Method for eliminating inter-symbol interference and nonlinear distortion by combining channel estimation

Abstract

The invention provides an inter-symbol interference and nonlinear distortion elimination method based on channel estimation. The inter-symbol interference and nonlinear distortion elimination method comprises the steps of setting system parameters, estimating an ultra-wideband channel, constructing an inter-symbol interference matrix and a nonlinear distortion matrix, estimating an original signal through a least square method and the like. According to the method, ultra-wideband channel estimation is realized by utilizing pilot signals, then an inter-symbol interference matrix and a nonlinear distortion matrix are constructed by adopting an iterative algorithm according to a channel estimation result, inter-symbol interference and nonlinear distortion contained in received signals are eliminated based on the constructed matrixes, and accurate estimation of original signals is realized. The bit error rate of a communication system is reduced, and the system performance is improved.

Description

Technical Field

[0001] The invention relates to a signal detection and estimation method, and in particular to an inter-symbol interference and nonlinear distortion elimination method for joint channel estimation. Background Art

[0002] As an indispensable component of the Orthogonal Frequency Division Multiplexing (OFDM) communication system, the nonlinear characteristics and memory characteristics of the power amplifier (PA) will cause nonlinear distortion of the transmitted signal. After the distorted signal is transmitted through the ultra-wideband channel, it will affect the detection and estimation of the received signal and reduce the system performance. In order to eliminate the influence of PA on the received signal, a variety of channel and nonlinear distortion joint estimation methods have been proposed, so as to eliminate the influence of nonlinear distortion at the receiving end and achieve accurate channel estimation (for specific methods, see Uehara's "Iterative nonlinear self-interference cancellation for in-band full-duplex wireless communications under mixer imbalance and amplifier nonlinearity" 2020 "IEEE Transactions on Wireless Communications"; Li Taoyong's "A sparse channel estimation scheme combined with distorted signal detection for UWB OFDM systems" 2022 "IEEE Wireless Communications Letters" and "Sparse channel estimation with distorted signals for UWB OFDM systems" 2023 "IEEE Wireless Communications Letters").

[0003] In order to avoid inter-symbol interference, these methods all assume that the length of the cyclic prefix (CP) of the OFDM system is always greater than the sum of the channel length and the maximum delay of the PA response, which will lead to two deficiencies: first, the length of the ultra-wideband channel impulse response is very long and the maximum delay of the PA response to the ultra-wideband signal is very large, so a very long CP is required to meet this requirement, which increases the complexity of system design; second, the CP length affects the system performance. The increase in CP will reduce the OFDM system throughput, thereby reducing the information transmission rate of the system. Summary of the invention

[0004] The present invention aims to overcome the shortcomings of the above-mentioned prior art without increasing the CP length and reducing the system throughput, and proposes a method for eliminating inter-symbol interference and nonlinear distortion of joint channel estimation.

[0005] The present invention provides a method for eliminating inter-symbol interference and nonlinear distortion of joint channel estimation, comprising the following steps:

[0006] Step 1: Set the number of transmission symbols N d , number of signal carriers N, cyclic prefix length υ, pilot signal X 1 =[X 1 (0),...,X 1 (N-1)] T , receiving signal where Y i =[Y i (0),...,Y i (N-1)] T ,i=1,...,N d , the number of iterations T, the number of nonlinear stages D, the number of delay stages Q, the nonlinear coefficient vector of the power amplifier α = [α 1,0 ,α 3,0 ,...,α 2D-1,0 ,...,α 1,1 ,α 3,1 ,...,α 2D-1,Q ];

[0007] Step 2: Based on the power amplifier nonlinear coefficient vector α and the pilot signal X 1 and its corresponding received signal Y 1 , construct the measurement matrix Φ, and use the compressed sensing method to measure the ultra-wideband channel h=[h(0),...,h(L-1)] T Make estimates;

[0008] Step 3: Channel estimation results obtained based on pilot signals An iterative algorithm is used to eliminate the inter-symbol interference and nonlinear distortion of the subsequent received signal and restore the original signal.

