CN111830477A - Time delay Doppler parameter joint estimation method based on FRI sampling - Google Patents

Time delay Doppler parameter joint estimation method based on FRI sampling Download PDF

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CN111830477A
CN111830477A CN202010550129.5A CN202010550129A CN111830477A CN 111830477 A CN111830477 A CN 111830477A CN 202010550129 A CN202010550129 A CN 202010550129A CN 111830477 A CN111830477 A CN 111830477A
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CN111830477B (en
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付宁
尉志良
乔立岩
彭喜元
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Harbin Institute of Technology
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Abstract

The invention discloses a time delay Doppler parameter joint estimation method based on FRI sampling. The invention belongs to the technical field of signal processing, and determines a pulse Doppler signal to be sampled according to a pulse repetition period; performing FRI sampling on the determined pulse Doppler signal to be sampled to obtain a Fourier coefficient; performing variable substitution on the obtained Fourier coefficients based on an SCS model, and determining a time delay parameter; determining Doppler parameters by adopting the same method according to the symmetry of the Fourier coefficients after variable substitution; and determining a complex coefficient of a calculation delay parameter equation according to the delay parameter and the Doppler parameter, determining an amplitude parameter according to the calculated complex coefficient and the Doppler parameter, and pairing the amplitude parameter, the delay parameter and the Doppler parameter to complete joint estimation. The invention utilizes the FRI sampling structure to directly sample the Fourier coefficient of the signal, breaks through the limitation of the signal bandwidth and greatly reduces the sampling rate of the signal.

Description

Time delay Doppler parameter joint estimation method based on FRI sampling
Technical Field
The invention relates to the technical field of signal processing, in particular to a time delay Doppler parameter joint estimation method based on FRI sampling.
Background
Pulse doppler signals are widely present in linear time-varying systems such as radar, sonar, wireless communication and the like, and estimating the time delay and doppler parameters thereof is a key task for identifying the linear time-varying systems. According to the Nyqiust sampling theorem, the sampling rate must be greater than or equal to twice the signal bandwidth in order to recover the analog signal from discrete sampled samples without distortion. With the development of modern information technology, the bandwidth of the pulse signal is gradually increased, and the pressure of the sampling device is increased. Therefore, the nyquist sampling theorem gradually becomes a bottleneck of the design of the pulse doppler signal sampling system, and the development of the signal processing technology is restricted.
Many under-Nyquist sampling schemes have been proposed for pulsed doppler signals. Some researchers put forward a method for simultaneously estimating time delay Doppler parameters based on a compressed sensing theory, however, the method has large high-dimensional optimization calculation amount and is not suitable for practical application. The national Nanjing university of science and engineering utilizes an orthogonal compressed sensing sampling structure to provide a sequential delay Doppler parameter estimation method, which can complete the under-Nyquist sampling of pulse Doppler signals, but the sampling rate setting of the method is still related to the signal bandwidth, and the required sampling rate is still higher.
In recent years, a brand-new sampling theory, namely a Finite Rate of Information (FRI) theory, appears in the field of vision of people, and the theory can break through the relation between sampling Rate and signal bandwidth and further reduce the signal utilization Rate. FRI signal is a signal that can be determined by a finite parameter per unit time, and its finite information rate is the number of free parameters per unit time. A pulsed doppler signal is a signal that can be characterized by a finite number of parameters, namely, delay, doppler and amplitude parameters. For pulse Doppler signals, Bajwa et al propose a time delay Doppler parameter joint estimation method based on FRI sampling theory, however, the method is sensitive to noise and poor in noise immunity. Bar-Ilan et al put forward a Doppler focusing parameter estimation method by using FRI sampling theory, which can complete the estimation of a delay Doppler parameter under a lower signal-to-noise ratio, however, as Doppler focusing requires a discrete gridding process, the estimation precision of the algorithm to the parameter deviating from a grid is rapidly reduced, and therefore, more pulse numbers are required in practical application to ensure the precision of the Doppler focusing process. Heretofore, an under-sampling scheme which simultaneously satisfies the requirements of a small number of pulses, a low sampling rate and good noise resistance for a pulse doppler signal has not been provided.
