{array}& \left(20\right)\end{array}$

To extract the desired FSCs from Eq. (20), the vector

[00040]${\left[{}_{t_1}^{t_2}y\left(t\right)\mathrm{dt},{}_{t_2}^{t_3}y\left(t\right)\mathrm{dt},.Math.,{}_{t_{N-1}}^{t_N}y\left(t\right)\mathrm{dt}\right]}^T$

is denoted by, where N is the number of time instants in the interval T; and {circumflex over (x)} is defined:

[00041]$\begin{array}{cc}\stackrel{}{x}={\left[-\frac{\stackrel{}{x}\left[-K\right]}{jK{}_0},.Math.,\stackrel{}{x}\left[0\right],.Math.,\frac{\stackrel{}{x}\left[K\right]}{jK{}_0}\right]}^T.& \left(21\right)\end{array}$

With this notation, Eq. (20) can be written in the following matrix form:

[00042]$\begin{array}{cc}y=A\stackrel{}{x},& \left(22\right)\end{array}$

where A is given in Eq. (19). Hence, this equation describes the relation between the IF-TEM measurements and the FSCs.

[0157] The goal is to determine these FSCs embedded in {circumflex over (x)} from which a perfect recovery of the FRI parameters may be performed. If the matrix A has full column rank, then the Fourier coefficients vector can be computed as

[00043]$\begin{array}{cc}\stackrel{}{x}=A^{}y,& \left(23\right)\end{array}$

where A.sup. denotes the Moore-Penrose inverse.

[0158] In [1], the matrix is assumed to be uniquely left-invertible. The authors showed via simulations that the matrix has full column rank, however, a proof is not presented. In the following, the inventors show that for an adequate number of firings, the matrix A is indeed uniquely left-invertible, that is, has full column rank.

[0159] In the following, some main results are presented: it is shown that for the sampling kernel choice Eq. (14), it is possible to uniquely identify the FSCs from the IF-TEM time instants. Specifically, it is shown that for a particular choice of the IF-TEM parameters, the matrix A defined in Eq. (19) is left-invertible. The results are summarized in the following theorems.

Theorem 1: Consider a positive integer K and a number T0. Let 0t.sub.1<t.sub.2< . . . <t.sub.N<T for an integer N, and

[00044]${}_0=\frac{2}{T}.$

Then the matrix A defined in Eq. (19) is left-invertible provided that N2K+2.

[0160] The proof is described further below Appendix.

[0161] Theorem 1 implies that there should be a minimum of 2K+2 IF-TEM time instants within an interval of T to enable recovery of the FSCs, and subsequent reconstruction of the FRI signal. To ensure this, the minimum firing rate

[00045]$\begin{array}{cc}\frac{b-c}{}& \left(\mathrm{cf}.\mathrm{Eq}.\left(5\right)\right)\end{array}$

should be chosen such that

[00046]$\begin{array}{cc}\frac{b-c}{}\frac{2K+2}{T}.& \left(24\right)\end{array}$

[0162] By combining Theorem 1, the result in Eq. (24), and the fact that 2 L FSCs are sufficient to recover the FRI parameters, the inventors have formulated the sampling and reconstruction of FRI signals using IF-TEM in the following theorem.

Theorem 2: Let x(t) be a T-periodic FRI signal of the following form

[00047]$x\left(t\right)=\underset{p}{.Math.}\underset{=1}{\overset{L}{.Math.}}{a}_{}h\left(t-{}_{}-pT\right),$

where [0, T), ||<, and L is known. It is assumed that the amplitudes

[00048]${\left\{a_{}\right\}}_{=1}^L$

are finite, and the pulse h(t) is known and absolutely integrable. Consider the sampling mechanism shown in FIG. 2A. Let the sampling kernel g(t) satisfy

[00049]$\stackrel{}{g}\left(k{}_0\right)=\{\begin{array}{cc}1& \mathrm{if}k=\left\{-K,.Math.,K\right\},\\ 0& \mathrm{otherwise},\end{array}$

and max.sub.t|(h*g)(t)|<. Choose the real positive TEM parameters {b, , } such that c<b<, where c is defined in (Eq. A-18), and

[00050]$\begin{array}{cc}\frac{b-c}{}\frac{2K+2}{T}.& \left(25\right)\end{array}$

Then, the parameters

[00051]${\left\{a_{},{}_{}\right\}}_{=1}^L$

can be perfectly recovered from the TEM outputs if KL.

[0163] Based on Theorem 2, a reconstruction algorithm to compute the FRI parameters from TEM firings is presented in Algorithm 1.

TABLE-US-00001 Algorithm 1: Reconstruction of a T-Periodic FRI Signal Using Theorem 2. [00052]$\mathrm{Input}\text{:}N2K+2\mathrm{spike}\mathrm{times}{\left\{t_n\right\}}_{n-1}^N\mathrm{in}a\mathrm{period}T.$ 1: Let n 1 2: while n N 1 do 3: Compute y.sub.n = b(t.sub.n+1 t.sub.n) + 4: n := n + 1. 5: end while 6: Compute Fourier coefficients vector {circumflex over (x)} =A.sup.y in Eq. (23). 7: [00053]$\mathrm{Estimate}{\left\{\left(a_{,}{}_{}\right)\right\}}_{-1}^L\mathrm{using}a\mathrm{spectral}\mathrm{analysis}\mathrm{method}.$ [00054]$\mathrm{Output}\text{:}{\left\{\left(a_{,}{}_{}\right)\right\}}_{-1^{*}}^L$

[0164] It should be noted that not all steps of this algorithm have to be implemented to be useful. For example, step (7) could be performed with a different method than a spectral analysis method, or not performed at all by a certain device, for example, a unit of ADC. Also, even one iteration calculation according to step (3) leads to a useful data.

[0165] The IF-TEM parameters may be selected such that there is a minimum of N2 L+2 time instants

[00055]${\left\{t_n\right\}}_{n=1}^N$

within a time interval T. Thus, the minimum firing rate that enables accurate reconstruction is

[00056]$\frac{2L+2}{T}.$

The maximum firing rate is bounded by

[00057]$\frac{b+c}{}.$

While the threshold , which is a parameter of the comparator, is easier to control, the integrator constant is a parameter of the integrator, and it is usually fixed. Thus, assuming a fixed value of b and , choosing small results in a large firing rate above the minimum desirable value of

[00058]$\frac{2L+2}{T}.$

In practice, both b and may be generated through a DC voltage source, and therefore large values of bias and threshold require high power. Hence, to minimize the power requirements, it is desirable for b and to be as small as possible.

[0166] Next, the inventors have numerically validated Theorem 2. FIGS. 5A-5F illustrate sampling and reconstruction of a stream of Dirac impulses using TEM by applying the SoS kernel, according to the present disclosure. Here, both and are chosen uniformly at random over (0,1). FIGS. 5A-5C show the input signal and its reconstruction for L=3, L=5, and L=10 respectively, and FIGS. 5D-5F show the filtered signal y(t) and the time instants t.sub.n for L=3, L=5, and L=10 respectively. In FIGS. 5A-5F, h(t) is considered as a Dirac impulse with time period T=1 seconds. Simulations for L=3, 5, and 10 have been performed. The time delays and amplitudes were selected uniformly at random over (0,1). The input signal x(t) was filtered using an SoS sampling kernel with ={K, . . . , K}, where K=L. The filtered output y(t) was sampled using an IF-TEM which has a threshold =0.07 and =1. The bias of the IF-TEM was set as b=0.9, 1.3, and 2.5 for L=3, 5, and 10 respectively. The parameters are chosen to satisfy the inequality in Eq. (25), and resulted in 13, 18, and 36 samples per period for L=3, 5, and 10. Per Theorem 2, 8, 12, and 22 samples per period were sufficient. The reconstruction was found to be stable even for a larger number of impulses. The choice of IF-TEM parameters and the resulting firing rate are presented in Table 1:

TABLE-US-00002 TABLE 1 L b .sub.0 F.sub.R (samples/s) 3 0.9 0.07 1 2 13 5 1.3 0.07 1 2 18 10 2.5 0.07 1 2 36

[0167] FIGS. 6A to 6B illustrate sampling and reconstruction of stream of pulses using TEM by applying the SoS kernel, according to the present disclosure, showing the input signal and its reconstruction for L=3 (FIG. 6A)), and the filtered signal y(t) and the time instants t.sub.n for L=3 (FIG. 6B). In FIGS. 6A to 6B, the estimation, using the same filter, of a stream of pulses is depicted with L=3, {}={0.5, 0.45, 0.4}, and {}={0.2, 0.33, 0.8}. The IF-TEM b, , which satisfy the inequality in Eq. (25), are 0.9, 0.07, 1 respectively. The resulting firing rate is as few as 13 samples/s.

