A method for canceling inter-symbol interference and nonlinear distortion in joint channel estimation

Abstract

This invention provides a method for eliminating inter-symbol interference (ISI) and nonlinear distortion in a joint channel estimation system. The method includes system parameter settings, ultra-wideband channel estimation, construction of ISI and nonlinear distortion matrices, and least-squares estimation of the original signal. This method utilizes pilot signals to achieve ultra-wideband channel estimation. Subsequently, based on the channel estimation results, an iterative algorithm is used to construct the ISI and nonlinear distortion matrices. Based on these constructed matrices, the ISI and nonlinear distortion in the received signal are eliminated, achieving accurate estimation of the original signal, reducing the bit error rate of the communication system, and improving system performance.

Description

TECHNICAL FIELD

[0001] The present application relates to a signal detection estimation method, in particular to a method for joint channel estimation, inter-symbol interference and nonlinear distortion cancellation. BACKGROUND

[0002] As an indispensable component in 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, various joint channel and nonlinear distortion estimation methods have been proposed, so as to eliminate the influence of nonlinear distortion at the receiving end while achieving accurate channel estimation (for specific methods, please refer to the following articles: X. Gao, et al., Iterative nonlinear self-interference cancellation for in-band full-duplex wireless communications under mixer imbalance and amplifier nonlinearity, IEEE Transactions on Wireless Communications, 2020; L. Li, et al., A sparse channel estimation scheme combined with distorted signal detection for UWB OFDM systems, IEEE Wireless Communications Letters, 2022; and L. Li, et al., Sparse channel estimation with distorted signals for UWB OFDM systems, IEEE Wireless Communications Letters, 2023).

[0003] In order to avoid inter-symbol interference, 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 result in two deficiencies: one is 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, and a long CP is needed to meet this requirement, which increases the complexity of system design; the other is that the CP length affects the system performance, and the increase of the CP will reduce the OFDM system throughput, thereby reducing the information transmission rate of the system. SUMMARY

[0004] The present application aims to overcome the deficiencies in the prior art without increasing the CP length and reducing the system throughput, and proposes a method for eliminating inter-symbol interference and nonlinear distortion based on joint channel estimation.

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

[0006] Step 1: Set the number of transmission symbols N d , the number of signal carriers N, the length of the cyclic prefix υ, the pilot signal X 1 =[X 1 (0),...,X 1 (N-1)] T , the received signal Y i =[Y i (0),...,Y i (N-1)] T , i = 1,...,N d , the number of iterations T, the nonlinear order D, the delay order Q, and 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 nonlinear coefficient vector of the power amplifier α, the pilot signal X 1 and the corresponding received signal Y 1 , construct the measurement matrix Φ, and use the compressed sensing method to estimate the ultra-wideband channel h = [h(0),...,h(L-1)] T ;

[0008] Step 3: Based on the channel estimation result obtained from the pilot signal The iterative algorithm is used to realize the elimination of the subsequent received signal inter-symbol interference and non-linear distortion, and to restore the original signal.

[0009] Further, the step one is specifically:

[0010] The super wideband OFDM system with N carriers adopts the packet transmission mode, and each time the transmitting end transmits a packet composed of N d OFDM symbols, that is wherein X 1 is a known pilot signal, is the transmitted data; the signal X i is subjected to inverse discrete Fourier transform and CP addition, and the time domain signal x i =[x i (0),...,x i (N+υ-1)] T The nth sampling signal is expressed as

[0011]

[0012] wherein υ is the length of the CP, and the signal is used as the input signal of the PA, the PA is modeled by using the memory polynomial model, and the output signal u(k) of the PA is expressed as

[0013]

[0014] wherein α 2d-1,q represents the coefficient at the non-linear order 2d-1 and the delay τ q , τ0<τ1<…<τ Q , D is the non-linear order, Q is the delay order, the output signal of the PA is subjected to the super wideband channel and reaches the receiving end, and at this time, the signal received by the receiving end is

[0015]

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

[0017]

[0018] Among them, W 1 (k) represents Gaussian white noise, and the signal energy... and They are respectively

[0019]

[0020] and

[0021]

[0022] Furthermore, step two specifically involves:

[0023] The received signal Y corresponding to the pilot signal 1 The k-th element is represented as

[0024]

[0025] Where, τ q W is the delay number of the q-th level delay. 1 (k) represents Gaussian white noise, and the signal energy... Calculated by the following formula

[0026]

[0027] At this point, the element Φ(k,l) in the k-th row and l-th column of the measurement matrix Φ is constructed according to the following formula.

