Fast DFT (Discrete Fourier Transform) approximation method based on sparse factor optimization

Abstract

The invention discloses a fast DFT (Discrete Fourier Transform) approximation method based on sparse factor optimization. The method comprises the following steps: decomposing a DFT matrix into a plurality of coefficient factor matrixes based on a special symmetric structure of the DFT matrix; the factor matrix is mapped to two feasible sets, and multiplication of elements in the feasible sets can be realized only by bit shift without a multiplier in actual hardware implementation; introducing a calibration matrix to carry out amplitude and phase calibration on the factor matrix subjected to quantization mapping; introducing a real-number overall scaling parameter, carrying out overall scaling on the approximation matrix to reduce an approximation error, and finally obtaining DFT (Discrete Fourier Transform) approximation; according to the method, the DFT matrix can be approximated to the form of multiplying a series of sparse factor matrixes through a low-complexity optimization algorithm, the hardware complexity of hardware implementation is reduced, and the approximation precision and orthogonality are high.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical calculation methods, and in particular to a fast processing DFT approximation method based on sparse factor optimization Background Art

[0002] DFT is a discrete Fourier transform (DFT) method, which is a basic tool widely used in various fields such as engineering, science and mathematics.

[0003] In the field of numerical analysis, Fast Fourier Transform (FFT) is a widely used algorithm that can convert time domain signals into frequency domain signals, or convert frequency domain signals into time domain signals.

[0004] The fast computing power of FFT makes it widely used in digital signal processing, image processing, audio processing and other fields. However, as the amount of data increases, the computational complexity of FFT also increases. Therefore, how to improve the computational efficiency of FFT has become an important research topic. In the field of fast algorithms, sparse matrix is ​​an important tool, which refers to a matrix with a small number of non-zero elements. Due to the characteristics of sparse matrix, many computational problems can be solved faster through sparse matrix. For example, in the field of numerical computing methods, sparse matrix is ​​widely used to solve linear equations, optimization problems and machine learning algorithms.

[0005] The existing DFT approximation methods are mainly based on the idea of ​​matrix decomposition and quantization mapping. First, the DFT matrix is ​​decomposed into several smaller matrices, then these matrices are quantized and mapped, and finally the approximate result of the DFT matrix is ​​obtained by optimizing the problem. Although this method can improve the computational efficiency to a certain extent, there are still some problems.

[0006] The existing DFT approximation methods still have high computational complexity when processing big data and cannot meet the requirements of real-time processing. In addition, the existing methods may cause some numerical errors in the quantization mapping process, which may accumulate and affect the accuracy of the final result. In addition, the existing methods do not fully utilize the characteristics of sparse matrices when processing sparse matrices, so there is room for further improvement in computational efficiency.

[0007] In VLSI processors, Fast Fourier Transform (FFT) is crucial because of its high computational efficiency in implementing DFT. In the hardware implementation of FFT, there are always computational errors due to the limited accuracy of implementing irrational matrix coefficients, which makes direct DFT approximation possible. Directly approximating the DFT matrix while ensuring that the approximation error is lower than the system error can achieve faster and more accurate DFT operations. Summary of the invention

[0008] The technical problem to be solved by the present invention is to provide a fast DFT approximation method based on sparse factor optimization, which can improve the problem of computational complexity: the existing DFT approximation method still has high computational complexity when processing big data and cannot meet the requirements of real-time processing. The present invention effectively reduces the computational complexity and improves the computational efficiency by introducing the reverse bit order sequence and sparse matrix, and utilizing the least squares problem and optimization problem. 2. The present invention solves the problem of numerical error: the existing DFT approximation method may cause some numerical errors in the quantization mapping process, and these errors may accumulate, thereby affecting the accuracy of the final result. The technical solution effectively reduces the numerical error and improves the accuracy of the result by introducing the calibration matrix and scaling parameters. 3. The present invention solves the problem of insufficient utilization of sparse matrix characteristics: the existing DFT approximation method does not fully utilize the characteristics of sparse matrix when processing sparse matrix, so there is room for further improvement in computational efficiency. The present invention approximates the DFT matrix as the product of a series of sparse matrices by utilizing the characteristics of sparse matrix, each sparse matrix can be implemented without a multiplier, and the overall accuracy of DFT approximation is maintained by optimization.