Disclosure of Invention
The invention provides a time delay Doppler parameter joint estimation method based on FRI sampling, which is used for simultaneously meeting the under-sampling scheme with less pulse number, low sampling rate and good noise immunity, and the invention provides the following technical scheme:
a time delay Doppler parameter joint estimation method based on FRI sampling comprises the following steps:
step 1: determining a pulse Doppler signal to be sampled according to the pulse repetition period;
step 2: performing FRI sampling on the determined pulse Doppler signal to be sampled to obtain a Fourier coefficient;
and step 3: performing variable substitution on the obtained Fourier coefficients based on an SCS model, and determining a time delay parameter;
and 4, step 4: determining Doppler parameters according to the Fourier coefficients after variable substitution;
and 5: and determining a complex coefficient of a delay parameter equation according to the delay parameter and the acquired Fourier coefficient, determining a pulse amplitude parameter according to the acquired complex coefficient and the Doppler parameter, and pairing the amplitude parameter, the delay parameter and the Doppler parameter to complete joint estimation.
Preferably, the step 1 specifically comprises: the pulse Doppler signal to be sampled is composed of P pulse periods, the pulse Doppler signal to be sampled is determined according to the pulse repetition period, and the pulse Doppler signal to be sampled x (t) is expressed by the following formula:
Figure BDA0002542220630000021
where τ is the pulse repetition period of the signal, P-0, 1H (t) is a pulse basis function with a known waveform, alFor amplitude parameters of the first target echo of the signal, τlTime delay parameter for the ith target echo, vlIs the ith target echo Doppler parameter.
Preferably, the step 2 specifically comprises: FRI sampling is carried out on a determined pulse Doppler signal x (t) to be sampled, the x (t) is divided and then is respectively mixed with K complex exponential signals, the frequency is divided by an integrator, the integrator is triggered by emission pulses, the integration is carried out once every tau time duration, the integration result is sampled at a time interval tau to obtain a Fourier coefficient, and the Fourier coefficient x is represented by the following formulap[k]：
Figure BDA0002542220630000022
Where H (k) is the Fourier coefficient of the function h (t).
Preferably, the step 3 specifically comprises:
step 3.1: performing variable substitution on the obtained Fourier coefficients based on an SCS model, and expressing the Fourier coefficients after variable substitution by the following formula:
Figure BDA0002542220630000023
Figure BDA0002542220630000024
wherein, blIs a complex coefficient;
step 3.2: let xp,k'＝[xp[k'],xp[k'+1],xp…,xp[k'+L]]Containing L +1 consecutive fourier coefficients, constructing an autocorrelation matrix R, which is represented by:
Figure BDA0002542220630000031
K'＝K-L
wherein K' is the sampling value in each PRIVector x of energy compositionp,k'The number of (2);
when the parameters are not the same, the sampling condition that P is more than or equal to L and K is more than or equal to L +1 can ensure that the rank of the autocorrelation matrix R is L, and the requirement of an ESPRIT algorithm on an input matrix is met;
when the time delay parameter or the Doppler parameter is the same, P is more than or equal to Sτ,K≥L+SvEnsuring that the rank of the autocorrelation matrix R meets the sufficient unnecessary conditions of the ESPRIT algorithm, solving the autocorrelation matrix R by using the ESPRIT algorithm to obtain a time delay parameter
Figure BDA0002542220630000032
Preferably, the determining the doppler parameter according to the fourier coefficient after the variable substitution in step 4 specifically includes:
if the parameters are not the same, the sampling condition that P is more than or equal to L +1 and K is more than or equal to L ensures that the rank of the autocorrelation matrix is L, and the requirement of an ESPRIT algorithm on an input matrix is met;
when the time delay or Doppler parameter is the same, P is more than or equal to L + Sτ,K≥SvThe sampling condition of (2) ensures that the autocorrelation matrix rank meets the sufficient unnecessary condition required by an ESPRIT algorithm;
for the condition that the parameters of the pulse which is not received are the same, P is more than or equal to L +1, K is more than or equal to L +1 to ensure the accurate reconstruction of the Doppler parameter, and for the condition that the time delay parameter or the Doppler parameter is the same, P is more than or equal to L + Sτ,K≥L+SvEnsuring Doppler parameters
Figure BDA0002542220630000033
The autocorrelation matrix R is solved by adopting an ESPRIT algorithm to obtain Doppler parameters
Figure BDA0002542220630000034
Preferably, the step 5 specifically comprises:
according to the solved time delay parameter
Figure BDA0002542220630000035
Using a Fourier transformed by a variableComplex coefficient b of time delay parameter equation determined by leaf coefficientlAccording to the value of b obtainedlValue and Doppler parameter
Figure BDA0002542220630000036
To determine a pulse amplitude parameter
Figure BDA0002542220630000037
For calculated blHas the value of (a) of the amplitude parameter a of the first echo targetlAnd Doppler parameter vlCorrespondingly, using blAnd the amplitude parameter a of the l-th echo targetlMatching the amplitude parameter, the delay parameter and the Doppler parameter to complete the matching of the combined parameters
Figure BDA0002542220630000038
Joint estimation of (1).
