WO2019043814A1 - Trigonometric function approximation device, trigonometric function approximation method, and trigonometric function approximation program - Google Patents

Abstract

In the present invention, a main generation unit (11) generates, by using a phase signal (51), a first sawtooth wave signal (501) synchronized with an ideal function. An adjustment unit (121) generates an adjustment signal (604) by multiplying the phase signal (51) by a constant and by further adding a constant thereto. A sub-generation unit (122) generates, by using the adjustment signal (604), a second triangular wave signal (602) which has the same phase and frequency as an error correction signal (603) for offsetting an error function representing an error with respect to an approximate triangular wave signal generated from the first sawtooth wave signal (501). A second correction unit (123) generates the error correction signal (603) by adjusting a value range of the waveform of the second triangular wave signal (602). An addition unit (113) generates a corrected sawtooth wave signal (501a) by adding the first sawtooth wave signal (501) and the error correction signal (603). An approximate signal generation unit (114) generates an approximate signal (52) by using the corrected sawtooth wave signal (501a).

Description

Trigonometric function approximation apparatus, trigonometric function approximation method, and trigonometric function approximation program

The present invention relates to a trigonometric function approximation device, a trigonometric function approximation method, and a trigonometric function approximation program that approximates a trigonometric function such as a cosine function or a sine function.

Trigonometric functions are fundamental functions widely used for signal processing such as wave signal processing or image processing. Wave signal processing is used as an example in the field of radar. Image processing is used in the field of 3DCG (3-Dimensional Computer Graphics) as a specific example. Trigonometric functions are transcendental functions. Thus, trigonometric functions can not be calculated by a finite number of algebraic operations such as addition and multiplication. Therefore, when the calculation of the trigonometric function is realized by computer implementation, an approximate calculation method or a table reference method using a pre-calculated value stored in the memory is used.

A specific example of the approximation calculation method is polynomial approximation such as Taylor expansion. In Non-Patent Document 1, CORDIC (Coordinate) is used as an approximate calculation method suitable for circuit implementation.
A rotational digital computer is disclosed. CORDIC is widely used in devices such as calculators.
Non-Patent Document 2 discloses a simple trigonometric function approximation method. As an example of a simple trigonometric function approximation method, a method of performing approximation with only three operations of remainder operation, subtraction, and absolute value operation is disclosed.
Further, Patent Document 1 discloses a technique of approximately outputting a sine wave based on a parabola signal and an error between a parabola signal and a sine wave signal recorded in advance.

JP 2005-045674 A

The approximate calculation method disclosed in Non-Patent Document 1 requires iterative calculation for approximation. Therefore, the latency of the result output and the circuit scale become problems.
Further, in the trigonometric function approximation method disclosed in Non-Patent Document 2, low approximation accuracy is a problem.
Further, in the technology disclosed in Patent Document 1, the size of the memory amount for storing the error of the sine wave signal with respect to the parabola signal is a mounting problem.

An object of the present invention is to provide a trigonometric function approximation device which can be implemented with a small circuit area and a small amount of memory, and which has high accuracy.

A trigonometric function approximating apparatus according to the present invention uses an output of a trigonometric function to which a phase signal is input as an ideal function and generates an approximating signal approximating the ideal function.
A main generation unit including a first sawtooth wave generator that generates a first sawtooth wave signal that is synchronized with the ideal function using the phase signal;
Error correction generation that generates an error correction signal that is generated from the ideal function and the first sawtooth signal and that represents an error between the ideal triangular wave signal whose phase, frequency, and amplitude are the same as the ideal function Equipped with
The error correction generation unit
An adjustment unit configured to generate an adjustment signal obtained by multiplying the phase signal by a constant and adding a constant;
A sub-generation unit that generates a second triangular wave signal whose phase and frequency are the same triangular wave signal as the error correction signal using the adjustment signal;
And a second correction unit configured to adjust a value range of the waveform of the second triangular wave signal and generate the error correction signal.
The main generation unit further includes
An addition unit that adds the first sawtooth signal and the error correction signal to generate a corrected sawtooth signal;
The correction sawtooth signal is used to provide an approximate signal generation unit that generates, as the approximate signal, a triangular wave signal having the same phase, frequency, and amplitude as the ideal function.

The trigonometric function approximation apparatus according to the present invention generates a triangular wave signal tuned to an ideal function by sequentially converting a sawtooth wave signal to a triangular wave signal with respect to a phase signal as an input of the trigonometric function. Then, the trigonometric function approximating apparatus superimposes an error correction signal generated from the same phase signal on the generated triangular wave signal. Therefore, according to the trigonometric function approximating apparatus of the present invention, the trigonometric function can be accurately approximated by a small-scale calculation circuit or calculation program.

