CN103065039A - High-precision sine/cosine function computing method based on Euler's formula - Google Patents

Abstract

The invention relates to a high-precision sine/cosine function computing method based on Euler's formula. The computing method utilizes Taylor's expansion of complex operation to obtain a high-precision sine/cosine function value and includes designing three submodules of a phase subdivision module, an initial value computation module and an iteration solution module; wherein the phase subdivision module divides a trigonometric function phase value to be computed into N (N is a positive integer) parts, the initial value computation module computes out a trigonometric function value as an iteration initial value after a phase subdivision, the iteration solution module plugs the iteration initial value into a computing equation to iterate, and the more the iterations, the higher the precision of a computed sine/cosine value; and lastly utilizing Euler's formula to transform a iteration result into a sine/cosine value of the corresponding phase to be computed.

Description

A kind of high precision sine/cosine function computing method based on Euler's formula

Technical field

The present invention relates to digital signal the field occurs, relate in particular to a kind of computing method of high precision sine/cosine function value.

Background technology

The Direct Digital frequency synthesis has now become the significant design method that signal occurs, and its major advantage is that output frequency, phase place and amplitude can accurately and rapidly conversion under the control of digital processing unit.Phase-Amplitude Converter is the important component part of Direct Digital Frequency Synthesizers, and its precision has directly determined precision and the purity of output sin/cos ripple, and therefore, the accurate Calculation of offset of sinusoidal/cosine value is the most important thing.

At present, the computing method of sine/cosine function numerical value mainly contain look-up table, method of interpolation and cordic algorithm.To phase place and frequency resolution and the exigent occasion of output accuracy, look-up table can consume a large amount of storage unit, and this has not only increased energy consumption, and has increased chip area.Cordic algorithm uses rotation of coordinate to ask for corresponding sine and cosine value, and it has solved the consumption problem of resource, and is highly suitable for that FPGA is upper to be realized, but cordic algorithm is in the situation of fixed number of iterations, and the precision of calculating changes with the variation of angle to be calculated.Therefore, cordic algorithm does not satisfy high-precision requirement in the larger occasion of frequency change.

The Direct Digital frequency synthesis is at high-end technology and military technology, and comparatively widely demand arranged in the communication technology, this is just to its atomic little frequency tuning and phase resolution, and " jump " ability between two frequencies, higher requirement has been proposed, thereby it is particularly important with the sin/cos computing method that the input angle has nothing to do to design a kind of computational accuracy.

Summary of the invention

The object of the invention is to propose a kind of computing method of the high precision sine/cosine function based on Euler's formula, realize the sin/cos computing method that computational accuracy and input angle are irrelevant.Described method can be in the hope of the high-precision sine/cosine of arbitrary phase between [0,2 π] according to the precise integration principle.

Technical scheme of the present invention is:

Described method is by segmenting phase place to be asked, and tries to achieve iterative initial value under the little phase place according to Euler's formula, and the corresponding times N of iteration (N is positive integer), the iteration result changed, and then obtain sine/cosine under the phase place described to be asked; Said method comprising the steps of:

Step 1, phase subdivision; Phase place η to be asked is subdivided into 2 ^NPart, the span of phase place η described to be asked is [0,2 π], the phase place τ after the segmentation ₀For:

τ ₀＝η/2 ^N (6)

Step 2, initial value calculates; Carry out the calculating of complex values for the phase place substitution Euler's formula after the segmentation, Taylor expansion has been adopted in the calculating of described complex values, can be launched in theory infinite multinomial, but when realizing, consider that high order power is very little to the contribution of complex values, participate in computing so get first five items:

$\cos {\mathrm{\τ}}_0+i\sin {\mathrm{\τ}}_0=e^{i{\mathrm{\τ}}_0}\≈1+i{\mathrm{\τ}}_0+\frac{{\left(i{\mathrm{\τ}}_0\right)}^2}{2}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^3}{6}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^4}{24}---\left(7\right)$

(7) in, order $T_0=i{\mathrm{\τ}}_0+\frac{{\left(i{\mathrm{\τ}}_0\right)}^2}{2}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^3}{6}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^4}{24},$

T ₀It is exactly described iterative initial value.

