LTI system circuit simulation system, method and device and storage medium

Abstract

The embodiment of the invention provides an LTI system circuit simulation system, method and device and a storage medium, and belongs to the technical field of experiment teaching. The system comprises a time domain analysis subsystem and a frequency domain analysis subsystem, the time domain analysis subsystem comprises a signal decomposition module, a linear system property proving module, a convolution theorem derivation module and an LTI system circuit signal processing module; according to the method and the device, the weighted sum of the plurality of unit impulse functions decomposed by the input signal is more intuitively and dynamically displayed, and the process that the input signal is gradually expanded by the unit impulse functions is displayed; and displaying the first result, the second result, the third result, the fourth result and the fifth result and an equivalence comparison result among the results step by step and sampling point by sampling point, so that a user is helped to understand the expansion process of each step and a final result thereof, and an abstract mathematical concept becomes visual and visible.

Description

Technical Field

[0001] The present application relates to the field of experimental teaching technology, and in particular to an LTI system circuit simulation system, method, device and storage medium. Background Art

[0002] The LTI system (Linear Time-Invariant) is a core concept in signal processing and system theory, and is widely used in communication, control, and audio processing. Traditional LTI system teaching systems usually require classroom lectures and textbook learning. The existing LTI system teaching effect is limited, and it is often difficult to establish an intuitive connection between mathematical formulas and actual system behaviors. Summary of the invention

[0003] In order to solve the above technical problems, the embodiments of the present application provide an LTI system circuit simulation system, method, device and storage medium.

[0004] In a first aspect, an embodiment of the present application provides an LTI system circuit simulation system, the system comprising: a time domain analysis subsystem and a frequency domain analysis subsystem; The time domain analysis subsystem includes: a signal decomposition module, a linear system property proof module, a convolution theorem derivation module and an LTI system circuit signal processing module; The signal decomposition module is used to decompose the input signal into a weighted sum of multiple unit impulse functions and display the process of the input signal being gradually expanded by the unit impulse functions; The linear system property proof module is used to obtain a first response signal after the weighted sum of each unit impulse function is processed by the LTI system circuit signal processing module, and use the first response signal as a first result, and, according to each unit impulse function and the corresponding signal component of each input signal as a weight, a plurality of second response signals are obtained after being processed by the LTI system circuit signal processing module, and use the sum of the second response signals as a second result; The linear system property proof module is further used to display the first result, the second result and the comparison result of the equivalence between the first result and the second result step by step and sampling point by sampling point; The convolution theorem derivation module is used to use the signal components of each of the input signals as weights, take the responses obtained after each of the unit impulse functions is processed by the LTI system circuit signal processing module as third response signals, perform weighted superposition of the signal components of each of the input signals on the third response signal, and take the obtained weighted superposition sum as the third result; The convolution theorem derivation module is further used to display the second result and the third result, and a comparison result of the equivalence between the second result and the third result; The frequency domain analysis subsystem is used to obtain the unit impulse response signal after the unit impulse function is processed by the LTI system circuit signal processing module, edit and change the frequency domain characteristics of the unit impulse response signal, convolve the changed unit impulse response signal with the input signal, and use the convolved signal with changed frequency domain characteristics as a fourth result, and send the input signal to the LTI system circuit signal processing module to obtain an output signal, and use the output signal as a fifth result; The frequency domain analysis subsystem is further used to display the fourth result and the fifth result, as well as a comparison result of the frequency domain difference between the fourth result and the fifth result.

[0005] In one embodiment, the signal decomposition module includes: a first signal generation submodule, a first impulse function generation submodule, a first step function control submodule, a first signal decomposition submodule and a first animation module; The first signal generating submodule is used to generate an input signal and specify a unit impulse function type; The first impulse function generating submodule is used to generate a unit impulse function according to the unit impulse function type and determine a corresponding time interval; The first signal decomposition submodule is used to decompose the input signal into a sum of multiple weighted unit impulse functions through a signal decomposition formula; The signal decomposition formula is: ; in, is the input signal, is the unit impulse function, Δt is the time interval, N is the number of discrete points of integration, and k is the index of the discrete point; The first step function control submodule is used to gradually increase the time interval through a unit step formula and superimpose a weighted unit impulse function; The unit step formula is: ; in, is a unit step function, is a signal component of the input signal expanded by a unit impulse function; The first animation module is used to dynamically display in steps according to the sampling points the various weighted impulse signals formed by the decomposition of the input signal by the unit impulse function at each sampling point, as well as the process of expanding the unit impulse function of the complete input signal based on each unit impulse function and the weighted sum of each unit impulse function.

[0006] In one embodiment, the linear system property proof module includes: a second signal generation submodule, a second impulse function generation submodule, a second signal decomposition submodule, a linear property reasoning module, and a second animation module; The second signal generating submodule is used to generate an input signal and specify a unit impulse function type; The second impulse function generating submodule is used to generate a unit impulse function according to the unit impulse function type and determine a corresponding time interval; The second signal decomposition submodule is used to decompose the input signal into a sum of multiple weighted unit impulse functions; The linear property reasoning module is used to obtain the first result through a first summation formula, and to obtain the second result through a second summation formula; The first summation formula is: ; Where x(t) is the input signal, is the impulse function, is the processing function of the signal processing module of the LTI system circuit, Δt is the time interval, N is the number of discrete points of integration, and k is the index of the discrete point; The second summation formula is: ; The second animation module is used to dynamically display the comparison results of each sampling point between the first result and the second result in steps according to the sampling points, and the comparison results of the equivalence between the first result and the second result.

[0007] In one embodiment, the convolution theorem derivation module includes: a third signal generation submodule, a third impulse function generation submodule, a third signal decomposition submodule, a convolution theorem proof module and a third animation module; The third signal generating submodule is used to generate an input signal and specify a unit impulse function type; The third impulse function generating submodule is used to generate a unit impulse function according to the unit impulse function type and determine a corresponding time interval; The third signal decomposition submodule is used to decompose the input signal into a sum of multiple weighted unit impulse functions; The convolution theorem proving module is used to obtain the second result through the second summation formula; The convolution theorem proving module is further used to obtain a third result through a third summation formula; The third summation formula is:

[0008] Where x(t) is the input signal, is the unit impulse function, is the processing function of the LTI system circuit signal processing module, h(t) is the impulse response obtained after the unit impulse signal is processed by the LTI system circuit signal processing module, Δt is the time interval, N is the number of discrete points of integration, and k is the index of the discrete point; The third animation module is used to dynamically display the comparison results between the second result and the third result of each sampling point in steps, and the comparison results of the equivalence between the second result and the third result, and, based on the weighted sum of the signal components of each impulse response signal and each input signal, finally form a complete convolution process of the impulse response and the input signal.

