A method for integrated co-sited interference cancellation of sensing

Abstract

The application relates to a co-sited interference cancellation method for integrated sensing and communication, and is characterized by comprising the following steps: step 1, processing a transmission signal x(t) after radio frequency modulation of a transmission end of an integrated sensing and communication base station; and step 2, reconstructing interference by constructing a multi-tap circuit at a local transceiver end of the integrated sensing and communication base station. The co-sited interference cancellation method for integrated sensing and communication has the superior technical effect that a variable step size least mean square algorithm is adopted to effectively solve the restrictive relationship between a step size factor and a steady-state error in the case of a fixed step size, co-sited interference can be cancelled in an integrated sensing and communication system, and sensing performance is improved; interference cancellation in a radio frequency domain is designed, the complexity of a hardware structure is lower; and a gradient descent method is used to optimize a root mean square value of a signal, so that an optimized value of a downlink communication system is realized.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of signal processing, and particularly relates to a co-sited interference cancellation method for integrated communication and sensing. BACKGROUND

[0002] In the integrated communication and sensing system, the signal strength of the signal received by the receiving end of the integrated communication and sensing base station from the local transmitting end is much higher than that of the signal from other nodes. Due to the limited dynamic range of the analog-to-digital converter (ADC), the receiving end is blocked by the ADC due to strong co-sited interference from the transmitting end, and thus cannot realize normal communication. In order to make the communication system run normally, co-sited interference cancellation is needed. In the integrated communication and sensing system, if the orthogonal frequency division multiplexing (OFDM) signal is used as the integrated communication and sensing signal, the integrated communication and sensing base station senses and communicates at the same time, and also works in duplex, the integrated communication and sensing transmitting end produces strong co-sited interference on the sensing receiving end. In addition, compared with the dedicated sensing system, the distance between the transmitting end and the receiving end of the integrated communication and sensing system is smaller, and thus the co-sited interference is stronger.

[0003] At present, the elimination of local co-site interference is generally divided into the following aspects: space domain, radio frequency domain and digital domain. Among them, the space domain co-site interference elimination refers to the case where the base station transceiver antenna is independently designed, in order to avoid the interference of the signal from the transmitting end to the receiving end of the base station, the transceiver antenna is physically isolated. By increasing the distance between the transceiver antennas of the base stations, the path loss of the signal in the free space is utilized, so that the interference power received by the receiving end antenna is smaller. The remaining problem is that although this natural isolation is simple to implement, in practice, the distance between the transceiver antennas cannot be too large due to the size of the equipment. Some scholars have proposed to increase the shielding material between the transceiver antennas to achieve isolation, but this not only increases the hardware cost but also the effect is not ideal. At present, there are mainly two methods of direct coupling and indirect coupling to realize the co-site interference elimination technology in the radio frequency domain. Among them, the direct coupling method directly uses the local transmitting antenna signal as the reference for interference reconstruction. By adjusting the phase, amplitude and time delay parameters of the reference signal, the error between the reconstructed signal and the received local co-site interference signal is minimized, and this method does not depend on the newly added communication link. Then, the receiving end is subtracted from the reconstructed signal to achieve the effect of eliminating interference. Although the indirect coupling co-site interference elimination also reconstructs the local co-site interference signal and then subtracts it to achieve interference elimination, the difference lies in that it needs to reconstruct an additional transmitting link to complete the interference reconstruction. These two methods have the problems of incomplete interference elimination, large steady-state error and slow iteration speed. The digital domain co-site interference elimination is generally used to eliminate residual interference. After the co-site interference elimination in the space domain and the analog domain, the co-site interference signal strength has been reduced to a certain level, and the input signal of the ADC can be within its dynamic range, but the co-site interference has not been completely eliminated, so it is still necessary to eliminate the residual interference through the powerful signal processing capability of the digital domain to reduce the interference below the noise floor, and realize normal communication. SUMMARY

[0004] Based on the technical problems existing in the prior art as described above, the present application provides a sensing and communication integrated co-site interference elimination method. The sensing and communication integrated co-site interference elimination method comprises:

[0005] Step 1, processing the transmitting signal x(t) of the sensing and communication integrated base station transmitting end after radio frequency modulation;

[0006] Step 2, constructing a multi-tap circuit at the local transceiver end of the sensing and communication integrated base station to realize interference reconstruction.