FIG. 2 is a functional configuration diagram of the trigonometric function approximation device according to the first embodiment. 6 shows an example of a hardware configuration of a trigonometric function approximation apparatus according to Embodiment 1. 7 shows another example of the hardware configuration of the trigonometric function approximation device according to the first embodiment. FIG. 6 is a flowchart of trigonometric function approximation processing according to the first embodiment. FIG. 6 is a diagram for explaining main generation processing according to the first embodiment. FIG. 6 is a diagram for explaining an error correction signal according to the first embodiment. FIG. 7 is a diagram for explaining an error correction generation process according to the first embodiment. FIG. 7 is a diagram for explaining an error correction generation process according to the first embodiment. FIG. 6 is a diagram for explaining trigonometric function approximation processing according to the first embodiment.

Hereinafter, embodiments of the present invention will be described with reference to the drawings. In the drawings, the same or corresponding parts are denoted by the same reference numerals. In the description of the embodiment, the description of the same or corresponding parts will be omitted or simplified as appropriate.

FIG. 1 shows an example of the functional configuration of the trigonometric function approximating apparatus 10 according to the present embodiment.
FIG. 2 shows an example of the hardware configuration of the trigonometric function approximating apparatus 10 according to the present embodiment.
FIG. 3 shows another example of the hardware configuration of the trigonometric function approximating apparatus 10 according to the present embodiment.

First, the outline of the trigonometric function approximating apparatus 10 will be described.
The trigonometric function approximating apparatus 10 according to the present embodiment receives the phase signal 51 input to the trigonometric function as an input, and outputs the output of the trigonometric function as the approximating signal 52. The phase signal 51 is an input signal to be input to the trigonometric function. That is, the trigonometric function approximating apparatus 10 generates an approximation signal 52 approximating the ideal function 61 by setting the output of the trigonometric function to which the phase signal 51 is input as the ideal function 61 (see FIG. 5).
The trigonometric function approximating apparatus 10 generates an approximate signal 52 by adding a simple arithmetic process to the phase signal 51. More specifically, the trigonometric function approximating apparatus 10 performs processing represented on the phase signal 51 by a combination of remainder operation, subtraction, multiplication, and absolute value operation. Thus, the trigonometric function approximating apparatus 10 can generate the approximate signal 52 that approximates the output of the trigonometric function with high accuracy.
The operations performed by the trigonometric function approximation device 10 correspond to a trigonometric function approximation method and a trigonometric function approximation program.

Embodiment 1
*** Description of the configuration ***
Next, a functional configuration example and a hardware configuration example of the trigonometric function approximating apparatus 10 will be described using FIGS. 1, 2 and 3.

As shown in FIG. 2, one of the hardware configuration examples of the trigonometric function approximating apparatus 10 is a computer. The trigonometric function approximating apparatus 10 in this configuration example includes hardware of a signal input / output interface 21, a processor 22, and a storage device 23. The processor 22 is connected to other hardware via signal lines to control these other hardware.

The signal input / output interface 21 connects an input / output device such as a signal generation source such as a sensor or a display device. As a specific example, the signal input / output interface 21 is a connection terminal of USB, HDMI (registered trademark), IEEE 1394, or NIC. USB is an abbreviation for Universal Serial Bus. HDMI (registered trademark) is an abbreviation of High-Definition Multimedia Interface. NIC is an abbreviation of Network Interface Card.

The processor 22 is an IC that performs processing. The processor 22 is, specifically, a CPU, a DSP, or a GPU. IC is an abbreviation for Integrated Circuit. CPU is an abbreviation of Central Processing Unit. DSP is an abbreviation of Digital Signal Processor. GPU is an abbreviation of Graphics Processing Unit.

The storage device 23 includes a memory 231 and a storage 232. A specific example of the memory 231 is a RAM. An example of the storage 232 is an HDD. The storage 232 may also be a portable storage medium such as an SD (registered trademark) memory card, a CF, a NAND flash, a flexible disk, an optical disk, a compact disk, a Blu-ray (registered trademark) disk, and a DVD. RAM is an abbreviation of Random Access Memory. HDD is an abbreviation of Hard Disk Drive. SD (registered trademark) Memory Card SD (registered trademark) is an abbreviation of Secure Digital. CF is an abbreviation of Compact Flash. DVD is Digital
Abbreviation for Versatile Disk.

Further, as shown in FIG. 3, one of the hardware configuration examples of the trigonometric function approximating apparatus 10 is a digital circuit. The trigonometric function approximating apparatus 10 in this configuration example is composed of a plurality of digital circuit blocks. All digital circuit blocks are connected by signal lines. The plurality of digital circuit blocks include bit extractors 701 and 708, adders 702 and 707, subtractors 703, 705, 709, 711, 712 and 714, multipliers 706, 713 and 715, and an exclusive OR 704, It is 710.

