Underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation joint optimal solution search

Abstract

The application discloses a kind of based on gradient estimation joint optimal solution search underwater acoustic OFDM system carrier synchronization algorithm, belong to underwater acoustic communication technical field.The specific method includes the following steps: step one: in underwater acoustic OFDM communication system, time domain method is used to underwater acoustic OFDM carrier synchronization.Step two: in an OFDM frame, first OFDM symbol is estimated based on optimal solution search carrier synchronization algorithm to carry out frequency offset estimation.Step three: from the second OFDM symbol, gradient descent-based carrier synchronization algorithm is used to estimate frequency offset value, and the frequency offset estimation result of the first OFDM symbol is used as the initial value of iteration.The present application can carry out more fine residual carrier frequency offset compensation by pilot symbol, improve the step of underwater acoustic system carrier synchronization algorithm using Adam method, reduce complexity while improving accuracy, realize reliable and effective underwater acoustic OFDM communication.

Description

Technical Field

[0001] The present invention relates to an underwater acoustic OFDM system carrier synchronization algorithm, in particular to an underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation and joint optimal solution search, belonging to the technical field of underwater acoustic communications. Background Art

[0002] The underwater acoustic channel is a doubly selective fading channel with severe multipath and Doppler effects. Orthogonal Frequency Division Multiplexing (OFDM) is a modulation technique with high spectrum efficiency and strong multipath resistance, making it a common technology for underwater acoustic communications.

[0003] Generally speaking, there are two causes of carrier frequency deviation. One is phase noise caused by the difference in fixed frequencies between the transmitter and receiver, and the other is carrier frequency offset (CFO) caused by Doppler shift. This carrier frequency offset causes the OFDM signal entering the Fast Fourier Transformation (FFT) window to no longer be a complete symbol period. Furthermore, it contains partial symbol periods of other subcarriers, which are multipath components and affect the decision of the current symbol. Whether it is an integer multiple of the subcarrier spacing or a fractional frequency offset, it will cause a loss of orthogonality between subcarriers, leading to inter-channel interference (ICI).

[0004] To ensure the performance of OFDM systems, receivers must utilize synchronization techniques—accurately estimating the start and end points of OFDM symbols to facilitate FFT processing. Carrier synchronization accuracy, estimation range, and algorithm complexity are typically considered key performance indicators, each of which influences and constrains the other. Therefore, when designing a carrier synchronization algorithm, it's important to comprehensively consider the algorithm's real-time requirements, the available hardware resources, and the application scenario to maximize performance in practical applications.

[0005] OFDM is a commonly used technology for high-speed underwater acoustic communication, but it is very sensitive to synchronization deviation, which can have a serious impact on OFDM communication systems. Therefore, research on improving the carrier synchronization algorithm for underwater acoustic OFDM communication is particularly important. Summary of the Invention

[0006] The application is to solve the problem of the influence of the carrier frequency offset caused by the Doppler frequency shift of the underwater acoustic OFDM channel on the channel interference, further improve the carrier synchronization accuracy, and reduce the calculation complexity.

[0007] The technical scheme adopted by the application to solve the above problem is:

[0008] The gradient estimation-based joint optimal solution search underwater acoustic OFDM system carrier synchronization algorithm is realized through the following steps:

[0009] S1: In the underwater acoustic OFDM communication system, the time domain method is used for underwater acoustic OFDM carrier synchronization;

[0010] S2: In an OFDM frame, the carrier synchronization algorithm based on optimal solution search is first used to estimate the frequency offset of the first OFDM symbol;

[0011] S3: From the second OFDM symbol, the carrier synchronization algorithm based on gradient descent is used to estimate the frequency offset value, and the frequency offset estimation result of the first OFDM symbol is used as the initial value of iteration.

