A method and apparatus for wireless positioning and multipath signal generation
By employing real-time data processing and buffer output methods, the problems of carrier phase variation and time delay in wireless positioning and multipath signal generation were solved, enabling accurate simulation of motion scenarios and stable signal output, thereby improving testing efficiency and the accuracy of system verification.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- XIDIAN UNIV
- Filing Date
- 2026-03-04
- Publication Date
- 2026-06-05
AI Technical Summary
Existing wireless positioning and multipath signal generation technologies cannot accurately simulate changes in the carrier phase level of modulated signals, and are not suitable for motion scenarios. They also suffer from accumulated errors caused by fixed time delay values and the inability to divide the frequency word evenly.
The system uses a host computer to send frequency, amplitude, modulation, and delay data in real time. The digital processing module performs linear interpolation and combines phase increment calculation and lookup table methods to generate instantaneous signal values. The output is buffered using a ping-pong buffer and continuously output through a digital-to-analog converter module.
It achieves accurate simulation of multipath signals in complex motion scenarios, taking into account carrier phase and time delay characteristics, improving simulation accuracy and signal generation stability. It is suitable for the testing needs of wireless positioning and communication receiving systems, and improves testing efficiency and accuracy.
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Figure CN122159983A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of wireless positioning technology, specifically a method and apparatus for wireless positioning and multipath signal generation. Background Technology
[0002] The rapid development of information technology has driven the technological advancement of wireless positioning technology. Wireless positioning systems are increasingly appearing in people's daily lives, playing a significant role in agriculture, transportation, power, and military fields, and making a tremendous contribution to my country's national economy and military defense.
[0003] Wireless positioning and multipath signal generation devices can simulate multiple wireless signals under different scenarios and can be widely used in the research and development and testing of wireless positioning systems and communication receiving systems. These devices can simulate complex electromagnetic environments and motion scenarios, enabling better testing and verification of the performance and reliability of wireless positioning systems and communication receivers. This significantly improves the research efficiency of wireless positioning systems and communication receivers; therefore, wireless positioning and multipath signal generation technology has high research significance and practical value.
[0004] Existing wireless positioning and multipath signal generation technologies cannot achieve the same level of accuracy as the carrier phase of the modulated signal, and are unsuitable for motion scenarios. The patent document "GNSS Multipath Signal Simulation Method and GNSS Multipath Signal Simulator" provides a GNSS multipath signal simulation method and a GNSS multipath signal simulator. The method includes: a digital signal processor receiving simulation parameters of the analog source and multipath satellite signal simulation parameters sent by a host computer; the digital signal processor calculating the multipath signal amplitude in each multipath channel and the time difference between the multipath signal and the direct signal arriving at the GNSS receiver front end based on the multipath satellite signal simulation parameters; the digital signal processor calculating the corresponding carrier and pseudocode frequency control word deviations and transmitting the frequency control words of the multipath signal and the direct signal to a digital signal synthesis module; and the digital signal synthesis module generating... The method generates a direct satellite signal and creates corresponding multipath signal component channels based on the number of GNSS multipath signals to be simulated. It also generates one or more GNSS multipath sinusoidal carrier and pseudocode signals based on the frequency control word of the multipath signals. The digital signal processor calculates GNSS satellite navigation message data every preset time interval and inputs it to the digital signal synthesis module. The digital signal synthesis module spreads and modulates the pseudocode signal with the GNSS satellite navigation message data, and then modulates the spread-spectrum modulated data onto the GNSS multipath sinusoidal carrier to obtain the GNSS multipath digital intermediate frequency signal. After digital combining, it outputs the signal to the satellite signal conversion and frequency conversion module. The satellite signal conversion and frequency conversion module converts the GNSS multipath digital intermediate frequency signal into a GNSS analog intermediate frequency signal, and after spectrum shifting, obtains the superimposed signal of the direct satellite signal and multipath satellite signals at the corresponding frequency points. This method has the following drawbacks: 1) This algorithm only focuses on the phase change of the pseudocode caused by the time delay, while ignoring the change of the carrier phase, and cannot simulate the time delay characteristics of the phase modulation signal; 2) The time delay value of this algorithm is a fixed value, which is only suitable for generating multipath signals and not suitable for positioning signals whose time delay changes due to motion; 3) The formula for generating signals by this algorithm does not take into account the problem of accumulated error caused by the frequency word not being divisible, which will cause the simulation accuracy to deteriorate as the running time increases. Summary of the Invention
[0005] This invention provides a method and apparatus for wireless positioning and multipath signal generation, solving the technical problem of real-time control of the carrier phase of modulated signals.