[0009] In one embodiment, the frequency domain analysis subsystem includes: a spectrum editing module; The spectrum editing module is used to click or drag the frequency on the spectrum graph to adjust it, and display the adjusted frequency in real time; The spectrum editing module is further used to edit and change the frequency domain characteristics of the unit impulse response signal, convolve the changed unit impulse response signal with the input signal, and use the convolved signal with the changed frequency domain characteristics as the fourth result.

[0010] In one embodiment, the frequency domain analysis subsystem further includes: a comparison display module; The comparison and display module is used to send the input signal to the LTI system circuit signal processing module to obtain an output signal, and use the output signal as the fifth result; The comparison display module is also used to display a comparison result signal generated by comparing the fifth result and the fourth result, and dynamically display the change process of the spectrum diagram.

[0011] In one embodiment, the system further comprises: a user interaction subsystem; The user interaction subsystem is used to control the sampling interval, total sampling duration, type of window function and windowing range in the signal processing and display process, and provides stepping, backtracking, comparison and zooming functions.

[0012] In a second aspect, an embodiment of the present application provides an LTI system circuit simulation method, wherein the LTI system circuit simulation method is applied to the LTI system circuit simulation system of the first aspect, wherein the method includes: Decomposing the input signal into a weighted sum of a plurality of unit impulse functions by the signal decomposition module, and displaying a process in which the input signal is gradually expanded by the unit impulse functions; Obtaining, through the linear system property proof module, a first response signal after the weighted sum of each of the unit impulse functions is processed by the LTI system circuit signal processing module, and taking the first response signal as a first result, and, according to each of the unit impulse functions and the corresponding signal components of each of the input signals as weights, obtaining a plurality of second response signals after being processed by the LTI system circuit signal processing module, and taking the sum of the second response signals as a second result; The first result, the second result, and the comparison result of the equivalence between the first result and the second result are also displayed step by step and sampling point by sampling point through the linear system property proof module; The convolution theorem derivation module uses the signal components of each input signal as a weight, uses the response obtained after each unit impulse function is processed by the LTI system circuit signal processing module as a third response signal, performs weighted superposition of the signal components of each input signal on the third response signal, and uses the obtained weighted superposition sum as a third result; The second result and the third result, as well as the comparison result of the equivalence between the second result and the third result, are also displayed through the convolution theorem derivation module; Obtaining, through the frequency domain analysis subsystem, a unit impulse response signal of the unit impulse function after being processed by the LTI system circuit signal processing module, editing and changing the frequency domain characteristics of the unit impulse response signal, convolving the changed unit impulse response signal with the input signal, and using the convolved signal with changed frequency domain characteristics as a fourth result, and sending the input signal to the LTI system circuit signal processing module to obtain an output signal, and using the output signal as a fifth result; The fourth result and the fifth result, as well as the comparison result of the frequency domain difference between the fourth result and the fifth result, are displayed by the frequency domain analysis subsystem.

[0013] In a third aspect, an embodiment of the present application provides an electronic device, including a memory and a processor, wherein the memory is used to store a computer program, and when the computer program is executed by the processor, the LTI system circuit simulation method provided in the second aspect is executed.

[0014] In a fourth aspect, an embodiment of the present application provides a computer-readable storage medium storing a computer program, which executes the LTI system circuit simulation method provided in the second aspect when running on a processor.

[0015] The LTI system circuit simulation system, method, device and storage medium provided by the present application more intuitively and dynamically display the decomposition of the input signal into the weighted sum of multiple unit impulse functions, and display the gradual expansion process of the input signal by the unit impulse function, and display the first result, the second result, the third result, the fourth result and the fifth result step by step sampling point as well as the comparison results of the equivalence between the results, so as to help users understand the expansion process of each step and its final result, and make abstract mathematical concepts intuitive and visible. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] In order to more clearly illustrate the technical solution of the present application, the following is a brief introduction to the drawings required for use in the embodiments. It should be understood that the following drawings only illustrate certain embodiments of the present application and should not be regarded as limiting the scope of protection of the present application. In each of the drawings, similar components are numbered similarly.

[0017] Figure 1 A schematic diagram of the structure of an LTI system circuit simulation system provided by an embodiment of the present application is shown; Figure 2 Another schematic diagram of the structure of the LTI system circuit simulation system provided by the embodiment of the present application is shown; Figure 3 A schematic diagram of a process of an LTI system circuit simulation method provided by an embodiment of the present application is shown; Figure 4 A schematic diagram of the structure of an electronic device provided in an embodiment of the present application is shown.

[0018] Icons: 10-LTI system circuit simulation system; 11-time domain analysis subsystem; 12-frequency domain analysis subsystem; 13-user interaction subsystem; 111-signal decomposition module; 112-linear system property proof module; 113-convolution theorem derivation module; 114-LTI system circuit signal processing module; 121-spectrum editing module; 122-comparison display module; 400-electronic device; 401-transceiver; 402-processor; 403-memory. DETAILED DESCRIPTION

[0019] The technical solutions in the embodiments of the present application will be described clearly and completely below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all of the embodiments.

[0020] The components of the embodiments of the present application generally described and shown in the drawings herein may be arranged and designed in various configurations. Therefore, the following detailed description of the embodiments of the present application provided in the drawings is not intended to limit the scope of the application claimed for protection, but merely represents the selected embodiments of the present application. Based on the embodiments of the present application, all other embodiments obtained by those skilled in the art without making creative work belong to the scope of protection of the present application.

[0021] Hereinafter, the terms "including", "having" and their cognates, which may be used in various embodiments of the present application, are intended only to indicate specific features, numbers, steps, operations, elements, components, or combinations of the foregoing items, and should not be understood as first excluding the existence of one or more other features, numbers, steps, operations, elements, components, or combinations of the foregoing items or adding the possibility of one or more features, numbers, steps, operations, elements, components, or combinations of the foregoing items.