[0012] Further, the time domain method for underwater acoustic OFDM carrier synchronization in S1 specifically includes:

[0013] According to the OFDM property, the signal emitted by the transducer is:

[0014]

[0015] Assuming that the noise of the underwater acoustic channel is an additive white Gaussian noise channel, the noise is w, T is the OFDM symbol length, N is the number of subcarriers, and the received signal is:

[0016]

[0017] Sampling y(t) with a sampling period of T s Then:

[0018]

[0019] There is a symbol synchronization deviation Δn in synchronization f At this time, it is equivalent to performing a discrete Fourier transform operation from the front of the cyclic prefix, that is:

[0020]

[0021] Let Δn f (i) is the starting point τ in the orthogonal regionmax / T s The signal after phase compensation is:

[0022] Y k (i) = S k (i)+w(i) (5)

[0023] By using the properties of the orthogonal region, we can change the starting position of the discrete Fourier transform to obtain multiple S k Estimate of Y k (1),Y k (2),…, for multiple Y k Differential processing can achieve diversity and smoothing effects, so the received signal with carrier frequency deviation is:

[0024]

[0025] Where, is the sampling frequency of the receiving end, N fft is the number of discrete Fourier transform points, T o is the OFDM symbol duration, S k is the spectrum of the subcarrier;

[0026] The baseband signal adopts rectangular wave and quadrature phase shift keying modulation mode, then S k is a SINC function with a main lobe width of 2 / T o , the sidelobe width is 1 / T o When:

[0027]

[0028] Taking the derivative of formula (7) with respect to m, we can get:

[0029]

[0030] By using high-power fast Fourier transform, in addition to the data at the subcarrier, the frequency data adjacent to the subcarrier can also be obtained. If the adjacent frequency points are within the allowable range, the inherent inter-channel interference they bring can be treated as noise and differentiated to achieve a differential effect. If adjacent subcarriers are used to transmit pilot signals, the inter-channel interference they bring can be removed. Other methods using interpolation or subtraction will introduce errors. The frequency domain signal at non-orthogonal frequency points is:

[0031]

[0032] Subscript j is the non-subcarrier frequency after high-multiple Fourier transformation, which should satisfy:

[0033]

[0034] In formula (10), the frequency resolution after high multiple Fourier transform is Δf', the frequency during original transmission is Δf, the orthogonal factor is λ, the total phase is π, and the influence of these non-orthogonal frequency points can be treated as noise.

[0035] Further, all the power difference is ≤0.25dB, and the sum of the maximum phase angles on the left and right sides is ≤π / 4.

[0036] Further, the carrier synchronization algorithm based on optimal solution search in S2 specifically comprises:

[0037] In the underwater acoustic OFDM communication system, the number of sub-channels is fixed as 1024, the signal bandwidth is B, the modulation frequency of the first sub-carrier is f0, the carrier frequency of the kth carrier is f k = f0+kΔf (k≤64), the sub-channel interval is Δf=B / K, K is the Kth sub-channel (K≤1024), the modulation mode of quadrature phase shift keying is adopted, and the OFDM signal s(t) can be represented as:

[0038]

[0039] In formula (11), T=1 / Δf represents the symbol length, T CP represents the duration of the cyclic prefix, d k represents the user information to be sent.

[0040] Suppose the length of the cyclic prefix is greater than or equal to the length of the channel delay spread, T CP >> T DS After passing through the channel, the impulse response h(τ,t) of the signal received at the receiving end can be represented as:

[0041]

[0042] In formula (12), h p (t) represents the amplitude of the channel, τ p (t) represents the delay of the pth channel, τ p (t)≈τ p -at, at is the time delay caused by multipath,

[0043] Because of small-scale fading, the channel amplitude will change due to multipath effect, h p (t)≈h p Further analysis of h(τ,t) can be represented as:

[0044]

[0045] The signal received at the receiving end after passing through the channel amplitude attenuation and time delay spread can be represented as:

[0046]

[0047] In formula (14), n(t) represents Gaussian white noise, whose power spectrum density is N0 / 2,

[0048] After the received signal y(t) undergoes packet synchronization estimation, symbol timing synchronization, and coarse frequency offset compensation,

[0049]

[0050] In formula (15), ε is the carrier frequency offset, H k is the frequency response of the kth subcarrier of the OFDM symbol, w(t) is the additive noise,

[0051] In the carrier synchronization algorithm based on optimal solution search, assuming a frequency offset estimation range and setting the search step, the frequency offset estimation value in each search interval is The signal is finely compensated in the time domain, and the compensated signal can be expressed as:

[0052]

[0053] pilot symbols d k , k∈κ p , and the resulting mean square error is:

[0054]

[0055] In formula (17), d k Represents the user data sent out by the transmitter after modulation. Represents the user data received by the receiving end after passing through the channel, Represents the user data after frequency offset compensation at the receiving end;

[0056] If there are multiple frequency offset results, the mean square error of each frequency offset estimate of the received signal can be calculated by the above formula. The accurate frequency offset value of the received signal can be obtained by finding the minimum mean square error.

