Faster-Nyquist signal sending method, receiving method and system based on frequency domain precoding

Abstract

The invention discloses a super-Nyquist signal sending method, receiving method and system based on frequency domain precoding in the field of information communication, and aims to solve the problems of high bit error rate and complex signal demodulation in the prior art. The method comprises the following steps: a sending end segments original bit data, performs constellation mapping through phase modulation / quadrature amplitude modulation, and generates a modulation symbol vector; and then, carrying out frequency domain pre-coding processing, inserting a cyclic prefix, and sending a signal through a super-Nyquist shaping filter. After a receiving end receives a signal, Gaussian white noise is added, matched filtering, whitening filtering and cyclic prefix removal are carried out in sequence, and a plurality of original modulation symbols are recovered through symbol-by-symbol detection. The method is suitable for a high-order modulation mode, inter-symbol interference is eliminated through frequency domain precoding, signal transmission is optimized, the bit error rate is remarkably reduced, and the demodulation efficiency is improved.

Description

Technical Field

[0001] The present invention relates to a super-Nyquist signal sending method, receiving method and system based on frequency domain precoding, and belongs to the technical field of information communication. Background Art

[0002] In the information and communications field, super-Nyquist signaling technology has garnered significant attention in recent years as a key means of improving spectrum efficiency. However, existing technologies often introduce inter-symbol interference (ISI) when implementing super-Nyquist transmission, negatively impacting signal quality and transmission performance. To address this challenge, researchers have proposed various precoding schemes, such as Tomlinson-Harrassima (TH) precoding and Trellis precoding. While these schemes can reduce the complexity of signal detection at the receiver to a certain extent, they often come at the expense of bit error performance. Furthermore, linear precoding techniques such as singular value decomposition (SVD) precoding and G-to-minus-half (GTMH) precoding have also been investigated, but they also fail to fully address the issue of bit error performance loss. Therefore, developing a super-Nyquist signaling method that can reduce signal demodulation complexity while minimizing bit error performance loss has become a critical and pressing issue in the field of communications technology. Summary of the Invention

[0003] The purpose of the present invention is to provide a super-Nyquist signal sending method, receiving method and system based on frequency domain precoding, which can solve the problems of large error performance loss and high signal demodulation complexity in super-Nyquist signal transmission in the prior art.

[0004] In order to solve the above technical problems, the present invention is implemented by adopting the following technical solutions.

[0005] In one aspect, the present invention provides a super-Nyquist signal transmission method based on frequency domain precoding, which is performed by a wireless channel transmitting end and includes: Obtaining original bit data, and performing segmentation processing on the bit data to obtain multiple bit segments; Perform constellation mapping on each bit segment through a phase modulation / orthogonal amplitude modulation module to obtain a number of modulation symbols, and compose the plurality of modulation symbols into a modulation symbol vector; Performing frequency domain precoding processing on the modulation symbol vector to obtain a precoded symbol vector; Taking samples of a fixed length from the end of the pre-coded symbol vector as a cyclic prefix, and inserting the samples into the front of the second symbol vector to form an extended signal vector; The extended signal vector is processed by a preset super-Nyquist shaping filter to obtain a shaped transmission signal.

[0006] In combination with the first aspect, further performing frequency domain precoding processing on the modulation symbol vector includes: Performing a fast Fourier transform on the modulation symbol vector to convert the modulation symbol vector from the time domain to the frequency domain; Multiply the modulation symbol vector transferred to the frequency domain by the inverse matrix of the diagonal matrix to obtain the frequency domain precoded signal; The frequency-domain precoded signal is then subjected to an inverse transform of the Fast Fourier Transform, thereby converting the precoded signal from the frequency domain back to the time domain and obtaining a precoded symbol vector.

[0007] In combination with the first aspect, further, the expression of the precoding symbol vector includes: ; in, represents the precoding symbol vector; represents the complex conjugate transpose of the Fourier matrix; represents the inverse matrix of a diagonal matrix; represents the normalized Fourier matrix; represents the modulation symbol vector.