As shown in FIG. 1, the trigonometric function approximating apparatus 10 includes a main generation unit 11, an error correction generation unit 12, and a first correction unit 13 as functional components. The main generation unit 11 includes a first sawtooth wave generation unit 111, an addition unit 113, and a first aliasing wave generation unit 112. The error correction generation unit 12 includes an adjustment unit 121, a sub generation unit 122, and a second correction unit 123. The sub-generation unit 122 includes a second sawtooth wave generation unit 1221 and a second second aliasing wave generation unit 1222. These functions are realized by software or digital circuits.

When the trigonometric function approximating apparatus 10 is realized by software, the storage 232 stores a program for realizing the functions of the respective units of the trigonometric function approximating apparatus 10. This program is read by the processor 22 into the memory 231 and executed by the processor 22. Thereby, the function of each part of the trigonometric function approximating apparatus 10 is executed. The addition unit 113 realizes a function by an adder.
When the trigonometric function approximating apparatus 10 is realized by a digital circuit, the first sawtooth wave generator 111 is configured by a bit extractor 701 that extracts only the decimal part of the input signal. The second sawtooth wave generator 1221 is configured of a bit extractor 708. The addition unit 113 is configured of an adder 702. The first return wave generation unit 112 is composed of subtractors 703 and 705 and an exclusive OR 704. The second second aliasing wave generator 1222 is configured by subtractors 709 and 711 and an exclusive OR 710. Adjustment unit 121 is configured of multiplier 706 and adder 707. The first correction unit 13 includes a subtractor 714 and a multiplier 715. The second correction unit 123 includes a subtractor 712 and a multiplier 713. The multipliers 706, 713, and 715 may be implemented by bit shift because they perform multiplication of powers of two.

Each of the processor and digital circuitry is also referred to as processing circuitry. That is, the function of each part of the trigonometric function approximating apparatus 10 is realized by processing circuitry.

The “parts” of each part of the trigonometric function approximating apparatus 10 may be read as “process”. In addition, the “process” of each process described below may be read as a “program”, a “program product”, or a “computer readable storage medium storing a program”.

**** Explanation of operation ****
FIG. 4 is a flowchart of the trigonometric function approximation process S100 according to the present embodiment.
FIG. 5 is a diagram for explaining main generation processing according to the present embodiment.
FIG. 6 is a diagram for explaining an error correction signal according to the present embodiment.
FIG. 7 is a diagram for explaining an error correction generation process according to the present embodiment.
FIG. 8 is a diagram for explaining an error correction generation process according to the present embodiment.
FIG. 9 is a diagram for explaining trigonometric function approximation processing according to the present embodiment.
In the trigonometric function approximation device 10, the phase signal 51 is input to the main generation unit 11 and the error correction generation unit 12. Then, in the trigonometric function approximating apparatus 10, the approximate signal 52 is output from the first correction unit 13.

<Main generation process>
The main generation unit 11 executes sawtooth wave generation processing S101, addition processing S106, and return wave generation processing S107 on the phase signal 51, and outputs a first triangular wave signal 502.

<< 1st sawtooth wave generation processing >>
In step S101, the first sawtooth wave generation unit 111 of the main generation unit 11 uses the phase signal 51 to generate a first sawtooth wave signal 501 synchronized with the ideal function 61. Specifically, the first sawtooth wave generation unit 111 acquires the phase signal 51 and executes the first sawtooth wave generation process. The first sawtooth wave generation unit 111 forms, for the phase signal 51, a first sawtooth wave signal 501 tuned to a desired trigonometric function frequency. A desired trigonometric function is an ideal function 61. The ideal function 61 is an output of a trigonometric function to which the phase signal 51 is input. In FIG. 5, the ideal function 61 is represented by a dotted line, and the first sawtooth signal 501 is represented by a solid line.
Here, it demonstrates using an example. The phase signal 51 is a floating point or fixed point signal represented by f = 2πθ. The first sawtooth wave generation unit 111 generates the first sawtooth wave signal 501 by processing for extracting the decimal point part of θ. That is, the first sawtooth wave generation unit 111 generates a first sawtooth wave signal 501 having a peak at the timing when θ reaches 1.0 and having a value range of 0.0 to 1.0.
This first sawtooth wave generation process is realized by extracting a bit string corresponding to the decimal part of the phase signal 51 when the trigonometric function approximating apparatus 10 is implemented by a digital circuit and the phase signal 51 is implemented in fixed point. it can. That is, the first sawtooth wave generation process of the first sawtooth wave generation unit 111 is realized by the bit extractor 701. As described above, the phase signal 51, ie, the phase signal f, is expressed by the multiplication of 2π and θ. Therefore, the timing at which θ reaches 1.0 coincides with the timing at which the phase signal f reaches an integral multiple of 2π. Therefore, the first sawtooth signal 501 output from the first sawtooth wave generator 111 is tuned to a trigonometric function of a period 2π.