[0006] To achieve the above objectives, the present invention provides the following technical solution: An apparatus for wireless positioning and multipath signal generation, comprising: It includes a host computer, a digital processing module, a driver module, and a digital-to-analog converter, which are connected in sequence via communication. The host computer is used for human-computer interaction, setting the frequency, amplitude, modulation data and delay data of each channel, and periodically sending the data to the digital processing module; The digital processing module is used to perform linear interpolation on the received transmission delay data, and is also used to calculate the instantaneous value of the signal by combining phase increment calculation with a lookup table based on the frequency, amplitude, modulation data and the interpolated delay data, and send the calculated digital signal stream into the buffer. The driving module is used for efficient conversion and transmission of high-speed data streams, and is used to read the digital signal stream in the buffer and convert it into data that conforms to the interface specification of the digital-to-analog converter module before outputting it. The digital-to-analog conversion module is used to convert the received digital signal into an analog signal and condition the signal to output the desired analog signal.
[0007] A method for wireless positioning and multipath signal generation includes: The system acquires frequency, amplitude, modulation, and delay data periodically sent by the host computer and sends them to the digital processing module. The digital processing module performs linear interpolation on the transmission delay data based on the received data; The digital processing module calculates the instantaneous value of the signal using a combination of phase increment calculation and lookup table based on the frequency, amplitude, modulation data, and interpolated delay data, and then stores the data in a buffer. The driver module reads the data from the buffer and sends it to the digital-to-analog converter module for digital-to-analog conversion and signal conditioning to achieve continuous output signal.
[0008] Preferably, the method of calculating the instantaneous value of the signal using phase increment calculation combined with a lookup table specifically includes: Pre-calculate and store the sample values of a complete cycle of sine wave in static memory, and construct a lookup table; Calculate the current phase value and map it to the integer field, then set the counter value to 0; Calculate the fixed phase increment within each high-precision time interval; Use the current phase value after shifting as an index to calculate the cosine value using a lookup table method; Calculate the phase value at the next point, and increment the counter. Determine if the counter value has reached the threshold. If not, jump to the next step and use the shifted value of the current phase value as an index to calculate the cosine value using a lookup table. If the condition is met, calculate the current phase value and map it to the integer field, setting the counter value to 0. Continue until the total number of calculated points reaches the required value, then end the algorithm.
[0009] Preferably, before the host computer sends data, the method further includes: the host computer providing a human-computer interaction page to configure the transmitter's operating parameters, loading a delay file, and encapsulating the configuration data into frames according to a predefined fixed communication frame format, and sending them via the Ethernet protocol.
[0010] Preferably, the buffer is a ping-pong buffer, which contains two buffers, one buffer outputting data and the other buffer writing new signal data.
[0011] Preferably, the digital processing module further includes the following functions: establishing a high-speed data connection with the host computer through the Ethernet controller integrated on the PS terminal, uploading system status information and processing results, and executing the overall control scheduling algorithm using the built-in ARM processor.
[0012] Preferably, the driving module includes an AXI DMA controller, an AXI FIFO buffer, and a custom controller based on a self-developed IP core; The AXI DMA controller is used to read signal data from the buffer of the digital processing module to the FIFO buffer at the PL end via direct memory access. The AXIS FIFO buffer is used to balance the difference in data throughput rate between the digital processing module and the digital-to-analog conversion module; The custom controller is used to monitor the FIFO status, read data at a fixed rate, convert it into a format that conforms to the interface specification of the digital-to-analog converter module, and then drive the output.
