JPH0630439B2 - Zero crossing point detector for sinusoidal signal - Google Patents

Description

Description: TECHNICAL FIELD The present invention relates to a sine wave zero-crossing point detection device capable of detecting a zero-crossing point of a sine wave with high accuracy from a sample value even if the sampling rate is long.

[Conventional technology]

Recent electronics technology is moving toward digitalization.

An apparatus for determining the distance to an object by measuring the beat signal period of the frequency-modulated continuous microwave transmission wave and the reflected wave from the object is disclosed in, for example, Japanese Patent Laid-Open No. 52-27.
It is disclosed in, for example, Japanese Patent Publication No. 395. Here, the beat signal is a sine wave, and it is necessary to detect the zero crossing point of the sine wave in order to obtain this cycle.

In this type of technique, the key is whether or not the period or zero-crossing point can be measured with high accuracy by lowering the sampling rate, that is, lengthening the sampling period only when the period of the beat signal is digitally measured.

Conventionally, it is generally performed to obtain a zero-crossing point by linearly approximating a sine wave from sample values before and after the zero-crossing point.

[Problems to be Solved by the Invention]

However, in this method, the range that can be approximated to a straight line is narrow, and in order to perform highly accurate detection, the sampling period must be shorter than the sine wave period.

When the sine wave sample data is taken, the data are discrete, and thus the zero-crossing points often exist between the sampling points.

In the conventional sine wave approximation, as shown in FIG. 1, two sample data (x ₀ , before and after the zero crossing point sampled at the period T, where x is the time axis and y is the sample value axis).
From y ₀ ), (x ₁ , y ₁ ), a linear approximation is performed between them, and it is assumed that the zero-crossing point is located from the point x ₀ to XT on the x-axis. XT / (− y ₀ ) = (T−XT) / y ₁ (1) X = (− y ₀ ) / (y ₁ −y ₀ ) ... (2) is calculated as X.

This method is easy to calculate, and high-accuracy calculation results can be obtained in the range where the approximation of sin x = x holds, but the range is narrow (0.35ra in the approximation of 2% accuracy).
d). Therefore, in order to obtain the zero crossing point with high accuracy, the sampling rate must be increased. That is, when detecting a high-precision zero-crossing point by the conventional first-order approximation method, the sampling rate must be increased, and when sampling with the A / D converter, a fast conversion processing time is required. It had to be used, and the device price had to be very expensive.

Therefore, an object of the present invention is to overcome the drawbacks of the prior art and to provide a zero-crossing point detection device for a sine wave signal which has a wide approximation range and is capable of highly accurate zero-crossing point detection even if the sampling period is lengthened. .

[Means for Solving the Problems]

In order to achieve the above object, the present invention spans before and after a zero crossing with a sampling period T from a sine wave signal on xy rectangular coordinates represented by y = Asin ωx, where A is an amplitude and ω is an angular frequency. three successive sample values _{y 0, y 1,} and sampling means for sampling the _{y 2,} the three sample values _{y 0, y 1,} three coordinates of the _{y 2 (x 0, y 0} ), (x 1 , Y ₁ ), (x ₂ , y ₂ ), and an increment on the x coordinate axis from the x ₀ point to the zero crossing point of the sine wave signal as XT, and x ₁ = (1-X ) T, x ₂ = (2-
X) T, y ₁ −y ₀ = ΔY, (y ₁ −y ₀ ) − (y ₂ −
y ₁ ) = Δ ² Y, X ′ = − y ₀ / {ΔY + Δ ² Y) / 6}, and a second calculation means for calculating X = X ′ [1 + X ′ (2Δ ² Y / 6) / {ΔT + Δ
² Y) / 6}] and a third arithmetic means for obtaining the zero crossing point detecting device for a sine wave signal.

[Action]

In the present invention, a sine wave is approximated by a cubic expression, and a zero-crossing point of a sine-wave signal is detected by solving a simultaneous equation with an amplitude period and a zero-crossing point as unknowns from three sample values. Wide range of accuracy, therefore
Even if the sampling rate is low, that is, the sampling period is long, the detection accuracy can be guaranteed, and therefore, it becomes possible to perform processing using an inexpensive low-speed A / D converter.

〔Example〕

 First, the principle of the present invention will be described.

In general, when the Taylor expansion of ^{sin x, sin x = x-} x 3/3! + ^X 5/5! -X ^7/7! +
………… It is expressed by (3).

