TWI426396B - A numerical differential method and a device for calculating the points of the signal using unequally spaced signal samples - Google Patents

Description

Numerical differential method for calculating points of the signal by using signal samples with unequal spacing and device thereof

Field of digital signal processing

The current numerical differential method can be summarized into the following three ways.

(1) Forward differential mode:

The mathematical form of this approach is:

f ⁽ⁿ⁾ (t _i ) is the nth derivative of the signal f(t) (t is a time variable) at t=t _i ; D ₁ ( n , j )( j =1, 2,... m) is a previously established constant of m; △ t for the f (t) to retrieve samples of the pitch.

It can be seen from equation (2)-1 that f ⁽ⁿ⁾ (t _i ) is calculated by using m samples taken from f(t) starting from t=t _i .

(2) Backward difference mode:

The mathematical form of this method is;

Where D ₂ ( n , j )( j =1, 2, . . . m ) is also the m constants established in advance, and Δ t is also the spacing between the samples of f(t)撷.

It can be seen from equation (2)-2 that f ⁽ⁿ⁾ (t _i ) is calculated by using m samples taken from f(t) starting from t=t _i .

(3) Intermediate differential mode:

The mathematical form of this method is;

Where D ₃ ( n , j ) and D ₄ ( n , j )( j =12,.... u ) are 2u constants established in advance, and Δ t is also the interval between the samples of f(t) .

It can be seen from equation (2)-3 that f ⁽ⁿ⁾ (t _i ) is calculated from t=t _i and forward and backward for 2u samples taken by f(t).

The above three mathematical formulas are mainly derived from the Gregory-Newton formula and Taylor’s. Due to space limitations, and the process of this deduction is well known to scholars of this class, I will not repeat them.

壹 For example, if we want to calculate the numerical differentiation of the signal f(t), we need to divide f(t) into a number of intervals in advance. As shown in Figure 1, we divide it equidistantly into the first one. , 2nd, ..... j-1, j, j+1, ... Wait a lot of intervals.

When calculating the numerical differentiation of each point of f(t) in the jth interval, if the (2)-1 type is used, except for the sample of the interval (ie, the jth interval), the next interval (ie, the jth) is still needed. Samples of +1 interval); if using (2)-2, you need to use the sample of the previous interval; if you use the (2)-3 formula, some samples of the previous and subsequent intervals need to be used.

In any case, we can't just use the samples in our own interval to calculate the numerical differentiation of each point in our own interval; this is the shortcoming of the current numerical differential calculation. In addition, it has another disadvantage, that is, the distance between all samples and the sample must be equal.

To this end, the author is proposing this latest numerical differential calculation method.

贰, the calculation method derived from Lagrange’s polynomial:

Assume that the signal f(t) (t is a time variable) at t [0, T ] ( T R ) can be approximated by the n-1 polynomial of t, then we can arbitrarily f find n samples f(t ₁ ), f(t ₂ in f(t) in this [0, T] ), ... f(t _n ), and use f Lagrange's polynomial to approximate f(t) to the following:

Where L(i,t) is represented as before (1)-3.

In equation (4)-1, the first differential is made for t and t = t _k ( t _k [0, T ]) can be obtained after substitution:

(4)-2 is to use only n samples f(t _i ) in [0,T] (i=1,2,...,n) to calculate their own intervals [0,T] One of t = t _k points has a differential value f'(t _k ); and, it is worth noting that any two adjacent sample spacings in f(t _i ) are not necessarily equal.

When we derive the mathematical formula (4)-2 of numerical differentiation above, we differentiate the t from the (4)-1 type on both sides, which makes f'(t) become the n-2 polynomial of t; therefore, if To make the accuracy of the formula (4)-2 meet our needs, the precondition is that f'(t) can also be approximated by the n-2 polynomial of t.

Now, suppose that we are the first to choose n and m (n, m N; and n > m), such that f ^(m) (t) can be approximated by the nm-1 polynomial of t, and f(t) can of course be more closely represented by the n-1 polynomial of t.

Therefore, we can make m differentials on both sides of (4)-1 and then let t=t _{k be} substituted:

The equation (4)-3 is an approximate mathematical expression of the differential value of the m-number of f(t) at t=t _k .

However, (4)-3 is not a good calculation method, and the author will propose a better calculation method:

If f(t), f'(t),...,f ^(m) (t) is at t [0, T ] can be approximated by the n-th order polynomial of t (in this case, (4)-1 is of course more approximationally established), then f(t) in (4)-1 is The f'(t) alternative is available:

Change i of the above formula to u ₁ , and then substitute t for t = t _k to obtain:

In the right side of the above formula Substituting (4)-2, you can get:

Now, f ( t ) in the formula (4)-1 is replaced by f "( t ), and is derived in the same manner as described above.