[0007] Further, in step 1, the transmitting signal x(t) of the sensing and communication integrated base station transmitting end after radio frequency modulation is processed:

[0008] The transmitting signal x(t) of the integrated sensing and communication base station after radio frequency modulation is known to the receiving end, and interference reconstruction is performed on the basis of x(t), and then radio frequency domain interference cancellation is performed. The signal e(t) after interference cancellation is used as the receiving signal for radio frequency demodulation. Since the transmitting signal of the downlink communication system is represented as x(t), the signal of the integrated sensing and communication base station receiving end is r(t), in order to eliminate the co-site interference signal and noise at the receiving end to complete the communication and sensing functions, and the co-site interference channel between the transmitting end and the receiving end of the integrated sensing and communication is a Rice channel. When the transmitting end signal x(t) reaches the receiving end through the channel, the amplitude, time delay and phase of the original signal change. In order to reflect these changes, x(t) needs to be processed.

[0009] Further, in step 2, a multi-tap circuit is constructed at the local transceiver of the integrated sensing and communication base station to reconstruct the interference, specifically including:

[0010] The OFDM integrated sensing and communication signal at the transmitting end is divided into two branches after the delay unit, wherein: i (t) is the quadrature component, and the other branch becomes the quadrature component x q (t) after passing through the phase shifter. The amplitudes of the two signals change after passing through the attenuator. If the attenuation parameter after parameter estimation is , the time delay parameter is , then the signals after the attenuator become and The two signals are superimposed and synthesized to obtain the reconstructed signal, and thus the reconstructed interference signal is obtained as:

[0011]

[0012] Let X(t) = [x i (t) x q (t)] T be the reference signal, [·] T denotes matrix transposition. Let the attenuator control the amplitude attenuation of the in-phase and quadrature branches, and let its weight vector be G(t) = [g i (t) g q (t)] T . The initial value of the weight vector is set as W(0) = [g i (0) g q (0)] T = [0 0] T . The reconstructed interference signal is denoted as:

[0013]

[0014] In the LMS algorithm, the update formula of G(t) is:

[0015]

[0016] wherein: J(t) = E[e 2 (t)] is the mean square of error signal, the signal r(t) received by the receiving end minus the interference reconstruction signal, i.e. the strong co-site interference signal from the local transmitting end is cancelled, and e(t) represents the local received signal minus the reconstruction signal, and the following signal expression is obtained:

[0017]

[0018] Δt is the step interval for each iteration, let t = nΔt, wherein: n = 0, …, N is the number of iterations; μ is the step factor, which is a fixed constant, the two-way round-trip delay is τ and the Doppler shift is f D . In order to achieve a balance between convergence speed and steady-state error, the update formula of G(t) in the radio frequency interference cancellation algorithm is:

[0019]

[0020] wherein: ρ is a very small number set to prevent the denominator from being zero, and the denominator

[0021] X T (t) = [x i (t) x q (t)].....(6),

[0022] At this time, the error signal mean square value in a step interval Δt is expressed as:

[0023]

[0024] wherein: P c is the power of in-phase and quadrature components, and are the reconstruction signal in-phase and quadrature component coefficients, h i and h q are the co-site interference signal in-phase and quadrature component coefficients, P s is the power of the perceived echo; σ 2 is the average power of additive white Gaussian noise, according to the above formula, the value of j is only related to h i , h q , When and , J reaches the minimum value, and this objective function is , the gradient descent method is used to optimize the root mean square value of the signal to achieve the goal of minimization. The gradient descent method is applied to the partial derivative of the loss function J with respect to and respectively:

[0025]

[0026] The weight vector update formula is further simplified as:

[0027]