[0013] Preferably, the digital-to-analog conversion module includes a DAC chip, an 8-stage active low-pass filter, a programmable attenuator, and an operational amplifier; The DAC chip is used to convert digital signals into initial analog signals; The 8-stage active low-pass filter is used to filter analog signals; The programmable attenuator is used to precisely adjust the amplitude of the analog signal; The operational amplifier is used to amplify the power of analog signals.
[0014] Preferably, the method further includes a periodic error correction mechanism to control the cumulative error of signal calculation within a preset range.
[0015] Preferably, the instantaneous signal value calculated by the digital processing module is applicable to the MSK signal, and the formula for calculating the instantaneous value of the MSK signal is as follows:
[0016] in, For the current moment, Let be the instantaneous value of the signal received by the receiver at time t. For the number of transmitters, Let be the number of paths for the signal from the i-th transmitter. Let be the amplitude of the signal from the j-th path of the i-th transmitter. Let be the transmission delay of the signal along the j-th path from the i-th transmitter. Let be the angular frequency of the signal from the i-th transmitter. It is the phase-modulated signal of the signal of the j-th path of the i-th transmitter.
[0017] Compared with existing technologies, the present invention has the following advantages: The present invention provides a method for wireless positioning and multipath signal generation. Frequency, amplitude, modulation, and delay data are transmitted in real time by a host computer. A digital processing module performs linear interpolation on the transmission delay data and generates instantaneous signal values using phase increment calculation combined with a lookup table. This accurately simulates dynamically changing multipath signals in wireless positioning scenarios while taking into account carrier phase and delay characteristics. It solves the problem of traditional methods focusing only on pseudocode phase changes and ignoring carrier phase delay characteristics, thus more realistically reproducing signal transmission patterns in complex motion scenarios. The interpolation processing and buffered output enable continuous and stable signal output, effectively avoiding accumulated errors caused by the indivisibility of the frequency word, improving simulation accuracy under long-term operating conditions, and ensuring signal generation stability and reliability. By directly outputting continuous signals through digital-to-analog conversion and signal conditioning, the overall solution adapts to the research and testing needs of wireless positioning and communication receiving systems, simulating complex electromagnetic environments and dynamic motion scenarios, significantly improving testing efficiency and system verification accuracy. Attached Figure Description
[0018] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly described below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.
[0019] Figure 1 This is a flowchart of a method for wireless positioning and multipath signal generation according to the present invention; Figure 2 This is a flowchart of the wireless positioning and multipath signal generation method in Embodiment 1 of the present invention; Figure 3 This is a schematic diagram illustrating the periodic updating and linear interpolation of the transmission delay by the digital processing module of the present invention. Figure 4 This is a flowchart of the algorithm for combining phase increment with lookup table in this invention; Figure 5 This is a structural block diagram of the wireless positioning and multipath signal generation device in Embodiment 2 of the present invention; Figure 6 This is a structural block diagram of the wireless positioning and multipath signal generation device implemented based on the ZYNQ7100 platform in Embodiment 2 of the present invention. Detailed Implementation
[0020] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.
[0021] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.
[0022] To enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings.
[0023] like Figure 1 As shown, the present invention provides a method for wireless positioning and multipath signal generation, comprising: The system acquires frequency, amplitude, modulation, and delay data periodically sent by the host computer and sends them to the digital processing module. The digital processing module performs linear interpolation on the transmission delay data based on the received data; The digital processing module calculates the instantaneous value of the signal using a combination of phase increment calculation and lookup table based on the frequency, amplitude, modulation data, and interpolated delay data, and then stores the data in a buffer. The driver module reads the data from the buffer and sends it to the digital-to-analog converter module for digital-to-analog conversion and signal conditioning to achieve continuous output signal.
