A method for multi-channel under-sampling of a multi-exponential decay sinusoidal signal

Abstract

The application discloses a multi-exponential decay sinusoidal signal (MEDS) multi-channel under-sampling hardware implementation method. Since under-sampling of the MEDS signal can cause frequency ambiguity, the application adds an auxiliary sampling channel to avoid the occurrence of the phenomenon. For P MEDS signals with a common support model in the frequency domain, the application needs at least P samples to accurately estimate the parameters of the signals. The application greatly reduces the number of samples and the number of sampling channels required for signal parameter measurement, and has higher reconstruction precision and noise robustness than the traditional method.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, specifically relating to a hardware implementation method for multi-channel undersampling of multi-exponentially decaying sinusoidal signals. Background Technology

[0002] Parameter estimation for multiple exponentially damped sinusoids (MEDS) signals is a widespread problem in signal processing fields such as magnetic resonance imaging (MRI) and source localization. Most existing methods for parameter estimation of MEDS signals are based on the well-known Nyquist sampling theory. It is well known that Nyquist sampling theory requires the sampling rate to be at least twice the signal bandwidth. However, with the development of science and technology, the bandwidth of signals to be processed in signal processing is increasing, and the required sampling rate is becoming increasingly higher, posing a significant challenge to traditional Nyquist sampling theory. Therefore, in recent years, many under-Nyquist sampling methods have emerged.

[0003] With the in-depth research of numerous researchers on under-Nyquist sampling methods, many undersampling methods have emerged in recent years. While undersampling methods can effectively reduce the sampling rate required for ultra-wideband signals, they often suffer from a drawback: frequency ambiguity. Therefore, effectively addressing frequency ambiguity is crucial for obtaining accurate estimation results when using undersampling methods for parameter estimation. Leveraging the sparsity of signals, many undersampling methods based on compressed sensing (CS) theory have made significant contributions to solving the parameter estimation problem for MEDS signals. Mignot et al. proposed a CS-based undersampling method in which indoor acoustic echoes are modeled as MEDS signals. However, this method requires the MEDS signal to conform to a sparse model, lacking universality. Therefore, CS-based undersampling methods have certain requirements regarding the sparsity and prior nature of the input signal, which also limits their versatility. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, this invention proposes a hardware implementation method for multi-channel undersampling of MEDS signals with a common frequency domain support model, addressing the frequency ambiguity and parameter estimation problems caused by undersampling. A hardware system for MEDS signal undersampling is designed, capable of receiving and processing P MEDS signals. These P signals have the same frequency and attenuation factor but different amplitudes, i.e., a common frequency domain support model. The proposed undersampling system has P+1 sampling channels: P main sampling channels and 1 sub-sampling channel. The sub-sampling channel has a time delay relative to the main sampling channel to resolve the frequency ambiguity problem, ultimately allowing for the estimation of signal amplitude, frequency, and attenuation factor parameters.

[0005] The technical solution adopted by this invention to solve its technical problem is:

[0006] A hardware implementation method for multi-channel undersampling of multi-exponentially decaying sinusoidal signals includes the following steps:

[0007] Step 1: Use LabVIEW software on a host computer to control an NI PXIe-6363 arbitrary waveform generator to generate P (P=3 in this invention) MEDS signals with a common frequency domain support model. These signals have the following form:

[0008]

[0009] Where t∈(0,T), T is the signal duration, and c i,k (c i,k ≠0, c i,k ∈C, i=1,2,…,P, k=1,2,…K) is the complex amplitude, s k =r k +j2πf k (k = 1, 2, ..., K) is a complex frequency, r k It is the attenuation factor, f k It represents the frequency, and K represents the number of frequency components;

[0010] Step 2: In the simulation preprocessing module, the MEDS signal to be tested with a frequency domain common support model is subjected to simulation preprocessing operation. In this process, the module performs analog-to-digital conversion, amplification, filtering and digital-to-analog conversion operations on the signal, and then inputs the signal into P+1 undersampled channels for sampling, including P main sampling channels and 1 sub-sampling channel.

[0011] Step 3: The main sampling channel samples each MEDS signal. In the main sampling channel, the sampling frequency is f... s The sampling rate is the MEDS signal x to be tested. i (t) is uniformly sampled, and its sampled value x i [n] is represented by the following formula:

[0012]

[0013] Where K is the number of known frequency components, c i,k It is a complex amplitude. s k =r k +j2πf k It is a complex frequency, r k It is the attenuation factor, f k It is the frequency, n∈Z + ;

[0014] Step 4, sample the value x i [n] is input into the improved rotation-invariant subspace algorithm. Running the improved rotation-invariant subspace algorithm can estimate the parameters of the MEDS signal with a frequency domain common support model.