[0008] In combination with the first aspect, further, the expression of the cyclic prefix extension signal vector is: ; in, represents the length of the cyclic prefix; b represents the extended signal vector inserted into the cyclic prefix; represents the precoding symbol vector.

[0009] In combination with the first aspect, further, the expression of the shaped transmission signal includes: s=Hb Wherein, s represents the shaped transmission signal; H represents the super-Nyquist shaping filter matrix; and b represents the extended signal vector into which the cyclic prefix is ​​inserted.

[0010] In a second aspect, a super-Nyquist signal receiving method based on frequency domain precoding is performed by a wireless channel receiving end, comprising: receiving a shaped transmit signal and adding Gaussian white noise to the shaped transmit signal to obtain a noisy receive signal vector; Performing matched filtering on the noisy received signal vector to obtain a first filtered signal; Whitening the first filtered signal through a whitening filter to obtain a second filtered signal; removing a cyclic prefix from the second filtered signal to extract a valid signal vector; The effective signal vector is subjected to symbol-by-symbol signal detection to determine and restore the original plurality of modulation symbols.

[0011] In combination with the second aspect, further, performing matched filtering on the noisy received signal vector to obtain a first filtered signal includes: Multiplying the noisy received signal vector by a preset matched filter matrix to obtain a first filtered signal; The expression of the first filtered signal includes: ; Wherein, z represents the first filtered signal; represents the matched filter matrix; r represents the noisy received signal.

[0012] In combination with the second aspect, further, processing the first filtered signal through a whitening filter to obtain a second filtered signal includes: multiplying the first filtered signal by a preset whitening filter matrix to convert colored noise in the first filtered signal into Gaussian white noise, thereby outputting a second filtered signal; The expression of the second filtered signal includes: ; in, represents the second filtered signal; D represents the whitening matched filter; z represents the first filtered signal.

[0013] In combination with the second aspect, further, the decision to restore the original expressions of the plurality of modulation symbols includes: ; in, represents the i-th sample point output by the whitening matched filter; A represents the constellation set; represents the i-th symbol in the constellation set; Represents the recovered modulation symbols.

[0014] In a third aspect, a super-Nyquist signal system based on frequency domain precoding includes: On the sending side: A data preprocessing module, configured to obtain original bit data and perform segment processing on the bit data to obtain a plurality of bit segments; A constellation mapping module is used to perform constellation mapping on each bit segment through a phase modulation / orthogonal amplitude modulation module to obtain a plurality of modulation symbols, and to form a modulation symbol vector from the plurality of modulation symbols; A frequency domain precoding module, configured to perform frequency domain precoding processing on the modulation symbol vector to obtain a precoded symbol vector; a cyclic prefix adding module, configured to take a fixed-length sample from the end of the pre-coded symbol vector and insert it as a cyclic prefix to the front of the second symbol vector to form an extended signal vector; a shaping filter module, configured to process the extended signal vector through a preset super-Nyquist shaping filter to obtain a shaped transmission signal; On the receiving end: a Gaussian noise adding module, receiving a shaped transmit signal and adding Gaussian white noise to the shaped transmit signal to obtain a noisy receive signal vector; a matched filtering module, configured to perform matched filtering on the noisy received signal vector to obtain a first filtered signal; A whitening matched filtering module, configured to perform whitening processing on the first filtered signal through a whitening filter to obtain a second filtered signal; a cyclic prefix removal module, configured to remove the cyclic prefix from the second filtered signal and extract a valid signal vector; The symbol-by-symbol detection module is used to perform symbol-by-symbol signal detection on the effective signal vector, and determine and restore the original modulation symbols.