[0022] Furthermore, the terms “first”, “second”, “third”, etc. are merely used for distinguishing descriptions and are not to be understood as indicating or implying relative importance.

[0023] Unless otherwise defined, all terms (including technical terms and scientific terms) used herein have the same meanings as those generally understood by those skilled in the art to which the various embodiments of the present application belong. The terms (such as those defined in generally used dictionaries) will be interpreted as having the same meanings as the contextual meanings in the relevant technical field and will not be interpreted as having idealized meanings or overly formal meanings unless clearly defined in the various embodiments of the present application.

[0024] LTI system (Linear Time Invariant System) is a core concept in signal processing and system theory, and is widely used in fields such as communication, control, and audio processing. Traditional LTI system teaching methods usually include: theoretical explanation, mathematical derivation, simple simulation, and laboratory practice. However, these traditional methods have the following shortcomings: high abstraction, pure mathematical derivation and theoretical explanation are often too abstract, and it is difficult for users to intuitively understand the system behavior; insufficient interactivity, static charts and traditional experimental equipment are usually difficult to adjust system parameters in real time and immediately observe their impact; low visualization, traditional methods are difficult to dynamically display the signal change process in both the time domain and the frequency domain, which limits the user's understanding of the comprehensive characteristics of the system; theory and practice are out of touch, and users often find it difficult to establish an intuitive connection between mathematical formulas and actual system behavior. Therefore, in order to solve the above problems, this application proposes an LTI system circuit simulation system 10, method, device, and storage medium.

[0025] Example 1 The present application embodiment provides a LTI system circuit simulation system 10, such as Figure 1 As shown, the system includes: a time domain analysis subsystem 11 and a frequency domain analysis subsystem 12; the time domain analysis subsystem 11 includes: a signal decomposition module 111, a linear system property proof module 112, a convolution theorem derivation module 113 and an LTI system circuit signal processing module 114; the signal decomposition module 111 is used to decompose the input signal into a weighted sum of multiple unit impulse functions, and to display the process of the input signal being gradually expanded by the unit impulse function; the linear system property proof module 112 is used to obtain the first response signal after the weighted sum of each unit impulse function is processed by the LTI system circuit signal processing module 114, and the first response signal is used as the first result, and, according to each unit impulse function and the corresponding signal component of each input signal as the weight, the LTI system circuit The signal processing module 114 obtains multiple second response signals after processing, and takes the sum of the second response signals as the second result; the linear system property proof module 112 is also used to display the first result, the second result and the comparison result of the equivalence between the first result and the second result step by step and sampling point by sampling point; the convolution theorem derivation module 113 is used to use the signal component of each input signal as a weight, and take the response of each unit impulse function obtained after being processed by the LTI system circuit signal processing module 114 as the third response signal, perform weighted superposition of the signal components of each input signal on the third response signal, and take the obtained weighted superposition sum as the third result; the convolution theorem derivation module 113 is also used to display the second result and the third result, as well as the comparison result of the equivalence between the second result and the third result.

[0026] The frequency domain analysis subsystem 12 is used to obtain the unit impulse response signal after the unit impulse function is processed by the LTI system circuit signal processing module 114, edit and change the frequency domain characteristics of the unit impulse response signal, convolve the changed unit impulse response signal with the input signal, and use the convolved signal with changed frequency domain characteristics as the fourth result, and send the input signal to the LTI system circuit signal processing module 114 to obtain the output signal, and use the output signal as the fifth result; the frequency domain analysis subsystem 12 is also used to display the fourth result and the fifth result, as well as the comparison results of the frequency domain differences between the fourth result and the fifth result.

[0027] It should be noted that the LTI system has a basic property, that is, in the LTI system, the behavior of the system can be completely described by the unit impulse response signal. For an LTI system, the input signal is x(t), the output signal is y(t), and the impulse response obtained by the unit impulse response signal input into the system is h(t). Then the output signal can be expressed as the convolution of the input signal and the impulse response signal: y(t) = (x * h)(t).

[0028] Therefore, if we use a specially selected unit pulse signal as input into the LTI system, obtain the impulse response h(t), and then convolve any input signal with the impulse response, the output signal of the system can be simulated through the convolution operation. Therefore, we set the original formula: y(t) = T[x(t)], and derive the convolution theorem according to the following steps: (1) x(t) = ∫x(τ)δ(t-τ) dτ; (2) y(t) = T[x(t)]= T[∫x(τ)δ(t-τ) dτ]; (3) y(t) = ∫T[x(τ)δ(t-τ)]dτ (using linear properties); (4) y(t) = ∫x(τ)T[δ(t-τ)] dτ (again using linear properties); (5) y(t) = ∫x(τ)h(t-τ) dτ, and the final convolution theorem formula is: y(t) = ∫x(τ)h(t-τ) dτ.

[0029] This solution uses a graphical visualization method in the experiment to dynamically display the sampling characteristics of the unit impulse signal and the convolution operation process of the unit impulse response and the input signal in step by step, so as to understand the steps of signal decomposition and convolution in detail. By visualizing the input signal and system impulse response in the time domain and frequency domain, users can see more clearly how to calculate the output signal of the system through convolution.

[0030] In addition, the frequency domain characteristics of the system impulse response signal are interactively changed on the spectrum diagram and convolved with the input signal to change the frequency domain characteristics of the system output signal. The original system output signal is further compared with the above convolution result in the frequency domain to show the consistency of the two output signals at the unmodified frequency point and the precise difference at the modified frequency point, intuitively showing that the behavior of the system is completely described by its impulse response. These visualization and interactive methods help users to have a deeper understanding of the behavior of the LTI system and the mathematical nature of convolution.

[0031] In one embodiment, the signal decomposition module 111 includes: a first signal generation submodule, a first impulse function generation submodule, a first step function control submodule, a first signal decomposition submodule and a first animation module; the first signal generation submodule is used to generate an input signal and specify a unit impulse function type; the first impulse function generation submodule is used to generate a unit impulse function according to the unit impulse function type and determine the corresponding time interval; the first signal decomposition submodule is used to decompose the input signal into a sum of multiple weighted unit impulse functions through a signal decomposition formula; The signal decomposition formula is: ; in, is the input signal, is the unit impulse function, Δt is the time interval, N is the number of discrete points of integration, and k is the index of the discrete point.