<< Addition processing >>
The addition unit 113 of the main generation unit 11 adds the first sawtooth signal 501 and the error correction signal 603 to generate a corrected sawtooth signal 501a. The addition process S106 will be described later.

<< Return wave generation processing >>
The first return wave generation unit 112 generates a first triangular wave signal 502 by performing return wave generation processing on the corrected sawtooth signal 501 a output from the addition unit 113. In FIG. 5, a state in which the error correction signal 603 is not added to the first sawtooth signal 501 will be described. That is, the case where the first return wave generation unit 112 performs the return wave generation processing on the first sawtooth signal 501 output from the first sawtooth generation unit 111 will be described. The first return wave generation process S107 for performing the return wave generation process on the corrected sawtooth signal 501a output from the adding unit 113 will be described later.

The return wave generation processing converts the first sawtooth wave signal 501 so as to be line symmetrical with respect to the signal of 0.5 or less on the signal waveform at the boundary of 0.5 as shown in FIG. It is a process.
Specifically, the first return wave generation unit 112 subtracts the value 0.5 from the first sawtooth signal 501 and further performs an absolute value operation to execute the return wave generation process.
When the trigonometric function approximating apparatus 10 is realized by a digital circuit, the subtraction of 0.5 is realized by the subtractor 703 in this return wave generation processing. Further, in the absolute value operation, a subtractor 705 having an output of an exclusive OR 704 having an output of the subtractor 703 and a sign bit as an input, a sign bit of an output of the subtractor 703 and an exclusive OR 704 as an input. Is realized by In the return wave generation process of the first return wave generation unit 112, the signal is returned at timing when θ reaches an integral multiple of 0.5. That is, the first triangular wave signal 502 which is the output of the first folded wave generation unit 112 is completely tuned to the ideal function 61 since it corresponds to folding at a timing when the phase signal f becomes an integral multiple of π. Further, the first triangular wave signal 502 which is the output of the first return wave generation unit 112 has a range of 0.0 to 0.5.

<< correction processing >>
The first correction unit 13 adds an offset to the first triangular wave signal 502 and multiplies it by a constant in order to adjust the value range of the first triangular wave signal 502 to -1.0 to 1.0 which is the same as the ideal function 61. . The first triangular wave signal 502, which is the output of the main generation unit 11, has a value range of 0.0 to 0.5, so here, −0.25 is added and 4 is multiplied. When the trigonometric function approximating apparatus 10 is implemented by a digital circuit, this operation is realized by the subtraction of 0.25 by the subtractor 714 and the multiplication of 0.5 by the multiplier 715. As described above, the first correction unit 13 adds the offset to the first triangular wave signal 502, and multiplies the first triangular wave signal 502 by a constant to generate the approximate triangular wave signal 61x of FIG. The approximate triangular wave signal 61 x in FIG. 6 is an approximate signal in a state where the error correction signal 603 is not added to the first sawtooth signal 501. The first correction processing S108 for outputting the approximation signal 52 when the error correction signal 603 is added to the first sawtooth signal 501 will be described later.

<Error correction signal generation processing>
As shown in FIG. 6, the error correction generation unit 12 generates an error correction signal 603 that cancels out an error function 63 representing an error between the ideal function 61 and the approximate triangular wave signal 61x not subjected to error correction. The approximate triangular wave signal 61 x is generated from the first sawtooth wave signal 501 and is a triangular wave signal having the same phase, frequency and amplitude as the ideal function 61. The approximate triangular wave signal 61x and the error function 63 will be described below using a specific example.

In FIG. 6, the dotted line is the ideal function 61. The thin dotted line is the output of the main generation unit 11 when the error correction signal 603 is not added, that is, the approximate triangular wave signal 61x not subjected to the error correction. The thin dashed-dotted line is an example of an error function 63 representing an error between the ideal function 61 and the approximate triangular wave signal 61x not subjected to error correction. Hereinafter, the error correction signal 603 generated by the error correction generation unit 12 will be described with reference to the example of FIG.

The output of the main generation unit 11 when the error correction signal 603 is not added, that is, the approximate triangular wave signal 61 x without error correction is f _a (θ), and the ideal function 61 is f _i (θ) = cos (2πθ) I assume. At this time, an error function ε (θ) between f _a (θ) and f _i (θ) is defined by ε (θ) = f _i (θ) −f _a (θ).