[0015] Compared with the prior art, the present invention has the following beneficial effects: This proposed super-Nyquist signal transmission method based on frequency domain precoding, which includes both transmission and reception, aims to reduce bit error rates (BERs) and simplify signal demodulation complexity. Frequency domain precoding eliminates intersymbol interference (ISI) and optimizes signal transmission characteristics. Furthermore, a cyclic prefix is ​​used to convert linear convolution into cyclic convolution, simplifying signal processing. Simulation results demonstrate that this method significantly reduces BERs under high-order modulation schemes, approaching theoretical performance. Compared to traditional methods, it offers higher transmission efficiency and reliability. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 FIG2 is a system block diagram of a super-Nyquist signal transmission method based on frequency domain precoding provided by an embodiment of the present invention; Figure 2 The figure shows the bit error performance of different precoding methods when the modulation method provided by the embodiment of the present invention is BPSK; Figure 3 FIG2 shows the bit error performance of different precoding modes when the modulation mode provided by the embodiment of the present invention is 8-QAM. DETAILED DESCRIPTION

[0017] The technical solution of the present invention is described in detail below through the accompanying drawings and specific embodiments. It should be understood that the embodiments of the present invention and the specific features in the embodiments are detailed descriptions of the technical solution of the present invention, rather than limitations on the technical solution of the present invention. In the absence of conflict, the embodiments of the present invention and the technical features in the embodiments can be combined with each other.

[0018] The term "and / or" simply describes a relationship between related objects, indicating that three possible relationships exist. For example, "A and / or B" can mean: A exists alone, A and B exist simultaneously, or B exists alone. Additionally, the character " / " generally indicates an "or" relationship between the related objects. Example 1

[0019] See also Figure 1 This embodiment introduces a super-Nyquist signal transmission method based on frequency domain precoding, which is performed by a wireless channel transmitter to reduce signal demodulation complexity and improve transmission performance, including: Step S1: Obtain original bit data and segment the bit data to obtain bit segments; During bit data preprocessing, the user segments the bit data to be transmitted into fixed lengths, with each segment containing the same number of bits. For example, if each segment contains 8 bits, the original bit data is divided into multiple 8-bit segments, making the segmented data easier to modulate.

[0020] Step S2: performing constellation mapping on each bit segment through a phase-shift / quadrature-shift modulation (PSK / QAM) module to generate a number of modulation symbols, and combining the modulation symbols into a modulation symbol vector; The processed bit segments are mapped to modulation symbols. Several modulation symbols form a specific constellation diagram on the complex plane. After constellation mapping, each bit segment is mapped to a specific modulation symbol, which is represented by the mathematical symbol {} Represented, and composed of modulation symbol vector a, expressed as , where N is the total number of modulation symbols.

[0021] Step S3: performing frequency domain precoding processing on the modulation symbol vector to obtain a precoded symbol vector; To eliminate inter-symbol interference (ISI) and optimize signal transmission, the modulation symbol vectors are precoded in the frequency domain. This process includes the following steps: Performing a fast Fourier transform (FFT) on the modulation symbol vector to convert it from the time domain to the frequency domain, thereby obtaining a frequency domain modulation symbol vector A; The frequency domain modulation symbol vector A is combined with the inverse matrix of the diagonal matrix Multiply them to obtain the frequency domain precoded signal; The precoded signal is converted from the frequency domain back to the time domain through the inverse fast Fourier transform (IFFT) to obtain the precoded symbol vector .

[0022] Furthermore, the precoding symbol vector Expressions include: ; in, represents the precoding symbol vector; represents the complex conjugate transpose of the Fourier matrix; represents the inverse matrix of a diagonal matrix; represents the normalized Fourier matrix; represents the modulation symbol vector.

[0023] Step S4: taking samples of a fixed length from the end of the pre-coded symbol vector as a cyclic prefix and inserting it to the front of the second symbol vector to form an extended signal vector; In signal processing, linear convolution is easily affected by boundary effects during the processing process, resulting in signal distortion. Therefore, the purpose of adding a cyclic prefix is ​​to convert linear convolution into circular convolution, thereby avoiding boundary effects, simplifying the signal processing process and reducing inter-symbol interference.

