JPS62602B2 - - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention relates to a method for generating a stepped pseudo sine wave, and particularly to a step pseudo sine wave generation method that can remove any harmonics from harmonic distortion caused by quantization errors when quantizing step levels. Concerning a sine wave generation method.

Divide the time axis of one period of the sine wave equally by a certain natural number,
It is well known that a sine wave can be simulated by a staircase wave in which each section is replaced with a certain level. However, in the prior art, an error (quantization error) occurs between the desired value and the actual value of the step level, resulting in a high frequency, which has the disadvantage of requiring the use of a high-performance filter. Another disadvantage is that in order to substantially reduce the quantization error, the cost of the quantization device increases.

An object of the present invention is to provide a stepped pseudo sine wave generation method that does not generate harmonics even if there is a quantization error, thereby providing a sine wave with low distortion even without a high-performance filter. It is to provide.

According to one embodiment of the present invention, the harmonic component is made zero by setting the number of time axis divisions to be a multiple of the prime number of the order of the harmonic whose harmonic component is desired to be made zero.
The present invention will be explained in detail below. First, the present invention will be explained using mathematical formulas to facilitate understanding of the present invention. The principle of the present invention and the following mathematical formulas can be found in, for example, Electrical Testing Institute Technical Report, 1974, Vo19, No. 4,
Pages 164 to 172 are entitled "About synchronous rectification that is not affected by harmonics." FIG. 1 is a characteristic diagram showing a stepped pseudo sine wave. In the figure, the phase angle at the center of the i-th section is θi, and the phase angle at which the sine wave is sampled is απ/M(-1α
1), the staircase level Ei and the phase angle θi
is expressed by the following equation.

Ei=sin(θi−απ/M) (1) θi=(2i−1)π/M=2π/M(i−1)+π/
M(2) However, M is the number of divisions in one period.

When α=0, the staircase wave becomes an odd function and the calculation becomes easy, so consider α=0. When α≠0, only the phase changes, but the harmonic content does not change.
The Fourier series of a stepped pseudo sine wave in which the time axis of one period is divided into M equal parts is expressed by the following equation.

Here, if the staircase level of the i-th section is Ei, the n-th Fourier coefficient is Here, when Ei=sin θi (α=0), And Σcos {(n±1)θi}=cos{π/M(n±1)+(M-1)(n±1)π/M}・sin(n±1)π/sin ^(n±
1) _M π Therefore, when n±1≠RM (where R is a natural number), Σcos {(n±1)θi}=0 (5) When n±1=RM, Σcos{(n±1 )θi}=Σcos {2Rπ(i-1)+Rπ}=(-1) ^R M (6) (5), Substitute equation (6) into equation (4), and then apply the result to equation (3). By substituting, we get the following equation.

When n≠RM±1, An=bn=0 (7) When n=RM±1, An=bnM/nπsinπ/M (8) As is clear from equations (7) and (8) above, there is a quantization error. If not, the harmonics will be only when n=RM±1, that is, the (RM±1) order. Therefore, the strain rate is and is inversely proportional to the number of divisions. Therefore, it can be seen that this harmonic can be removed by increasing the number of divisions. Theoretically, this (RM±1) order harmonic cannot be completely removed at any time. 7th A
In the figure, R=1 and M=12, so n=11, 13
This shows that harmonics of .

However, the above discussion assumes an ideal state without quantization errors, and quantization errors occur when step levels are quantized. The quantization error here refers to the level given by equation (1) as the true value. When such a quantization error exists, harmonics other than the (RM±1)th order are generated. Figure 7B shows the 11th and 13th harmonics because there is a quantization error with respect to the theoretical values a, b, and c (it is extremely difficult to actually set such theoretical values in an actual circuit). In addition to this, 3rd, 5th, and 7th harmonics are also generated.

Next, we will discuss harmonics caused by the quantization error and the method for removing them according to the present invention. In general, a staircase wave is generated by making it possible to set a staircase level between 0° and 90°, and creating a waveform with a full cycle by folding back and reversing the polarity. Therefore, errors in the phase relationship as shown in FIG. 2 are most likely to occur. FIG. 2 is a characteristic diagram showing quantization errors. Here, it is assumed that there is an error of εi in absolute value in the i-th interval. That is, Ei in equation (1) is expressed by the following equation.

Ei=sin(θi−απ/M)+εi (9) Here, the harmonics due to Σi are equal to the harmonics due to the square wave of amplitude εi and include all harmonics. The amplitude of the fundamental wave also changes depending on εi, but when εi is small, the change is small, so harmonics due to εi can be considered. As for the harmonics, the even harmonics are canceled and only the odd harmonics are left, which is expressed by the following equation from equation (3). (However, α=0).

