Dual-channel FRI under-sampling and parameter measurement method based on spectrum spreading

Abstract

A kind of double-channel FRI under-sampling and parameter measurement method based on spectrum spreading, comprising the following steps: step one, mathematical modeling is carried out to pulse sequence and base pulse;Step two, the sampling structure of signal is designed in combination with the characteristics of pulse sequence;Step three, the spectrum of pulse sequence is moved;Step four, the spectrum of base pulse is moved;Step five, the sampling kernel filtering and low-speed sampling process of pulse sequence and base pulse;Step six, the non-ideal effect of filter is eliminated, and the pulse sequence is recovered.The sampling rate of the sampling system is reduced, and the flexibility of the under-sampling system is improved.According to the correlation between the sampling samples obtained by the two channels, the influence of non-ideal LPF on parameter recovery accuracy is eliminated, and the parameter reconstruction performance of the system is effectively improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of signal processing, and particularly relates to a double-channel FRI under-sampling and parameter measurement method based on spectrum expansion. BACKGROUND

[0002] Pulse sequence is widely used in radar and other detection fields. According to the traditional Nyquist sampling theorem, when the pulse sequence is sampled, the original parameters must be perfectly reconstructed from the sampling samples, and the sampling rate must be twice the signal bandwidth. However, with the development of technology and our increasing demand for signal resolution, the bandwidth of the pulse sequence increases sharply, which poses a huge challenge to the traditional Nyquist sampling theorem. At the same time, high-precision high-sampling-rate analog-to-digital converters are difficult to implement, and it is difficult to build hardware that runs at a very high rate. The finite rate of innovation (FRI) sampling theory provides a new method for under-sampling the pulse sequence. The finite rate of innovation was first proposed by Vetterli et al. This theory shows that a signal with a limited rate of innovation can be sampled at a rate much lower than the Nyquist rate, and the characteristic parameters of the pulse sequence can be accurately measured.

[0003] Vetterli et al. have proposed various FRI sampling structures, including a single-channel sampling structure using a sinc sampling kernel function, filtering and sampling the input signal using an ideal sampling kernel function, obtaining the samples needed to reconstruct the parameters, and taking the discrete Fourier transform of the samples to obtain the discrete Fourier coefficients of the signal, and then using spectral estimation algorithms to reconstruct the free parameters of the signal. Due to the limited frequency spectrum information obtained by the single-channel, in order to further reduce the sampling rate, Gedalyahu et al. first proposed a multi-channel sampling scheme that can obtain most of the Fourier coefficients distributed in the signal spectrum. Eldar et al. proposed a four-channel crystal receiver, which can sample signals in different frequency bands. However, in order to avoid spectral aliasing, the system is relatively redundant and complex. Dragotti et al. proposed a multi-channel sampling method that combines multiple sampling channels to improve the detection accuracy of the sampling system. Hojjat Akhondi Asl et al. proposed a multi-channel sampling structure based on exponential reproducing kernel E-spline, which not only reduces the order of the sampling kernel of each channel, but also enhances the anti-noise performance of the system.

[0004] However, due to the individual differences of the signal spectrum, the existing FRI sampling system needs to be improved for different pulse shapes, and a suitable sampling core is designed according to the spectrum of the signal, and at present there is still a lack of a general and stable sub-Nyquist sampling method for a variety of pulse streams. At the same time, the non-ideal effect of the sampling core will affect the signal reconstruction, reduce the reconstruction accuracy of the characteristic parameters of the pulse, and the system error is large. In summary, there are still many limitations in the current undersampling system. Therefore, how to effectively improve the universality of the system and eliminate the non-ideal effect of the physical components becomes a key challenge. The solution to these problems is of great significance to the undersampling and parameter measurement of the pulse sequence. SUMMARY

[0005] In order to overcome the shortcomings of the existing undersampling system, which needs to be specially designed according to the pulse spectrum characteristics, the universality is poor, and the influence of the non-ideal effect of the physical components on the parameter reconstruction, the present application proposes a double-channel FRI undersampling and parameter measurement method based on spectrum expansion, which provides an effective idea for the sampling of pulse sequence. The method uses a double-channel parallel sampling structure, and uses the spectrum spreading technology in the random demodulation theory to move the spectrum of the input signal to the entire frequency band. This technology allows the use of baseband to restore the spectrum of any pulse signal, reduces the sampling rate of the sampling system, and improves the flexibility of the undersampling system. According to the correlation between the sampling samples obtained by the two channels, the influence of the non-ideal LPF on the parameter recovery accuracy is eliminated, and the parameter reconstruction performance of the system is effectively improved.

