Ultrasonic detection signal processing method based on synchronous time reversal algorithm

Abstract

The application discloses an ultrasonic detection signal processing method based on a synchronization time reversal algorithm, and steps are as follows: a reinforced concrete two-dimensional simulation model is established; ultrasonic array detection is carried out on the reinforced concrete two-dimensional simulation model, imaging data collection is carried out, full matrix data are obtained; signal transmission and reception are carried out, the signal is retransmitted after time reversal processing, the signal is received again after focusing and enhancement at the original sound source position; specific grouping processing is carried out on the signal, combination calculation is carried out, and the result is normalized; the processed signal is enhanced by using an aggregation function, and a defect reflection echo signal with a higher amplitude is obtained. The application has the advantages that: by simultaneously considering time and space reversal, the wave field is more effectively focused, the problem of insufficient focusing of the ultrasonic beam in the concrete is solved, the synchronization focusing and enhancement of the sound wave at the defect are realized, the signal-to-noise ratio of the detection signal is improved, and the imaging quality is effectively improved.

Description

TECHNICAL FIELD

[0001] The application relates to the field of ultrasonic nondestructive testing, and particularly relates to an ultrasonic detection signal processing method based on a synchronism time reversal algorithm. BACKGROUND

[0002] Compared with other conventional detection technologies, ultrasonic detection technology has the advantages of non-destructiveness, high sensitivity and strong penetration capacity, and is widely applied in concrete structure detection. The application research of ultrasonic detection technology in concrete detection has a history of more than 80 years. Early research contents are limited to concrete material properties such as ultrasonic wave speed, attenuation and compressive strength. Since the 1970s, thanks to the rapid development of integrated circuit technology and signal processing technology, the practicability and application range of ultrasonic detection technology have been developed and expanded. In addition, with the rapid development of intelligent sensing technology, the fusion of ultrasonic detection technology and advanced technical means has become a key path to improve the efficiency and accuracy of engineering detection. From the perspective of technical development, the application of ultrasonic detection technology in the field of concrete structure has evolved from traditional technology to digitization and intelligentization, indicating that more innovative technologies will be integrated in the future to further enhance the accuracy and efficiency of concrete structure safety evaluation and maintenance work.

[0003] The ultrasonic pulse echo method is an important means of concrete nondestructive testing. This method is suitable for evaluating the type and size of defects in a known area, and is particularly suitable for rapid quantitative detection of defects such as concrete thickness, cracks and cavities. The ultrasonic pulse echo method is widely used due to its simple operation and high detection efficiency. At present, this method tends to be portable and intelligent, making on-site operation more convenient and data analysis more automated. By increasing the frequency of sound waves and improving data processing algorithms, the clarity and resolution of imaging are improved, making it possible to detect small defects. In addition, the combination of sound waves of different frequencies and other advanced technologies can improve the accuracy and reliability of concrete structure detection.

[0004] Time reversal technique is a physical phenomenon based on wave propagation, which realizes the control and focusing of wave field by reversing the wave signal in time domain and retransmitting it. It is widely used in the fields of acoustics, electromagnetics, optics and quantum physics. Time reversal technique was first proposed in the late 1970s, and research found that it could achieve acoustic focusing, laying the foundation for subsequent research. After 1990, computer technology promoted its further development. Researchers began to explore its application in the control of electromagnetic waves and light waves. In the early 21st century, time reversal technique began to be applied in the field of non-destructive testing, using it to achieve high-precision target positioning and imaging, and in the field of wireless communication, to improve the transmission quality of signals and the capacity of the system. In recent years, with the in-depth study of time reversal physics and the continuous emergence of new technologies, it has been applied in the fields of quantum information processing and acoustic control. Time reversal technique has higher focusing accuracy, stronger anti-interference ability, higher flexibility and practicality than traditional focusing methods, and can realize effective focusing in complex media, improve the resolution and contrast of ultrasonic imaging, and improve the imaging quality.

[0005] Time reversal algorithm is a signal processing method based on the symmetry of wave equation, which realizes high-precision wave field reconstruction by using the time reversal invariance of wave motion in medium. The traditional time reversal algorithm realizes the principle as follows: the excitation signal is emitted by the sound source and received by the sensors at different positions and distances from the sound source. The closer ones arrive first, and the farther ones arrive later. The time sequence of the received signal is inverted, and the time reversal signal is re-emitted, which can generate a signal that is reversed in time sequence, i.e. the signal that arrives first is emitted last, and the signal that arrives last is emitted first. In this way, these signals will converge to the sound source position at the same time and phase, thus forming a focusing effect at that position.