[0057]

[0058] Furthermore, assuming that the residual frequency offset of the i-th symbol is The residual frequency offset of the next i+1 symbol in the same frame is Taking advantage of the fact that the frequency offsets of two adjacent symbols are similar, the residual frequency offset of the current symbol in the same frame is approximately equal to the residual frequency offset of the previous symbol, so:

[0059]

[0060] Furthermore, the frequency offset estimation method is as follows: based on the known frequency offset value of the current symbol, the next symbol is compensated with the current frequency offset estimation value, so that the detection area of ​​the next symbol frequency offset estimation can be reduced. When the search step size remains unchanged, the number of detections of the next symbol is reduced, and the running time is also reduced accordingly.

[0061] Furthermore, the gradient descent-based carrier synchronization algorithm described in S3 specifically includes:

[0062] The error gradient can be expressed as

[0063]

[0064] In formula (20), Then we have:

[0065] y k =∫ T tv(t)e -iβt e -2πikΔft dt (21)

[0066] Through the error gradient, the frequency deviation β can be iteratively calculated as:

[0067]

[0068] In formula (22), K β The step size of the frequency deviation iteration and the initial value of the frequency deviation are set

[0069] for

[0070] Furthermore, the Adam method is used to improve the selection of step size, which can be expressed as:

[0071]

[0072] In formula (23), is the estimated exponential moving mean of the gradient.

[0073] The beneficial effects of the present invention are:

[0074] 1. According to the frame structure characteristics of the OFDM system, the residual carrier frequency offset compensation of the pilot symbol in the carrier synchronization algorithm based on optimal solution search is more refined, and the overall operation complexity is reduced. By applying the stochastic gradient descent algorithm in the carrier synchronization of the receiver of the underwater acoustic OFDM communication system, the local minimum error value is crossed, and the algorithm has higher synchronization accuracy. By the method of gradient estimation combined with optimal solution search, the frequency offset estimation accuracy is improved, the problem that the gradient descent algorithm is easy to fall into local minimum error is avoided, the complexity of the algorithm is effectively reduced, and the optimal mean square error performance is realized.

[0075] 2. The underwater acoustic system carrier synchronization algorithm can improve the step length by pilot symbol for more refined residual carrier frequency offset compensation, and the Adam method is used, so that the precision is improved while the complexity is reduced, and reliable and effective underwater acoustic OFDM communication is realized.

[0076] 3. The present application has excellent synchronization accuracy, and the running time of the algorithm is short, which guarantees the real-time requirement of communication. BRIEF DESCRIPTION OF DRAWINGS

[0077] Figure 1 is a flowchart of the underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation combined with optimal solution search of the present application;

[0078] Figure 2 is a flowchart of the carrier synchronization algorithm based on optimal solution search of the present application. DETAILED DESCRIPTION

[0079] Specific implementation mode one: combining Figure 1 and Figure 2 The present application is described, and the underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation combined with optimal solution search is realized by the following steps:

[0080] S1: In the underwater acoustic OFDM communication system, the time domain method is used for underwater acoustic OFDM carrier synchronization; that is, the underwater acoustic OFDM carrier synchronization algorithm based on the time domain method specifically includes:

[0081] According to the OFDM property, the signal emitted by the transducer is:

[0082]

[0083] Assuming that the noise of the underwater acoustic channel is an additive white Gaussian noise (AWGN) channel, the noise is w, T is the OFDM symbol length, N is the number of subcarriers, since the signal modulation used does not need carrier modulation and demodulation, the received signal is:

[0084]

[0085] Sampling y(t) with a sampling period of T s Then:

[0086]

[0087] There is a symbol timing offset Δn in synchronization f , which is equivalent to a Discrete Fourier Transform (DFT) operation from the front of the cyclic prefix, i.e.

[0088]

[0089] The symbol timing offset only causes a rotation of the phase of the transmitted signal and does not destroy the orthogonality. In actual communication, the length of the cyclic prefix is greater than the maximum time delay spread of the channel, and there is an orthogonality zone, i.e. max < t < t CP As long as the starting point of the DFT operation falls within the orthogonality zone, the orthogonality of the received signal will not be destroyed. Only the phase rotation caused by the different starting points needs to be compensated to achieve data demodulation.