[0168] The results in [1] and the previous section were based on the approximation that there is no measurement noise. However, in practice, the signals are contaminated by noise. In the presence of noise, the IF-TEM outputs or time instants are perturbed. While using Algorithm 1, this results in a perturbation in the matrix A as well as the measurements y in Eq. (22). In this case, when computing the FSCs using Eq. (22), the stability of A, which is measured by the condition number of the matrix, impacts the results. Next, it is shown that, according to the present disclosure, by excluding zero from , perfect recovery is possible, and in the noisy scenario, the resulting method is more robust.

[0169] As it is shown below, when excluding the zero vector, a recovery problem becomes similar to Eq. (23) but with the matrix B defined in Eq. (29) replacing A. This matrix has a better condition number than A. To gain intuition as to why this is the case, the intersection of a function (t) and a straight line r(t) as shown in FIGS. 7A and 7B is considered. The function (t) is a K-th order trigonometric polynomial whose coefficients are FSCs of the FRI signal and the slope of r(t) is equal to x[0]={circumflex over (x)}[0](see the Appendix for more details). As it is discussed in detail in the Appendix, the closer the values of (t) to r(t) at the time encoding instants, the poorer the condition number of the corresponding matrices.

[0170] As shown in the Appendix, the matrices A and B have full column rank provided that the straight line r(t)(t) at all time-encoding instants

[00059]${\left\{t_n\right\}}_{n=1}^N,$

and this condition holds for N>2K+1. Note that x[0]0 when we consider matrix A and x[0]=0 in the context of matrix B. Furthermore, it can be shown that the smaller the values of

[00060]${\left\{.Math.r\left(t\right)-f\left(t\right).Math.\right\}}_{t_n=1}^N$

the poorer the condition number. For N>2K+1 and x[0]0, it is possible that there exists a set of time encodings

[00061]${\left\{t_n\right\}}_{n=1}^N$

such that the straight line r(t) is close to the trigonometric polynomial (t) at

[00062]${\left\{t_n\right\}}_{n=1}^N.$

In such a case, from Eq. (44), a set of Fourier coefficients can be determined such that A{circumflex over (x)}0.sub.N, where 0.sub.N is a zero vector of length N. Hence for that particular set of

[00063]${\left\{t_n\right\}}_{n=1}^N,$

A becomes ill-conditioned. However, when x[0]=0, the inventors have observed that it is less likely that r(t) with zero-slope becomes close to (t) at the time-encoding instants.

[0171] Several illustrative examples depicting this intuition are shown in FIGS. 7A to 7B where K=2 and N=7. In FIG. 7A, the condition number of matrices A and B are 30 and 3, respectively. The condition number of A is ten times higher than that of B. This is because the straight line corresponding to A (shown in red) is closer to the trigonometric polynomial at the time-encodings than that of B (in magenta). In the example shown in FIG. 7B, the condition number of A is 3000 as |r(t)(t) is small for

[00064]$t{\left\{t_n\right\}}_{n=1}^N,$

whereas |r(t)(t) is relatively large for x[0]=0 and consequently B has lower condition number.

[0172] Next, a perfect recovery guarantee for FRI signals is presented by using IF-TEM without the zero frequency in the SoS kernel.

[0173] In the following, it is shown that by excluding the zero frequency in the inventors have achieved perfect reconstruction for FRI signals of the form of Eq. (6). In this case, the resulting matrix has a much more stable structure compared to A of Eq. (19). Suppose k=0 is removed; then, following Eq. (20)

[00065]$\begin{array}{cc}\begin{array}{c}{y}_n={}_{t_n}^{t_{n+1}}y\left(t\right)\mathrm{dt}\\ ={}_{t_n}^{t_{n+1}}{.Math.}_{k\left\{0\right\}}\stackrel{}{x}\left[k\right]e^{jk{}_{0}t}\mathrm{dt}\\ ={.Math.}_{k\left\{0\right\}}\stackrel{}{x}\left[k\right]\frac{\left(e^{\mathrm{jk}{}_0{t}_{n+1}}-e^{\mathrm{jk}{}_0{t}_n}\right)}{\mathrm{jk}{}_0}\end{array}.& \left(26\right)\end{array}$

To extract the FSCs from Eq. (26), by y.sub.0 the vector

[00066]${\left[{}_{t_1}^{t_2}y\left(t\right)\mathrm{dt},{}_{t_2}^{t_3}y\left(t\right)\mathrm{dt},...,{}_{t_{N-1}}^{t_N}y\left(t\right)\mathrm{dt}\right]}^T$

is denoted, where N is the number of time instants in the interval T. The measurements y.sub.0 and the FSCs

[00067]$\begin{array}{cc}{\stackrel{}{x}}_0={\left[-\frac{\stackrel{}{x}\left[-K\right]}{\mathrm{jK}{}_0},...,-\frac{\stackrel{}{x}\left[-1\right]}{j{}_0},\frac{\stackrel{}{x}\left[1\right]}{j{}_0},...,\frac{\stackrel{}{x}\left[K\right]}{\mathrm{jK}{}_0}\right]}^T& \left(27\right)\end{array}$

are now related as

[00068]$\begin{array}{cc}{y}_0=B{\stackrel{}{x}}_0,& \left(28\right)\end{array}$

where B is given as in Eq. (29). Next, it is shown that the matrix B has full column rank and is uniquely left invertible (and Eq. (29) is shown below).
Theorem 3: Consider a positive integer K and a number T>0. Let 0t.sub.1<t.sub.2< . . . <t.sub.N<T for an integer N, and

[00069]${}_0=\frac{2}{T}.$

Then the matrix B defined in Eq. (29) is left-invertible provided that N>2K+1.

[0174] Proof: The proof follows the same line as that of Theorem 1 with the constraint {circumflex over (x)}[0]=0 as detailed in Case-1 in the Appendix.

[00070]$\begin{array}{cc}B=\left[\begin{array}{cccccc}{e}^{-\mathrm{jK}{}_0{t}_2}-e^{-\mathrm{jK}{}_0{t}_1}& .Math.& e^{-j{}_0{t}_2}-e^{-j{}_0{t}_1}& e^{j{}_0{t}_2}-e^{j{}_0{t}_1}& .Math.& e^{\mathrm{jK}{}_0{t}_2}-e^{\mathrm{jK}{}_0{t}_1}\\ e^{-\mathrm{jK}{}_0{t}_3}-e^{-\mathrm{jK}{}_0{t}_2}& .Math.& e^{-j{}_0{t}_3}-e^{-j{}_0{t}_2}& e^{j{}_0{t}_3}-e^{j{}_0{t}_2}& .Math.& e^{\mathrm{jK}{}_0{t}_3}-e^{\mathrm{jK}{}_0{t}_2}\\ .Math.& & .Math.& .Math.& & .Math.\\ e^{-\mathrm{jK}{}_0{t}_N}-e^{-\mathrm{jK}{}_0{t}_{N-1}}& .Math.& e^{-j{}_0{t}_N}-e^{-j{}_0{t}_{N-1}}& e^{j{}_0{t}_N}-e^{j{}_0{t}_{N-1}}& .Math.& e^{\mathrm{jK}{}_0{t}_N}-e^{\mathrm{jK}{}_0{t}_{N-1}}\end{array}\right]& \left(29\right)\end{array}$

Since the left-inverse of B exists, the Fourier coefficients vector is computed as

[00071]$\begin{array}{cc}{\hat{x}}_0=B^{}{y}_0.& \left(30\right)\end{array}$

[0175] Although the FSCs are computed uniquely, they are not consecutive unlike the FSCs computed in Theorem 2. Since high resolution spectral estimation techniques such as the annihilating filter requires 2 L consecutive FSCs, to uniquely determine the FRI parameters, it is needed that K2 L. This results in twice the firing rate compared to that in Theorem 2.