[0006] The technical scheme adopted by the present application to solve its technical problems is:

[0007] A double-channel FRI undersampling and parameter measurement method based on spectrum expansion, comprising the following steps:

[0008] Step 1, mathematical modeling of pulse sequence and basis pulse: assuming that the signal to be measured is composed of a series of pulse signals, wherein the pulse shape of the basis pulse signal h(t) is known, L is the number of basis pulse signals contained in the signal s(t), is the time delay and amplitude of the pulse sequence in the signal to be measured, is unknown and needs to be recovered by the sampling structure and reconstruction algorithm, and then the pulse sequence is reconstructed;

[0009] Step 2, design the sampling structure of the signal in combination with the characteristics of the pulse sequence, in order to improve the universality of the system and avoid the relative limitation of the single-channel sampling structure; a double-channel parallel sampling structure is designed, and the corresponding parameter settings of the proposed sampling system are made;

[0010] Step three, the spectrum shift process of the pulse sequence, according to the proposed sampling structure, the pulse sequence needs to be spectrum shifted using a pseudo-random sequence, which uniformly distributes the information on the main frequency band of the signal to the entire frequency domain axis;

[0011] Step four, the spectrum shift process of the base pulse, according to the proposed sampling structure, the base pulse is spectrum shifted using a cosine signal q(t) = cos(2πft), which shifts the information on the main frequency band of the signal to the low frequency;

[0012] Step five, the sampling kernel filtering and low-speed sampling process of the pulse sequence and the base pulse, according to the proposed sampling structure, the pulse sequence and the base pulse need to be sampled kernel filtered to obtain the frequency domain information of the signal baseband, and finally the discrete Fourier transform (DFT) is performed on the samples obtained to obtain the corresponding discrete Fourier coefficients;

[0013] Step six, eliminating the non-ideal effect of the filter and recovering the pulse sequence, the purpose of eliminating the non-ideal effect of the LPF can be achieved through the obtained samples s2[n] and h2[n], and then the pulse sequence is reconstructed.

[0014] Further, in the step one, the pulse sequence is represented as:

[0015]

[0016] The discrete Fourier transform (DFT) is performed on the pulse sequence s(t) to obtain the Fourier coefficient expression of the signal:

[0017]

[0018] Further, the process of the step two is as follows:

[0019] Step 2.1, generating a pseudo-random sequence through a pseudo-random sequence generator, the pseudo-random sequence is artificially set based on a specific algorithm or rule and has approximate random statistical characteristics, the pseudo-random sequence is used to modulate the to-be-tested signal, which uniformly shifts the frequency domain information of the to-be-tested signal s(t) to the entire frequency spectrum, and the frequency spectrum of any pulse signal can be recovered through the baseband signal. The pseudo-random sequence is represented in the following form:

[0020]

[0021] Wherein T is the duration length of the pseudo-random sequence p(t);

[0022] The multiplication of two signals in the frequency domain means convolution in the frequency domain, the convolution of the to-be-tested signal and the pseudo-random sequence is equivalent to uniformly smearing the frequency domain on the entire frequency domain axis, and the frequency domain representation of the pseudo-random sequence p(t) is:

[0023]

[0024] where T p = T / Q,

[0025] Step 2.2, non-ideal LPF modeling, a double-channel parallel sampling structure is adopted, and non-ideal LPF is used as a sampling kernel for filtering both the pulse sequence and the base pulse. Compared with the original sampling structure, both assume that the sampling kernel is ideal, and the non-ideal sampling kernel has a great influence on the signal reconstruction accuracy. Therefore, the pulse signal and the base signal are required to pass through the non-ideal LPF, so that they both contain non-ideal effects, and the connection between the sampling samples is used to eliminate them in the signal reconstruction stage. The unit impulse response of the sampling kernel is represented as g(t), and its frequency domain expression is G(Ω);

[0026] Step 2.3, a series of Fourier coefficients are obtained, and a parameter estimation algorithm is used to recover the pulse sequence. After the pulse sequence is filtered by the sampling kernel, the signal spectrum located in the baseband can be obtained. The obtained samples are subjected to discrete Fourier transform (DFT) to obtain the related Fourier series coefficients.

[0027] Further, in step three, the shifted signal is represented as s1(t):

[0028] s1(t) = s(t) p(t) (5)

[0029] The frequency domain conversion of the above formula is obtained:

[0030]

[0031] The diagonal frequency variable Ω is subjected to a discrete transformation, so that Ω = mΩ0, where Ω0 = 2π / T, m ∈ Z, and formula (6) is further represented as:

[0032]

[0033] where S1[m] = S1(mΩ0), S[m] = S(mΩ0) and P[m] = P(mΩ0) are the Fourier coefficients of the signals s1(t), s(t) and p(t) respectively, and are further simplified as:

[0034]

[0035] In step four, the shifted signal is represented as h1(t):

[0036] h1(t) = h(t) q(t) (9)

[0037] The frequency domain conversion of the above formula can be obtained:

[0038]

[0039] Discretization transformation is made to the diagonal frequency variable Ω, so that Ω = mΩ0, where Ω0= 2π / T, m∈Z, then formula (10) is further expressed as:

[0040]

[0041] Where H1[m], H[m] and Q[m] are Fourier coefficients of signals h1(t), h(t) and q(t), which are further simplified as:

[0042]

[0043] The process of step five is as follows:

[0044] Step 5.1, sampling kernel filtering process of the measured pulse signal, after the signal modulation, the main frequency domain information of the signal is uniformly distributed on the entire frequency domain axis, so LPF can be used as a sampling kernel to intercept the frequency domain information of the signal baseband, and a non-ideal LPF is used as a sampling kernel for filtering, and the signal after filtering is expressed as:

[0045] s2(t) = s1(t) * g(t) (13)

[0046] The signal s1(t) is expressed in the frequency domain, and the following is obtained:

[0047] S2(Ω) = S1(Ω) · G(Ω) (14)

[0048] Discretization processing is made to the diagonal frequency variable Ω, and formula (14) is further expressed as:

[0049] S2(mΩ0) = S1(mΩ0) · G(mΩ0) (15)

[0050] Where S2[m] = S2(mΩ0), S1[m] = S1(mΩ0) and G[m] = G(mΩ0) are Fourier coefficients of signals s2(t), s1(t) and g(t), which are further simplified as:

[0051]

[0052] Step 5.2, sampling kernel filtering process of the base pulse, using LPF as a sampling kernel to filter the base signal to intercept the frequency domain information of the base signal in the baseband, but since there is no ideal LPF in practice, we use a non-ideal LPF as a sampling kernel for filtering, which can make the base signal contain the frequency spectrum response of the LPF, thereby achieving the purpose of eliminating the frequency domain response of the LPF. The signal after filtering is expressed as:

[0053] h2(t) = h1(t) * g(t) (17)

[0054] The signal h2(t) is represented in the frequency domain, and the following is obtained:

[0055] H2(Ω) = H1(Ω) · G(Ω) (18)

[0056] The diagonal frequency variable Ω is discretely transformed, and the following is further represented:

[0057] H2(mΩ0) = H1(mΩ0) · G(mΩ0) (19)

[0058] Where H2[m], H1[m] and G[m] are the Fourier coefficients of the signals h2(t), h1(t) and g(t), and the following is further simplified:

[0059] H2[m] = H1[m] · G[m] (20)

[0060] Step 5.3, low-speed sampling process of pulse sequence and base pulse. After the pulse sequence and the base pulse are filtered by the sampling kernel, the frequency domain information of the signal baseband can be obtained, and a set of Fourier coefficients is needed to restore the original signal, so the discrete Fourier transform DFT is performed on the obtained samples to obtain the required discrete Fourier coefficients s2[m] and h2[m].

[0061] The process of the sixth step is as follows:

[0062] Step 6.1, eliminating the non-ideal effect of LPF. By utilizing the relationship between the sampling samples of the to-be-tested signal and the base signal, the frequency domain response of the LPF can be effectively eliminated, and by combining formulas (16) and (20), the following is further obtained:

[0063]

[0064] Where h(t) and q(t) are known, and therefore B = 1 / H1[k] is a constant;

[0065] Step 6.2, constructing an observation vector z. Formula (21) is converted into a matrix-vector form, and the observation vector is constructed as follows:

[0066] z = [Z[-M], Z[1-M], …, Z[M]] T (22)

[0067] Observation matrix:

[0068]

[0069] In formula (23), the observation matrix P is a matrix with a dimension of (2M+1) x (2K+1), and the lmth value is P[m-k] (m=-M,..., M, k=-K,..., K), so formula (21) is written as:

[0070] z = B * P * s (24)

[0071] In the above formula, s = [S[-K], S[1-K],..., S[K]] T is a (2K+1) x 1 vector composed of Fourier series of the pulse sequence s(t);

[0072] Step 6.3, quantize the simulation time axis t e [0, T) into N uniform grids, and the size of the grid is set as δ = T / N, so the simulation time quantity t is approximately expressed as t = nδ, n = {0, 1,..., N-1}, and the unknown time delay parameter t is approximately expressed as t l ≈ n l δ, where n k e {0, 1,..., N-1} represents the quantized value of the time delay parameter t l , and after quantization, formula (2) is converted into:

[0073]

[0074] where k = -K, 1-K,..., K, and in order to further operate, the above formula (25) is converted into a matrix-vector form:

[0075]

[0076] where s = [S[-K], S[1-K],..., S[K]] T is a vector with a size of (2K+1) x 1,

[0077]

[0078] is a diagonal matrix with a dimension of (2K+1) x (2K+1);

[0079] Step 6.4, construct the observation matrix Φ, where Φ = PHΨ, P is constructed according to formula (23) in step 6.2, H is constructed according to formula (27) in step 6.3, and then the Ψ matrix needs to be constructed to construct the observation matrix Φ;

[0080] The time value of the pulse sequence s(t) is t e [0, T), and after quantizing the time region according to a fixed interval δ, a complete set representation η = {0, δ, 2δ,..., (N-1)δ} about the simulation time quantity t can be constructed, N = T / δ, t lThe set of γ = {n1δ, n2δ,..., n L δ} and L < N; therefore, after the quantization process, the obtained time delay parameter t l The set of γ constitutes a subset of the complete set of η of the characteristic parameter t, i.e. The set of γ is represented by the complete set of η, i.e.:

[0081]

[0082] wherein Ψ is a matrix of (2K+1)×N dimensions, the knth element (k=-K, 1-K,..., K, n=0, 1,..., N-1) is composed of Formula (25) can be converted into the expression of a sparse matrix:

[0083] s = HΨa (29)

[0084] wherein a∈R N×1 is an L sparse vector, and the position of the non-zero element represents and the non-zero element value is a, combining formula (24) and (29), it is further converted into:

[0085] z = BPHΨa = Φa (30)

[0086] In the above formula, Φ = BPHΨ is an observation matrix, and the dimension is (2K+1)×N;

[0087] Step 6.5, solving z = Φa, by solving the non-zero element value of the sparse vector a from the observation vector y, calculating the expression, converting it into an L0 norm minimization problem, i.e.

