Dual-channel FRI undersampling and parameter measuring method based on spectrum expansion

Abstract

The invention discloses a dual-channel FRI undersampling and parameter measurement method based on spectrum expansion. The method comprises the following steps: step 1, carrying out mathematical modeling on a pulse sequence and a base pulse; step 2, designing a sampling structure of the signal in combination with the characteristics of the pulse sequence; step 3, shifting the frequency spectrum of the pulse sequence; step 4, carrying out a frequency spectrum shifting process of the base pulse; 5, sampling kernel filtering and low-speed sampling of the pulse sequence and the base pulse; and 6, eliminating the non-ideal effect of the filter, and recovering the pulse sequence. According to the invention, 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 samples obtained by the two channels, the influence of the non-ideal LPF on the parameter recovery precision is eliminated, and the parameter reconstruction performance of the system is effectively improved.

Description

Technical Field

[0001] The invention belongs to the technical field of signal processing, and in particular relates to a dual-channel FRI undersampling and parameter measurement method based on spectrum expansion. Background Art

[0002] Pulse sequences are widely used in radar and other detection fields. According to the traditional Nyquist sampling theorem, when sampling a pulse sequence, in order to perfectly reconstruct the original parameters from the sampled samples, the sampling must be performed at a rate twice the signal bandwidth. However, with the development of technology and our increasing requirements for signal resolution, the bandwidth of the pulse sequence has increased dramatically, which poses a huge challenge to the traditional Nyquist sampling theorem. At the same time, high-precision and high-sampling-rate analog-to-digital converters are also difficult to achieve, and it is difficult to build hardware that runs at very high rates. The finite innovation rate (FRI) sampling theory provides us with a new method for undersampling pulse sequences. The finite innovation rate was first proposed by Vetterli et al. This theory shows that a signal with a finite innovation rate can be sampled at a rate much smaller 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. The ideal sampling kernel function is used to filter and sample the input signal to obtain the samples required for the reconstruction parameters. The discrete Fourier transform of the signal can obtain the discrete Fourier coefficients of the signal, and then the spectrum estimation algorithm is used to reconstruct the free parameters of the signal. Since the spectrum information obtained by a single channel is limited, in order to further reduce the sampling rate, Gedalyahu et al. first proposed a multi-channel sampling scheme, which can obtain most of the Fourier coefficients distributed on the signal spectrum. Eldar et al. proposed a four-channel crystal receiver. This sampling method can sample signals in different frequency bands. However, in order to avoid spectrum aliasing, the system is 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 the exponential reproducing kernel E-spline. This method not only reduces the sampling kernel order of each channel, but also enhances the system's anti-noise performance.

[0004] However, due to the individual differences in signal spectra, the existing FRI sampling system needs to be improved for different pulse shapes, and a suitable sampling kernel needs to be designed according to the signal spectrum. At present, there is still a lack of a universal and stable sub-Nyquist sampling method for a variety of pulse streams. At the same time, the non-ideal effects of the sampling kernel will affect the signal reconstruction, reduce the reconstruction accuracy of the characteristic parameters of the pulse, and cause large system errors. In summary, there are still many limitations in the current undersampling system. Therefore, how to effectively improve the versatility of the system while eliminating the non-ideal effects of physical components has become a key challenge. The solution to these problems is of great significance to the undersampling and parameter measurement of pulse sequences. Summary of the invention

[0005] In order to overcome the shortcomings of existing undersampling systems, such as the need for special design based on pulse spectrum characteristics, poor versatility, and the influence of non-ideal effects of physical components on parameter reconstruction, the present invention proposes a dual-channel FRI undersampling and parameter measurement method based on spectrum expansion, which provides an effective idea for sampling pulse sequences. This method utilizes a dual-channel parallel sampling structure and uses the spread spectrum technology in 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 sampled samples obtained from the two channels, the influence of non-ideal LPF on the accuracy of parameter recovery is eliminated, effectively improving the parameter reconstruction performance of the system.