[0176] An alternative approach to reduce the firing rate is to assume that the time-delays are on a grid. In this case, determination of time-delays and amplitudes of the FRI signal from FSCs is cast as a compressive sensing problem [3]. This problem is efficiently solved from 2 L FSCs, that are not necessarily consecutive, by using sparse recovery approaches such as orthogonal matching pursuit (OMP). Hence, by assuming that the time-delays of the FRI signal are on a grid, it is required that KL.

[0177] The minimum firing rate for the IF-TEM is

[00072]$\begin{array}{cc}\frac{b-c}{}\frac{2K+2}{T},& \left(31\right)\end{array}$

where K2 L for off-grid time-delays and KL for time-delays on-grid. By combining Theorem 3 with the result in Eq. (31), the sampling and reconstruction of FRI signals using IF-TEM is described in the following theorem.
Theorem 4: Let x(t) be a T-periodic FRI signal of the following form

[00073]$x\left(t\right)=\underset{p}{.Math.}\underset{=1}{\overset{L}{.Math.}}{a}_{}h\left(t-{}_{}-pT\right),$

where the number of FRI signals L is known, and h(t) is a signal with known pulse shape. Consider the sampling mechanism shown in FIG. 2A. Let the sampling kernel g(t) satisfy

[00074]$\stackrel{}{g}\left(k{}_0\right)=\{\begin{array}{cc}1& \mathrm{if}k=\left\{-,.Math.,-1,1,.Math.,\right\}\\ 0& \mathrm{otherwise}\end{array},$

and max.sub.t|(h*g)(t)|<. Choose the real positive TEM parameters {b, , } such that c<b<, where c is defined in (Eq. A-18), and

[00075]$\begin{array}{cc}\frac{b-c}{}\frac{2K+2}{T}.& \left(32\right)\end{array}$

Then, the parameters

[00076]${\left\{a_{},{}_{}\right\}}_{=1}^L$

can be perfectly recovered from the TEM outputs if [0178] 1) K2 L when

[00077]${\left\{{}_{}\right\}}_{=1}^L$

are off-grid, [0179] 2) KL when

[00078]${\left\{{}_{}\right\}}_{=1}^L$

are on-grid.
Since the time delays and amplitudes, of the FRI signals are estimated uniquely, without any constant offset, from any set of 2 L or more consecutive FSCs, excluding the zero frequency does not result in any offset in the reconstruction. An algorithm to perfectly recover the FRI parameters from IF-TEM samples is summarized in Algorithm 2.

TABLE-US-00003 Algorithm 2: Reconstruction of a T-Periodic FRI Signal. [00079]$\mathrm{Input}:N2K+2\mathrm{spike}\mathrm{times}{\left\{t_n\right\}}_{n=1}^N\mathrm{in}a\mathrm{period}T.$ 1: Let n 1 2: while n N 1 do 3: Compute y.sub.n = b(t.sub.n+1 t.sub.n) + 4: n := n + 1. 5: end while 6: Compute Fourier coefficients vector {circumflex over (x)}.sub.0 = B.sup.y.sub.0 in Eq. (30). 7: [00080]$\mathrm{Estimate}{\left\{\left(a_{},{}_{}\right)\right\}}_{=1}^L\mathrm{from}{x}_0\mathrm{by}\mathrm{using}\mathrm{CS}\mathrm{methods}\mathrm{for}KL$ [00081]$\mathrm{Output}:{\left\{\left(a_{},{}_{}\right)\right\}}_{=1}^L.$

[0180] It should be noted that not all steps of this algorithm have to be implemented to be useful. For example, step (7) could be performed with a different method than a CS method, or not performed at all by a certain device, for example, a unit of ADC. Also, even one iteration calculation according to step (3) leads to a useful data.

[0181] In many practical systems, the time instants can only be recorded with finite precision, i.e., in practical circumstances, the recorded times are effective time instances

[00082]$\left\{t_n^{}\right\}$

which differ from the real-time instances {t.sub.n}, and perfect reconstruction may no longer be possible.

[0182] The inventors have compared the robustness of the Algorithms 1 and 2 in the presence of perturbation to the measured time instants. Algorithm 2 has provided better recovery than Algorithm 1.

[0183] In the above Algorithms, the first step is to estimate the Fourier samples from the TEM measurements by taking pseudo inverses of A (cf. Eq. (22)) and B (cf. Eq. (28)), respectively. Both the matrices are functions of the measured time instants and the sampling kernel. In FIG. 8, the condition numbers of the matrices with perturbed firing instants are compared, as a function of the number of FRI signals L. To that aim, 5000 random sets of monotonic sequences

[00083]${\{t_n[0,T)\}}_{n=1}^N$

were used. As shown in FIG. 8, the condition number of the matrix B.sup.(4L+2)(2L) is substantially smaller than the condition number of the matrix A.sup.(4L+2)(2L+1).

[0184] Next, the inventors have performed the reconstruction of a T-periodic FRI signal from non-uniform noisy samples (time instants) using the two reconstruction algorithms of Theorems 2 and 4. A periodic FRI signal x(t) of the form of Eq. (6) was created. The signal x(t) with period T=1 consists of L=3 pulses with h(t)=.sup.(3)(20t), where .sup.(3)(t) is a third-order cubic B-spline, with

[00084]${\left\{a_{}\right\}}_{=1}^3=\left\{0\text{.5},-0.45,0.4\right\},$

and

[00085]${\left\{{}_{}\right\}}_{=1}^3=\left\{0.2,0.4,0.8\right\}.$

The TEM parameters are b=1.2, =1, and changes from 0.04 to 0.09 resulting in 13 to 24 time samples. The parameters are chosen to satisfy the condition of Eq. (32). We consider a sum-of-sincs kernel with ={K, . . . ,0, . . . , K} for Theorem 2, and ={K, . . . , 1,1, . . . , K} for Theorem 4. For both kernels, the time instances {t.sub.n} were perturbed by a zero-mean white Gaussian noise with variance 0.001. For both the methods it was set K=2 L, and OMP was applied to recover the time-delays and amplitudes.

[0185] The reconstruction accuracy of the two algorithms is compared in terms of relative mean square error (MSE), given by

[00086]$\begin{array}{cc}MSE=\frac{{.Math.x\left(t\right)-\stackrel{}{x}\left(t\right).Math.}_{L_2\left[0,T\right]}}{{.Math.x\left(t\right).Math.}_{L_2\left[0,T\right]}},& \left(33\right)\end{array}$

where x(t) is the reconstructed signal. In FIG. 9, the MSE of the two algorithms is shown as a function of the number of noisy time instances. The MSE of Algorithm 2 is 2-6 dB lower compared to that of Algorithm 1 for different firing rates.