Now, the sine wave signal y = Asin .omega.x (3) taking those Taylor expansion up to the third order terms according to equation, y = A (ωx-ω 3 x 3/6) is denoted by ... (4).

In terms of accuracy, the higher the order, the better. However, in practice, the calculation of the solution becomes complicated when the order becomes high, so
In consideration of the fact that the sampling rate is increased, the 3rd order is the limit at present, and it is considered appropriate.

In the equation (4), three sample values (x ₀ , y ₀ ), (x ₁ ,
_{y 1), (x 2,} y 2) by _{substituting, y 0 = A (ωx 0} -ω 3 x 0 3/6) ...... (5) y 1 = A (ωx 1 -ω 3 x 1 3 _{/ 6) ...... (6) y} 2 = a (ωx 2 -ω 3 x 2 3/6) ...... (7) thereof (5) to (7) the amplitude a and angular frequency ω by solving equation as simultaneous equations Is deleted, y ₂ x ₀ x ₁ (x ₀ ² −x ₁ ² ) + y ₀ x ₁ x ₂ (x _{1
^{2 -x 2 2) + y 1}} x 2 x 0 (x 2 2 -x 0 2) = 0 ...
(7 ') As shown in FIG. 1, the zero-crossing point is taken as the origin from x ₀ to XT, and x ₀ , x ₁ , x ₂ , y ₀ , y ₁ ,
Replace y ₂ with the following numbers:

_{x 0 = -XT ... (8)} x 1 = (1-X) T ... (9) x 2 = (2-X) T ... (10) y 1 -y 0 = ΔY ... (11) (y 1 - y ₀ ) − (y ₂ −y ₁ ) = Δ ² Y (12) Here, the expressions (8) to (12) are substituted into the expression (7 ′), and x
Erase ₀ , x ₁ , x ₂ . Moreover, in general, (a ² −b ² )
Since = (a−b) (a + b), the formula (7 ′) can be modified as follows.

_{y 2 (-X) T (1} -X) T (-T) (1-2X) T +
y ₀ (1-X) T (2-X) T (-T) (3-2X) T
_{+ Y 1 (-X) T (} 2-X) T (2T) {2 (1-X)
T} = 0 Here, (1-X) T ⁴ is common to each term, and ≠ 0. Therefore, if this is deleted, y ₂ X (1-2X) -y ₀ (2-X) ( 3-2X) -4
To summarize this _{y 1 X (2X) = 0} , y 1 (4X 2 -8X) -y 0 (2X 2 -7X + 6) -y
₂ ^(2X ₂ -X) = 0 by transforming ^{_{it, 2y 1 (2X 2 -X)}} -y 0 (2X 2 -X) -y 2 (2
^{_{X 2 -X) 6y 1 X-}} 6y 0 (x + 1) = 0 , where, _y 1 using (11) and (12) below, y
Clearing the ^{_{2, Δ 2 Y (2X 2}} -X) -6ΔYX-6y 0 = 0 ^{i.e., 2Δ 2 YX 2 - (6ΔY} + Δ 2 Y) X-6y 0 = 0 ...
(13) can be obtained.

Solving for X in equation (13), X = [√ {(6ΔY + Δ ² Y) ² + 48y ₀ Δ ² Y} +
(6ΔY + Δ ² Y)] / 4Δ ² Y (14) From this equation (14), the zero-crossing point can be known. However, this equation has a square root and the calculation is complicated, and the denominator is (12 As is clear from the equation), it is a difference in difference, and an error is likely to occur in practical use.

Therefore, the equation (13) is modified as follows.

X = −y ₀ / {ΔY + (Δ ² Y) / 6} + X ² {2 (Δ
² Y) / 6} / {ΔY + (Δ ² Y) / 6} ... (15) where X ′ = − y ₀ / {ΔY + (Δ ² Y) / 6} ... (16) If the second term on the right side of the equation) is sufficiently smaller than the first term, a large error does not occur even if X in the second term is set to X≈X '.

Therefore, X≈X ′ [1 + X ′ {2 (Δ ² Y) / 6} / {ΔY +
X can be approximated as (Δ ² Y) / 6}] (17).

Equation (17) has two correction terms for linear approximation, one is (Δ ² Y) / 6 in equation (15), and the other is (17 equation).
Since the second term X '{2 (Δ ² Y) / 6} / {ΔY + (Δ ² Y) / 6} in parenthesis [] on the right side is added, an error is less likely to occur in the process of calculation, Further, there is a feature that the calculation is simplified as compared with the formula (14).