The invention has the following beneficial effects:
according to the invention, the FRI sampling structure is utilized, the Fourier coefficient of the signal is directly sampled, the limitation of signal bandwidth is broken through, and the sampling rate of the signal is greatly reduced;
by utilizing the SCS model, time delay and Doppler parameters can be solved by combining Fourier coefficients sampled in a plurality of PRI, and the noise immunity and the estimation precision of the algorithm are greatly improved. At the same time, the use of the SCS model allows the design method to have the ability to simultaneously reduce the number of transmit pulses and the number of Fourier coefficients sampled per channel.
Drawings
FIG. 1 is a flow chart of Fourier coefficient acquisition by an FRI sampling structure;
FIG. 2 is a schematic diagram of a time-domain SCS model construction of a received signal;
FIG. 3 is a diagram of the result of parameter estimation when the parameters of echo-free pulses are the same;
FIG. 4 is a diagram showing the result of parameter estimation when the parameters of the echo pulses are the same;
FIG. 5 is a time delay parameter T of different echo numberlEstimating an NMSE profile;
FIG. 6 is a drawing showingDoppler parameter v of different echo numberlThe NMSE profile is estimated.
Detailed Description
The present invention will be described in detail with reference to specific examples.
The first embodiment is as follows:
the invention aims to solve the problem of time delay Doppler parameter joint estimation under the condition of pulse Doppler signal undersampling, and aims to provide a parameter joint estimation method based on an FRI undersampling structure and a Sparse Common Support (SCS for short). Under the condition that the waveform of a transmitted pulse is known, the system directly acquires Fourier coefficients X of specific positions of a signal through an FRI undersampling structurep[k]And by utilizing the symmetry of the delay parameter and the Doppler parameter in the Fourier coefficient, the delay parameter and the Doppler parameter can be solved by a similar method. By utilizing the SCS model, the parameters can be jointly solved by utilizing Fourier coefficients acquired in a plurality of pulse repetition intervals (PRI for short), so that the noise immunity and the parameter estimation precision of the algorithm are improved, and the minimum PRI number required by parameter recovery and the sampling point number required by each PRI can be effectively reduced. If each emission pulse corresponds to L echo signals, P is larger than or equal to L +1 emission pulses under the condition that L echoes have different parameters, and each PRI sampling K is larger than or equal to L +1 Fourier coefficients, so that the accurate estimation of 3L unknown parameters can be ensured. For the case where L echoes have the same delay or Doppler parameters, with at most SτThe echoes have the same time delay parameter and at most LvThe echoes have the same Doppler parameter, then P ≧ L + SτEach emission pulse has PRI sampling K ≥ L + SvThe Fourier coefficients can ensure accurate estimation of unknown delay, Doppler and amplitude parameters.
A time delay Doppler parameter joint estimation method based on FRI sampling comprises the following steps:
step 1: determining a pulse Doppler signal to be sampled according to the pulse repetition period;
the step 1 specifically comprises the following steps: the pulse Doppler signal to be sampled is composed of P pulse periods, the pulse Doppler signal to be sampled is determined according to the pulse repetition period, and the pulse Doppler signal to be sampled x (t) is expressed by the following formula:
Figure BDA0002542220630000041
where τ is the pulse repetition period of the signal, P-0, 1.., P-1 is the index of the pth PRI, h (t) is the pulse basis function with a known waveform, and alFor amplitude parameters of the first target echo of the signal, τlTime delay parameter for the ith target echo, vlIs the ith target echo Doppler parameter.