[0032] A first step function control submodule is used to gradually increase the time interval through a unit step formula and superimpose a weighted unit impulse function; The unit step formula is: ; in, is a unit step function, It is the signal component of the input signal expanded by the unit impulse function; The first animation module is used to dynamically display in steps according to the sampling points the various weighted impulse signals formed by the decomposition of the input signal by the unit impulse function at each sampling point, and the process of expanding the unit impulse function that finally forms the complete input signal according to the unit impulse function and the weighted sum of the unit impulse functions.

[0033] It should be noted that the impulse function expansion formula of the input signal x(t) δ(t) has an important property: the sampling property, which indicates that the impulse function can extract the value of the function x(t) at t=τ, as shown in the following formula:

[0034] Any continuous-time signal x(t) can be expressed as a weighted sum of an infinite number of unit impulse functions, as shown in the following basic formula:

[0035] The integral form shows that x(t) can be regarded as the superposition of the weighted sum of unit impulse functions at all time points, where x(τ) is the weighting coefficient.

[0036] In order to better demonstrate the expansion process, this solution introduces the window function To limit the integral range, so that it gradually expands from a limited range to an infinite range. Among them, the window function is defined as follows:

[0037] Use window functions to decompose the underlying formula into finite and infinite parts:

[0038] As T approaches infinity, this formula approaches the basic formula:

[0039] For the convenience of step-by-step display, the input signal is decomposed into the sum of multiple weighted unit impulse functions through the signal decomposition formula. At the same time, for the convenience of step-by-step control in the experimental device to trigger and display the superposition of each component, the time interval is gradually increased through the unit step formula, and the weighted unit impulse functions are superimposed, where the unit step function of the unit step formula has the following function: when t≥kΔt, =1, allowing the signal to pass through. When t<kΔt, =0, blocking the signal from passing through.

[0040] Through the unit step function control the start time of each component, so as to facilitate the display of the superposition of each component by controlling the trigger of the unit step function in the experimental device.

[0041] Furthermore, this scheme dynamically displays step by step the weighted impulse signals formed by the decomposition of the input signal by the unit impulse function at each sampling point according to the sampling points. First, determine the total duration Ts of the input signal. Then, gradually increase the value of T, such as T = Δt, T = 2Δt, T = 3Δt, ……, until T is greater than or equal to Ts. The specific process is as follows: Step 1: Initial display. At the beginning, a smaller T = Δt can be selected, which only contains one unit impulse function. The specific formula is as follows:

[0042] Step 2: Gradually increase the integration range, gradually increase the value of T, increasing by one Δt each time; when T = 2Δt, display by controlling the unit step function. The specific formula is as follows:

[0043] When T = 3Δt, display by controlling the unit step function, and continue to increase the value of T until the entire signal range is covered. Finally, when T is infinite, the integration covers the entire time axis. The specific formula is as follows:

[0044] It should be understood that the user freely generates a path of original signal through the first signal generation sub-module, and specifies the type of unit impulse function and the time interval Δt, where the type of unit impulse function includes: rectangular function or normal distribution function. The first impulse function generation sub-module generates the unit impulse function according to the type of unit impulse function, and uses the graphical interface to display a single impulse function and its corresponding weight .

[0045] Gradually increase the integration range by controlling the step function. The specific process is as follows: (1) The user gradually increases the value of T by Δt each time, and the increase of the T value is dynamically displayed. (2) The graph is updated in real time to show the new impulse function and its weight. (3) The new impulse function is superimposed on the graph to gradually construct the entire signal.

[0046] Gradually increase the time interval through the unit step formula, and superimpose the weighted unit impulse functions, where the unit step function of the unit step formula has the following function: when t ≥ kΔt, = 1, allowing the signal to pass through. When t < kΔt, = 0, blocking the signal from passing through.

[0047] It is further understood that when k = 0, only x(0)δ(t) is allowed to pass through. As the value of k increases, more components are gradually introduced. This process simulates the gradual expansion of the integration range. The user can adjust the value of k through the control interface. As the value of k increases, the LTI system circuit simulation system 10 calculates and displays the cumulative effect of the passed components in real time, and the user can intuitively see how the signal is gradually constructed.

[0048] In one embodiment, the linear system property proof module 112 includes: a second signal generation sub-module, a second impulse function generation sub-module, a second signal decomposition sub-module, a linear property reasoning module, and a second animation module; the second signal generation sub-module is used to generate an input signal and specify the type of the unit impulse function; the second impulse function generation sub-module is used to generate a unit impulse function according to the type of the unit impulse function and determine the corresponding time interval; the second signal decomposition sub-module is used to decompose the input signal into the sum of multiple weighted unit impulse functions; the linear property reasoning module is used to obtain a first result through a first summation formula, and obtain a second result through a second summation formula; The first summation formula is: ; where x(t) is the input signal, is the impulse function, is the processing function of the LTI system circuit signal processing module 114, Δt is the time interval, N is the number of discrete points of the integration, and k is the index of the discrete point; The second summation formula is: ; The second animation module is used to dynamically display step by step the comparison results of each sampling point between the first result and the second result, and the comparison result of the equivalence between the first result and the second result according to the sampling points.

[0049] It should be noted that in the above implementation, the following formula is derived after windowing the continuous integral function:

[0050] Assume that the processing function of the LTI system circuit signal processing module 114 is F, that is, y(t) = F[x(t)], and obtain the first result through the first summation formula, which is as follows:

[0051] The second summation formula is derived through the LTI system as follows:

[0052] In order to facilitate the step-by-step control triggering and display of the superposition of each component in the experimental setup, the unit step function u(t) is used to explicitly represent the delay operation. The above formula is further rewritten as follows:

[0053] pass Controlling the start time of each component makes it easier to demonstrate the superposition of each component in the experimental setup by controlling the triggering of the step function.

[0054] In this embodiment, by selecting T=Δt, only one impulse function is included:

[0055] Display separately and Compare the results.

[0056] Gradually increase the value of T, each time increasing by one Δt, and compare the obtained results with each sampling point until the entire signal range is covered. For example, when T=2Δt:

[0057] The comparison results of the following two formulas are shown by controlling the step function.

