JPH0644300B2 - Real-time Fourier transform method and apparatus - Google Patents

Description

The present invention relates to a Fourier transform method and apparatus, and more particularly to a real-time Fourier transform method capable of continuously outputting the result of Fourier transform of a signal while inputting the signal. Regarding

[Conventional technology]

The Fourier transform is an extremely effective method for analyzing a periodic function and is used for analyzing various measured values. Generally, in the Fourier transform, a signal F (t) having a fundamental frequency that is periodic with respect to time t is represented by the sum of a plurality of periodic functions having a period that is an integral multiple of this fundamental frequency. That is, the input signal F (t) is Is approximated by the following function G (t). Where i = 1, 2,
, N, and the larger N is, the more accurate the Fourier transform is, and G (t) becomes more faithful to F (t). Note that C _i represents the amplitude of each periodic function, and ψ _i represents the phase.

[Problems to be solved by the invention]

In the conventional Fourier transform method, the input signal F (t) is sampled at a predetermined cycle for a predetermined period, and each periodic function is determined based on the sampling value obtained by this sampling. Therefore, even if the signal F (t) is input, there is a problem that the result of the Fourier transform cannot be obtained before the predetermined sampling period and calculation period have elapsed.

Therefore, an object of the present invention is to provide a real-time Fourier transform method capable of continuously outputting a Fourier transform result while inputting a signal.

[Means for solving problems]

According to the present invention, a signal F (t) that changes with respect to time t is defined as the sum of N periodic functions having frequencies that are integral multiples of the fundamental frequency. (However, C _i is the amplitude, ψ _i is the phase) In a Fourier transform method and apparatus for transforming so that it can be expressed by a function G (t) defined by the formula, a signal F (t) is input and The period of the basic cycle T given as the reciprocal number is divided into k parts, and the values at the respective divided points of the input signal F (t) are obtained as sample values, and the average value of the sample values is k-divided average value Fk (t). The procedure to obtain
= 1, 2, ..., N, to obtain N divided average values F1 (t), F2 (t), ..., FN (t), N periodic functions and the N divided averages. The N-ary simultaneous linear equations obtained from the relationship with the values are solved to continuously obtain the instantaneous values of the N periodic functions at time t.

[Action]

First, assume that the signal to be converted is F (t), and that this signal is a periodic signal having a fundamental frequency. And
This signal F (t) is expressed as a function G (t) having a form as shown in equation (1) as the sum of N periodic functions having frequencies that are integral multiples of the fundamental frequency.

Here, C _i is the amplitude and ψ _i is the phase.

Now, the signal F (t) is a function as shown in FIG.
It is assumed that the basic cycle T (T = 1 /) is a period as shown in the figure. Figures (a), (b), and (c) of the same figure show the case where the basic period T is divided into one, two, and three, respectively. Here, the value at each divided time point is called the sample value in that division, the sample value in the case of 1 division is X ₁₁ , the sample value in the case of 2 division is X ₂₁ and X ₂₂ , and the sample in the case of 3 division. The values are X ₃₁ , X ₃₂ , and X ₃₃ . If the average value of the sample values is calculated and the average value of the sample values when k-divided is generally expressed by the k-divided average value F (t), F1 (t) = X ₁₁ (2) F2 (t) = (X ₂₁ + X ₂₂ ) / 2 (3) F3 (t) = (X ₃₁ + X ₃₂ + X ₃₃ ) / 3 (4).

The basic principle of the present invention is based on the finding of a mathematical law that, if G (t) = F (t), this k-divided average value Fk (t) can be expressed by the following equation.

However, i = kn, m is an integer part of N / k. In addition, the mathematical proof that the equation (5) holds will be given later.

In the present invention, the input signal F (t) is divided into k (k = 1,
2, ..., N), and N divided average values F1 (t), F2
(T), ..., FN (t) are measured and obtained at time t. Now, if the matrices A, S, and F are defined as follows, Equation (9) is obtained from equation (5).

AS = F (9) Therefore, if F1 (t) to FN (t) are obtained, S is obtained by the following equation using the inverse matrix A ⁻¹ of A.

