CN104330779B - Airborne synthetic aperture radar kinematic error compensation method - Google Patents

Airborne synthetic aperture radar kinematic error compensation method Download PDF

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CN104330779B
CN104330779B CN201410713789.5A CN201410713789A CN104330779B CN 104330779 B CN104330779 B CN 104330779B CN 201410713789 A CN201410713789 A CN 201410713789A CN 104330779 B CN104330779 B CN 104330779B
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synthetic aperture
aperture radar
airborne
error compensation
radar
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CN104330779A (en
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解嘉宇
郑义成
李勇
徐新林
刘喆
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NO 91635 TROOPS OF PEOPLES LIBERATION ARMY
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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
    • 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
    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
    • G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. synthetic aperture radar [SAR] techniques
    • G01S13/9004—SAR image acquisition techniques
    • G01S13/9019—Auto-focussing of the SAR signals
    • 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
    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
    • G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. synthetic aperture radar [SAR] techniques
    • G01S13/904—SAR modes

Abstract

The invention discloses a kind of airborne synthetic aperture radar kinematic error compensation method；Its step includes: initiation parameter, Range compress, distance dimension fast Fourier transform, and world subdivision, each block reference point compensates, each block nonuniform fast Fourier transform and each block joining image-forming.The invention has the beneficial effects as follows: the airborne synthetic aperture radar kinematic error compensation method of the present invention utilizes known movement locus error, fluctuating large scene target area is carried out piecemeal process, use nonuniform fast Fourier transform that each piecemeal scene is carried out kinematic error compensation, effectively integrate the imaging results of each segmented areas, enable the airborne synthetic aperture radar kinematic error compensation method of the present invention to carry out efficient kinematic error compensation process for large scene fluctuating target imaging region.

Description

Airborne synthetic aperture radar kinematic error compensation method
Technical field
The invention belongs to Radar Technology field, particularly relate to a kind of airborne synthetic aperture radar kinematic error compensation method.
Background technology
Synthetic aperture radar (synthetic aperture radar) is exactly to utilize radar and the relative motion of target less for size true The method that real antenna aperature data process synthesizes the radar in bigger equivalent aerial aperture, also referred to as synthetic aperature radar.Synthesis hole The feature of footpath radar is that resolution is high, and energy all weather operations can efficiently identify camouflage and penetrate cloak.Obtained Gao Fang Position resolving power is equivalent to the azimuth resolution that a wide aperture antenna can be provided by.Synthetic aperture radar can be divided into focus type and non-poly- Burnt type two class.With aboard or can have several different mode of operation on spacecraft, it is most commonly that positive side-looking mode, is referred to as Synthetic aperture side-looking radar；In addition with strabismus mode, Doppler beam sharpening pattern and fixed point irradiation mode etc..If radar Keep geo-stationary, make target travel imaging, then become ISAR, also referred to as Range-Doppler Imaging system.Synthesis hole Footpath radar is in military surveillance, mapping, fire control, guidance, and the aspect such as environmental remote sensing and resource exploration has extensive use.
Airborne synthetic aperture radar (Airborne synthetic aperture radar) be transmitter, receiver be placed in helicopter, Two-dimentional high-resolution imaging system on the motion platforms such as unmanned plane, can apply to round-the-clock, round-the-clock target reconnaissance and knowledge The field such as not, development and national security for national economy play an important role.Stability yet with airborne platform self Poor and easily affected by factors such as air agitations, cause airborne synthetic aperture radar movement locus to be typically off preferable straight line Track, causes image deformation occur and obscure, has a strong impact on image quality.Therefore, for obtaining high-quality imaging results, must Airborne synthetic aperture radar kinematic error correctly must be compensated.