[0028] where: G *T is the weight of the actual co-site interference signal, I is the unit matrix, and the mean square error is rewritten as:

[0029]

[0030] As above, the signal after interference cancellation is e(t), and the essence of reducing the error by the gradient descent method is to converge the error vector constructed by the model to the expected value set. To determine whether the algorithm converges and whether the signal can be successfully converged to the expected target, it is necessary to analyze whether the error vector can be converged to the zero vector. For the convergence of the algorithm, the domain of G(t) is continuous, and there exists a constant L in the domain, which satisfies:

[0031] ||G(k1)-G(k2)||≤L||k1-k2||……(11),

[0032] where: k1, k2 are greater than 0 and are integers, the function G(t) satisfies Lipschitz continuity, and the following formula is derived:

[0033]

[0034]

[0035] From the above formula, when the step size is

[0036]

[0037] Therefore, the convergence speed of the function is O(·) represents the complexity. When the step size is greater than 0 and satisfies , the function G(t) converges, and the true value of the parameters h i and h j is represented as G(0), the difference between J and it is represented as E(η), and the time constant τ eThe convergence speed of the adaptive interference suppression algorithm is defined as the total iteration time before the size of each component in the error vector is attenuated to 1 / e of the initial value, wherein e is the base of the natural logarithm, the iteration time is equal to the number of iterations multiplied by the duration of each iteration, and τ is derived from the iteration time of the signal e = k e T, wherein k e is the number of iterations set, which is approximately equal to is a signal period, then:

[0038]

[0039] As can be seen from the above formula, when the step size and the signal period are set as above, the convergence speed of the algorithm is fast, and in the algorithm used, when , J reaches the minimum value L = σ 2 + P s , wherein L is the Lipschitz constant in the algorithm, and when the step size is used, the gradient descent method used for convergence of the residual interference can converge to the expected value set, the variable step size least mean square (LMS) algorithm proposed in the present application can converge and achieve the required steady-state error, and the relationship between J(t) after the nth iteration and the initial J(t) is:

[0040]

[0041] It can be seen that when for any n, there is , the above formula converges, the value of p is very small, and in order to ensure that the denominator is not 0, the convergence condition is approximately | μ(t) | < 1, wherein h i = kg i , h q = kg q From the above formula, it can be seen that the value of J is only related to h i , h q , and When and , J reaches the minimum value, the gradient descent method is used to optimize the root mean square value of the signal to achieve the goal of minimization, and the gradient descent method is applied to obtain the partial derivatives of the loss function J with respect to and , and the following is obtained: When the value of the loss function does not reach the expected target, multiple iterations are performed to continuously modify the parameters until the optimal solution is reached, and in this process h i , h q , is a function of iteration number k, and the gradient of J is also a function of iteration number k, the step size factor in the algorithm is set to a variable step size, the step size is changed according to the change of the time factor, and three parameters are introduced to control the step size factor, and the step size updating formula is as follows:

[0042]

[0043] The step interval is adjusted by using the autocorrelation of the error signal, and in the formula, alpha, beta, epsilon and gamma are control parameters, alpha+epsilon+gamma=1. It should be noted that, in order to realize the convergence speed as ideal as possible under the condition that the steady-state error of the fixed step size LMS algorithm is as small as possible, the variable step size LMS algorithm is adopted, the step size factor is updated by using the predicted error signal value, and the first part of the step size updating formula adopts the arctangent function to establish a new nonlinear relationship with the time parameter and alpha and beta coefficients to adjust the influence degree of the arctangent function.

[0044] The beneficial effects of the present application are:

[0045] 1. The method for eliminating co-site interference in the integrated sensing and communication system adopts the variable step size least mean square algorithm, effectively solves the restrictive relationship between the step size factor and the steady-state error in the fixed step size case, and can eliminate the co-site interference in the integrated sensing and communication system and improve the sensing performance.

[0046] 2. The method for eliminating co-site interference in the integrated sensing and communication system eliminates interference in the radio frequency domain, and the complexity of the hardware structure is lower.

