JPH0248941B2 - TAJIGENRISANGATAFUURIEHENKANKEISANHOSHIKI - Google Patents

Description

[Detailed Description of the Invention] (1) Technical Field of the Invention The present invention relates to a multi-element discrete Fourier transform (hereinafter referred to as
This is a multidimensional discrete Fourier transform calculation method that efficiently processes MDFT (abbreviated as MDFT) on a parallel processing computer.

(2) Prior art and problems Figure 1 shows an overview of parallel processing computers. 1 is a pipeline type parallel processing computer;
2 is a unified memory, 3 is a control unit, 4 is a parallel processing device, 5 is an addition pipeline, and 6 is a multiplication pipeline. Vector data is stored in the unified memory 2. The control unit decodes the vector instruction and sends it to the addition pipeline 5.
It controls various calculation pipelines such as the multiplication pipeline 6 and the multiplication pipeline 6. Addition pipeline 5
adds two pieces of vector data in a pipeline manner, and a multiplication pipeline 6 processes two pieces of vector data in a pipeline manner. Figure 2 shows an overview of a parallel processing computer, in which 1' is a parallel processing computer, 2' is a unified memory, 3' is a control unit, and 4' is a parallel processing computer.
is a parallel processing unit, and 5'-0 to 5'-n are arithmetic units of the same type. In the parallel processing computer shown in FIG. 2, the element a _i of the operand vector read from the one-dimensional memory 2' and the element b _i of the operation vector are input to the arithmetic unit 5'-j, and the next element a _{i+ 1} and b _i+1 are input to the calculation unit 5'-+1.

The parallel processing computer shown in FIGS. 1 and 2 aims to improve computational efficiency by regularly processing data regularly arranged in memory in parallel using a parallel processing device. Specifically, for example, the addition of two vectors a→, b→ c _i =a _i +b _i (i=1, 2,...n) where a→=(a ₁ , a ₂ ,... ...a _o ) b → = (b ₁ , b ₂ , ......b _o ) c → = (c ₁ , c ₂ , ......c _o ) When performing vector length, n also has a parallel processing function. In a computer with no data, it is executed by n addition instructions, whereas
In a parallel processing computer, this is executed using one or fewer than n vector addition instructions.

Now, the processing efficiency (number of instructions or number of data processed within a certain amount of time) of these types of computers
Although it depends on the hardware characteristics of the computer, it also greatly depends on the specific architecture of the computer and the software that describes the data processing contents and procedures. However, B. Make the length of the data to be processed in parallel (vector length) as long as possible.

○B Keep the interval in memory of data to be processed in parallel as short as possible. That is, data is arranged consecutively on the memory.

This is the basic requirement for maximum efficiency. This is called parallel processing requirement. The present invention
In parallel processing computers such as those shown in FIGS. 1 and 2, MDFT is processed to satisfy the requirements of ○A and ○B above.

Before explaining the present invention, a one-dimensional discrete Fourier transform (hereinafter referred to as DFT), which is the basis for calculating MDFT, will be described. Figure 3 A is
It shows the procedure and data flow for calculating DFT, and Figure 3B explains the notation in Figure 3A. To explain according to the diagram, the O marks at both ends of each stage indicate the data in memory and their order (order), the O mark at the left end of each stage indicates input data, and the O mark at the right end indicates output data. There is. In the first stage (Stage) processing, a vector V ₁ consisting of data at positions 1 to 4,
Calculate vector addition V ₁ +V ₂ for vector V ₂ consisting of data from positions 5 to 8,
The results are stored from position 1 to every other position 7 (solid line portion in the figure). Also, vector subtraction V ₁
-V ₂ is calculated, each element of the result is multiplied by a twiddle factor, and the results are stored from position 2 to every other position 8. Stage 2 and Stage 3
Similarly, processing is performed based on vectors V ₁ and V ₂ , but the twiddle factors are updated each time the stage advances.

The processing procedure when the number of data is generalized to n is shown in FIG. Here, for simplicity, it is assumed that n can be expressed as a power of 2 (n= ^2l , l is a positive integer), and data is expressed as {a _i }. The contents of the twiddle factor table are as ω=exp(−2πi/n), It is. This generalized procedure can be easily extended to cases where n cannot be expressed as a power of two.

Now, to explain MDFT, we will consider a two-dimensional case for simplicity. The data to be converted is expressed as a _i,j . Here, it is assumed that i=1, 2, ......, n ₁ j=1, 2, ......, n ₂ . This data a _i,j is represented in a two-dimensional array as shown in FIG. 5A. In Figure 5 A, i direction is column direction, j
The direction of is called the row direction. Two dimensions
MDFT is basically formulated as follows.