[0006] The technical solution adopted by the present invention to solve its technical problem is:

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

[0008] Step 1: Mathematically model the 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, and L is the number of base pulse signals contained in the signal s(t). is the time delay and amplitude of the pulse train in the signal to be measured, are unknown, and these parameters need to be recovered through sampling structure and reconstruction algorithm to reconstruct the pulse sequence;

[0009] Step 2: Combined with the characteristics of the pulse sequence, the sampling structure of the signal is designed. In order to improve the versatility of the system and avoid the relative limitations of the single-channel sampling structure, a dual-channel parallel sampling structure is designed, and the corresponding parameters of the proposed sampling system are set;

[0010] Step 3: Spectrum shifting process of the pulse sequence. According to the proposed sampling structure, a pseudo-random sequence needs to be used to shift the spectrum of the pulse sequence, so as to evenly distribute the information on the main frequency band of the signal to the entire frequency domain axis.

[0011] Step 4: Spectrum shifting process of base pulse: according to the proposed sampling structure, the spectrum of base pulse is shifted using cosine signal q(t)=cos(2πft), so as to shift the information on the main frequency band of the signal to the low frequency.

[0012] Step 5: Sampling kernel filtering and low-speed sampling process of pulse sequence and base pulse. According to the proposed sampling structure, it is necessary to perform sampling kernel filtering on the pulse sequence and base pulse to obtain the frequency domain information of the signal baseband, and finally perform discrete Fourier transform (DFT) on the sampled samples to obtain the corresponding discrete Fourier coefficients.

[0013] Step six, eliminate the non-ideal effects of the filter and restore the pulse sequence. The purpose of eliminating the non-ideal effects of the LPF can be achieved by obtaining samples s2[n] and h2[n], and then the pulse sequence can be reconstructed.

[0014] Further, in step 1, the pulse sequence is expressed as:

[0015]

[0016] The discrete Fourier transform (DFT) of the pulse sequence s(t) is obtained, and the Fourier coefficient expression of the signal is obtained as follows:

[0017]

[0018] Furthermore, the process of step 2 is as follows:

[0019] Step 2.1, generate 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 a nearly random statistical characteristic. Use the pseudo-random sequence to modulate the signal to be tested, and evenly move the frequency domain information of the signal to be tested s(t) to the entire spectrum. The spectrum of any pulse signal can be restored through the baseband signal. The pseudo-random sequence is expressed as follows:

[0020]

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

[0022] Multiplying two signals in the frequency domain means convolution in the frequency domain. The signal to be tested is convolved with the pseudo-random sequence, which is equivalent to evenly smearing the frequency domain on the entire frequency domain axis. 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, using a dual-channel parallel sampling structure, using non-ideal LPF as the sampling kernel for filtering both the pulse sequence and the base pulse. Compared with the original sampling structure, it is assumed that the sampling kernel is ideal, and the non-ideal sampling kernel has a great influence on the signal reconstruction accuracy. Therefore, it is necessary to pass the pulse signal and the base signal through the non-ideal LPF to make them both contain non-ideal effects, so as to eliminate them by using the connection between the sampling samples in the signal reconstruction stage. The unit impulse response of the sampling kernel is expressed as g(t), and its frequency domain expression is G(Ω);

[0026] Step 2.3: Obtain a series of Fourier coefficients and use the parameter estimation algorithm to recover the pulse sequence. After the pulse sequence is filtered by the sampling kernel, the signal spectrum at the baseband can be obtained. Perform discrete Fourier transform DFT on the obtained samples to obtain the relevant Fourier series coefficients.

[0027] Furthermore, in step 3, the signal after the shift is expressed as s1(t):

[0028] s1(t)=s(t)·p(t) (5)

[0029] Convert the above formula to frequency domain and get:

[0030]

[0031] Discretize the angular frequency variable Ω so that Ω=mΩ0, where Ω0=2π / T,m∈Z, then formula (6) can be further expressed as:

[0032]

[0033] Where S1[m]=S1(mΩ0), S[m]=S(mΩ0) and P[m]=P(mΩ0) are the Fourier coefficients of signals s1(t), s(t) and p(t), respectively, which can be further simplified to:

[0034]

[0035] In step 4, the signal after the shift is expressed as h1(t):

[0036] h1(t)=h(t)·q(t) (9)

[0037] By converting the above formula into frequency domain, we can get:

[0038]

[0039] The angular frequency variable Ω is discretized 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 the Fourier coefficients of signals h1(t), h(t) and q(t), which can be further simplified to:

[0042]

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

[0044] Step 5.1, sampling kernel filtering process of the pulse signal to be tested. After the signal to be tested is modulated, the main frequency domain information of the signal is evenly distributed on the entire frequency domain axis. Therefore, LPF can be used as the sampling kernel to intercept the frequency domain information of the signal baseband, and non-ideal LPF is used as the sampling kernel for filtering. The signal after filtering is expressed as:

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

[0046] Representing the signal s1(t) in the frequency domain, we get:

[0047] S2(Ω)=S1(Ω)·G(Ω) (14)

[0048] By discretizing the angular frequency variable Ω, formula (14) can be 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 the Fourier coefficients of signals s2(t), s1(t) and g(t), respectively, which can be further simplified to:

[0051]

[0052] Step 5.2, the sampling kernel filtering process of the base pulse, uses LPF as the sampling kernel to filter the base signal, and intercepts the frequency domain information of the base signal in the baseband. However, since there is no ideal LPF in practice, we use a non-ideal LPF as the sampling kernel for filtering, so that the base signal can contain the 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] Representing the signal h2(t) in the frequency domain, we get:

[0055] H2(Ω)=H1(Ω)·G(Ω) (18)

[0056] The angular frequency variable Ω is discretized and further expressed as:

[0057] H2(mΩ0)=H1(mΩ0)·G(mΩ0) (19)

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

[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 base pulse are filtered by the sampling kernel, the frequency domain information of the signal baseband can be obtained. A set of Fourier coefficients is required 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].

[0061] The process of step six is ​​as follows:

[0062] Step 6.1: Eliminate the non-ideal effect of LPF. By utilizing the relationship between the sampled samples of the measured signal and the base signal, the frequency domain response of LPF can be effectively eliminated. Combining formulas (16) and (20), we can further obtain:

[0063]

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

[0065] Step 6.2: Construct the observation vector z. Convert formula (21) into a matrix-vector form to construct the observation vector:

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

[0067] Observation matrix:

[0068]

[0069] In formula (23), the observation matrix P is a matrix of dimension (2M+1)×(2K+1), and the lmth value is P[mk] (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 It is a (2K+1)×1 vector composed of the Fourier coefficients of the pulse sequence s(t);

[0072] Step 6.3: Quantize the simulation time axis t∈[0,T) and divide it into N uniform grids. The size of the grid is set to δ=T / N, so the simulation time quantity t is approximately expressed as t=nδ, n={0,1,…,N-1}, and the unknown delay parameter is approximately expressed as t l ≈n l δ, where n k ∈{0,1,…,N-1} represents the delay parameter t l After quantization, formula (2) is transformed into:

[0073]

[0074] Where k = -K, 1-K, ..., K. For further calculation and processing, 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 of size (2K+1)×1,

[0077]

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

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

[0080] The time value of the pulse sequence s(t) is t∈[0,T). When the time region is quantized according to a fixed interval δ, the complete set representation of the simulated time quantity t can be constructed as η={0,δ,2δ,…,(N-1)δ}, N=T / δ, t lThe set of is represented as γ={n1δ,n2δ,…,n L δ}, and L<<N; therefore, after the quantization process, the delay parameter t l The set γ of t constitutes a subset of the complete set η of characteristic parameters t, that is, The complete set η is used to represent the delay parameter set γ, that is:

[0081]

[0082] Where Ψ is a (2K+1)×N dimensional matrix, and the knth element (k=-K,1-K,…,K,n=0,1,…,N-1) is Composition, formula (25) can be transformed into the expression of sparse matrix:

[0083] s=HΨa (29)

[0084] Where a∈R N×1 is an L sparse vector, and the position of non-zero elements represents Then it is a non-zero element value. Combining formulas (24) and (29), it is further transformed into:

[0085] z=BPHΨa=Φa (30)

[0086] In the above formula, Φ = BPHΨ is a measurement matrix with dimension (2K+1)×N;

[0087] Step 6.5, solve z = Φa, by solving the non-zero element values ​​of the sparse vector a from the observation vector y, calculate the expression and convert it into an L0 norm minimization problem, that is:

[0088]

[0089] Wherein the L0 norm ||a||0 represents the number of non-zero elements in the sparse vector a. According to the research on random demodulation theory, the random measurement matrix Φ constructed by using a pseudo-random sequence can be made to satisfy the RIP constraint condition. On the basis that the measurement matrix Φ satisfies the RIP condition, the OMP algorithm is used as an effective method to solve formula (31), thereby recovering the characteristic parameters of the pulse sequence.