However, INT(M/4) represents only the integer part of M/4.

Also, if Di is the level after quantization (see Figure 7B), then from equation (3), For quantization, the amplitude is set to 1 (normalization), this is divided into L, L quantization steps are virtually set, and Ei is rounded to an appropriate quantization step. The quantization staircase is numbered 1, 2, ......j...L, counting from the bottom. The number (level) j of the quantization staircase in interval i is a function of i, Di=j(i)/L Therefore, Looking at equations (10) to (12) macroscopically, if the number L of quantization steps is increased and the quantization error is decreased, An decreases. However, it does not decrease monotonically with the size of L. The same can be said about M. Therefore, attention is paid to nθi. Now n is an odd number, so nθ
It can be inferred that An can be made zero by making i equal to the central phase angle θh of the h-th section.

Therefore, a method of concentrating nθi on θh will be considered. We have already set nθi to a specific angle θh
For example, in the case of n=3 in FIG. 3 (M=12), nθi=3θ ₁ , 3θ ₂ ,...
...Let's consider that 3θ ₈ etc. is represented by a certain angle.

If nθi=θh, nθi=(2i-1)nπ/M=θh=(2h-1)π
/M Therefore, h=ni-n-1/2 When h>M, h may be replaced with h' due to the periodicity of the phase angle.

Therefore h'=ni-n-1/2-M=n(i-M/n)-n-1
/2 = ni′−n−1/2 (13) Therefore, if h′ has i′ that satisfies equation (13),
nθi is concentrated at a specific θh. That is, if M/n is an integer from i'=i-M/n, sin nθi is concentrated at a specific sin θh for i=1 to M. If M/n is an integer, i is an integer, so i' is an integer such that i'<i, and nθi is concentrated at a specific θh. In this way, if M/n is not an integer, nθi is h=1 to M for i=1 to M.
Although it is distributed over the entire interval of M, if M/n is an integer, nθi is concentrated in a specific θh (plurality). Here, there are a plurality of θhs, but these θhs can be replaced by phase angles in the interval from 0° to 90° due to the characteristics of trigonometric functions. Therefore, using this property that M/n is an integer, we can set the number of divisions M on the time axis to 3,
By selecting a multiple of a prime number (a number multiplied by a prime number) such as 5, 7, etc., harmonics of these prime numbers and their multiples can be easily removed.

Next, a specific example will be shown in which an arbitrary harmonic is actually reduced to zero using the above method.

(b) Reduce the third harmonic to zero.

M may be 3 or a multiple thereof, but as mentioned above, normally a staircase wave is a method in which the staircase level can be set only between 0° and 90°, and a waveform of the entire period is created by folding and polarity reversal. The case where M=12, where one period can be divided into four, will be shown. FIG. 3 is a characteristic diagram showing the relationship of nθi when M=12. θi=(2i−1)π/
From M=(2i-1)π/12, θ ₁ = 15°, θ ₂
= 45°, θ ₅ = 135°, θ ₈ = 225°, θ ₁₂ =
It is 345°. Therefore, 3θ ₁ =θ ₂ , 3θ ₂ =
θ ₅ =π−θ ₂ , 3θ ₃ =θ ₈ =π+θ ₂ . FIG. 3 is a characteristic diagram showing the relationship of nθi. Therefore, from equation (12), That is, 3θi is concentrated at _θ2 . That is,
3θ〓 can be represented by _θ2 . Therefore, if j(1)+j(2)-j(3)=0 (14) is satisfied, the third harmonic becomes zero. α=
0 and mechanically rounding Di to the nearest quantization step is shown in FIG. Also the fifth
4 and 7C are diagrams showing quantized signals when M=12 and L=1 in FIG. 4. As is clear from FIG. 4, even in the extreme case of L=1, if equation (14) is satisfied, A ₃ =0. In the case of mechanical quantization when M=12 and L=15 shown in FIGS. 4 and 7B (7A, 7B),
(mechanical allocation to the quantization level closest to the theoretical values a, b, and c shown in Figure B) is not satisfied, and a third harmonic is generated due to a quantization error. In this case, it is easy to change the quantization to satisfy equation (14), for example, j(2)
set to 10, or set j(3) to 15. However, even with such operations, the 5th and 7th harmonics cannot be eliminated.

(b) Reduce the 3rd and 5th harmonics to zero.

As an example, consider the case where one period is divided into M=3×5=15. From θi=(2i-1)π/15,
θ ₁ = 12°, θ ₂ = 36°, θ ₃ = 60°, ......
θ ₁₄ = 324°, θ ₁₅ = 348°. Figure 8 shows sin3θ when the number of sections M = 15 and the number of divisions L = 15.
It is a characteristic diagram showing the values in each section of i, sin5θi, and j(i).

