CN109213960B - Method and device for reconstructing periodic non-uniform sampling band-limited signal - Google Patents
Method and device for reconstructing periodic non-uniform sampling band-limited signal Download PDFInfo
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Abstract
The invention discloses a method and a device for reconstructing a periodic non-uniform sampling band-limited signal, which relate to the technical field of communication] ‑1 The conventional frequency spectrum of the signal is determined, and the complexity of reconstruction of the periodic non-uniform sampling band-limited signal is greatly reduced.
Description
Technical Field
The present disclosure relates generally to the field of communications technologies, and in particular, to communication signal reconstruction, and in particular, to a method and an apparatus for reconstructing a periodic non-uniform sampling band-limited signal.
Background
The problem of periodic non-uniform sampling of band-limited signals (i.e., signals with limited frequency domain bandwidth) is encountered in many research fields, such as tomography, magnetic field calculation, synthetic aperture radar imaging processing in an azimuth multi-channel mode, and the like. How to recover a uniformly sampled signal from a periodically non-uniformly sampled signal is the key to subsequent processing.
It is known that there exists a signal reconstruction algorithm based on the periodic non-uniform sampling theorem. However, when the number of samples is large, this algorithm involves a large matrix calculation and is therefore very time consuming.
Disclosure of Invention
In view of the above-mentioned drawbacks and disadvantages of the prior art, it is desirable to provide a method and an apparatus for reconstructing a periodic non-uniform sampling band-limited signal, so as to reduce the complexity of reconstruction of the periodic non-uniform sampling band-limited signal.
In a first aspect, an embodiment of the present invention provides a method for reconstructing a periodic non-uniform sampling band-limited signal, where the method includes:
determining a non-uniform frequency spectrum of a periodic non-uniform sampling band-limited signal;
determining a first row and a first column of an inverse matrix of a reconstruction matrix between a conventional frequency spectrum and a non-uniform frequency spectrum;
determining the conventional frequency spectrum of the signal according to the non-uniform frequency spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix;
the signal is reconstructed from the conventional spectrum of the signal.
Further, the determining the non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal specifically includes:
the non-uniform frequency spectrum of the periodic non-uniformly sampled band-limited signal is determined by non-uniform fast fourier transform.
Further, the determining a non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal through the non-uniform fast fourier transform specifically includes:
using a Gaussian function as a weight kernel function, assume a periodic non-uniform sampling band-limited signal sequence { x (t) n ) When the length is MN, determining the non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal as follows:
wherein: m is the number of samples of a non-uniform sampling period, N is the number of non-uniform sampling periods, T is the equivalent sampling interval,
b is a parameter of a Gaussian function [ ·]For rounding operations, p is an oversampling factor, ω 0 ＝2π/N，c n ＝t n /T。
Preferably, the determining the first row and the first column of the inverse matrix of the reconstruction matrix between the conventional spectrum and the non-uniform spectrum specifically includes:
determining the reconstruction matrix by non-uniform fast Fourier transform;
determining a first row of an inverse matrix of the reconstruction matrix through a half division;
determining a first column of an inverse matrix of a reconstruction matrix through hermitian symmetry of the reconstruction matrix.
Further, the determining the conventional spectrum of the signal according to the non-uniform spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix specifically includes:
the conventional spectrum of the signal is determined as:
wherein:
m is the sampling number of a non-uniform sampling period;
[r]＝[r 0 ,r 1 ,…,r M-1 ]the first row of an inverse matrix of the reconstruction matrix is obtained;
[c]＝[c 0 ,c 1 ,…,c M-1 ] T the first column of an inverse matrix of the reconstruction matrix is obtained;
preferably, the weight kernel function of the non-uniform fast fourier transform is any one of: gaussian function, B-curve function, kessel-bessel function.