[0186] Next, the inventors have compared the two frameworks when the input FRI signal has noise. To simulate a continuous-time noise effect on the input to IF-TEM, perturbation to the FSCs at the output of the SoS filter was added. In the noisy case, the output of the SoS filter is given by

[00087]$\begin{array}{cc}{}^{}\left(t\right)={.Math.}_k\left(\stackrel{}{x}\left[k\right]+\left[k\right]\right)e^{jk{}_{0}t},& \left(34\right)\end{array}$

where are independent and identically distributed circular complex Gaussian random variables with zero mean and variance .sub.{circumflex over (x)}. In this simulation, SNR was defined as

[00088]$\begin{array}{cc}10{\log }_{10}\left(\frac{{.Math.}_k{.Math.\stackrel{}{x}\left[k\right].Math.}^2}{.Math..Math.{}_{\hat{x}}^2}\right).& \left(35\right)\end{array}$

The inventors have compared Algorithm 1 and 2 in terms of MSE in the estimation of time delays as:

[00089]$\begin{array}{cc}MSE=10\log \left({.Math.}_{=1}^L{\left({}_{}-{\stackrel{}{}}_{}\right)}^2\right).& \left(36\right)\end{array}$

For both approaches, the inventors have used the same number of firing rates and measurements. The time delays were estimated by the OMP algorithm. The MSEs in the time delay estimation are shown in FIG. 10 for different SNRs and for K=L, 2 L. The inventors have observed that for SNR>20 dB, Algorithm 2 has lower error compared to Algorithm 1.

[0187] The following experiments consider off-grid time-delays with h(t) as a third-order cubic B-spline. The inventors have considered L=3 pulses where time-delays and amplitudes are generated uniformly at random over intervals (0, T] and [1,5], respectively. For IFTEM, t =1 and b=2.5c were set where c is computed as in Eq. (18). The threshold is chosen to satisfy Eq. (24). To compare the methods with zero and without-zero frequency, the inventors have considered sampling SoS kernels with ={K, . . . , K} and {K, . . . , 1,1, . . . , K}, respectively. The inventors have used an annihilating filter with Cadzow denoising to estimate the time-delays in the presence of noise. Since Cadzow denoising requires more than 2 L consecutive samples of FSCs, the inventors have considered K2 L+1 while excluding the zero. Based on the fact that Algorithm 2 estimates

[00090]${\left\{\stackrel{}{x}\left[k\right]\right\}}_{k=-K}^{-1}\mathrm{and}{\left\{\stackrel{}{x}\left[k\right]\right\}}_{k=1}^K,$

Cadzow denoising on each of these sequences independently was applied; and then block annihilation to determine the time-delays jointly was applied. In addition, the inventors have shown results for estimation of signal parameters via rotational invariance technique (ESPRIT) for with zero approach as well as for without zero approach.

[0188] In this simulation, the inventors have compared the two approaches in terms of MSEs (cf. Eq. (36)) when the time encodings are perturbed. The perturbed time encodings are given as

[00091]$t_n^{}=t_n+{}_n$

where t.sub.0 is the actual time encoding and En is a random variable uniformly distributed over [/2, /2]. The MSEs in the estimation of time-delays for different numbers of FSCs and perturbation levels are shown in FIGS. 11A to 11D illustrating comparison of with zero (Algorithm 1) and without zero (Algorithm 2) approaches for off-grid time delays with perturbation in the time encodings. The inventors used 500 independent noise and FRI signal realizations to compute each MSE value. In FIGS. 11A, 11B, and 11C MSEs for with zero, without zero, and ESPRIT approaches are shown. In FIG. 11D, one-dimensional MSE plots for K=2 L+1 and 5 L are shown for further clarity. In these experimental settings, the time encodings had values between [0, T] where T=1 sec. In this case, adding a perturbation with 0.04 had severe effect on the recovery with MSE in the range of 5 dB for all the approaches. The inventors have observed that the zero approach and ESPRIT have similar MSEs. While comparing these two approaches with the proposed without zero approach, the inventors have noted a gain of 3-10 dB for <0.04. Since perturbation in the time encoding is also equivalent to quantization noise, a lower MSE indicates that the proposed without zero approach can operate at lower bits compared to with zero method.

[0189] The accuracy of time-delay estimations of any FRI recovery approach is strongly dependent on the minimum distance between two consecutive FRI pulses in the presence of noise. To analyze the resolution abilities of the two approaches, an FRI signal with two pulses of equal amplitudes has been considered. By treating the difference between the time delays, .sub.2.sub.1, the inventors have computed MSEs (as in Eq. (36)) in their estimation when the time encodings are perturbed with =0.008 (low noise) and =0.014 (high noise). The results are shown in FIG. 12. For both the noise levels without zero approach has a lower MSE compared to the with zero approach. The gain in the MSE is more prominent (up to 5 dB) for the high noise level. The results show that without zero approach has better resolution compared to the with zero approach.

[0190] Unlike conventional sampling, the number of measurements per unit time in the IF-TEM is not fixed. For a class of c-bounded FRI signals, parameters {K, b, (} were chosen to achieve a desired minimum firing rate. However, the firing rate may vary from signal to signal in the class considered. The bias b plays an important role in designing the rate as it should always be kept above c (which is maximum amplitude of x in absolute value, i.e., |x(t)|c). In this experiment, the inventors have analyzed the effect of b on the firing rate for a fixed c. To this end, it was set c=1; and b was varied. For each b, the inventors have generated 100 FRI signals with randomly chosen amplitudes and time-delays such that the signals are bounded by c. The average firing rates for both with zero (denoted as FR-A) and without zero (denoted as FR-B) approaches were computed and are shown in FIG. 13. Also the maximum firing (maxFR) and minimum firing (minFR) rates,

[00092]$\frac{b+c}{}\mathrm{and}\frac{b-c}{}$

together with the rate of innovation (Rol) 2 L are plotted. Here L=5 was chosen and it was kept fixed for all b. For each b, was set to be 1 and was selected to satisfy Eq. (24). For both methods, the inventors have had identical mean firing rates.

[0191] The inventors have observed that for small values of b, the firing rates are large, and they reduce as b increases. For large values of b, the gap between maxFR and min FR reduces, and IF-TEM operates at a rate closer to Rol (i.e., the rate of innovation). However, a large b amounts to high power consumption. Hence, there is a trade-off between power consumption and the optimum firing rate.

[0192] Consider a nonperiodic FRI signal of the form

[00093]$\begin{array}{cc}\tilde{x}\left(t\right)={.Math.}_{=1}^L{a}_{}h\left(t-{}_{}\right),& \left(37\right)\end{array}$

where h(t) is a known pulse and the amplitudes and delays {()|[0, T),

[00094]${a_{}\}}_{=1}^L$

are unknown parameters. It is assumed that the pulse h(t) has finite support R, namely

[00095]$\begin{array}{cc}h\left(t\right)=0,.Math.t.Math.\frac{R}{2}.& \left(38\right)\end{array}$

[0193] Given that according to the present disclosure the interest is in pulses with a very wide or even infinite spectrum, i.e. very short pulses, traditional sampling techniques are expected to be ineffective in this case (see [2]).

[0194] The inventors have designed a sampling kernel g(t) such that

[00096]$\begin{array}{cc}\tilde{y}\left(t\right)=\tilde{x}\left(t\right)*\tilde{g}\left(t\right)=y\left(t\right),t[0,T),& \left(39\right)\end{array}$

where y(t) defined in Eq. (11). Specifically, g(t) is compactly supported and defined by

[00097]$\begin{array}{cc}\tilde{g}\left(t\right)={.Math.}_{s=-S}^{S}g\left(t+\mathrm{sT}\right),& \left(40\right)\end{array}$

where S is determined by R and T (more details are available in [2]). Since both time instants and the time delays of the FRI signals are within the interval [0, T), i.e., t.sub.n[0, T) and [0, T), =1, . . . , L, the time instants taken in the nonperiodic case using IF-TEM are the same as in the periodic case. Therefore, the recovery guarantees developed for the periodic case in Theorems 4 and 2 are applicable to the non-periodic case as well.

APPENDIX

[0195] In this Appendix Theorems 1 and 3 are proven. The inventors have considered a unified approach to prove both theorems by using the fact that the proof of Theorem 3 is a special case of that of Theorem 1 with {circumflex over (x)}[0]=0.