[0006] The basic principles of traditional time-reversal algorithms indicate that when there is only one defect target in the ultrasonic inspection target area, it is very easy to focus the retransmitted signals from different transducers at the target location. However, when there are two or more defect targets in the target area (for example, two targets), each transducer receives and time-reverses the sum of the two target signals. The first target signal, time-reversed by N transducers, will reach its maximum at the first target location at time t1. Similarly, the second target signal will reach its maximum at the second target location at time t2, but t1 may not equal t2. Therefore, imaging using the signal at time t1 will accurately locate the first target, while the position of the second target will be affected. Conversely, imaging using the signal at time t2 will affect the position of the first target. Due to coherent superposition, the synchronicity of the two target locations remains approximately the same, but their synchronicity may differ. Even at their respective optimal times, the field values ​​at the two target locations may differ, potentially drowning out or missing the weaker target. Summary of the Invention

[0007] A method for processing ultrasonic detection signals based on a synchronous time reversal algorithm, comprising:

[0008] Step S1: constructing a two-dimensional reinforced concrete simulation model, arranging a plurality of air defects inside the two-dimensional reinforced concrete simulation model, and setting a plurality of ultrasonic probes on the high-order surface and low-order surface of the concrete of the two-dimensional reinforced concrete simulation model;

[0009] Step S2: stimulating the ultrasonic probe to perform ultrasonic array detection on the reinforced concrete two-dimensional simulation model and collect imaging data, wherein the imaging data is signal data in a full matrix format;

[0010] Wherein, step S2 is specifically as follows:

[0011] Step S21: Each ultrasonic probe transmits an ultrasonic signal in sequence. After receiving the transmitted ultrasonic signal, each ultrasonic probe acts as a receiving point and retransmits the ultrasonic signal based on a time-space inversion algorithm. During the retransmission process, the position of the air defect is focused to obtain a signal at the position of the air defect.

[0012] Re-transmit once more to enhance the signal at the air defect location and obtain an enhanced signal;

[0013] Performing excitation enhancement on the enhanced signal, and further obtaining a spatiotemporal signal through spatiotemporal conversion;

[0014] Step S22: performing synchronization operation and normalization processing on the spatiotemporal signal; obtaining normalized signal data;

[0015] Step S23: performing amplitude enhancement on the normalized signal data through an aggregation function to obtain final imaging data.

[0016] Furthermore, step S1 is specifically as follows:

[0017] Step S11: constructing a two-dimensional reinforced concrete simulation model, setting three air defects inside the two-dimensional reinforced concrete simulation model, the three air defects being a first square air defect, a second square defect, and a circular air defect; and further setting a plurality of steel bars with a diameter of 10 mm inside the two-dimensional reinforced concrete simulation model;

[0018] The dimensions of the reinforced concrete 2D simulation model are 800 mm long, 400 mm wide, and 800 mm high. The first square air defect has a side length of 20 mm and a depth of 205 mm in the model. The second square air defect has a side length of 30 mm and a depth of 260 mm in the model. The circular air defect has a diameter of 50 mm and a depth of 475 mm in the model.

[0019] Step S12: Setting an ultrasonic detection array consisting of 6 ultrasonic probes, with the center of each ultrasonic probe simultaneously transmitting and receiving ultrasonic signals; using the ultrasonic detection array to detect the high-order surface and low-order surface of concrete in the reinforced concrete two-dimensional simulation model;

[0020] The ultrasonic probes are arranged at intervals, and the intervals between the ultrasonic probes are equal to the diameters of the ultrasonic probes, both being 40 mm.

[0021] Furthermore, step S2 is specifically as follows:

[0022] The six ultrasound probes are stimulated to transmit ultrasound signals in sequence. Each ultrasound probe receives and stores the sound pressure data of the ultrasound signal until the full matrix sound pressure data in a 6×6 signal format is obtained, based on which the imaging data is obtained.

