Method of using samples with unequal spacing within a signal range to calculate first-order or higher-order numerical integration of the signal within the signal range and device thereof

A means for calculating one or more numerical integrations of the signal in the interval may be utilized using unequal spacing samples in a signal interval.

The field of digital signal processing technology for numerical integration calculation.

At present, there are four common types of numerical integration methods, which are outlined below:

(1) Trapezoidal Integral Method: This integration method is derived directly from the definition of the integral. It divides f(t) into n sub-intervals [a, a + Δt], [a + Δt, a + 2 ], ---, [a + (n) in the interval [a, b] to be integrated. -1) Δt, a+n], then treat each subinterval and the graph enclosed by the function f(t) as a small trapezoid, and finally calculate the sum of the area of these small ladders as f(t) t<img file="TWI426397B_D0001.tif" wi="32" he="39" img-format="tif" img-content="character" orientation="portrait" inline="no" />Approximate integral value between [a, b]. The trapezoidal integral approximation formula is:<maths><img alt="" file="TWI426397B_D0002.tif" he="154" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1797" /></maths>

among them,<img file="TWI426397B_D0003.tif" wi="237" he="130" img-format="tif" img-content="character" orientation="portrait" inline="no" />,n<img file="TWI426397B_D0004.tif" wi="34" he="41" img-format="tif" img-content="character" orientation="portrait" inline="no" />N.

In general, the (2)-1 formula is only suitable for simpler functions (such as lower frequency, smaller curve waviness, etc.). In the case of more complex functions, the sample to be taken must be sufficient to make it accurate. More (that is, n needs to be large enough, Δt needs to be small enough); therefore, it is not suitable for the calculation of all functions.

(2) Simpson's Integral Method and Newton-Cotes Integral Method: In the above trapezoidal integral method, the interval integral to the function f(t) is divided into n sub-segments. The interval is integrated as the line of each sub-interval is a straight line. Now if we integrate the graph of each successive two subintervals into the curve of the quadratic function in the subinterval divided by [a, b], then this integral method is the Simpson integral method, and the integral of its integral The formula is:<maths><img alt="" file="TWI426397B_D0005.tif" he="152" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1810" /></maths>

Further, if we integrate the graph of each successive three subintervals into the curve of the cubic function in the subinterval divided by [a, b], the integral is the Newton-Kaz integral method, which approximates The product formula is:<maths><img alt="" file="TWI426397B_D0006.tif" he="259" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1972" /></maths>

(2) in the -2 and (2)-3<img file="TWI426397B_D0007.tif" wi="222" he="122" img-format="tif" img-content="character" orientation="portrait" inline="no" />。

(2)-2 and (2)-3 are obvious. Under the same n (ie, the same number of samples), the result of the integral will be correcter than the formula (2)-1.

However, the calculations of (2)-2 and (2)-3 are still difficult to calculate correctly for more complex functions.

(3) Romberg Integral Method: The Nongberg Integral Method is derived from the nature of the Extrapolated Derivative. We can first find the subintervals by the trapezoidal integral method. The trapezoidal integral approximation is further extrapolated to approximate the Simpson integral by these approximate integral values, and finally the approximate Simpson integral values are used to extrapolate the approximate Newton-Kaz The integral value; so repeatedly derived, the final one is the Nongberg integral value.

The Nongberger integral formula can be expressed in the following way:<maths><img alt="" file="TWI426397B_D0008.tif" he="1040" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1804" /></maths>

Where kn;n is the number of subintervals into which the integration interval [a, b] is divided;<img file="TWI426397B_D0009.tif" wi="223" he="119" img-format="tif" img-content="character" orientation="portrait" inline="no" />。

(2) In the formula -4, when k=n, A(n,n) is the Ingrid integral value, namely:<maths><img alt="" file="TWI426397B_D0010.tif" he="110" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1788" /></maths>

The advantage of the Nongberger integration method is that it is more accurate than the trapezoidal integral method, the Simpson integral method and the Newton-Kazs integral method. The disadvantage is that the calculation time is long and can only be applied to the calculation of a simpler function.

(4) Gaussian Integration: The Gaussian integration method has a deeper theory and must be discussed in a special chapter. Due to space limitations, the author will only use a few words.

