CN104462017A - I/O number-unequal quick non-uniform discrete Fourier transform method and system - Google Patents
I/O number-unequal quick non-uniform discrete Fourier transform method and system Download PDFInfo
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Abstract
The invention discloses an I/O number-unequal quick non-uniform discrete Fourier transform method and system. Compared with the prior art, frequency modulation Z transform is adopted to replace an FFT in a traditional NUFFT method, and I/O number-unequal quick non-uniform discrete Fourier transform is achieved; according to the method, the quick transform error is small and can be ignored, and the advantages of high precision and high efficiency are kept; by means of the method, the application field of the traditional NUFFT method is greatly expanded, and the applicability of the traditional NUFFT method is greatly improved.
Description
Technical Field
The invention belongs to the field of numerical calculation, and particularly relates to a fast non-uniform discrete Fourier transform method and system with unequal I/O (input/output) numbers.
Background
A Non-uniform discrete Fourier transform (NUDFT) is a discrete Fourier transform in which input/output (I/O) sampling positions are Non-uniform real numbers. Because the I/O positions are non-uniform, the traditional Fast Fourier Transform (FFT) method cannot be directly used for improving the NUDFT realization efficiency; and a Non-uniform fast fourier transform (NUFFT) method performs oversampling and interpolation operations at an I/O Non-uniform sampling position, and then implements the NUDFT by using the FFT. The NUFFT method has the advantages of high efficiency, accuracy and the like, and is widely applied to the fields of spectrum methods (spectral methods), magnetic resonance imaging (magnetic resonance imaging), radar signal processing and the like.
Currently, there are representative NUFFT implementations proposed by American scholars A.Dutt et al (see document 1: A.Dutt, V.Rokhlin, "Fast Fourier transforms for non-required data", SIAM J.SCI.COMPUT, Vol.14, No.6, pp.1368-1393, Nov.1993), J.A.Fessler et al (see document 2: J.A.Fessler, B.P.Sutton, IEEE Trans.Signal Processing, Vol.51, No.2, pp.560-574,2003), Q.Liu et al (see document 3: Q.H.Liu, N.Nguyen, Anhm ac algorithm for non-required transform of the family of the United states, Vol.84, and IEEE J.1. Ser.18). These methods differ in that a different interpolation method is used. However, none of these methods can achieve fast NUDFT processing with unequal numbers of I/os. In practical applications, such as when the synthetic aperture radar signal processes large scene data (see document 4: France schetti, G., R.Lanari, and E.S. Marzouk, "A new two-dimensional acquisition mode SAR processor," IEEE Transactions on Aero space and electronic Systems, vol.32, pp.854-863,1996.), it is often necessary to efficiently calculate NUFFT processing with unequal I/O numbers. Therefore, the existing NUFFT implementation method is difficult to meet the practical application requirement.
Disclosure of Invention
In order to solve the technical problem, the invention provides a fast non-uniform discrete Fourier transform method with unequal I/O numbers.