[0196] Proof: The matrix A in (Eq. A-22) is decomposed as

[00098]$\begin{array}{cc}A=\mathrm{DV},& \left(41\right)\end{array}$$\mathrm{where}$$\begin{array}{cc}D=\left(\begin{array}{ccccc}-1& 1& 0& .Math.& 0\\ 0& -1& 1& .Math.& 0\\ 0& 0& -1& .Math.& 0\\ .Math.& .Math.& .Math.& & .Math.\\ 0& 0& 0& .Math.& 1\end{array}\right){}^{\left(N-1\right)N}& \left(42\right)\end{array}$

and V is given as in (Eq. A-43). To determine {circumflex over (x)} uniquely from y=DV {circumflex over (x)}, the matrix A should not have a non-zero null space vector. The matrix D is a difference operator which has null space vector C1.sub.N where c\{0}. Hence, if there exists a non-zero vector x as in Eq. A(21) whose components satisfy Eq. (9), such that Vx=C1.sub.N for some arbitrary c, then there does not exist a unique solution. However, according to the present disclosure, it can be shown that for N2K+22 L+1, uniqueness is guaranteed. Specifically, it can be shown that there does not exist an x satisfying Eq. (9), a set

[00099]${\left\{t_n\right\}}_{n=1}^N,$

and c0 such that Vx=c1.sub.N for N>2K+1, Eq. (43), and Eq. (44).

[00100]$\begin{array}{cc}V=\left(\begin{array}{ccccccc}{e}^{-\mathrm{jK}{}_0{t}_1}& .Math.& e^{-j{}_0{t}_1}& t_1& e^{j{}_0{t}_1}& .Math.& e^{\mathrm{jK}{}_0{t}_1}\\ e^{-\mathrm{jK}{}_0{t}_2}& .Math.& e^{-j{}_0{t}_2}& t_2& e^{j{}_0{t}_2}& .Math.& e^{\mathrm{jK}{}_0{t}_2}\\ .Math.& & .Math.& .Math.& .Math.& & .Math.\\ e^{-\mathrm{jK}{}_0{t}_N}& .Math.& e^{-j{}_0{t}_N}& t_N& e^{j{}_0{t}_N}& .Math.& e^{\mathrm{jK}{}_0{t}_N}\end{array}\right){}^{N2K+1}& \left(43\right)\end{array}$$\begin{array}{cc}\underset{w}{\underset{}{\left(\begin{array}{ccccccc}{e}^{-\mathrm{jK}{}_0{t}_1}& .Math.& e^{-j{}_0{t}_1}& 1& e^{j{}_0{t}_1}& .Math.& e^{\mathrm{jK}{}_0{t}_1}\\ e^{-\mathrm{jK}{}_0{t}_2}& .Math.& e^{-j{}_0{t}_2}& 1& e^{j{}_0{t}_2}& .Math.& e^{\mathrm{jK}{}_0{t}_2}\\ .Math.& & .Math.& .Math.& .Math.& & .Math.\\ e^{-\mathrm{jK}{}_0{t}_N}& .Math.& e^{-j{}_0{t}_N}& 1& e^{j{}_0{t}_N}& .Math.& e^{\mathrm{jK}{}_0{t}_N}\end{array}\right)}}\underset{\stackrel{}{\stackrel{\_}{x}}}{\left(\begin{array}{c}\stackrel{\_}{x}\left[-K\right]\\ .Math.\\ \stackrel{\_}{x}\left[-1\right]\\ -c\\ \stackrel{\_}{x}\left[1\right]\\ .Math.\\ \stackrel{\_}{x}\left[K\right]\end{array}\right)}\underset{t}{\underset{}{\left(\begin{array}{c}{t}_1\\ t_2\\ .Math.\\ t_N\end{array}\right)}}& \left(44\right)\end{array}$

[0197] For simplicity of discussion, it is defined:

[00101]$\stackrel{\_}{x}\left[k\right]=\frac{\hat{x}\left[k\right]}{\mathrm{jk}{}_0}$

for k0 and x[0]={circumflex over (x)}[0]. The modulus and angle of the complex-valued coefficient x[k] is denoted by |x[k]| and <x[k], respectively. The equation Vx=C1.sub.N is re-written as in Eq. (44) or alternatively as

[00102]$\begin{array}{cc}f\left(t\right)=r\left(t\right),\mathrm{for}t=t_1,...,t_N,& \left(45\right)\end{array}$$\mathrm{where}$$\begin{array}{cc}f\left(t\right)=c-2{.Math.}_{k=1}^K.Math.\stackrel{\_}{x}\left[k\right].Math.\cos \left(k{}_{0}t+\stackrel{}{x}\left[k\right]\right)& \left(46\right)\end{array}$

is a K-th order trigonometric polynomial and r(t)=x[0]t is a straight line with slope x[0]. If there exist a c and

[00103]${\left\{\stackrel{}{x}\left[k\right]\right\}}_{k=0}^K$

such that (t) and r(t) intersect each other N-times within an interval [0,T), that is, there exist a set ={t.sub.n[0,T),n=1, . . . , N} satisfying (t)=r(t) then uniqueness is not guaranteed.

[0198] Two mutually exclusive cases can be considered: (1) x[0]={circumflex over (x)}[0]=0 and (2) x[0]={circumflex over (x)}[0]0.

[0199] Case-1: For x[0]=0, the slope of the straight line r(t) is zero and hence, Eq. (45) is equivalent to determining zeros of (t) in the interval [0, T). Since (t) is a trigonometric polynomial of order K with

[00104]${}_0=\frac{2}{T},$

it will have a maximum of 2K zeros within the interval [0, T). Hence, for N>2K, there does not exist a c0 and a feasible

[00105]${\left\{\stackrel{\_}{x}\left[k\right]\right\}}_{k=0}^K$

such that Eq. (45) holds true.

[0200] Case-2: Consider the case when x[0]0. It is possible to determine the maximum number of intersections of a trigonometric polynomial of order K with a straight line. To this end, let t.sub.1 be the first intersection point. Further, it can be assumed that the slope of (t) at t=t.sub.1 is positive (or negative), that is, (t.sub.1)>0 (or (t.sub.1)<0) where (t) denotes derivative of (t). This implies that there may exist a minimum (or maximum) of (t) for t[0, t.sub.1) and a maximum (or minimum) for t(t.sub.1, T). An illustrative example is shown in FIG. 14. Since r(t) is a monotone function, to have a second intersection t.sub.2, it is necessary that (t) changes its sign. In essence, if there exist a t.sub.2, then there should be a maximum (or minimum) of (t) in the interval (t.sub.1, t.sub.2). Applying this argument to the remaining intersection points in it is inferred that for N intersection points there should be at least N1 extrema. Alternatively; a function with N1 extrema can intersect a monotone function at a maximum of N points. As (t) too is a K-th order trigonometric polynomial, it has a maximum of 2K zeros. This implies that (t) can have a maximum of 2K extrema. Hence, (t) can intersect r(t) at a maximum of 2K+1 points within interval [0, T). This implies that for N>2K+1, the equation (t)=r(t) can not have any solution and hence, matrix A does not have a non-zero null-vector and uniqueness is guaranteed.

[0201] Hence, in the above sampling and reconstruction frameworks for periodic FRI signals by using IF-TEMs have been described. A robust recovery approach has been provided with the sampling kernel different than in [1]. In the presence of perturbations of time encodings, the modified approach outperforms the method in [1] for the same number of measurements. Compared to conventional amplitude-based sampling for FRI signals the proposed TEM-based method is less power consuming and hence, more cost effective.

[0202] Furthermore, as it was mentioned in the background, ADC are used also in neuromorphic-based hardware. Building neuromorphic computing systems usually requires the ability to efficiently emulate brain capabilities in hardware. Several types of neuro-mimetic devices were researched, but it still has been desirable to develop a compact device that can simulate spiking neurons.

[0203] In the conventional ADC system, the hardware cost and size increase with the sampling rate. To allow for low-rate sampling, the inventors again have considered using sub-Nyquist signal architectures, for FRI signals. Sub-Nyquist systems operate at lower sampling rates and hence can reduce the hardware cost and complexity.