Step 2: performing FRI sampling on the determined pulse Doppler signal to be sampled to obtain a Fourier coefficient;
the step 2 specifically comprises the following steps: FRI sampling is carried out on a determined pulse Doppler signal x (t) to be sampled, the x (t) is divided and then is respectively mixed with K complex exponential signals, the frequency is divided by an integrator, the integrator is triggered by emission pulses, the integration is carried out once every tau time duration, the integration result is sampled at a time interval tau to obtain a Fourier coefficient, and the Fourier coefficient x is represented by the following formulap[k]：
Figure BDA0002542220630000051
Where H (k) is the Fourier coefficient of the function h (t).
And step 3: performing variable substitution on the obtained Fourier coefficients based on an SCS model, and determining a time delay parameter;
the step 3 specifically comprises the following steps:
step 3.1: performing variable substitution on the obtained Fourier coefficients based on an SCS model, and expressing the Fourier coefficients after variable substitution by the following formula:
Figure BDA0002542220630000052
Figure BDA0002542220630000053
wherein, blIs a complex coefficient;
step 3.2: let xp,k'＝[xp[k'],xp[k'+1],xp…,xp[k'+L]]Containing L +1 consecutive fourier coefficients, constructing an autocorrelation matrix R, which is represented by:
Figure BDA0002542220630000054
K'＝K-L
wherein K' is a vector x which can be formed by sampling values in each PRIp,k'The number of (2);
when the parameters are not the same, the sampling condition that P is more than or equal to L and K is more than or equal to L +1 can ensure that the rank of the autocorrelation matrix R is L, and the requirement of an ESPRIT algorithm on an input matrix is met;
when the time delay parameter or the Doppler parameter is the same, P is more than or equal to Sτ,K≥L+SvEnsuring that the rank of the autocorrelation matrix R meets the sufficient unnecessary conditions of the ESPRIT algorithm, solving the autocorrelation matrix R by using the ESPRIT algorithm to obtain a time delay parameter
Figure BDA0002542220630000055
And 4, step 4: determining Doppler parameters according to the Fourier coefficients after variable substitution;
in the step 4, the determining the doppler parameter according to the fourier coefficient after the variable substitution specifically includes:
if the parameters are not the same, the sampling condition that P is more than or equal to L +1 and K is more than or equal to L ensures that the rank of the autocorrelation matrix is L, and the requirement of an ESPRIT algorithm on an input matrix is met;
when the time delay or Doppler parameter is the same, P is more than or equal to L + Sτ,K≥SvThe sampling condition of (2) ensures that the autocorrelation matrix rank meets the sufficient unnecessary condition required by an ESPRIT algorithm;
for the condition that the parameters of the pulse which is not received are the same, P is more than or equal to L +1, K is more than or equal to L +1, the accurate reconstruction of the Doppler parameter is ensured, and for the condition that the time delay is not the sameThe parameter or Doppler parameter is the same, P is more than or equal to L + Sτ,K≥L+SvEnsuring Doppler parameters
Figure BDA0002542220630000061
The autocorrelation matrix R is solved by adopting an ESPRIT algorithm to obtain Doppler parameters
Figure BDA0002542220630000062
And 5: and determining a complex coefficient of a delay parameter equation according to the delay parameter and the acquired Fourier coefficient, determining a pulse amplitude parameter according to the acquired complex coefficient and the Doppler parameter, and pairing the amplitude parameter, the delay parameter and the Doppler parameter to complete joint estimation.
The step 5 specifically comprises the following steps:
according to the solved time delay parameter
Figure BDA0002542220630000063
Determining a complex coefficient b of a time delay parameter equation by using a Fourier coefficient after variable substitutionlAccording to the value of b obtainedlValue and Doppler parameter
Figure BDA0002542220630000064
To determine a pulse amplitude parameter
Figure BDA0002542220630000065
For calculated blHas the value of (a) of the amplitude parameter a of the first echo targetlAnd Doppler parameter vlCorrespondingly, using blAnd the amplitude parameter a of the l-th echo targetlMatching the amplitude parameter, the delay parameter and the Doppler parameter to complete the matching of the combined parameters
Figure BDA0002542220630000066
Joint estimation of (1).