The technical scheme of the invention is as follows: the fast non-uniform discrete Fourier transform method with unequal I/O numbers is characterized in that after oversampling and interpolation operations are carried out at I/O non-uniform sampling positions, non-uniform fast Fourier transform with unequal I/O numbers is realized by utilizing frequency modulation Z transform; the method specifically comprises the following steps:
s1: initializing NUDFT parameters, including: input data x to be transformed, I/O nonuniform position alphamAnd ωkThe number of I/O sampling points M and K;
where x denotes the input data to be transformed, i.e. the discrete sequence of inputs, alphamRepresents the mth input sample point position, ωkDenotes the kth output sample point position, M denotes the number of input samples, K denotes the number of output samples, K is 0,1m∈[-M/2,M/2-1]，ωk∈[-K/2,K/2-1]；
Initializing NUFFT parameters, including: interpolation parameters G, b and Q;
wherein G is an oversampling multiple, b is a real number greater than 1/2, and Q is a total number of interpolation points;
s2: according to the NUDFT parameters and the NUFFT parameters and the formula in step S1:
Wherein q is1,q2Are all integers, q1＝-Q/2,...,Q/2，q2＝-Q/2,...,Q/2；
S3: obtaining an interpolated weighting factor p (α) for the I/O location according to step S2m,q1) And using a formula <math>
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</math> Calculating data to be subjected to FM Z-conversion
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s4: according to the data obtained in step S3 and subjected to frequency modulation Z transformationFor the formula:
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</math> the frequency sampling discrete point locations representing the frequency modulation Z-transform,the position of an output sampling discrete point of frequency modulation Z transformation is shown, a (-) shows data to be subjected to frequency modulation Z transformation, and j shows an imaginary number unit;
s5: according to the frequency modulation Z transformation result A (k, q) obtained in step S42) And the interpolation weighting factor ρ (ω) obtained in step S2k,q2) From the formula:carrying out weighted summation to obtain a result Xk of non-uniform discrete Fourier transform;
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the invention also provides a fast non-uniform discrete Fourier transform system with unequal I/O numbers, which comprises: the device comprises an initialization unit, a weighting factor calculation unit, a frequency modulation Z conversion data calculation unit to be implemented, a frequency modulation Z conversion unit and a weighting summation unit;
the initialization unit initializes the NUDFT parameter and the NUFFT parameter;
the weighting factor calculation unit calculates an interpolation weighting factor according to the NUDFT parameter and the NUFFT parameter initialized by the initialization unit;
the to-be-implemented frequency modulation Z conversion data calculation unit obtains an interpolation weighting factor of an I/O position according to the weighting factor calculation unit, and calculates to-be-implemented frequency modulation Z conversion data;
the frequency modulation Z conversion unit calculates a frequency modulation Z conversion result by utilizing frequency modulation Z conversion according to the data to be subjected to frequency modulation Z conversion obtained by the frequency modulation Z conversion data calculation unit;
and the weighted summation unit carries out weighted summation according to the frequency modulation Z conversion result obtained by the frequency modulation Z conversion unit and the interpolation weighting factor obtained by the weighting factor calculation unit to obtain the result of the non-uniform discrete Fourier transform.
The invention has the beneficial effects that: 1. compared with the prior art, the fast non-uniform discrete Fourier transform method and the system with unequal I/O numbers adopt frequency modulation Z transform to replace FFT in the traditional NUFFT method, and realize the non-uniform fast Fourier transform with unequal I/O numbers;
2. the method has the advantages that the error of rapid conversion is very small and can be ignored, and the advantages of high precision, high efficiency and the like are kept;
3. the method can be applied to the condition that the number of I/O is not equal;
4. the method greatly expands the application field and the applicability of the traditional NUFFT method.
Drawings
FIG. 1 is a schematic flow diagram of the present invention.
FIG. 2 is a schematic diagram of an input data sequence to be transformed for use in verifying feasibility in accordance with the present invention.
FIG. 3 is a schematic diagram of input sequence sample locations used by the present invention to verify feasibility.
FIG. 4 is a schematic diagram of output sequence sample locations used by the present invention to verify feasibility.
FIG. 5 is a graph showing the error of the results obtained after the treatment by the method of the present invention.
Detailed Description
In order to facilitate the understanding of the technical solutions of the present invention for those skilled in the art, the technical contents of the present invention are further described below with reference to the accompanying drawings.
To facilitate understanding of the context of the present invention, the following terms are first defined:
definitions 1, non-uniform discrete Fourier transform (NUDFT)
The one-dimensional non-uniform discrete fourier transform is defined as:
wherein, M is 0, K is M-1, K is 0, K is I/O sampling point number, M, K is I/O sampling point number, alpha is alpham,ωkRespectively the m-th input sample point and the k-th output sample point position, alpham∈[-M/2,M/2-1],ωk∈[-K/2,K/2-1]And j is an imaginary unit.
Three types of non-uniform discrete Fourier transform, Definitions 2
Non-uniform discrete fourier transforms can be classified into three types according to the characteristics of I/O locations:
(a) type 1-non-uniform discrete Fourier transform (NUDFT of type1)
When the input of the non-uniform discrete Fourier transform is a non-uniform real number and the output is a uniform integer, i.e. when alpha ism≠-M/2,...,M/2-1,ωkK/2-1, formula (1) is defined as type 1-non-uniform discrete fourierAnd (6) transforming.