[0024] Furthermore, the expression of the cyclic prefix extension signal vector is:

[0025] in, represents the length of the cyclic prefix; b represents the extended signal vector inserted into the cyclic prefix.

[0026] Step S5: Process the extended signal vector through a preset super-Nyquist shaping filter to obtain a shaped transmission signal.

[0027] Since the super-Nyquist transmission rate is higher than the Nyquist rate, inter-symbol interference may be introduced, so specific shaping filtering is required to optimize the signal. The super-Nyquist shaping filter is used to further process the signal to make it meet the super-Nyquist transmission requirements.

[0028] In an embodiment of the present invention, a super-Nyquist shaping filter adopts a root raised cosine filter, and flexibly adjusts the time domain compression factor τ and the root raised cosine filter roll-off coefficient α according to actual transmission requirements. The time domain compression factor is used to control the degree of compression of the signal in the time domain, and its value range is between 0 and 1. A smaller compression factor can further improve the transmission rate, but may increase the signal bandwidth and the risk of inter-symbol crosstalk. The root raised cosine filter roll-off coefficient α determines the frequency roll-off characteristics of the filter, and its value range is also between 0 and 1. A larger roll-off coefficient can provide better anti-inter-symbol crosstalk capability, but will sacrifice a certain amount of bandwidth utilization. By reasonably selecting the time domain compression factor and the root raised cosine filter roll-off coefficient, an optimal balance can be achieved between signal transmission rate, anti-interference capability and bandwidth utilization, thereby optimizing the transmission performance of the super-Nyquist signal and ultimately obtaining a shaped transmission signal s.

[0029] Furthermore, the expression of the shaped transmission signal includes: s=Hb Wherein, s represents the shaped transmission signal; H represents the super-Nyquist shaping filter matrix.

[0030] In summary, the present invention effectively eliminates intersymbol interference (ISI) through frequency-domain precoding, reduces signal demodulation complexity at the receiving end, and significantly improves signal transmission reliability and error performance. Simulation results show that this method can approach theoretical performance under high-order modulation schemes, offering significant advantages over traditional precoding methods. Example 2

[0031] The shaped transmitted signal propagates in free space and reaches the receiver through the wireless channel. The wireless channel then introduces various noise interferences, among which Gaussian white noise is a common one. By adding Gaussian white noise, the system performance in a noisy environment can be evaluated, such as bit error rate and signal-to-noise ratio.

[0032] Therefore, the present invention also introduces a super-Nyquist signal receiving method based on frequency domain precoding, which is executed by a wireless channel receiving end to efficiently recover the signal sent by the transmitting end. The method includes: (1) receiving a shaped transmission signal and adding Gaussian white noise to the shaped transmission signal to simulate noise interference in an actual transmission environment, thereby obtaining a noisy received signal vector; The expression of the noisy received signal vector is: r=s+w Where r represents the noisy received signal vector; w represents the additive white Gaussian noise vector.

[0033] (2) performing matched filtering on the noisy received signal vector to obtain a first filtered signal; Performing matched filtering on the noisy received signal vector using a root raised cosine filter to reduce the impact of out-of-band noise and interference on the valid signal, and obtaining a first filtered signal z by multiplying the noisy received signal vector with a preset matched filter matrix; The expression of the first filtered signal includes: ; Wherein, z represents the first filtered signal; represents the matched filter matrix; r represents the noisy received signal.

[0034] It should be noted that matched filtering can effectively reduce the impact of out-of-band noise and interference on useful signals, improve the signal-to-noise ratio of the signal, and the signal after matched filtering is easier to perform subsequent processing, such as whitening matched filtering and symbol-by-symbol detection.