At present, after representational airborne synthetic aperture radar kinematic error compensation method has University of Electronic Science and Technology Shi Jun etc. to propose To projection time domain approach (J.Shi, L.Ma, X.Zhang, " Streaming BP for non-linear motion compensation SAR imaging based on GPU,”IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing,vol.6,no.4,pp.2035-2050,2013.)；Frequency domain post-processing approach (Stefano P., Virginia Z., Antonio P.,and Gianfranco F.,Azimuth-to-frequency mapping in airborne SAR data corrupted by uncompensated motion errors,IEEE Geosci.Lett.,Vol.10,no.6,pp.1493-1497,2013；Karlus A.C. M.,Rolf S.,Precise topography-and aperture-dependent motion compensation for airborne SAR, IEEE Geosci.Lett.,Vol.2,no.2,pp.172-176,2005.)；And frequency domain blocks processing method (Zhe L., Huan H., Yongjiang Y.,Motion Compensation for High Resolution Airborne SAR,EUSAR 2014,pp.1-4) Deng.But rear orientation projection's time domain approach operand is big, the target scene scope that frequency domain post-processing approach can effectively process is less；Frequently Domain partitioning processing method cannot process the imaging of height relief scene (Topography fluctuant scenario).Therefore, existing Kinematic error compensation method cannot meet airborne synthetic aperture radar fluctuating large scene, the practical application request of efficient imaging.
Summary of the invention
The goal of the invention of the present invention is: in order to solve problem above, and the present invention proposes a kind of airborne synthetic aperture radar motion Error compensating method, to solving existing airborne synthetic aperture radar motion compensation process for temporal motion error compensating method Operand is big, the effective scene domain of frequency domain kinematic error compensation is little and cannot process the problems such as fluctuating scene imaging.
The technical scheme is that a kind of airborne synthetic aperture radar kinematic error compensation method, comprise the following steps:
A, airborne synthetic aperture radar systematic parameter is carried out initialization process,
Wherein, airborne synthetic aperture radar systematic parameter includes: the signal center frequency η that radar system is launched0, radar system Transmitted signal bandwidth B, radar system launches signal chirp rate μ, radar system pulse recurrence frequency PRF, radar system Distance is to sampling number M, and radar system orientation is to sampling number N, and airborne platform is at the real space of each orientation moment t Position vector [xa(t),ya(t),za(t)], airborne synthetic aperture radar raw radar data matrix s and scene each point mesh to be imaged Absolute altitude degree DEM schemes；
B, use conventional synthesis aperture radar gauged distance compression method to original time of the airborne synthetic aperture radar in step A Wave datum matrix s is compressed processing, and obtains the data matrix s after Range compressRC；
C, to the data matrix s after Range compress in step BRCCarry out fast Fourier transform, when obtaining distance frequency domain-orientation Numeric field data matrix SRC；
D, utilize the scene each point object height DEM to be imaged in step A to scheme, target scene to be imaged is divided into I The block of non-overlapping copies,
Wherein, I >=1, the height of each point target of each block is identical；
E, according to each block center reference point target location in step D, during to the distance frequency domain-orientation obtained in step C Numeric field data matrix SRCCarry out piecemeal reference point compensation, obtain the data matrix S after each block center reference point compensatesΔ,i；
F, each scene block center reference point of obtaining in step E is compensated after data matrix SΔ,iBe respectively adopted non-homogeneous soon Speed Fourier transformation method processes, and obtains the data matrix S after each block motion compensatesMoCo,i；
G, each block motion of obtaining in step F is compensated after data matrix SMoCo,iCarry out splicing, obtain to be imaged Imaging results S after scene motion error compensationMoCo,all。
Further, in described airborne synthetic aperture radar raw radar data matrix s, row data are that orientation is adopted to echo-signal Sample, column data is that distance is to echo signal sample.
Further, described conventional synthesis aperture radar gauged distance compression method is particularly as follows: launch according to synthetic aperture radar Parameter, uses matched filtering technique to be filtered processing to signal to the distance of synthetic aperture radar.
Further, described matched filtering technique matched filtering reference information particularly as follows:
F (τ)=exp (j π μ τ2)
Wherein, j is imaginary unit, and μ is the chirp rate of radar signal, τ be distance to fast time variable,
Further, the data matrix s after described Range compressRCIn, row data are that orientation is to echo signal sample, column data For distance to echo signal sample.
