CN104462017B - The unequal Fast Inhomogeneous discrete Fourier transform method and system of I/O numbers - Google Patents

The unequal Fast Inhomogeneous discrete Fourier transform method and system of I/O numbers Download PDF

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CN104462017B
CN104462017B CN201410673810.3A CN201410673810A CN104462017B CN 104462017 B CN104462017 B CN 104462017B CN 201410673810 A CN201410673810 A CN 201410673810A CN 104462017 B CN104462017 B CN 104462017B
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刘喆
俞咏江
张晓玲
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a kind of unequal Fast Inhomogeneous discrete Fourier transform method and system of I/O numbers, replace the FFT in tradition NUFFT methods using Chirp-Z Transform compared with prior art, realize the unequal nonuniform fast Fourier transform of I/O numbers；The error of method of the present invention Fast transforms is little and can ignore, the advantages of maintain high accuracy, high efficiency；The method of the present invention have greatly expanded the application of traditional NUFFT methods and the suitability.

Description

The unequal Fast Inhomogeneous discrete Fourier transform method and system of I/O numbers
Technical field
The invention belongs to numerical arts, and in particular to a kind of unequal Fast Inhomogeneous direct computation of DFT of I/O numbers Leaf transformation method and system.
Background technology
(Non-uniform discrete Fourier transform, are abbreviated as non-uniform discrete Fourier transformation NUDFT) it is discrete Fourier transform that input/output (I/O) sampling location is non-homogeneous real number.Due to I/O positions it is non-homogeneous, Conventional fast Fourier transform method (Fast Fourier transform, write a Chinese character in simplified form FFT) cannot directly to improve NUDFT realities Existing efficiency；And Fast Inhomogeneous Fourier transformation method (Non-uniform fast Fourier transform are abbreviated as NUFFT over-sampling and interpolation operation are carried out in I/O nonuniform samplings position), then recycles FFT to realize NUDFT.NUFFT side Method has the advantages that efficient, accurate, therefore is widely used in spectral method (spectral method), nuclear magnetic resonance The field such as (magnetic resonance imaging), Radar Signal Processing.
At present, representational NUFFT implementation methods have the propositions such as American scholar A.Dutt NUFFT implementation methods (see Document 1：A.Dutt,V.Rokhlin,“Fast Fourier transforms for nonequispaced data”,SIAM J.SCI.COMPUT., Vol.14, no.6, pp.1368-1393, Nov.1993), the NUFFT of the proposition such as J.A.Fessler is realized Method is (see document 2：J.A.Fessler,B.P.Sutton,IEEE Trans.Signal Processing,Vol.51,no.2, Pp.560-574,2003), the NUFFT implementation methods of the proposition such as Q.Liu are (see document 3：Q.H.Liu,N.Nguyen,An accurate algorithm for nonuniform fast Fourier transforms(NUFFT’s).IEEE Microwave and Guided Wave Letters, Vol.8, no. (1), pp.18-20,1998) etc..These methods are different Part is to have used different interpolation methods.However, these methods cannot all realize the unequal quick NUDFT of I/O numbers Process.In actual applications, as SAR signal processing large scene data when (see document 4：Franceschetti, G.,R.Lanari,and E.S.Marzouk,“A new two-dimensional squint mode SAR processor,"IEEE Transactions on Aerospace and Electronic Systems,vol.32, Pp.854-863,1996.), often needs efficiently to calculate the unequal NUFFT process of I/O numbers.Therefore, existing NUFFT is realized Method is difficult to meet practical application request.
The content of the invention
To solve above-mentioned technical problem, the present invention proposes a kind of unequal Fast Inhomogeneous discrete fourier of I/O numbers Alternative approach.