In a second aspect, an embodiment of the present invention further provides an apparatus for reconstructing a band-limited signal with non-uniform sampling period, where the apparatus includes:
the non-uniform frequency spectrum determining unit is used for determining a non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal;
the device comprises a reconstruction matrix determining unit, a reconstruction processing unit and a processing unit, wherein the reconstruction matrix determining unit is used for determining the first row and the first column of an inverse matrix of a reconstruction matrix between a conventional frequency spectrum and a non-uniform frequency spectrum;
a regular spectrum determination unit, configured to determine a regular spectrum of a signal according to the non-uniform spectrum and a first row and a first column of an inverse matrix of the reconstruction matrix;
a signal reconstruction unit for reconstructing a signal from a conventional spectrum of said signal.
Further, the non-uniform spectrum determination unit is specifically configured to:
the non-uniform frequency spectrum of the periodic non-uniformly sampled band-limited signal is determined by non-uniform fast fourier transform.
Further, the non-uniform spectrum determination unit is specifically configured to:
using a Gaussian function as the weight kernel function, assume a periodic non-uniform sampling band-limited signal sequence { x (t) } n ) When the length is MN, determining the non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal as follows:
wherein: m is the number of samples of a non-uniform sampling period, N is the number of non-uniform sampling periods,t is the equivalent sampling interval of the sample,
b is a parameter of a Gaussian function [ ·]For rounding operations, p is an oversampling factor, ω 0 ＝2π/N，c n ＝t n /T。
Preferably, the reconstruction matrix determining unit is specifically configured to:
determining the reconstruction matrix by non-uniform fast Fourier transform;
determining a first row of an inverse matrix of the reconstruction matrix through a half division;
determining a first column of an inverse matrix of a reconstruction matrix through hermitian symmetry of the reconstruction matrix.
Further, the conventional spectrum determination unit is specifically configured to:
the conventional spectrum of the signal is determined as:
wherein:
m is the sampling number of a non-uniform sampling period;
[r]＝[r 0 ,r 1 ,…,r M-1 ]the first row of the inverse matrix of the reconstruction matrix;
[c]＝[c 0 ,c 1 ,…,c M-1 ] T the first column of the inverse matrix of the reconstruction matrix;
preferably, the weight kernel function of the non-uniform fast fourier transform is any one of: gaussian function, B-curve function, kessel-bessel function.
In a third aspect, an embodiment of the present invention further provides a device for reconstructing a periodic non-uniform sampling band-limited signal, including a processor and a memory; the method is characterized in that:
the memory includes instructions executable by the processor to cause the processor to perform:
determining a non-uniform frequency spectrum of a periodic non-uniform sampling band-limited signal;
determining a first row and a first column of an inverse matrix of a reconstruction matrix between a conventional frequency spectrum and a non-uniform frequency spectrum;
determining the conventional frequency spectrum of the signal according to the non-uniform frequency spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix;
the signal is reconstructed from the conventional spectrum of the signal.
In a fourth aspect, the embodiments of the present invention also provide a computer-readable storage medium, on which a computer program is stored, where the computer program is configured to:
determining a non-uniform frequency spectrum of a periodic non-uniform sampling band-limited signal;
determining a first row and a first column of an inverse matrix of a reconstruction matrix between a conventional frequency spectrum and a non-uniform frequency spectrum;
determining the conventional frequency spectrum of the signal according to the non-uniform frequency spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix;
the signal is reconstructed from the regular spectrum of the signal.
According to the method and the device for reconstructing the periodic non-uniform sampling band-limited signal, the conventional frequency spectrum of the signal is determined through the non-uniform frequency spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix, and the method can obtain the [ A ] only through less calculation amount] -1 The conventional frequency spectrum of the signal is determined, and the complexity of reconstruction of the periodic non-uniform sampling band-limited signal is greatly reduced.