Step 3, iterative; Described iterations is the N that defines in the step 1, and iteration result carries out obtaining after the conversion sine/cosine of phase place η to be asked, according to (1) as can be known, and η=2 ^N* τ ₀, described conversion is carried out according to Euler's formula; Each iterative phase value that participates in all is last time to participate in 2 times of iterative phase value in the iterative process, and iterative formula is based on polynomial square of formula, i+1 (i=0,1,2,3 ... N-1) the phase place τ of inferior iteration _I+1Complex values and the i time iterative phase τ _iComplex values have the identical form of expression:

$\left\{\begin{array}{c}{e}^{i{\mathrm{\τ}}_i}=1+T_i\\ e^{i{\mathrm{\τ}}_{i+1}}=e^{i(2\×{\mathrm{\τ}}_i)}={(1+T_i)}^2=1+2T_i+T_i^2=1+T_{i+1}\end{array}\right.---\left(8\right)$

Described iterative formula is

T _i+1＝2T _i+T _i ² (9)

Gained sin/cos calculated value is

$\left\{\begin{array}{c}\cos \mathrm{\η}=\mathrm{img}\left(e^{\mathrm{i\η}}\right)=\mathrm{img}\left(e^{i{\mathrm{\τ}}_N}\right)=\mathrm{img}(1+T_N)\\ \sin \mathrm{\η}=\mathrm{real}\left(e^{\mathrm{i\η}}\right)=\mathrm{real}\left(e^{i{\mathrm{\τ}}_N}\right)=\mathrm{real}(1+T_N)\end{array}\right.---\left(10\right)$

Further, the computing method of described a kind of high precision sine/cosine function based on Euler's formula, for any one phase value, the error of calculation all is independently, does not have the cumulative errors between the phase value.

Further, the computing method of described a kind of high precision sine/cosine function based on Euler's formula, iterations is more, and the sine/cosine function value is more accurate.

Description of drawings

Fig. 1 is algorithm block diagram of the present invention;

Fig. 2 is the iterative computation process flow diagram;

Fig. 3 is the corresponding relation of iterations and precision.

Embodiment

Referring to Fig. 1, a kind of algorithm block diagram of computing method of the high precision sine/cosine function based on Euler's formula.

Said method comprising the steps of:

Step 1, phase subdivision; Phase place η to be asked is subdivided into 2 ^NPart, the span of phase place η described to be asked is [0,2 π], the phase place τ after the segmentation ₀For:

τ ₀＝η/2 ^N (11)

Step 2, initial value calculates; Carry out the calculating of complex values for the phase place substitution Euler's formula after the segmentation, Taylor expansion has been adopted in the calculating of described complex values, can be launched in theory infinite multinomial, but when realizing, consider that high order power is very little to the contribution of complex values, participate in computing so get first five items:

$\cos {\mathrm{\τ}}_0+i\sin {\mathrm{\τ}}_0=e^{i{\mathrm{\τ}}_0}\≈1+i{\mathrm{\τ}}_0+\frac{{\left(i{\mathrm{\τ}}_0\right)}^2}{2}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^3}{6}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^4}{24}---\left(12\right)$

(7) in, order $T_0=i{\mathrm{\τ}}_0+\frac{{\left(i{\mathrm{\τ}}_0\right)}^2}{2}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^3}{6}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^4}{24},$

T ₀It is exactly described iterative initial value.

Step 3, iterative; Described iterations is the N that defines in the step 1, and iteration result carries out obtaining after the conversion sine/cosine of phase place η to be asked, according to (1) as can be known, and η=2 ^N* τ ₀, described conversion is carried out according to Euler's formula; Each iterative phase value that participates in all is last time to participate in 2 times of iterative phase value in the iterative process, and iterative formula is based on polynomial square of formula, i+1 (i=0,1,2,3 ... N-1) the phase place τ of inferior iteration _I+1Complex values and the i time iterative phase τ _iComplex values have the identical form of expression:

$\left\{\begin{array}{c}{e}^{i{\mathrm{\τ}}_i}=1+T_i\\ e^{i{\mathrm{\τ}}_{i+1}}=e^{i(2\×{\mathrm{\τ}}_i)}={(1+T_i)}^2=1+2T_i+T_i^2=1+T_{i+1}\end{array}\right.---\left(13\right)$

Described iterative formula is

T _i+1＝2T _i+T _i ² (14)

The process flow diagram of iterative computation is referring to Fig. 2.