[0009] Further, step one is specifically as follows:

[0010] The ultra-wideband OFDM system with N carriers adopts packet transmission. Each time the transmitter transmits a packet, it consists of N d OFDM symbols, that is Where X 1 is a known pilot signal, is the transmitted data; signal X i After inverse discrete Fourier transform and adding CP, the time domain signal x i =[x i (0),...,x i (N+υ-1)] T The nth sampling signal is expressed as

[0011]

[0012] Where υ is the length of CP, signal Take as the input signal of PA, and model PA using memory polynomial model, then the output signal u(k) of PA is expressed as

[0013]

[0014] Among them, α 2d-1,q Indicates that in the nonlinear series 2d-1, the delay τ q The coefficient on τ0<τ1<…<τ Q , D is the nonlinear series, Q is the delay series, and the output signal of PA reaches the receiving end after passing through the ultra-wideband channel. At this time, the signal received by the receiving end is

[0015]

[0016] Where 0≤m≤N d (N+υ)-1, h(l) is the ultra-wideband channel h=[h(0),...,h(L-1)] T The lth element of , and when l<0 or l≥L, h(l)=0, w(m) is additive Gaussian white noise with mean 0; without loss of generality, assume that the system CP length υ<L+τ Q , and L+τ Q ≤N, received signal y=[y(0),...,y(N d (N+υ)-1)] T After discrete Fourier transform and CP removal, the transmitted signal X i The corresponding received signal Y i=[Y i (0),...,Y i (N-1)] the kth element Y i (k) is expressed as

[0017]

[0018] Among them, W 1 (k) is Gaussian white noise, signal energy and They are

[0019]

[0020] and

[0021]

[0022] Further, step 2 is specifically as follows:

[0023] The received signal Y corresponding to the pilot signal 1 The kth element in is represented by

[0024]

[0025] Among them, τ q is the delay number of the qth stage delay, W 1 (k) is Gaussian white noise, signal energy Calculated by the following formula

[0026]

[0027] At this time, the element Φ(k,l) in the kth row and lth column of the measurement matrix Φ is constructed according to the following formula:

[0028] Φ(k,l)=RX 1

[0029] The pth element R(p) in the vector R = [R(0), ..., R(N-1)] is formed by the following formula

[0030]

[0031] Based on the measurement matrix Φ, the received signal Y 1 With the pilot signal X 1 The relationship is expressed as

[0032] Y 1 =Φh+W 1

[0033] Among them, W 1 =[W 1 (0),...,W 1(N-1)] T is Gaussian white noise W 1 (k) is a vector composed of , since the ultra-wideband channel is sparse, the channel estimation can be achieved by using the Compressive Sampling Matching Pursuit (CoSaMP) algorithm to solve the following formula:

[0034]

[0035] in, is the channel estimation result, ||h||1 represents the 1-norm of h.

[0036] Further, the step three is specifically as follows:

[0037] Receive signal Y i ,i=2,...,N d Expressed as

[0038] Y i =A i X i-1 +B i X i +W i

[0039] Among them, X i-1 For the i-1th transmitted signal, X i =[X i (0),...,X i (N-1)] is the i-th transmitted signal, W i =[W i (0),...,W i (N-1)] is the Gaussian white noise added to the i-th received signal, and the inter-symbol interference matrix A i and the nonlinear distortion matrix B i They are

[0040]

[0041] and

[0042]

[0043] Matrix A i The element a in the kth row and pth column i (k,p) is expressed as

[0044]

[0045] in, is the channel estimation result The lth element in the signal energy for

[0046]

[0047] The matrix B i The element b in the kth row and pth column i (k,p) is expressed as

[0048]

[0049] Among them, the signal energy for

[0050]

[0051] At this time, when the i-th original signal X i The least squares method can be used for estimation, that is,

[0052] X i =((B i ) H B i ) -1 (B i ) H (Y i -A i X i-1 )

[0053] The original signal can be restored by using an iterative algorithm, which specifically includes the following steps:

[0054] Input: Received OFDM signal Pilot signal X 1 , cyclic prefix length υ, number of iterations T, number of nonlinear levels D, number of delay levels Q, nonlinear coefficient vector of power amplifier α = [α 1,0 ,α 3,0 ,...,α 2D -1,0,...,α 1,1 ,α 3,1 ,...,α 2D-1,Q ], channel estimation result

[0055] Output: The restored original signal

[0056] Step 1) Initialization: symbol counter i=1, maximum number of iterations T;

[0057] Step 2) i = i + 1;

[0058] Step 3) According to X i-1 ,calculate And construct the matrix Ai ;

[0059] Step 4) Calculate the scaling factor And get the i-th original signal X i The initial estimate of

[0060] Step 5) Iteration counter t = 1;

[0061] Step 6) Based on the estimated value Calculate signal energy And construct the matrix B i ;