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Other features, objects and advantages of the present application will become more apparent upon reading of the following detailed description of non-limiting embodiments thereof, made with reference to the accompanying drawings in which:
fig. 1 is a schematic diagram of periodic non-uniform sampling and a corresponding uniform sampling timing sequence when M =3 in the prior art;
fig. 2 is a flowchart of a method for reconstructing a periodic non-uniform sampling band-limited signal according to an embodiment of the present invention;
FIG. 3 shows that when the number of sampling points provided by the embodiment of the present invention is different, X is determined by using the method of the prior art and the method provided by the embodiment of the present invention non (ω) a comparative plot of the time taken;
FIG. 4 is a direct solution [ A ] provided by embodiments of the present invention] -1 [X non (ω)]And the method provided by the embodiment of the invention is adopted to determine [ X (omega) ]]A schematic of the time ratios used;
fig. 5 is a schematic structural diagram of a device for reconstructing a periodic non-uniform sampling band-limited signal according to an embodiment of the present invention;
fig. 6 is a schematic structural diagram of a computer system suitable for implementing the terminal device or the server according to the embodiment of the present application.
Detailed Description
The present application will be described in further detail with reference to the drawings and examples. It is to be understood that the specific embodiments described herein are merely illustrative of the relevant invention and not restrictive of the invention. It should be noted that, for convenience of description, only the portions related to the present invention are shown in the drawings.
It should be noted that the embodiments and features of the embodiments in the present application may be combined with each other without conflict. The present application will be described in detail below with reference to the accompanying drawings in conjunction with embodiments.
The periodic non-uniform signal reconstruction problem can be abstracted as follows: let the continuous time domain signal x (t) be a band-limited signal, the frequency band range of its Fourier transform is [ - π B/2, π B/2](B in Hz). Sequence { x (t) n ) Is for x (t) at non-uniform point columns { t } n The samples at, with the corresponding equivalent uniform sampling interval T satisfying the nyquist sampling theorem: t is less than 1/B. Non-uniform dot column t n The deviation between { n } and the uniform dot column { n, T } exhibits a cyclic structure with period M, i.e., t n Can be expressed as follows:
t n ＝nT+r n T＝(kM+m)T+r kM+m T＝(kM+m)T+r m T
wherein r is n T is T n And nT, M ≡ n (mod M).
Fig. 1 illustrates periodic non-uniform sampling and its corresponding uniform sampling timing when M = 3.
It can be seen that the problem of periodic non-uniform signal reconstruction is actually how to reconstruct the signal from { x (t) } n ) Where x (nT) is recovered.
For the reconstruction problem of periodic Non-Uniform signals, the literature provides a reconstruction Algorithm (PNUA) based on periodic Non-Uniform sampling theory. The algorithm principle is as follows:
defining a non-uniform frequency spectrum X non (ω) is as follows:
then X non (ω) is related to the conventional spectrum X (ω) of the signal as follows:
[X non (ω)]＝[A][X(ω)] (2)
wherein
[X non (ω)]＝[X non (ω 0 ),X non (ω 1 ),…,X non (ω M-1 )] T
ω k ＝ω+kω MT ,0≤k≤M-1,
The element A (k) in the matrix [ A ] is
Pair formula (2) two sides are multiplied by [ A ]] -1 Then, can obtain [ X (omega)]And in turn a uniform sample sequence of the signal x (nT) can be reconstructed by inverse fourier transform.
One of the main drawbacks of the PNUA algorithm is that the amount of computation is huge:
non-uniform spectrum X non The expression of (ω) is similar to the conventional Discrete Fourier Transform (DFT), which means that if the sequence length is N, then X is directly calculated non (omega) requires O (N) 2 ) The amount of calculation of (a). In addition, when the number of channels is large, the sequence { A (k) } also requires a large amount of computation (because of its expression and X) non (ω) similarly).
When the number of non-uniform sampling points is large, the scale of the matrix [ A ] is increased, and if the [ X (omega) ] is solved by the formula (2) by adopting a conventional method for solving a linear equation set, the operation of large matrix inversion is involved, and the operation amount is extremely large.