[0009] The technical solution adopted by the present invention to solve the above technical problems is: a fast DFT approximation method based on sparse factor optimization, characterized by comprising the following steps:

[0010] A fast DFT approximation method based on sparse factor optimization includes the following steps:

[0011] S1. Construct the sparse factor matrix F of the N-point DFT matrix based on the symmetric structure of the DFT matrix N ;

[0012]

[0013] Among them, ω N =e -j2π / N is the Nth root of unity, is an imaginary unit, Represents ω N (k-1)(i-1) times, F N,(k,i)Denotes the matrix F N The element in the kth row and the ith column;

[0014] S2, sparse factor matrix F for N-point DFT matrix N Decompose according to the following formula;

[0015]

[0016] Where t = log2N, P N is the reverse bit order permutation matrix, and the reverse bit sequence is r N , the reverse bit order permutation matrix can be obtained by Get, where e l represents the l+1th column of an N×N identity matrix, r N,(l) Represents vector r N The lth element of i It can be expressed as:

[0017]

[0018] Where blkdiag(·) represents the matrix A i is a diagonal block matrix, and the block matrices on its diagonal are composed of the matrices in brackets. represents the Kronecker product;

[0019] S3. Map the factor matrix to the feasible set Obtain the first quantization factor matrix; that is:

[0020]

[0021] S4. Map the factor matrix to the second feasible set Obtain the second quantization factor matrix; that is:

[0022]

[0023] S5, performing an amplitude and phase calibration process on the factor matrix of the first quantization mapping and the factor matrix of the second quantization mapping; comprising:

[0024] 501. Obtain a least square calibration matrix by solving the following linear equation;

[0025]

[0026] Where: ||·|| F represents the F norm of the matrix; H i is the calibration matrix;

[0027] 502. The least square calibration matrix obtained by the first quantization factor matrix is ​​scaled by factor α iAfter scaling and then mapping, we get the first calibration matrix that obeys the feasible set constraint:

[0028]

[0029] 503. The least square calibration matrix obtained by the second quantization factor matrix is ​​scaled by factor α i After scaling and then mapping, we get the second calibration matrix that obeys the feasible set constraint:

[0030]

[0031] in:

[0032] S6. Scaling by real number factor For the sparse factor matrix F N The optimized output approximate DFT matrix is:

[0033]

[0034] in: Beneficial Effects

[0035] Compared with the prior art, the advantages of the present invention are:

[0036] 1. Improve computational efficiency: The present invention greatly reduces the computational complexity of the DFT matrix by introducing sparse matrices and quantized mapping methods, so that real-time DFT calculations can be achieved in big data processing. Existing DFT approximation methods that are not targeted at specific points are usually implemented using methods such as exhaustive search. However, when the number of DFT points is large, the computational complexity of the algorithm increases sharply, making it difficult to apply to large-scale data processing. This algorithm only needs to calculate several closed-form expressions to obtain the DFT approximation matrix. Even for large-scale data processing problems, the algorithm still maintains the characteristic of low computational complexity.

[0037] At the same time, the present invention also makes full use of the characteristics of sparse matrices, so that the calculation efficiency is further improved.

[0038] 2. Improve the accuracy of the results: In the process of quantization mapping, the present invention introduces a calibration matrix and a scaling parameter, which effectively reduces the numerical error, thereby improving the accuracy of the results. In addition, the present invention further optimizes the results by optimizing the problem, making the results more accurate.

[0039] 3. Wide application scope: The DFT approximation method of the present invention is not only applicable to general DFT calculations, but also to DFT matrices with special structures, such as inverse bit order DFT matrices. Existing DFT approximation methods are mostly designed for DFT matrices with a specific number of points. The method of the present invention is applicable to DFT matrix approximation with any number of points (integer powers of 2), and has stronger universality. Therefore, the application scope of the present invention is wider and can meet more practical needs.

[0040] 4. Easy to implement: The DFT approximation method of the present invention makes the calculation process simpler and easier to implement by introducing sparse matrix and quantization mapping. At the same time, the present invention also provides detailed algorithm steps so that developers can easily implement the present invention. The DFT approximation result given by the method of the present invention does not contain irrational numbers, and no multiplier is required in the actual hardware implementation, which greatly reduces the hardware complexity of the DFT implementation.