[0062] Step 7) Update the estimate

[0063] Step 8) t = t + 1;

[0064] Step 9) If t≤T, go to Step 6), otherwise

[0065] Step 10) If i≤N d , go to Step 2), otherwise output

[0066] The beneficial effect of the present invention is that: in view of the fact that the existing joint channel and nonlinear distortion estimation method does not take into account the scenario that the length of the ultra-wideband channel impulse response is long and the maximum delay of the PA response to the ultra-wideband signal is large, resulting in the system requiring a large number of CPs to avoid inter-symbol interference, thereby reducing the system throughput and affecting the transmission rate, a joint channel estimation inter-symbol interference and nonlinear distortion elimination method is proposed. This method integrates the inter-symbol interference caused by insufficient system CP and the nonlinear distortion caused by PA into the detection of received signals, adopts compressed sensing technology and iterative estimation algorithm, and on the basis of realizing accurate channel estimation, eliminates inter-symbol interference and nonlinear distortion at the same time, realizes the detection and estimation of the original signal, reduces the bit error rate of the communication system, and improves the performance of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 is a flow chart of the method of the present invention;

[0068] Figure 2 Comparison diagram of normalized mean square error of channel estimation between the proposed algorithm and the comparison algorithm;

[0069] Figure 3 The figure is a comparison chart of the data bit error rate obtained by the proposed algorithm and the comparison algorithm. DETAILED DESCRIPTION

[0070] The present invention will be further described below in conjunction with the accompanying drawings and examples of the present invention.

[0071] The present invention is achieved by the following steps: Figure 1 As shown, first set the system parameters according to actual needs, including the number of transmission symbols N d , signal carrier number N, pilot signal X 1 =[X 1 (0),...,X 1 (N-1)] T , cyclic prefix length υ, received signal where Y i =[Y i (0),...,Y i (N-1)] T ,i=1,...,N d , the number of iterations T, the number of nonlinear stages D, the number of delay stages Q, the nonlinear coefficient vector of the power amplifier α = [α 1,0 ,α 3,0 ,...,α 2D -1,0,...,α 1,1 ,α 3,1 ,...,α 2D-1,Q ] and other parameters, and then based on the pilot signal and its corresponding received signal, the ultra-wideband channel estimation is realized by using the compressed sensing technology. Finally, according to the ultra-wideband channel estimation result, the inter-symbol interference matrix and the nonlinear distortion matrix are constructed by the iterative algorithm, and the detection and estimation of the original signal are realized by using the least squares method. The specific description is as follows:

[0072] Step 1: System parameter settings are as follows:

[0073] The ultra-wideband OFDM system with N carriers adopts packet transmission. Each time the transmitter transmits a packet, it consists of N d OFDM symbols, that is Where X 1 is a known pilot signal, Is the data to be transmitted. Signal X i After inverse discrete Fourier transform and adding CP, the time domain signal x i =[x i (0),...,x i (N+υ-1)] T The nth sampling signal is expressed as

[0074]

[0075] Where υ is the length of CP. Signal The PA is modeled using a memory polynomial model, and the PA output signal u(k) can be expressed as

[0076]

[0077] Among them, α 2d-1,q Indicates that in the nonlinear series 2d-1, the delay τ q The coefficient on τ0<τ1<…<τ Q , D is the nonlinear series, Q is the delay series. The output signal of PA reaches the receiving end after passing through the ultra-wideband channel. At this time, the signal received by the receiving end is

[0078]

[0079] Where 0≤m≤N d (N+υ)-1, h(l) is the ultra-wideband channel h=[h(0),...,h(L-1)] T The lth element of , and when l<0 or l≥L, h(l)=0, w(m) is additive Gaussian white noise with mean 0. Without loss of generality, assume that the system CP length υ<L+τ Q , and L+τ Q ≤N, received signal y=[y(0),...,y(N d (N+υ)-1)] T After discrete Fourier transform and CP removal, the transmitted signal X i The corresponding received signal Y i =[Y i (0),...,Y i (N-1)] the kth element Y i (k) can be expressed as

[0080]

[0081] Among them, W 1 (k) is Gaussian white noise, signal energy and They are

[0082]

[0083] and

[0084]

[0085] Step 2: Ultra-wideband channel estimation is as follows:

[0086] The system uses pilot signal X 1To realize ultra-wideband channel estimation, since the pilot signal is located at the head of the data packet, its corresponding received signal Y 1 There is no interference from the previous OFDM symbol. At this time, Y 1 The kth element Y in 1 (k) is expressed as