[0023] Furthermore, step S21 is specifically as follows:

[0024] Step S211: The ultrasonic probe transmits an ultrasonic signal a(ω) for detection, and the i-th ultrasonic probe as the receiving point obtains the reflected signal , expressed as:

[0025] ;

[0026] ;

[0027] Where ω represents the frequency of the ultrasonic signal, represents the path transfer function;

[0028] Step S212: The ultrasonic probe re-emits the signal by time reversal in the time domain through conjugate processing, focusing and enhancing at the air defect position of the model, denoted as:

[0029]

[0030] wherein, represents the time reversal signal of the defect echo signal received by the i-th ultrasonic probe as a receiving point, represents the ultrasonic signal after conjugate processing, represents the path transfer function after conjugate processing;

[0031] Step S213: After the ultrasonic signal is re-emitted, the signal of the air defect position is obtained , denoted as:

[0032]

[0033] Step S214: The enhanced signal reflected by the defect position is received again by the ultrasonic probe as a receiving point , denoted as:

[0034]

[0035] wherein, Gj(ω) represents the path transfer function received by the j-th probe;

[0036] Step S215: The time reversal enhanced signal is obtained, denoted as:

[0037]

[0038] wherein, bj(ω) represents the defect signal received by the j-th probe; D’j(ω) represents the time reversal enhanced signal with a*(ω)‧a(ω)‧a(ω) as the excitation signal;

[0039] Step S216: The time reversal enhanced signal is converted to the time domain to obtain the space-time signal :

[0040]

[0041] wherein, represents a complex exponential function.

[0042] Further, step S22 is specifically:

[0043] Step S221: Synchronization operation;

[0044] The ultrasonic detection array containing N elements is divided into M sub-arrays, each sub-array consisting of L elements, then the time reversal signal of the m-th sub-array is​​​​​ is represented as:

[0045] ;

[0046] in the formula, denotes the time reversal signal of the m+1th probe; the first subarray is from the first array element to the Lth array element, and the last subarray is from the Mth array element to the (M+L-1)th array element, that is, the Nth array element;

[0047] Step S222: normalizing the obtained signal to allow all potential targets in the target domain to be imaged and displayed with the same visibility, to obtain a subarray time reversal signal , that is, the normalized signal data, is represented as:

[0048] ;

[0049] wherein, denotes the moment when the target position reaches the maximum value.

[0050] Further, step S23 is specifically:

[0051] a aggregation function is defined to process the normalized signal data, to obtain final imaging data, which is represented as:

[0052] ;

[0053] wherein, denotes the final imaging data, denotes the product.

[0054] The beneficial effects of the present application are:

[0055] The synchronicity time reversal algorithm can more effectively focus the wave field by simultaneously considering the reversal of time and space, achieve higher accuracy in spatial resolution, enhance the ability of coherent recognition of target positions and exclusion of non-target positions, and make the target display under the same visibility. The present application solves the focusing problem of ultrasonic beams in concrete by using the synchronicity time reversal algorithm, which is beneficial to enhance the focusing effect of sound waves at defects. BRIEF DESCRIPTION OF DRAWINGS

[0056] Fig. 1 is the overall step flowchart of the ultrasonic detection signal processing method based on the synchronicity time reversal algorithm of the present application.

[0057] Fig. 2 is the focusing step flowchart of the ultrasonic detection signal processing method based on the synchronicity time reversal algorithm of the present application.

[0058] Fig. 3 Schematic diagram of a concrete model of the ultrasonic detection signal processing method based on the synchronous time reversal algorithm of the present invention.

[0059] Fig. 4 This is a diagram of the aperture signal amplitude when the ultrasonic detection signal processing method based on the synchronous time reversal algorithm of the present invention is not adopted.

[0060] Fig. 5 This is a diagram of the aperture signal amplitude after the ultrasonic detection signal processing method based on the synchronous time reversal algorithm of the present invention is adopted.

[0061] Fig. 6 This is a normalized amplitude diagram at an air defect when the ultrasonic detection signal processing method based on the synchronous time reversal algorithm of the present invention is not adopted.