Technical solution of the present invention is：The unequal Fast Inhomogeneous discrete Fourier transform method of I/O numbers, it is non-in I/O After uniform sampling position carries out over-sampling and interpolation operation, realize that I/O numbers are unequal non-homogeneous quick using Chirp-Z Transform Fourier transformation；Specifically include following steps：
S1：Initialization NUDFT parameters, including：Input data x to be transformed, the non-homogeneous position alphas of I/OmAnd ωk, I/O samplings Point number M and K；
Wherein, x represents input data to be transformed, that is, the discrete serieses being input into, αmRepresent m-th input sample point position Put, ωkK-th output sampling point position is represented, M represents input sample point number, and K is represented and exported sampled point number, k=0, 1 ..., K-1, m=0,1 ..., M-1, αm∈ [- M/2, M/2-1], ωk∈[-K/2,K/2-1]；
Initialization NUFFT parameters, including：Interpolation parameter G, b and Q；
Wherein, G is over-sampling multiple, and b is greater than 1/2 real number, and Q is interpolation point total number；
S2：According to NUDFT parameters and NUFFT parameters and formula in step S1：
To each input sample point m, calculate corresponding to each I/O position alpham, ωkInterpolation weighter factor ρ (αm,q1) and ρ (ωk,q2)；
Wherein, q1,q2It is integer, the q1=-Q/2 ..., Q/2, q2=-Q/2 ..., Q/2；
S3：The interpolation weighter factor ρ (α of I/O positions are obtained according to step S2m,q1), and utilize formulaCalculate the data of Chirp-Z Transform to be performed
Wherein,It is integer,[] represents rounding operation；
S4：The data of the Chirp-Z Transform to be performed obtained according to step S3To formula：
Using Chirp-Z Transform, A is calculated (k,q2)；
Wherein, Represent the frequency of Chirp-Z Transform Sample variance point position,The output sample variance point position of Chirp-Z Transform is represented, a () represents Chirp-Z Transform to be performed Data, j represents imaginary unit；
S5：According to Chirp-Z Transform result A (k, q that step S4 is obtained2), and the interpolation weighter factor ρ that step S2 is obtained (ωk,q2), by formula：Summation is weighted, is obtained in non-uniform discrete Fu Result Xk of leaf transformation；
Wherein,
The present invention also provides I/O numbers unequal Fast Inhomogeneous discrete Fourier transform system, including：Initialization is single Unit, weighter factor computing unit, Chirp-Z Transform Data Computation Unit to be performed, Chirp-Z Transform unit and weighted sum unit；
The initialization unit is initialized to NUDFT parameters and NUFFT parameters；
The weighter factor computing unit is according to the NUDFT parameters and NUFFT parameters after the initialization of initialized unit, meter Calculate interpolation weighter factor；
The Chirp-Z Transform Data Computation Unit to be performed obtains the interpolation of I/O positions according to weighter factor computing unit Weighter factor, is calculated the data of Chirp-Z Transform to be performed；
The Chirp-Z Transform to be performed that the Chirp-Z Transform unit is obtained according to Chirp-Z Transform Data Computation Unit to be performed Data, using Chirp-Z Transform, be calculated Chirp-Z Transform result；
The Chirp-Z Transform result that the weighted sum unit is obtained according to Chirp-Z Transform unit, and weighter factor calculating The interpolation weighter factor that unit is obtained, is weighted summation, obtains the result of non-uniform discrete Fourier transformation.
Beneficial effects of the present invention：1st, the unequal Fast Inhomogeneous discrete Fourier transform side of I/O numbers of the invention Method and system, replace the FFT in tradition NUFFT methods using Chirp-Z Transform compared with prior art, realize I/O numbers not Equal nonuniform fast Fourier transform；
2nd, the error of method of the present invention Fast transforms is very little and can ignore, and maintains high accuracy, high efficiency etc. excellent Point；
3rd, the method for the present invention can be applied to the unequal situation of I/O numbers；
4th, the method for the present invention have greatly expanded application and the suitability of traditional NUFFT methods.
Description of the drawings
Fig. 1 is the schematic flow sheet of the present invention.
Fig. 2 is the used input data sequence schematic diagram to be transformed of present invention checking feasibility.
Fig. 3 is the list entries sampling location schematic diagram used by present invention checking feasibility.
Fig. 4 is the output sequence sampling location schematic diagram used by present invention checking feasibility.
Fig. 5 is resultant error schematic diagram resulting after the method for the present invention is processed.
Specific embodiment
For convenience of skilled artisan understands that the technology contents of the present invention are made by the solution of the present invention below in conjunction with the accompanying drawings Further illustrate.
Present disclosure is understood for convenience, defines following term first：
Define 1, non-uniform discrete Fourier transformation (NUDFT)
One-dimensional Inhomogeneous discrete Fourier transform is defined as：
Wherein, m=0 ..., M-1, k=0 ..., K-1 are I/O sampled points numbering respectively, and M, K are I/O sampled points respectively Number, αm,ωkIt is that m-th input sample point and k-th export sampling point position, α respectivelym∈[-M/2,M/2-1],ωk∈[- K/2, K/2-1], j is imaginary unit.