[0024] The system transmits frequency, amplitude, modulation, and delay data in real time to a host computer. The digital processing module performs linear interpolation on the transmission delay data and generates instantaneous signal values using phase increment calculation combined with a lookup table. This accurately simulates dynamically changing multipath signals in wireless positioning scenarios while taking into account carrier phase and delay characteristics. It solves the problem of traditional methods that only focus on pseudocode phase changes and ignore carrier phase delay characteristics, thus more realistically reproducing the signal transmission patterns in complex motion scenarios. By employing interpolation processing and buffered output, continuous and stable signal output can be achieved, effectively avoiding accumulated errors caused by the indivisibility of the frequency word, improving simulation accuracy under long-term operating conditions, and ensuring signal generation stability and reliability. Through digital-to-analog conversion and signal conditioning, continuous signals are directly output. The overall solution is adaptable to the research and testing needs of wireless positioning systems and communication receiving systems, and can simulate complex electromagnetic environments and dynamic motion scenarios, significantly improving testing efficiency and system verification accuracy.
[0025] Example 1 like Figure 2 and Figure 3 As shown, the wireless positioning and multipath signal generation method provided in Embodiment 1 of the present invention includes the following steps: S1. The host computer provides a human-machine interface and is responsible for configuring the transmitter's operating parameters, including but not limited to key parameters such as carrier frequency, signal power, and symbol rate.
[0026] S2. The host computer loads the delay file and periodically sends out delay data signal transmission delay data. The signal transmission delay is defined as the delay from the transmitter to the receiver at a specific moment.
[0027] S3. The host computer encapsulates and frames the above data according to a predefined, fixed communication frame format, and finally sends the complete configuration data frame to the digital processing module via the Ethernet protocol.
[0028] S4. The digital processing module receives configuration data frames from the host computer via the Ethernet interface and performs parsing and error checking on these frames to ensure data transmission integrity. The parsed valid parameters include, but are not limited to, carrier frequency, signal power, symbol rate, and signal transmission delay, which are independently set for each channel.
[0029] S5. This algorithm uses a dynamic delay simulation mechanism. The host computer periodically calculates the transmission delay of all signals, and the digital processing module periodically acquires new delay data and processes it accordingly. Figure 3 The diagram shows linear interpolation of two adjacent time delay data points, and the time delay data will change in real time as required.
[0030] S6. The digital processing module generates the instantaneous value s(t) of the signal according to the formula given in the algorithm steps.
[0031] like Figure 4 As shown, this method calculates instantaneous values using a combination of phase increment calculation and lookup table: S601. Pre-calculate and store the sine wave sample values of a complete cycle in static memory, and construct a lookup table; S602. Calculate the current phase value using the formula. And map it to the integer field, setting the value of counter i to 0; S603. Calculate the fixed phase increment within each high-precision time interval Ts. ; S604, put The shifted value is used as an index to calculate the cosine value cos(Φ) using a lookup table. S605. Calculate the phase value of the next point, and increment the value of counter i. S605. Determine whether the value of counter i has reached the threshold n. If not, jump to step S604; if the condition is met, jump to step S602 until the total number of points calculated reaches the required value, and end the algorithm.
[0032] S7. After calculating data of a certain length according to the above steps, the digital processing module writes the data into the ping-pong buffer. This method uses a data generation and output mode. To ensure continuous signal output, a ping-pong buffer is used for processing. The ping-pong buffer has two buffers. When buffer 1 is full, the data in buffer 1 is output, and newly generated data is written to buffer 2. When the data generation rate is greater than the output data rate, buffer 2 can be filled before the data in buffer 1 is completely output, achieving continuous signal output. When the amount of data in the buffer reaches a predetermined threshold, the digital processing module stably outputs the ready data to the subsequent driver module.
[0033] S8. The digital processing module outputs the data in the buffer to the driver module. The driver module converts the data into data that conforms to the interface specification of the digital-to-analog converter module, and then outputs it to the digital-to-analog converter module. The digital-to-analog converter module performs digital-to-analog conversion and signal conditioning on the data to achieve the desired output signal.