[0062] Fig. 7 This is a normalized amplitude diagram at the air defect after adopting the ultrasonic detection signal processing method based on the synchronous time reversal algorithm of the present invention. DETAILED DESCRIPTION

[0063] The present invention works and is implemented in the following manner: an ultrasonic detection signal processing method based on a synchronous time reversal algorithm is provided: establishing a two-dimensional reinforced concrete simulation model; conducting ultrasonic array detection on the two-dimensional reinforced concrete simulation model, collecting imaging data, and obtaining full matrix data; transmitting and receiving signals, performing time reversal processing on the signals and then retransmitting the signals, so that the signals are focused and enhanced at the original sound source position and then received again; performing specific grouping processing and then combining calculations, and normalizing the results; using an aggregation function to enhance the processed signals to obtain a defect reflection echo signal with a higher amplitude; extracting the aperture signal amplitude before and after processing to verify the amplitude enhancement effect; comparing the full-focus imaging of the reinforced concrete model and the imaging effect after processing by the synchronous time reversal algorithm to verify the effectiveness of the algorithm in concrete detection imaging; and using the signal-to-noise ratio to evaluate the imaging quality.

[0064] Reference Figs. 1-3 , an ultrasonic detection signal processing method based on a synchronous time reversal algorithm, comprising:

[0065] Step S1: constructing a two-dimensional reinforced concrete simulation model, arranging a plurality of air defects inside the two-dimensional reinforced concrete simulation model, and setting a plurality of ultrasonic probes on the high-order surface and low-order surface of the concrete of the two-dimensional reinforced concrete simulation model;

[0066] Step S2: stimulating the ultrasonic probe to perform ultrasonic array detection on the reinforced concrete two-dimensional simulation model and collect imaging data, wherein the imaging data is signal data in a full matrix format;

[0067] Wherein, step S2 is specifically as follows:

[0068] Step S21: Each ultrasonic probe transmits an ultrasonic signal in sequence. After receiving the transmitted ultrasonic signal, each ultrasonic probe acts as a receiving point and retransmits the ultrasonic signal based on a time-space inversion algorithm. During the retransmission process, the position of the air defect is focused to obtain a signal at the position of the air defect.

[0069] Re-transmit once more to enhance the signal at the air defect location and obtain an enhanced signal;

[0070] Performing excitation enhancement on the enhanced signal, and further obtaining a spatiotemporal signal through spatiotemporal conversion;

[0071] Step S22: performing synchronization operation and normalization processing on the spatiotemporal signal; obtaining normalized signal data;

[0072] Step S23: performing amplitude enhancement on the normalized signal data through an aggregation function to obtain final imaging data.

[0073] Furthermore, step S1 is specifically as follows:

[0074] Step S11: constructing a two-dimensional reinforced concrete simulation model, setting three air defects inside the two-dimensional reinforced concrete simulation model, the three air defects being a first square air defect, a second square defect, and a circular air defect; and further setting a plurality of steel bars with a diameter of 10 mm inside the two-dimensional reinforced concrete simulation model;

[0075] The dimensions of the reinforced concrete 2D simulation model are 800 mm long, 400 mm wide, and 800 mm high. The first square air defect has a side length of 20 mm and a depth of 205 mm in the model. The second square air defect has a side length of 30 mm and a depth of 260 mm in the model. The circular air defect has a diameter of 50 mm and a depth of 475 mm in the model.

[0076] Step S12: Setting an ultrasonic detection array consisting of 6 ultrasonic probes, with the center of each ultrasonic probe simultaneously transmitting and receiving ultrasonic signals; using the ultrasonic detection array to detect the high-order surface and low-order surface of concrete in the reinforced concrete two-dimensional simulation model;

[0077] The ultrasonic probes are arranged at intervals, and the intervals between the ultrasonic probes are equal to the diameters of the ultrasonic probes, both being 40 mm.

[0078] Furthermore, step S2 is specifically as follows:

[0079] The six ultrasound probes are stimulated to transmit ultrasound signals in sequence. Each ultrasound probe receives and stores the sound pressure data of the ultrasound signal until the full matrix sound pressure data in a 6×6 signal format is obtained, based on which the imaging data is obtained.