[0047] 3. The method for eliminating co-site interference in the integrated sensing and communication system uses the gradient descent method to optimize the root mean square value of the signal, so as to realize the optimization value of the downlink communication system.

[0048] 4. The improved step size updating formula of the method for eliminating co-site interference in the integrated sensing and communication system can solve the problem that when the error factor is near 0, the step size factor produces a sharp mutation, that is, a small error change will cause a large step size change, so that the communication and sensing performance of the integrated sensing and communication system is unstable, and when the error signal is close to 0, the step size change is relatively gentle, so that the adaptive weight vector value of the algorithm when reaching the steady state is closer to the actual value. BRIEF DESCRIPTION OF DRAWINGS

[0049] Figure 1 The integrated sensing and communication system block diagram in the embodiment of the present application is shown in the figure;

[0050] Figure 2A 、 Figure 2B The interference performance of the variable step size least mean square algorithm used in the embodiment of the present application and the performance comparison with other common channel estimation algorithms are shown in the figure. DETAILED DESCRIPTION

[0051] In order to enable a more clear understanding of the above-mentioned purposes, features and advantages of the present application, the following will be combined with the description of the accompanying drawings to make a further detailed introduction of the present application. Figure 1 The present application further discloses a general sense integration co-site interference cancellation method.

[0052] Embodiment:

[0053] As shown in the general sense integration co-site interference cancellation method includes: Figure 1

[0054] Step A: Transceiving signals of the 5G NR general sense integration system;

[0055] As shown in the general sense integration co-site interference cancellation method includes: Figure 1 The transmitting end adopts OFDM general sense integration signals, wherein a frame structure of transmitting 14 OFDM symbols in a single time slot with a subcarrier spacing of 30 kHz is adopted, each frame is 10 ms, the bandwidth is 100 MHz, the number of Fast Fourier Transform (FFT) points is 4096, the proportion of Cyclic Prefix (CP) is 6.67%, each subframe contains 2 slots, and each slot is composed of 14 OFDM symbols (under normal CP); the 5G NR general sense integration system transmits K=MxNxlog24 bits in one time-frequency resource block, N is the number of subcarriers, M is the number of symbols, the bit stream information b(k)∈{0,1} represents the kth transmission bit, k=1,…,K, through digital mapping, the serial bit stream is quadrature amplitude modulated (QAM), the modulation order is 4, and the modulated bits are mapped to p(k)∈{-1,1}, wherein the length of k is MxN, and there are P target objects in the scene, and the transmitting signal of the pth target object with an azimuth angle of θ p and a slant distance of R p is represented as:

[0056]

[0057] The signal is converted from serial to parallel, and the continuous serial signal is converted into N rows of parallel signals. The modulated symbol signal after conversion is transmitted in a matrix A Tx :

[0058]

[0059] ​Then, the Inverse Fast Fourier Transform (IFFT) is performed to convert the frequency domain signal into parallel time domain transmission signals, the parallel time domain signals are converted into serial signals d(t) again, and after the CP processing, the baseband signals are modulated by a Digital to Analog Converter (DAC) and other radio frequency modulators, and finally the integrated time domain signals x(t) are emitted by the transmitting antenna:

[0060]

[0061] wherein a m，n is the modulation symbol data of the nth subcarrier on the mth symbol, is the frequency of the nth subcarrier, Δf is the subcarrier spacing, and T is the symbol duration, rect(·) is a rectangular window function.

[0062] After the radio frequency domain interference cancellation, the received signals are demodulated from the radio frequency signals to the baseband signals, and then the baseband signals are divided into communication and sensing parts for processing: for the communication processing, the integrated signals are demodulated from the radio frequency signals to the baseband signals by the ADC and other radio frequency demodulators, the CP is removed, and after the CP removal, the serial signals are converted into parallel signals, the FFT is performed to convert the time domain signals into the frequency domain signals, the parallel frequency domain signals are converted into the serial frequency domain signals of MxN, the channel equalization is performed, the noise interference is removed after the channel equalization, the digital demodulation is performed, and finally the signals b n (k) ∈ {0, 1};

[0063] For the sensing processing, the CP is also removed, the serial signals are converted into parallel signals after the CP removal, and then the symbol-by-symbol division is performed according to the sensing processing algorithm for the sensing processing.