[0090] The beneficial effects of the present invention are mainly manifested in: the traditional Nyquist sampling rate is relatively high, and the undersampling technology can effectively reduce the sampling rate of the signal. However, due to the large differences in the spectral characteristics of the signal, it is necessary to design a suitable sampling kernel according to the needs for the signals to be measured in different frequency domains, which brings great difficulties to the design of the sampling system, and it is also necessary to consider eliminating the non-ideal effects of the sampling kernel. The method of the present invention is based on the technology of spectrum expansion, adopts a dual-channel sampling structure, and eliminates the non-ideal effects of the filter in the frequency domain through the correlation between the pulse sequence and the base pulse frequency domain. At the same time, a pseudo-random sequence is used to modulate the pulse sequence, and the main frequency band of the signal is evenly distributed on the entire frequency domain axis, and then a low-pass filter is used as the sampling kernel to intercept the frequency domain information of the baseband and restore the characteristic parameters in the signal. While improving the parameter estimation performance, the versatility of the system is greatly improved. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0092] Figure 2 This is a graph of robustness experiment results. DETAILED DESCRIPTION

[0093] The present invention will be further described below in conjunction with the accompanying drawings.

[0094] Reference Figure 1 and Figure 2 , a dual-channel FRI undersampling and parameter measurement method based on spectrum expansion, the system structure is as follows Figure 1 As shown, the method comprises the following steps:

[0095] Step 1: Mathematically model the 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, and 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. is unknown, and these parameters need to be restored through sampling structure and reconstruction algorithm to reconstruct the pulse sequence. The pulse sequence is expressed as:

[0096]

[0097] The discrete Fourier transform (DFT) of the pulse sequence s(t) is obtained, and the Fourier coefficient expression of the signal is obtained as follows:

[0098]

[0099] Step 2: Combined with the characteristics of the pulse sequence, the sampling structure of the signal is designed. In order to improve the versatility of the system and avoid the relative limitations of the single-channel sampling structure, a dual-channel parallel sampling structure is designed, such as Figure 1 As shown, the corresponding parameters of the proposed sampling system are set as follows:

[0100] Step 2.1, generate 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 a statistical characteristic that is approximately random. Use the pseudo-random sequence to modulate the signal to be tested, and evenly move the frequency domain information of the signal to be tested s(t) to the entire spectrum. The spectrum of any pulse signal can be restored through the baseband signal. The pseudo-random sequence is expressed as follows:

[0101]

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

[0103] Multiplying two signals in the frequency domain means convolution in the frequency domain. The signal to be tested is convolved with the pseudo-random sequence, which is equivalent to evenly smearing the frequency domain on the entire frequency domain axis. The frequency domain representation of the pseudo-random sequence p(t) is:

[0104]

[0105] Where T p =T / Q,

[0106] Step 2.2: Non-ideal LPF modeling, such as Figure 1 As shown in the figure, a dual-channel parallel sampling structure is adopted, and non-ideal LPF is used as the sampling kernel for filtering the pulse sequence and the base pulse. Compared with the original sampling structure, it is assumed that the sampling kernel is ideal, and the non-ideal sampling kernel has a great influence on the signal reconstruction accuracy. Therefore, it is necessary to pass the pulse signal and the base signal through the non-ideal LPF so that they both contain non-ideal effects, so as to eliminate them by using the connection between the sampling samples in the signal reconstruction stage. The unit impulse response of the sampling kernel is expressed as g(t), and its frequency domain expression is G(Ω);

[0107] Step 2.3, by obtaining a series of Fourier coefficients, the pulse sequence is restored using a parameter estimation algorithm. After the pulse sequence is filtered by the sampling kernel, the signal spectrum at the baseband can be obtained. The obtained samples are subjected to discrete Fourier transform (DFT) to obtain the relevant Fourier series coefficients;

[0108] Step 3: Spectrum shifting process of the pulse sequence. According to the proposed sampling structure, a pseudo-random sequence is needed to shift the spectrum of the pulse sequence, so that the information on the main frequency band of the signal is evenly distributed on the entire frequency domain axis. The signal after shifting is expressed as s1(t):

[0109] s1(t)=s(t)·p(t) (36)

[0110] Convert the above formula to frequency domain and get:

[0111]

[0112] Discretize the angular frequency variable Ω so that Ω=mΩ0, where Ω0=2π / T,m∈Z, then formula (6) can be further expressed as:

[0113]

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

[0115]