[0035] (3) performing whitening processing on the first filtered signal through a whitening filter to obtain a second filtered signal; In a super-Nyquist transmission system, due to the artificial introduction of inter-symbol interference, colored noise will appear in the signal. In order to reduce the influence of colored noise, the first filtered signal is multiplied by a preset whitening filter matrix to convert the colored noise in the first filtered signal into Gaussian white noise (the characteristics of Gaussian white noise are easier to process), thereby outputting a second filtered signal. ; The second filtered signal Expressions include: ; in, represents the second filtered signal; D represents the whitening matched filter.

[0036] It should be noted that the whitening filtering process simplifies the signal detection process and improves the detection performance, especially in the presence of colored noise.

[0037] (4) removing the cyclic prefix from the second filtered signal and extracting a valid signal vector; The operation process is as follows: The second filtered signal Split into two parts, namely cyclic prefix , and the effective signal part , whose expression is:

[0038] in, represents a cyclic prefix; The valid part of the first filtered signal.

[0039] It should be noted that the cyclic prefix is ​​redundant information. After removing the cyclic prefix, the data can be reduced, the effective part of the signal can be more concentrated, and symbol-by-symbol detection can be performed directly, which simplifies the signal processing process and improves the detection performance.

[0040] (5) For the effective signal vector Perform symbol-by-symbol signal detection to determine and restore the original modulation symbols.

[0041] Since there is no inter-symbol interference between the valid signal parts after removing the cyclic prefix, at this time, the valid signal part The signal is detected symbol by symbol instead of the entire signal jointly. Finally, the original modulation symbols are restored by the decision of the minimum Euclidean distance.

[0042] The decision restores the original expressions of the modulation symbols, including: ; in, represents the i-th sample point output by the whitening matched filter; A represents the constellation set; represents the i-th symbol in the constellation set; Represents the recovered modulation symbols.

[0043] It should be noted that symbol-by-symbol detection significantly reduces the complexity of signal detection, and is particularly suitable for high-order modulation schemes. It also improves the accuracy of detection, thereby effectively reducing the bit error rate. Example 3

[0044] See also Figure 3 ,The figure shows the comparison of the error performance of different precoding methods under binary phase shift keying (BPSK) modulation. Figure 3 The horizontal axis is the bit signal-to-noise ratio (Eb / No), and the vertical axis is the bit error rate (BER).

[0045] As can be seen from the figure, traditional Nyquist transmission systems perform well near the theoretical bit error performance curve, while existing precoding methods such as singular value decomposition (SVD), Tomlinson-Halasma (TH), and G-to-minus-half (GTMH) precoding methods cannot achieve satisfactory bit error performance in super-Nyquist transmission systems. In contrast, the frequency-domain precoding method proposed in this invention is able to significantly approach the theoretical performance curve, demonstrating its significant advantage in bit error performance. This is because the proposed frequency-domain precoding method effectively eliminates inter-symbol interference and reduces the complexity of signal demodulation by optimizing the frequency-domain characteristics of the signal, thereby approaching ideal transmission performance under high signal-to-noise ratio conditions.

[0046] See also Figure 3, the figure shows the error performance comparison of different precoding methods under the 8-order quadrature amplitude modulation (8-QAM) modulation mode. Similarly, the horizontal axis is the bit signal-to-noise ratio (Eb / No) and the vertical axis is the bit error rate (BER). The frequency domain precoding method proposed in the embodiment of the present invention performs better than other traditional precoding methods when the time domain compression factor τ takes a specific value. This shows that under high-order modulation modes, the precoding method of the present invention can better adapt to the needs of super-Nyquist transmission systems, and effectively improves the signal's anti-interference ability and transmission efficiency by optimizing the frequency domain characteristics of the signal. Especially in high-order modulation, the Euclidean distance between symbols is small, and the optimization of the error performance is more critical. The frequency domain precoding method of the present invention significantly reduces the error rate by detecting the symbol by symbol, verifying its superior performance under complex modulation modes.