FIG. 6 shows an example where θ (x) = x for the sake of explanation. When {θ | 0 ≦ θ <0.5} is considered as a representative of a section in which the error function is continuous, f _a (θ) in the section is expressed by Equation 1 below.
(Formula 1) f _a (θ) = − 4θ + 1
Therefore, the error function ε (θ) is a sine function represented by Equation 2 below.
(Equation 2) ε (θ) = cos (2πθ) + 4θ-1

From Expression 2, θ satisfying θ (θ) = 0 in the same section is three points of 0, 0.25, and 0.5. Equation 2 takes an extreme value at θ shown in Equation 3 and Equation 4 below. And in the relevant range, when n = 0, that is, the extreme value is taken at θ ≒ 0.11, 0.39, and ε (θ) | _{θ = 0.11, 0.39} ≒ ± 0.21. . Therefore, the error function ε (θ) is a function having a value range of approximately −0.21 to 0.21 in the same section and having a waveform similar to a sine wave. The error function ε (θ) is phase-shifted by π / 2 with respect to the ideal function 61, that is, a function having a waveform similar to a cosine function with respect to the sine function, and has a double frequency. Further, as shown in FIG. 6, in the section adjacent to the same section (for example, {θ | 0.5 ≦ θ <1.0}), the waveform is symmetrical with respect to the connection point. This characteristic is the same in other sections. Therefore, the error correction generator 12 generates an error correction signal 603 that cancels out the error function of Equation 2 having this characteristic.

(Expression 3) θ = (2nπ + π-sin ^-1 (2 / π)) / 2π, n∈Z
(Expression 4) θ = (2nπ + sin ⁻¹ (2 / π)) / 2π, n∈Z

Next, an error correction generation process by the error correction generation unit 12 will be described with reference to FIGS. 7 and 8.

<< Adjustment process >>
The adjustment unit 121 of the error correction generation unit 12 generates an adjustment signal 604 obtained by multiplying the phase signal 51 by a constant and adding a constant. The adjustment unit 121 performs adjustment processing for realizing phase and frequency modulation of the output of the subsequent stage by performing constant multiplication and constant addition on the phase signal 51.
In step S102, the adjustment unit 121 doubles the phase signal 51 as shown in FIG. 7 and further adds 0.25 to generate an adjustment signal 604. The adjustment processing by the adjustment unit 121 is realized by multiplication of constant 2 using the multiplier 706 and addition of 0.25 using the adder 707 when the trigonometric function approximation device 10 is realized by a digital circuit. .

<< Secondary processing >>
Next, using the adjustment signal 604, the sub-generation unit 122 generates a second triangular wave signal 602 that is a triangular wave signal whose phase and frequency are the same as those of the error correction signal 603. The configuration and function of the sub-generation unit 122 are the same as those of the main generation unit 11. The second sawtooth wave generation unit 1221 and the second return wave generation unit 1222 have the same configuration and function as the first sawtooth wave generation unit 111 and the first return wave generation unit 112 in the main generation unit 11.

<<<< 2nd sawtooth wave generation process >>>
In step S103, the second sawtooth wave generation unit 1221 of the sub-generation unit 122 obtains the adjustment signal 604 generated by the adjustment unit 121. As shown in FIG. 7, the second sawtooth wave generation unit 1221 uses the adjustment signal 604 to shift the phase from the phase of the first sawtooth wave signal 501, and the frequency is 2 of the first sawtooth wave signal 501. A second sawtooth signal 601 is generated which is doubled. The second sawtooth wave generation unit 1221 generates a second sawtooth wave signal 601 by extracting only the decimal part of the adjustment signal 604. Similar to the main generation unit 11, the sub-generation unit 122 receives the phase signal doubled by the adjustment unit 121 in order to output the triangular wave signal folded back at the timing when the input signal reaches 0.5. . Thus, the sub-generation unit 122 can obtain an output having a cycle that is twice that of the output of the main generation unit 11. Further, the sub-generation unit 122 can obtain an output that is phase-shifted by 1⁄4 cycle by inputting a phase signal to which 0.25 corresponding to 1⁄4 of the amplitude is added. The second sawtooth wave generation process is similar to the sawtooth wave generation process described above.

<<< Second return wave generation process >>>
In step S104, as shown in FIG. 8, the second return wave generation unit 1222 of the sub-generation unit 122 outputs the second triangular wave signal 602. The second folded wave generation unit 1222 uses the second sawtooth signal 601 and has a frequency equal to that of the second sawtooth signal 601 and having an amplitude that is a half of that of the second sawtooth signal 601. A triangular wave signal 602 is generated. Specifically, the second folded wave generation unit 1222 subtracts 0.5 from the second sawtooth signal 601 and performs absolute value calculation to generate the second triangular wave signal 602. The second return wave generation process is similar to the above-described return wave generation process.
The operation of the sub-generation unit 122 when the trigonometric function approximation device 10 is realized by a digital circuit is similar to the operation of the main generation unit 11.