[0058]

[0059]

[0060] Continue to increase the value of T until the entire signal range is covered. Eventually, when T is infinite, the integral covers the entire time axis, as shown in the following formula:

[0061] It should be noted that the user performs the initial display in advance, and the steps of generating the input signal and the unit impulse signal and display are shown above, which will not be repeated here. Then, the integral range needs to be gradually increased by controlling the step function. The specific steps are as follows: (1) The user gradually increases the T value, increasing one Δt each time, and dynamically displays the increase of the T value. (2) The graph is updated in real time to display the new impulse function and its weight. (3) The sum of the impulse function and its weight in each step is sent to the LTI system, and the first result is obtained through the first summation formula. (4) Each impulse function and its weight in each step is sent to the linear property reasoning module, and the second result is obtained through the second summation formula. (5) The first result and the second result are compared, and the comparison results of each sampling point between the first result and the second result, as well as the comparison results of the equivalence between the first result and the second result are displayed, and the important conclusion of exchanging the order of integration and signal processing is obtained.

[0062] It is further explained that the present application adopts a dynamic step-by-step display formula to gradually display y(t) = F[x(t)] = F[∫x(τ)δ(t-τ) dτ]= ∫F[x(τ)δ(t-τ)] dτ. When an impulse signal is expanded by the impulse function, the sum of the results of each impulse function and its weight after being processed by the linear signal system is equal to the sum of each impulse function and its weight after being processed by the linear system. The basic formula is transformed equivalently to show this process more clearly step by step. Specifically, the integral of the basic formula is decomposed into finite and infinite parts, and a window function is introduced to gradually show the expansion of each step.

[0063] In one embodiment, the convolution theorem derivation module 113 includes: a third signal generation submodule, a third impulse function generation submodule, a third signal decomposition submodule, a convolution theorem proving module and a third animation module; the third signal generation submodule is used to generate an input signal and specify a unit impulse function type; the third impulse function generation submodule is used to generate a unit impulse function according to the unit impulse function type and determine the corresponding time interval; the third signal decomposition submodule is used to decompose the input signal into a sum of multiple weighted unit impulse functions; the convolution theorem proving module is used to obtain a second result through a second summation formula; the convolution theorem proving module is also used to obtain a third result through a third summation formula; The third summation formula is:

[0064] Where x(t) is the input signal, is the unit impulse function, is the processing function of the LTI system circuit signal processing module 114, h(t) is the impulse response obtained after the unit impulse signal is processed by the LTI system circuit signal processing module 114, Δt is the time interval, N is the number of discrete points of integration, and k is the index of the discrete point; the third animation module is used to dynamically display the comparison results between the second result and the third result of each sampling point in steps, and the comparison results of the equivalence between the second result and the third result, and, according to the weighted sum of the signal components of each impulse response signal and each input signal, finally form a complete convolution process of the impulse response and the input signal.

[0065] It should be noted that the following discrete point formula is obtained by windowing the integral function:

[0066] Due to the linear nature of the LTI system, x(kΔt) can be regarded as a discrete point that is independent of the independent variable t. , which can be regarded as a scalar x(kΔt) multiplied by δ(t−kΔt). According to the linear properties of the LTI system, we have: ]=x(kΔt)F[ ].

[0067] Due to the time-invariant nature of the LTI system, if the system's response to a unit impulse function δ(t) is h(t), that is, F[δ(t)]=h(t), then the impulse function with time offset is , the response is .

[0068] Therefore, combining these two points, we have F[ ]=x(kΔt)h[ ], so we get the following step-by-step formula of the convolution theorem:

[0069] in represents the input signal, is the impulse response obtained after the unit impulse function is processed by the linear signal processing system, is a processing function of the LTI system circuit signal processing module 114 .

[0070] In order to facilitate the step-by-step control triggering and display of the superposition of each component in the experimental setup, the unit step function u(t) will be used to explicitly represent the delay operation. The above formula is further rewritten as follows:

[0071] pass Controlling the start time of each component makes it easier to demonstrate the superposition of each component in the experimental setup by controlling the triggering of the step function.

[0072] In this embodiment, by selecting a smaller T=Δt, only one impulse function is included:

[0073] Display separately and Compare the results.

[0074] Gradually increase the value of T, each time increasing by one Δt, and compare the obtained results with each sampling point until the entire signal range is covered. For example, when T=2Δt:

[0075] The comparison results of the following two formulas are shown by controlling the step function.

[0076]

[0077]

[0078] Continue to increase the value of T until the entire signal range is covered. Eventually, when T is infinite, the integral covers the entire time axis, as shown in the following formula:

[0079] It should be noted that the result of infinite integration of step-by-step superposition is The convolution results are compared and displayed, and finally the step-by-step derivation of the complete convolution theorem is demonstrated.

[0080] It should be noted that the user performs the initial display in advance, and the steps of generating the input signal and the unit impulse signal and display are shown above, which will not be repeated here. Then, the integral range needs to be gradually increased by controlling the step function. The specific steps are as follows: (1) The user gradually increases the T value, increasing it by one Δt each time, and dynamically displays the increase in the T value. (2) The graph is updated in real time to display the new impulse function and its weight. (3) Each impulse function and its weight in each step are sent to the signal processing system to display the results of each component , and then sum each component through the second summation formula to obtain the second result. (4) Send each impulse function in each step to the signal processing system, multiply the obtained impulse response with the value corresponding to the input signal in this step, display the result of each component, and obtain the third result through the third summation formula (5) Dynamically display the comparison results between the second result and the third result of each sampling point in steps, and the comparison results of the equivalence between the second result and the third result, and, according to the weighted sum of the signal components of each impulse response signal and each input signal, finally form a complete convolution process of the impulse response and the input signal.

[0081] It is further explained that the present application uses a fixed window to derive a dynamic step-by-step display formula, which gradually presents that due to the superposition and homogeneity of the LTI system, the response of the system to the input signal can be expressed as a weighted superposition of the unit impulse response of each input component as a weight, and further determines that the output y(t) can be regarded as the unit impulse response T[δ ] to prove the convolution theorem. By using a fixed window to transform the original formula equivalently, at each time step, the system processes the input signal component, calculates the weighted sum of the component to the output, and then superimposes all these weighted sums to obtain the system's response to the entire input signal. This interactive and dynamic display method can help users deeply understand the specific process of convolution, as well as the additive and homogeneous characteristics of the LTI system in an intuitive and vivid way, effectively reducing the difficulty of understanding and improving learning effects.