[0087]

[0088] Among them, the signal energy Calculated by the following formula

[0089]

[0090] The relationship between channels is expressed in the form of vectors, as shown below:

[0091] Y 1 =Φh+W 1

[0092] Among them, W 1 =[W 1 (0),...,W 1 (N-1)] T is Gaussian white noise W 1 (k), Φ is the measurement matrix. Based on the pilot signal X 1 and its corresponding received signal Y 1 , the element Φ(k,l) in the kth row and lth column of the measurement matrix Φ is expressed as

[0093] Φ(k,l)=RX 1

[0094] The pth element R(p) in the vector R = [R(0), ..., R(N-1)] is formed by the following formula

[0095]

[0096] Since the UWB channel is sparse, the channel estimation can be achieved by using the Compressive Sampling Matching Pursuit (CoSaMP) algorithm to solve the following equation:

[0097]

[0098] in, is the channel estimation result, ||h||1 represents the 1-norm of h.

[0099] Step 3: Use iterative algorithm to construct inter-symbol interference matrix and nonlinear distortion matrix to realize the detection and estimation of original signal. Specifically:

[0100] Except for the first OFDM symbol of the data packet, other OFDM symbols are unknown to the receiving end. In order to restore the original signals of these symbols at the receiving end, the received signal Y i ,i=2,...,N d Expressed as

[0101] Y i =A i X i-1 +B i X i +W i

[0102] Among them, X i-1 For the i-1th transmitted signal, X i =[X i (0),...,X i (N-1)] is the i-th transmitted signal, W i =[W i (0),...,W i (N-1)] is the Gaussian white noise added to the i-th received signal, and the inter-symbol interference matrix A i and the nonlinear distortion matrix B i They are

[0103]

[0104] and

[0105]

[0106] Matrix A i The element a in the kth row and pth column i (k,p) is expressed as

[0107]

[0108] in, is the channel estimation result The lth element in the signal energy for

[0109]

[0110] The matrix B i The element b in the kth row and pth column i (k,p) is expressed as

[0111]

[0112] Among them, the signal energy for

[0113]

[0114] From Y i It can be seen from the expression that the signal contains the inter-symbol interference generated by the i-1th transmitted signal. Therefore, in order to achieve X i To detect and estimate, it is necessary to eliminate the inter-symbol interference first. 1 It is known that based on this formula, the present invention can use an iterative method to recover subsequent signals. Assuming that the signal X has been recovered i-1 , using the second step to obtain the channel estimation result The inter-symbol interference matrix A can be constructed i , but the nonlinear distortion matrix B i The construction of also needs to calculate the signal energy of the ith signal, and the signal energy of the ith signal also needs to be estimated. To this end, the present invention proposes an iterative algorithm to realize the estimation of the original signal, and the algorithm specifically includes the following steps:

[0115] Input: Received OFDM signal Pilot signal X 1 , cyclic prefix length υ, number of iterations T, number of nonlinear levels D, number of delay levels Q, nonlinear coefficient vector of power amplifier α = [α 1,0 ,α 3,0 ,...,α 2D-1,0 ,...,α 1,1 ,α 3,1 ,...,α 2D-1,Q ], channel estimation result

[0116] Output: The restored original signal

[0117] Step 1) Initialization: symbol counter i=1, maximum number of iterations T;

[0118] Step 2) i = i + 1;

[0119] Step 3) According to X i-1 ,calculate And construct the matrix A i ;

[0120] Step 4) Calculate the scaling factor And get the i-th original signal X i The initial estimate of

[0121] Step 5) Iteration counter t = 1;

[0122] Step 6) Based on the estimated value Calculate signal energy And construct the matrix B i ;

[0123] Step 7) Update the estimate

[0124] Step 8) t = t + 1;

[0125] Step 9) If t≤T, go to Step 6), otherwise

[0126] Step 10) If i≤N d , go to Step 2), otherwise output

[0127] The proposed algorithm fully considers the influence of ultra-wideband channel, inter-symbol interference and nonlinear distortion of power amplifier on the received signal. The channel influence is incorporated into the construction of the inter-symbol interference matrix and the nonlinear distortion matrix. Then, an iterative algorithm is used to iteratively update the original signal estimation result, which improves the accuracy of channel estimation and reduces the bit error rate of the communication system.