[0047] By referring to Figure 2 and Figure 3 The simulation results clearly demonstrate that the proposed super-Nyquist signal transmission method based on frequency-domain precoding performs well under various modulation schemes, significantly reducing bit error rates and simplifying signal processing. These technical results fully demonstrate the practicality and advancement of the present invention in super-Nyquist transmission systems, providing an efficient and reliable signal transmission solution for the information and communications field. Example 4

[0048] A super-Nyquist signal system based on frequency domain precoding, comprising: On the sending side: A data preprocessing module, configured to obtain original bit data and perform segment processing on the bit data to obtain a plurality of bit segments; A constellation mapping module is used to perform constellation mapping on each bit segment through a phase modulation / orthogonal amplitude modulation module to obtain a plurality of modulation symbols, and to form a modulation symbol vector from the plurality of modulation symbols; A frequency domain precoding module, configured to perform frequency domain precoding processing on the modulation symbol vector to obtain a precoded symbol vector; a cyclic prefix adding module, configured to take a fixed-length sample from the end of the pre-coded symbol vector and insert it as a cyclic prefix to the front of the second symbol vector to form an extended signal vector; a shaping filter module, configured to process the extended signal vector through a preset super-Nyquist shaping filter to obtain a shaped transmission signal; On the receiving end: a Gaussian noise adding module, receiving a shaped transmit signal and adding Gaussian white noise to the shaped transmit signal to obtain a noisy receive signal vector; a matched filtering module, configured to perform matched filtering on the noisy received signal vector to obtain a first filtered signal; A whitening matched filtering module, configured to perform whitening processing on the first filtered signal through a whitening filter to obtain a second filtered signal; a cyclic prefix removal module, configured to remove the cyclic prefix from the second filtered signal and extract a valid signal vector; The symbol-by-symbol detection module is used to perform symbol-by-symbol signal detection on the effective signal vector, and determine and restore the original modulation symbols.

[0049] The embodiments of the present invention are described above in conjunction with the accompanying drawings, but the present invention 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 invention, ordinary technicians in this field can also make many forms without departing from the scope of protection of the purpose of the present invention and the claims, which are all protected by the present invention.

Claims

1. A super-Nyquist signal transmission method based on frequency domain precoding, characterized in that: Executed by the wireless channel transmitter, including: Obtaining original bit data, and performing segmentation processing on the bit data to obtain multiple bit segments; Perform constellation mapping on each bit segment through a phase modulation / orthogonal amplitude modulation module to obtain a number of modulation symbols, and compose the plurality of modulation symbols into a modulation symbol vector; Performing frequency domain precoding processing on the modulation symbol vector to obtain a precoded symbol vector; Taking samples of a fixed length from the end of the pre-coded symbol vector as a cyclic prefix, and inserting the samples into the front of the second symbol vector to form an extended signal vector; The extended signal vector is processed by a preset super-Nyquist shaping filter to obtain a shaped transmission signal.

2. The method for transmitting super-Nyquist signals based on frequency domain precoding according to claim 1, wherein: Performing frequency domain precoding processing on the modulation symbol vector includes: Performing a fast Fourier transform on the modulation symbol vector to convert the modulation symbol vector from the time domain to the frequency domain; Multiply the modulation symbol vector transferred to the frequency domain by the inverse matrix of the diagonal matrix to obtain the frequency domain precoded signal; The frequency-domain precoded signal is then subjected to an inverse transform of the Fast Fourier Transform, thereby converting the precoded signal from the frequency domain back to the time domain and obtaining a precoded symbol vector.

3. The method for transmitting super-Nyquist signals based on frequency domain precoding according to claim 2, wherein: The expression of the precoding symbol vector includes: ； in, represents the precoding symbol vector; represents the complex conjugate transpose of the Fourier matrix; represents the inverse matrix of a diagonal matrix; represents the normalized Fourier matrix; represents the modulation symbol vector.