Further, the data matrix S after each block center reference point compensates in described step EΔ,iParticularly as follows:
S Δ , i ( t , η ) = S R C ( t , η ) · exp [ j 4 π · ( η 0 + η ) · R ( t ; x 0 , i , y 0 , i , z 0 , i ) c ]
Wherein, i is the serial number of each scene block spatially position, i=1,2 ..., I, I are block number, η0For radar The signal center frequency that system is launched, t is the orientation moment, and η is frequency of distance, (x0,i,y0,i,z0,i) it is i-th block center ginseng Examination point locus, R (t；x0,i,y0,i,z0,i) it is relative to the i-th block center reference space of points at each orientation moment airborne platform The oblique distance history of position, c is light velocity size.
Further, particularly as follows: at each orientation moment t, there is motion by mistake in described airborne synthetic aperture radar system oblique distance history The oblique distance of the airborne platform relative target of difference, is expressed as
R ( t ; x , y , z ) = ( x a ( t ) - x ) 2 + ( y a ( t ) - y ) 2 + ( z a ( t ) - z ) 2
Wherein, (x, y z) represent the coordinate of scene reference point.
Further, in described step F nonuniform fast Fourier transform method transformation for mula particularly as follows:
X k = Σ m = 0 M - 1 x m exp ( j 2 π m M ω k )
Wherein, m=0 ..., M-1 is input sample point numbering, k=0 ..., K-1 is output sampled point numbering, and M is that input is adopted Sampling point number, K is output sampled point number, and m is m-th input sample point position, and k is that kth exports sampling optimization Put,J is imaginary unit.
The invention has the beneficial effects as follows: the airborne synthetic aperture radar kinematic error compensation method of the present invention utilizes known motion Trajectory error, carries out piecemeal process to fluctuating large scene target area, uses nonuniform fast Fourier transform to each piecemeal Scene carries out kinematic error compensation, has effectively integrated the imaging results of each segmented areas；Combined by scene partitioning block-splicing The Airborne SAR Raw Signal comprising kinematic error is processed by Nonuniform fast Fourier transform method, efficiently, flexibly in fact Show kinematic error compensation and focal imaging, expanded the target scene scope of effective exercise error compensation simultaneously, make the present invention Airborne synthetic aperture radar kinematic error compensation method efficiently can move by mistake for large scene fluctuating target imaging region Difference compensation deals.
Accompanying drawing explanation
Fig. 1 is the airborne synthetic aperture radar kinematic error compensation method schematic flow sheet of the present invention.
Fig. 2 is the airborne platform space position parameter figure that the embodiment of the present invention is used.
Detailed description of the invention
In order to make the purpose of the present invention, technical scheme and advantage clearer, below in conjunction with drawings and Examples, to this Invention is further elaborated.Should be appreciated that specific embodiment described herein only in order to explain the present invention, not For limiting the present invention.
The present invention mainly uses the method for emulation experiment to carry out verifying the feasibility of the program, and institute is in steps, conclusion all exists On MATLAB R2012a, checking is correct.As it is shown in figure 1, be the airborne synthetic aperture radar kinematic error compensation side of the present invention Method schematic flow sheet.A kind of airborne synthetic aperture radar kinematic error compensation method, comprises the following steps:
A, airborne synthetic aperture radar systematic parameter is carried out initialization process.
The all known quantities of airborne synthetic aperture radar systematic parameter of the present invention.Airbome synthetic aperture thunder in the embodiment of the present invention Reach systematic parameter to specifically include: the signal center frequency η that radar system is launched0, η0=5.5GHz；Radar system launches signal Bandwidth B, B=200MHz；Radar system launches signal chirp rate μ, μ=4.9e+7Hz/s；Radar system pulse weight Complex frequency PRF, PRF=1100Hz；Radar system distance is to sampling number M, M=5000；Radar system orientation to Sampling number N, N=4096；Airborne platform is in the real space position vector of each orientation moment t [xa(t),ya(t),za(t)], airborne synthetic aperture radar raw radar data matrix s and scene each point object height to be imaged DEM (Digital Elevation Map) figure.As in figure 2 it is shown, the airborne platform space bit used by the embodiment of the present invention Put Parameter Map.Row data in airborne synthetic aperture radar raw radar data matrix s are that orientation is to echo signal sample, columns According to for distance to echo signal sample.