[0088]

[0089] wherein the L0 norm ||a||0 represents the number of non-zero elements in the sparse vector a, according to the research of the random demodulation theory, by using the pseudo-random sequence to construct the random observation matrix Φ, it can satisfy the RIP constraint condition, on the basis of the observation matrix Φ satisfying the RIP condition, the OMP algorithm is an effective method for solving formula (31), thereby the characteristic parameter of the pulse sequence is restored.

[0090] The beneficial effects of the present application mainly manifest in that the traditional Nyquist sampling rate is high, the undersampling technology can effectively reduce the sampling rate of the signal, but due to the great difference in the spectral characteristics of the signal, for different frequency domain signals to be measured, a suitable sampling core needs to be designed according to the needs, which brings great difficulty to the design of the sampling system, and at the same time, the non-ideal effect of the sampling core also needs to be considered. The method of the present application is based on the technology of spectral expansion, adopts a double-channel sampling structure, eliminates the non-ideal effect of the filter in the frequency domain through the correlation between the pulse sequence and the base pulse frequency domain. At the same time, the pulse sequence is modulated by using a pseudo-random sequence, the main frequency band of the signal is uniformly distributed on the entire frequency domain axis, and then a low-pass filter is used as a sampling core to intercept the frequency domain information of the base band and restore the characteristic parameters in the signal. While improving the parameter estimation performance, the universality of the system is greatly improved. BRIEF DESCRIPTION OF DRAWINGS

[0091] Figure 1 is a general undersampling system structure diagram in a non-ideal environment.

[0092] Figure 2 is a robustness experimental result graph. DETAILED DESCRIPTION

[0093] The present application will be further described below in combination with the drawings.

[0094] Reference Figure 1 and Figure 2 , a double-channel FRI undersampling and parameter measurement method based on spectral expansion, the system structure is shown in Figure 1 , the method comprises the following steps:

[0095] Step one, mathematical modeling of the pulse sequence and the base pulse: it is assumed that the signal to be measured is composed of a series of pulse signals, the pulse shape of the base pulse signal h(t) is known, L is the number of base pulse signals contained in the signal s(t), is the time delay and amplitude of the pulse sequence in the signal to be measured. Of course is unknown, and needs to be recovered through the sampling structure and the reconstruction algorithm, and then the pulse sequence is reconstructed. The pulse sequence is represented as:

[0096]

[0097] Discrete Fourier transform (DFT) is performed on the pulse sequence s(t), and the Fourier coefficient expression of the signal is:

[0098]

[0099] Step two, according to the characteristics of the pulse sequence, the sampling structure of the signal is designed, in order to improve the universality of the system and avoid the problem of relative limitation of single channel sampling structure, a double channel parallel sampling structure is designed, as shown in Figure 1 The corresponding parameter settings of the proposed sampling system are as follows:

[0100] Step 2.1, generate pseudo-random sequence through pseudo-random sequence generator, pseudo-random sequence is artificially set based on specific algorithm or rule, has approximate random statistical characteristics, use pseudo-random sequence to modulate the signal to be measured, the frequency domain information of the signal to be measured s(t) is uniformly moved to the whole frequency spectrum, the frequency spectrum of any pulse signal can be recovered through baseband signal, the pseudo-random sequence is represented as follows:

[0101]

[0102] Among them T is the duration length of the pseudo-random sequence p(t);

[0103] The multiplication of two signals in the frequency domain means convolution in the frequency domain, the convolution of the signal to be measured and the pseudo-random sequence is equivalent to uniformly smear the frequency domain on the whole frequency domain axis, the frequency domain representation of the pseudo-random sequence p(t) is:

[0104]

[0105] Among them T p =T / Q,

[0106] Step 2.2, non-ideal LPF modeling, as shown in Figure 1 The double channel parallel sampling structure is adopted, and non-ideal LPF is used as the sampling core for filtering the pulse sequence and the base pulse. Compared with the original sampling structure, the sampling core is ideal, and the non-ideal sampling core has a great influence on the signal reconstruction accuracy. Therefore, the pulse signal and the base signal are filtered through the non-ideal LPF, so that the non-ideal effect is included, and the unit impulse response of the sampling core is represented as g(t), and its frequency domain expression is G(Ω);

[0107] Step 2.3, a series of Fourier coefficients are obtained, and the parameter estimation algorithm is used to recover the pulse sequence. After the pulse sequence is filtered by the sampling core, the signal spectrum located in the baseband can be obtained. Discrete Fourier transform (DFT) is performed on the obtained samples, and the related Fourier series coefficients can be obtained.

[0108] Step three, the spectrum shifting process of the pulse sequence, according to the proposed sampling structure, a pseudo-random sequence is used to shift the spectrum of the pulse sequence, and information on the main frequency band of the signal is uniformly distributed on the entire frequency domain axis. The signal after shifting is represented as s1(t):

[0109] s1(t) = s(t) p(t) (36)

[0110] The frequency domain conversion of the above formula is obtained:

[0111]

[0112] The diagonal frequency variable Ω is discretely transformed, so that Ω = mΩ0, where Ω0 = 2π / T, m∈Z, and formula (6) is further represented as:

[0113]

[0114] Where S1[m] = S1(mΩ0), S[m] = S(mΩ0) and P[m] = P(mΩ0) are the Fourier coefficients of the signals s1(t), s(t) and p(t) respectively, and the above formula is further simplified as:

[0115]