(b) Type 2-non-uniform discrete Fourier transform (NUDFT of type2)
When the input of the non-uniform discrete Fourier transform is a uniform integer and the output is a non-uniform real number, i.e. when alpha ism＝-M/2,...,M/2-1,ωkNot equal to-K/2.., K/2-1, equation (1) is defined as type 2-non-uniform discrete fourier transform.
(c) Type 3-non-uniform discrete Fourier transform (NUDFT of type3)
When the input and output of the non-uniform discrete Fourier transform are both non-uniform real numbers, i.e. when alpha ism≠-M/2,...,M/2-1,ωkNot equal to-K/2.., K/2-1, equation (1) is defined as type 3-non-uniform discrete fourier transform.
From the above definitions, both type1 and type2 non-uniform discrete fourier transforms can be considered as special cases of type3 non-uniform discrete fourier transforms, so the present invention considers the fast implementation of type3 non-uniform discrete fourier transforms in the following.
Defining non-uniform discrete Fourier transforms with unequal numbers of 3's and I/O' s
When the I/O sampling number is not equal, namely when M is not equal to K, the formula (1) is the non-uniform discrete Fourier transform with the unequal I/O number.
Definition 4, FM Z Transform (Chirp Z-Transform, CZT for short)
The frequency modulation Z transformation is a fast method for solving the Fourier transformation of any frequency domain resolution of a discrete function, and the specific implementation can be realized by two times of phase multiplication and one time of convolution, and the detailed reference can be found in the following documents: rabiner L.R., Schafer R.W., Rader C.M.the chirp z-transform algorithm and its application. IEEETransmission on Audio electronics Vol.17:86-92,1969.
Definition 5, parameters used in interpolation
The interpolation weighting factors of the I/O nonuniform positions are respectively as follows:
where G is an oversampling multiple which is a positive integer greater than or equal to 2, b is a real number greater than 1/2, q1,q2Is located in [ -Q/2, Q/2]The uniform integers of the interval are used to number the interpolation positions, Q is the total number of interpolation points [ · C]Representing a rounding operation.
The I/O interpolation point positions are respectively:
the precision of the transformation can be changed by adjusting parameters G, Q, b in equations (2) and (3), and the specific implementation method can be found in the literature: dutt, V.Rokhlin, "Fast Fourier transforms for needed data", SIAM J.SCI.COMPUT, Vol.14, No.6, pp.1368-1393, Nov.1993.
The feasibility of the scheme is verified mainly by adopting a simulation experiment method, and all steps and conclusions are verified to be correct on MATLAB R2012 a.
As shown in fig. 1, the specific implementation steps of the present invention are as follows:
s1: initializing parameters;
initializing NUDFT parameters, including: the input data x to be transformed is shown in fig. 2, the I/O non-uniform positions α and ω are shown in fig. 3 and fig. 4, respectively, and the number of I/O sampling points of the present invention is: m is 64, K is 13;
initializing NUFFT parameters, and respectively setting interpolation parameters G, b and Q as follows: g-2, b-1 and Q-14;
s2: calculating an I/O position interpolation weighting factor; using the NUDFT parameter and the NUFFT parameter in S1 and equation (2):
for each m, m in the present invention is 0,1m,ωkIs interpolated with a weighting factor p (alpha)m,q1) And ρ (ω)k,q2) Wherein k, m, q1,q2Are integers, and in the present invention, k is 0,1,.., 12, m is 0,1,.., 63, q is 0,11,q2＝-7,...,7，αmAnd ωkRespectively the m-th input sampling point and the k-th output sampling point, alpha in the inventionm∈[-32,31],ωk∈[-6.5,5.5]See, respectively, fig. 3 and 4;
s3: calculating data to be subjected to frequency modulation Z conversion; interpolated weighting factor p (alpha) from the I/O location obtained from S2m,q1) And using the formula: <math>
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</math> calculating data to be subjected to FM Z-conversionIn the inventionIs an integer, αm∈[-32,31]The position of the mth input sampling point is shown in figure 3 and is an integer;
s4: performing frequency modulation Z conversion; data obtained in S3 to be subjected to FM Z conversionAnd according to the formula:
S5: weighting and summing; using frequency modulation Z from S4Transformation result A (k, q)2) Interpolation weighting factor ρ (ω) obtained in S2k,q2) And according to the formula:carrying out weighted summation to obtain the result X of the non-uniform discrete Fourier transformkWhereink＝0,1,...,12，ωk∈[-K/2,K/2-1]it is the kth output sample point position that is shown in figure 4.
