CN113050043A - Ground penetrating radar ultra wide band Gaussian pulse FRI sampling method based on non-ideal LPF - Google Patents

Ground penetrating radar ultra wide band Gaussian pulse FRI sampling method based on non-ideal LPF Download PDF

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CN113050043A
CN113050043A CN202110323328.7A CN202110323328A CN113050043A CN 113050043 A CN113050043 A CN 113050043A CN 202110323328 A CN202110323328 A CN 202110323328A CN 113050043 A CN113050043 A CN 113050043A
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陈林林
黄国兴
张世铭
卢为党
彭宏
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Zhejiang University of Technology ZJUT
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/28—Details of pulse systems
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/02—Systems using reflection of radio waves, e.g. primary radar systems; Analogous systems
    • G01S13/0209—Systems with very large relative bandwidth, i.e. larger than 10 %, e.g. baseband, pulse, carrier-free, ultrawideband
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/88—Radar or analogous systems specially adapted for specific applications
    • G01S13/885—Radar or analogous systems specially adapted for specific applications for ground probing

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Abstract

A ground penetrating radar ultra wide band Gaussian pulse FRI sampling method based on a non-ideal LPF comprises the following steps: generating an original signal; step two, constructing a sampling kernel function; step three, low-speed sampling modeling; step four, sampling the sample y [ n ]]And obtaining and processing; step five, sampling a sample h [ n ]]And obtaining and processing; step six, representing the signal relation; step seven, in the data processing stage, the formula (9) only contains unknown parameters
Figure DDA0002993615120000011
Is a typical parameter estimation problem, solving equation (9) yields the result. The invention provides a ground penetrating radar ultra wide band Gaussian pulse FRI sampling method based on a non-ideal LPF, which reconstructs the processed sampling informationThe non-ideal effects brought by the filter can be eliminated, thereby improving the reconstruction accuracy of the system.