[0116] Step 4: Spectrum shifting of base pulses. According to the proposed sampling structure, the 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 shifted signal is represented by h1(t):

[0117] h1(t)=h(t)·q(t) (40)

[0118] Performing continuous-time Fourier transform (CTFT) on the above equation yields:

[0119]

[0120] The angular frequency variable Ω is discretized so that Ω=mΩ0, where Ω0=2π / T,m∈Z, then formula (10) is further expressed as:

[0121]

[0122] Where H1[m], H[m] and Q[m] are the Fourier coefficients of signals h1(t), h(t) and q(t). The above formula can be further simplified to:

[0123]

[0124] Step 5: Sampling kernel filtering and low-speed sampling process of pulse sequence and base pulse. According to the proposed sampling structure, it is necessary to perform sampling kernel filtering on the pulse sequence and base pulse to obtain the frequency domain information of the signal baseband. Finally, the sampled samples are subjected to discrete Fourier transform (DFT) to obtain the corresponding discrete Fourier coefficients. The process is as follows:

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

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

[0127] Representing the signal s1(t) in the frequency domain, we get:

[0128] S2(Ω)=S1(Ω)·G(Ω) (45)

[0129] By discretizing the angular frequency variable Ω, formula (14) can be further expressed 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. The above formula is further simplified to:

[0132]

[0133] Step 5.2, the sampling kernel filtering process of the base pulse uses LPF as the sampling kernel to filter the base signal and intercept the frequency domain information of the base signal in the baseband. However, since there is no ideal LPF in practice, we use a non-ideal LPF as the sampling kernel for filtering. In this way, the base signal can contain the 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:

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

[0135] Representing the signal h2(t) in the frequency domain, we get:

[0136] H2(Ω)=H1(Ω)·G(Ω) (49)

[0137] Discretize the angular frequency variable Ω, then the above formula 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 signals h2(t), h1(t) and g(t). The above formula can be further simplified to:

[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 are filtered by the sampling kernel, the frequency domain information of the signal baseband can be obtained. A set of Fourier coefficients is required 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 6: Eliminate the non-ideal effects of the filter and restore the pulse sequence. The non-ideal effects of the LPF can be eliminated by obtaining samples s2[n] and h2[n], and then the pulse sequence can be reconstructed. The process is as follows:

[0143] Step 6.1: Eliminate the non-ideal effect of LPF. By utilizing the relationship between the sampled samples of the measured signal and the base signal, the frequency domain response of LPF can be effectively eliminated. Combining formulas (16) and (20), we can further obtain:

[0144]

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

[0146] Step 6.2: Construct the observation vector z. For ease of processing, convert formula (21) into a matrix-vector form to construct the observation vector:

[0147] z=[Z[-M],Z[1-M],…,Z[M]] T (53)

[0148] Observation matrix:

[0149]

[0150] In formula (23), the observation matrix P is a matrix of dimension (2M+1)×(2K+1), and the lmth value is P[mk] (m=-M,…,M,k=-K,…,K), so formula (21) is written as:

[0151] z=B·P·s (55)

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

[0153] Step 6.3: Quantize the simulation time axis t∈[0,T) and divide it into N uniform grids. The size of the grid is set to δ=T / N. Therefore, the simulation time quantity t is approximately expressed as t=nδ, n={0,1,…,N-1}, and the unknown delay parameter is approximately expressed as t l ≈n l δ, where n k ∈{0,1,…,N-1} represents the delay parameter t l After quantization, formula (2) is transformed into:

[0154]

[0155] Where k = -K, 1-K, ..., K. For further calculation and processing, 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 vector of size (2K+1)×1,

[0158]

[0159] is a diagonal matrix of dimension (2K+1)×(2K+1);

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

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

[0162]

[0163] Where Ψ is a (2K+1)×N dimensional matrix, and the knth element (k=-K,1-K,…,K,n=0,1,…,N-1) is Composition, formula (25) can be transformed into sparse matrix

[0164] The expression of array:

[0165] s=HΨa (60)

[0166] Where a∈R N×1 is an L sparse vector, and the position of non-zero elements represents Then it is a non-zero element value. Combining formulas (24) and (29), it is further transformed into:

[0167] z=BPHΨa=Φa (61)

[0168] In the above formula, Φ = BPHΨ is a measurement matrix with dimension (2K+1)×N;