Define the 2, three types of non-uniform discrete Fourier transformation
According to the characteristics of I/O positions, non-uniform discrete Fourier transformation can be divided into three types：
(a) Class1-non-uniform discrete Fourier transformation (NUDFT of type1)
When the input of non-uniform discrete Fourier transformation be non-homogeneous real number and export be uniform integer when, that is, work as αm≠- M/2,...,M/2-1,ωk=-K/2 ..., K/2-1, formula (1) are defined as Class1-non-uniform discrete Fourier transformation.
(b) type 2- non-uniform discrete Fourier transformation (NUDFT of type2)
When the input of non-uniform discrete Fourier transformation be uniform integer and export be non-homogeneous real number when, that is, work as αm=- M/2,...,M/2-1,ωk≠-K/2 ..., K/2-1, formula (1) are defined as type 2- non-uniform discrete Fourier transformation.
(c) type 3- non-uniform discrete Fourier transformation (NUDFT of type3)
When the input and output of non-uniform discrete Fourier transformation are non-homogeneous real numbers, that is, work as αm≠-M/2,..., M/2-1,ωk≠-K/2 ..., K/2-1, formula (1) are defined as type 3- non-uniform discrete Fourier transformation.
Understood according to defined above, Class1 and 2 non-uniform discrete Fourier transformation of type can be considered that type 3 is non-homogeneous 3 non-uniform discrete Fourier transformation of type is considered in the special circumstances of discrete Fourier transform, therefore herein below of the present invention Quick realization.
Define 3, the unequal non-uniform discrete Fourier transformation of I/O numbers
When I/O number of samples is unequal, i.e., as M ≠ K, formula (1) is exactly the unequal non-uniform discrete Fourier of I/O numbers Conversion.
Define 4, Chirp-Z Transform (Chirp Z-Transform, write a Chinese character in simplified form CZT)
Chirp-Z Transform is a kind of fast method for solving any frequency domain resolution Fourier transformation of discrete function, concrete real Apply and can be realized by phase multiplication twice and a convolution, refer to document in detail：Rabiner L.R.,Schafer R.W.,Rader C.M.The chirp z-transform algorithm and its application.IEEE Transacation on Audio electroacoust.Vol.17:86-92,1969。
The parameter used in defining 5, interpolation
The interpolation weighter factor of the non-homogeneous positions of I/O is respectively：
Wherein, G is over-sampling multiple, and it is greater than the positive integer equal to 2, and b is greater than 1/2 real number, q1,q2It is to be located at To be numbered to location of interpolation, Q is interpolation point total number to [- Q/2, Q/2] interval uniform integer, and [] represents and round fortune Calculate.
I/O interpolation points position is respectively：
Parameter G in adjustment type (2) and formula (3) can be passed through, changing the precision of conversion, specific implementation method can for Q, b With referring to document：A.Dutt,V.Rokhlin,“Fast Fourier transforms for nonequispaced data”, SIAM J.SCI.COMPUT.,Vol.14,no.6,pp.1368-1393,Nov.1993。
The main method using emulation experiment of the invention carries out verifying the feasibility of the program that all steps, conclusion all exist Verify on MATLAB R2012a correct.
