JPS5925253B2 - Cosine/sine calculation method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention relates to a method for determining the cosine and sine values of a predetermined angle used in Fourier transform processing, and particularly to a method for quickly replanting the cosine and sine values using pipeline processing.

In the discrete Fourier transform, when the data is a complex number, calculation processing of the transformation of equation (1) or the inverse transformation of equation (2) is required.

ln21 α to −1−2−Xjexp(−2 nj=0
, j-0, ..., n-1... (2) In the Fourier transform of equation (1), Xj is complex data and αk is a Fourier coefficient, which will be explained below according to the transformation of equation (1). .

Now, if the number of data is 8 (XO..Xl, . . . 7θ (however, θ−2π
/8) is required. by the way,
In order to obtain the cosine and sine of θ, 2θ, .

However, K-011, 2,...7 That is, the cosine and sine values COsθ, Sin for θ
θ is determined in advance, and using these COsθ and Sinθ, COs(K+1)θ, Sin(K+1)θ(K-1,
2,...7) one after another.

However, this method requires a very long calculation time,
Fast Fourier transform cannot be performed. That is, COsθ,
From Sinθ, first calculate COs2θ and Sin2θ,
Next, use these values to calculate COs3θ and Sin3θ, and thereafter similarly calculate COs(K+1)θ and Sin(K+1
) θ.In other words, based on equations (3) and (4) above, 0sKθ, Sin
In order to obtain Kθ, pipeline processing, which is a known method of so-called high-speed arithmetic processing, cannot be applied. Therefore, the present invention can quickly calculate COsθ and Sin based on pipeline processing.
From θ, COs(K+1)θ, Sin(K+1)θ(K
-1, 2,...7), and this purpose is to provide a new method that can calculate COsKθ, S from COsθ, Sinθ.
Each angle θ, 2θ, 3θ, ... nθ calculated in the calculation method of calculating inKθ (K-2, 3, ... n)
Storage means for storing cosine and sine values corresponding to COsθ
Based on the formula of Sinθ and double angle, COS2mθ, Si
a calculation means for sequentially calculating n2mθ (m1, 2c, . . .) and storing it in the storage means, and shifting cosine and sine data sequentially inputted from a predetermined address of the storage means to another address of the storage means; a first pipeline processing means; a second pipeline processing means for executing an addition operation process using the cosine and sine data manually inputted from the storage means after the shift processing, and storing the operation result in the storage means; and a storage area of the storage means {COS2
mθ, Sin2mθ} storage area T1; {CO
S2nl-1θ, Sin2nl-1θ}, {COS2n
lθ, Sin2mθ} storage area T2; {CO
s2m-2θ, Sin2m-2θ}, {COs2m-1
θ, Sin2m−1θ}, {COs(2m−2+2m−
1) Storage area T3 for storing θ, {COS2mθ, Sin2mθ}; {COs2m−3θ, Sin2m−3θ}
, {COS2rrl-2θ, Sin2nl-2θ}, {
COs(2m-3+2m-2)θ, Sin(2m-3+
2m-2) θ}, {COs2m-1θ, Sin2m-1
θ}, {COs(2m-3+2n1-1)θ, Sin(
2n1-3+2m-riθ}, {COS2mθ, Sin2
mθ} storage area T4; ...; {COsθ, S
inθ}, {COs2θ, Sin2θ}, {COs3θ
, Sin3θ}...{COsnθ, Sinnθ} into a storage area T1 for storing the data, and the first pipeline processing means stores the contents of each address in the storage area Ti into the storage area Ti
The second pipeline processing means sequentially performs an addition operation process using the contents of the first address and the even numbered address of the memory Ti, and sequentially transfers the operation result to the even number area of Ti. The process of writing to odd-numbered addresses is executed, and the arithmetic means calculates COS2mθ, Sin2
After calculating nlθ (m-1, 2, ゜゜゜), Ti to Ti
The shift process to +1 and the addition process are repeated alternately to obtain COsKθ, SinKθ(K-2, 3, . . .
This is achieved by using the cosine and sine calculation method to find n).

Hereinafter, the present invention will be explained in detail with reference to the drawings.

Now, if equations (3) and (4) are written in terms of matrices and vectors, then if we set (here, the twiddle factor of θ), equation (5) can be further expressed as follows.