Fig. 2 illustrates a method for reconstructing a periodically non-uniformly sampled band-limited signal according to an embodiment of the present invention. Referring to fig. 2, a method for reconstructing a periodic non-uniform sampling band-limited signal according to an embodiment of the present invention includes:
step S201, determining a non-uniform frequency spectrum of a periodic non-uniform sampling band-limited signal;
step S202, determining a first row and a first column of an inverse matrix of a reconstruction matrix between a conventional frequency spectrum and a non-uniform frequency spectrum;
step S203, determining the conventional frequency spectrum of the signal according to the non-uniform frequency spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix;
and step S204, reconstructing the signal according to the conventional frequency spectrum of the signal.
The method determines the conventional frequency spectrum of the signal through the non-uniform frequency spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix. It can be seen that a large matrix needs to be inverted compared to the prior art methods described above, by which a [ A ] is obtained compared to only with a smaller amount of computation] -1 The conventional frequency spectrum of the signal is determined, and the complexity of reconstruction of the periodic non-uniform sampling band-limited signal is greatly reduced.
It should be noted that while the operations of the method of the present invention are depicted in the drawings in a particular order, this does not require or imply that the operations must be performed in this particular order, or that all of the illustrated operations must be performed, to achieve desirable results. Rather, the steps depicted in the flowcharts may change the order of execution. For example, the execution order of step S201 and step S202 may be changed. Additionally or alternatively, certain steps may be omitted, multiple steps combined into one step execution, and/or one step broken down into multiple step executions. For example, step S201 and step S202 may be performed in one step in combination.
Further, in step S201, determining a non-uniform spectrum of the periodic non-uniform sampling band-limited signal specifically includes:
the non-uniform frequency spectrum of the periodic non-uniformly sampled band-limited signal is determined by non-uniform fast fourier transform.
Formula (3) can be regarded as formula (1) at x (t) n )＝1/(MT)，ω＝kω MT Special case of time. While equation (1) is actually a non-uniform discrete Fourier transform (NUDFT) form, which differs from the conventional Discrete Fourier Transform (DFT) in the uniform case in that t n ＝(n+r n ) T is the equivalentA non-integer multiple of the sampling interval T. However, just as the DFT corresponds to the fast algorithm FFT, the NUDFT also has a corresponding fast algorithm-non-uniform fast Fourier transform (NUFFT). NUFFT is with e -jωTn The sequence is approximated by a corresponding weighted sum
The sum of the NUDFT is then converted into a form that can take an FFT.
wherein phi (-) and
respectively, a weight kernel function and Fourier transform thereof, p is an oversampling factor, and q is a kernel length. The weight kernel function can be a Gaussian function, a B curve function, a Kaiser-Bessel function, and the like.
Taking the weight kernel function as a Gaussian function as an example, then e -jcω Can be approximately expressed as
Wherein b is a parameter of a Gaussian function [. Degree]For rounding operations, the weighting coefficients ρ l Is composed of
Assume sequence { x (t) n ) The length is MN, where M is the number of samples of a non-uniform sampling period and N is the number of non-uniform sampling periods. Consider formula (1) in
The values of (A) and (B) are reduced in sign by
Wherein X non (k) Is composed of
ω 0 ＝2π/N，c n ＝t n and/T. In the formula (4), let ω = -k ω 0 /p，c＝p·c n And substituting into formula (5) to arrange into the following form
Non-uniform frequency spectrum
Now, the FFT can be used for subsequent calculations.
As can be seen, when a Gaussian function is used and the sequence { x (t) n ) When the length is MN, in step S101, determining a non-uniform spectrum of a periodic non-uniform sampling band-limited signal through non-uniform fast fourier transform, specifically including:
determining a non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal as follows:
wherein: m is the period of non-uniform sampling, T is the equivalent sampling interval,
b is a parameter of a Gaussian function [ ·]For rounding operations p to be oversampling factors, omega 0 ＝2π/N，c n ＝t n /T。
Calculating τ in equation (6) l qMN real-to-complex multiplications and qMN complex additions are required, i.e., 2qMN real-to-real multiplications and 2qMN real additions.