Through the processing of the steps, the quick realization of the I/O non-uniform discrete Fourier transform is completed; FIG. 5 is a schematic diagram illustrating the error result between the non-uniform discrete fast Fourier transform obtained by the processing from the above steps S1 to S5 and the actual non-uniform discrete Fourier transform; as can be seen from FIG. 5, the error of the fast transformation is very small and can be ignored, so the method provided by the invention can realize the non-uniform discrete Fourier transformation under the condition of unequal I/O numbers with high precision and high efficiency.
The invention also provides a fast non-uniform discrete Fourier transform system with unequal I/O numbers, which comprises: the device comprises an initialization unit, a weighting factor calculation unit, a frequency modulation Z conversion data calculation unit to be implemented, a frequency modulation Z conversion unit and a weighting summation unit;
the initialization unit initializes the NUDFT parameter and the NUFFT parameter;
initializing NUDFT parameters, including: input data x to be transformed, I/O nonuniform position alphamAnd ωkThe number of I/O sampling points M and K;
where x denotes the input data to be transformed, i.e. the discrete sequence of inputs, alphamRepresents the mth input sample point position, ωkDenotes the kth output sample point position, M denotes the number of input samples, K denotes the number of output samples, and K is 0,1m∈[-M/2,M/2-1]，ωk∈[-K/2,K/2-1]；
Initializing NUFFT parameters including interpolation parameters G, b and Q;
wherein G is an oversampling multiple, b is a real number greater than 1/2, and Q is a total number of interpolation points;
the weighting factor calculation unit calculates an interpolation weighting factor according to the NUDFT parameter and the NUFFT parameter initialized by the initialization unit;
the weighting factor calculation unit is used for calculating the weighting factor according to the NUDFT parameters and the NUFFT parameters initialized by the initialization unit and a formula:
Wherein q is1,q2Are all integers, q1＝-Q/2,...,Q/2，q2＝-Q/2,...,Q/2；
The data calculation unit to be subjected to frequency modulation Z conversion calculates data to be subjected to frequency modulation Z conversion according to the interpolation weighting factor of the I/O position obtained by the weighting factor calculation unit;
the to-be-implemented frequency modulation Z conversion data calculation unit obtains an interpolation weighting factor rho (alpha) of the I/O position according to the weighting factor calculation unitm,q1) And using a formula <math>
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</math> Calculating to obtain Z conversion data to be subjected to frequency modulation
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the frequency modulation Z conversion unit calculates a frequency modulation Z conversion result by utilizing frequency modulation Z conversion according to the frequency modulation Z conversion data to be implemented, which is obtained by the frequency modulation Z conversion data calculation unit;
the frequency modulation Z transformation unit obtains data to be subjected to frequency modulation Z transformation according to the Fourier coefficient calculation unitFor the formula:
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</math> indicating the location of the input interpolation point,representing the position of an output interpolation point, and j represents an imaginary unit;
the weighted summation unit carries out weighted summation according to the frequency modulation Z conversion result obtained by the frequency modulation Z conversion unit and the interpolation weighting factor obtained by the weighting factor calculation unit to obtain the result of non-uniform discrete Fourier transform;
the weighted summation unit obtains A (k, q) according to the frequency modulation Z conversion unit2) And an interpolated weighting factor rho (omega) obtained by the weighting factor calculation unitk,q2) From the formula:carrying out weighted summation to obtain the result X of the non-uniform discrete Fourier transformk；
Wherein, <math>
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the invention provides a fast non-uniform discrete Fourier transform method and system with unequal I/O numbers, which have the following advantages:
1. compared with the prior art, the fast non-uniform discrete Fourier transform method and system with unequal I/O numbers adopt frequency modulation Z transform to replace FFT in the traditional NUFFT method, and realize the non-uniform fast Fourier transform with unequal I/O numbers;
2. the method has the advantages that the error of rapid conversion is very small and can be ignored, and the advantages of high precision, high efficiency and the like are kept;
3. the method can be applied to the condition that the number of I/O is not equal;
4. the method greatly expands the application field and the applicability of the traditional NUFFT method.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Various modifications and alterations to this invention will become apparent to those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the claims of the present invention.