[0169] Step 6.5, solve z = Φa. The non-zero element values ​​of the sparse vector a can be solved from the observation vector y. The most classic way to calculate this expression is to convert it into an L0 norm minimization problem, that is:

[0170]

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

[0172] In order to verify the feasibility of the method of the present invention, relevant simulation experiments are carried out in this section to verify the pulse sequence. Indicates that the unknown parameters in the signal are set, and the amplitude parameter is set to a l =[0.83,0.64,0.35,0.17,0.97], the delay parameter is set to t l =[0.31, 0.42, 0.55, 0.67, 0.89]. The base signal of the pulse sequence is known. The sampling rate of the initial signal is set to 20kHz, and the method of the present invention is used to undersample it, and the sampling can be performed at a minimum of 2000Hz.

[0173] Experiment 1: Idealized experiment. The method of the present invention is compared with the classic FRI sampling method. The ideal LPF is used as the sampling kernel for sampling, and no noise is introduced in the process. The original signal is restored by the parameter recovery algorithm. The experimental results are shown in Table 1. It can be seen from the experimental results that when the ideal sampling kernel is used, all methods are feasible. Table 1 shows the parameter recovery results.

[0174]

[0175] Table 1

[0176] Experiment 2: Non-idealized experiment. Compared with Experiment 1, a non-ideal LPF is used as the sampling kernel for the experiment, and noise is added during the experiment. The experimental results are shown in Table 2. From the results, it can be seen that the method of the present invention can accurately recover the free parameters of the signal under test in a non-ideal environment. At the same time, the classic FRI sampling method has a low reconstruction accuracy because it does not eliminate the non-ideal effect of the filter.

[0177]

[0178] Table 2

[0179] Experiment 3: Robustness experiment. Compared with Experiment 2, robustness experiments were conducted on the classic FRI sampling method, the method of the present invention, and the multi-channel sampling method. Gaussian white noise was added to the sampling system, and the signal-to-noise ratio increased from -20dB to 60dB, with a step of nearly 10dB. In order to more intuitively compare the differences between the systems, the normalized mean square error was introduced for evaluation. The experiment was conducted 1000 times, and the experimental results are displayed on Figure 2 As can be seen from the figure, the method of the present invention can effectively improve the noise resistance and robustness of the entire sampling system, improve the accuracy of the system in recovering free parameters, and is a universal sub-Nyquist sampling method.

[0180] The contents described in the embodiments of this specification are merely enumerations of implementation forms of the inventive concept and are for illustrative purposes only. The protection scope of the present invention should not be considered to be limited to the specific forms described in this embodiment, and the protection scope of the present invention also extends to equivalent technical means that can be thought of by ordinary technicians in this field based on the inventive concept.

Claims

1. A dual-channel FRI undersampling and parameter measurement method based on spectrum expansion, characterized in that: The method comprises the following steps: Step 1: Mathematically model the 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, and L is the number of base pulse signals contained in the signal s(t). is the time delay and amplitude of the pulse train in the signal to be measured, are unknown, and these parameters need to be recovered through sampling structure and reconstruction algorithm to reconstruct the pulse sequence; Step 2: Combined with the characteristics of the pulse sequence, the sampling structure of the signal is designed. In order to improve the versatility of the system and avoid the relative limitations of the single-channel sampling structure, a dual-channel parallel sampling structure is designed, and the corresponding parameters of the proposed sampling system are set; Step 3: Spectrum shifting process of the pulse sequence. According to the proposed sampling structure, a pseudo-random sequence needs to be used to shift the spectrum of the pulse sequence, so as to evenly distribute the information on the main frequency band of the signal to the entire frequency domain axis. Step 4: Spectrum shifting process of base pulse: according to the proposed sampling structure, the spectrum of base pulse is shifted using cosine signal q(t)=cos(2πft), so as to shift the information on the main frequency band of the signal to the low frequency. Step 5: Sampling kernel filtering and low-speed sampling process of pulse sequence and base pulse. According to the proposed sampling structure, it is necessary to perform sampling kernel filtering on the pulse sequence and base pulse to obtain the frequency domain information of the signal baseband, and finally perform discrete Fourier transform (DFT) on the sampled samples to obtain the corresponding discrete Fourier coefficients. Step six, eliminate the non-ideal effects of the filter and restore the pulse sequence. The purpose of eliminating the non-ideal effects of the LPF can be achieved by obtaining samples s2[n] and h2[n], and then the pulse sequence can be reconstructed.