Let f(t) be t<img file="TWI426397B_D0011.tif" wi="33" he="39" img-format="tif" img-content="character" orientation="portrait" inline="no" />The integral function between [a, b], the Gauss integral method is:

1. Take n samples for f(t) in [a, b]<i>f</i>(<i>t</i>₁)、<i>f</i>(<i>t</i>₂)、<i>f</i>(<i>t</i>₃)............f(t_n)。

2. with f(t_i) Use Lagrange's polynomial.

3. The above Lagrange's polynomial is represented by the Legendre's polynomial.

4. Calculate n weighting coefficients using the orthogonal properties of Legendre's polynomial W₁, W₂, W₃,....W_nTo make the following formula:<maths><img alt="" file="TWI426397B_D0012.tif" he="147" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1830" /></maths>

The (2)-6 formula is the Gaussian integral formula.

Gaussian integration methods generally can accumulate more difficult and complex functions. Weight coefficient W_iThe value of (i=1,2,...n) cannot be all zero, its size is the same as it is<i>t</i>=<i>t</i>_<i>i</i>Location, and<i>f</i>(<i>t</i>_<i>i</i>)、<i>f</i>(<i>t</i>_<i>i</i>The neighboring sample value is related; therefore it is an adjustment factor.

As shown in Figure 1,<i>t</i>=<i>t</i>₄Due to the large slope,<i>W</i>₄Will obviously become bigger, in<i>t</i>=<i>t</i>_<i>i</i>Time slope is close to zero (but negative), which makes<i>W</i>_<i>i</i>Also approaching zero (also negative), but<i>t</i>=<i>t</i>_<i>i</i>+2Time<i>W</i>_<i>i</i>+2It will become a larger negative value.

Although the Gaussian integral method is the most accurate method in the current numerical integration method, it still has a blind spot. As shown in Figure 2, when the sampling rate does not meet the change of the slope of the curve, no matter how the weight coefficient is taken, it cannot be calculated. The correct value (unless the sample rate is increased, ie the number of samples is increased).

The above is the current numerical integration method.

It is an object of the present invention to provide a more accurate integration method than the current numerical integration method described above.

The integration method of the present invention, which samples the function f(t), is unequal. The distance between two adjacent samples is related to the slope of f(t) at the same point as before. The larger the slope, the smaller the spacing, otherwise the opposite. Therefore, before calculating the numerical integration, the decision must be made by a circuit one.

As shown in Figure 3c, it is the judgment circuit block for the judgment of the sample selection. The following points are explained:

1. We must know in advance the highest frequency of the components contained in the integral f(t).

2. Predetermine a positive number m (see the precision required for integration)<i>m</i><img file="TWI426397B_D0013.tif" wi="30" he="45" img-format="tif" img-content="character" orientation="portrait" inline="no" /><i>N</i>), if f(t) contains the highest frequency<i>f</i>_<i>n</i>, then circuit block b<img file="TWI426397B_D0014.tif" wi="89" he="89" img-format="tif" img-content="character" orientation="portrait" inline="no" />The sampling frequency of the converter is set to<i>Mf</i>_<i>n</i>. This represents the frequency<i>f</i>_<i>n</i>At the time, it was taken m samples per week.

3. Under the same precision requirements, as long as it can maintain m samples taken every week.

4. But, b square<img file="TWI426397B_D0015.tif" wi="89" he="89" img-format="tif" img-content="character" orientation="portrait" inline="no" />Is fixed at a sampling rate of m<i>f</i>_<i>n</i>In the sampled, the input signal to be integrated f(t) sometimes looks like [T in Figure 4]₁,T₂Between the two shows a relatively smooth (ie low frequency), sometimes like [T₂,T₃The more varied (ie, higher frequency) is shown between the two; therefore, it is not necessary to do the integral calculation for the samples taken at a fixed sampling rate.

5. The output of each instant of the differential circuit block a represents the slope of the function f(t) at that instant, and the slope determines the timing of the next sample.

6. Located in<i>t</i>=<i>t</i>_<i>x</i>When b provides a sample<i>f</i>(<i>t</i>_<i>x</i>) Give c, c to keep it. c is accepting and storing<i>f</i>(<i>t</i>_<i>x</i>), at the same time, I also got the information from a<i>f</i>'(<i>t</i>_<i>x</i>) information, the<i>f</i>'(<i>t</i>_<i>x</i>) is<i>f</i>(<i>t</i>)in<i>t</i>=<i>t</i>_<i>x</i>The slope of time, so c can calculate the time based on this<i>f</i>(<i>t</i>"instantaneous frequency", if this "instantaneous frequency" is<i>f</i>_<i>x</i>, its "instantaneous sampling rate" should be<img file="TWI426397B_D0016.tif" wi="91" he="132" img-format="tif" img-content="character" orientation="portrait" inline="no" />, ie "instantaneous sample spacing" should be<img file="TWI426397B_D0017.tif" wi="97" he="134" img-format="tif" img-content="character" orientation="portrait" inline="no" />. So, c is accepting the sample<i>f</i>(<i>t</i>_<i>x</i>After waiting for the time interval to<img file="TWI426397B_D0018.tif" wi="89" he="134" img-format="tif" img-content="character" orientation="portrait" inline="no" />The first sample that appears afterwards is accepted again, and those appearing in the meantime are discarded.