[0090] Let Δn f (i) be the distance from the starting point τ max / T s of the orthogonality zone, and the signal after phase compensation be:

[0091] Y k (i) = S k (i) + w(i)

[0092] Using the properties of the orthogonality zone, the starting position of the DFT is changed to obtain multiple estimates Y k (1), Y k (2), …, of S k (i). Difference processing of multiple Y k can achieve diversity and smoothing effects.

[0093] The received signal containing the carrier frequency offset is:

[0094]

[0095] In the formula, is the sampling frequency of the receiving end, N fft is the number of DFT points, T o is the OFDM symbol duration, and S k is the frequency spectrum of the subcarrier.

[0096] The baseband signal uses a rectangular wave and a QPSK modulation method, and Sk is a SINC function with a main lobe width of 2 / T o , the sidelobe width is 1 / T o When:

[0097]

[0098] Taking the derivative of this formula with respect to m, we can get:

[0099]

[0100] Divide the above formula Except for , all other terms are constants, and the result is only related to the sampling frequency, from which the sampling frequency deviation can be estimated. When using a high-power FFT, in addition to the data at the subcarrier, the frequency point data adjacent to the subcarrier can also be obtained. The adjacent frequency points are within the allowable range, and the inherent ICI they bring can be treated as noise. By performing a differential operation, a differential effect can be obtained. If adjacent subcarriers are used to transmit pilot signals, the ICI they bring can be removed. Other methods that use interpolation or subtraction will introduce errors. The frequency domain signal at non-orthogonal frequency points is:

[0101]

[0102] Subscript j is the non-subcarrier frequency after high-multiple DFT and should satisfy:

[0103]

[0104] The impact of these non-orthogonal frequencies can be treated as noise.

[0105] The frequency resolution after high-power DFT is Δf', the original transmission frequency Δf, the orthogonality factor λ, the interval Δf between adjacent subcarriers, the total phase is π, and the phase interval of each frequency point is π / (N fftnew / N fftold ), in order to utilize non-orthogonal subcarriers, the point that can be used for QPSK is that when there is no noise, the maximum can only use λ = 1 / 8, but there is noise in the channel. In order to ensure the effect, all power differences do not exceed 0.25dB, and the sum of the maximum phase angles on the left and right sides does not exceed π / 4.

[0106] S2: In an OFDM frame, frequency offset estimation is performed on the first OFDM symbol using a carrier synchronization algorithm based on optimal solution search. By introducing optimal solution search, pilot symbols are used to perform more precise residual carrier frequency offset compensation.

[0107] The carrier synchronization algorithm based on optimal solution search specifically includes:

[0108] In the underwater acoustic OFDM communication system, the number of subchannels is fixed to 1024, the signal bandwidth is B, the modulation frequency of the first subcarrier is f0, and the carrier frequency of the kth carrier is f k =f0+kΔf(k≤64), then the subchannel spacing is Δf=B / K, K is the K-th subchannel (K≤1024), and using QPSK modulation, the OFDM signal s(t) can be expressed as:

[0109]

[0110] Where, T = 1 / Δf represents the symbol length; T CP Represents the duration of the cyclic prefix; d k Represents the user information to be sent.

[0111] Assuming that the length of the cyclic prefix is ​​greater than or equal to the delay spread length of the channel, T CP >>T DS , after passing through the channel, the impulse response h(τ,t) of the signal received by the receiver can be expressed as:

[0112]

[0113] Where h p (t) represents the amplitude of the channel; τ p (t) represents the delay of the p channel, τ p (t)≈τ p -at, at is the delay caused by multipath,

[0114] Because of small-scale fading, the channel amplitude will change due to multipath effects, with h p (t)≈h p . Further analysis of h(τ,t)

[0115]

[0116] The signal received by the receiver after channel amplitude attenuation and delay spread can be expressed as:

[0117]

[0118] Where n(t) represents Gaussian white noise, and its power spectral density is N0 / 2.

[0119] When the received signal y(t) undergoes packet synchronization estimation, symbol timing synchronization, and coarse frequency offset compensation,

[0120]

[0121] Among them, ε is the carrier frequency deviation, H kw(t) is additive noise.