[0034] Example 2 like Figures 3-4 As shown, based on the same inventive concept as Embodiment 1 above, the present invention also provides a device for wireless positioning and multipath signal generation, implemented on the ZYNQ7100 platform, specifically divided into four core functional modules: host computer, digital processing module, driver module, and digital-to-analog conversion module, as shown in the module block diagram below: (1) The host computer is responsible for human-computer interaction. Users use the host computer to set the parameters of each channel and then send them to the digital processing module.
[0035] (2) The digital processing module is the core of the system's control and calculation, implemented by the PS terminal of the ZYNQ7100 platform. Its functions include: Communication and interaction: Through the Ethernet controller integrated on the PS terminal, a high-speed data connection is established with the host computer. It is responsible for receiving parameters such as carrier frequency, symbol rate, power, and latency, and uploading system status information and processing results.
[0036] Scheduling and Computation: The built-in ARM processor is responsible for executing the overall scheduling algorithm and, based on the received parameters, performs calculations according to the formula.
[0037] Calculate the instantaneous value of the MSK signal, and then... It is the symbol sequence of the k-th transmitter. It is the symbol period of the k-th transmitter; Data management: The calculated digital signal stream is sent to the ping-pong buffer, waiting for the driver module to read it; (3) The driver module is responsible for the efficient conversion and transmission of high-speed data streams. It is implemented by the PL terminal of the ZYNQ7100 platform, and its functions include: Data Acquisition: Using the AXIDMA controller, signal data can be efficiently read from the ping-pong buffer at the PS end to the FIFO buffer at the PL end through direct memory access without consuming the computing power of the ARM processor.
[0038] FIFO Buffer: AXIS FIFO is used as a FIFO buffer to balance the possible differences in data throughput rates between the PS end and the subsequent digital-to-analog conversion module, ensuring the continuity of the data stream.
[0039] Custom controller: Using self-developed IP cores, it continuously monitors the FIFO status. When the FIFO is not empty, it reads data at a fixed rate that meets the requirements of the digital-to-analog conversion module, converts the data into a format that conforms to its interface specifications, and then drives the output.
[0040] (4) The digital-to-analog converter module is responsible for accurately converting digital signals into high-quality analog signals. Its functions include: Digital-to-analog conversion: The received digital signal is converted into the initial analog signal using the LTC1668IG high-performance DAC chip.
[0041] Signal conditioning: The analog signal is filtered using an 8-stage active low-pass filter, the amplitude is precisely adjusted using a programmable attenuator, and the power is amplified using an operational amplifier to finally generate the desired signal that meets the design requirements.
[0042] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the scope of the invention. No reference numerals in the claims should be construed as limiting the scope of the claims.
[0043] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can be appropriately combined to form other embodiments that can be understood by those skilled in the art. The above content is only for illustrating the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made based on the technical concept proposed in this invention shall fall within the scope of protection of the claims of this invention.
Claims
1. A device for wireless positioning and multipath signal generation, characterized in that, include: It includes a host computer, a digital processing module, a driver module, and a digital-to-analog converter, which are connected in sequence via communication. The host computer is used for human-computer interaction, setting the frequency, amplitude, modulation data and delay data of each channel, and periodically sending the data to the digital processing module; The digital processing module is used to perform linear interpolation on the received transmission delay data, and is also used to calculate the instantaneous value of the signal by combining phase increment calculation with a lookup table based on the frequency, amplitude, modulation data and the interpolated delay data, and send the calculated digital signal stream into the buffer. The driving module is used for efficient conversion and transmission of high-speed data streams, and is used to read the digital signal stream in the buffer and convert it into data that conforms to the interface specification of the digital-to-analog converter module before outputting it. The digital-to-analog conversion module is used to convert the received digital signal into an analog signal and condition the signal to output the desired analog signal.