[0116] Step four, the spectrum shifting process of the base pulse, according to the proposed sampling structure, a cosine signal q(t) = cos(2πft) is used to shift the spectrum of the base signal, and the information on the main frequency band of the signal is shifted to the low frequency. The signal after shifting is represented as h1(t):

[0117] h1(t) = h(t) q(t) (40)

[0118] The continuous time Fourier transform (CTFT) of the above formula is obtained:

[0119]

[0120] The diagonal frequency variable Ω is discretely transformed, so that Ω = mΩ0, where Ω0 = 2π / T, m∈Z, and formula (10) is further represented as:

[0121]

[0122] Where H1[m], H[m] and Q[m] are the Fourier coefficients of the signals h1(t), h(t) and q(t), and the above formula is further simplified as:

[0123]

[0124] Step five, the sampling kernel filtering of pulse sequence and base pulse, according to the proposed sampling structure, the sampling kernel filtering of pulse sequence and base pulse is needed to obtain the frequency domain information of signal baseband, and finally the discrete Fourier transform (DFT) is performed on the samples obtained by sampling to obtain the corresponding discrete Fourier coefficients, the process is as follows:

[0125] Step 5.1, the sampling kernel filtering process of the pulse signal to be measured. After the signal modulation, the main frequency domain information of the signal is uniformly distributed on the entire frequency domain axis, so the LPF can be used as the sampling kernel to intercept the frequency domain information of the signal baseband. The non-ideal LPF is used as the sampling kernel for filtering, and the signal after filtering is represented as:

[0126] s2(t)=s1(t)*g(t) (44)

[0127] The signal s1(t) is represented in the frequency domain to obtain:

[0128] S2(Ω)=S1(Ω)·G(Ω) (45)

[0129] The angle frequency variable Ω is discretized, and formula (14) is further represented as:

[0130] S2(mΩ0)=S1(mΩ0)·G(mΩ0) (46)

[0131] Where S2[m]=S2(mΩ0), S1[m]=S1(mΩ0) and G[m]=G(mΩ0) are the Fourier coefficients of signals s2(t), s1(t) and g(t) respectively, and the above formula is further simplified as:

[0132]

[0133] Step 5.2, the sampling kernel filtering process of the base pulse, the LPF is used as the sampling kernel to filter the base signal and intercept the frequency domain information of the base signal in the baseband. However, there is no ideal LPF in practice, so a non-ideal LPF is used as the sampling kernel for filtering, so that the base signal contains the frequency spectrum response of the LPF, thereby achieving the purpose of eliminating the frequency domain response of the LPF. The signal after filtering is represented as:

[0134] h2(t)=h1(t)*g(t) (48)

[0135] The signal h2(t) is represented in the frequency domain to obtain:

[0136] H2(Ω)=H1(Ω)·G(Ω) (49)

[0137] Discretizing the diagonal frequency variable Ω, the above equation can be further expressed as:

[0138] H2(mΩ0) = H1(mΩ0) · G(mΩ0) (50)

[0139] where H2[m], H1[m] and G[m] are the Fourier coefficients of the signals h2(t), h1(t) and g(t), the above equation can be further simplified as:

[0140] H2[m] = H1[m] · G[m] (51)

[0141] Step 5.3, low-speed sampling process of pulse sequence and base pulse, after the pulse sequence and base pulse pass through the sampling kernel filter, the frequency domain information of the signal baseband can be obtained, a set of Fourier coefficients are needed to restore the original signal, therefore, the obtained samples are subjected to discrete Fourier transform (DFT) to obtain the required discrete Fourier coefficients s2[m] and h2[m];

[0142] Step six, eliminating the non-ideal effect of the filter and restoring the pulse sequence, the purpose of eliminating the non-ideal effect of the LPF can be achieved through the obtained samples s2[n] and h2[n], and then the pulse sequence is reconstructed, the process is as follows:

[0143] Step 6.1, eliminating the non-ideal effect of the LPF. By utilizing the relationship between the sampling samples of the to-be-tested signal and the base signal, the frequency domain response of the LPF can be effectively eliminated, combining equations (16) and (20), the following equation is further obtained:

[0144]

[0145] where h(t) and q(t) are known, so B = 1 / H1[k] is a constant;

[0146] Step 6.2, constructing an observation vector z, in order to facilitate processing, equation (21) is converted into a matrix-vector form to construct an observation vector:

[0147] z = [Z[-M], Z[1-M], …, Z[M]] T (53)

[0148] Observation matrix:

[0149]

[0150] In equation (23), the observation matrix P is a matrix with a dimension of (2M+1) x (2K+1), and the l m value is P[m-k] (m = -M, …, M, k = -K, …, K), so equation (21) can be written as:

[0151] z = B - P - s (55)

[0152] In the above formula, s = [S[-K], S[1-K],..., S[K]] T is a (2K+1) x 1 vector composed of Fourier series of the pulse sequence s(t);

[0153] Step 6.3, quantize the simulation time axis t e [0, T) into N uniform grids, and the size of the grid is set as δ = T / N. Therefore, the simulation time quantity t is approximately expressed as t = nδ, n = {0, 1,..., N-1}, and the unknown time delay parameter t is approximately expressed as t l ≈ n l δ, where n k e {0, 1,..., N-1} represents the quantized value of the time delay parameter t l , and after quantization, formula (2) is converted into:

[0154]

[0155] Where k = -K, 1-K,..., K, in order to further operate, the above formula (25) is converted into a matrix-vector form:

[0156]

[0157] Where s = [S[-K], S[1-K],..., S[K]] T is a (2K+1) x 1 vector,

[0158]

[0159] is a (2K+1) x (2K+1) dimensional diagonal matrix;

[0160] Step 6.4, construct the observation matrix Φ, where Φ = PHΨ, P is constructed according to formula (23) in step 6.2, H is constructed according to formula (27) in step 6.3, and then the Ψ matrix needs to be constructed to construct the observation matrix Φ;

[0161] The time value of the pulse sequence s(t) is t e [0, T), and after quantizing the time region according to a fixed interval δ, a complete set representation η = {0, δ, 2δ,..., (N-1)δ} about the simulation time quantity t can be constructed, N = T / δ, and the set of t l can be expressed as γ = {n1δ, n2δ,..., n L δ}, and L << N, therefore, after quantization, the set γ of the obtained time delay parameter t l constitutes a subset of the complete set η of the characteristic parameter t, that is We use a complete set η to represent the set of time delay parameters γ, that is:

[0162]

[0163] Here, Ψ is a (2K+1)×N dimension matrix, and the kn-th element (k=-K,1-K,…,K,n=0,1,…,N-1) is determined by… The composition, formula (25) can be transformed into sparse moments.

[0164] The form of array representation:

[0165] s=HΨa (60)

[0166] In the formula, a∈R N×1 It is an L sparse vector, where the positions of the non-zero elements are represented. The value of the non-zero element is then obtained by combining formulas (24) and (29), which further transforms into:

[0167] z=BPHΨa=Φa (61)

[0168] In the above formula, Φ=BPHΨ is an observation matrix with dimensions (2K+1)×N;

[0169] Step 6.5, Solve for z = Φa. This can be achieved by finding the non-zero elements of the sparse vector a from the observed vector y. The most classic method for calculating this expression is to transform it into an L0 norm minimization problem, i.e.:

[0170]

[0171] Where L0 norm ||a||0 represents the number of non-zero elements in sparse vector a. According to the study of random demodulation theory, the random observation matrix Φ constructed by using pseudo-random sequence can satisfy the RIP constraint condition. On the basis that the observation matrix Φ satisfies the RIP condition, the OMP algorithm can be used as an effective method to solve formula (31) and thus recover the characteristic parameters of the pulse sequence.

[0172] To verify the feasibility of the method of this invention, relevant simulation experiments were conducted in this section. The pulse sequence uses... This indicates that the unknown parameters in the signal are set, with the amplitude parameter set to a. l = [0.83, 0.64, 0.35, 0.17, 0.97], with the time delay parameter set to t. l = [0.31, 0.42, 0.55, 0.67, 0.89]. The base signal of the pulse sequence is known. Setting the initial signal sampling rate to 20 kHz, undersampling it using the method of this invention allows sampling at a minimum of 2000 Hz.

[0173] Experiment 1: Ideal experiment. The method of the present application and the classical FRI sampling method are compared, and an ideal LPF is used as a sampling kernel for sampling, and no noise is introduced in the process, and the original signal is recovered by the parameter recovery algorithm, and the experimental results are shown in Table 1. From the experimental results, it can be seen that when an ideal sampling kernel is used, all methods are feasible. Table 1 is the parameter recovery result,

[0174]

[0175] Table 1

[0176] Experiment 2: Non-ideal experiment. Compared with experiment 1, a non-ideal LPF is used as a sampling kernel for experiment, and noise is added in the experiment process, and the experimental results are shown in Table 2. From the results, it can be seen that the method of the present application can accurately recover the free parameters of the signal to be measured in a non-ideal environment. At the same time, the classical FRI sampling method cannot eliminate the non-ideal effect of the filter, resulting in low reconstruction accuracy.

[0177]

[0178] Table 2

[0179] Experiment 3: Robustness experiment. Compared with experiment 2, the classical FRI sampling method, the method of the present application and the multi-channel sampling method are subjected to robustness experiment, and Gaussian white noise is added to the sampling system, and the signal-to-noise ratio is from -20dB to 60dB, with a step of 10dB. In order to more intuitively compare the differences between the systems, the normalized mean square error is introduced for evaluation, and the experimental results are shown in Figure 2 From the figure, it can be seen that the method of the present application can effectively improve the anti-noise performance and robustness of the entire sampling system, and improve the accuracy of the system in recovering free parameters, and is a universal sub-Nyquist sampling method.

[0180] The embodiments of the present application are only a list of implementation forms of the inventive concept, and are only for illustrative purposes. The protection scope of the present application should not be regarded as being limited to the specific forms described in the embodiments, and the protection scope of the present application also extends to equivalent technical means that can be thought of by those skilled in the art according to the inventive concept.