[0028] Φ(k,l)=RX 1

[0029] The p-th 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 pilot signal X 1 The relationship is represented as

[0032] Y 1 =Φh+W 1

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

[0034]

[0035] wherein, is a channel estimation result, and ||h||1 represents a 1-norm of h.

[0036] Further, the step three is specifically:

[0037] a received signal Y i , i = 2,..., N d is represented as

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

[0039] wherein, X i-1 is an i-1th transmission signal, X i = [X i (0),..., X i (N-1)] is an i-th transmission signal, W i = [W i (0),..., W i (N-1)] is a Gaussian white noise added to an i-th received signal, an inter-symbol interference matrix A i and a nonlinear distortion matrix B i are respectively

[0040]

[0041] and

[0042]

[0043] an element a i (k, p) in a kth row and a pth column of the matrix A i is represented as

[0044]

[0045] wherein, is a channel estimation result in the lth element, and a signal energy For

[0046]

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

[0048]

[0049] Where, the signal energy is

[0050]

[0051] At this time, when the ith original signal X i The least square 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 recovered by using an iterative algorithm, which specifically includes the following steps:

[0054] Input: received OFDM signal Pilot signal X 1 , cyclic prefix length υ, iteration number T, nonlinear order D, delay order 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: recovered original signal

[0056] Step 1) initialization: symbol counter i = 1, maximum iteration number T;

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

[0058] Step 3) according to X i-1 , calculate and construct matrix Ai ;

[0059] Step 4) calculating the scale factor and obtaining the initial estimated value of the i-th original signal X i ;

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

[0061] Step 6) calculating the signal energy according to the estimated value ; and constructing the matrix B i ;

[0062] Step 7) updating the estimated value

[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 present application has the beneficial effect that: for the existing channel and nonlinear distortion joint estimation method does not consider the length of the ultra wideband channel impulse response is long and the maximum delay of PA response to the ultra wideband signal is large, resulting in the system needs a large amount of CP to avoid inter-symbol interference, thereby reducing the system throughput, affecting the transmission rate, a method for eliminating inter-symbol interference and nonlinear distortion of joint channel estimation is proposed. The method integrates the inter-symbol interference caused by insufficient CP of the system and the nonlinear distortion caused by the PA into the received signal detection, uses the compression sensing technology and the iterative estimation algorithm, eliminates the inter-symbol interference and the nonlinear distortion on the basis of realizing the accurate channel estimation, 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 DRAWINGS

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

[0068] Figure 2 is the comparison chart of the normalized mean square error of channel estimation of the proposed algorithm and the comparative algorithm;

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

[0070] The present invention will be further described below with reference to the accompanying drawings and examples.

[0071] This invention is achieved through the following steps: Figure 1 As shown, first, set the system parameters according to actual needs, including the number of transmission symbols N. d The number of carrier waves N, and the 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 iteration number T, the nonlinear series D, the delay series Q, and the nonlinear coefficient vector α of the power amplifier are [α...]. 1,0 ,α 3,0 ,...,α 2D -1,0,...,α 1,1 ,α 3,1 ,...,α 2D-1,Q Parameters such as […] are used, and then, based on the pilot signal and its corresponding received signal, compressed sensing technology is used to estimate the ultra-wideband channel. Finally, based on the ultra-wideband channel estimation results, an iterative algorithm is used to construct the inter-symbol interference matrix and the nonlinear distortion matrix, and the least squares method is used to detect and estimate the original signal. Detailed explanations are as follows:

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

[0073] An ultra-wideband OFDM system with N carriers uses packet transmission, where each packet transmitted by the transmitter consists of N carriers. d Composed of OFDM symbols, i.e. Where X 1 Given the pilot signal, For transmitted data. Signal X i After the inverse discrete Fourier transform and the addition of CP, its time-domain signal x i =[x i (0),...,x i (N+υ-1)] T The nth sampled signal is represented as

[0074]

[0075] Where υ is the length of CP. This will be used as the input signal to the PA. Modeling the PA using a memory polynomial model, the output signal u(k) of the PA can be expressed as:

[0076]

[0077] Where, α 2d-1,q This indicates that in a nonlinear series 2d⁻¹ with a delay of τ q The coefficients on τ0 < τ1 < ... < τ Q D is the nonlinear series, and Q is the delay series. The output signal of PA reaches the receiver after passing through an ultra-wideband channel. At this time, the signal received by the receiver 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 l-th element is given, and h(l) = 0 when l < 0 or l ≥ L, and w(m) is additive white Gaussian noise with zero mean. 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 removal of CP, the transmitted signal X is sent. i The corresponding received signal Y i =[Y i (0),...,Y i The k-th element Y in [N-1] i (k) can be expressed as

[0080]

[0081] Among them, W 1 (k) represents Gaussian white noise, and the signal energy... and They are respectively