The above derivation will continue, and finally:

Above Move to the front and replace u _m with i , you can get:

The above formula is the previous (1)-1 type, and the former formula is rewritten below:

计算, the calculation method derived from Taylor’s expansion:

If f ( t ) is at t In [0, T ], it can be approximated by the nth degree polynomial of t, then f(t) can be used t = t _k ( t _k [0, T ]) is the expansion point, and is approximated by Taylor's expansion:

Now, in the [0, T ] arbitrarily select the n points t ₁ , t ₂ .... t _{n and} then into the above equation one by one, and move f ( t _k ) to the left, you can get the following simultaneous equations :

(5)-2 can be expressed as a matrix equation:

Because the function set {( t - t _k ) ⁱ | i =1, 2,... n } is at t R must be linearly independent of each other, so the coefficient determinant of (5)-2 must not be zero, so that (5)-3 must have a unique set of solutions, which is:

It can be clearly seen from equation (5)-4 that any point in the interval [0, T ] t = t _k , the j-order value of f ( t ) is differential f ^{( j )} ( t _k ) ( j =1 , 2,... n ) can be calculated by using any n samples f ( t _i ) ( i =1, 2, ... n ) pre-fetched in the same interval [0, T ], and the distance between the sample and the sample It does not have to be equal.

肆, analysis of calculation accuracy:

The accuracy of the above numerical differentiation is directly related to the following factors:

1. The highest frequency f _{m of the} signal components contained in f ( t ).

2. Determine the size of T to be calculated for the interval [0, T ].

3. Let f ( t ) denote the size of n by the polynomial of n times t.

Because the signals processed in general, especially the signals transmitted in communication, are a combination of string signals; therefore, we will analyze the accuracy of numerical differentiation with a string signal.

Let f ( t ) be:

Where: ω _i = 2 πf _i ; C _i , θ _i are constants; f _i is the frequency of each signal component, i = 1, 2, ....

If the largest of f _i is f _m , then the corresponding signal component is: g _m ( t ) = C _m cos( ω _m t + θ _m ). The signal of the highest frequency f _m is set to t =0 as the expansion point. After Taylor's expansion, it can be written as:

g _m ( t ) If the 2n+1 polynomial of t can be used, the first series on the right side of the above formula can be taken to n items, and the first series is taken to n+1, and becomes:

(6) The absolute value of the ratio of the n+1th term of the first series to the right of the previous term And the absolute value of the ratio of the n+2th item of the second series to the previous item .

Because t [0, T ], that is, the maximum value of t is T, so the t of the above two ratios can be substituted by T and .

We can determine a small number ε according to the precision required by the prior, so that the following two formulas are established:

Solving two inequalities is available:

Since n>0, the solution is:

among them:

Because (6)-2-1 is taken to 2n+1, it is:

Equation (6)-5 provides the direction of analysis of the required accuracy when calculating numerical differentiation.

Figure 2 shows a simple embodiment of the present invention. Here are the points to illustrate:

(1) In the figure, 1 is an A/D converter. The rate of picking up signal samples in a typical A/D converter is mostly fixed; however, it can be non-fixed here.

(2) Figure 2 is in t In the [0, T] interval, it is used as a temporary storage device for temporarily storing the sample f(t _i ) captured by the A/D.

(3) We have previously assigned the parameter SL(m,i,t _k )(m,i,k=1,2,...n,n N) Calculated one by one and stored in the memory, this memory is shown as 3 in the figure.

(4) 4 of the figure is The calculator, the output of the calculation result is the differential value of the m-time value of f(t) at t=t _k .

Moreover, the various circuit blocks in the figures are all well-known techniques, and thus the inventors will be omitted from this description.

1‧‧‧A/D converter

2‧‧‧ register

3‧‧‧ memory

4‧‧‧Plus calculator

Figure 1: The differential signal curve is divided into equal interval graphs.

Figure 2: Block diagram of an embodiment of the m-number value differential.

1‧‧‧A/D converter

2‧‧‧ register

3‧‧‧ memory

4‧‧‧Plus calculator

Claims (3)

A sample spacing does not need to be equal to m times (m is a fixed positive integer defined in advance, m is the same as m described below). The numerical differential calculation device mainly includes: one as an interval [ 0,T]( T R ) The signal f(t) to be differentiated is arbitrarily arbitrarily taken n (n is a fixed positive integer defined in advance, n is the same as n), and sample f(t _n ) is used for A/ D converter; one as a sample memory for temporarily storing the n samples f(t _n ) captured by the above A/D converter; n as a calculation Lagrange polynomial L(i, t) at t =t _n is a differential calculator for the differential L'(i,t _n ); where L(i,t) is: t is a time variable; m is used as a calculator for calculating SL ( m , i , t _k ); ; and one as computable A calculator is used; wherein f(t _i ) is the sample data read from the sample memory. As for the device described in claim 1, the technical feature contained therein is that the same set of n samples f is obtained by f(t) in the interval [0, T] ( t _n ), it is possible to calculate the differential of the value of the first, second, ..., m times of t = t _i (i = 1, 2, ..., n). As for the device described in claim 1, the technical feature contained therein is that n samples taken from f(t) in the interval [0, T] are between the sample and the sample. The spacing may not necessarily be equal.