[0041] In general, compared with the prior art, the present invention has the advantages of higher calculation efficiency, higher result accuracy, wider application range and easier implementation, and is a more superior DFT approximation method.

[0042] The DFT given by the method of the present invention has good performance in terms of approximation accuracy and orthogonality. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a general implementation block diagram of the method of the present invention;

[0044] Figure 2 To use the feasible set The obtained 8-DFT approximate signal manifold diagram;

[0045] Figure 3 To use the feasible set The resulting 8-DFT approximation of the signal manifold. DETAILED DESCRIPTION

[0046] The present invention is further described in detail below with reference to the accompanying drawings.

[0047] The present invention proposes a fast DFT approximation method based on sparse factor optimization, and its overall implementation block diagram is as follows: Figure 1 As shown, it includes the following steps:

[0048] Step 1: Construct the sparse factor matrix F of the N-point DFT matrix based on the symmetric structure of the DFT matrix N : An N (N is an integer of 2) point DFT matrix F N The elements in ; represented by Among them, ω N =e -j2π / Nis the Nth root of unity, is an imaginary unit, Represents ω N (k-1)(i-1) times, F N,(k,i) Denotes the matrix F N The element in the kth row and the ith column in the time domain is x = [x(0), x(1), ..., x(N-1)] T The DFT samples X = [X(0), X(1), ..., X(N-1)] T It can be expressed as X = F N x;

[0049] Step 2: First, the N-point DFT matrix F N Decompose into In the form of; where t = log2N, P N is the reverse bit order permutation matrix.

[0050] Taking N=8 as an example, the natural order sequence is {0,1,2,3,4,5,6,7}, and its binary representation is {000,001,010,011,100,101,110,111}. The binary representation of the reverse bit order is {000,100,010,110,001,101,011,111}, and then the reverse bit order sequence is {0,4,2,6,1,5,3,7}. Based on this reverse bit order sequence, the reverse bit order permutation matrix can be obtained as P8=[e0,e4,e2,e6,e1,e5,e3,e7]; where e l Represents the l+1th column of an N×N identity matrix. i is a sparse matrix of dimension N×N, which is a full rank matrix. Put the reverse bit order sequence into an N×1 vector to get r N , then A i It can be expressed as

[0051] Where blkdiag(·) represents the matrix A i is a diagonal block matrix, and the block matrices on its diagonal are composed of the matrices in brackets. represents the Kronecker product, r N,(l) Represents vector r N The lth element of .

[0052] Step 3: Define two feasible sets and Where q is a non-negative integer;

[0053] Mapping factor matrix to feasible set Obtain the factor matrix of the first quantization mapping; that is:

[0054]

[0055] Mapping factor matrix to feasible set Obtain the factor matrix of the second quantization mapping; that is:

[0056]

[0057] Among them, except A t-1 and A t According to the above definition, A t-1 and A t All elements in satisfy the feasible set constraint, so no mapping operation is required. The remaining matrix A i The real or imaginary part of the element may not be in the feasible set. In this case, it needs to be mapped and replaced with the closest value in the feasible set. Define the feasible set The mapping function is And the feasible set The mapping function is Then we get A i The result after approximation or

[0058] Step 5: performing an amplitude and phase calibration process on the first quantization factor matrix and the second quantization factor matrix; comprising:

[0059] By introducing the calibration matrix H i The matrix after quantization mapping Perform phase and amplitude calibration, calibration matrix H i The elements in must still satisfy the feasible set constraint; since the feasible set constraint is a discrete constraint, it is difficult to process directly. First, discard the constraint and construct the least squares problem The calibrated matrix Close to the original matrix A i ; Among them, ||·|| F represents the F norm of the matrix; the optimal solution to this least squares problem can be obtained by Directly obtain; it is worth noting that the matrix A i and are all sparse matrices, and their special structure makes The positions of the non-zero elements in Consistent, so it is still a sparse matrix with no more than two non-zero elements in each row and column;

[0060] A may contain non-feasible set elements, so it needs to be mapped; i The non-zero elements in are always inside the unit circle, and The non-zero elements in are always outside or on the unit circle, such that The elements in are always on or inside the unit circle. When Mapping When using feasible sets If you directly The non-zero elements in are directly mapped, and the elements obtained after mapping will be limited to -1, 0 or 1, resulting in a large approximation error. To solve this problem, A scaling parameter α is introduced before mapping i , the scaling parameter is given by The non-zero element with the smallest absolute value in is determined. The non-zero element with the smallest absolute value is ε i ,but Then Mapping To ensure The smallest non-zero element in is mapped to ±1, making it Each element in the system can play an effective calibration role;