Therefore, once W1, W2, . . . , Wk are determined, Xk+1 can be easily determined by multiplying by X1.

Next, an algorithm for obtaining twiddle factors up to W-W8 as K-8 will be explained with reference to FIG.

Note that this algorithm is suitable for pipeline processing as described later. In the figure, w is a twiddle factor, W2,...W
8 is the width of the twiddle factor w, and T1 to T4 are storage areas.

Step 1... First, 8 ws are stored in the storage area T1.
Prepare.

Step 2...Next, four W2's are created by multiplying each adjacent two of the eight w's above and stored in the storage area T2.

Step 3... Out of these four W2, each adjacent two
By multiplying two W2s, two W4s are created and stored in the storage area T3.

Step 4...Next, W8 is created by multiplying these two W4s and stored in the storage area T4.

Above, in steps 1 to 4, the storage area T
This is a process of multiplying two adjacent twiddle factors of l or their widths and storing them in the storage area Ti+1. For this reason, pipeline processing, which is a known method for high-speed calculation processing, can be applied. Step 5: Shift the contents of the first area of the storage area T4, ie, W8, to the second area of the storage area T3. As a result, W4 and W8 are stored in the storage area T3. Step 6...Second area and fourth area of storage area T2
The contents W of the first and second areas of the storage area T3 are respectively stored in the areas.
4. After shifting W8, the content of the first area and the second area, that is, the seat, and W4 are multiplied to create 1, which is stored in the third area of T2.

Note that 1 can also be created by dividing these from W8 output ψ.

As a result, the first, second, third, fourth areas of the storage area T2 are stored in W2, W4, W6, . This means that W8 has been stored. Step 7...Finally, the second
The contents W2, W4, W6 . After shifting W8, the contents w and W2 of the first and second areas are multiplied, the contents w and W4 of the first and fourth areas are multiplied, and the contents w and W4 of the first and fourth areas are multiplied, and then the contents w and W4 of the first and fourth areas are The contents w sho ψ are multiplied and the respective multiplication results are stored in the third, fifth, and seventh areas of T1. As a result, the W.
W2, . . . W8 are obtained. As described above, in steps 5 to 7, after the contents of each area of the storage area Ti are sequentially stored in the even-numbered areas of the storage area Ti-1 (shift processing), the contents of the first area of Ti-1 and the contents of the first area of Ti-1 are Multiplication with the contents of each even number area is performed sequentially, and the multiplication result is T1
This is a process of sequentially storing data in an odd-numbered area of −1 (arithmetic processing), and is suitable for pipeline processing, which is a well-known method of high-speed arithmetic processing.

However, even if pipeline processing is possible in the above processing, each multiplication is a matrix multiplication, and therefore eight multiplications and four additions between matrix elements are required, making high-speed processing impossible.

By the way, the multiplication results of W2 and W3 are as shown in the following equations.

That is, W2 and W3 are COs2θ and Sin2θ, respectively.
: COs3θ and Sin3θ, and generally Wk can be easily found if COsKθ and SinKθ can be calculated.

The present invention provides COsKθ, SinKθ (K=2, 3,...
n) is determined using almost the same method as the algorithm for determining Wk described above, and will be explained below with reference to FIGS. 2 and 3.

In FIG. 2, MEM is a main memory that stores cosine and sine values, PLU1 is a double angle calculation unit that performs double angle calculations of trigonometric functions described below, PLU2 is a first pipeline processing unit that performs shift processing that will be described later, PLU3 is a second pipeline processing unit that performs addition calculation processing of trigonometric functions, and G
M is a gate means, COT is 1 reading and writing order of cosine and sine data from the main memory, 2 PLU1, PLU2,
This is a control circuit that controls PLU3 to enable processing, and controls three gate means GM to control which output data of PLU1, PLU2, and PLU3 is written to the main memory.

FIG. 3 is an explanatory diagram to make it easier to understand the processing according to the present invention. In the diagram, MEM is the main memory, MC
A storage area for storing sKθ, MS is a storage area for storing SinKθ, T1 to T4 are storage areas with the same symbols as in FIG. 1, PLU1 is a double angle calculation unit, PL
U2 is a pipeline processing device for shift processing, and PLU3 is a pipeline processing device for addition processing. Furthermore, the first
In the figure, in step 1, the first...
・After preparing 8 w in the 8th area, proceed to steps 2 and 3.
, 4, it has been explained that W2, W4, and W8 are sequentially obtained and stored in each storage area, but even if pipeline processing is applied to such processing, the effect of high-speed processing will not be improved, so FIG. Now, the processing from Step 1 to Step 4 is performed using the double angle calculation unit PLU1 without using pipeline processing.
COs2x, . sin2x; COs4xl
sin4x; COs8xl8ln8x is sought.