Description

Ground penetrating radar ultra wide band Gaussian pulse FRI sampling method based on non-ideal LPF
Technical Field
The invention relates to the technical field of signal processing, in particular to a ground penetrating radar ultra wide band Gaussian pulse FRI sampling method based on a non-ideal LPF.
Background
With the development of communication technology in recent years, the bandwidth of signals used in the field of radar is continuously increasing. According to the traditional Nyqiust sampling theorem, the sampling rate of a signal is required to be more than twice of the highest frequency of the signal, so that the original signal can be reconstructed without distortion. In order to effectively reduce the sampling frequency of the signal and effectively reconstruct the original signal, many experts and scholars have conducted a series of researches on the undersampling method. The conventional under-sampling methods include a Compressed Sensing (CS) method and a Finite Innovation Rate (FRI) method. FRI sampling theory was first proposed in 2002 by Vetterli et al. According to the theory, various parameters of the parameterized signal can be effectively reconstructed by acquiring partial frequency domain information of the parameterized signal and processing the frequency domain information by using a specific algorithm, so that the signal is reconstructed. The classic FRI sampling structure is shown in fig. 1.
In fig. 1, the most important part is the sampling kernel g (t), which is the unit impulse response of the filter in practical application. After the signal is processed by the sampling kernel function, partial frequency domain information of the signal can be obtained, and then the sampling sample y [ n ] can be obtained by low-speed sampling, wherein the sampling sample comprises partial frequency domain information of the original signal x (t). The frequency domain information may then be processed according to existing nulling filter methods or subspace estimation methods, which may complete the reconstruction of the original signal. The existing common sampling kernel functions include a sinc sampling kernel, an SOS sampling kernel, a B spline function and the like.
Taking a basic sinc sampling kernel as an example, modeling is carried out on the process of carrying out FRI sampling reconstruction on the ultra-wideband Gaussian pulse of the ground penetrating radar. Using a ground penetrating radar ultra wide band gaussian pulse as an original signal x (t), in a radar system, a complete echo signal can be represented by superposition of received echoes, and the radar echo signal is represented without considering noise and other interference as follows:
Figure BDA0002993615100000021
where h (t) is a known Gaussian pulse shape,
Figure BDA0002993615100000022
respectively corresponding to the amplitude parameter and the time delay parameter of the detected pulse, wherein T is the pulse repetition interval of the radar, and after the original signal is processed by a sampling kernel function, namely y (T) is expressed as the convolution process of the original signal and the sampling kernel function:
Figure BDA0002993615100000023
the sampled values y [ n ] can be obtained through low-speed sampling, and y [ n ] is expressed as:
Figure BDA0002993615100000024
wherein T isSFor the sampling period of the signal, it can be seen from the formula that the sampling samples contain partial fourier coefficients of the original signal, and then a typical parameter estimation problem can be obtained by simplification, and by solving the problem, the amplitude and delay parameters of the signal can be obtained, i.e. the signal is successfully reconstructed.
A key component in FRI sampling systems is the selection of the sampling kernel, i.e., the selection of the filter. However, in the hardware implementation process, because the filter cannot achieve the effect of simulation, a non-ideal effect is brought, and reconstruction accuracy is influenced. How to eliminate the non-ideal effect of the filter is an important problem influencing the reconstruction precision of the ground penetrating radar ultra wide band Gaussian pulse FRI sampling system.
Disclosure of Invention
In order to overcome the defects of the prior art and solve the problem of non-ideal effect of a filter in a ground penetrating radar ultra wide band Gaussian pulse FRI sampling system, the invention provides a non-ideal LPF-based ground penetrating radar ultra wide band Gaussian pulse FRI sampling method, an original signal can be expressed as a combination of base signals with different time delays and amplitudes, the original signal and the base signals are processed by the method of the invention, so that two groups of sampling information containing the non-ideal effect of the filter can be obtained, the sampling information without the non-ideal effect of the filter can be obtained after processing, the non-ideal effect brought by the filter can be eliminated by reconstructing the processed sampling information, and the reconstruction precision of the system is improved.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a ground penetrating radar ultra wide band Gaussian pulse FRI sampling method based on a non-ideal LPF comprises the following steps:
step one, generating an original signal: parameter setting of an original signal, the original signal being represented as
Figure BDA0002993615100000031
Wherein T ∈ [0, T) is the observation time,
Figure BDA0002993615100000032
Figure BDA0002993615100000033
for unknown delay and amplitude parameters, L is the number of signals x (t) including base signals h (t), which are expressed as:
Figure BDA0002993615100000034
step two, constructing a sampling kernel function: the basic sinc kernel is taken as a sampling kernel function, and non-ideal effects existing in the hardware implementation process are considered, so that the sampling kernel function is expressed as
Figure BDA0002993615100000035
Its continuous time Fourier transform can be expressed as
Figure BDA0002993615100000036
Step three, low-speed sampling modeling: using a delay interval of t-nTsThe continuous impulse string as the sampling function needs to satisfy the requirement that the low-speed sampling frequency domain needs more than twice the cut-off frequency of the filter, namely 1/TsF is more than 2f, and f is the cut-off frequency of the non-ideal filter;
step four, sampling the sample y [ n ]]And acquiring and processing: the original signal x (T) is sampled at a low speed after passing through a non-ideal filter, the sampling rate being 1/Ts> 2f, obtaining sample y [ n ]]Then, fourier transform is performed to obtain:
Figure BDA0002993615100000037
from equation (2)
Figure BDA0002993615100000038
For a signal extended by one period, one period is taken, that is, the signal has complete frequency domain information, and if n is 0, the signal is expressed as:
Figure BDA0002993615100000041
step five, sampling a sample h [ n ]]And acquiring and processing: the base signal h (T) is passed through a non-ideal filter and then sampled at a low rate of 1/Ts> 2f, obtaining a sample h [ n ]]Then, fourier transform is performed to obtain:
Figure BDA0002993615100000042
in the same step four, let n be 0, formula (4) is expressed as:
Figure BDA0002993615100000043
step six, representing the signal relation: between the original signal x (t) and the base signal h (t)In at rest
Figure BDA0002993615100000044
The relationship, in the frequency domain, is expressed as:
Figure BDA0002993615100000045
taking k samples of the sample, i.e. the order
Figure BDA0002993615100000046
Is expressed as
Figure BDA0002993615100000047