Here, further explanation is added.

The present invention uses the sine wave signal y = Asin which is the target of zero crossing detection.
Three consecutive sample values y ₀ , y ₁ , y ₂ sampled before and after the zero crossing point of ωx, that is, a sampling point such that y ₀ or y ₂ has a different sign from the other two sample values. Coordinates (x ₀ , y ₀ ),
Are approximated by ^{y = A (ωx-ω 3} x 3/6) ( see Equation _{4): (x 1, y 1} ), (x 2, y 2) a third order approximation equation of each sinusoidal signal.

Therefore, in order to ensure accuracy, it is assumed that three consecutive sample values sampled before and after the zero crossing point are used as described above. Although FIG. 1 shows a case where there is a zero crossing point between the coordinates (x ₀ , y ₀ ) and (x ₁ , y ₁ ) of two sampling points, such a zero crossing point has coordinates ( x ₁ , y ₁ ) and (x
₂ , y ₂ ). This is because if a sine wave signal is approximated by a cubic equation, it should be as close to the zero crossing point as possible.
This is because it is easier to obtain a highly accurate calculation result by using sampled values of points.

Further, the zero-crossing point is expressed by XT on the x-axis on the basis of the point x ₀ , but X has a special constraint, for example, 0 ≦ X.
There is no restriction such as ≤1, and the equations detailed above hold true no matter where the zero-crossing points are.

FIG. 2 shows a linear approximation error curve 201 at the zero crossing point when calculated by the linear approximation by the equation (2) when the sampling rate is 0.8 radian, and 3 by the equation (17).
Cubic curve approximation error curve 20 calculated by cubic curve approximation
2 and the linear approximation error curve 2 when the sampling rate is reduced from 0.8 radians to 0.52 radians
03 is shown.

The error in FIG. 2 is obtained as follows, for example. That is, the sampling rate of 0.8 radian is normalized to 1.0, and the horizontal axis scale is taken. For example, the horizontal scale 0.25 is 0.2 radian. Three sample values (x ₀ , y ₀ ), (x ₁ , y ₁ ), (x ₂ , y ₂ ) are respectively (−0.2, −0.19867), (0.6, 0.56464). ), (1.4, 0.98545).

From the equation (2), X = y ₀ / (y ₁ −y ₀ ) = 0.26027, and the error E (%) is E = (0.26027−0.25) × 100 = 0.33.
(%)

FIG. 2 shows that the error of the linear approximation which is almost the same as that of the cubic curve approximation error curve 202 is 0.52 radian (the linear approximation error curve 20
It also shows that it becomes 3).

As can be seen from FIG. 2, when the zero crossing point is obtained with the same accuracy as the cubic curve approximation in the case of the linear approximation, it is 0.8
The sampling period must be shortened by about 1.5 times, which is 0.52 radians with respect to radians.

An embodiment of the zero crossing point detecting device according to the present invention constructed according to the above-described principle is shown in FIGS. 3 and 4. FIG. 3 shows a structural example of the second and third calculating means in the zero crossing detecting device according to the present invention, and FIG. 4 shows a structural example of the sampling means and the first calculating means used in the same device. Is.

The apparatus of FIG. 3 is provided with adders 1, 2, 3, 4, and 10, coefficient units 5 and 6, dividers 7 and 8, and multipliers 9 and 11. Each of the adders 1 to 4 and 10 multiplies the input signal by the code attached to the input end, adds the input signal, inverts the code, and outputs the inverted signal. Coefficient multipliers 5 and 6 multiply the input signal by the coefficient displayed in the block and output it. The dividers 7 and 8 divide the input signal from the left side by the input signal from the side part and output it.
The multipliers 9 and 11 multiply the input signal from the left side by the input signal from the side part and output it.

Now, as shown in FIG. 1, the values in the vicinity of the zero crossings of the sample values y _i sampled at the sampling period T are sequentially y
_{If 0} , y ₁ and y ₂ , then ΔY of the equation (11) is calculated by the adder 1 and Δ of the equation (12) is calculated by the adders 2 and 3.
² Y is obtained. In FIG. 3, those values are represented by negative signals. Further, in order to obtain the denominator of the equation (16), the adder 4 produces ΔY + Δ ² Y / 6. Using this as a denominator, -y ₀ obtained through the coefficient unit 6 in the divider 7 is used as the numerator, and X'in the equation (16) is created. Thus, arithmetic element 1
˜4, 6 and 7 constitute the second arithmetic means referred to in the present invention.