[0015] Step 5: Run the frequency estimation algorithm to calculate the frequency.

[0016] Furthermore, the process of step four is as follows:

[0017] Step 4.1, firstly, obtain at least from each master sampling channel Sample x i [n] represents the number of sampled values ​​for each main channel. And make If l = 1, 2, ..., L, and L + K = N, then formula (2) can be written in the following matrix form:

[0018]

[0019] in,

[0020]

[0021] c i =[c i,1 ,c i,2 ,…,c i,K (5)

[0022] Step 4.2, using samples from each channel The covariance matrix is ​​constructed as follows: m×m (m≥K+1):

[0023]

[0024] Step 4.3, according to the rotation-invariant subspace algorithm, let E represent a matrix of size m×K, from R... xThe eigenvectors corresponding to the K largest eigenvalues ​​of E are used to form the eigenvectors. Let E1 represent the first m-1 rows of E, and let E2 represent the last m-1 rows of E. Let E1 be the pseudo-inverse matrix, then eigenvalues Therefore, matrix D is now a known quantity. Furthermore, the amplitude is calculated according to formula (3), as follows:

[0025]

[0026] in, Denotes the pseudo-inverse matrix of D;

[0027] Step 4.4, according to The minimum possible solutions for the attenuation factor and frequency are calculated using the following formulas:

[0028]

[0029]

[0030] in, express The principal value of the argument.

[0031] Furthermore, the process of step five is as follows:

[0032] Step 5.1, the sub-sampling channel samples signal x i (t) Sampling is performed, and the auxiliary sampling channel has a delay time ΔT relative to the main sampling channel. d The sampling rate is f s The obtained sampled values ​​are expressed as follows:

[0033]

[0034] Where, n d ∈Z + T s =1 / f s ΔT d T represents the delay time. s It is the sampling period;

[0035] Step 5.2 addresses the frequency ambiguity problem caused by undersampling. Since trigonometric functions are periodic, the estimated normalized frequency... and There exists a difference, described by the following formula:

[0036]

[0037] Right now

[0038]

[0039] in, express The principal value of the argument, Because of d k (d k For any k ∈ Z, k ∈ {1, 2, ..., K}, there are infinitely many values, so the frequency This results in an infinite number of solutions, which is the frequency ambiguity problem;

[0040] Step 5.3, rewrite the delayed sample value represented by formula (10) as follows:

[0041]

[0042] in,

[0043] Step 5.4, let x be... d =[x d [0],x d [1],…,x d [N′-1]] T b = [b1, b2, ..., b K ] T Then formula (13) can be written in the following matrix form:

[0044] x d =VAb (14)

[0045] in,

[0046]

[0047]

[0048] Step 5.5, when N′≥K, formula (14) has a unique solution:

[0049]

[0050] If the delay time meets the requirement Then d k There is a unique solution:

[0051]

[0052] Among them, ∠b k b k The principal value of the argument, 0 ≤ ∠b k <2π, finally, solve formula (12) to calculate the frequency f. k .

[0053] The main advantages of this invention are: it greatly reduces the number of samples and sampling channels required for signal parameter measurement, and has higher reconstruction accuracy and noise robustness than traditional methods. Attached Figure Description

[0054] Figure 1 This is a structural diagram of a multi-channel undersampling system for multi-exponentially decaying sinusoidal signals.

[0055] Figure 2 This is the reconstruction result diagram under noisy conditions.

[0056] Figure 3 This is a graph showing the frequency parameter recovery results for different sample numbers when there is noise. Detailed Implementation

[0057] The present invention will now be further described with reference to the accompanying drawings.

[0058] Reference Figures 1-3 A hardware implementation method for multi-channel undersampling of multi-exponentially decaying sinusoidal signals, the system structure is as follows: Figure 1 As shown, it includes the following steps:

[0059] Step 1: Use LabVIEW software on a host computer to control an NI PXIe-6363 arbitrary waveform generator to generate P (P=3 in this invention) MEDS signals with a common frequency domain support model. These signals have the following form:

[0060]

[0061] Where t∈(0,T), T is the signal duration, and c i,k (c i,k ≠0, c i,k ∈C, i=1,2,…,P, k=1,2,…K) is the complex amplitude, s k =r k +j2πf k (k = 1, 2, ..., K) is a complex frequency, r k It is the attenuation factor, f k It represents the frequency, and K represents the number of frequency components;

[0062] Step 2: In the simulation preprocessing module, the MEDS signal to be tested with a frequency domain common support model is subjected to simulation preprocessing operation. In this process, the module performs analog-to-digital conversion, amplification, filtering and digital-to-analog conversion operations on the signal, and then inputs the signal into P+1 undersampled channels for sampling, including P main sampling channels and 1 sub-sampling channel.