[0082] In one embodiment, if Figure 2 As shown, the frequency domain analysis subsystem 12 includes: a spectrum editing module 121; the spectrum editing module 121 is used to click or drag the frequency on the spectrum graph to adjust it, and display the adjusted frequency in real time; the spectrum editing module 121 is also used to edit and change the frequency domain characteristics of the unit impulse response signal, convolve the changed unit impulse response signal with the input signal, and use the convolved signal with the changed frequency domain characteristics as the fourth result.

[0083] In one embodiment, the frequency domain analysis subsystem 12 also includes: a comparison display module 122; the comparison display module 122 is used to send the input signal to the LTI system circuit signal processing module 114 to obtain an output signal, and use the output signal as the fifth result; the comparison display module 122 is also used to display the comparison result signal generated by comparing the fifth result and the fourth result, and dynamically display the change process of the spectrum diagram.

[0084] It should be noted that the spectrum editing module 121 can realize the following functions: (1) frequency selection and adjustment, the user can click or drag the frequency component on the spectrum graph to adjust and change its amplitude and phase. (2) real-time feedback, real-time display of the spectrum and time domain signal of the adjusted system output signal, so that the user can immediately see the effect of the adjustment.

[0085] It is further explained that in order to make the user's operation more convenient and intuitive, this module uses some interactive means to simplify the adjustment process of amplitude and phase, such as: the user can drag the point on the polar coordinates to adjust the amplitude and phase at the same time, where polar coordinates are used to intuitively represent the amplitude and phase in the spectrum diagram; each frequency component is represented as a vector, and the user can adjust it by dragging the endpoint of the vector. If dragging radially, the length of the vector is changed to adjust the amplitude. If dragging at an angle, the vector is rotated to adjust the phase; the user can also directly adjust the formula and mapping function.

[0086] Users only need to perform simple drag operations to directly and clearly adjust the frequency components in the visual interface of the spectrum graph, thereby successfully achieving the desired signal modification goals, greatly improving the convenience and efficiency of operations, and effectively reducing the difficulty of understanding and improving learning effects.

[0087] In one embodiment, the system also includes: a user interaction subsystem 13; the user interaction subsystem 13 is used to control the sampling interval, total sampling time, type of window function and windowing range in the signal processing and display process, and provide stepping, backtracking, comparison and zooming functions.

[0088] In this embodiment, the first animation module, the second animation module and the third animation module are used to generate an animation sequence based on the output of the signal processing module, and create animation frames for the processing process of each impulse function and its weight.

[0089] Each animation module includes the following parts: a graphics rendering engine, which is responsible for converting mathematical data into visual graphics, drawing impulse functions, weights, and cumulative signals; a multi-view manager, which is used to manage multiple views, such as time domain view and frequency domain view, and allows the simultaneous display of signal representations from different angles; an animation controller, which is used to control the playback, pause, and speed adjustment of animations, etc., to achieve smooth animation transition effects.

[0090] Each animation module is controlled by the user interaction subsystem 13 to allow the user to control the animation playback, and provide interactive functions such as stepping, backtracking, comparing and zooming to observe specific parts of the signal in detail, and support fast playback and jumping of the animation by storing intermediate calculation results.

[0091] The present application more intuitively and dynamically displays the decomposition of an input signal into a weighted sum of multiple unit impulse functions, and displays the process of gradual expansion of the input signal by the unit impulse function, and displays the first result, the second result, the third result, the fourth result and the fifth result step by step sampling point, as well as the comparison results of the equivalence between the results, to help users understand the expansion process of each step and its final result, making abstract mathematical concepts intuitive and visible.

[0092] Example 2 An embodiment of the present application provides a method for simulating an LTI system circuit.

[0093] See also Figure 3 , LTI system circuit simulation methods include: S301 , decomposing an input signal into a weighted sum of a plurality of unit impulse functions through the signal decomposition module 111 , and displaying a process in which the input signal is gradually expanded by the unit impulse functions.

[0094] S302, obtaining the first response signal after the weighted sum of each unit impulse function is processed by the LTI system circuit signal processing module 114 through the linear system property proof module 112, and taking the first response signal as the first result, and, according to each unit impulse function and the corresponding signal component of each input signal as the weight, obtaining multiple second response signals after being processed by the LTI system circuit signal processing module 114, and taking the sum of the second response signals as the second result.

[0095] S303, the linear system property proof module 112 is also used to display the first result, the second result, and the comparison result of the equivalence between the first result and the second result step by step and sampling point by sampling point.

[0096] S304, using the convolution theorem derivation module 113 to take the signal components of each input signal as a weight, taking the response of each unit impulse function after being processed by the LTI system circuit signal processing module 114 as a third response signal, performing weighted superposition of the signal components of each input signal on the third response signal, and taking the obtained weighted superposition sum as the third result.

[0097] S305 , the second result and the third result, as well as the comparison result of the equivalence between the second result and the third result, are also displayed through the convolution theorem derivation module 113 .

[0098] S306, obtaining the unit impulse response signal after the unit impulse function is processed by the LTI system circuit signal processing module 114 through the frequency domain analysis subsystem 12, editing and changing the frequency domain characteristics of the unit impulse response signal, convolving the changed unit impulse response signal with the input signal, and taking the convolved signal with changed frequency domain characteristics as the fourth result, and sending the input signal to the LTI system circuit signal processing module 114 to obtain the output signal, and taking the output signal as the fifth result.

[0099] S307 , displaying the fourth result and the fifth result, as well as the comparison result of the frequency domain difference between the fourth result and the fifth result, through the frequency domain analysis subsystem 12 .

[0100] The present application more intuitively and dynamically displays the decomposition of an input signal into a weighted sum of multiple unit impulse functions, and displays the process of gradual expansion of the input signal by the unit impulse function, and displays the first result, the second result, the third result, the fourth result and the fifth result step by step sampling point, as well as the comparison results of the equivalence between the results, to help users understand the expansion process of each step and its final result, making abstract mathematical concepts intuitive and visible.