[0204] Asynchronous ADCs (AADCs) present an alternative to conventional ADCs. An output sample is generated only when the signal characteristics change, resulting in considerable power and bandwidth savings. Since AADCs lack a global clock and a steady output data rate, they may outperform traditional clock-based ADCs and be used in the sub-Nyquist systems.

[0205] Time encoding, an event-driven approach, allows switching from acquiring the values of the observed signals to storing informative time. This is achieved by recording time instances in which a specific phenomenon is observed as the digital representation of the analog signal. As above, no global clock is then required; and the change in the sampling rate is signal dependent. Furthermore, compared to its amplitude-based homonyms, TEMs have simple and perhaps entirely analogue encoder stages that consume very little power and have smaller die sizes.

[0206] There is a similarity between how neurons encode information and TEM or IF-TEM. TEM-based systems that prioritize acquisition times are the basis of neuromorphic computing. In an effort to boost energy efficiency and resource requirements by orders of magnitude, neuromorphic computing seeks to create novel computational frameworks that mimic the brain's operation. The IF-TEM mechanism can be used to interpret the spike times of spiking neural networks (SNNs), leading to a better knowledge of how to utilize neuromorphic hardware and replace power hungry ADCs.

[0207] The inventors have prepared a hardware prototype demonstrating the applicability of efficiently emulating an integrate-and-fire (IF) neuron, that is have designed an ADC which encodes the data using an IF-TEM and allows the data to be obtained without using a global clock. Such time encoding hardware, mainly due to its asynchronous mechanism, has an advantage compared to traditional ADCs in that the circuit does not need to be synchronized to a global clock which entails high engineering and other costs. The inventors have implemented possibly the first sub-Nyquist time-based sampler using IF-TEM.

[0208] Also, the present disclosure relates to robust recovery of FRI signals. The inventors also have found that in practice, the IF-TEM circuit introduces noise into the signal, which can affect the time instances {t.sub.n}. Even in the absence of noise, it is only possible to determine time instances with limited precision. Hence, the IF-TEM hardware needs a robust nonlinear recovery technique.

[0209] The inventors have developed a robust reconstruction algorithm to recover the FRI parameters more accurately. The hardware prototype can accurately recover the FRI signals for FRI pulses with a minimum separation of 4 s between them. The tested hardware and reconstruction method can recover the FRI parameters at up to 25 dB SNR while operating at 10-12 times less than the Nyquist rate. This novel reconstruction method is more noise-resistant than Algorithms 1 and 2 to the noise in the hardware-measured data.

[0210] Thus, in this disclosure, a robust reconstruction technique is introduced for the sub-Nyquist sampling technique disclosed above, as well as the first analog time-based hardware implementation of sub-Nyquist sampling of FRI signals. Prior to acquiring timing information with the IF-TEM, the signal is prefiltered using the sampling kernel. As above, the sampling kernel eliminates the zeroth frequency component of the signal for robust recovery. However, during recovery, in view of the noise in the hardware data, a different forward model with a T-periodic bandlimited function is utilized than in Algorithm 2: a partial summation is used so that the noise in each element of matrix C from Eq. (53) below is expected to be smaller than in matrix B of Eq. (29).

[0211] As for the FRI-TEM hardware prototype, it can be employed in low-power time-of-flight applications. It has been used to demonstrate the accurate reconstruction of FRI signals using the novel algorithm.

[0212] To this end, the hardware components have been designed or selected to accommodate a broad spectrum of FRI signal frequencies without modification, assuming a minimum separation of 4 s between FRI pulses. The separation was assumed just based on the specific filter used, it could be reduced by choosing a different filter. In particular, before the IF-TEM sampler, a bandpass filter (BPF) has been utilized as the sampling kernel. The filter removes unnecessary information from the signal and allows sampling below the Nyquist rate. For robust recovery, the sampling kernel eliminates the signal's zeroth frequency component. The IF-TEM hardware generates a sequence of spike timings by sampling the filtered signal each time the threshold is reached. As an immediate, i.e., very quick, reset mechanism is desired, the inventors have built an integrator and a reset function in which the integrator's capacitor operates in its linear region and discharges rapidly and substantially fully. Then, it is possible to accurately recover the FRI parameters using sub-Nyquist samples using the novel robust reconstruction method. In particular, an example of the system functions at eight times the rate of innovation, which is significantly lower than the Nyquist rate of the signal.

[0213] Referring to FIG. 15, it schematically shows an example of IF-TEM according to the present disclosure. IF-TEM 340 includes an integrator 360-1 with a discharge circuit 360A, a comparator 370, and a switch 362 (which in general may be a part of the integrator 360-1 or a component external to integrator 360-1). The switch 362 is positioned to receive a signal, transmitted from the comparator, for initiating reset of the integrator 360-1, and for switching on the discharge circuit 360A. The discharge circuit 360A includes a capacitor configured to operate in its linear zone to provide rapid and complete discharge of the integrator 360-1.

[0214] Referring to FIG. 16, it schematically shows another example of IF-TEM according to the present disclosure. IF-TEM 340A includes an integrator 360-2 with a discharge circuit 360A, a comparator 370, and a switch 362, which is positioned to receive a signal, transmitted from the comparator, for initiating reset of the integrator 360-2 and transmitted from the comparator 370, and to switch on the discharge circuit 360A.

[0215] In some examples, as in FIG. 16, the switch 362 includes at least three terminals, 362-1, 362-2, and 362-3, and is configured to receive a control signal for initiating reset of the integrator 360-2 at a first terminal 362-1, and to switch on the discharge circuit 360A by allowing a current flow between the second (362-2) and third (362-3) terminals.

[0216] The switch 362 may be presented by or include a FET. The first terminal 362-1 may be a gate of the FET.

[0217] In a further example, schematically shown in FIG. 17, an IF-TEM 340B, according to the present disclosure, includes at least one of a differentiator 372 located on a path from the comparator 370 to the switch 362 and an amplifier 374 located on a path from the differentiator 372 and upstream of the switch 362. The differentiator 372, which is optional, generates spikes at time instances {t.sub.n}. The amplifier 374 allows matching the output of the differentiator 372 (or, when it is absent, the output of the comparator 370) with the switch 362 input and resetting through a discharge the integrator 360-2. The differentiator may be before the amplifier.

[0218] The discharge circuit 360A in any example or modification of the integrator 360-2 may include a resistor and a capacitor. An equivalent resistance of the discharge circuit may be selected in a range 1-10 kOhm and an equivalent capacitance of the discharge circuit 360A may be selected to make an RC parameter of the discharge circuit to be in a range 1-10 nF.

[0219] IF-TEMs of FIGS. 16 and 17 present examples also of IF-TEMs including a comparator 370 and a differentiator 372, positioned to receive a signal from an output of the comparator 370.

[0220] IF-TEMs of FIGS. 15 to 17 and their modifications may be used to implement IF-TEM 140 of FIG. 3A. Consequently, any of the sampling systems according to the present disclosure may be adapted to use the TEM as the IF-TEM described with reference to FIGS. 15 to 17. An ADC, for example the ADC 200 of FIG. 18, may include such a sampling system and a reconstruction system 180.

[0221] FIG. 18 schematically presents an ADC 200 including the signal processing system 100 according to any of the above examples. The ADC 200 may also include a reconstruction unit/system 180 (which may be configured as system 180A or 180B) for reconstructing a signal being a continuous time, finite rate of innovation signal sampled by the signal processing system 100 according to any of the above-described methods.

[0222] In general, the reconstruction unit 180 is configured as described above to receive and process the data indicative of the series of time-encodings provided by the signal processing system 100 (sampling system) to reconstruct the input signal by defining the linear relation between the vector representation y of the kernel-filtered signal and vector representation of the FSCs.