[0122] In the carrier synchronization algorithm based on optimal solution search, a frequency offset estimation range is assumed, a search step is set, and the frequency offset estimation value in each search interval is The signal is finely compensated in the time domain, and the compensated signal can be represented as

[0123]

[0124] Pilot symbols with equal intervals d k , k∈κ p The obtained mean square error (MSE)

[0125]

[0126] In the formula, d k represents the user data sent out after modulation at the transmitting end; represents the user data received after the channel at the receiving end; represents the user data compensated for frequency offset at the receiving end.

[0127] If there are multiple frequency offset results, the MSE of each frequency offset estimation value of the received signal can be calculated from the above formula, and the accurate frequency offset value of the received signal can be obtained by finding the minimum mean square error

[0128]

[0129] When estimating each symbol, according to the set frequency offset range and frequency search step, if all symbols are sequentially calculated repeatedly according to each assumed frequency offset value, a large amount of calculation time will be consumed, because in addition to the frequency offset estimation of the first symbol, the remaining symbols also need to be calculated again. In order to reduce the complexity of the carrier synchronization algorithm, the relationship between the frequency offset estimation values of two adjacent symbols is analyzed. It is assumed that the residual frequency offset of the i-th symbol is The residual frequency offset of the next i+1 symbol in the same frame is By using the feature that the frequency offsets of two adjacent symbols are similar, the residual frequency offset of the current symbol in the same frame is approximately equal to the residual frequency offset of the previous symbol, Therefore, the frequency offset estimation method is: on the basis of the known current symbol frequency offset value, the next symbol is compensated by using the current frequency offset estimation value, so that the detection range of the next symbol frequency offset estimation is reduced, the detection times of the next symbol are reduced under the condition that the search step is unchanged, and the running time is also reduced. This improved method will improve the synchronization accuracy while reducing the overall operation complexity.

[0130] According to the above method, after the received signal v(t) is processed, the frequency offset range is first set. For each assumed frequency offset estimate value according to the detection interval, the received signal is compensated for the frequency offset in the time domain. Then, the pilot symbol is used to estimate the phase rotation caused by the residual frequency offset. Finally, the MSE of the compensated signal is calculated under different initial frequency offset values, and the frequency offset estimate with the minimum MSE is found. The specific process of the algorithm is as follows: Figure 2 shown.

[0131] S3: Starting from the second OFDM symbol, a gradient descent-based carrier synchronization algorithm is used to estimate the frequency offset, using the frequency offset estimate from the first OFDM symbol as the initial value for the iteration. By applying the stochastic gradient descent algorithm to carrier synchronization at the receiving end of an underwater acoustic OFDM communication system, the algorithm achieves higher synchronization accuracy while maintaining the same algorithm complexity.

[0132] The carrier synchronization algorithm based on gradient descent specifically includes:

[0133] If the gradient had to be calculated for every sample at each iteration, the overhead would be extremely high. Using the principle of stochastic gradient descent, the weights are updated based on each individual training sample. By continuously adjusting the weights by inputting a large number of samples, a set of weights is obtained that allows the algorithm to obtain results that are as close as possible for newly added samples, greatly reducing the amount of computation required. The essence of the gradient algorithm is to use the first-order derivative of a function to select the direction of descent, allowing for rapid identification of the minimum point. However, the limitation of the gradient descent algorithm is that it cannot cross the local minimum error value. Once trapped in the local minimum error, it is difficult to reach the global minimum error.

[0134] If frequency offset estimation is performed on all sample sets, it will take a lot of time. The stochastic gradient descent method uses a single (or a small number of) samples to estimate the current gradient, which can effectively reduce the amount of computation of the synchronization algorithm.

[0135] The error gradient can be expressed as:

[0136]

[0137] Where, have:

[0138] y k =∫ T tv(t)e -iβt e -2πikΔft dt

[0139] Through the error gradient, the frequency deviation β can be iteratively calculated as:

[0140]

[0141] Where, K β is the step size of frequency deviation iteration. The initial value of frequency deviation is set to

[0142] The step size in the gradient descent-based carrier synchronization algorithm has a significant impact on the synchronization results and is crucial to the algorithm's iteration speed and operational stability. An iterative step size with a constant interval will prevent the algorithm from taking into account convergence speed, CFO tracking speed, and steady-state misalignment. To balance operating speed, frequency offset estimation accuracy, and stability, the Adam method is used to improve step size selection. The Adam algorithm adaptively optimizes step size selection by calculating the first-order moment estimate and second-order moment estimate of the gradient. Compared with traditional algorithms, it can achieve faster convergence speed and requires only a few parameter adjustments, allowing for efficient calculations. The step size selection can be expressed as:

[0143]

[0144] Where, is the estimated exponential moving mean of the gradient.