[0101] Example 3 In addition, an embodiment of the present application provides an electronic device, including a memory and a processor, wherein the memory stores a computer program, and when the computer program runs on the processor, the LTI system circuit simulation method provided in Embodiment 2 is executed.

[0102] For details, see Figure 4The electronic device 400 includes: a transceiver 401, a bus interface and a processor 402, wherein the processor 402 is used to: decompose the input signal into a weighted sum of multiple unit impulse functions through the signal decomposition module 111, and display the process of the input signal being gradually expanded by the unit impulse function; obtain the first response signal after the weighted sum of each unit impulse function is processed by the LTI system circuit signal processing module 114 through the linear system property proof module 112, and use the first response signal as the first result; and, obtain multiple second response signals after the signal components of each input signal as weights are processed by the LTI system circuit signal processing module 114 according to each unit impulse function and the corresponding second response signals, and use the sum of the second response signals as the second result; also display the first result, the second result and the comparison result of the equivalence between the first result and the second result step by step sampling point by sampling point through the linear system property proof module 112; use the signal components of each input signal as weights through the convolution theorem derivation module 113, and compare the equivalence between each unit impulse The response obtained after the excitation function is processed by the LTI system circuit signal processing module 114 is used as the third response signal, and the signal components of each input signal are weighted superimposed on the third response signal, and the obtained weighted superposition sum is used as the third result; the second result and the third result, as well as the comparison result of the equivalence between the second result and the third result are also displayed through the convolution theorem derivation module 113; the unit impulse response signal after the unit impulse function is processed by the LTI system circuit signal processing module 114 is obtained through the frequency domain analysis subsystem 12, the frequency domain characteristics of the unit impulse response signal are edited and changed, the changed unit impulse response signal is convolved with the input signal, and the convolution signal with changed frequency domain characteristics is used as the fourth result, and the input signal is sent to the LTI system circuit signal processing module 114 to obtain the output signal, and the output signal is used as the fifth result; the fourth result and the fifth result are displayed through the frequency domain analysis subsystem 12, as well as the comparison result of the frequency domain difference between the fourth result and the fifth result.

[0103] In the embodiment of the present application, the electronic device 400 further includes a memory 403. Figure 4 In the embodiment, the bus architecture may include any number of interconnected buses and bridges, specifically connecting various circuits of one or more processors represented by processor 402 and memory represented by memory 403. The bus architecture may also connect various other circuits such as peripherals, voltage regulators, and power management circuits, which are well known in the art and are therefore not further described herein. The bus interface provides an interface. The transceiver 401 may be a plurality of components, i.e., including a transmitter and a receiver, providing a unit for communicating with various other devices on a transmission medium. The processor 402 is responsible for managing the bus architecture and general processing, and the memory 403 may store data used by the processor 402 when performing operations.

[0104] The electronic device 400 provided in the embodiment of the present application can execute the steps of the LTI system circuit simulation method provided in the above method embodiment 2, which will not be described again here to avoid repetition.

[0105] The electronic device provided in this embodiment dynamically displays the expansion process of the impulse function of the input signal x(t) in a more intuitive way, decomposes the integral of the original formula into finite and infinite parts, and introduces a window function to gradually display the expansion of each step, thereby helping users understand the expansion process of each step and its final result, making abstract mathematical concepts intuitive and visible.

[0106] Example 4 The present application also provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the LTI system circuit simulation method provided in Example 2 is implemented.

[0107] In this embodiment, the computer-readable storage medium may be a read-only memory (ROM), a random access memory (RAM), a magnetic disk, or an optical disk.

[0108] The computer-readable storage medium provided in this embodiment can implement the LTI system circuit simulation method provided in Embodiment 2, and will not be described again here to avoid repetition.

[0109] The embodiments of the present application are described above in conjunction with the accompanying drawings, but the present application is not limited to the above-mentioned specific implementation methods. The above-mentioned specific implementation methods are merely illustrative and not restrictive. Under the guidance of the present application, ordinary technicians in this field can also make many forms without departing from the purpose of the present application and the scope of protection of the claims, all of which are within the protection of the present application.

Claims

1. An LTI system circuit simulation system, characterized in that: The system comprises: a time domain analysis subsystem and a frequency domain analysis subsystem; The time domain analysis subsystem includes: a signal decomposition module, a linear system property proof module, a convolution theorem derivation module and an LTI system circuit signal processing module; The signal decomposition module is used to decompose the input signal into a weighted sum of multiple unit impulse functions and display the process of the input signal being gradually expanded by the unit impulse functions; The linear system property proof module is used to obtain a first response signal after the weighted sum of each unit impulse function is processed by the LTI system circuit signal processing module, and use the first response signal as a first result, and, according to each unit impulse function and the corresponding signal component of each input signal as a weight, a plurality of second response signals are obtained after being processed by the LTI system circuit signal processing module, and use the sum of the second response signals as a second result; The linear system property proof module is further used to display the first result, the second result and the comparison result of the equivalence between the first result and the second result step by step and sampling point by sampling point; The convolution theorem derivation module is used to use the signal components of each of the input signals as weights, take the responses obtained after each of the unit impulse functions is processed by the LTI system circuit signal processing module as third response signals, perform weighted superposition of the signal components of each of the input signals on the third response signal, and take the obtained weighted superposition sum as the third result; The convolution theorem derivation module is further used to display the second result and the third result, and a comparison result of the equivalence between the second result and the third result; The frequency domain analysis subsystem is used to obtain the unit impulse response signal after the unit impulse function is processed by the LTI system circuit signal processing module, edit and change the frequency domain characteristics of the unit impulse response signal, convolve the changed unit impulse response signal with the input signal, and use the convolved signal with changed frequency domain characteristics as a fourth result, and send the input signal to the LTI system circuit signal processing module to obtain an output signal, and use the output signal as a fifth result; The frequency domain analysis subsystem is further used to display the fourth result and the fifth result, as well as a comparison result of the frequency domain difference between the fourth result and the fifth result.