DO 20 I=1,2 DO 10 J=1,n ₁・n ₂ /n _I DFT of size n _I (one-dimensional) 10 CONTINUE 20 CONTINUE To state this specifically, when I=1 5th DFT (one-dimensional) is applied to a vector (column vector) of size n ₁ in the column direction as shown in Figure B. This DFT (one-dimensional) is applied to n ₂ sets of column vectors for j.

When I = 2, DFT (one-dimensional) for a vector (row vector) with a size n ₂ in the row direction as shown in Figure 5 C.
apply. This DFT (one-dimensional) is applied to n ₁ sets of row vectors for i. Here, means that a has been converted by the process.

A two-dimensional MDFT is realized by sequentially performing the above processing. here,
The processing order of is commutative. in this way,
MDFT is for subvectors of input data.
It can be calculated by applying DFT multiple times.

Next, consider parallel processing of this MDFT. The sizes of this two-dimensional MDFT are assumed to be n ₁ =4 and n ₂ =8 for ease of explanation. Also, data
It is assumed that a _i,j is on a continuous memory with respect to i. In other words, the order of data a _i,j in memory is
a ₁ , ₁ , a ₂ , ₁ , ......, a ₄ , ₁ , a ₁ , ₂ , a ₂ , ₂ , ......
，
a ₄ , ₂ , a ₁ , ₃ , ………, a ₁ , ₄ , ……… a ₁ , ₈ , a ₂ , ₈ ,
...Assume that they are stored in the order of change starting from the first subscript, such as a ₄ and ₈ . The processing procedure and data flow in the prior art are shown in FIG.
The notation in FIG. 6 is the same as in FIG. To explain according to FIG. 6, in Stages 1 and 2, it is assumed that one-dimensional DFTs of size 4 are sequentially processed, and eight sets of identical one-dimensional DFTs are processed.
Process in parallel. Also, stages 3, 4 and 5
In this, a one-dimensional DFT of size 8 is processed in parallel according to FIG. 3, and four sets of processing are performed sequentially.

Generalizing the size of this two-dimensional MDFT, n ₁ ,
When n is ₂ , one of the following two methods is used at each stage. That is, assuming that a one-dimensional DFT of size n ₁ (or n ₂ ) is processed sequentially, n ₂ sets (or n ₁ sets) are processed in parallel.

A one-dimensional DFT of size n ₂ (or n ₁ ) is processed in parallel, and this is repeated n ₁ times (or n ₂ times).

One of these two methods is selected so that the vector length during parallel processing is as long as possible depending on the values of n ₁ and n ₂ . However, in such conventional technology, the length of the vector is the height max (n ₁ , n ₂ ), and the arrangement of the data to be converted in memory is not continuous, but the interval is at least min (n ₁ , n ₂ ) Therefore, the parallel processing condition mentioned above is not satisfied, and the parallel processing efficiency is not sufficiently improved.

(3) Purpose of the invention The present invention is based on the above consideration, and
When executing MDFT on a parallel processing computer, make sure to satisfy the parallel processing requirements mentioned above.
The purpose of this paper is to provide a multidimensional discrete Fourier transform calculation method that can perform MDFT.

(4) Structure of the invention For this purpose, the multidimensional discrete Fourier transform calculation method of the present invention uses a parallel processing computer having a memory, a parallel processing device, and a control unit to calculate M dimensions (M is an integer greater than 1). In a multidimensional discrete Fourier transform calculation method that performs a discrete Fourier transform, each of 2N data a _{i1, i2,} ...... _iM belonging to a data group of an M-dimensional array is stored in the above memory in the order of change from the first index. In addition to storing in advance in a predetermined data area of
(b) Processing to set the control variable I to the numerical value “1”; (c) Processing to create a twiddle factor table based on n _L ; (c) processing to create a twiddle factor table based on the 2N data stored in the memory above; A process of adding vectors V 1 and V 2 by using _{the data strings up to the th data string as a vector V and the N+1st to 2Nth data strings as a vector V 2, and (d) adding the vectors V 1 and V 2 to} each other. (e) Processing to subtract the vectors V ₁ and V ₂ determined as above, (f) Subtraction result vector (g) processing to sequentially store each data of the multiplication result vector from the second data storage position in the above data area by a twiddle factor vector; (h) control variable I is log ₂ Process to check whether it is smaller than n _L , and (li) If Yes in (h) above, revise the twiddle factor table based on the value of numerical value N, numerical value n _L , and control variable I. , After that, the control variable I is increased by the unit amount and the process returns to the process in (c) above. This method is characterized by performing the following steps: checking whether the variable L satisfies a predetermined condition, and if so, returning to the process in (a) above.