[0064] It should be noted that the uplink transmitted signal y(t) is a DFT-S-OFDM signal, and therefore the differences between the OFDM signal demodulation need to be considered when the receiving end is demodulated. The model considered in this embodiment is a single integrated base station and multiple user terminals, the signals transmitted by the multiple user terminals are converged into an OFDM transmission signal, the OFDM demodulation processing is performed, and in the receiving end, it is assumed that the time synchronization and signal sampling are ideal conditions. The base station receiving end receives the signals transmitted by the multiple users, and the channel impulse between the lth user and the receiving end is h y (t, l), the uplink transmitted signal is y(t, l), and the uplink signals transmitted by all users are y(t).

[0065] From the above analysis of the interference of a single base station, the received signal y(t) of the integrated base station is:

[0066]

[0067] where b m，n is the complex amplitude factor of the time-frequency resource block, a m，n is the complex amplitude of the co-sited interference signal, τ SI is the time delay of the co-sited interference signal. From the interference analysis in the previous section, the main interference between the transmitter and receiver is the direct path interference, and the positions of the transmitter and receiver antennas do not change, so the influence of the Doppler shift is not considered. The received signal is divided into three parts: the sensing echo signal, the local co-sited interference signal, and the noise. The received modulated signal is represented as:

[0068] d m，n = a m，n exp(-j2πnΔfτ)exp(j2πf D t),

[0069] The received symbol matrix is:

[0070]

[0071] The above formula is rewritten as:

[0072]

[0073] Step B, received signal processing:

[0074] Compared with the traditional sensing algorithm, the phase shift on the kth subcarrier when the time delay is τ is Then the echo signal on the kth subcarrier on the lth symbol becomes:

[0075]

[0076] where b0 is the two-way path loss, is the matrix representation of the Additive White Gaussian Noise (AWGN), which follows a Gaussian distribution with a mean of 0 and a variance of σ 2 , f0 is a constant, and F Rx Now contains the parameters τ and f D , b0 to be estimated, and F Tx , which has no effect on the estimation problem. Therefore, the echo signal on the kth subcarrier on the lth symbol is removed from the equation by symbol-by-symbol division:

[0077]

[0078] Let Since before is zero mean Gaussian white noise, using symbol by symbol division does not change its statistical properties, so Z k，l is still zero mean Gaussian white noise, and the final processed signal is expressed in the following form:

[0079]

[0080] After the above processing, the first term of the received signal is in the form of two exponential multiplications except for a constant, and the FFT transform is performed on M symbols:

[0081]

[0082] The two exponential terms cancel each other out, and only the kth term of the signal will obtain a peak value, and then the IFFT is performed on the signal,

[0083]

[0084] Similarly, the two exponential terms cancel each other out, and only the lth term of the signal obtains a peak value, and the peak values obtained by combining the above two in the speed-distance three-dimensional graph are the speed and distance of the signal, and the target distance can be estimated from the above formula:

[0085]

[0086] Given that the transmitted signal frequency is f C , the radial velocity is obtained according to the Doppler shift change:

[0087]

[0088] According to the signal round-trip delay information to determine the target distance and the relative speed determined by the Doppler frequency, for angle estimation, an independent spectrum estimation method is used for estimation, and in the above formula, τ and f D are extracted, and the FFT transform is performed on the signal after element by element elimination, and the speed and distance information of the target is contained in the phase of the N subcarriers corresponding to a single OFDM symbol and the phase difference of multiple OFDM symbols, and through the FFT transform, the signal matrix row and column correspond to the distance and radial velocity of the target.