7. The above 6 actions are continued. Finally, some samples with unequal spacing can be collected for the d circuit blocks to be integrated.

8. As for the circuit blocks a, b and c in Fig. 3, we can design according to the functions required above, which are well-known techniques, and therefore are omitted.

Now, we begin to explain how to use the above-mentioned samples of unequal spacing to do numerical integration calculations. The description will be divided into two parts, one of which is a numerical integration, and the other is a numerical integration of more than two times.

(1) One-time numerical integration:<i>f</i>(<i>t</i>)in<i>t</i><img file="TWI426397B_D0019.tif" wi="39" he="42" img-format="tif" img-content="character" orientation="portrait" inline="no" />[<i>a</i>,<i>b</i>n was taken n samples<i>f</i>(<i>t</i>₁),<i>f</i>(<i>t</i>₂),.......<i>f</i>(<i>t</i>_<i>n</i>), where the sample spacing between each two adjacent points is<i>t</i>_<i>i</i>+1-<i>t</i>_<i>i</i>(<i>i</i>=1,2,.....<i>n</i>) are not necessarily equal.

According to Lagrange's polynomial,<i>f</i>(<i>t</i>)in<i>t</i><img file="TWI426397B_D0020.tif" wi="39" he="42" img-format="tif" img-content="character" orientation="portrait" inline="no" />[<i>a</i>,<i>b</i>Sample available<i>f</i>(<i>t</i>_<i>i</i>)(<i>i</i>=1,2,.....<i>n</i>) is approximately expressed by the n-1 power polynomial of t:<maths><img alt="" file="TWI426397B_D0021.tif" he="157" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1790" /></maths>

among them<i>L</i>(<i>i</i>,<i>t</i>) is shown in (1)-4, and is rewritten as follows:<maths><img alt="" file="TWI426397B_D0022.tif" he="509" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1817" /></maths>

Now, do the integration between [a, b] directly on both sides of (3)-1:<maths><img alt="" file="TWI426397B_D0023.tif" he="146" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1804" /></maths>

because<i>L</i>(<i>i</i>,<i>t</i>) is a polynomial of t, so it can be directly applied to the current polynomial integral formula<img file="TWI426397B_D0024.tif" wi="225" he="101" img-format="tif" img-content="character" orientation="portrait" inline="no" />,make:<maths><img alt="" file="TWI426397B_D0025.tif" he="106" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1819" /></maths>

Then (3)-2 can be written as:<maths><img alt="" file="TWI426397B_D0026.tif" he="150" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1815" /></maths>

If used<i>Sf</i>(<i>a</i>,<i>b</i>) indicates that f(t) is<i>t</i><img file="TWI426397B_D0027.tif" wi="42" he="42" img-format="tif" img-content="character" orientation="portrait" inline="no" />[<i>a</i>,<i>b</i>The first time between the points, the mathematical formula is as shown in (1)-2-1, and is rewritten in the following:<maths><img alt="" file="TWI426397B_D0028.tif" he="150" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1808" /></maths>

The above formula is a numerical integration mathematics that we want.

(2) Two or more numerical integrals: The numerical integrals referred to herein as two or more times are not the numerical integrals of the above-mentioned two or more weights; generally referred to as two or more weights. , its integrand has two or more variables: but the invention<i>f</i>(<i>t</i>There is only one variable t, so the so-called one, two, ---.. numerical integration refers to the function<i>f</i>(<i>t</i>t change One time, two times, ------. points. This is especially clarified first.