However, it is of course possible to obtain it according to the algorithm shown in FIG. Step 1...Main memory MEM
COsθ and Sinθ are prepared at address T1(1) of the cosine storage area and sine storage area MC, MS.

Note that the shaded area in the figure displays data related to processing. Step 2: Read the contents of address T1(1) of MC and MS to double angle arithmetic unit PLU1.

If the contents read from MC and MS are (MC) and (MS), the double angle calculation unit PLUl executes the calculation of the following equation,
A process of storing the result of the calculation at a predetermined address of the MC and MS separately instructed is executed. That is, from formula (substitute), CO
s2θ and Sin2θ are calculated from the equation (self) and stored at address T2(1) of cosine and sine storage areas MC and MS, respectively.

Step 3...The contents of address T2(1) of MC and MS are read to the double angle calculation unit PLUl, and the calculation of (substitute) and σD is executed, and the calculation results are COs4θ and Sin4θ.
is stored in address T3(1) of MC and MS.

Step 4...The contents of address T3(1) of MC and MS are read out to the double angle arithmetic processing unit PLU1, and the (alternative) and (self) calculations are executed, and the calculation results COs8θ, Sin
8θ is stored at address T4(1) of MC and MS. As described above, through steps 1 to 4, the contents of the cosine and sine storage areas become as shown in FIG. 2A. Step 5... Shift processing... The contents of address T4(1) in the storage area T4 of MC and MS are read to the pipeline processing unit PLU2 for shift processing.

The pipeline processing unit PLU2 processes each area Ti( of the storage area Ti for the cosine and sine storage areas MC, MS,
1), Ti(2), Ti(3)... are stored in the even number areas Ti-1(2), Ti-1(4), and the storage area Ti-1.
Pipeline processing is executed to shift to Ti-1(6)... Therefore, the content COs8θ of the MC address T4(1) is shifted to the MC address T3(2), and the content Sin8θ of the MS address T4(1) is shifted to the MS address T3(2).

(Refer to Figure 3) Step 6... Shift processing...Similar to step 5, contents T3(1), T3(2) of storage area T3 of MC and MS, ie, COs4x
．． lsin4x; COx8x. sin8x is sequentially input to the pipeline processing unit PLU2, and each
C, MSf) Shift to T2(2), T2(4).

As a result, the contents of the cosine and sine area become as shown in FIG. 3C. Step 7 Additive operation processing...Once the above shift processing is completed, the MC,
For the MS, the storage areas T2(1) and T2(2) are each paired to form a pipeline processing unit PLU for addition operations.
3.

The content read from the first area Ti(1) of the storage area Ti of the MC is {MCTi(1)}, and the content read from the even area Ti(2n) of the storage area Ti of the MC is {MCTi(2n).
}, the contents read from the first area Ti(1) of the storage area Ti of the MS are {MSTi(1)}, and the contents read from the even area Ti(2n) of the storage area Ti of the MS are {MSTi(2n)}.
}, the pipeline processing unit PLU3 executes the calculation of the following equation, and stores the calculation results in the odd number area Ti(2n+1) of the storage area Ti of MC and MS, respectively. That is, COs6θ according to the (self) formula, and Sin6 according to the (self) formula.
θ is calculated and stored in the third storage area T2 of MC and MS.
Each is stored in area T2 (3) (see Fig. 3 2)
.

Step 8... Shift processing...Same as step 6, the storage areas T2(1), T2(2), T3(2), T3(1),
That is, COs2x, . sin2x; COs4x,. si
n4x; COs6x, . sin6x; COs8x,. s
in8x are sequentially input to the pipeline processing device PLU2, and the pipeline processing device sequentially processes them into MC,
Even-numbered areas of the storage area T1 of the MS, that is, the second, fourth, and sixth
, eighth region Ti(2), T1(4), T1(6), T1
(8) is shifted and stored (see Figure 3 E).