2. The LTI system circuit simulation system according to claim 1, characterized in that: The signal decomposition module includes: a first signal generation submodule, a first impulse function generation submodule, a first step function control submodule, a first signal decomposition submodule and a first animation module; The first signal generating submodule is used to generate an input signal and specify a unit impulse function type; The first impulse function generating submodule is used to generate a unit impulse function according to the unit impulse function type and determine a corresponding time interval; The first signal decomposition submodule is used to decompose the input signal into a sum of multiple weighted unit impulse functions through a signal decomposition formula; The signal decomposition formula is: ； in, is the input signal, is the unit impulse function, Δt is the time interval, N is the number of discrete points of integration, and k is the index of the discrete point; The first step function control submodule is used to gradually increase the time interval through a unit step formula and superimpose a weighted unit impulse function; The unit step formula is: ； in, is a unit step function, is a signal component of the input signal expanded by a unit impulse function; The first animation module is used to dynamically display in steps according to the sampling points the various weighted impulse signals formed by the decomposition of the input signal by the unit impulse function at each sampling point, as well as the process of expanding the unit impulse function of the complete input signal based on each unit impulse function and the weighted sum of each unit impulse function.

3. The LTI system circuit simulation system according to claim 1, characterized in that: The linear system property proof module includes: a second signal generation submodule, a second impulse function generation submodule, a second signal decomposition submodule, a linear property reasoning module and a second animation module; The second signal generating submodule is used to generate an input signal and specify a unit impulse function type; The second impulse function generating submodule is used to generate a unit impulse function according to the unit impulse function type and determine a corresponding time interval; The second signal decomposition submodule is used to decompose the input signal into a sum of multiple weighted unit impulse functions; The linear property reasoning module is used to obtain the first result through a first summation formula, and to obtain the second result through a second summation formula; The first summation formula is: ； Where x(t) is the input signal, is the impulse function, is the processing function of the signal processing module of the LTI system circuit, Δt is the time interval, N is the number of discrete points of integration, and k is the index of the discrete point; The second summation formula is: ； The second animation module is used to dynamically display the comparison results of each sampling point between the first result and the second result in steps according to the sampling points, and the comparison results of the equivalence between the first result and the second result.

4. The LTI system circuit simulation system according to claim 3, characterized in that: The convolution theorem derivation module includes: a third signal generation submodule, a third impulse function generation submodule, a third signal decomposition submodule, a convolution theorem proof module and a third animation module; The third signal generating submodule is used to generate an input signal and specify a unit impulse function type; The third impulse function generating submodule is used to generate a unit impulse function according to the unit impulse function type and determine a corresponding time interval; The third signal decomposition submodule is used to decompose the input signal into a sum of multiple weighted unit impulse functions; The convolution theorem proving module is used to obtain the second result through the second summation formula; The convolution theorem proving module is further used to obtain a third result through a third summation formula; The third summation formula is: Where x(t) is the input signal, is the unit impulse function, is the processing function of the LTI system circuit signal processing module, h(t) is the impulse response obtained after the unit impulse signal is processed by the LTI system circuit signal processing module, Δt is the time interval, N is the number of discrete points of integration, and k is the index of the discrete point; The third animation module is used to dynamically display the comparison results between the second result and the third result of each sampling point in steps, and the comparison results of the equivalence between the second result and the third result, and, based on the weighted sum of the signal components of each impulse response signal and each input signal, finally form a complete convolution process of the impulse response and the input signal.

5. The LTI system circuit simulation system according to claim 1, characterized in that: The frequency domain analysis subsystem includes: a spectrum editing module; The spectrum editing module is used to click or drag the frequency on the spectrum graph to adjust it, and display the adjusted frequency in real time; The spectrum editing module is further used to edit and change the frequency domain characteristics of the unit impulse response signal, convolve the changed unit impulse response signal with the input signal, and use the convolved signal with the changed frequency domain characteristics as the fourth result.

6. The LTI system circuit simulation system according to claim 5, characterized in that: The frequency domain analysis subsystem also includes: a comparison and display module; The comparison and display module is used to send the input signal to the LTI system circuit signal processing module to obtain an output signal, and use the output signal as the fifth result; The comparison display module is also used to display a comparison result signal generated by comparing the fifth result and the fourth result, and dynamically display the change process of the spectrum diagram.

7. The LTI system circuit simulation system according to any one of claims 1 to 6, characterized in that: The system further comprises: a user interaction subsystem; The user interaction subsystem is used to control the sampling interval, total sampling duration, type of window function and windowing range in the signal processing and display process, and provides stepping, backtracking, comparison and zooming functions.

8. A LTI system circuit simulation method, characterized in that: Applied to the LTI system circuit simulation system according to any one of claims 1 to 7, the method comprising: Decomposing the input signal into a weighted sum of a plurality of unit impulse functions by the signal decomposition module, and displaying a process in which the input signal is gradually expanded by the unit impulse functions; Obtaining, through the linear system property proof module, a first response signal after the weighted sum of each of the unit impulse functions is processed by the LTI system circuit signal processing module, and taking the first response signal as a first result, and, according to each of the unit impulse functions and the corresponding signal components of each of the input signals as weights, obtaining a plurality of second response signals after being processed by the LTI system circuit signal processing module, and taking the sum of the second response signals as a second result; The first result, the second result, and the comparison result of the equivalence between the first result and the second result are also displayed step by step and sampling point by sampling point through the linear system property proof module; The convolution theorem derivation module uses the signal components of each input signal as a weight, uses the response obtained after each unit impulse function is processed by the LTI system circuit signal processing module as a third response signal, performs weighted superposition of the signal components of each input signal on the third response signal, and uses the obtained weighted superposition sum as a third result; The second result and the third result, as well as the comparison result of the equivalence between the second result and the third result, are also displayed through the convolution theorem derivation module; Obtaining, through the frequency domain analysis subsystem, a unit impulse response signal of the unit impulse function after being processed by the LTI system circuit signal processing module, editing and changing the frequency domain characteristics of the unit impulse response signal, convolving the changed unit impulse response signal with the input signal, and using the convolved signal with changed frequency domain characteristics as a fourth result, and sending the input signal to the LTI system circuit signal processing module to obtain an output signal, and using the output signal as a fifth result; The fourth result and the fifth result, as well as the comparison result of the frequency domain difference between the fourth result and the fifth result are displayed by the frequency domain analysis subsystem.

9. An electronic device, characterized in that: The method comprises a memory and at least one processor, wherein the memory stores a computer program, and the processor is configured to execute the computer program to implement the LTI system circuit simulation method according to claim 8.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program, and when the computer program is executed, the LTI system circuit simulation method according to claim 8 is implemented.

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