Consider b in (1)-2-1 as a variable and replace it with t, then:<maths><img alt="" file="TWI426397B_D0029.tif" he="135" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="651" /></maths>

And at<i>t</i><img file="TWI426397B_D0030.tif" wi="41" he="41" img-format="tif" img-content="character" orientation="portrait" inline="no" />[<i>a</i>,<i>b</i>In the middle<i>f</i>(<i>t</i>) sampling time point, right<i>Sf</i>(<i>a</i>,<i>t</i>) draw n samples<i>Sf</i>(<i>a</i>,<i>t</i>₁),<i>Sf</i>(<i>a</i>,<i>t</i>₂)………<i>Sf</i>(<i>a</i>,<i>t</i>_<i>n</i>), and then use these samples to represent Lagrange's polynomial<i>Sf</i>(<i>a</i>,<i>t</i>),which is:<maths><img alt="" file="TWI426397B_D0031.tif" he="161" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1799" /></maths>

among them:<i>L</i>(<i>j</i>,<i>t</i>) is shown as (1)-4.

in<b><i>t</i></b><img file="TWI426397B_D0032.tif" wi="32" he="39" img-format="tif" img-content="character" orientation="portrait" inline="no" />[<i>a</i>,<i>b</i>] The pair of (3)-4 sides are integrated:<maths><img alt="" file="TWI426397B_D0033.tif" he="145" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="888" /></maths>

Substituting (1)-3 into the above formula gives:<maths><img alt="" file="TWI426397B_D0034.tif" he="161" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1788" /></maths>

If order:<maths><img alt="" file="TWI426397B_D0035.tif" he="117" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="577" /></maths>

then:<maths><img id="" he="156" wi="1841" file="TWI426397B_D0036.tif" alt="" img-content="drawing" img-format="tif" orientation="portrait" inline="no" /></maths>

(3)-5 or (3)-6 are all f(t)<i>t</i><img file="TWI426397B_D0037.tif" wi="34" he="45" img-format="tif" img-content="character" orientation="portrait" inline="no" />[<i>a</i>,<i>b</i>The second numerical integral mathematical formula between the two.

Repeat the above behavior k-1 times, you can get:<maths><img alt="" file="TWI426397B_D0038.tif" he="156" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1804" /></maths>

now:<maths><img alt="" file="TWI426397B_D0039.tif" he="117" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="642" /></maths>

Then you can get (1)-2-2, rewritten in the following:<maths><img alt="" file="TWI426397B_D0040.tif" he="145" id="" img-content="drawing" img-format="tif" inline="no" orientation="portrait" wi="1823" /></maths>

(3)-7 or (1)-2-2 are all f(t)<i>t</i><img file="TWI426397B_D0041.tif" wi="39" he="41" img-format="tif" img-content="character" orientation="portrait" inline="no" />[<i>a</i>,<i>b</i>The mathematical formula of the k-time value integral in the middle.

The description of the present invention is now complete!

As mentioned earlier, an important key to the present invention is how to change its sampling rate in a timely manner, as illustrated in the description of Figure 3.

In this embodiment, we will provide another method, which is to directly convert the signal frequency value into a pulse wave of an equal or equal frequency, and the pulse wave directly goes to the signal sample.

Figure 5 shows a simple embodiment of the present invention, which is illustrated by the following examples:

(1) The "frequency/pulse" converter of the above signal is shown in Figure 1.

(B) 2 shows the A/D converter whose sampling rate is directly controlled by 1.

(3) The signal sample f(t) captured by the A/D converter in the interval [a, b] to be integrated_i) (i = 1, 2, .... n) must be temporarily stored; Figure 3 is to store f (t_i) The scratchpad.

(4) We must prioritize the parameters<img file="TWI426397B_D0042.tif" wi="218" he="104" img-format="tif" img-content="character" orientation="portrait" inline="no" />Once calculated, it is stored in memory 4; where L(i, x) is the Lagrange's polynomial of x, i = 1, 2, ....

(5) 5 is shown<img file="TWI426397B_D0043.tif" wi="431" he="148" img-format="tif" img-content="character" orientation="portrait" inline="no" />The final output of the calculator is the numerical integral of f(t) between [a, b].

The detailed structural drawings in the various circuit blocks in the drawings are well-known techniques, and the detailed description thereof will be omitted.

<p>1frequency/pulse converter</p><p>2A/D converter</p><p>3 register</p><p>4 memory</p><p>5Plus calculator</p>

Figure 1: Explanatory diagram of the weight coefficient of each point of the curve and the slope of the point

Figure 2: Illustration of the sampling rate that does not match the slope change demand

Figure 3: Judgment icon for sample capture

Figure 4: Comparison of lower and higher curve frequencies

Figure 5: Block diagram of the embodiment of numerical integration