4. The method for transmitting super-Nyquist signals based on frequency domain precoding according to claim 1, wherein: The expression of the cyclic prefix extension signal vector is: ； in, represents the length of the cyclic prefix; b represents the extended signal vector inserted into the cyclic prefix; represents the precoding symbol vector.

5. The method for transmitting super-Nyquist signals based on frequency domain precoding according to claim 1, wherein: The expression of the shaped transmission signal includes: s=Hb; Wherein, s represents the shaped transmission signal; H represents the super-Nyquist shaping filter matrix; and b represents the extended signal vector into which the cyclic prefix is ​​inserted.

6. A super-Nyquist signal receiving method based on frequency domain precoding, characterized in that: Executed by the wireless channel receiving end, including: receiving a shaped transmit signal and adding Gaussian white noise to the shaped transmit signal to obtain a noisy receive signal vector; Performing matched filtering on the noisy received signal vector to obtain a first filtered signal; Whitening the first filtered signal through a whitening filter to obtain a second filtered signal; removing a cyclic prefix from the second filtered signal to extract a valid signal vector; The effective signal vector is subjected to symbol-by-symbol signal detection to determine and restore the original plurality of modulation symbols.

7. The super-Nyquist signal receiving method based on frequency domain precoding according to claim 6, characterized in that: Performing matched filtering on the noisy received signal vector to obtain a first filtered signal includes: Multiplying the noisy received signal vector by a preset matched filter matrix to obtain a first filtered signal; The expression of the first filtered signal includes: ； Wherein, z represents the first filtered signal; represents the matched filter matrix; r represents the noisy received signal.

8. The super-Nyquist signal receiving method based on frequency domain precoding according to claim 6, characterized in that: Processing the first filtered signal through a whitening filter to obtain a second filtered signal, comprising: multiplying the first filtered signal by a preset whitening filter matrix to convert colored noise in the first filtered signal into Gaussian white noise, thereby outputting a second filtered signal; The expression of the second filtered signal includes: ； in, represents the second filtered signal; D represents the whitening matched filter; z represents the first filtered signal.

9. The super-Nyquist signal receiving method based on frequency domain precoding according to claim 6, characterized in that: The decision restores the original expressions of the modulation symbols, including: ； in, represents the i-th sample point output by the whitening matched filter; A represents the constellation set; represents the i-th symbol in the constellation set; Represents the recovered modulation symbols.

10. A super-Nyquist signal system based on frequency domain precoding, characterized in that: include: On the sending side: A data preprocessing module, configured to obtain original bit data and perform segment processing on the bit data to obtain a plurality of bit segments; A constellation mapping module is used to perform constellation mapping on each bit segment through a phase modulation / orthogonal amplitude modulation module to obtain a plurality of modulation symbols, and to form a modulation symbol vector from the plurality of modulation symbols; A frequency domain precoding module, configured to perform frequency domain precoding processing on the modulation symbol vector to obtain a precoded symbol vector; a cyclic prefix adding module, configured to take a fixed-length sample from the end of the pre-coded symbol vector and insert it as a cyclic prefix to the front of the second symbol vector to form an extended signal vector; a shaping filter module, configured to process the extended signal vector through a preset super-Nyquist shaping filter to obtain a shaped transmission signal; On the receiving end: a Gaussian noise adding module, receiving a shaped transmit signal and adding Gaussian white noise to the shaped transmit signal to obtain a noisy receive signal vector; a matched filtering module, configured to perform matched filtering on the noisy received signal vector to obtain a first filtered signal; A whitening matched filtering module, configured to perform whitening processing on the first filtered signal through a whitening filter to obtain a second filtered signal; a cyclic prefix removal module, configured to remove the cyclic prefix from the second filtered signal and extract a valid signal vector; The symbol-by-symbol detection module is used to perform symbol-by-symbol signal detection on the effective signal vector, and determine and restore the original modulation symbols.

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