2. The dual-channel FRI undersampling and parameter measurement method based on spectrum expansion according to claim 1, characterized in that: In the step 1, the pulse sequence is expressed as: By performing discrete Fourier transform (DFT) on the pulse sequence s(t), the Fourier coefficient expression of the signal is obtained as follows:

3. The dual-channel FRI undersampling and parameter measurement method based on spectrum expansion as claimed in claim 2, characterized in that: The process of step 2 is as follows: Step 2.1, generate 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 a statistical characteristic that is approximately random. Use the pseudo-random sequence to modulate the signal to be tested, and evenly move the frequency domain information of the signal to be tested s(t) to the entire spectrum. The spectrum of any pulse signal can be restored through the baseband signal. The pseudo-random sequence is expressed as follows: in T is the duration of the pseudo-random sequence p(t); Multiplying two signals in the frequency domain means convolution in the frequency domain. The signal to be tested is convolved with the pseudo-random sequence, which is equivalent to evenly smearing the frequency domain on the entire frequency domain axis. The frequency domain representation of the pseudo-random sequence p(t) is: Where T p =T / Q, Step 2.2, non-ideal LPF modeling, using a dual-channel parallel sampling structure, using non-ideal LPF as the sampling kernel for filtering both the pulse sequence and the base pulse. Compared with the original sampling structure, it is assumed that the sampling kernel is ideal, and the non-ideal sampling kernel has a great influence on the signal reconstruction accuracy. Therefore, it is necessary to pass the pulse signal and the base signal through the non-ideal LPF to make them both contain non-ideal effects, so as to eliminate them by using the connection between the sampling samples in the signal reconstruction stage. The unit impulse response of the sampling kernel is expressed as g(t), and its frequency domain expression is G(Ω); Step 2.3: Obtain a series of Fourier coefficients and use the parameter estimation algorithm to recover the pulse sequence. After the pulse sequence is filtered by the sampling kernel, the signal spectrum at the baseband can be obtained. Perform discrete Fourier transform DFT on the obtained samples to obtain the relevant Fourier series coefficients.

4. The dual-channel FRI undersampling and parameter measurement method based on spectrum expansion as claimed in claim 3, characterized in that: In step 3, the signal after the shift is represented by s1(t): s1(t)=s(t)·p(t) (5) Convert the above formula to frequency domain and get: Discretize the angular frequency variable Ω so that Ω=mΩ0, where Ω0=2π / T,m∈Z, then formula (6) can be further expressed as: Where S1[m]=S1(mΩ0), S[m]=S(mΩ0) and P[m]=P(mΩ0) are the Fourier coefficients of signals s1(t), s(t) and p(t), respectively, which can be further simplified to:

5. The dual-channel FRI undersampling and parameter measurement method based on spectrum expansion as claimed in claim 4, characterized in that: In step 4, the signal after the shift is expressed as h1(t): h1(t)=h(t)·q(t) (9) By converting the above formula into frequency domain, we can get: The angular frequency variable Ω is discretized so that Ω=mΩ0, where Ω0=2π / T,m∈Z, then formula (10) is further expressed as: Where H1[m], H[m] and Q[m] are the Fourier coefficients of signals h1(t), h(t) and q(t), which can be further simplified to:

6. The dual-channel FRI undersampling and parameter measurement method based on spectrum expansion as claimed in claim 5, characterized in that: The process of step five is as follows: Step 5.1, sampling kernel filtering process of the pulse signal to be tested. After the signal to be tested is modulated, the main frequency domain information of the signal is evenly distributed on the entire frequency domain axis. Therefore, LPF can be used as the sampling kernel to intercept the frequency domain information of the signal baseband, and non-ideal LPF is used as the sampling kernel for filtering. The signal after filtering is expressed as: s2(t)=s1(t)*g(t) (13) Representing the signal s1(t) in the frequency domain, we get: S2(Ω)=S1(Ω)·G(Ω) (14) By discretizing the angular frequency variable Ω, formula (14) can be further expressed as: S2(mΩ0)=S1(mΩ0)·G(mΩ0) (15) 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, which can be further simplified to: Step 5.2, the sampling kernel filtering process of the base pulse uses LPF as the sampling kernel to filter the base signal and intercept the frequency domain information of the base signal in the baseband. However, since there is no ideal LPF in practice, we use a non-ideal LPF as the sampling kernel for filtering. In this way, the base signal can contain the 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: h2(t)=h1(t)*g(t) (17) Representing the signal h2(t) in the frequency domain, we get: H2(Ω)=H1(Ω)·G(Ω) (18) The angular frequency variable Ω is discretized and further expressed as: H2(mΩ0)=H1(mΩ0)·G(mΩ0) (19) Where H2[m], H1[m] and G[m] are the Fourier coefficients of signals h2(t), h1(t) and g(t), which can be further simplified to: H2[m]=H1[m]·G[m] (20) Step 5.3, the low-speed sampling process of the pulse sequence and the 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. A set of Fourier coefficients is required to restore the original signal. Therefore, the acquired samples are subjected to discrete Fourier transform DFT to obtain the required discrete Fourier coefficients s2[m] and h2[m].