B, use conventional synthesis aperture radar gauged distance compression method to original time of the airborne synthetic aperture radar in step A Wave datum matrix s is compressed processing, and obtains the data matrix s after Range compressRC。
The conventional synthesis aperture radar gauged distance compression method that the present invention uses is particularly as follows: launch ginseng according to synthetic aperture radar Number, uses matched filtering technique to be filtered processing to signal to the distance of synthetic aperture radar.The coupling of matched filtering technique Filter function particularly as follows:
F (τ)=exp (j π μ τ2)
Wherein, j is imaginary unit, and μ is the chirp rate of radar signal, τ be distance to fast time variable,
Data matrix s after Range compress in the present inventionRCFor two-dimensional matrix, row data are that orientation is to echo signal sample, columns According to for distance to echo signal sample.
C, to the data matrix s after Range compress in step BRCCarry out fast Fourier transform, when obtaining distance frequency domain-orientation Numeric field data matrix SRC。
Adjust the distance the data matrix s after compressingRC, use fast Fourier transform (Fast Fourier transform, FFT) by column Process, obtain distance frequency domain-orientation time domain data matrix SRC.Fast Fourier transform method is those skilled in the art Conventional processing method, the present invention does not repeats.
D, utilize the target scene each point object height DEM to be imaged in step A to scheme, target scene to be imaged is divided into The block of I non-overlapping copies.
The present invention schemes according to scene each point object height DEM to be imaged, and target scene to be imaged is divided into I non-overlapping copies Block, I >=1 here, divide block method be that point target identical for height is divided into same block.The present invention is real Execute example and specifically target scene to be imaged is divided into I=4 block.
E, according to each block center reference point target location in step D, to the distance frequency domain-orientation obtained in step C Time domain data matrix SRCCarry out piecemeal reference point compensation, obtain the data matrix S after each block center reference point compensatesΔ,i。
Data matrix S after the compensation of the present invention each block center reference pointΔ,iBy distance frequency domain-orientation time domain data matrix SRCWith Reference point penalty functionIt is multiplied and obtains, particularly as follows:
S Δ , i ( t , η ) = S R C ( t , η ) · exp [ j 4 π · ( η 0 + η ) · R ( t ; x 0 , i , y 0 , i , z 0 , i ) c ]
Wherein, i is the serial number of each scene block spatially position, i=1,2 ..., I, I are block number, η0For radar system The signal center frequency launched, t is the orientation moment, and η is frequency of distance, reference point (x0,i,y0,i,z0,i) it is i-th block center The locus of point, R (t；x0,i,y0,i,z0,i) it is relative to the i-th block center reference space of points at each orientation moment airborne platform The oblique distance history of position, c is light velocity size.Here airborne synthetic aperture radar system oblique distance history is particularly as follows: in each orientation , there is the oblique distance of the airborne platform relative target of kinematic error, be expressed as in moment t
R ( t ; x , y , z ) = ( x a ( t ) - x ) 2 + ( y a ( t ) - y ) 2 + ( z a ( t ) - z ) 2
Wherein, (x, y z) represent the coordinate of scene reference point.
In the embodiment of the present invention, target scene to be imaged is divided into 4 blocks；The signal center frequency that radar system is launched η0=5.5e+9GHz；Each block reference space of points position is respectively [x0,1,y0,1,z0,1]=[362.5,462.5,0] m, [x0,2,y0,2,z0,2]=[557.1,462.5,0] m, [x0,3,y0,3,z0,3]=[751.6,462.5,0] m, [x0,4,y0,4,z0,4]=[946.2,462.5,0] m.