The second embodiment is as follows:
the invention provides a time delay Doppler parameter joint estimation method based on FRI sampling, which comprises the following specific implementation steps:
step 1: assuming that the pulsed doppler signal x (t) to be sampled consists of P pulse transmission cycles, each PRI contains L target echoes, which can be expressed as:
Figure BDA0002542220630000067
where τ is the pulse repetition period of the signal, i.e., a PRI, P0, 1, P-1 is the index of the pth PRI, and h (t) is the pulse basis function for which the waveform is known
Figure BDA0002542220630000068
An unknown amplitude parameter representing the ith target echo of the signal,
Figure BDA0002542220630000069
respectively, unknown time delay and Doppler parameters of the ith target echo of the signal.
And 2, sampling by FRI to obtain a Fourier coefficient of the Doppler signal. The pulse doppler signal is sampled by an FRI sampling structure as shown in fig. 1, x (t) is split and then mixed with K complex exponential signals, and then passes through an integrator triggered by a transmission pulse, the integration is performed once every tau time duration, and then the integration result is sampled at a time interval tau as well. The sample value for the kth lane in the pth PRI may be represented as:
Figure BDA0002542220630000071
where H (k) is the Fourier coefficient of the basis function h (t). It can be seen that the delay parameter τ is the fourier coefficient obtainedlAnd Doppler parameter vlAre fully symmetric about index k and index p, respectively, and therefore can be solved in a similar way.
And 3, writing the formula (2) into the following form through variable substitution based on the SCS model:
Figure BDA0002542220630000072
wherein the content of the first and second substances,
Figure BDA0002542220630000073
it can be seen that the acquired fourier coefficients have the same time support τ for different pulse periods plBut different complex coefficients
Figure BDA0002542220630000074
This satisfies the time-domain SCS model as shown in fig. 2.
Let xp,k'＝[xp[k'],xp[k'+1],xp…,xp[k'+L]]Containing L +1 continuous Fourier coefficients, and constructing autocorrelation matrix
Figure BDA0002542220630000075
Where K' ═ K-L is the vector x that can be made up of the samples in each PRIp,k'The number of (2). If the parameters are not the same, the sampling condition that P is larger than or equal to L and K is larger than or equal to L +1 can ensure that the rank of the autocorrelation matrix R is L, and the requirement of an ESPRIT algorithm on an input matrix is met. If the time delay or Doppler parameter is the same, P is more than or equal to Sτ,K≥L+SvIs a sufficient unnecessary condition to ensure that the rank of the autocorrelation matrix R meets the requirements of the ESPRIT algorithm. The autocorrelation matrix R is then solved using the ESPRIT algorithm. The unknown time delay parameter can be obtained
Figure BDA0002542220630000076
Step 4, utilizing Fourier coefficient Xp[k]The same method as step 3 can be used to solve the Doppler parameters due to the symmetry of the middle delay parameter and the Doppler parameters
Figure BDA0002542220630000077
At this time, if the parameters are not the same, the sampling condition that P is greater than or equal to L +1 and K is greater than or equal to L can ensure that the rank of the autocorrelation matrix is L, so that the requirement of L is metThe ESPRIT algorithm requires an input matrix. If the time delay or Doppler parameter is the same, P is more than or equal to L + Sτ,K≥SvIs a sufficient unnecessary condition to ensure that the autocorrelation matrix rank meets the requirements of the ESPRIT algorithm.
In summary, for the condition that the parameters of the pulse not received are the same, P is more than or equal to L +1, K is more than or equal to L +1, and the delay Doppler parameter can be ensured
Figure BDA0002542220630000078
Accurate reconstruction. For the condition of time delay or same Doppler parameter, P is more than or equal to L + Sτ,K≥L+SvCan guarantee the delay Doppler parameter
Figure BDA0002542220630000079
Accurate reconstruction.