As shown in figure 1, the specific implementation step of the present invention is as follows：
S1：Initiation parameter；
Initialization NUDFT parameters, including：Input data x to be transformed is as shown in Fig. 2 the non-homogeneous position alphas of I/O and ω point Not as shown in Figure 3 and Figure 4, I/O sampled point numbers of the invention are respectively：M=64, K=13；
Initialization NUFFT parameters, the present invention are respectively provided with interpolation parameter G, and b and Q are：G=2, b=1 and Q=14；
S2：Calculate I/O positions interpolation weighter factor；Using NUDFT parameters and NUFFT parameters and formula (2) in S1：
For each m, m=0,1 ..., 63 in the present invention, calculate corresponding to each I/O position alpham,ωkInterpolation weighting Factor ρ (αm,q1) and ρ (ωk,q2), wherein k, m, q1,q2Be integer, k=0,1 ..., 12, m=0 in the present invention, 1 ..., 63, q1,q2=-7 ..., 7, αmAnd ωkBe respectively m-th input sample point and k-th output sampling point position, the present invention in αm ∈[-32,31],ωk∈ [- 6.5,5.5], respectively as shown in Fig. 3 and Fig. 4；
S3：Calculate the data of Chirp-Z Transform to be performed；According to the interpolation weighter factor ρ (α that I/O positions are obtained by S2m, q1), and utilize formula：Calculate the data of Chirp-Z Transform to be performedIn the present inventionIt is integer, αm∈ [- 32,31] be m-th input sample point position as shown in Figure 3, It is integer；
S4：Chirp-Z Transform；The data of the Chirp-Z Transform to be performed obtained by S3And according to formula：
Calculated using Chirp-Z Transform (CZT) Obtain A (k, q2), whereink,m,q1,q2It is integer, k=0, 1 ..., 12, m=0,1 ..., 63, q1,q2=-7 ..., 7, αm,ωkIt is m-th input sample point and k-th output respectively Sampling point position respectively as shown in Fig. 3 and Fig. 4, αm∈[-32,31],ωk∈[-6.5,5.5]；
S5：Weighted sum；Using the Chirp-Z Transform result A (k, the q that are obtained by S42), the interpolation weighter factor that obtained by S2 ρ(ωk,q2), and according to formula：Summation is weighted, non-uniform discrete is obtained Result X of Fourier transformationk, wherein,K=0,1 ..., 12, ωk∈[-K/2,K/2-1] Be k-th output sampling point position as shown in Figure 4.
Through above-mentioned steps process, the quick realization of I/O non-uniform discrete Fourier transformations is accomplished；Fig. 5 is to use Jing Cross non-uniform discrete fast Fourier transform and actual non-uniform discrete Fourier that above-mentioned steps S1 are obtained to the process of step S5 Error result schematic diagram between conversion；As shown in Figure 5, the error of Fast transforms is very little and can ignore, therefore, the present invention The method of proposition can high accuracy, expeditiously realize I/O numbers it is unequal in the case of non-uniform discrete Fourier transformation.
The present invention also provides a kind of I/O numbers unequal Fast Inhomogeneous discrete Fourier transform system, including：Initially Change unit, weighter factor computing unit, Chirp-Z Transform Data Computation Unit to be performed, Chirp-Z Transform unit and weighted sum list Unit；
The initialization unit is initialized to NUDFT parameters and NUFFT parameters；
Initialization NUDFT parameters, including：Input data x to be transformed, the non-homogeneous position alphas of I/OmAnd ωk, I/O sampled points Number M and K；
Wherein, x represents input data to be transformed, that is, the discrete serieses being input into, αmRepresent m-th input sample point position Put, ωkK-th output sampling point position is represented, M represents input sample point number, and K represents output sampled point number, and k= 0,1 ..., K-1, m=0,1 ..., M-1, αm∈ [- M/2, M/2-1], ωk∈[-K/2,K/2-1]；
Initialization NUFFT parameters, including interpolation parameter G, b and Q；
Wherein, G is over-sampling multiple, and b is greater than 1/2 real number, and Q is interpolation point total number；
The weighter factor computing unit is according to the NUDFT parameters and NUFFT parameters after the initialization of initialized unit, meter Calculate interpolation weighter factor；
Weighter factor computing unit is according to the initialized NUDFT parameters of initialization unit and NUFFT parameters and formula：
To each input sample point m, calculate corresponding to each I/O position alpham, ωkInterpolation weighter factor ρ (αm,q1) and ρ (ωk,q2)；
Wherein, q1,q2It is integer, the q1=-Q/2 ..., Q/2, q2=-Q/2 ..., Q/2；
The Chirp-Z Transform Data Computation Unit to be performed obtains the interpolation of I/O positions according to weighter factor computing unit Weighter factor, calculates the data of Chirp-Z Transform to be performed；