[0089] In order to verify the effectiveness of the algorithm of the present application, the performance of the variable step size LMS algorithm under different interference is simulated in Figure 2, and the common channel estimation algorithm is used as a comparison under the same conditions. Figure 2AThe change of the communication error code rate with the signal-to-noise ratio in the integrated scene is simulated, the signal-to-noise ratio is set to 0:40dB, the interference noise ratio is set to 40dB, 60dB and 80dB, as a comparison object, the error code rate under different interference noise ratios is compared with that without interference, and it can be known that the interference cancellation performance of the improved variable step size LMS algorithm is better under low signal-to-noise ratio, with the signal-to-noise ratio being larger, the interference cancellation effect is worse, the interference is smaller, the interference cancellation ability of the algorithm is stronger, when the interference power is lower than 60dB, the algorithm can better eliminate the interference. Figure 2B The variable step size least mean square algorithm is simulated and compared with traditional channel estimation algorithms such as least squares (LS), minimum mean square error (MMSE), and it can be seen that the mean square error (MSE) of the variable step size least mean square algorithm is obviously smaller than that of the traditional channel estimation algorithm under the same signal-to-noise ratio, and the interference cancellation ability is stronger.

[0090] The application is not limited by the above-mentioned embodiments, the above-mentioned embodiments and the description are only to illustrate the principles of the application, and various changes and improvements can be made without departing from the spirit and scope of the application, and these changes and improvements all fall within the scope of the application. The scope of protection of the application is defined by the appended claims.