Claims (8)
- The fast non-uniform discrete Fourier transform method with unequal I/O numbers is characterized in that after oversampling and interpolation operations are carried out at I/O non-uniform sampling positions, non-uniform fast Fourier transform with unequal I/O numbers is realized by frequency modulation Z transform.
- 2. The fast non-uniform discrete fourier transform method with unequal number of I/os as claimed in claim 1, comprising the steps of:s1: initializing NUDFT parameters, including: to be changedTransformed input data x, I/O nonuniform position alphamAnd ωkThe number of I/O sampling points M and K;where x denotes the input data to be transformed, i.e. the discrete sequence of inputs, alphamRepresents the mth input sample point position, ωkDenotes the kth output sample point position, M denotes the number of input samples, K denotes the number of output samples, and K is 0,1m∈[-M/2,M/2-1]，ωk∈[-K/2,K/2-1]；Initializing NUFFT parameters, including: interpolation parameters G, b and Q;wherein G is an oversampling multiple, b is a real number greater than 1/2, and Q is a total number of interpolation points;s2: according to the NUDFT parameters and the NUFFT parameters and the formula in step S1:for each input sample point m, calculate the corresponding I/O location αm，ωkIs interpolated with a weighting factor p (alpha)m,q1) And ρ (ω)k,q2)；Wherein q is1,q2Are all integers []Represents a rounding operation, and q1＝-Q/2,...,Q/2，q2＝-Q/2,...,Q/2；S3: obtaining an interpolated weighting factor p (α) for the I/O location according to step S2m,q1) And using a formulaCalculating data to be subjected to FM Z-conversionWherein,is an integer which is a function of the number,[·]representing a rounding operation;s4: according to the data obtained in step S3 and subjected to frequency modulation Z transformationFor the formula:using frequency modulation Z transformation, calculating to obtain A (k, q)2)；Wherein, the frequency sampling discrete point locations representing the frequency modulation Z-transform,the position of an output sampling discrete point representing frequency modulation Z transformation, a (-) represents data to be subjected to frequency modulation Z transformation, and j represents an imaginary number unit;s5: according to the frequency modulation Z transformation result A (k) obtained in step S4,q2) And the interpolation weighting factor ρ (ω) obtained in step S2k,q2) From the formula:carrying out weighted summation to obtain the result X of the non-uniform discrete Fourier transformk；Wherein,
- a fast non-uniform discrete fourier transform system with unequal number of I/os, comprising: the device comprises an initialization unit, a weighting factor calculation unit, a frequency modulation Z conversion data calculation unit to be implemented, a frequency modulation Z conversion unit and a weighting summation unit;the initialization unit initializes the NUDFT parameter and the NUFFT parameter;the weighting factor calculation unit calculates an interpolation weighting factor according to the NUDFT parameter and the NUFFT parameter initialized by the initialization unit;the to-be-implemented frequency modulation Z conversion data calculation unit obtains an interpolation weighting factor of an I/O position according to the weighting factor calculation unit, and calculates to-be-implemented frequency modulation Z conversion data;the frequency modulation Z conversion unit calculates a frequency modulation Z conversion result by utilizing frequency modulation Z conversion according to the data to be subjected to frequency modulation Z conversion obtained by the frequency modulation Z conversion data calculation unit;and the weighted summation unit carries out weighted summation according to the frequency modulation Z conversion result obtained by the frequency modulation Z conversion unit and the interpolation weighting factor obtained by the weighting factor calculation unit to obtain the result of the non-uniform discrete Fourier transform.