[0063] Step 3: The main sampling channel samples each MEDS signal. In the main sampling channel, the sampling frequency is f... sThe sampling rate is the MEDS signal x to be tested. i (t) is uniformly sampled, and its sampled value x i [n] is represented by the following formula:

[0064]

[0065] Where K is the number of known frequency components, c i,k It is a complex amplitude. s k =r k +j2πf k It is a complex frequency, r k It is the attenuation factor, f k It is the frequency, n∈Z + ;

[0066] Step 4, sample the value x i [n] is input into the improved rotation-invariant subspace algorithm. Running the improved rotation-invariant subspace algorithm can estimate the parameters of the MEDS signal with a frequency domain common support model. The process is as follows:

[0067] Step 4.1, firstly, obtain at least from each master sampling channel Sample x i [n] represents the number of sampled values ​​for each main channel. And make If l = 1, 2, ..., L, and L + K = N, then formula (2) can be written in the following matrix form:

[0068]

[0069] in,

[0070]

[0071] c i =[c i,1 ,c i,2 ,…,c i,K (5)

[0072] Step 4.2, using samples from each channel The covariance matrix is ​​constructed as follows: m×m (m≥K+1):

[0073]

[0074] Step 4.3, according to the rotation-invariant subspace algorithm, let E represent a matrix of size m×K, from R... x The eigenvectors corresponding to the K largest eigenvalues ​​of E are used to form the eigenvectors. Let E1 represent the first m-1 rows of E, and let E2 represent the last m-1 rows of E. Let E1 be the pseudo-inverse matrix, then eigenvalues Therefore, matrix D is now a known quantity. Furthermore, the amplitude is calculated according to formula (3), as follows:

[0075]

[0076] in, Denotes the pseudo-inverse matrix of D;

[0077] Step 4.4, according to The minimum possible solutions for the attenuation factor and frequency are calculated using the following formulas:

[0078]

[0079]

[0080] in, express The principal value of the argument;

[0081] Step 5: Run the frequency estimation algorithm. The process is as follows:

[0082] Step 5.1, the sub-sampling channel samples signal x i (t) Sampling is performed, and the auxiliary sampling channel has a delay time ΔT relative to the main sampling channel. d The sampling rate is f s The obtained sampled values ​​are expressed as follows:

[0083]

[0084] Where, n d ∈Z + T s =1 / f s ΔT d T represents the delay time. s It is the sampling period;

[0085] Step 5.2 addresses the frequency ambiguity problem caused by undersampling. Since trigonometric functions are periodic, the estimated normalized frequency... and There exists a difference, described by the following formula:

[0086]

[0087] Right now

[0088]

[0089] in, express The principal value of the argument, Because of d k (d k For any k ∈ Z, k ∈ {1, 2, ..., K}, there are infinitely many values, so the frequency This results in an infinite number of solutions, which is the frequency ambiguity problem;

[0090] Step 5.3, rewrite the delayed sample value represented by formula (10) as follows:

[0091]

[0092] in,

[0093] Step 5.4, let x be... d =[x d [0],x d [1],…,x d [N′-1]] T b = [b1, b2, ..., b K ] T Then formula (13) can be written in the following matrix form:

[0094] x d =VAb (14)

[0095] in,

[0096]

[0097]

[0098] Step 5.5, when N′≥K, formula (14) has a unique solution:

[0099]

[0100] If the delay time meets the requirement Then d k There is a unique solution:

[0101]

[0102] Among them, ∠b k b k The principal value of the argument, 0 ≤ ∠b k <2π, finally, solve formula (12) to calculate the frequency f. k .

[0103] In the first hardware experiment, we verified that the method of this invention can reconstruct the original signal with a small number of samples in the absence of noise. First, we used an NI PXIe-6363 arbitrary waveform generator to generate the MEDS signal to be tested, setting the number of frequency components K=3 and the amplitude c... 1,k =[0.3,0.2,0.9]V,c 2,k = [0.1, 0.2, 0.85]V, c 3,k = [0.99, 0.15, 0.6]V, frequency f k = [1700, 3900, 6800] Hz; Then, we input this signal into the simulation program and display the result on the LabVIEW platform, as shown in the following figure. Figure 2 As shown. We use from the sampling system The original signal was recovered from the sampled values, from Figure 2 It can be seen that the frequency and amplitude values ​​can be accurately recovered.

[0104] Then, to further demonstrate the effectiveness of the method of this invention, we compared the frequency parameter recovery results under noisy conditions with different sample numbers. The experiment was conducted 1000 times, and the recovery results are as follows: Figure 3 As shown in the figure. The experimental results show that, under the same signal-to-noise ratio, the more samples acquired, the more accurate the frequency parameter estimation; under the same sample number, the higher the signal-to-noise ratio, the more accurate the frequency parameter estimation.