<< Second correction process >>
In step S105, the second correction unit 123 adjusts the value range of the waveform of the second triangular wave signal 602, and executes a second correction process that is generated as an error correction signal 603. The second correction unit 123 adjusts the range of the waveform by performing multiplication and subtraction on the second triangular wave signal 602. Specifically, the second correction unit 123 aligns the second triangular wave signal 602, which has a value range of 0.0 to 0.5, with the value range of the error function ε (θ). The second correction unit 123 adds an offset and multiplies a constant to adjust the second triangular wave signal 602 to the value range of the error function ε (θ). The second correction unit 123 generates the second triangular wave signal 602 as the error correction signal 603 by adding the offset and multiplying the constant.
Specifically, the second correction unit 123 can make the second triangular wave signal 602 a signal with a range of −0.0625 to 0.0625 by subtracting 0.25 and multiplying by 0.25. In this case, although the value does not exactly match the range of the error function ε (θ), bit shift operation can be performed by performing processing using a power of two, and the operation cost can be suppressed. When the trigonometric function approximating apparatus 10 is implemented by a digital circuit, the second correction processing is realized by the subtraction of 0.25 by the subtractor 712 and the multiplication of 0.25 by the multiplier 713.
The error function ε (θ) is expressed by the difference between the ideal function in the range of −1.0 to 1.0 and the approximate signal 52 without correction fitted by the first correction unit 13. On the other hand, before the error correction signal 603 passes through the first correction unit 13, that is, when generating the first triangular wave signal 502 whose range is 0 to 0.5 (the width is a quarter of the ideal function), It is added to the first sawtooth signal. Therefore, the width of the value range of the error correction signal 603 needs to be 1⁄4 of 0.5. Specifically, the second correction unit 123 causes the subtractor 712 to set the value range 0 to 0.5 of the second triangular wave signal 602 to −0.25 to 0.25. Then, the second correction unit 123 sets the value range to 1⁄4 by the multiplier 713, and sets the value range to −0.0625 to 0.0625.

Next, the addition process of step S106 and the first return wave generation process of step S107 will be described using FIG.

<< Add process (step S106) >>
In step S106, the addition unit 113 adds the first sawtooth signal 501 and the error correction signal 603, and performs addition processing to generate a corrected sawtooth signal 501a. As shown in FIG. 9, the error correction signal 603 has a phase which is about one-fourth of the ideal function 61, a frequency of which is about twice that of the ideal function 61, and an amplitude of the ideal function 61. It is about a quarter.

<Approximate signal generation processing>
The first return wave generation unit 112 and the first correction unit 13 are referred to as an approximation signal generation unit 114. The approximation signal generation unit 114 generates a triangular wave signal having the same phase, frequency, and amplitude as the ideal function 61 as the approximation signal 52 using the corrected sawtooth signal 501 a by the first return wave generation processing and the first correction processing. .

<< First Returning Wave Generation Process (Step S107) >>
In step S107, the first return wave generation unit 112 performs the above-described return wave generation processing on the corrected sawtooth signal 501a output from the addition unit 113. The first return wave generation unit 112 generates the first triangular wave signal 502 of FIG. 9 by performing the return wave generation processing on the corrected sawtooth signal 501 a.

<First correction processing>
In step S108, the first correction unit 13 adds an offset to the first triangular wave signal 502 in order to adjust the value range of the first triangular wave signal 502 to -1.0 to 1.0 which is the same as the ideal function 61, Multiply a constant. The first triangular wave signal 502 which is the output of the main generation unit 11 has a value range of 0.0 to 0.5. Therefore, the first correction unit 13 adds −0.25 to the first triangular wave signal 502 and multiplies it by 4. When the trigonometric function approximating apparatus 10 is implemented by a digital circuit, this operation is realized by the subtraction of 0.25 by the subtractor 714 and the multiplication of 0.5 by the multiplier 715. Thus, the first correction unit 13 adds the offset to the first triangular wave signal 502 in FIG. 9 and multiplies the first triangular wave signal 502 in FIG. 9 by a constant to generate the approximate signal 52 in FIG.

Note that in the range of 0.5 ≦ θ <1.0 in FIG. 8 and the range of the latter half of one cycle after 0.5 ≦ θ <1.0, the error correction function 603 with respect to the error function ε (θ) Are out of phase. Therefore, it seems that the error is amplified in the above range. However, in the present embodiment, the error correction signal 603 is added to the first sawtooth signal so as not to amplify the error even in the above range. The error function is in antiphase every half cycle because of the aliasing process by the first aliasing wave generator. Therefore, the error correction signal is added to the first sawtooth signal before the first folded wave generation unit, and is then folded back together with the first sawtooth signal, so that the error correction signal and the first sawtooth signal are simultaneously in antiphase. And

**** Explanation of the effect of the present embodiment ****
As described above, in the trigonometric function approximating apparatus 10 according to the present embodiment, the main generation unit sequentially generates the sawtooth signal and the triangular wave signal with respect to the input phase signal. Then, the main generation unit generates a triangular wave signal tuned to a desired trigonometric function. The sub-generation unit receives the same phase signal as input, performs multiplication and addition, and generates an adjustment signal. And a sub production | generation part produces | generates a sawtooth wave signal and a triangular wave signal sequentially similarly to an adjustment signal. Then, the sub-generation unit generates an error correction signal in which the value range of the triangular wave signal is adjusted. Then, the main generation unit adds the error correction signal to the sawtooth signal in the process of generating the triangular wave signal. Therefore, according to the trigonometric function approximating apparatus 10 according to the present embodiment, it is possible to output an approximation signal of a trigonometric function which is tuned to a desired trigonometric function and whose error is more similar.