(5) Examples of the invention Hereinafter, the present invention will be explained with reference to the drawings.

The parallel processing procedure of the present invention for a two-dimensional DFT as shown in FIG. 5A is shown in FIG. 7th
The notation of the figure is the same as in the case of FIG. Referring to FIG. 7, in stages 1, 2, and 3, four sets of one-dimensional DFTs of size 8 are processed in parallel at the same time. Furthermore, in stages 4 and 5, eight sets of one-dimensional DFTs of size 4 are processed in parallel at the same time. The feature _of the present invention lies in the method of creating and revising the twiddle factors.According to the present invention, the length of the vector is always _n1・_n2 / ₂ ( This is the limit value) The arrangement of the data to be converted in memory (data interval) is always continuous (the interval is always 1), the parallel processing conditions mentioned above are completely satisfied, and parallel processing The effect will be maximized.

The present invention will be generalized and explained for an M-dimensional case. Let the size of each element be n ₁ , n ₂ , ......n _M
Then, data a _i1 , a _i2 , ......, a _iM (i1 = 1, 2...
...i ₁ , i ₂ = 1, 2...... n ₂ , ......, i _M = 1,
2......n _M ) are stored in memory in the order of change starting from the first subscript. The processing procedure of this M-dimensional MDFT is shown in FIG.

The method for creating the twiddle factor table will be explained in more detail. First, a DFT twiddle factor table of n _L /2 is created for a certain level L (L=M, M-1, . . . , or 1). That is, as ω=exp(2πi/n _L ) Create a twiddle factor table. For simplicity, this table is denoted as ξ, and ξ={ξ ₁ , ξ ₂ , ......, ξ _oL /2}.

Next, ξ is expanded to create a twiddle factor table for MDFT (denoted as ξ〓) of size N. This method expands each element of ξ so that each element is 2N/n _L consecutively. That is, ξ〓={ξ ₁ , ξ ₁ , ......ξ ₁ , ξ ₂ , ξ ₂ , ......,
ξ ₂ ,
………ξ _oL/2 , ξ _oL/2 , ………, ξ _oL/2 } Create a table. However, ξ ₁ , ξ ₂ , ...
...The number of ξ _oL/2 is 2N/n _L , respectively.

The method of revising the twiddle factor table will be explained in more detail. The revision is done at a certain level L (L=M, M-1,
......, 1) is performed (log ₂ n _L ) -1 times. Now, let's talk about the Pth revision.
Repeat the following procedure S times (S = 1, 2, ......, n _L /4P). In other words, consecutive numbers starting from (4PN/n _L ) * (S-1) + 1st in the twiddle factor table
Continuously copy 2PN/n _L elements from the (4PN/n _L )*(S-1)+2PN/n _L +1st of the twiddle factor table.

As a concrete example, the two-dimensional MDFT shown in Figure 7
Figure 9 shows the actual creation and revision of the twiddle factor table in the case of . Here, N=16 (n=n ₁・n ₂ /
2). To explain along Figure 9, first, L
=2, a DFT twiddle factor table ξ={0,1,2,3} with a size of 4 (=n ₂ /2) is created. Note that each numerical value in the table indicates a power to ω. Next, the size of ξ is 16
Create a twiddle factor table ξ〓 for MDFT (=N). That is, ξ〓={0, 0, 0, 0, 1, 1, 1, 1, 2, 2,
2, 2, 3, 3, 3, 3}. This is the stage 1 twiddle factor table.

Next, the number of revisions is 2 for L=2.
(=log ₂ n ₂ -1) times. The first time (P=1)
In the revision of , a total of two copies will be made as follows. That is, first, with S=1, 4 (=2PN/n ₂ ) consecutive elements from the first in the twiddle factor table are divided into 5 (=4PN/n ₂ *(S-1)+2PN/n _{2 )} elements in the twiddle factor table.
Continuously copy from +1). Next, with S=2, four consecutive elements from the 9th place in the twiddle factor table are successively copied from the 13th place in the twiddle factor table. The result is a stage 2 twiddle factor table. Similarly, in the second revision (P=2), one copy is performed as follows. That is, S=1
8 consecutive from the first in the twiddle factor table (=
2PN/n ₂ ) elements of the twiddle factor table 9 (=
4PN/n ₂ *(S-1)+2PN/n ₂ +1) Copy consecutively from the th. This result is the stage 3 twiddle factor table.