Step 5, according to the solved time delay parameter
Figure BDA00025422206300000710
B can be calculated by the formula (3)lThe value of (c). Using calculated blValue and Doppler parameter
Figure BDA00025422206300000711
Can calculate the pulse amplitude parameter
Figure BDA00025422206300000712
Wherein, as can be seen from the formula (3), for the calculated
Figure BDA00025422206300000713
Has the value of (a) of the amplitude parameter a of the first echo targetlAnd Doppler parameter vlCorresponding to it. So that we can use blAnd subsequent amplitude parameter alThe solution process of (2) completes the pairing of the amplitude, the time delay and the Doppler parameters, namely, completes the final 3L parameters
Figure BDA00025422206300000714
Joint estimation and pairing.
Results of the experiment
1. Noiseless experiment
And setting the number L of each PRI echo of the signal to be detected to be 4, wherein the pulse basis function is an ideal dirac pulse. When the parameters of the non-receiving pulse are the same, the number of the transmitting pulses is set to be P-5, and the number of Fourier coefficients sampled per PRI is set to be K-5. Fig. 3 shows the signal delay parameter τlAnd Doppler parameter vlThe reconstruction effect of (1). When 2 echo pulses have the same time delay and doppler parameters, the number of sampling channels is set to be P-6, and the number of fourier coefficients sampled in each PRI is K-6. Fig. 4 shows the signal delay parameter τlAnd Doppler parameter vlThe reconstruction effect of (1). It can be seen that the sampling structure and method reconstruct the delay parameter and the doppler parameter of the signal without error under the condition that the parameters of the received pulse are not the same and the parameters of the received echo pulse are the same.
2. Noise experiment
The number L of each PRI echo of the signal to be detected is set to be 2, 3, 4 and 5 respectively, the pulse basis function is an ideal dirac pulse, and parameters of the pulses which are not received are the same. The signal-to-noise ratio of the received signal is set to-10 dB-10dB, the number of the transmitted pulses is set to P as 100, and the number of Fourier coefficients of each PRI sample is set to K as 100. Fig. 5 and 6 show the signal delay parameter τ respectivelylAnd Doppler parameter vlAnd (3) reconstructing a Normalized Mean Square Error (NMSE) curve. The experimental result shows that the effect of estimating the signal parameter by the sampling structure and the method is better and better along with the improvement of the signal-to-noise ratio. Meanwhile, for different target echo numbers, the reconstruction accuracy of the method is higher and higher along with the reduction of the echo number.
The above is only a preferred embodiment of the delay-doppler parameter joint estimation method based on FRI sampling, and the protection range of the delay-doppler parameter joint estimation method based on FRI sampling is not limited to the above embodiments, and all technical solutions belonging to the idea belong to the protection range of the present invention. It should be noted that modifications and variations which do not depart from the gist of the invention will be those skilled in the art to which the invention pertains and which are intended to be within the scope of the invention.

Claims (6)

1. A time delay Doppler parameter joint estimation method based on FRI sampling is characterized in that: the method comprises the following steps:
step 1: determining a pulse Doppler signal to be sampled according to the pulse repetition period;
step 2: performing FRI sampling on the determined pulse Doppler signal to be sampled to obtain a Fourier coefficient;
and step 3: performing variable substitution on the obtained Fourier coefficients based on an SCS model, and determining a time delay parameter;
and 4, step 4: determining Doppler parameters according to the Fourier coefficients after variable substitution;
and 5: and determining a complex coefficient of a delay parameter equation according to the delay parameter and the acquired Fourier coefficient, determining a pulse amplitude parameter according to the acquired complex coefficient and the Doppler parameter, and pairing the amplitude parameter, the delay parameter and the Doppler parameter to complete joint estimation.
2. The method of claim 1 for jointly estimating delay-doppler parameters based on FRI sampling, wherein: the step 1 specifically comprises the following steps: the pulse Doppler signal to be sampled is composed of P pulse periods, the pulse Doppler signal to be sampled is determined according to the pulse repetition period, and the pulse Doppler signal to be sampled x (t) is expressed by the following formula:
Figure FDA0002542220620000011
where τ is the pulse repetition period of the signal, P-0, 1.., P-1 is the index of the pth PRI, h (t) is the pulse basis function with a known waveform, and alFor amplitude parameters of the first target echo of the signal, τlTime delay parameter for the ith target echo, vlIs the ith target echo Doppler parameter.