[0080] Further, step S21 is specifically:

[0081] Step S211: the ultrasonic probe transmits an ultrasonic signal a(ω) for detection, and the i-th ultrasonic probe as a receiving point acquires a reflected signal , which is expressed as:

[0082] ;

[0083] ;

[0084] In the formula, ω represents the frequency of the ultrasonic signal, represents a path transfer function;

[0085] Step S212: the ultrasonic probe re-emits the signal through conjugate processing in the time domain for time reversal, and focuses and enhances at the air defect position of the model, which is expressed as:

[0086] ;

[0087] wherein, represents a time reversal signal of a defect echo signal received by the i-th ultrasonic probe as a receiving point, represents an ultrasonic signal after conjugate processing, represents a path transfer function after conjugate processing;

[0088] Step S213: after the ultrasonic signal is re-emitted, a signal at the air defect position is acquired , which is expressed as:

[0089] ;

[0090] Step S214: the enhanced signal reflected from the defect position is received again by the ultrasonic probe as a receiving point , which is expressed as:

[0091] ;

[0092] wherein Gj(ω) represents a path transfer function received by the j-th probe;

[0093] Step S215: a time reversal enhanced signal is acquired, which is expressed as:

[0094] ;

[0095] wherein bj(ω) represents a defect signal received by the j-th probe; D’j(ω) represents a time reversal enhanced signal with a*(ω)‧a(ω)‧a(ω) as an excitation signal;

[0096] Step S216: converting the time reversal enhanced signal into time domain to obtain a space-time signal :

[0097] ;

[0098] wherein, represents a complex exponential function.

[0099] Further, the step S22 is specifically:

[0100] Step S221: synchronization operation;

[0101] Divide the ultrasonic detection array containing N elements into M sub-arrays, each sub-array consisting of L elements, then the time reversal signal of the mth sub-array is represented as:

[0102] ;

[0103] wherein, represents the time reversal signal of the m+1th probe; the first sub-array is from the first element to the Lth element, and the last sub-array is from the Mth element to the (M+L-1)th element, i.e. the Nth element;

[0104] Step S222: normalizing the obtained signal to allow all potential targets in the target domain to be imaged and displayed with the same visibility, to obtain a sub-array time reversal signal , i.e. normalized signal data, represented as:

[0105] ;

[0106] wherein, represents the moment when the target position reaches the maximum value.

[0107] Further, the step S23 is specifically:

[0108] Define an aggregation function to process the normalized signal data to obtain the final imaging data, represented as:

[0109] ;

[0110] wherein, represents the final imaging data, represents the product.

[0111] As Figs. 4-5As shown in the amplitude graph of the aperture signal before and after the processing of the synchronous time reversal algorithm, the amplitude of the defect wave and the amplitude of the bottom wave are both enhanced after the processing of the synchronous time reversal algorithm. The original defect wave amplitude is about 0.61, which is increased to about 0.77 after processing. The original bottom wave amplitude is about 0.78, which is increased to 0.91 after processing. The results show that the synchronous time reversal algorithm can effectively improve the signal amplitude, thereby improving the imaging quality.

[0112] By extracting the horizontal pixel amplitude before and after the signal processing of the three defects of the reinforced concrete model, it can be seen from the figure that the synchronous time reversal algorithm can effectively improve the signal-to-noise ratio of the air defect. Figs. 6-7

[0113] The normalized amplitude change difference of the signal-to-noise ratio after the processing of the synchronous time reversal algorithm is calculated, and the signal-to-noise ratio of the reinforced concrete defect is improved by 3.21 dB. The results show that by using the synchronous time reversal algorithm, the influence of structural noise on the imaging effect can be significantly reduced, and the signal-to-noise ratio can be improved, which shows good application potential in the field of concrete defect detection imaging.​