Chirp-Z Transform Data Computation Unit to be performed obtains the interpolation weighting of I/O positions according to weighter factor computing unit Factor ρ (αm,q1), and utilize formulaIt is calculated frequency modulation Z to be performed Conversion data
Wherein,It is integer,[] represents rounding operation；
The Chirp-Z Transform to be performed that the Chirp-Z Transform unit is obtained according to Chirp-Z Transform Data Computation Unit to be performed Data, using Chirp-Z Transform, are calculated Chirp-Z Transform result；
The data of the Chirp-Z Transform to be performed that Chirp-Z Transform unit is obtained according to Fourier coefficient computing unitIt is right Formula：
Using Chirp-Z Transform, it is calculated A(k,q2)；
Wherein, Input interpolation point position is represented,Output interpolation point position is represented, j represents imaginary unit；
The Chirp-Z Transform result that the weighted sum unit is obtained according to Chirp-Z Transform unit, and weighter factor calculating The interpolation weighter factor that unit is obtained, is weighted summation, obtains the result of non-uniform discrete Fourier transformation；
A (k, q that weighted sum unit is obtained according to Chirp-Z Transform unit2), and weighter factor computing unit obtains Interpolation weighter factor ρ (ωk,q2), by formula：Summation is weighted, obtains non- Result X of uniform discrete Fourier transformk；
Wherein,
A kind of unequal Fast Inhomogeneous discrete Fourier transform method and system of I/O numbers proposed by the present invention, tool Have the following advantages：
1st, the unequal Fast Inhomogeneous discrete Fourier transform method and system of a kind of I/O numbers of the invention, it is and existing There is technology to compare the FFT replaced using Chirp-Z Transform in tradition NUFFT methods, realize I/O numbers unequal non-homogeneous fast Fast Fourier transformation；
2nd, the error of method of the present invention Fast transforms is very little and can ignore, and maintains high accuracy, high efficiency etc. excellent Point；
3rd, the method for the present invention can be applied to the unequal situation of I/O numbers；
4th, the method for the present invention have greatly expanded application and the suitability of traditional NUFFT methods.
One of ordinary skill in the art will be appreciated that embodiment described here is to aid in reader and understands this Bright principle, it should be understood that protection scope of the present invention is not limited to such especially statement and embodiment.For ability For the technical staff in domain, the present invention can have various modifications and variations.It is all within the spirit and principles in the present invention, made Any modification, equivalent substitution and improvements etc., should be included within scope of the presently claimed invention.

Claims (7)

  1. The unequal Fast Inhomogeneous discrete Fourier transform method of 1.I/O numbers, it is characterised in that in I/O nonuniform samplings After position carries out over-sampling and interpolation operation, realize that the unequal nonuniform fast Fourier of I/O numbers becomes using Chirp-Z Transform Change；
    Specifically include following steps：
    S1：Initialization NUDFT parameters, including：Input data x to be transformed, the non-homogeneous position alphas of I/OmAnd ωk, I/O sampled points Number M and K；
    Wherein, x represents input data to be transformed, that is, the discrete serieses being input into, αmRepresent m-th input sample point position, ωk Represent k-th output sampling point position, M represents input sample point number, K represents output sampled point number, and k=0,1 ..., K-1, m=0,1 ..., M-1, αm∈ [- M/2, M/2-1], ωk∈[-K/2,K/2-1]；
    Initialization NUFFT parameters, including：Interpolation parameter G, b and Q；
    Wherein, G is over-sampling multiple, and b is greater than 1/2 real number, and Q is interpolation point total number；
    S2：According to NUDFT parameters and NUFFT parameters and formula in step S1：
    ,
    To each input sample point m, calculate corresponding to each I/O position alpham, ωkInterpolation weighter factor ρ (αm,q1) and ρ (ωk, q2)；
    Wherein, q1,q2Integer is, [] represents rounding operation, and q1=-Q/2 ,-Q/2+1 ..., Q/2, q2=-Q/2 ,-Q/2 +1,...,Q/2；
    S3：The interpolation weighter factor ρ (α of I/O positions are obtained according to step S2m,q1), and utilize formulaCalculate the data of Chirp-Z Transform to be performed
    Wherein,It is integer,[] represents rounding operation；
    S4：The data of the Chirp-Z Transform to be performed obtained according to step S3Using formula：
    Carry out Chirp-Z Transform, be calculated A (k, q2)；