2. A method for wireless positioning and multipath signal generation, characterized in that, An apparatus for wireless positioning and multipath signal generation according to claim 1, comprising: The system acquires frequency, amplitude, modulation, and delay data periodically sent by the host computer and sends them to the digital processing module. The digital processing module performs linear interpolation on the transmission delay data based on the received data; The digital processing module calculates the instantaneous value of the signal using a combination of phase increment calculation and lookup table based on the frequency, amplitude, modulation data, and interpolated delay data, and then stores the data in a buffer. The driver module reads the data from the buffer and sends it to the digital-to-analog converter module for digital-to-analog conversion and signal conditioning to achieve continuous output signal.
3. The method for wireless positioning and multipath signal generation according to claim 2, characterized in that, The method of calculating the instantaneous value of the signal using phase increment calculation combined with a lookup table specifically includes: Pre-calculate and store the sample values of a complete cycle of sine wave in static memory, and construct a lookup table; Calculate the current phase value and map it to the integer field, then set the counter value to 0; Calculate the fixed phase increment within each high-precision time interval; Use the current phase value after shifting as an index to calculate the cosine value using a lookup table method; Calculate the phase value at the next point, and increment the counter. Determine if the counter value has reached the threshold. If not, jump to the next step and use the shifted value of the current phase value as an index to calculate the cosine value using a lookup table. If the condition is met, calculate the current phase value and map it to the integer field, setting the counter value to 0. Continue until the total number of calculated points reaches the required value, then end the algorithm.
4. The method for wireless positioning and multipath signal generation according to claim 2, characterized in that, Before the host computer sends data, the process also includes: the host computer providing a human-computer interaction page to configure the transmitter's operating parameters, loading a delay file, and encapsulating the configuration data into frames according to a predefined fixed communication frame format, and sending them via the Ethernet protocol.
5. A method for wireless positioning and multipath signal generation according to claim 2, characterized in that, The buffer is a ping-pong buffer, which contains two buffers. When one buffer outputs data, the other buffer writes new signal data.
6. A method for wireless positioning and multipath signal generation according to claim 2, characterized in that, The digital processing module also includes the following functions: establishing a high-speed data connection with the host computer through the Ethernet controller integrated on the PS terminal, uploading system status information and processing results, and executing the overall control scheduling algorithm using the built-in ARM processor.
7. A method for wireless positioning and multipath signal generation according to claim 2, characterized in that, The driver module includes an AXI DMA controller, an AXIS FIFO buffer, and a custom controller based on a self-developed IP core; The AXI DMA controller is used to read signal data from the buffer of the digital processing module to the FIFO buffer at the PL end via direct memory access. The AXIS FIFO buffer is used to balance the difference in data throughput rate between the digital processing module and the digital-to-analog conversion module; The custom controller is used to monitor the FIFO status, read data at a fixed rate, convert it into a format that conforms to the interface specification of the digital-to-analog converter module, and then drive the output.
8. A method for wireless positioning and multipath signal generation according to claim 2, characterized in that, The digital-to-analog conversion module includes a DAC chip, an 8-stage active low-pass filter, a programmable attenuator, and an operational amplifier. The DAC chip is used to convert digital signals into initial analog signals; The 8-stage active low-pass filter is used to filter analog signals; The programmable attenuator is used to precisely adjust the amplitude of the analog signal; The operational amplifier is used to amplify the power of analog signals.
9. A method for wireless positioning and multipath signal generation according to claim 2, characterized in that, The method also includes a periodic error correction mechanism to control the cumulative error of signal calculation within a preset range.
10. A method for wireless positioning and multipath signal generation according to claim 2, characterized in that, The instantaneous signal value calculated by the digital processing module is applicable to MSK signals. The formula for calculating the instantaneous value of an MSK signal is as follows: in, For the current moment, Let be the instantaneous value of the signal received by the receiver at time t. For the number of transmitters, Let be the number of paths for the signal from the i-th transmitter. Let be the amplitude of the signal from the j-th path of the i-th transmitter. Let be the transmission delay of the signal along the j-th path from the i-th transmitter. Let be the angular frequency of the signal from the i-th transmitter. It is the phase-modulated signal of the signal of the j-th path of the i-th transmitter.