The calculation of equation (6) involves FFT and weighting of the pMN points. pMNlog is required for FFT operation 2 (pMN) timesComplex multiplication sum pMNlog 2 (pMN) times of repeated addition. The weighting process requires MN multiplications of real and complex numbers. The total required real number multiplication and addition operations are thus 4pMNlog, respectively 2 (pMN) +2MN and 3pMNlog 2 (pMN)。
As can be seen, the total real number addition and real number multiplication operations required by the above procedure are (2q +3p log respectively 2 p)MN+3pMNlog 2 MN and (2q +2+4p log 2 p)MN+4pMN log 2 And (4) MN. Compared with the direct calculation formula (1), the calculation amount is O (M) 2 N 2 ) Reduced to O (MNlog) 2 MN)。
Fig. 3 is a schematic diagram illustrating a comparison of time used by the method of formula (1) and the method provided in the embodiment of the present invention when the number of sampling points is different, which shows that the method provided in the embodiment of the present invention greatly reduces complexity and improves operation efficiency.
Further, in step S202, determining a first row and a first column of an inverse matrix of a reconstruction matrix between the normal spectrum and the non-uniform spectrum specifically includes:
determining the reconstruction matrix by non-uniform fast Fourier transform;
determining a first row of an inverse matrix of a reconstruction matrix through a half division method;
the first column of the inverse of the reconstruction matrix is determined by the hermitian symmetry of the reconstruction matrix.
Further, in step S203, determining the conventional spectrum of the signal according to the non-uniform spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix, specifically including:
determining the conventional frequency spectrum of the signal as in equation (7)
The first M elements of (a):
wherein:
m is the number of samples of a non-uniform sampling period;
[r]＝[r 0 ,r 1 ,…,r M-1 ]the first row of an inverse matrix of the reconstruction matrix is obtained;
[c]＝[c 0 ,c 1 ,…,c M-1 ] T the first column of an inverse matrix of the reconstruction matrix is obtained;
the conventional algorithm for solving the linear equation system is a Gaussian elimination method, and the required operation amount is O (M) 3 ). If the method provided by the embodiment of the invention is adopted, only the method needs to be carried out
The calculated amount of [ A ] is obtained] -1 First row and first column of (D), then can be at O (Mlog) 2 M) is solved under the operation complexity [ X (omega) ]]。
Direct solution of [ A] -1 [X non (ω)]And determining [ X (omega) ] by using the formula (7)]The time ratio curve is shown in fig. 4, and it can be seen that the method provided by the embodiment of the invention greatly reduces the complexity and improves the operationAnd (4) calculating efficiency.
With further reference to fig. 5, an exemplary block diagram of an apparatus 500 for reconstruction of a periodic non-uniformly sampled band-limited signal according to an embodiment of the present application is shown, the apparatus comprising:
a non-uniform spectrum determination unit 501, configured to determine a non-uniform spectrum of a periodic non-uniform sampling band-limited signal;
a reconstruction matrix determination unit 502 for determining a first row and a first column of an inverse matrix of a reconstruction matrix between a normal spectrum and a non-uniform spectrum;
a regular spectrum determination unit 503, configured to determine a regular spectrum of the signal according to the non-uniform spectrum and a first row and a first column of an inverse matrix of the reconstruction matrix;
a signal reconstruction unit 504 for reconstructing the signal from the conventional spectrum of the signal.
Further, the non-uniform spectrum determining unit 501 is specifically configured to:
and determining the non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal through non-uniform fast Fourier transform.