[0145] At the end of the set number of iterations, if the system reaches a stable state, a correct frequency offset estimate can be obtained. If the system does not reach a stable state, an accurate frequency offset estimate cannot be obtained. Furthermore, blindly increasing the number of iterations in pursuit of a stable state increases the system's computational burden and thus the system's runtime. Furthermore, the nature of the stochastic gradient descent algorithm indicates that incorrectly setting the initial value can cause the system to fall into a local error minimum.

[0146] To address the problem of inter-channel interference caused by carrier frequency offset due to Doppler shift in the underwater acoustic OFDM channel, and to further improve carrier synchronization accuracy while reducing computational complexity, a time-domain method is first introduced to complete carrier synchronization. Then, in an OFDM frame, the frequency offset is estimated using the carrier synchronization algorithm based on optimal solution search for the first OFDM symbol. Starting from the second OFDM symbol, a gradient descent-based carrier synchronization algorithm is used to estimate the frequency offset, and the frequency offset estimate for the first OFDM symbol is used as the initial value for iteration. As a result, the joint synchronization algorithm not only achieves excellent synchronization accuracy, but also requires a short runtime, ensuring real-time communication requirements.

[0147] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment as above, it is not intended to limit the present invention. Any technician familiar with the present profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical content disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement of the above embodiments made according to the technical essence of the present invention, within the spirit and principles of the present invention, without departing from the content of the technical solution of the present invention, shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. A carrier synchronization algorithm for underwater acoustic OFDM systems based on gradient estimation and optimal solution search, characterized by: The underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation and optimal solution search is implemented by the following steps: S1: In the underwater acoustic OFDM communication system, the time domain method is used to synchronize the underwater acoustic OFDM carrier; S2: In an OFDM frame, the frequency offset is estimated for the first OFDM symbol using a carrier synchronization algorithm based on optimal solution search. S3: Starting from the second OFDM symbol, a carrier synchronization algorithm based on gradient descent is used to estimate the frequency offset value, and the frequency offset estimation result of the first OFDM symbol is used as the initial value of the iteration.

2. The underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation and optimal solution search according to claim 1 is characterized by: The time domain method for underwater acoustic OFDM carrier synchronization described in S1 specifically includes: According to the properties of OFDM, the signal sent by the transducer is: Assuming that the noise of the underwater acoustic channel is an additive white Gaussian noise channel, the noise is w, T is the OFDM symbol length, N is the number of subcarriers, and the receiving end signal is: Sample y(t) with a sampling period of T s but: There is a symbol synchronization deviation Δn in synchronization f When , it is equivalent to performing a discrete Fourier transform operation in front of the cyclic prefix, that is: Let Δn f (i) is the starting point τ in the orthogonal region max / T s The signal after phase compensation is: Y k (i)=S k (i)+w(i) (5) By using the properties of the orthogonal region, we can change the starting position of the discrete Fourier transform to obtain multiple S k Estimate of Y k (1),Y k (2),…, for multiple Y k Differential processing can achieve diversity and smoothing effects, so the received signal with carrier frequency deviation is: Where, is the sampling frequency of the receiving end, N fft is the number of discrete Fourier transform points, T o is the OFDM symbol duration, S k is the spectrum of the subcarrier; The baseband signal adopts rectangular wave and quadrature phase shift keying modulation mode, then S k is a SINC function with a main lobe width of 2 / T o , the sidelobe width is 1 / T o When: Taking the derivative of formula (7) with respect to m, we can get: By using high-power fast Fourier transform, in addition to the data at the subcarrier, the frequency data adjacent to the subcarrier can also be obtained. If the adjacent frequency points are within the allowable range, the inherent inter-channel interference they bring can be treated as noise and differentiated to achieve a differential effect. If adjacent subcarriers are used to transmit pilot signals, the inter-channel interference they bring can be removed. Other methods using interpolation or subtraction will introduce errors. The frequency domain signal at non-orthogonal frequency points is: Subscript j is the non-subcarrier frequency after high-multiple Fourier transformation, which should satisfy: In formula (10), the frequency resolution after high-power Fourier transformation is Δf', the original transmission frequency is Δf, the orthogonality factor is λ, the interval between adjacent subcarriers is Δf, and the total phase is π. The impact of these non-orthogonal frequency points can be treated as noise.