[0105] The embodiments described in this specification are merely examples of implementations of the inventive concept and are for illustrative purposes only. The scope of protection of this invention should not be considered limited to the specific forms described in these embodiments; rather, it extends to equivalent technical means conceived by those skilled in the art based on the inventive concept.

Claims

1. A hardware implementation method for multi-channel undersampling of multi-exponentially decaying sinusoidal signals, characterized in that, The method includes the following steps: Step 1: Use LabVIEW software on the host computer to control the NIPXIe-6363 arbitrary waveform generator to generate P MEDS signals with a common frequency domain support model. These signals have the following form: Where t∈(0,T), T is the signal duration, and c i,k (c i,k ≠0, c i,k ∈C, i=1,2,…,P, k=1,2,…K) is the complex amplitude, s k =r k +j2πf k (k = 1, 2, ..., K) is a complex frequency, r k It is the attenuation factor, f k It represents the frequency, and K represents the number of frequency components; Step 2: In the simulation preprocessing module, the MEDS signal to be tested with a frequency domain common support model is subjected to simulation preprocessing operation. In this process, the module performs analog-to-digital conversion, amplification, filtering and digital-to-analog conversion operations on the signal, and then inputs the signal into P+1 undersampled channels for sampling, including P main sampling channels and 1 sub-sampling channel. Step 3: The main sampling channel samples each MEDS signal. In the main sampling channel, the sampling frequency is f... s The sampling rate is the MEDS signal x to be tested. i (t) is uniformly sampled, and its sampled value x i [n] is represented by the following formula: Where K is the number of known frequency components, c i,k It is a complex amplitude. s k =r k +j2πf k It is a complex frequency, r k It is the attenuation factor, f k It is the frequency, n∈Z + ; Step 4, sample the value x i [n] is input into the improved rotation-invariant subspace algorithm. Running the improved rotation-invariant subspace algorithm can estimate the parameters of the MEDS signal with a frequency domain common support model. Step 5: Run the frequency estimation algorithm to calculate the frequency.

2. The hardware implementation method for multi-channel undersampling of a multi-exponentially decaying sinusoidal signal as described in claim 1, characterized in that, The process of step four is as follows: Step 4.1, firstly, obtain at least from each master sampling channel Sample x i [n] represents the number of sampled values ​​for each main channel. And make If L+K=N, then formula (2) can be written in the following matrix form: in, c i =[c i,1 ,c i,2 ,…,c i,K ] (5) Step 4.2, using samples from each channel The covariance matrix is ​​constructed as follows: m×m (m≥K+1): Step 4.3, according to the rotation-invariant subspace algorithm, let E represent a matrix of size m×K, from R... x The eigenvectors corresponding to the K largest eigenvalues ​​of E are used to form the eigenvectors. Let E1 represent the first m-1 rows of E, and let E2 represent the last m-1 rows of E. Let E1 be the pseudo-inverse matrix, then eigenvalues Therefore, matrix D is now a known quantity, and the magnitude is calculated according to formula (3) as follows: in, Denotes the pseudo-inverse matrix of D; Step 4.4, according to The minimum possible solutions for the attenuation factor and frequency are calculated using the following formulas: in, express The principal value of the argument.

3. The hardware implementation method for multi-channel undersampling of a multi-exponentially decaying sinusoidal signal as described in claim 2, characterized in that, The process of step five is as follows: Step 5.1, the sub-sampling channel samples signal x i (t) Sampling is performed, and the auxiliary sampling channel has a delay time ΔT relative to the main sampling channel. d The sampling rate is f s The obtained sampled values ​​are expressed as follows: Where, n d ∈Z + T s =1 / f s ΔT d T represents the delay time. s It is the sampling period; Step 5.2 addresses the frequency ambiguity problem caused by undersampling. Since trigonometric functions are periodic, the estimated normalized frequency... and There exists a difference, described by the following formula: Right now in, express The principal value of the argument, Because of d k (d k For any k ∈ Z, k ∈ {1, 2, ..., K}, there are infinitely many values, so the frequency This results in an infinite number of solutions, which is the frequency ambiguity problem; Step 5.3, rewrite the delayed sample value represented by formula (10) as follows: in, Step 5.4, let x be... d =[x d [0],x d [1],…,x d [N′-1]] T b = [b1, b2, ..., b K ] T Then formula (13) can be written in the following matrix form: x d =VAb (14) in, Step 5.5, when N′≥K, formula (14) has a unique solution: If the delay time meets the requirement Then d k There is a unique solution: Among them, ∠b k b k The principal value of the argument, 0 ≤ ∠b k <2π, finally, solve formula (12) to calculate the frequency f. k .

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