Further, the non-uniform spectrum determination unit 501 is specifically configured to:
using a Gaussian function as a weight kernel function, assume a periodic non-uniform sampling band-limited signal sequence { x (t) n ) When the length is MN, determining the non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal as follows:
wherein: m is the number of samples for a non-uniform sampling period, N is the number of non-uniform sampling periods, T is the equivalent sampling interval,
b is a parameter of a Gaussian function [ ·]For the rounding operation, p is the oversampling factor, ω 0 ＝2π/N，c n ＝t n /T。
Preferably, the reconstruction matrix determining unit 502 is specifically configured to:
determining the reconstruction matrix by non-uniform fast Fourier transform;
determining a first row of an inverse matrix of a reconstruction matrix through a half division;
the first column of the inverse of the reconstruction matrix is determined by the hermitian symmetry of the reconstruction matrix.
Further, the conventional spectrum determination unit 503 is specifically configured to:
the conventional spectrum of the signal is determined as:
wherein:
m is the number of samples of a non-uniform sampling period;
[r]＝[r 0 ,r 1 ,…,r M-1 ]the first row of the inverse matrix of the reconstruction matrix;
[c]＝[c 0 ,c 1 ,…,c M-1 ] T the first column of an inverse matrix of the reconstruction matrix is obtained;
the weight kernel function of the non-uniform fast Fourier transform is any one of the following: gaussian function, B-curve function, kessel-bessel function.
It should be understood that the units or modules recited in the apparatus 500 correspond to the various steps in the method described with reference to fig. 2. Thus, the operations and features described above with respect to the method are equally applicable to the apparatus 500 and the units included therein and will not be described again here. The apparatus 500 may be implemented in a browser or other security applications of the electronic device in advance, or may be loaded into the browser or other security applications of the electronic device by downloading or the like. Corresponding elements in apparatus 500 may cooperate with elements in an electronic device to implement aspects of embodiments of the present application.
Referring now to FIG. 6, shown is a block diagram of a computer system 600 suitable for use in implementing a terminal device or server of an embodiment of the present application.
As shown in fig. 6, the computer system 600 includes a Central Processing Unit (CPU) 601 that can perform various appropriate actions and processes according to a program stored in a Read Only Memory (ROM) 602 or a program loaded from a storage section 608 into a Random Access Memory (RAM) 603. In the RAM603, various programs and data necessary for the operation of the system 600 are also stored. The CPU 601, ROM 602, and RAM603 are connected to each other via a bus 604. An input/output (I/O) interface 605 is also connected to bus 604.
The following components are connected to the I/O interface 605: an input portion 606 including a keyboard, a mouse, and the like; an output portion 607 including a display such as a Cathode Ray Tube (CRT), a Liquid Crystal Display (LCD), and the like, and a speaker; a storage section 608 including a hard disk and the like; and a communication section 609 including a network interface card such as a LAN card, a modem, or the like. The communication section 609 performs communication processing via a network such as the internet. A driver 610 is also connected to the I/O interface 605 as needed. A removable medium 611 such as a magnetic disk, an optical disk, a magneto-optical disk, a semiconductor memory, or the like is mounted on the drive 610 as necessary, so that a computer program read out therefrom is mounted in the storage section 608 as necessary.
In particular, the process described above with reference to fig. 2 may be implemented as a computer software program, according to an embodiment of the present disclosure. For example, embodiments of the present disclosure include a computer program product comprising a computer program tangibly embodied on a machine-readable medium, the computer program comprising program code for performing the method of fig. 2. In such an embodiment, the computer program may be downloaded and installed from a network through the communication section 609, and/or installed from the removable medium 611.
The flowchart and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present invention. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions.
The units or modules described in the embodiments of the present application may be implemented by software or hardware. The described units or modules may also be provided in a processor, and may be described as: a processor includes an XX unit, a YY unit, and a ZZ unit. Where the name of such unit or module does not in some cases constitute a definition of the unit or module itself, for example, the XX unit may also be described as "unit for XX".