S = A ⁻¹ F (10) Here, S is a matrix having N periodic functions to be obtained as elements, as shown in the equation (7), so that N instantaneous Fourier expansion functions at time t The value will be obtained. If the above operation is continuously performed in time, it is possible to continuously output the Fourier transform result while inputting a signal.

〔Example〕

Hereinafter, the present invention will be described based on examples. FIG. 1 is a flow chart showing a conversion procedure of a real-time Fourier transform method according to the present invention. First, in step S1, the signal F (t) to be converted is input. The method according to the present invention is
Since it is possible to perform Fourier transform in real time, if an instantaneous value of the input signal F (t) at time t is given, some Fourier transform output corresponding to this is obtained, but in reality, This output is delayed by at least the period of the basic cycle T and then output.

In step S2, the reference frequency is obtained based on this signal F (t). This is the input signal F
If the reference frequency of (t) is known, its value may be used, and if it is not known, the value measured using an appropriate frequency measuring means may be used.

Next, in step S3, the parameter k is set to the initial value 1. This parameter k corresponds to the number of divisions of the basic period T, and as shown in steps S7 and S8 to be described later, k = 1, 2, ... Repeated.

When the value of the parameter k is set, the signal F (t) input in step S4 is divided into k in the period of the basic cycle T. If k = 1, 2, 3, then the second
It is as shown in Figures (a), (b), and (c).

In step S5, the sample value at each divided time point is obtained. For example, when k = 3, three sample values X ₃₁ , X ₃₂ , and X ₃₃ are obtained.

In step S6, the divided sample values Fk (t) are obtained by summing the obtained sample values and averaging them.
For example, when k = 3, F3 (t) = (X ₃₁ + X ₃₂
+ X ₃₃ ) / 3.

As shown in steps S7 and S8, k =
Repeat for 1-N. Where N is the input signal F
It is a parameter indicating how many sums of (t) are expressed as a sum of periodic functions. The larger the number, the more accurate the Fourier transform can be performed. As a result, the N divided average values F1
(T) to FN (t) are obtained.

Here, in step S9, the N-ary simultaneous linear equations are solved based on the N values. That is, as described above, by ^obtaining the matrix S using the inverse matrix A ⁻¹ ,
As shown in step S10, the instantaneous values of the N periodic functions at time t are extracted.

As shown in steps S11 and S12, by continuing the above procedure in time, the Fourier transform result, that is, the instantaneous values of the N periodic functions is sequentially output while the signal F (t) is input. be able to.

FIG. 3 is a circuit diagram showing an example of a real-time Fourier transform circuit for executing the real-time Fourier transform method according to the present invention. In this circuit, N = 9, that is, the input signal F
(T) is Fourier-transformed into the sum of nine periodic functions as shown in equation (11).

In the circuit diagram of FIG. 3, the block labeled A is an inverting addition operation element, the block labeled B is a delay element, and the block labeled C is an inverting element. The inverting addition arithmetic element A and the inverting element C can be configured by, for example, operational amplifiers, and the delay element B is, for example, a BBD (Bucket Brigate Decoder).
vice) element. With this circuit, when an input signal F (t) is applied to the input terminal at the left end of the figure, nine periodic functions (constituting the function G (t)) are output from the nine output terminals at the right end of the figure. Instantaneous value is output.

Five delay element arrays B9, B6, B5, B8, B7 are formed in the front stage of this circuit, that is, in the left half of the figure. The delay element array B9 is eight delay elements cascade-connected between the nodes n90 to n98, the delay element array B6 is three delay elements cascade-connected between the nodes n60 to n63, and the delay element array B5 is Four delay elements cascaded between the nodes n50 to n54, the delay element array B8,
The seven delay elements cascaded between the nodes n80 to n87 and the delay element array B7 are each formed of six delay elements cascaded between the nodes n70 to n76. The delay time of each delay element B depends on the input signal F
Based on (t), it is set as shown in FIG. For example, each of the delay elements in the delay element array B9 has a delay time of a time T / 9 obtained by dividing the fundamental cycle T of the input signal F (t) by 9, and the delay elements in the delay element array B7 similarly have the delay time. It has a delay time of only T / 7. For example, node n91
Is delayed by a time T / 9 compared to the output of node n90. By setting in this way, the sampled value of the signal F (t) at each time point is measured by measuring the voltage at each of the following nodes in all cases where the basic cycle T is divided into k (k = 1 to 9). You can ask at the same time.