3. The carrier synchronization algorithm for underwater acoustic OFDM system based on gradient estimation and optimal solution search according to claim 2 is characterized by: The power difference is ≤0.25dB, and the sum of the maximum phase angles on the left and right sides is ≤π / 4.

4. The underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation and optimal solution search according to claim 2 is characterized by: The carrier synchronization algorithm based on optimal solution search described in S2 specifically includes: In the underwater acoustic OFDM communication system, the number of subchannels is fixed to 1024, the signal bandwidth is B, the modulation frequency of the first subcarrier is f0, and the carrier frequency of the kth carrier is f k =f0+kΔf(k≤64), then the subchannel spacing is Δf=B / K, K is the K-th subchannel (K≤1024), and the modulation method of orthogonal phase shift keying is adopted. The OFDM signal s(t) can be expressed as: In formula (11), T = 1 / Δf represents the symbol length, T CP represents the duration of the cyclic prefix, d k Represents the user information to be sent; Assuming that the length of the cyclic prefix is ​​greater than or equal to the delay spread length of the channel, T CP >>T DS , after passing through the channel, the impulse response h(τ,t) of the signal received by the receiver can be expressed as: In formula (12), h p (t) represents the amplitude of the channel, τ p (t) represents the delay of the p channel, τ p (t)≈τ p -at, at is the delay caused by multipath, Because of small-scale fading, the channel amplitude will change due to multipath effects, with h p (t)≈h p , further analysis h(τ,t) can be expressed as: The signal received by the receiver after channel amplitude attenuation and delay spread can be expressed as: In formula (14), n(t) represents Gaussian white noise, whose power spectrum density is N0 / 2, After the received signal y(t) undergoes packet synchronization estimation, symbol timing synchronization, and coarse frequency offset compensation, In formula (15), ε is the carrier frequency offset, H k is the frequency response of the kth subcarrier of the OFDM symbol, w(t) is the additive noise, In the carrier synchronization algorithm based on optimal solution search, assuming a frequency offset estimation range and setting the search step, the frequency offset estimation value in each search interval is The signal is finely compensated in the time domain, and the compensated signal can be expressed as: pilot symbols d k , k∈κ p , and the resulting mean square error is: In formula (17), d k Represents the user data sent out by the transmitter after modulation. Represents the user data received by the receiving end after passing through the channel, Represents the user data after frequency offset compensation at the receiving end; If there are multiple frequency offset results, the mean square error of each frequency offset estimate of the received signal can be calculated by the above formula. The accurate frequency offset value of the received signal can be obtained by finding the minimum mean square error. 。 5. The carrier synchronization algorithm for underwater acoustic OFDM system based on gradient estimation and optimal solution search according to claim 4 is characterized by: Assume that the residual frequency offset of the i-th symbol is The residual frequency offset of the next i+1 symbol in the same frame is Taking advantage of the fact that the frequency offsets of two adjacent symbols are similar, the residual frequency offset of the current symbol in the same frame is approximately equal to the residual frequency offset of the previous symbol, so: 。 6. The underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation and optimal solution search according to claim 5 is characterized by: The frequency offset estimation method is as follows: based on the known frequency offset value of the current symbol, the current frequency offset estimation value is used to compensate the next symbol, so that the detection area of ​​the frequency offset estimation of the next symbol can be narrowed. When the search step size remains unchanged, the number of detections of the next symbol is reduced, and the running time is also reduced accordingly.

7. The underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation and optimal solution search according to claim 6 is characterized by: The carrier synchronization algorithm based on gradient descent described in S3 specifically includes: The error gradient can be expressed as In formula (20), Then we have: y k =∫ T tv(t)e -iβt e -2πikΔft dt (21) Through the error gradient, the frequency deviation β can be iteratively calculated as: In formula (22), K β is the step size of frequency deviation iteration, and the initial value of frequency deviation is set to 8. The underwater acoustic OFDM system carrier synchronization algorithm based on gradient estimation and optimal solution search according to claim 7 is characterized by: The Adam method is used to improve the selection of step size. The selection of step size can be expressed as: In formula (23), is the estimated exponential moving mean of the gradient.

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