As another aspect, the present application also provides a computer-readable storage medium, which may be the computer-readable storage medium included in the apparatus in the above embodiment; or it may be a separate computer readable storage medium not incorporated into the device. The computer readable storage medium stores one or more programs for use by one or more processors in performing the formula input methods described herein.
The foregoing description is only exemplary of the preferred embodiments of the application and is illustrative of the principles of the technology employed. It will be appreciated by a person skilled in the art that the scope of the invention according to the present application is not limited to the specific combination of the above-mentioned features, but also covers other embodiments where any combination of the above-mentioned features or their equivalents is made without departing from the inventive concept. For example, the above features may be replaced with (but not limited to) features having similar functions disclosed in the present application.
Claims (12)
1. A method of reconstructing a periodic non-uniformly sampled band-limited signal, the method comprising:
determining a non-uniform frequency spectrum of a periodic non-uniform sampling band-limited signal;
determining a first row and a first column of an inverse matrix of a reconstruction matrix between the uniformly sampled spectrum and the non-uniformly sampled spectrum;
determining a uniform sampling frequency spectrum of a signal according to the non-uniform frequency spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix, specifically comprising:
determining a uniformly sampled spectrum of the signal as:
wherein:
m is the sampling number of a non-uniform sampling period;
[r]＝[r 0 ,r 1 ,…,r M-1 ]the first row of the inverse matrix of the reconstruction matrix;
[c]＝[c 0 ,c 1 ,…,c M-1 ] T the first column of the inverse matrix of the reconstruction matrix;
and reconstructing the signal according to the uniformly sampled frequency spectrum of the signal.
2. The method according to claim 1, wherein the determining the non-uniform spectrum of the periodic non-uniformly sampled band-limited signal specifically comprises:
the non-uniform frequency spectrum of the periodic non-uniformly sampled band-limited signal is determined by non-uniform fast fourier transform.
3. The method according to claim 2, wherein the determining the non-uniform spectrum of the periodic non-uniformly sampled band-limited signal by non-uniform fast fourier transform comprises:
using a Gaussian function as a weight kernel function, assume a periodic non-uniform sampling band-limited signal sequence { x (t) n ) When the length is MN, determining the non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal as follows:
wherein: m is the number of samples of a non-uniform sampling period, N is the number of non-uniform sampling periods, T is the equivalent sampling interval,
b is a parameter of a Gaussian function [ ·]For rounding operations, p is the oversampling factor, w 0 ＝2π/N，c n ＝t n /T。
4. The method according to claim 1, wherein the determining a first row and a first column of an inverse matrix of a reconstruction matrix between the uniformly sampled spectrum and the non-uniformly sampled spectrum specifically comprises:
determining the reconstruction matrix by non-uniform fast Fourier transform;
determining a first row of an inverse matrix of the reconstruction matrix through a half division;
determining a first column of an inverse matrix of a reconstruction matrix through hermitian symmetry of the reconstruction matrix.
5. The method according to claim 2 or 4, wherein the weight kernel function of the non-uniform fast Fourier transform is any of: gaussian function, B-curve function, kessel-bessel function.
6. An apparatus for reconstructing a periodic non-uniformly sampled band-limited signal, the apparatus comprising:
the non-uniform frequency spectrum determining unit is used for determining a non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal;
the device comprises a reconstruction matrix determining unit, a sampling unit and a sampling unit, wherein the reconstruction matrix determining unit is used for determining the first row and the first column of an inverse matrix of a reconstruction matrix between a uniform sampling frequency spectrum and a non-uniform frequency spectrum;
the uniform sampling frequency spectrum determining unit is used for determining a uniform sampling frequency spectrum of a signal according to the non-uniform frequency spectrum and the first row and the first column of the inverse matrix of the reconstruction matrix;
the method specifically comprises the following steps:
determining a uniformly sampled spectrum of the signal as:
wherein:
m is the sampling number of a non-uniform sampling period;
[r]＝[r 0 ,r 1 ,…,r M-1 ]the first row of the inverse matrix of the reconstruction matrix;
[c]＝[c 0 ,c 1 ,…,c M-1 ] T the first column of an inverse matrix of the reconstruction matrix is obtained;
and the signal reconstruction unit is used for reconstructing a signal according to the uniformly sampled frequency spectrum of the signal.