Step 9...Additional operation processing...Similar to the processing in step 7, MC, M
First area T1(1) and second area T1 of storage area T1 of S
(2); The first area T1(1), the fourth area T1(4); The first area T1(1) and the sixth area T1(6) are sequentially read out to the pipeline processing device PLU3, and A2). 0
3) The calculation of the expression is performed and the calculation result is sequentially stored in the storage area T.
1 third, fifth, and seventh regions T1(3), T1(5), T
1 (7) and as shown in Figure 2, COsθ, Si
nθ; COs2θ, Sin2θ...; COs7θ, S
in7θ, COs8θ, and sin8θ are obtained in the storage area T1 of MC and MS.

(Go to Figure 3) As described above, in the present invention, COsKθ, SinKθ
By executing shift pipeline processing and addition operation pipeline processing on (K-1, 2,...n),
COsKθ,
SinKθ can be obtained.

The twiddle factor w and its width can be easily obtained from COsKθ and SinKθ, and the effect is great. In the above explanation, up to COs8θ and Sin8θ were obtained, but COsKθ and SinKθ (K is an integer greater than 8) can be obtained using exactly the same method.

[Brief explanation of drawings]

FIG. 1 is a diagram for explaining an algorithm for determining the twiddle factor w, and FIG. 2 is an explanatory diagram for obtaining COsKθ and SinKθ according to the present invention. MEM...Main memory, MC...Cosine storage area, MS...Sine storage area, Tl, T2
, T3, T4...Storage area, PLU1...
...Double angle calculation unit, PLU2...Pipeline processing device for shift, PLU3...Pipeline processing device for addition calculation.

Claims (1)

[Claims] 1 From coSθ, sinθ, cosKθ, sinKθ(
Storage means for storing cosine and sine values corresponding to each angle θ, 2θ, 3θ…nθ calculated in the calculation method of calculating K=2, 3,...n), based on the formula of cosθ, sinθ, and double angle. cos2^m'θ, sinn2^m'θ(
a calculation means for sequentially calculating m'=1, 2,...m where n=2^m) and storing it in the storage means, and storing the cosine and sine data sequentially inputted from a predetermined address of the storage means; A first pipeline processing means for shifting to another address of the means, performing an addition operation process using the cosine and sine data input from the storage means after the shift processing, and storing the operation result in the storage means. A second pipeline processing means is provided, and the storage area of the storage means is {
Storage area T_m_+_1 for storing cos2^mθ, sin2^mθ}; {cos2^m^-^1θ, sin2
^m^-^1θ}, {cos2mθ, sin2^mθ}
Storage area Tm; {cos＾m＾−＾2θ,s
in2^m^-^2θ}, {cos2^m^-^1θ・
sin2^m^-^1θ}, {cos(2^m^-^2
+2^m^-^1) θ, sin(2^m^-^2+2^
m^-^1) θ}, {cos2^m^θ, sin2^m
θ} storage area T_m_-_1; {cos2^
m^-^3θ, sin2^m^-^3θ}, {cos2
^m^-^2θ, sin2^m^-^2θ}, {cos
(2^m^-^3+2^m^-^2) θ, sin(2^
m^-^3+2^m^-^2)θ}, {cos2^m^
-^1θ, sin2^m^-^1θ}, {cos(2^
m^-^3+2^m^-^1) θ, sin(2^m^-
^3+2^m^-^1)θ}, {cos2^mθ, si
Storage area T_m_-_2 for storing n2^mθ}:...;{
cosθ, sinθ}, {cos2θ, sin2θ},
{cos3θ, cos4θ}…{cosnθ, sinn
θ} into a storage area T_1 for storing θ}, the first pipeline processing means sequentially shifts the contents of each address of the storage area Ti to an even numbered area of the storage area Ti+1, and the second
The pipeline processing means sequentially performs an addition operation process using the contents of the first address and the even-numbered addresses of the memory Ti, and executes the process of sequentially writing the operation results to the odd-numbered addresses of Ti, cos2^ by the arithmetic means
After calculating mθ, sin2^mθ (m=1, 2,...), T
The shift processing and addition operation processing from i to T_i_+_1 are alternately repeated and cosKθ, sinKθ(K
A cosine and sine calculation method characterized by finding the following equations (=2, 3,...n).