Step seven, a data processing stage: and D, processing the frequency domain information obtained in the fourth step and the fifth step to obtain:
Figure BDA0002993615100000048
after simplification, it is written as:
Figure BDA0002993615100000049
the formula (9) only contains unknown parameters
Figure BDA0002993615100000051
Is a typical parameter estimation problem, solving equation (9) yields the result.
Further, in the sixth step, equation (9) is solved by using a nulling filter method or a subspace estimation method.
The method of the invention is different from the traditional FRI undersampling system, and considers and solves the influence of non-ideal effect on reconstruction precision. The method of the invention processes the base signal h (t) as the original signal, and the original signal and the base signal both contain non-ideal effects. Then the non-ideal effect is eliminated through processing, and finally the signal is reconstructed by using the frequency domain information with the non-ideal effect eliminated. The method eliminates the non-ideal effect caused by sampling kernel function, and can improve the reconstruction precision to a great extent.
The invention has the following beneficial effects: the reconstruction precision of the system is improved, and the anti-noise capability is strong.
Drawings
Fig. 1 is a diagram of a classic FRI sampling architecture.
Fig. 2 is a block diagram of the FRI sampling system structure of the method of the present invention.
Fig. 3 is a graph showing the reconstruction effect of experiment two, in which (a) shows the reconstruction effect of the FRI system and (b) shows the reconstruction effect of the system according to the method of the present invention.
Fig. 4 is a comparison graph of system noise resistance.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 2 to 4, a structural block diagram of a ground penetrating radar ultra wide band gaussian pulse FRI sampling method based on a non-ideal LPF is shown in fig. 2, and the method comprises the following steps:
step one, generating an original signal: parameter settings of the original signal, which can be represented as
Figure BDA0002993615100000061
Wherein T ∈ [0, T) is the observation time,
Figure BDA0002993615100000062
Figure BDA0002993615100000063
for unknown delay and amplitude parameters, L is the number of signals x (t) including base signals h (t), which are expressed as:
Figure BDA0002993615100000064
step two, constructing a sampling kernel function: by the basic sincThe kernel is a sampling kernel function, and takes into account non-ideal effects existing during hardware implementation, so the sampling kernel function can be expressed as
Figure BDA0002993615100000065
Its continuous time Fourier transform is expressed as
Figure BDA0002993615100000066
Step three, low-speed sampling modeling: using a delay interval of t-nTsAs a sampling function, so that the requirement of a low-speed sampling frequency domain for more than twice the cut-off frequency of the filter, i.e. 1/T, is satisfiedsF is more than 2f, and f is the cut-off frequency of the non-ideal filter;
step four, sampling the sample y [ n ]]And acquiring and processing: the original signal x (T) is sampled at a low speed after passing through a non-ideal filter, the sampling rate being 1/Ts> 2f, obtaining sample y [ n ]]Then, fourier transform is performed to obtain:
Figure BDA0002993615100000067
the formula (1.14) shows
Figure BDA0002993615100000068
For a signal extended by one period, one period is taken, that is, the signal has complete frequency domain information, and if n is 0, the signal is expressed as:
Figure BDA0002993615100000069
step five, sampling a sample h [ n ]]And acquiring and processing: the base signal h (T) is passed through a non-ideal filter and then sampled at a low rate of 1/Ts> 2f, obtaining a sample h [ n ]]Then, fourier transform is performed to obtain:
Figure BDA00029936151000000610
in the same step four, let n be 0, and formula (4) is expressed as:
Figure BDA0002993615100000071
step six, representing the signal relation: between the original signal x (t) and the base signal h (t)
Figure BDA0002993615100000072
The relationship, in the frequency domain, is expressed as:
Figure BDA0002993615100000073
taking k samples of the sample, i.e. the order
Figure BDA0002993615100000074
Is expressed as
Figure BDA0002993615100000075
Step seven, a data processing stage: and D, processing the frequency domain information obtained in the fourth step and the fifth step to obtain:
Figure BDA0002993615100000076
after simplification, it is written as:
Figure BDA0002993615100000077
the formula (8) contains only unknown parameters
Figure BDA0002993615100000078
Is a typical parameter estimation problem, using a nulling filter approach or subspace estimationAnd (6) counting and solving.
In order to verify the reconstruction effect and the anti-noise performance of the method, MATLAB software is used for simulating the method. Use signal
Figure BDA0002993615100000079
As the original signal, the amplitude parameter is set to al＝[0.9,0.6,0.8,0.5]With time delay parameter set to tl＝[0.2,0.4,0.6,0.8]Using the formula
Figure BDA00029936151000000710
Generating a base signal, t0For the initial delay of the base signal, set to 0.5, σ is set to 0.001.
Experiment one: the use of an idealized sampling kernel, i.e. the impulse effect of the filter, is represented as a typical cut-off characteristic. By using the above signals to perform experiments on the reconstruction effect of the FRI sampling system and the method of the present invention, it can be observed that the reconstructed amplitude parameters and time delay parameters are shown in table 1.
Figure BDA0002993615100000081
TABLE 1
It can be seen from table 1 that both the FRI sampling system and the method of the present invention can reconstruct the original signal with high accuracy when using an idealized filter.
And secondly, performing simulation experiments on the FRI sampling system and the method by using the non-ideal sampling kernel function. The non-ideal filter used in this experiment was a Chebyshev type I filter, and the results of the experiment are shown in fig. 3(a) and 3 (b).
It can be seen from fig. 3(a) that the reconstruction effect is adversely affected when the non-ideal sampling kernel is used, and the original signal can be reconstructed with high precision by using the method of the present invention.
And thirdly, performing an anti-noise experiment on the FRI sampling system and the method. Gaussian white noise was added to both systems, gradually increasing from 0dB to 100dB in steps of 5 dB. For the convenience of comparison, two different parameter reconstruction methods are introduced here to reconstruct the signal, including a nulling filter method and a subspace estimation method. Reconstruction accuracy the reconstruction delay accuracy is measured using Normalized Mean-Square Error (NMSE), which is expressed as follows:
Figure BDA0002993615100000082
the experimental result is shown in fig. 4, and it can be seen from the graph that the anti-noise performance of the method of the present invention is significantly better than that of the FRI sampling system.
The embodiments described in this specification are merely illustrative of implementations of the inventive concepts, which are intended for purposes of illustration only. The scope of the present invention should not be construed as being limited to the particular forms set forth in the examples, but rather as being defined by the claims and the equivalents thereof which can occur to those skilled in the art upon consideration of the present inventive concept.