Claims

1. A dual-channel FRI under-sampling and parameter measurement method based on spectrum spreading, characterized in that, The method comprises the following steps: Step one, mathematical modeling of pulse sequence and base pulse: assume that the signal to be measured is composed of a series of pulse signals, where the pulse shape of the base pulse signal h(t) is known, L is the number of base pulse signals contained in the pulse sequence signal s(t), is the amplitude and time delay of the pulse sequence in the signal to be measured, is unknown and needs to be recovered by the sampling structure and reconstruction algorithm, and then the pulse sequence is reconstructed; Step two, according to the characteristics of the pulse sequence, the sampling structure of the signal is designed, in order to improve the universality of the system and avoid the problem of relative limitation of single channel sampling structure; A kind of double channel parallel sampling structure is designed, and the corresponding parameter setting of the proposed sampling structure is carried out; Step three, the frequency spectrum moving process of pulse sequence, according to the proposed sampling structure, the pulse sequence needs to be frequency spectrum moved using pseudo-random sequence, and the information of the main frequency band of the signal is uniformly distributed on the whole frequency domain axis; Step four, the frequency spectrum moving process of base pulse, according to the proposed sampling structure, the base pulse is frequency spectrum moved using cosine signal q(t)=cos(2πft), and the information of the main frequency band of the signal is moved to low frequency; Step five, the sampling kernel filtering and low-speed sampling process of pulse sequence and base pulse, according to the proposed sampling structure, the pulse sequence and the base pulse need to be sampled kernel filtered to obtain the frequency domain information of the signal baseband, and finally the discrete Fourier transform (DFT) is carried out on the samples obtained to obtain the corresponding discrete Fourier coefficients s2[n] and h2[n]; Step six, the non-ideal effect of the filter is eliminated, and the pulse sequence is recovered, the purpose of eliminating the non-ideal effect of the LPF can be realized through the obtained samples s2[n] and h2[n], and then the pulse sequence is reconstructed.

2. The dual-channel FRI under-sampling and parameter measurement method based on spectrum spreading of claim 1, wherein, In step one, the pulse sequence is represented as: The discrete Fourier transform (DFT) of the pulse sequence s(t) is obtained, and the Fourier coefficient expression of the signal is:

3. The double-channel FRI under-sampling and parameter measurement method based on spectrum expansion according to claim 2, characterized in that, In step three, a pseudo-random sequence is generated by a pseudo-random sequence generator, which is artificially set based on a specific algorithm or rule and has approximate random statistical characteristics. The pseudo-random sequence is used to modulate the measured signal, so that the frequency domain information of the measured signal s(t) is uniformly moved to the whole frequency spectrum. The spectrum of any pulse signal is recovered through the baseband signal. The pseudo-random sequence is represented as follows: wherein T is the duration length of the pseudo-random sequence p(t); The multiplication of two signals in the frequency domain means convolution in the frequency domain. The convolution of the measured signal and the pseudo-random sequence is equivalent to uniformly smearing the frequency domain on the whole frequency domain axis. The frequency domain representation of the pseudo-random sequence p(t) is as follows: where T p = T / Q, Ω is the angular frequency variable; In step five, the non-ideal LPF modeling, a double-channel parallel sampling structure is adopted, and a non-ideal LPF is used as a sampling kernel to filter the pulse sequence and the base pulse. Compared with the original sampling structure, the sampling kernel is assumed to be ideal. The non-ideal sampling kernel has a great influence on the signal reconstruction accuracy. Therefore, the pulse signal and the base signal are required to pass through the non-ideal LPF, so that they contain non-ideal effects, and the unit impulse response of the sampling kernel is represented as g(t), and its frequency domain expression is G(Ω). In step five, the pulse sequence is recovered by obtaining a series of Fourier coefficients using a parameter estimation algorithm, and when the pulse sequence is filtered by a sampling kernel, the signal spectrum at the baseband can be obtained, and the obtained samples are subjected to a discrete Fourier transform (DFT) to obtain relevant discrete Fourier coefficients.

4. The spectrum spread-based dual-channel FRI undersampling and parameter measurement method of claim 3, wherein, In step three, the signal after shifting is represented as s1(t): s1(t)=s(t)·p(t) (5) The frequency domain conversion of the above formula is obtained: Discretize the angular frequency variable Omega, so that Omega=mOmega0, where Omega0=2pi / T, m∈Z, and then formula (6) is further represented as: Where S1[m]=S1(mOmega0), S[m]=S(mOmega0) and P[m]=P(mOmega0) are the Fourier coefficients of signals s1(t), s(t) and p(t) respectively, and are further simplified as:

5. The spectrum spread-based dual-channel FRI undersampling and parameter measurement method of claim 4, wherein, In step four, the signal after shifting is represented as h1(t): h1(t)=h(t)·q(t) (9) The frequency domain conversion of the above formula can be obtained: Discretize the angular frequency variable Omega, so that Omega=mOmega0, where Omega0=2pi / T, m∈Z, and then formula (10) is further represented as: Where H1[m], H[m] and Q[m] are the Fourier coefficients of signals h1(t), h(t) and q(t), and are further simplified as:

6. The spectrum spread-based dual-channel FRI undersampling and parameter measurement method of claim 5, wherein, The process of step five is as follows: Step 5.1, the sampling kernel filtering process of the measured pulse signal, after the signal is modulated, the main frequency domain information of the signal is uniformly distributed on the entire frequency domain axis, therefore, an LPF is used as a sampling kernel to intercept the frequency domain information of the signal at the baseband, a non-ideal LPF is used as a sampling kernel for filtering, and the signal after filtering is represented as: s2(t)=s1(t)*g(t) (13) The signal s1(t) is represented in the frequency domain, and the following is obtained: S2(Ω)=S1(Ω)·G(Ω) (14) Discretize the angular frequency variable Omega, and then formula (14) is further represented as: S2(mOmega0)=S1(mOmega0)·G(mOmega0) (15) Where S2[m]=S2(mOmega0), S1[m]=S1(mOmega0) and G[m]=G(mOmega0) are the Fourier coefficients of signals s2(t), s1(t) and g(t) respectively, and are further simplified as: Step 5.2, the sampling kernel filtering process of the base pulse, an LPF is used as a sampling kernel to filter the base signal and intercept the frequency domain information of the base signal at the baseband, but since there is no ideal LPF in practice, a non-ideal LPF is used as a sampling kernel for filtering, so that the base signal contains the frequency spectrum response of the LPF, thereby achieving the purpose of eliminating the frequency domain response of the LPF, and the signal after filtering is represented as: h2(t)=h1(t)*g(t) (17) The signal h2(t) is represented in the frequency domain, and the following is obtained: H2(Ω)=H1(Ω)·G(Ω) (18) Discretize the angular frequency variable Omega, and further represent as: H2(mΩ0) = H1(mΩ0) G(mΩ0) (19) where H2[m], H1[m] and G[m] are Fourier coefficients of the signals h2(t), h1(t) and g(t), and are further simplified as: H2[m] = H1[m] G[m] (20) Step 5.3, low-speed sampling process of pulse sequence and base pulse, after the pulse sequence and the base pulse pass through the sampling kernel filter, the frequency domain information of the signal baseband can be obtained, and a set of Fourier coefficients are required to restore the original signal, therefore, discrete Fourier transform (DFT) is performed on the obtained samples to obtain the required discrete Fourier coefficients s2[m] and h2[m].

7. The spectrum spread-based dual-channel FRI undersampling and parameter measurement method of claim 6, wherein, The process of the step six is as follows: Step 6.1, eliminating the non-ideal effect of the LPF, by utilizing the relationship between the sampling samples of the to-be-tested signal and the base signal, the frequency domain response of the LPF can be effectively eliminated, combining formulas (16) and (20), and further obtaining: where h(t) and q(t) are known, and therefore B = 1 / 2πH1[m] is a constant; Step 6.2, constructing an observation vector z, converting formula (21) into a matrix-vector form to construct an observation vector: z = [Z[-M], Z[1-M],..., Z[M]] T (22) The observation matrix: In formula (23), the observation matrix P is a matrix with a dimension of (2M+1)×(2K+1), and the lm value is P[m-k] (m = -M, …, M, k = -K, …, K), so formula (21) is written as: z = B P s (24) In the above equation, s = [S[-K], S[1-K],..., S[K]] T is a (2K + 1) x 1 vector consisting of the Fourier series coefficients of the pulse sequence s(t); Step 6.3, quantize the analog time axis t e [0, T) into N uniform grids, the size of the grid is set as δ = T / N, so the analog time quantity t is approximately expressed as t = nδ, n = {0, 1, …, N-1}, and the unknown time delay parameter t is approximately expressed as t l ≈ n l δ, wherein n l e {0, 1, …, N-1} represents the quantized value of the time delay parameter t l After quantization, formula (2) is converted into: where k = -K, 1-K, …, K, in order to further operate and process, the above formula (25) is converted into a matrix-vector form: where s = [S[-K], S[1-K],..., S[K]] T is a vector of size (2K + 1) x 1, is a diagonal matrix with a dimension of (2K+1)×(2K+1); Step 6.4, constructing an observation matrix Φ, where Φ = PHΨ, P is constructed according to formula (23) in step 6.2, H is constructed according to formula (27) in step 6.3, and then the matrix Ψ needs to be constructed, so as to construct the observation matrix Φ; The time values of the pulse sequence s(t) are t e [0, T). When the time region is quantized with a fixed interval δ, the complete set representation of the analog time quantity t can be constructed as η = {0, δ, 2δ, …, (N-1)δ}, N = T / δ, t e η l The set representation of the time delay parameter t is γ = {n1δ, n2δ, …, n L δ}, and L << N; therefore, after the quantization process, the obtained set γ of the time delay parameter t l is a subset of the complete set η of the characteristic parameter t, i.e. The complete set η is used to represent the set γ of the time delay parameter, i.e.: where Ψ is a (2K+1) x N matrix, the knth element (k = -K, 1-K,..., K, n = 0, 1,..., N-1) is given by Equation (25) can be converted into a sparse matrix representation as follows: s = H Ψ a (29) where a ∈ R N×1 is an L-sparse vector, the positions of nonzero elements represent and the nonzero element value is Then, combining equations (24) and (29), we further transform it into: z = BPHΨa = Φa (30) In the above formula, Φ = BPHΨ is an observation matrix with a dimension of (2K+1)×N; Step 6.5, solving z = Φa, by solving the non-zero element value of the sparse vector a from the observation vector y, the expression is calculated, which is converted into an L0 norm minimization problem, that is: where the L0 norm ||a||0 represents the number of non-zero elements in the sparse vector a, according to the research of the random demodulation theory, by utilizing the pseudo-random sequence to construct the random observation matrix Φ, the RIP constraint condition can be met, on the basis that the observation matrix Φ meets the RIP condition, the OMP algorithm is an effective method for solving formula (31), so as to restore the characteristic parameters of the pulse sequence.

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