Further, in the trigonometric function approximating apparatus 10 according to the present embodiment, the approximate triangular wave signal is obtained by the main generating unit adding simple arithmetic processing to the phase signal. Further, the error correction generation unit is configured by the adjustment unit for the same phase signal input, the sub generation unit having the same configuration as the main generation unit, and the waveform correction unit. In the trigonometric function approximating apparatus 10, the adding unit adds the output signal of the error correction generating unit to the first sawtooth signal. Thereby, in the trigonometric function approximating apparatus 10, the error with respect to the desired trigonometric function of the main generation unit is reduced, and the approximation accuracy is improved. Further, in the trigonometric function approximating apparatus 10, in the implementation of a trigonometric function (cosine function, sine function) having a high computational load, a highly accurate approximate output can be obtained with a small computational load.

As described above, in the trigonometric function approximating apparatus 10 according to the present embodiment, it is possible to obtain, from the phase signal, a signal output that cancels out the error signal. Therefore, according to the trigonometric function approximating apparatus 10 according to the present embodiment, the calculation load involved in the error correction can be reduced, and the circuit scale can be reduced. In addition, in the trigonometric function approximating apparatus 10 according to the present embodiment, depending on the implementation, the main generation unit can be used recursively, and the additional circuit involved in the error correction can be made smaller.

**** Other configurations ****
In the trigonometric function approximating apparatus according to the present embodiment, a first correction unit is disposed downstream of the main generation unit. Then, the first correction unit outputs an approximation signal having a value range of −1.0 to 1.0 and similar to the ideal function. However, since the output waveform of the main generation unit and the approximate shape of the output waveform of the first correction unit are similar to each other, the output of the main generation unit may be used as an approximation signal when an absolute value as a trigonometric function is not required. As a specific example, as in the case of obtaining the phase distribution of complex amplitude distribution generated by a plurality of wave sources in wave simulation, it is effective when relative values between values derived using a trigonometric function are required.

In the present embodiment, the value range of the ideal function has been described as −1.0 to 1.0. However, depending on the implementation system, a value range may use a function normalized to a positive number, such as 0.0 to 1.0. In the field of using functions normalized to positive numbers, there is, for example, attitude control of robotics. Moreover, in the case of the problem of finding the phase of the wavefront formed by a plurality of light sources in optical simulation, the range of the trigonometric function is not important because the phase is calculated from the relative values of the real part and imaginary part of the wavefront represented by complex numbers. .

In the first embodiment, each unit of the trigonometric function approximating apparatus has been described as an independent functional block. However, the configuration of the trigonometric function approximation device may not be the configuration as in the above-described embodiment. The functional block of the trigonometric function approximating device may have any configuration as long as the function described in the above-described embodiment can be realized.

A plurality of parts in the first embodiment may be combined and implemented. Alternatively, one part of the first embodiment may be implemented. In addition, Embodiment 1 may be implemented in any combination in whole or in part.
The embodiments described above are essentially preferable examples, and are not intended to limit the scope of the present invention, the scope of the application of the present invention, and the scope of the application of the present invention. The embodiment described above can be variously modified as needed.

DESCRIPTION OF SYMBOLS 10 Trigonometric function approximation apparatus, 11 Main production | generation part, 111 1st sawtooth wave production | generation part, 1221 2nd sawtooth wave production | generation part, 112 1st return wave production | generation part, 1222 2nd return wave production | generation part, 113 Addition part, 114 approximation signal Generation unit, 12 error correction generation unit, 121 adjustment unit, 122 sub-generation unit, 13 first correction unit, 123 second correction unit, 21 signal input / output interface, 22 processor, 23 storage device, 231 memory, 232 storage, 701 Bit extractor, 702 adder, 703 subtractor, 704 exclusive OR, 705 subtractor, 706 multiplier, 707 adder, 708 bit extractor, 709 subtractor, 710 exclusive OR, 711 subtractor, 712 Subtractor, 713 multiplier, 714 subtractor, 715 multiplier, 51 phase signal, 5 Approximate signal, 61x approximate triangular wave signal, 501 first sawtooth signal, 601 second sawtooth signal, 501a corrected sawtooth signal, 502 first triangular wave signal, 602 second triangular wave signal, 603 error correction signal, 604 adjustment signal, 61 Ideal function, 63 error function.