Claims (2)

1. A ground penetrating radar ultra wide band Gaussian pulse (FRI) sampling method based on a non-ideal LPF is characterized by comprising the following steps:
step one, generating an original signal: parameter setting of an original signal, the original signal being represented as
Figure FDA0002993615090000011
Wherein T ∈ [0, T) is the observation time,
Figure FDA0002993615090000012
for unknown delay and amplitude parameters, L is the number of signals x (t) including base signals h (t), which are expressed as:
Figure FDA0002993615090000013
step two, constructing a sampling kernel function: taking a basic sinc kernel as a sampling kernel function,and takes into account non-ideal effects that exist during hardware implementation, the sampling kernel is expressed as
Figure FDA0002993615090000014
Its continuous time Fourier transform can be expressed as
Figure FDA0002993615090000015
Step three, low-speed sampling modeling: using a delay interval of t-nTsThe continuous impulse string as the sampling function needs to satisfy the requirement that the low-speed sampling frequency domain needs more than twice the cut-off frequency of the filter, namely 1/TsF is more than 2f, and f is the cut-off frequency of the non-ideal filter;
step four, sampling the sample y [ n ]]And acquiring and processing: the original signal x (T) is sampled at a low speed after passing through a non-ideal filter, the sampling rate being 1/Ts> 2f, obtaining sample y [ n ]]Then, fourier transform is performed to obtain:
Figure FDA0002993615090000016
from equation (2)
Figure FDA0002993615090000017
For a signal extended by one period, one period is taken, that is, the signal has complete frequency domain information, and if n is 0, the signal is expressed as:
Figure FDA0002993615090000018
step five, sampling a sample h [ n ]]And acquiring and processing: the base signal h (T) is passed through a non-ideal filter and then sampled at a low rate of 1/Ts> 2f, obtaining a sample h [ n ]]Then, fourier transform is performed to obtain:
Figure FDA0002993615090000019
in the same step four, let n be 0, formula (4) is expressed as:
Figure FDA00029936150900000110
step six, representing the signal relation: between the original signal x (t) and the base signal h (t)
Figure FDA00029936150900000111
The relationship, in the frequency domain, is expressed as:
Figure FDA0002993615090000021
taking k samples of the sample, i.e. the order
Figure FDA0002993615090000022
Is expressed as
Figure FDA0002993615090000023
Step seven, a data processing stage: and D, processing the frequency domain information obtained in the fourth step and the fifth step to obtain:
Figure FDA0002993615090000024
after simplification, it is written as:
Figure FDA0002993615090000025
the formula (9) only contains unknown parameters
Figure FDA0002993615090000026
Is a typical parameter estimation problem, solving equation (9) yields the result.
2. The FRI sampling method for ultra-wideband Gaussian pulses of ground penetrating radar based on non-ideal LPF as claimed in claim 1, wherein in the sixth step, equation (9) is solved by using the nulling filter method or subspace estimation method.
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