F, each scene block center reference point of obtaining in step E is compensated after data matrix SΔ,iBe respectively adopted non-homogeneous soon Speed Fourier transformation method processes, and obtains the data matrix S after each block motion compensatesMoCo,i。
The transformation for mula of nonuniform fast Fourier transform method of the present invention particularly as follows:
X k = Σ m = 0 M - 1 x m exp ( j 2 π m M ω k )
Wherein, m=0 ..., M-1 is input sample point numbering, k=0 ..., K-1 is output sampled point numbering, and M is that input is adopted Sampling point number, K is output sampled point number, and m is m-th input sample point position, and k is that kth exports sampling optimization Put,J is imaginary unit.
The embodiment of the present invention is to the data matrix S after the compensation of each block referenceΔ,1,SΔ,2,SΔ,3,SΔ,4It is respectively adopted non-homogeneous quick Fu In leaf transformation method process, obtain each block motion compensate after data matrix SMoCo,1,SMoCo,2,SMoCo,3,SMoCo,4。
G, each block motion of obtaining in step F is compensated after data matrix SMoCo,iCarry out splicing, obtain to be imaged Imaging results S after scene motion error compensationMoCo,all。
The embodiment of the present invention is to the data matrix S after the compensation of each block motionMoCo,1,SMoCo,2,SMoCo,3,SMoCo,4Splice, Obtain imaging results S after target scene kinematic error compensation to be imagedMoCo,all。
Wherein, imaging results matrix SMoCo,allIt is by data matrix SMoCo,iStructure is spliced in the locus corresponding according to each block The data matrix become, SMoCo,all=[SMoCo,1SMoCo,2SMoCo,3SMoCo,4]。
The present invention is directed to that existing airborne synthetic aperture radar motion compensation process operand is big, useful effect scene domain is little, cannot Process the shortcomings such as fluctuating image scene, propose a kind of high efficiency, be suitable for large scene, the high performance motion of height relief target by mistake Difference compensation method.The present invention utilizes scene partitioning block-splicing to combine Nonuniform fast Fourier transform method to comprising kinematic error Airborne SAR Raw Signal processes, and achieves kinematic error compensation and focal imaging efficiently, flexibly.Present invention profit With Nonuniform fast Fourier transform, it is achieved that the high-performance kinematic error compensation in each block, and combine block splicing, expand The target scene scope to be imaged of effective exercise error compensation；Compared with existing airborne synthetic aperture radar motion compensation process, The present invention can overcome that temporal motion error compensating method operand is big, the effective scene domain of frequency domain kinematic error compensation is little, with And the limitations such as fluctuating scene imaging cannot be processed, it is possible to carry out efficient kinematic error benefit for large scene fluctuating target imaging region Repay process.
Those of ordinary skill in the art is it will be appreciated that embodiment described here is to aid in the reader understanding present invention's Principle, it should be understood that protection scope of the present invention is not limited to such special statement and embodiment.This area common It is various specifically that technical staff can make various other without departing from essence of the present invention according to these technology disclosed by the invention enlightenment Deformation and combination, these deformation and combination are the most within the scope of the present invention.