- 4. The unequal-number-of-I/O fast non-uniform discrete fourier transform system of claim 3 wherein the NUDFT parameters comprise: input data x to be transformed, I/O nonuniform position alphamAnd ωkThe number of I/O sampling points M and K;where x denotes the input data to be transformed, i.e. the discrete sequence of inputs, alphamRepresents the mth input sample point position, ωkDenotes the kth output sample point position, M denotes the number of input samples, K denotes the number of output samples, and K is 0,1m∈[-M/2,M/2-1]，ωk∈[-K/2,K/2-1]；Initializing NUFFT parameters including interpolation parameters G, b and Q;where G is the oversampling multiple, b is a real number greater than 1/2, and Q is the total number of interpolation points.
- 5. The non-uniform fast discrete fourier transform system with unequal I/O numbers according to claim 3, wherein the weighting factor calculating unit calculates the weighting factor based on the NUDFT parameters and the NUFFT parameters initialized by the initializing unit and the formula:for each input sample point m, calculate the corresponding I/O location αm，ωkIs interpolated with a weighting factor p (alpha)m,q1) And ρ (ω)k,q2)；Wherein q is1,q2Are all integers, q1＝-Q/2,...,Q/2，q2＝-Q/2,...,Q/2。
- 6. The fast non-uniform discrete Fourier transform system with unequal I/O numbers according to claim 3, wherein the Z-transform data calculating unit to be applied frequency modulation obtains an interpolated weighting factor p (α) for the I/O location according to the weighting factor calculating unitm,q1) And using a formulaCalculating to obtain Z conversion data to be subjected to frequency modulationWherein,is an integer which is a function of the number,[·]representing a rounding operation.
- 7. Number of I/Os according to claim 3The unequal fast non-uniform discrete Fourier transform system is characterized in that the frequency modulation Z transform unit obtains data to be subjected to frequency modulation Z transform according to the Fourier coefficient calculation unitFor the formula:using frequency modulation Z conversion, calculating to obtain A (k, q)2)；Wherein, indicating the location of the input interpolation point,indicating the output interpolation point location and j indicates the imaginary unit.
- 8. The fast non-uniform discrete Fourier transform system with unequal number of I/O according to claim 3, wherein said weighted summing unit derives A (k, q) from FM Z transform unit2) And an interpolated weighting factor rho (omega) obtained by the weighting factor calculation unitk,q2) From the formula:carrying out weighted summation to obtain the result X of the non-uniform discrete Fourier transformk；Wherein,
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|---|---|---|---|---|
| CN108279406A (en) * | 2018-04-16 | 2018-07-13 | 北京无线电测量研究所 | A kind of decoupling method, apparatus, system and the storage medium of radar |
Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN102652680A (en) * | 2011-03-01 | 2012-09-05 | 贺宁 | Ultrasound computed tomography method on basis of non uniform fast Fourier transform (NUFFT) |
-
2014
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Patent Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN102652680A (en) * | 2011-03-01 | 2012-09-05 | 贺宁 | Ultrasound computed tomography method on basis of non uniform fast Fourier transform (NUFFT) |
Non-Patent Citations (4)
| Title |
|---|
| CHRISTOPHER K. TURNES ETAL: "Efficient Calculations of 3-D FFTs on Spiral Contours", 《JOURNAL OF SCIENTIFIC COMPUTING》 * |
| JEFFREY A.FESSLER ETAL: "Nonuniform Fast Fourier Transforms Using Min-Max Interpolation", 《IEEE TRANSACTIONS ON SIGNAL PROCESSING》 * |
| 薛会等: "非标准快速傅里叶变换算法综述", 《CT理论与应用研究》 * |
| 陈柯等: "基于 NUFFT 算法的综合孔径辐射计图像反演方法", 《微波学报》 * |
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| CN108279406B (en) * | 2018-04-16 | 2020-06-05 | 北京无线电测量研究所 | Decoupling method, device and system of radar and storage medium |
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