Gained sin/cos calculated value is

$\left\{\begin{array}{c}\cos \mathrm{\η}=\mathrm{img}\left(e^{\mathrm{i\η}}\right)=\mathrm{img}\left(e^{i{\mathrm{\τ}}_N}\right)=\mathrm{img}(1+T_N)\\ \sin \mathrm{\η}=\mathrm{real}\left(e^{\mathrm{i\η}}\right)=\mathrm{real}\left(e^{i{\mathrm{\τ}}_N}\right)=\mathrm{real}(1+T_N)\end{array}\right.---\left(15\right)$

The computing method of described a kind of high precision sine/cosine function based on Euler's formula, for any one phase value, the error of calculation all is independently, does not have the cumulative errors between the phase value.Iterations is more, and the sine/cosine function value is more accurate.Relation between iterations and the sine/cosine computational accuracy is referring to Fig. 3.

More than to the description of the present invention and embodiment thereof, be not limited to this, only be one of embodiments of the present invention shown in the accompanying drawing.In the situation that does not break away from the invention aim, without designing and the similar structure of this technical scheme or embodiment, all belong to protection domain of the present invention with creating.

Claims (2)

1. the computing method based on the high precision sine/cosine function of Euler's formula is characterized in that the method comprises the steps:

Step 1, phase subdivision; Phase place η to be asked is subdivided into 2 ^NPart, the span of phase place η described to be asked is [0,2 π], the phase place τ after the segmentation ₀For:

τ ₀＝η/2 ^N (1)

Step 2, initial value calculates; Carry out the calculating of complex values for the phase place substitution Euler's formula after the segmentation, Taylor expansion has been adopted in the calculating of described complex values, can be launched in theory infinite multinomial, but when realizing, consider that high order power is very little to the contribution of complex values, participate in computing so get first five items:

$\cos {\mathrm{\τ}}_0+i\sin {\mathrm{\τ}}_0=e^{i{\mathrm{\τ}}_0}\≈1+i{\mathrm{\τ}}_0+\frac{{\left(i{\mathrm{\τ}}_0\right)}^2}{2}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^3}{6}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^4}{24}---\left(2\right)$

(2) in, order $T_0=i{\mathrm{\τ}}_0+\frac{{\left(i{\mathrm{\τ}}_0\right)}^2}{2}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^3}{6}+\frac{{\left(i{\mathrm{\τ}}_0\right)}^4}{24},$ T ₀It is exactly described iterative initial value.

Step 3, iterative; Described iterations is the N that defines in the step 1, and iteration result carries out obtaining after the conversion sine/cosine of phase place η to be asked, according to (1) as can be known, and η=2 ^N* τ ₀, described conversion is carried out according to Euler's formula; Each iterative phase value that participates in all is last time to participate in 2 times of iterative phase value in the iterative process, and iterative formula is based on polynomial square of formula, i+1 (i=0,1,2,3 ... N-1) the phase place τ of inferior iteration _I+1Complex values and the i time iterative phase τ _iComplex values have the identical form of expression:

$\left\{\begin{array}{c}{e}^{i{\mathrm{\τ}}_i}=1+T_i\\ e^{i{\mathrm{\τ}}_{i+1}}=e^{i(2\×{\mathrm{\τ}}_i)}={(1+T_i)}^2=1+2T_i+T_i^2=1+T_{i+1}\end{array}\right.---\left(3\right)$

Described iterative formula is

T _i+1＝2T _i+T _i ² (4)

Gained sin/cos calculated value is

$\left\{\begin{array}{c}\cos \mathrm{\η}=\mathrm{img}\left(e^{\mathrm{i\η}}\right)=\mathrm{img}\left(e^{i{\mathrm{\τ}}_N}\right)=\mathrm{img}(1+T_N)\\ \sin \mathrm{\η}=\mathrm{real}\left(e^{\mathrm{i\η}}\right)=\mathrm{real}\left(e^{i{\mathrm{\τ}}_N}\right)=\mathrm{real}(1+T_N)\end{array}\right.---\left(5\right)$

2. according to the computing method of the described a kind of high precision sine/cosine function based on Euler's formula of claims 1, it is characterized in that the iterations of iteration submodule is more, the precision of the sine/cosine of gained is higher.

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