Claims

1. A method for processing ultrasonic detection signals based on a synchronous time reversal algorithm, characterized in that: include: Step S1: constructing a two-dimensional reinforced concrete simulation model, arranging a plurality of air defects inside the two-dimensional reinforced concrete simulation model, and setting a plurality of ultrasonic probes on the high-order surface and low-order surface of the concrete of the two-dimensional reinforced concrete simulation model; Step S2: stimulating the ultrasonic probe to perform ultrasonic array detection on the reinforced concrete two-dimensional simulation model and collect imaging data, wherein the imaging data is signal data in a full matrix format; Wherein, step S2 is specifically as follows: Step S21: Each ultrasonic probe transmits an ultrasonic signal in sequence. After receiving the transmitted ultrasonic signal, each ultrasonic probe acts as a receiving point and retransmits the ultrasonic signal based on a time-space inversion algorithm. During the retransmission process, the position of the air defect is focused to obtain a signal at the position of the air defect. Re-transmit once more to enhance the signal at the air defect location and obtain an enhanced signal; Performing excitation enhancement on the enhanced signal, and further obtaining a spatiotemporal signal through spatiotemporal conversion; Step S22: performing synchronization operation and normalization processing on the spatiotemporal signal; obtaining normalized signal data; Step S23: performing amplitude enhancement on the normalized signal data by using an aggregation function to obtain final imaging data; Step S22 is specifically as follows: Step S221: Synchronous operation; The ultrasonic detection array containing N array elements is divided into M sub-arrays, each sub-array consists of L array elements, then the time-reversed signal of the m-th sub-array is Expressed as: ； Where, represents the time-reversed signal of the m+1th probe; the first subarray is from the first array element to the Lth array element, and the last subarray is from the Mth array element to the (M+L-1)th array element, that is, the Nth array element; Step S222: normalize the obtained signal to allow all potential targets in the target area to be imaged and displayed with the same visibility, and obtain the sub-array time-reversed signal. , that is, the normalized signal data, expressed as: ； in, express At the moment when the target position reaches its maximum value; Step S23 is specifically as follows: Define an aggregation function to process the normalized signal data and obtain the final imaging data, which is expressed as: ； in, represents the final imaging data, Represents the product.

2. The ultrasonic detection signal processing method based on synchronous time reversal algorithm according to claim 1 is characterized in that: Step S1 is specifically as follows: Step S11: constructing a two-dimensional reinforced concrete simulation model, setting three air defects inside the two-dimensional reinforced concrete simulation model, the three air defects being a first square air defect, a second square defect, and a circular air defect; and further setting a plurality of steel bars with a diameter of 10 mm inside the two-dimensional reinforced concrete simulation model; The dimensions of the reinforced concrete 2D simulation model are 800 mm long, 400 mm wide, and 800 mm high. The first square air defect has a side length of 20 mm and a depth of 205 mm in the model. The second square air defect has a side length of 30 mm and a depth of 260 mm in the model. The circular air defect has a diameter of 50 mm and a depth of 475 mm in the model. Step S12: Setting an ultrasonic detection array consisting of 6 ultrasonic probes, with the center of each ultrasonic probe simultaneously transmitting and receiving ultrasonic signals; using the ultrasonic detection array to detect the high-order surface and low-order surface of concrete in the reinforced concrete two-dimensional simulation model; The ultrasonic probes are arranged at intervals, and the intervals between the ultrasonic probes are equal to the diameters of the ultrasonic probes, both being 40 mm.

3. The ultrasonic detection signal processing method based on synchronous time reversal algorithm according to claim 2, characterized in that: Step S2 is specifically as follows: The six ultrasound probes are stimulated to transmit ultrasound signals in sequence. Each ultrasound probe receives and stores the sound pressure data of the ultrasound signal until the full matrix sound pressure data in a 6×6 signal format is obtained, based on which the imaging data is obtained.

4. The ultrasonic detection signal processing method based on synchronous time reversal algorithm according to claim 3 is characterized in that: Step S21 is specifically as follows: Step S211: The ultrasonic probe transmits an ultrasonic signal a(ω) for detection, and the i-th ultrasonic probe as the receiving point obtains the reflected signal , expressed as: ； ； Where ω represents the frequency of the ultrasonic signal, represents the path transfer function; Step S212: The ultrasonic probe performs time reversal and re-transmits the signal in the time domain through conjugate processing, and performs focus enhancement at the air defect position of the model, which is expressed as: ； in, represents the time-reversed signal of the defect echo signal received by the i-th ultrasonic probe as the receiving point, represents the ultrasonic signal after conjugate processing, represents the path transfer function after conjugate processing; Step S213: After re-reflecting the ultrasonic signal, obtain the signal of the air defect position , expressed as: ； Step S214: The ultrasonic probe serving as the receiving point receives the enhanced signal reflected from the defect position again. , expressed as: ； Among them, G j (ω) represents the path transfer function received by the jth probe; Step S215: Obtain the time-reversed enhanced signal, expressed as: ； Among them, b j (ω) represents the defect signal received by the jth probe; D ’ j (ω) represents the * (ω)‧a(ω)‧a(ω) is the time-reversed enhancement signal of the excitation signal; Step S216: Convert the time-reversed enhanced signal into the time domain to obtain the space-time signal : ； in, represents the complex exponential function.

Images