(1) k = 1: For example, node n90 (2) k = 2: nodes n80, n84 (3) k = 3: nodes n90, n93, n96 (4) k = 4: nodes n80, n82, n84, n86 ( 5) k = 5: nodes n50 to n54 (6) k = 6: nodes n60, n61, n93, n62, n96, n63 (7) k = 7: nodes n70 to n76 (8) k = 8: nodes n80 to n87 (9) k = 9: nodes n90 to n98 Therefore, the divided average value F in each case of k division
k (t) is obtained by averaging the voltage values of the above-mentioned nodes. This averaging is performed by the inverting addition computing element A2.
~ A9. For F1 (t), for example, the voltage value of the node n90 may be used as it is. Thus, F1 (t) to F9 (t) are obtained. After that, by calculation using the inverse matrix A ⁻¹ , C _i sin (2πt +
The instantaneous value (i = 1 to 9) of ψ _i ) at time t is output from the right end of the figure. Inverting element _C 1 is provided after the flip operation element A2~A9 ~C ₄ and processing element A10~A13
Is a circuit that performs an operation using this inverse matrix.

The matrix A when N = 9 is And its inverse matrix A ⁻¹ is Becomes Therefore, for example, from the expression (10), C ₃ sin (2π3t + ψ ₃ ) which is the third element of the matrix S is expressed as follows.

_{C 3 sin (2π3t + ψ 3} ) = F3 Therefore (t) -F6 (t) -F9 (t) (12), the third terminal of the right end of the output terminal of FIG. 3, -C a calculation device A11 _{A value of 3} sin (2π3t + ψ ₃ ) is output. That is, the calculation element A11 is performing the calculation of −F3 (t) + F6 (t) + F9 (t). In this embodiment, the output value is obtained with a minus sign because of the configuration of the arithmetic circuit.

As described above, in this circuit, the signal F is input to the leftmost input terminal.
When (t) is given, nine periodic functions C _{i obtained} by Fourier expanding the input signal F (t) from the nine output terminals at the right end
The instantaneous value of sin (2πt + ψ _i ) is output, which enables real-time Fourier transform. FIG. 5 shows the result of actual real-time Fourier transform using this circuit.
Here, the figure (a) is the waveform of the input signal F (t),
(b) shows nine periodic functions obtained by Fourier transform.

[Mathematical proof of basic principles]

 Finally, the mathematical proof of Eq. (5) is briefly shown here.

Now consider a component wave C _i sin (2πit + ψ _i ) (14) having a frequency i times the fundamental frequency f when the function F (t) is Fourier expanded into N component waves. With respect to this function, the l-th division point (l = 0, 1,
..., k-1) is from the starting point of the basic period T Only with a time delay. Therefore, the function value of the l-th division point is Is. From the relation of T = 1 / To get

Therefore, the k-divided average value Fik (t) for the component wave having the frequency of i is Becomes Summarizing the equation using the addition theorem, Becomes here, If put and, using a synthetic theorem of the trigonometric _{function, Fik (t) = C i} c ik sin (2πit + ψ i + θ
_ik ) (23) is obtained.

Now consider the solution of the equation x ^k = 1. If the l-th solution (l = 0, 1, ..., K−1) is ω _l , Is known to hold (however, Now consider equation (26), which is equivalent to the equation x ^k = 1.

In this equation (26), when x ≠ 0, Is. This shows that the solution of equation x ^k = 1 other than 0 satisfies the equation (27). Therefore, the first solution ω _{1 in} Eq. (27)
Substituting To get From equation (25), That is, it can be seen that the sum of k solutions of the equation x ^k = 1 becomes zero. Considering the solution of this equation on the complex plane,
The coordinates are the vertices of a regular k-gon inscribed in the unit circle.

Now, the coordinate value of the l-th vertex of this regular k-gon is Let's express it with. Then, from the geometric symmetry of the regular k-gon, Is established. Therefore, from equations (29) and (30), Will also hold.