7. The dual-channel FRI undersampling and parameter measurement method based on spectrum expansion according to claim 6, characterized in that: The process of step six is ​​as follows: Step 6.1: Eliminate the non-ideal effect of LPF. By utilizing the relationship between the sampled samples of the measured signal and the base signal, the frequency domain response of LPF can be effectively eliminated. Combining formulas (16) and (20), we can further obtain: Among them, h(t) and q(t) are known, so B = 1 / H1[k] is a constant; Step 6.2: Construct the observation vector z. Convert formula (21) into a matrix-vector form to construct the observation vector: z=[Z[-M],Z[1-M],…,Z[M]] T (22) Observation matrix: In formula (23), the observation matrix P is a matrix of dimension (2M+1)×(2K+1), and the lmth value is P[mk] (m=-M,…,M,k=-K,…,M,K), so formula (21) is written as: z=B·P·s (24) In the above formula, s=[S[-K],S[1-K],…,S[K]] T It is a (2K+1)×1 vector composed of the Fourier coefficients of the pulse sequence s(t); Step 6.3: Quantize the simulation time axis t∈[0,T) and divide it into N uniform grids. The size of the grid is set to δ=T / N, so the simulation time quantity t is approximately expressed as t=nδ, n={0,1,…,N-1}, and the unknown delay parameter is approximately expressed as t l ≈n l δ, where n k ∈{0,1,…,N-1} represents the delay parameter t l After quantization, formula (2) is transformed into: Where k = -K, 1-K, ..., K. For further calculation and processing, 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)×1, is a diagonal matrix of dimension (2K+1)×(2K+1); Step 6.4: construct the measurement matrix Φ, where Φ = PHΨ. In step 6.2, P is constructed according to formula (23), and in step 6.3, H is constructed according to formula (27). Next, the Ψ matrix needs to be constructed to construct the measurement matrix Φ. The time value of the pulse sequence s(t) is t∈[0,T). When the time region is quantized according to a fixed interval δ, the complete set representation of the simulated time quantity t can be constructed as η={0,δ,2δ,…,(N-1)δ}, N=T / δ, t l The set of is represented as γ={n1δ,n2δ,…,n L δ}, and L<<N; therefore, after the quantization process, the delay parameter t l The set γ of t constitutes a subset of the complete set η of characteristic parameters t, that is, The complete set η is used to represent the delay parameter set γ, that is: Where Ψ is a (2K+1)×N dimensional matrix, and the knth element (k=-K,1-K,…,K,n=0,1,…,N-1) is Composition, formula (25) can be transformed into the expression of sparse matrix: s=HΨa (29) Where a∈R N×1 is an L sparse vector, and the position of non-zero elements represents Then it is a non-zero element value. Combining formulas (24) and (29), it is further transformed into: z=BPHΨa=Φa (30) In the above formula, Φ = BPHΨ is a measurement matrix with dimension (2K+1)×N; Step 6.5, solve z = Φa, by solving the non-zero element values ​​of the sparse vector a from the observation vector y, calculate the expression and convert it into an L0 norm minimization problem, that is: Wherein the L0 norm ||a||0 represents the number of non-zero elements in the sparse vector a. According to the research on random demodulation theory, the random measurement matrix Φ constructed by using a pseudo-random sequence can be made to satisfy the RIP constraint condition. On the basis that the measurement matrix Φ satisfies the RIP condition, the OMP algorithm is used as an effective method to solve formula (31), thereby recovering the characteristic parameters of the pulse sequence.

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