[0128] Example: Simulation experiment of inter-symbol interference and nonlinear distortion elimination method for joint channel estimation

[0129] Experiment: In the simulation experiment, an orthogonal frequency division multiplexing system with orthogonal phase shift coding is used, and the number of carriers N of the system is set to 512, and the cyclic prefix length υ=N / 4=128. The parameters of the power amplifier used in the simulation are shown in Table 1, with the number of nonlinear levels D=2, the number of delay levels Q=3, and the delay τ q The values ​​of are 0, 10, 50 and 100 respectively. In order to avoid the randomness of the test results, the present invention conducts a total of 500 random experiments. In each experiment, the present invention randomly generates a sparse channel h with a length of L=64 and a sparsity of K=5 (i.e., only 5 of the 64 elements are non-zero), and in each experiment, the present invention randomly generates a data packet consisting of 5000 OFDM symbols for transmission.

[0130] Table 1: Nonlinear coefficients of power amplifiers

[0131]

[0132] Simulation 1: In order to verify that the proposed method can achieve accurate estimation of ultra-wideband channels, the present invention conducts the following simulation experiments.

[0133] The present invention compares the proposed method with Uehara's method (for specific methods, see Uehara's "Iterative nonlinear self-interference cancellation for in-band full-duplex wireless communications under mixer imbalance and amplifier nonlinearity"), and statistically analyzes the normalized mean square error (NMSE) of channel estimation of 500 experiments under different signal-to-noise ratios (SNRs). The specific results are as follows: Figure 2 As shown. Figure 2 It can be seen that as the SNR increases, the NMSE obtained by the proposed method and the comparison method decreases, but the NMSE obtained by the proposed method is smaller than that of the comparison method, indicating that the proposed method can obtain more accurate channel estimation results.

[0134] Simulation 2: In order to verify that the proposed method can eliminate inter-symbol interference and nonlinear distortion, the present invention conducts the following simulation experiments.

[0135] Similarly, the present invention compares the proposed method with Uehara's method (for specific methods, see Uehara's "Iterative nonlinear self-interference cancellation for in-band full-duplex wireless communications under mixer imbalance and amplifier nonlinearity"), and statistically analyzes the bit error rate (BER) of 5000 OFDM symbols transmitted in 500 experiments under different SNR conditions. The specific results are as follows: Figure 3 As shown. Figure 3 It can be seen that the BER performance obtained by the comparison method is extremely poor. This is because the cyclic prefix length υ is smaller than the channel length L and the maximum delay τ of the power amplifier. Q The sum of, that is, υ<L+τ Q =164, the received signal contains inter-symbol interference, but the comparison method cannot accurately detect and estimate the original signal because it fails to eliminate the inter-symbol interference. However, for this scenario, the BER obtained by the proposed method decreases with the increase of SNR, indicating that the proposed method can effectively eliminate the inter-symbol interference and nonlinear distortion in the received signal and achieve accurate estimation of the original signal.

[0136] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, ordinary technicians in the field should understand that the specific implementation methods of the present invention can still be modified or some technical features can be replaced by equivalents without departing from the spirit of the technical solution of the present invention, which should be included in the scope of the technical solution for protection of the present invention.

Claims

1. A method for eliminating inter-symbol interference and nonlinear distortion in joint channel estimation, characterized in that: The following steps are involved: Step 1: Set the number of transmitted symbols N d , number of signal carriers N, cyclic prefix length υ, pilot signal X 1 =[X 1 (0),...,X 1 (N-1)] T , receiving signal where Y i =[Y i (0),...,Y i (N-1)] T ,i=1,...,N d , the number of iterations T, the number of nonlinear stages D, the number of delay stages Q, the nonlinear coefficient vector of the power amplifier α = [α 1,0 ,α 3,0 ,...,α 2D-1,0 ,...,α 1,1 ,α 3,1 ,...,α 2D-1,Q ]; Step 2: Based on the power amplifier nonlinear coefficient vector α and the pilot signal X 1 and its corresponding received signal Y 1 , construct the measurement matrix Φ, and use the compressed sensing method to measure the ultra-wideband channel h=[h(0),...,h(L-1)] T Make estimates; Step 3: Channel estimation results obtained based on pilot signals An iterative algorithm is used to eliminate the inter-symbol interference and nonlinear distortion of the subsequent received signal and restore the original signal.