7. The apparatus according to claim 6, wherein the non-uniform spectrum determination unit is specifically configured to:
the non-uniform frequency spectrum of the periodic non-uniformly sampled band-limited signal is determined by non-uniform fast fourier transform.
8. The apparatus according to claim 7, wherein the non-uniform spectrum determination unit is specifically configured to:
using a Gaussian function as a weight kernel function, assume a periodic non-uniform sampling band-limited signal sequence { x (t) n ) When the length is MN, determining the non-uniform frequency spectrum of the periodic non-uniform sampling band-limited signal as follows:
wherein: m is the number of samples of a non-uniform sampling period, and N is the number of non-uniform sampling periodsTo this end, T is the equivalent sampling interval,
b is a parameter of a Gaussian function [ ·]For rounding operations, p is the oversampling factor, w 0 ＝2π/N，c n ＝t n /T。
9. The apparatus according to claim 6, wherein the reconstruction matrix determination unit is specifically configured to:
determining the reconstruction matrix by non-uniform fast Fourier transform;
determining a first row of an inverse matrix of the reconstruction matrix through a half division;
determining a head column of an inverse matrix of a reconstruction matrix through hermitian symmetry of the reconstruction matrix.
10. The apparatus according to claim 7 or 9, wherein the weight kernel function of the non-uniform fast fourier transform is any one of: gaussian function, B-curve function, kessel-bessel function.
11. A reconstruction device of a periodic non-uniform sampling band-limited signal comprises a processor and a memory; the method is characterized in that:
the memory contains instructions executable by the processor to cause the processor to perform:
determining a non-uniform frequency spectrum of a periodic non-uniform sampling band-limited signal;
determining a first row and a first column of an inverse matrix of a reconstruction matrix between the uniformly sampled spectrum and the non-uniformly sampled spectrum;
determining a uniformly sampled spectrum of the signal according to the non-uniform spectrum and a first row and a first column of an inverse matrix of the reconstruction matrix,
the method specifically comprises the following steps:
determining a uniformly sampled spectrum of the signal as:
wherein:
m is the number of samples of a non-uniform sampling period;
[r]＝[r 0 ,r 1 ,…,r M-1 ]the first row of an inverse matrix of the reconstruction matrix is obtained;
[c]＝[c 0 ,c 1 ,…,c M-1 ] T the first column of an inverse matrix of the reconstruction matrix is obtained;
and reconstructing the signal according to the uniformly sampled frequency spectrum of the signal.
12. A computer-readable storage medium having stored thereon a computer program for:
determining a non-uniform frequency spectrum of a periodic non-uniform sampling band-limited signal;
determining a first row and a first column of an inverse matrix of a reconstruction matrix between the uniformly sampled spectrum and the non-uniformly sampled spectrum;
determining a uniformly sampled spectrum of the signal according to the non-uniform spectrum and a first row and a first column of an inverse matrix of the reconstruction matrix,
the method specifically comprises the following steps:
determining a uniformly sampled spectrum of the signal as:
wherein:
m is the sampling number of a non-uniform sampling period;
[r]＝[r 0 ,r 1 ,…,r M-1 ]the first row of the inverse matrix of the reconstruction matrix;
[c]＝[c 0 ,c 1 ,…,c M-1 ] T the first column of the inverse matrix of the reconstruction matrix;
and reconstructing the signal according to the uniformly sampled frequency spectrum of the signal.
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