Further, the coefficient unit 5 and the divider 8 form 2 (Δ ² Y / 6) / (ΔY + Δ ² Y / 6) as a coefficient of X ′ of the second term in the right-hand bracket of the expression (17), and multiply it by Multiply X'with container 9 (17)
Get the second term on the right-hand side in parentheses in the expression. The adder 10 adds 1 to this, and the operation in the parentheses on the right side of the expression (17) is performed. This addition result can be multiplied by X ′ obtained by the divider 7 in the multiplier 11 to derive X in the equation (17). Thus, the arithmetic elements 5 and 8 to 11 constitute the third arithmetic means referred to in the present invention.

The sample values y ₀ , y ₁ , y ₂ used as input signals in the apparatus of FIG. 3 are determined as follows.

Since the above-mentioned (4) cubic equation are represented by the formula ^{y = A (ωx-ω 3} x 3/6) is approximate expression, affects the accuracy by how to take the sample values Oyobi. When the value of x becomes large, that is, when the coordinates of the sample points are away from the origin (zero crossing point) from which the coordinates are to be obtained, the accuracy becomes poor. Therefore, the interval [x ₀ , x ₂ ]
Select a sampling point that includes the origin in. That is, in order to obtain the zero-crossing point when the sine wave changes from positive to negative, three consecutive sample values are positive / positive / negative or positive / negative / negative in order of sampling. Select a sample value. Similarly, when the zero crossing point when the sine wave changes from negative to positive is obtained, the sign of the sampled value should be negative / negative / positive or negative / positive / positive in the order of sampling.
Select a point.

A concrete circuit example for realizing the above is shown in FIG.

For example, the sine wave voltage V extracted from both ends of the resistor 12 is sampled by the sampling circuit 13 at the timing when the sampling rate generator 14 generates, and a negative value of 2 is sampled.
Obtain the data represented by the complement of. This data is sent to the D flip-flop 16 which clocks the same timing.
The D flip-flop 16 holds the sample data at the rising edge of the clock CP. At the next clock CP, the data held by the D flip-flop 16 until now is held in the D flip-flop 17, and the D flip-flop 16 is made to hold new data from the sampling circuit 13.

Similarly, in the D flip-flop 18, the value of the D flip-flop 17 is held at the next clock CP, the data of the D flip-flop 16 is held in the D flip-flop 17, and the data of the sampling circuit 13 is held in the D flip-flop 16. It The above operation continues until the conditions for the zero crossing point calculation are satisfied.

The condition for the zero crossing point operation is determined by putting the most significant bit (sign bit) of the data in the code verification circuit 24. The calculation of the zero crossing points may be arbitrary three points at equal intervals, but since the solution is obtained by the cubic approximation, the accuracy is improved by calculating the sample values of the three points as close to zero as possible. That is, minus
When there is a zero-cross in the process of shifting from the positive to the positive (positive), the D flip-flops 16, 17, 18 when the sign becomes (0, 1, 1) or (0, 0, 1). The transfer of ~ 18 is stopped, and the data of the D flip-flops 16, 17, 18 are calculated in correspondence with y ₂ , y ₁ , y ₀ , respectively. This y ₂ ,
The values of x corresponding to y ₁ and y ₀ are x ₂ , x ₁ and x, respectively.
It is ₀ .

It is to be noted that in the code verification circuit 24 constituted by the exclusive NOR circuits 19 to 21 and the NAND circuit 23, the outputs D _{1n of} the D flip-flops 16, 17 and 18 in FIG.
Although 0, 1, 1 and exclusive NOR are taken for D _2n and D _3n respectively, in order to stop the clock CP at the code (0, 0, 1), take 0, 0, 1 and exclusive NOR respectively. Good.

Thus, when the outputs D _1n , D _2n , D _3n of the D flip-flops 16, 17, 18 become (0, ₁ , ₁ ),
Since the outputs of the exclusive NOR circuits 19, 20, and 21 are all 1, the output of the NAND circuit 23 is 0.
The clock CP for the D flip-flops 16-18
Is stopped by the AND circuit 15, the values of the D flip-flops (registers) 16 to 18 are not updated even after the subsequent sampling. The output of the D flip-flop 16 is y ₂ ,
The output of the D flip-flop 17 is calculated as y ₁ and the output of the D flip-flop 18 is calculated as y ₀ .