Claims (8)

1. an airborne synthetic aperture radar kinematic error compensation method, it is characterised in that comprise the following steps:
A, airborne synthetic aperture radar systematic parameter is carried out initialization process,
Wherein, airborne synthetic aperture radar systematic parameter includes: the signal center frequency η that radar system is launched0, radar system Transmitted signal bandwidth B, radar system launches signal chirp rate μ, radar system pulse recurrence frequency PRF, radar system Distance is to sampling number M, and radar system orientation is to sampling number N, and airborne platform is at the real space of each orientation moment t Position vector [xa(t),ya(t),za(t)], wherein, xa(t), ya(t), zaT () be airborne platform in vertical movement direction respectively, Along the direction of motion, the real space position of short transverse, airborne synthetic aperture radar raw radar data matrix s and mesh to be imaged Mark scene each point object height DEM figure；
B, use conventional synthesis aperture radar gauged distance compression method to original time of the airborne synthetic aperture radar in step A Wave datum matrix s is compressed processing, and obtains the data matrix s after Range compressRC；
C, to the data matrix s after Range compress in step BRCCarry out fast Fourier transform, when obtaining distance frequency domain-orientation Numeric field data matrix SRC；
D, utilize the target scene each point object height DEM to be imaged in step A to scheme, target scene to be imaged is divided into The block of I non-overlapping copies,
Wherein, I >=1, the height of each point target of each block is identical；
E, according to each block center reference point target location in step D, during to the distance frequency domain-orientation obtained in step C Numeric field data matrix SRCCarry out piecemeal reference point compensation, obtain the data matrix S after each block center reference point compensatesΔ,i；
F, each scene block center reference point of obtaining in step E is compensated after data matrix SΔ,iBe respectively adopted non-homogeneous soon Speed Fourier transformation method processes, and obtains the data matrix S after each block motion compensatesMoCo,i；
G, each block motion of obtaining in step F is compensated after data matrix SMoCo,iCarry out splicing, obtain to be imaged Imaging results S after scene motion error compensationMoCo,all。
2. airborne synthetic aperture radar kinematic error compensation method as claimed in claim 1, it is characterised in that: described airborne In synthetic aperture radar raw radar data matrix s, row data be orientation to echo signal sample, column data is that distance is to echo Signal sampling.
3. airborne synthetic aperture radar kinematic error compensation method as claimed in claim 1, it is characterised in that: described tradition Synthetic aperture radar gauged distance compression method is particularly as follows: according to synthetic aperture radar emission parameter, use matched filtering technique It is filtered processing to signal to the distance of synthetic aperture radar.
4. airborne synthetic aperture radar kinematic error compensation method as claimed in claim 3, it is characterised in that: described coupling The matched filtering function of filtering technique particularly as follows:
F (τ)=exp (j π μ τ2)
Wherein, j is imaginary unit, and μ is the chirp rate of radar signal, τ be distance to fast time variable,
5. airborne synthetic aperture radar kinematic error compensation method as claimed in claim 1, it is characterised in that: described distance Data matrix s after compressionRCIn, row data be orientation to echo signal sample, column data is that distance is to echo signal sample.
6. airborne synthetic aperture radar kinematic error compensation method as claimed in claim 1, it is characterised in that: described step Data matrix S after each block center reference point compensates in EΔ,iParticularly as follows:
S Δ , i ( t , η ) = S R C ( t , η ) · exp [ j 4 π · ( η 0 + η ) · R ( t ; x 0 , i , y 0 , i , z 0 , i ) c ]
Wherein, i is the serial number of each scene block spatially position, i=1,2 ..., I, I are block number, η0For radar The signal center frequency that system is launched, t is the orientation moment, and η is frequency of distance, (x0,i,y0,i,z0,i) it is i-th block center ginseng Examination point locus, R (t；x0,i,y0,i,z0,i) it is relative to the i-th block center reference space of points at each orientation moment airborne platform The oblique distance history of position, c is light velocity size.
7. airborne synthetic aperture radar kinematic error compensation method as claimed in claim 6, it is characterised in that: described airborne Particularly as follows: at each orientation moment t, there is the airborne platform relative target of kinematic error in polarization sensitive synthetic aperture radar system oblique distance history Oblique distance, is expressed as
R ( t ; x , y , z ) = ( x a ( t ) - x ) 2 + ( y a ( t ) - y ) 2 + ( z a ( t ) - z ) 2
Wherein, (x, y z) are the coordinate of scene reference point.
8. airborne synthetic aperture radar kinematic error compensation method as claimed in claim 1, it is characterised in that: described step In F nonuniform fast Fourier transform method transformation for mula particularly as follows:
X k = Σ m = 0 M - 1 x m exp ( j 2 π m M ω k )
Wherein, xmFor inputting data to be transformed, m=0 ..., M-1 is input sample point numbering, XkFor exporting data after conversion, K=0 ..., K-1 is output sampled point numbering, and M is input sample point number, and K is output sampled point number,J is imaginary unit.
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