    Wherein, Represent Chirp-Z Transform frequency sampling from Scatterplot position,The output sample variance point position of Chirp-Z Transform is represented, a () represents the number of Chirp-Z Transform to be performed According to j represents imaginary unit；
    S5：According to Chirp-Z Transform result A (k, q that step S4 is obtained2), and the interpolation weighter factor ρ (ω that step S2 is obtainedk, q2), by formula：Summation is weighted, non-uniform discrete Fourier transformation is obtained Result Xk；
    Wherein,
  2. The unequal Fast Inhomogeneous discrete Fourier transform system of 2.I/O numbers, including：Initialization unit, weighter factor meter Calculate unit, Chirp-Z Transform Data Computation Unit to be performed, Chirp-Z Transform unit and weighted sum unit；
    The initialization unit is initialized to NUDFT parameters and NUFFT parameters；
    The weighter factor computing unit is calculated and is inserted according to NUDFT parameters and NUFFT parameters after the initialization of initialized unit Value weighter factor；
    The Chirp-Z Transform Data Computation Unit to be performed obtains the interpolation weighting of I/O positions according to weighter factor computing unit The factor, is calculated the data of Chirp-Z Transform to be performed；
    The number of the Chirp-Z Transform to be performed that the Chirp-Z Transform unit is obtained according to Chirp-Z Transform Data Computation Unit to be performed According to using Chirp-Z Transform, being calculated Chirp-Z Transform result；
    The Chirp-Z Transform result that the weighted sum unit is obtained according to Chirp-Z Transform unit, and weighter factor computing unit The interpolation weighter factor for obtaining, is weighted summation, obtains the result of non-uniform discrete Fourier transformation.
  3. 3. the unequal Fast Inhomogeneous discrete Fourier transform system of I/O numbers according to claim 2, its feature exist In, the NUDFT parameters, including：Input data x to be transformed, the non-homogeneous position alphas of I/OmAnd ωk, I/O sampled points number M and K；
    Wherein, x represents input data to be transformed, that is, the discrete serieses being input into, αmRepresent m-th input sample point position, ωk K-th output sampling point position is represented, M represents input sample point number, and K represents output sampled point number, and k=0, 1 ..., K-1, m=0,1 ..., M-1, αm∈ [- M/2, M/2-1], ωk∈[-K/2,K/2-1]；
    Initialization NUFFT parameters, including interpolation parameter G, b and Q；
    Wherein, G is over-sampling multiple, and b is greater than 1/2 real number, and Q is interpolation point total number.
  4. 4. the unequal Fast Inhomogeneous discrete Fourier transform system of I/O numbers according to claim 2, its feature exist In the weighter factor computing unit is according to the initialized NUDFT parameters of initialization unit and NUFFT parameters and formula：
    ρ ( α m , q 1 ) = 1 2 b π exp ( - 1 4 b ( [ Gα m ] + q 1 - Gα m ) 2 ) , ρ ( ω k , q 2 ) = 1 2 b π exp ( - 1 4 b ( [ Gω k ] + q 2 - Gω k ) 2 ) ,
    To each input sample point m, calculate corresponding to each I/O position alpham, ωkInterpolation weighter factor ρ (αm,q1) and ρ (ωk, q2)；
    Wherein, q1,q2It is integer, the q1=-Q/2 ,-Q/2+1 ..., Q/2, q2=-Q/2 ,-Q/2+1 ..., Q/2.
  5. 5. the unequal Fast Inhomogeneous discrete Fourier transform system of I/O numbers according to claim 2, its feature exist In, the Chirp-Z Transform Data Computation Unit to be performed according to weighter factor computing unit obtain the interpolation weighting of I/O positions because Sub- ρ (αm,q1), and utilize formulaIt is calculated frequency modulation Z to be performed to become Change data
    Wherein,It is integer,[] represents rounding operation.
  6. 6. the unequal Fast Inhomogeneous discrete Fourier transform system of I/O numbers according to claim 2, its feature exist In the data of the Chirp-Z Transform to be performed that the Chirp-Z Transform unit is obtained according to Fourier coefficient computing unitIt is right Formula：
    Using Chirp-Z Transform, be calculated A (k, q2)；
    Wherein, Input interpolation point position is represented, Output interpolation point position is represented, j represents imaginary unit.
  7. 7. the unequal Fast Inhomogeneous discrete Fourier transform system of I/O numbers according to claim 2, its feature exist In A (k, q that the weighted sum unit is obtained according to Chirp-Z Transform unit2), and weighter factor computing unit obtains Interpolation weighter factor ρ (ωk,q2), by formula：Summation is weighted, obtains non- Result X of uniform discrete Fourier transformk；
    Wherein,
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