Next, the phase difference measurement according to the present invention is performed.

As shown in FIG. 5, the phase difference between the _two sine wave signals Y ₁ and Y ₂ is 0 at the negative sampling point before Y ₁ changes from negative to positive, and the sampling points are sequentially counted, and Y Count up to the sampling point before ₂ turns positive. If the number is I and the zero-crossing points of Y ₁ and Y ₂ calculated from the equation (17) are X ₁ T and X ₂ T, respectively, the phase difference φ is φ = (I−X ₁ + X ₂ ) T ... (18)

Further, the period measurement according to the present invention is performed as follows.

Similar to the phase difference, the period of the sine wave signal Y _{3 in} FIG. 6 is measured by counting the number of sampling points from the sampling point before Y ₃ changes to plus to the sampling point before the next change to plus (N , And each (1
If the zero crossing point obtained from 7) _X S T, and _{X E} T, the period P becomes _{P = (N-X S +} X E) T ... (19).

The above-mentioned details can be summarized as follows.

The present invention is not a commonly used method of simply obtaining a zero-crossing point from a third-order approximation of a sine wave, but in order to minimize the detection error, the parents of formula (14) to the parents of formula (15) are used. It is characterized by the fact that the transformation of the arithmetic expression was studied, analyzed and devised.

That is, instead of simply finding the solution of the equation (14) from the equation (13), x which is a recursive polynomial like the equation (15)
Introducing an equation in which the nth power of x is added to the solution of, the first term is a constant (approximate solution of 0), and the second and subsequent terms are smaller than the previous terms. The feature is that the final solution is derived by converting into a polynomial that becomes likewise by a subtle technique and substituting the (0th-order) approximate solution of the solution to be obtained. Although the zero-crossing point detection device according to the present invention has been described above based on an embodiment, the present invention is not limited to the illustrated circuit configuration shown as an embodiment, and may be appropriately changed without departing from the gist of the present invention. Changes can be implemented.

〔The invention's effect〕

Thus, according to the zero-crossing point detection device for a sine wave signal according to the present invention, the approximation range is wide, and highly accurate detection can be performed even with a long sampling period. Therefore, it is possible to perform highly accurate zero-crossing point detection using a low-speed and inexpensive A / D converter.

[Brief description of drawings]

FIG. 1 is an explanatory diagram for sampling a sine wave, FIG. 2 is an error curve diagram at a normalized sampling rate of linear approximation and cubic curve approximation, and FIG. 3 shows an embodiment of a zero crossing point detection device according to the present invention. Block diagram, FIG. 4 is a block diagram showing details of a part thereof, FIG. 5 is an explanatory diagram of phase difference measurement according to the present invention, and FIG. 6 is an explanatory diagram of period measurement according to the present invention. 1, 2, 3, 4, 10 ... Adder, 5, 6 ... Coefficient multiplier 7, 8 ... Divider, 9, 11 ... Multiplier, 12 ... Resistor 13 ... Sampling circuit, 14 ... Sampling rate Nelerator, 15 ... AND circuit 16, 17, 18 ... D flip-flop 19, 20, 21 ... Exclusive NOR circuit 22 ... Reset signal, 23 ... NAND circuit 24 ... Code verification circuit, 25, 26, 27 ... OR circuit.

Claims (1)

[Claims] 1. A = amplitude, ω = angular frequency, y = Asin
Sampling means for sampling three consecutive sample values y ₀ , y ₁ , y ₂ extending before and after the zero-crossing point at a sampling period T from a sine wave signal on the xy rectangular coordinates represented by ωx; From the x ₀ point, a first calculation means for obtaining three coordinates (x ₀ , y ₀ ), (x ₁ , y ₁ ), (x ₂ , y ₂ ) related to the values y ₀ , y ₁ , y ₂ Let XT be the increment on the x-coordinate axis to the zero crossing point of the sine wave signal, and x ₁ = (1-X) T, x ₂ = (2-
X) T, y ₁ −y ₀ = ΔY, (y ₁ −y ₀ ) − (y ₂ −
y ₁ ) = Δ ² Y, X ′ = − y ₀ / {ΔY + Δ ² Y) / 6}, and a second calculation means for calculating X = X ′ [1 + X ′ (2Δ ² Y / 6) / {ΔY + (Δ ²
Y) / 6}] and a third calculation means for calculating the zero crossing point of the sine wave signal.