[0223] As exemplified above, this can be implemented by the configuration of the reconstruction system 180B (described above with reference to FIG. 3C) which creates the characteristic matrix C via performing the consecutive partial summations of discrete representations of the input signal, as described in detail above. The partial summations are derived from the series of time-encodings and at least three predetermined positive real numbers parametrizing the TEM of system 100 comprising: a bias b satisfying a condition c<b< where c is a maximum real value of the input signal, a scaling factor x, and a threshold ; to thereby provide noise-resilient reconstruction of the input signal.

[0224] The reconstruction unit 180 may include a hardware and may implement any one of Algorithm 1, Algorithm 2, or Algorithm 3 as described above, or another reconstruction or recovery algorithm.

[0225] In the following the above embodiments, examples and developments are further explained, based on some specific examples presenting at times further suitable, but optional features, as well as their combinations and sub-combinations, which all have to be considered as disclosed.

[0226] An unstable recovery occurred when Algorithm 2 described above, associated with matrix B (see Eq. (29) was utilized in the process of reconstructing the data from the hardware measurements. Therefore, a reconstruction strategy that is more robust to noise is desired.

[0227] Consider a T-periodic FRI signal of the form of Eq. (6) and a sampling mechanism as shown in FIG. 2A. The signal x(t) is passed through the sampling kernel g(t) as defined in Eq. (13) and the resulting signal y(t) is sampled using an IF-TEM. Both the time encodings

[00106]${[t_n\}}_{n=1}^N$

and the amplitude measurements

[00107]${\left\{y_n\right\}}_{n=1}^N$

correspond to a discrete representation of y(t)=(x*g)(t). In other words, {t.sub.n} encodes information of the FRI signal. As the objective is to create robust hardware, the FRI parameters

[00108]${\left\{a_{},{}_{}\right\}}_{=1}^L$

need to be accurately estimated from the IF-TEM firings. To this end, together with the hardware implementation, a robust recovery algorithm is needed. In the following, the robust recovery mechanism of the present disclosure is introduced, that perfectly recovers the Fourier series coefficients from IF-TEM observations in the absence of noise with as few as 4 L+2 pulses inside an interval T. Then, the resilience of this method is illustrated in the presence of noise and it is demonstrated that it outperforms Algorithm 2 described above.

[0228] The IF-TEM circuit can introduce noise into the signal, which in turn can perturbs the time instances {t.sub.n}. Even in the absence of noise, the time instances can only be determined with limited precision. The modeled jittered time instances are expressed as

[00109]$\begin{array}{cc}{t}_n^{}=t_n+{}_n& \left(47\right)\end{array}$

where t.sub.n are the ideal time instances and

[00110]${}_n\stackrel{\mathrm{iid}}{~}\left[-\frac{}{2},\frac{}{2}\right]$

is the noise jitter. The hardware experiments performed by the inventors have shown that noise level fluctuates up to 70 ms. However, when an attempt was made to reconstruct FRI signals from IF-TEM measurements using the method presented above, inconsistent recovery was observed with hardware data. Therefore, in the following, the inventors propose a more noise-tolerant reconstruction method that addresses this limitation.

[0229] To assess the effectiveness of the new noise-tolerant reconstruction method, the inventors compare it with the approach presented above, under the presence of perturbations in the measured time instances. Both methods use a sample kernel without the zeroth frequency. However, while the method presented above employs the forward equation specified in Eq. (26), the new algorithm utilizes an alternative formulation introduced in Eq. (50) below.

[0230] In the following a method for determining the Fourier coefficients of the FRI signal that is more robust and improves recovery is presented. The recovery approach described above is based on computing the FSCs {circumflex over (x)} of the FRI signal x(t) using Eq. (30). In the case of noise, this leads to a perturbation in the measurements y.sub.n as well as the matrix B defined in Eq. (26) and Eq. (29), respectively. In this case, while computing the FSCs, the stability of B, which is measured by the condition number of the matrix, impacts the results. Next, it is shown that by utilizing a partial summation of y.sub.n, perfect recovery is achieved similarly to Theorem 2. In the noisy scenario, the resulting method is more robust. As is shown further below, when employing a partial summation of y.sub.n, a recovery problem similar to Eq. (30) is obtained, but with the matrix C defined in Eq. (53) replacing B. This matrix has a better condition number than B.

[0231] To gain intuition as to why this is the case, the inventors demonstrate that for every k, utilizing the partial summation for the measurements reduces the noise in each element of C by half compared to its corresponding element in B. This result is summarized in the following lemma.

[0232] Lemma 1. Let [B].sub.nk=e.sup.jk.sub.0t.sub.n+1-e.sup.jk.sub.0t.sub.n be the entries of matrix B, where n=1, . . . , N1, k. Let [C].sub.nk=e.sup.jk.sub.0t.sub.n+1 be the entries of matrix C, where n=1, . . . , N1, k{0}. The jittered time instances are modeled as

[00111]$t_n^{}=t_n+{}_n,$

where to are the ideal time instances and the jitter noise is modeled as

[00112]${}_n\stackrel{\mathrm{iid}}{~}\left[-\frac{}{2},\frac{}{2}\right],$

i.i.d. uniformly distributed. For every and k,

[00113]$\begin{array}{cc}\mathrm{var}\left({\left[B\right]}_{nk}\right)=2\mathrm{var}\left({\left[C\right]}_{nk}\right),& \left(48\right)\end{array}$

where var is the variance.
Proof By utilizing the fact that

[00114]$t_n^{}=t_n+{}_n,$

and using Eq. (29) and Eq. (53), it follows that,

[00115]$\begin{array}{cc}\begin{array}{c}\mathrm{var}\left({\left[B\right]}_{nk}\right)=\left(e^{jk{}_0\left(t_{n+1}+{}_{n+1}\right)}-e^{jk{}_0\left(t_n+{}_n\right)}\right)\\ ={.Math.e^{jk{}_0{t}_{n+1}}.Math.}^2\mathrm{var}\left(e^{jk{}_0{}_{n+1}}\right)+\\ {.Math.e^{jk{}_0{t}_n}.Math.}^2\mathrm{var}\left(e^{jk{}_0{}_n}\right)\\ =2\mathrm{var}\left(e^{jk{}_0{}_n}\right)\\ =2\mathrm{var}\left({\left[C\right]}_{nk}\right),\end{array}& \left(49\right)\end{array}$

establishing the lemma.

[0233] It can be intuitively inferred that by utilizing the partial summation, the noise in each element of [C].sub.nk becomes smaller than the corresponding noise in [B].sub.nk. Consequently, the matrix C has a better condition number than B. This can be explained by the fact that the condition number of a matrix is a measure of the sensitivity of the matrix to small perturbations in its elements, and a smaller condition number indicates that the matrix is less sensitive to such perturbations. Therefore, by reducing the noise in the elements of C using the partial summation, the inventors improve its condition number.

[0234] In the next step, the inventors employ the partial summation of y.sub.n to present a perfect recovery guarantee for FRI signals by using IF-TEM. Instead of recovering the FSCs from y.sub.n through the forward model Eq. (26) with in Eq. (16), which defines the relation between y.sub.n and the FSCs {circumflex over (x)}[k], the inventors propose an alternative model that is based on z.sub.n. These are the partial sums of the measurements y.sub.n defined as

[00116]$\begin{array}{cc}{z}_n={.Math.}_{i=1}^{n-1}{y}_i={.Math.}_k\frac{\stackrel{}{x}\left[k\right]}{jk{}_0}\left(e^{jk{}_0{t}_n}-e^{jk{}_0{t}_1}\right),& \left(50\right)\end{array}$

where n=2, . . . , N. Note that (20) can be written as

[00117]$$\begin{gathered} \begin{array}{cc}{z}_n={.Math.}_k\frac{\stackrel{}{x}\left[k\right]}{jk{}_0}{e}^{jk{}_0{t}_n}+c \\ & \left(51\right)\end{array} \end{gathered}$$$\begin{array}{cc}c=-{.Math.}_k\frac{\stackrel{}{x}\left[k\right]}{jk{}_0}{e}^{jk{}_0{t}_1}.& \left(52\right)\end{array}$