First, considering the third harmonic, sin3θi
can be represented by sinθ ₂ , sinθ ₅ , and sinθ ₈ (=0). Therefore, in equation (12), in order to set i=1 to 15 and A ₃ =0, j(1)−j(5)+j(6)=0 (15) j(2)−j( 4)+j(7)=0 (16) Similarly, considering the fifth harmonic, sin5θi
can be represented by sinθ ₃ and sinθ ₈ =(=0).
Therefore, from equation (12), in order for A ₅ = 0, j(1)-j(3)+j(4)-j(6)+j(7)=0 (17) Therefore, the above If formulas (15), (16), and (17) are satisfied, the third and fifth harmonics are not included. Let L=15 and determine the quantization level j(i) as shown in FIG. After mechanically assigning the quantization levels, it is sufficient to correct j(4) and j(12) to 15. As a result, the above equations (15), (16), and (17) are satisfied, and the third and fifth harmonics become zero. In addition, M=
Of course, 2×3×5=30 may also be used.

FIG. 6 is a block diagram of a specific apparatus in which the method according to the invention described above may be used. Reference numeral 1 denotes a digital quantization storage device, which stores digital quantities corresponding to sine wave amplitude values for each section obtained by dividing the time axis of the sine wave into M equal parts. As mentioned above, this stored value is normally a value in the 0° to 90° section of the sine wave. This stored value is converted into digital/analog (D/A) in a predetermined order by the control signal of the control circuit 9.
It is applied to converter 3 and converted into an analog value.
The output analog signal of the D/A converter is appropriately folded back and polarized in accordance with the control signal of the control circuit 9 by the polarity switching circuit 5 and applied to the filter 7 .
A staircase signal representing one period is therefore applied to the filter 7, and the staircase signal is applied to the filter 7.
A sinusoidal output signal is then smoothed and sent out. Here, according to the method according to the present invention, the number of divisions on the time axis in the digital quantization storage device 1 is a number multiplied by a prime factor of the harmonic order whose harmonic components are to be made zero, or a multiple of the number. selected to be,
A digital amount is stored according to the number of selections (number of divisions). For example, 2nd, 3rd, 4th, 5th, 7th, 8th,
If you want to make the components of 9 and 10 harmonics and their harmonics zero, these prime numbers are 2, 3, 5, and 7, so divide their multiples 2 x 3 x 5 x 7 = 210 or multiples of 210. number. At this time, the accuracy of the storage device (word length)
The required harmonic components will be zero even if the value is not increased.

As is clear from the above description, by using the method according to the present invention, it is possible to remove any harmonic components that would be caused by quantization errors when generating a stepped pseudo sine wave. In particular, since the number of sections is small and the quantization accuracy does not need to be low, both the number of words and the word length of the storage device can be small, which is economical.

[Brief explanation of the drawing]

Figure 1 is a characteristic diagram showing a stepped pseudo sine wave, Figure 2 is a characteristic diagram showing quantization error when generating a stepped pseudo sine wave, and Figure 3 is a characteristic diagram showing the number of time axis divisions M.
Figure 4 is a characteristic diagram of nθi when = 12, and Figure 5 is a diagram showing the content rate of each harmonic when the number of time axis divisions M = 12 and quantization is performed to remove the third harmonic. is a diagram showing a quantized signal in the case of L=1 in FIG. 4, FIG. 6 is a block diagram showing a specific example of a device to which the stepped pseudo sine wave generation method according to the present invention can be applied, and FIG. 7A is a diagram showing a quantized signal when L=1 in FIG. is the characteristic diagram of the ideal staircase wave, the seventh
Figures B and 7C are characteristic diagrams of quantized staircase waves generated by the present invention, Figures 7D, 7E, and 7
Figure F is a diagram showing harmonics generated in the cases of Figures 7A to 7C, and Figure 8 is a diagram showing the number of time axis divisions M.
= 12, sin3θi, sin5θ when the number of divisions L = 15
FIG. 3 is a diagram showing values of i and j(i). 1: Digital quantization storage device, 3: Digital
Analog converter, 5: polarity switch, 7: filter 9: control circuit.

Claims (1)

[Claims] 1. In a staircase pseudo sine wave generation method that simulates a sine wave with a staircase signal in which the time axis of one period of a sine wave is equally divided by a certain natural number to determine the number of sections, and each section is replaced with a certain quantization level, The number of divisions is set as a number multiplied by a prime number for the order of the desired plurality of harmonic components or a multiple of the number, and the combination of quantization levels of the staircase signal in a specific plurality of sections among the sections is determined. set to a specific value,
A step-shaped pseudo sine wave generation method in which the total sum is set to zero.