Then, in equations (30) and (31), il instead of l
Considering the case where (i is an integer), Is obtained. That is, when i = kl, Is always 1, so However, in the other cases, the sum becomes 0 because the vertex coordinates of the regular polygon are formed. On the other hand, for sine, equation (32) always holds due to the geometrical symmetry.

Here, let us return to equation (23). Equation (23) gives the coordinate values on the complex plane. Equation (19) is an equation, and coefficients a _ik and b _ik represented by equation (20) are from equations (32) and (33). Becomes Therefore, in equation (23), θ _ik = 0 holds for any positive integer k, i, and if i is an integer multiple of k, c _ik = k, and otherwise c _ik = 0 holds.

Now (However, i = kn, m is an integer part of N / k).

〔The invention's effect〕

As described above, according to the present invention, the N divided average values are obtained from the input signal F (t), and the instantaneous values of the N periodic functions are obtained based on the divided average values. However, it becomes possible to perform the real-time Fourier transform that continuously outputs the Fourier transform result.

[Brief description of drawings]

FIG. 1 is a flow chart showing a procedure of a real-time Fourier transform method according to the present invention, FIG. 2 is an explanatory view of a basic period dividing method in the method according to the present invention, and FIG. 3 is a real-time Fourier transform according to the present invention. FIG. 4 is a circuit diagram of an example of an arithmetic circuit for realizing the device, FIG. 4 is a conceptual diagram for explaining the configuration of the circuit shown in FIG. 3, and FIG. 5 is a real-time Fourier transform by the circuit shown in FIG. It is a graph which shows the performed result. S1 to S12 ... Each step in the flowchart, A2 to A13 ...
Inverting and adding operation element, B ... Delay element, B9, B6, B
5, B8, B7 ... Delay element array, C1 to C4 ... Inversion element, n ... Node.

Claims (6)

[Claims] 1. A signal F (t) varying with time t,
As the sum of N periodic functions with frequencies that are integer multiples of the fundamental frequency, (Where C _i is the amplitude and ψ _i is the phase) A Fourier transform method for transforming so that it can be expressed by a function G (t) defined by the equation: a step of inputting a signal F (t), The period of the basic cycle T given as the reciprocal of the reference frequency is divided into k parts, and the values at the respective divided points of the input signal F (t) are obtained as sample values, and the average value of the sample values is divided into k parts The procedure for obtaining Fk (t) is performed for each of k = 1, 2, ..., N, and N
The divided average values F1 (t), F2 (t), ..., FN
(T), and solving an N-ary simultaneous linear equation obtained from the relationship between the N periodic functions and the N divided average values to obtain instantaneous values at time t of the N periodic functions. A real-time Fourier transform method comprising the steps of: continuously obtaining. 2. An input signal F (t) is applied to a plurality of stages of delay circuits, and a sample value at each division time point is obtained based on the output of each stage of the delay circuits. The real-time Fourier transform method according to item 1. 3. The real-time Fourier transform method according to claim 2, wherein a BBD element is used as the delay circuit. 4. A signal F (t) varying with time t,
As the sum of N periodic functions with frequencies that are integer multiples of the fundamental frequency, (However, C _i is the amplitude, ψ _i is the phase) A Fourier transform device for transforming so as to be expressed by a function G (t) defined by the formula: Input means for inputting a signal F (t), The period of the basic cycle T given as the reciprocal of the reference frequency is divided into k parts, and the values at the respective divided points of the input signal F (t) are obtained as sample values, and the average value of the sample values is divided into k parts The procedure for obtaining the value Fk (t) is performed for each of k = 1, 2, ..., N, and N
The divided average values F1 (t), F2 (t), ..., FN
The instant at time t of the N periodic functions is solved by solving an N-ary simultaneous linear equation obtained from the relationship between the divided sampling means for obtaining (t) and the N periodic functions and the N divided average values. A real-time Fourier transform device comprising: an arithmetic unit that continuously obtains a value. 5. The divided sampling means has a plurality of stages of delay circuits for delaying the input signal F (t), and the sample value at each division time is obtained based on the output of each stage of the delay circuit. The real-time Fourier transform device according to claim 4, characterized in that 6. The real-time Fourier transform device according to claim 5, wherein a BBD element is used as the delay circuit.