3. According to the rightThe method for jointly estimating delay-doppler parameters based on FRI sampling according to claim 1, wherein: the step 2 specifically comprises the following steps: FRI sampling is carried out on a determined pulse Doppler signal x (t) to be sampled, the x (t) is divided and then is respectively mixed with K complex exponential signals, the frequency is divided by an integrator, the integrator is triggered by emission pulses, the integration is carried out once every tau time duration, the integration result is sampled at a time interval tau to obtain a Fourier coefficient, and the Fourier coefficient x is represented by the following formulap[k]：
Figure FDA0002542220620000012
Where H (k) is the Fourier coefficient of the function h (t).
4. The method of claim 1 for jointly estimating delay-doppler parameters based on FRI sampling, wherein: the step 3 specifically comprises the following steps:
step 3.1: performing variable substitution on the obtained Fourier coefficients based on an SCS model, and expressing the Fourier coefficients after variable substitution by the following formula:
Figure FDA0002542220620000013
Figure FDA0002542220620000021
wherein, blIs a complex coefficient;
step 3.2: let xp,k'＝[xp[k'],xp[k'+1],xp…,xp[k'+L]]Containing L +1 consecutive fourier coefficients, constructing an autocorrelation matrix R, which is represented by:
Figure FDA0002542220620000022
K'＝K-L
wherein K' is eachVector x of possible combinations of values sampled within PRIp,k'The number of (2);
when the parameters are not the same, the sampling condition that P is more than or equal to L and K is more than or equal to L +1 ensures that the rank of the autocorrelation matrix R is L, and the requirement of an ESPRIT algorithm on an input matrix is met;
when the time delay parameter or the Doppler parameter is the same, P is more than or equal to Sτ,K≥L+SvEnsuring that the rank of the autocorrelation matrix R meets the sufficient unnecessary conditions of the ESPRIT algorithm, solving the autocorrelation matrix R by using the ESPRIT algorithm to obtain a time delay parameter
Figure FDA0002542220620000023
5. The method of claim 1 for jointly estimating delay-doppler parameters based on FRI sampling, wherein: in the step 4, the determining the doppler parameter according to the fourier coefficient after the variable substitution specifically includes:
when the parameters are not the same, the sampling condition that P is more than or equal to L +1 and K is more than or equal to L ensures that the rank of the autocorrelation matrix is L, and the requirement of an ESPRIT algorithm on an input matrix is met;
when the time delay or Doppler parameter is the same, P is more than or equal to L + Sτ,K≥SvThe sampling condition of (2) ensures that the autocorrelation matrix rank meets the sufficient unnecessary condition required by an ESPRIT algorithm;
for the condition that the parameters of the pulse which is not received are the same, P is more than or equal to L +1, K is more than or equal to L +1 to ensure the accurate reconstruction of the Doppler parameter, and for the condition that the time delay parameter or the Doppler parameter is the same, P is more than or equal to L + Sτ,K≥L+SvEnsuring Doppler parameters
Figure FDA0002542220620000024
The autocorrelation matrix R is solved by adopting an ESPRIT algorithm to obtain Doppler parameters
Figure FDA0002542220620000025
6. The method of claim 1 for jointly estimating delay-doppler parameters based on FRI sampling, wherein: the step 5 specifically comprises the following steps:
according to the solved time delay parameter
Figure FDA0002542220620000026
Determining a complex coefficient b of a time delay parameter equation by using a Fourier coefficient after variable substitutionlAccording to the value of b obtainedlValue and Doppler parameter
Figure FDA0002542220620000027
To determine a pulse amplitude parameter
Figure FDA0002542220620000028
For calculated blHas the value of (a) of the amplitude parameter a of the first echo targetlAnd Doppler parameter vlCorrespondingly, using blAnd the amplitude parameter a of the l-th echo targetlMatching the amplitude parameter, the delay parameter and the Doppler parameter to complete the matching of the combined parameters
Figure FDA0002542220620000029
Joint estimation of (1).
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