Claims

1. A method for eliminating co-located interference using integrated inductive communication, characterized in that, Includes the following steps: Step 1: Process the transmitted signal x(t) after radio frequency modulation from the transmitter of the integrated sensing base station: The transmitted signal x(t) after radio frequency modulation at the transmitter of the integrated sensing base station is known to the receiver. Interference reconstruction is performed on x(t), followed by radio frequency domain interference cancellation. The signal e(t) after interference cancellation is used as the received signal for radio frequency demodulation. Since the transmitted signal of the known downlink communication system is represented as x(t), and the received signal of the integrated sensing base station is r(t), in order to eliminate the co-located interference signal and noise at the receiver to complete the communication and sensing functions respectively, the co-located interference channel between the integrated sensing transmitter and receiver is a Ricean channel. When the transmitted signal x(t) reaches the receiver through the channel, the amplitude, delay, and phase of the original signal all change. In order to reflect these changes, x(t) needs to be processed. Step 2: By constructing a multi-tap circuit at the local transceiver end of the integrated sensing base station, interference reconstruction is achieved. The OFDM integrated sensing signal at the transmitting end is split into two branches after passing through a delay unit, where: x i (t) is the quadrature component, and the other branch becomes the quadrature component x after passing through a phase shifter. q (t), the amplitude of the two signals changes after passing through the attenuator. If the attenuation parameter after parameter estimation is... The delay parameter is The signal after passing through the attenuator becomes and The reconstructed signal is obtained by superimposing and combining the two signals, and the reconstructed interference signal is designed based on this. for: Let X(t) = [x i (t) x q (t)] T For reference signal, [·] T The matrix is ​​transposed, and the attenuator of the in-phase orthogonal branch controls the amplitude attenuation. Let its weight vector be G(t) = [g i (t) g q (t)] T The initial value of the weight vector is set to W(0) = [g i (t) g q (t)] T =[0 0] T The reconstructed interference signal is denoted as: In the LMS algorithm, the update formula for G(t) is: Where: J(t) = E[e 2 [(t)] is the mean square of the error signal. The signal received at the receiver, r(t), minus the reconstructed interference signal, cancels out the strong co-location interference signal from the local transmitter. Let e(t) represent the local received signal minus the reconstructed signal, and we get the following signal expression: Δt is the step interval for each iteration, let t = nΔt, where: n = 0, ..., N is the number of iterations; μ is the step size factor, which is a fixed constant; the round-trip time delay is τ and the Doppler frequency shift is f. D To achieve a balance between convergence speed and steady-state error, the update formula for G(t) in the radio frequency interference cancellation algorithm is adopted as follows: Where: ρ is a very small number set to prevent the denominator from being 0, and in the denominator... X T (t)=[x i (t) x q (t)]......(6), At this point, the mean square value of the error signal within a step interval Δt is expressed as: Where: P c It is the power of the in-phase and quadrature components. and These are the coefficients of the in-phase and quadrature components of the reconstructed signal, h i and h q These are the in-phase and quadrature component coefficients of the co-located interference signal, P s It is the power of the sensed echo; σ 2 It is the average power of additive white Gaussian noise. From the above equation, it can be seen that the value of J is only related to h. i h q , Related, when and When J reaches its minimum value, the objective function is: Given a convex function, gradient descent is used to optimize the root mean square value of the signal to minimize it. Gradient descent is applied to the loss function J with respect to... and Find the partial derivatives respectively: The weight vector update formula can be further simplified to: Where: G* T Let I be the weight of the actual co-located interference signal, and let I be the identity matrix. Combining the weight vector update formula and J(t), the mean square error can be rewritten as: As shown above, the signal after interference cancellation is e(t). The essence of reducing error through gradient descent is to converge the error vector constructed by the model to the set expected value. To determine whether the algorithm converges and whether it can successfully converge the signal to the expected target, we need to analyze whether the error vector converges to the zero vector. Regarding the convergence of the algorithm, the domain of G(k) is continuous, and there exists a constant L within the domain such that for any two points in the set, it satisfies: ||G(k1)-G(k2)||≤L||k1-k2||......(11), Where k1 and k2 are both greater than 0 and are integers, the function G(k) satisfies Lipschitz continuity, and thus the following equation is derived: From the above formula, we know that when the step size... When, the following relationship exists: Therefore, the convergence rate of the function is O(·) represents the time complexity, when the step size is greater than 0 and satisfies the condition... When the function G(k) converges, the parameter h is... i and h j The truth value of is denoted as G(0), the difference between J and is denoted as E(η), and the time constant τ e The convergence speed of an adaptive interference suppression algorithm is measured by τ, defined as the total iteration time before the magnitude of each component in the error vector decays to 1 / e of its initial value, where e is the base of the natural logarithm, and the iteration time equals the number of iterations multiplied by the duration of each iteration. τ is derived from the iteration time of the signal. e =k e T, where: k e It is the set number of iterations, and its value is equal to... If it is one signal cycle, then: From the above formula, when the step size and signal period are set as described above, the algorithm converges faster. In the algorithm used, when... When J reaches its minimum value: L = σ 2 +P s Where L is the Lipschitz constant in the algorithm, which is used for iteration when using gradient descent, as long as the step size is satisfied. The gradient descent method used converges to the residual disturbance and can converge to the set expected value. The proposed variable step size least mean square algorithm can converge and achieve the required steady-state error. The relationship between J(t) after the nth iteration and the initial J(t) is as follows: Therefore, when for any n, there exists When the above equation converges, the value of ρ is extremely small. To ensure that the divisor is not zero, the convergence condition is |μ(t)| < 1, where: h i =kg i h q =kg q From the above formula, the value of J depends only on h. i h q , Related, when and When J reaches its minimum value, gradient descent is used to optimize the root mean square value of the signal to achieve the minimization objective. Gradient descent is applied to the loss function J with respect to... and Taking the partial derivatives separately, we get: When the value of the loss function does not reach the expected target, multiple iterations are performed to continuously modify the parameters until the optimal solution is reached. During this process, h... i h q , The step size is a function of the iteration number k, and the gradient of J is also a function of the iteration number k. In this algorithm, the step size factor is set to a variable step size, which changes according to the change of the time factor. Three parameters are introduced to control the step size factor. The step size update formula is as follows: In the above formula, α, β, ∈ and γ are control parameters, α+∈+γ=1. The variable step size LMS algorithm is adopted, and the prediction error signal value is used to control the update iteration of the step size factor. The step size update formula (17) uses the arctangent function and time parameter to establish a new nonlinear relationship and α and β coefficients to adjust the influence of the arctangent function.

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