2. The method for eliminating inter-symbol interference and nonlinear distortion of joint channel estimation according to claim 1, characterized in that: The step 1 is specifically as follows: The ultra-wideband OFDM system with N carriers adopts packet transmission. Each time the transmitter transmits a packet, it consists of N d OFDM symbols, that is Where X 1 is a known pilot signal, is the transmitted data; signal X i After the inverse discrete Fourier transform and the addition of a cyclic prefix (CP), the time domain signal x i =[x i (0),...,x i (N+υ-1)] T The nth sampling signal is expressed as Where υ is the length of CP, signal Take as the input signal of PA, and model PA using memory polynomial model, then the output signal u(k) of PA is expressed as Among them, α 2d-1,q Indicates that in the nonlinear series 2d-1, the delay τ q The coefficient on τ0<τ1<…<τ Q , D is the nonlinear series, Q is the delay series, and the output signal of PA reaches the receiving end after passing through the ultra-wideband channel. At this time, the signal received by the receiving end is Where 0≤m≤N d (N+υ)-1, h(l) is the ultra-wideband channel h=[h(0),...,h(L-1)] T The lth element of , and when l<0 or l≥L, h(l)=0, w(m) is additive Gaussian white noise with mean 0; without loss of generality, assume that the system CP length υ<L+τ Q , and L+τ Q ≤N, received signal y=[y(0),...,y(N d (N+υ)-1)] T After discrete Fourier transform and CP removal, the transmitted signal X i The corresponding received signal Y i =[Y i (0),...,Y i (N-1)] i (k) is expressed as Among them, W 1 (k) is Gaussian white noise, signal energy and They are and 3. The method for eliminating inter-symbol interference and nonlinear distortion in joint channel estimation according to claim 1, characterized in that: The step 2 is specifically as follows: The received signal Y corresponding to the pilot signal 1 The kth element in is represented by Among them, τ q is the delay number of the qth stage delay, W 1 (k) is Gaussian white noise, signal energy Calculated by the following formula At this time, the element Φ(k,l) in the kth row and lth column of the measurement matrix Φ is constructed according to the following formula: Φ(k,l)=RX 1 The pth element R(p) in the vector R = [R(0), ..., R(N-1)] is formed by the following formula Based on the measurement matrix Φ, the received signal Y 1 With the pilot signal X 1 The relationship is expressed as Y 1 =Φh+W 1 Among them, W 1 =[W 1 (0),...,W 1 (N-1)] T is Gaussian white noise W 1 (k) is a vector composed of , since the ultra-wideband channel is sparse, the channel estimation can be achieved by using the compressed sampling matching pursuit algorithm to solve the following formula in, is the channel estimation result, ||h||1 represents the 1-norm of h.

4. The method for eliminating inter-symbol interference and nonlinear distortion in joint channel estimation according to claim 1, characterized in that: The step three is specifically as follows: Receive signal Y i ,i=2,...,N d Expressed as Y i =A i X i-1 +B i X i +W i Among them, X i-1 For the i-1th transmitted signal, X i =[X i (0),...,X i (N-1)] is the i-th transmitted signal, W i =[W i (0),...,W i (N-1)] is the Gaussian white noise added to the i-th received signal, and the inter-symbol interference matrix A i and the nonlinear distortion matrix B i They are and Matrix A i The element a in the kth row and pth column i (k,p) is expressed as in, is the channel estimation result The lth element in the signal energy for The matrix B i The element b in the kth row and pth column i (k,p) is expressed as Among them, the signal energy for At this time, when the i-th original signal X i The least squares method can be used for estimation, that is, X i =((B i ) H B i ) -1 (B i ) H (Y i -A i X i-1 ) The original signal can be restored by using an iterative algorithm, which specifically includes the following steps: Input: Received OFDM signal Pilot signal X 1 , cyclic prefix length υ, number of iterations T, number of nonlinear levels D, number of delay levels Q, nonlinear coefficient vector of power amplifier α = [α 1,0 ,α 3,0 ,...,α 2D-1,0 ,...,α 1,1 ,α 3,1 ,...,α 2D-1,Q ], channel estimation result Output: The restored original signal Step 1) Initialization: symbol counter i=1, maximum number of iterations T; Step 2) i = i + 1; Step 3) According to X i-1 ,calculate And construct the matrix A i ; Step 4) Calculate the scaling factor And get the i-th original signal X i The initial estimate of Step 5) Iteration counter t = 1; Step 6) Based on the estimated value Calculate signal energy And construct the matrix B i ; Step 7) Update the estimate Step 8) t = t + 1; Step 9) If t≤T, go to Step 6), otherwise Step 10) If i≤N d , go to Step 2), otherwise output

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