Claims (8)

1. In a trigonometric function approximating apparatus which uses an output of a trigonometric function to which a phase signal is input as an ideal function and generates an approximation signal approximating the ideal function,
   A main generation unit including a first sawtooth wave generator that generates a first sawtooth wave signal that is synchronized with the ideal function using the phase signal;
   Error correction generation that generates an error correction signal that is generated from the ideal function and the first sawtooth signal and that represents an error between the ideal triangular wave signal whose phase, frequency, and amplitude are the same as the ideal function Equipped with
   The error correction generation unit
   An adjustment unit configured to generate an adjustment signal obtained by multiplying the phase signal by a constant and adding a constant;
   A sub-generation unit that generates a second triangular wave signal whose phase and frequency are the same triangular wave signal as the error correction signal using the adjustment signal;
   And a second correction unit configured to adjust a value range of the waveform of the second triangular wave signal and generate the error correction signal.
   The main generation unit further includes
   An addition unit that adds the first sawtooth signal and the error correction signal to generate a corrected sawtooth signal;
   An approximation signal generator configured to generate, as the approximation signal, a triangular wave signal having the same phase, frequency, and amplitude as the ideal function, using the correction sawtooth signal.
2. The sub-generation unit is
   Second sawtooth wave generation using the adjustment signal to generate a second sawtooth wave that is out of phase with the phase of the first sawtooth wave and whose frequency is twice that of the first sawtooth wave Department,
   A second folded wave that generates the second triangular wave signal having a frequency equal to that of the second sawtooth signal and having an amplitude that is half that of the second sawtooth signal using the second sawtooth signal The trigonometric function approximation device according to claim 1, comprising: a generation unit.
3. The error correction signal is
   The trigonometric function approximation according to claim 2, wherein the phase is quartered from the ideal function, the frequency is twice the ideal function, and the amplitude is a quarter of the ideal function. apparatus.
4. The second sawtooth wave generator is
   The trigonometric function approximation device according to claim 2 or 3, wherein the second sawtooth signal is generated by extracting a decimal part of the adjustment signal.
5. The second return wave generation unit
   The trigonometric function approximation device according to any one of claims 2 to 4, wherein the second triangular wave signal is generated by subtracting 0.5 from the second sawtooth signal and performing an absolute value operation.
6. The second correction unit is
   The trigonometric function approximation device according to any one of claims 2 to 5, wherein a value range of a waveform is adjusted by performing multiplication and subtraction on the second triangular wave signal.
7. In a trigonometric function approximation method of a trigonometric function approximating apparatus, wherein an output of a trigonometric function to which a phase signal is input is an ideal function, and an approximation signal approximating the ideal function is generated;
   A main generation unit generates a first sawtooth signal in phase with the ideal function using the phase signal;
   An adjusting unit generates an adjusting signal obtained by multiplying the phase signal by a constant and adding a constant;
   An error in which a sub-generation unit generates a phase and a frequency from the first sawtooth signal using the adjustment signal, and an error from the approximate triangular wave signal having the same phase, frequency, and amplitude as the ideal function. Generate a second triangular wave signal that is the same triangular wave signal as the error correction signal that cancels out the function,
   A second correction unit adjusts a value range of the waveform of the second triangular wave signal to generate the error correction signal;
   An adder adds the first sawtooth signal and the error correction signal to generate a corrected sawtooth signal,
   An approximation signal generation unit that generates, using the corrected sawtooth signal, a triangular wave signal having the same phase, frequency, and amplitude as the ideal function as the approximation signal.
8. In a trigonometric function approximation program of a trigonometric function approximating device, wherein an output of a trigonometric function to which a phase signal is input is an ideal function, and an approximation signal approximating the ideal function is generated,
   A first sawtooth wave generation process for generating a first sawtooth wave signal synchronized with the ideal function using the phase signal;
   An adjustment process for generating an adjustment signal obtained by multiplying the phase signal by a constant and adding a constant;
   An error that uses the adjustment signal to generate a phase and frequency from the first sawtooth signal and cancels an error function that represents an error from the approximate triangular wave signal whose phase, frequency, and amplitude are the same as the ideal function. A sub-generation process for generating a second triangular wave signal that is the same triangular wave signal as the correction signal;
   A second correction process of adjusting the value range of the waveform of the second triangular wave signal to generate the error correction signal;
   Addition processing for adding the first sawtooth signal and the error correction signal to generate a corrected sawtooth signal;
   Trigonometric function approximation that causes the trigonometric function approximation device, which is a computer, to execute approximation signal generation processing that generates, as the approximation signal, a triangular wave signal whose phase, frequency, and amplitude are the same as the ideal function using the corrected sawtooth signal program.

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