Next, with L=1, the size is 2 (=n ₁ /2)
Create a twiddle factor table ξ={0,1} for DFT. Next, expand ξ to size 16 (=N)
Create a twiddle factor table ξ for MDFT. That is, ξ ^~ = {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1
,1,1,1,1,1}. This is the stage 4 twiddle factor table. Next, revision is performed, and the number of revisions is 1 (log ₂ n ₁ -1) times at level L=1. In this revision, P=1, and one copy is performed as follows. That is, with S=1, 8 (=2PN/n ₁ ) consecutive elements from the 1st of the twiddle factor table are extracted from the 9th (4PN/n ₁ *(S-1)+2PN/n ₁ +1) of the twiddle factor table. Copy continuously. The result is a stage 5 twiddle factor table.

FIG. 10 shows an example of a hardware configuration for implementing the present invention. In FIG. 10, 12 is a memory device, 13 is a control device, 1
4 is a vector calculation device, 15 is a data transfer device,
16 indicates a twiddle factor table generation and revision device for MDFT. The memory device 12 includes
Vector data and twiddle factor table to be calculated are stored. The control device 13 controls each device according to the MDFT processing procedure shown in FIG. The data transfer device 15 transfers data between the memory device 12 and the vector arithmetic device 14 and between the memory device 12 and the MDFT twiddle factor table generation and revision device 16. The vector calculation device 14 has a function of performing various calculations in parallel on the transmitted vector data and twiddle factor table. The MDFT twiddle factor table generating/revising device 16 creates the MDFT twiddle factor table based on the DFT twiddle factor table as described above, and also generates the MDFT twiddle factor table based on the DFT twiddle factor table, and also generates the MDFT twiddle factor table based on the DFT twiddle factor table.
It has a function to revise the twiddle factor table for MDFT.

(6) Effects of the invention As is clear from the above explanation, the multidimensional discrete Fourier transform calculation method of the present invention allows parallel processing to make the length of data to be processed in parallel (vector length) as long as possible. Since it completely satisfies the parallel processing requirement of minimizing the interval between data in memory (data interval), it is possible to significantly improve processing efficiency.

[Brief explanation of drawings]

Figures 1 and 2 are diagrams showing an overview of a parallel processing computer, Figure 3 is a diagram for explaining the calculation method of DFT, Figure 4 is a diagram showing the processing procedure of parallel processing DFT, and Figure 5 is a diagram showing the processing procedure of parallel processing DFT. A diagram showing an example of a two-dimensional array, a column vector, and a row vector, FIG. 6 is based on the prior art
FIG. 7 is a diagram explaining the calculation method of MDFT, FIG. 7 is a diagram explaining the process for performing two-dimensional DFT according to the present invention, FIG. 8 is a diagram showing the processing procedure for M-dimensional MDFT according to the present invention, and FIG. Two-dimensional according to the invention
FIG. 10, which is a diagram for explaining the creation and revision of a twiddle factor table for MDFT, is a diagram showing an example of a hardware configuration for implementing the present invention. 12...Memory device, 13...Control device, 14
...vector calculation device, 15...data transfer device, 16...twiddle factor table generation for MDFT,
Revision device.

Claims (1)

[Claims] 1. In a multidimensional discrete Fourier transform calculation method that performs M-dimensional (M is an integer greater than 1) discrete Fourier transform using a parallel processing computer having a memory, a parallel processing device, and a control unit. , 2N pieces of data a _{i1, i2, .} value, and then (a) the size of the dimension indicated by the numerical value N and the control variable L.
(b) Processing to set the control variable I to the numerical value “1”; (c) Processing to create a twiddle factor table based on n _L ; (c) processing to create a twiddle factor table based on the 2N data stored in the memory above; A process of adding vectors V _{1 and} V _{2 by setting the data string up to the th data string as a vector V 1 and} the data strings from N+ _1st to _{2Nth as a vector V 2 ;} A process of storing each data every other data from the first data storage position of the data area, (e) a process of subtracting the vectors V ₁ and V ₂ determined as above, and (f) the result of the subtraction. (g) processing to sequentially store each data of the multiplication result vector from the second data storage position in the data area every other data storage position, and (h) processing to multiply the vector by the twiddle factor vector; Processing to check whether it is smaller than log ₂ n _L , and (li) If Yes in (h) above, revise the twiddle factor table based on the value of numerical value N, numerical value n _L , and control variable I. However, after that, the control variable I is increased by a unit amount, and the process returns to the process in (c) above. A multidimensional discrete Fourier transform calculation method characterized by: checking whether the control variable L satisfies a predetermined condition, and if so, returning to the process in (a) above; .