Let z=[z.sub.2, . . . , Z.sub.N].sup.T.sup.N1 be the vector of partial sums,

[00118]$\stackrel{}{z}={\left[-\frac{\stackrel{}{x}\left[-K\right]}{jK{}_0},.Math.,-\frac{\stackrel{}{x}\left[-1\right]}{j{}_0},c,\frac{\stackrel{}{x}\left[1\right]}{j{}_0},.Math.,\frac{\stackrel{}{x}\left[K\right]}{jK{}_0}\right]}^T{}^{\left(2K+1\right)}$

be the vector of FSCs with c in the zeroth place, and C.sup.(N1)(2K+1) be the matrix defined as

[00119]$\begin{array}{cc}C=\left[\begin{array}{ccc}{e}^{-\mathrm{jK}{}_0{t}_2}& .Math.1.Math.& e^{\mathrm{jK}{}_0{t}_2}\\ e^{-\mathrm{jK}{}_0{t}_3}& .Math.1.Math.& e^{\mathrm{jK}{}_0{t}_3}\\ .Math.& & .Math.\\ e^{-\mathrm{jK}{}_0{t}_N}& .Math.1.Math.& e^{\mathrm{jK}{}_0{t}_N}\end{array}\right]& \left(53\right)\end{array}$

Then, Eq. (51) can be expressed in matrix form as follows:

[00120]$\begin{array}{cc}z=C\stackrel{}{z}.& \left(54\right)\end{array}$

[0235] Since the set of time instants

[00121]${\left\{t_n\right\}}_{n=2}^N$

are distinct, and C is a Vandermonde matrix, it has full column rank provided that N12K+1. This means that the matrix C has linearly independent columns. Therefore, the vector of FSCs {circumflex over (z)} can be perfectly recovered via

[00122]$\begin{array}{cc}\hat{z}=C^{}z& \left(55\right)\end{array}$

where C.sup. denotes the Moore-Penrose inverse of C. Once {circumflex over (z)} is obtained, the FSCs {circumflex over (x)}[k] can be uniquely determined. Using

[00123]$\begin{array}{cc}\hat{z}\left[k\right]=\{\begin{array}{cc}\frac{\hat{x}\left[k\right]}{j{}_{0}k},& \mathrm{if}k,\\ -{.Math.}_{k^{}}\left(\frac{\hat{x}\left[k^{}\right]}{j{k}^{}{}_0}\right)e^{{\mathrm{jk}}^{}{}_0{t}_1}& \mathrm{if}k=0\end{array}.& \left(56\right)\end{array}$

The vector of FSCs z and the vector of FSCs {circumflex over (x)} are related y:

[00124]$\begin{array}{cc}\hat{x}={\left[\hat{z}\left[-K\right],.Math.,\hat{z}\left[-1\right],\hat{z}\left[1\right],.Math.,\hat{z}\left[K\right]\right]}^T{}^{2K}.& \left(57\right)\end{array}$

This equation allows one to obtain {circumflex over (x)} by selecting the appropriate elements of {circumflex over (z)}, which is the vector obtained from the partial sums of the measurements. Note that the resulting vector {circumflex over (x)} has dimensions 2K, which implies that it only contains the FSCs for positive and negative frequencies.

[0236] Using the vector {circumflex over (z)} and Eq. (57), the vector of FSCs {circumflex over (x)} is uniquely determined. This indicates that, in the modified kernel setup, without zero frequency, the set of FSCs {circumflex over (x)}[k] can be uniquely determined from time encodings if N12K+1. This condition implies that there should be a minimum of 2K+2 firing instants within an interval T. For an IF-TEM, the minimum firing rate is given as

[00125]$\frac{b-c}{}$

(see e.g., Eq. (5). Hence, for uniqueness recovery, the IF-TEM parameters must satisfy the inequality

[00126]$\frac{b-c}{}\frac{2K+2}{T}.$

[0237] A reconstruction algorithm to compute the FRI parameters from IF-TEM firings is presented in Algorithm 3. Compared to the technique presented above, the method described here requires the same number of FSCs in the absence of noise. However, in the presence of noise, as is typically the case in real-world hardware, the proposed approach yields a lower error for the same number of measurements.

TABLE-US-00004 Algorithm 3: Reconstruction of a T-periodic FRI signal. [00127]$\mathrm{Input}:N2K+2\mathrm{spike}\mathrm{times}{\left\{t_n\right\}}_{n=1}^N\mathrm{in}a\mathrm{period}T.$ 1. 1: Let n 1 2. while n N 1 do 3. Compute y.sub.n = b(t.sub.n+1 t.sub.n) + 4. [00128]$\mathrm{Compute}{z}_{n+1}={}_{i=1}^n{y}_i$ 5. n := n + 1. 6. end while 7. Compute the vector {circumflex over (z)} = C.sup.z, where C is defined in Eq. (53). 8. Compute the Fourier coefficients vector {circumflex over (x)} from {circumflex over (z)} using Eq. (57). 9. [00129]$\mathrm{Estimate}{\left\{\left(a_{},{}_{}\right)\right\}}_{=1}^L\mathrm{using}a\mathrm{spectral}\mathrm{analysis}\mathrm{method}\mathrm{for}K2L.$ [00130]$\mathrm{Output}:{\left\{\left(a_{},{}_{}\right)\right\}}_{=1}^L.$

[0238] In this section, the inventors provide numerical evidence of the validity of Algorithm 3 through simulations. It is also shown that the proposed reconstruction technique improves the conditioning of the forward transformation, leading to a significant reconstruction improvement necessary for precise recovery using real hardware. To validate Theorem 4, the inventors consider h(t) as a Dirac impulse with a time period of T=1 second, and L=5. The amplitudes are selected randomly over the range [1,1]. The time delays are chosen randomly between (0,1) such that they lie on a grid with a resolution of 0.05. The input signal x(t) is filtered using an SoS sampling kernel with =K, . . . 1,1, . . . , K, where K=L. The filtered output y(t) is sampled using an IF-TEM where the IF-TEM parameters are chosen to satisfy the inequality Eq. (24). In this particular case, the IF-TEM sampler resulted in 16 firing instants in one time period, as shown in FIG. 19B. It can be seen in FIG. 19A that the inventors achieve perfect recovery of the FRI signal by using a kernel without zero frequency.

[0239] As the IF-TEM circuit produces noise that perturbs the time instances t.sub.n, the jittered time instances:

[00131]$t_n^{}=t_n+{}_n$

are considered, as defined in Eq. (47). The inventors compare the proposed recovery method with the reconstruction algorithm described in above (Algorithm 2) in the presence of perturbations to the measured time instances. Each approach employs a sample kernel lacking a zeroth frequency. While the reconstruction approach proposed in Algorithm 2 uses the forward method defined in Eq. (26), the method described here uses a different forward method defined in Eq. (50).

[0240] The inventors use the forward operators or matrices B and C (see Eq. (28) and Eq. (54)) to recover the FRI signals in each of the previously discussed methods. The matrices B and C are functions of the measured time instants and sampling kernel. In FIG. 20, the condition numbers of matrices with the same number of 4 L+2 perturbed firing instants are compared as a function of the number of FRI signals L. To this aim, 5000 random sets of monotonic sequences

[00132]${t_n[0,T)}_{n=1}^N$

were generated. As depicted in FIG. 20, the condition number of matrix C is smaller than that of matrix B. This demonstrates that the reconstruction algorithm (Algorithm 3) enhances stability and noise resilience.

[0241] The inventors have evaluated and compared the relative mean square error (MSE) in the estimation of time delays performance for the reconstruction accuracy of the presented algorithm (Algorithm 3) with Algorithm 2 presented above, where the MSE is given by

[00133]$\begin{array}{cc}MSE=10\log \left({.Math.}_{=1}^L{\left({}_{}-{\hat{}}_{}\right)}^2\right),& \left(58\right)\end{array}$