[0061] Step 6: Scaling by a real number factor For the sparse factor matrix F N Optimize the output approximate DFT matrix and introduce a real number scaling factor β to optimize the overall result. Constructing the optimization problem The optimal solution of this optimization problem can be expressed by the closed-form expression Finally, we get the approximate result of the N-point DFT matrix

[0062] In order to verify the feasibility and effectiveness of the method of the present invention, a simulation test was carried out on the method of the present invention.

[0063] Taking N=8 as an example, the reverse bit order permutation matrix is ​​P8=[e0,e4,e2,e6,e1,e5,e3,e7]. The DFT matrix after reverse bit order permutation is It is not difficult to see that the 4×4 submatrix on the upper right of the matrix is ​​the same as the 4×4 submatrix on the upper left, and the 4×4 submatrix on the lower left and lower right has the opposite sign. Therefore, it can be decomposed into Thus, we get the factor matrix A3. When observing the first submatrix after decomposition, it is not difficult to find that its upper left and lower right submatrices are similar to the structure of the original P8F8 matrix, which can be further decomposed into Obtain factor matrices A1 and A2; using the property of the permutation matrix P8P8=I8, multiply both sides of the above equation by P8 to obtain the sparse decomposition of 8-DFT as F8=P8A1A2A3; where the factor matrix contains complex elements Among them: ω8 and Not in the feasible set, is in the feasible set, and ω8, Only exists in the factor matrix A1; next, we use the feasible set and Approximate the matrix A1: When using the feasible set When A1 is mapped to Its corresponding calibration matrix according to The smallest factor in the equation gives the scaling factor Then, according to the scaling factor, the calibration matrix that obeys the feasible set constraint is obtained. Scaling Factor Depend on Get, where Finally, the approximate decomposition of 8-DFT is obtained as The approximation error of this approximation is 0.0082, and its orthogonality index is 0.0030; Figure 2 As shown;

[0064] When using feasible sets When A1 is mapped to Scaling Factor for in Finally, the approximate decomposition of 8-DFT is obtained as The approximation error of this approximation is 0.0184, and its orthogonality index is 0.0194; according to the expression of 8-DFT approximation, the corresponding signal flow diagram can be drawn as follows Figure 3 shown.

[0065] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A fast DFT approximation method based on sparse factor optimization, characterized in that: The steps include: S1. Construct the sparse factor matrix F of the N-point DFT matrix based on the symmetric structure of the DFT matrix N ; Among them, ω N =e -j2π / N is the Nth root of unity, is an imaginary unit, Represents ω N (k-1)(i-1) times, F N,(k,i) Denotes the matrix F N The element in the kth row and the ith column; S2, the sparse factor matrix F of the N-point DFT matrix N Decompose according to the following formula; Where t = log2N, P N is the reverse bit order permutation matrix; let the reverse bit sequence be r N , the reverse bit order permutation matrix can be obtained by Get, where e l represents the l+1th column of an N×N identity matrix, r N,(l) Represents vector r N The lth element of i It can be expressed as: Where blkdiag(·) represents the matrix A i is a diagonal block matrix, and the block matrices on its diagonal are composed of the matrices in brackets. represents the Kronecker product, r N,(l) Represents vector r N The lth element of S3. Map the factor matrix to the feasible set Obtain the first quantization factor matrix; that is: S4. Map the factor matrix to the feasible set Obtain the second quantized factor matrix; that is: S5, performing an amplitude and phase calibration process on the factor matrix of the first quantization mapping and the factor matrix of the second quantization mapping; comprising:

501. Construct a least square calibration matrix using the following formula; Where: ||·‖ F represents the F norm of the matrix; H i is the calibration matrix; 502. Map the factor matrix of the first quantization mapping to the least square calibration matrix to obtain a first calibration factor matrix:

503. Map the factor matrix of the second quantization mapping to the least square calibration matrix to obtain a second calibration factor matrix: S6. Scaling by real number factor For the sparse factor matrix F N Optimize the output approximate DFT matrix. in:

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