[0082]

[0083] and

[0084]

[0085] Step 2: Ultra-wideband channel estimation specifically involves:

[0086] The system uses pilot signal X 1To realize the UWB 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 1 (k) is expressed as

[0087]

[0088] Where the signal energy Is calculated by the following formula

[0089]

[0090] The relationship between the vector and the channel is expressed as follows

[0091] Y 1 = Φh + W 1

[0092] Where W 1 = [W 1 (0),..., W 1 (N-1)] T Is a vector composed of Gaussian white noise W 1 (k), and 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 the lth column of the measurement matrix Φ is expressed as

[0093] Φ(k, l) = RX 1

[0094] Where the pth element R(p) in the vector R = [R(0),..., R(N-1)] is composed of the following formula

[0095]

[0096] Since the UWB channel is sparse, the channel estimation can be solved by using the Compressive Sampling Matching Pursuit (CoSaMP) algorithm to realize the following formula

[0097]

[0098] Where, Is the channel estimation result, and ||h||1 represents the 1-norm of h.

[0099] Third step: using an iterative algorithm to construct the inter-symbol interference matrix and the non-linear distortion matrix to realize the detection and estimation of the original signal, which is specifically:

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

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

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

[0103]

[0104] and

[0105]

[0106] Matrix A i The element a in the k-th row and p-th column i (k,p) is represented as

[0107]

[0108] in, For channel estimation results The l-th element in the signal energy for

[0109]

[0110] And matrix B i The element b in the k-th row and p-th column i (k,p) is represented as

[0111]

[0112] Among them, signal energy for

[0113]

[0114] From the expression of Y i , it can be seen that the signal contains the intersymbol interference of the (i-1)th transmitted signal. Therefore, in order to realize the detection and estimation of X i , it is necessary to eliminate the intersymbol interference first. Since the first pilot signal X 1 is known, based on the formula, the present application can use an iterative method to recover the subsequent signals. Assuming that the signal X i-1 has been recovered, the channel estimation result is obtained by using the second step. The intersymbol interference matrix A i can be constructed, but the construction of the nonlinear distortion matrix B i also needs to calculate the signal energy of the ith signal, and the signal energy of the ith signal also needs to be estimated. Therefore, the present application proposes an iterative algorithm to realize the estimation of the original signal, which specifically comprises the following steps:

[0115] Input: received OFDM signal Pilot signal X 1 , cyclic prefix length υ, iteration number T, nonlinear order D, delay order 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: recovered original signal

[0117] Step 1) initialization: symbol counter i = 1, maximum iteration number 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 obtain the initial estimation value i of the ith original signal X

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

[0122] Step 6) according to the estimation value Compute signal energy and construct 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 the ultra-wideband channel, inter-symbol interference and power amplifier nonlinear distortion on the received signal, and the channel influence is integrated into the construction of the inter-symbol interference matrix and the nonlinear distortion matrix. Then, the iterative algorithm is used to iteratively update the original signal estimation result, thereby improving the accuracy of channel estimation and reducing the bit error rate of the communication system.

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

[0129] Experiment: In the simulation experiment, the orthogonal frequency division multiplexing system using 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, the nonlinear order D = 2, the delay order Q = 3, and the delay τ q values are 0, 10, 50 and 100 respectively. In order to avoid the contingency of the test results, the present application performs 500 random experiments. In each experiment, the present application 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 application randomly generates a data packet composed of 5000 OFDM symbols for transmission.

[0130] Table 1: Nonlinear coefficients of power amplifier

[0131]

[0132] Simulation 1: In order to verify that the proposed method can realize accurate estimation of the ultra-wideband channel, the present application performs the following simulation experiment.

[0133] This invention compares the proposed method with Uehara's method (for details, please refer to 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 in 500 experiments under different signal-to-noise ratios (SNR). The specific results are as follows: Figure 2 As shown. From Figure 2 As can be seen, the NMSE obtained by the proposed method and the comparison method decreases with the increase of SNR, but the proposed method has a smaller NMSE than the comparison method, indicating that the proposed method can obtain more accurate channel estimation results.

[0134] Simulation 2: To verify that the proposed method can eliminate inter-symbol interference and nonlinear distortion, the following simulation experiment was conducted.

[0135] Similarly, this invention compares the proposed method with Uehara's method (for details, 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. From 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 less than the channel length L and the maximum delay τ of the power amplifier. Q The sum, i.e., υ < L + τ Q =164, the received signal contains inter-symbol interference (ISI). However, the comparison method fails to eliminate ISI, thus it cannot accurately detect and estimate the original signal. But in this scenario, the BER achieved by the proposed method decreases as the SNR increases, indicating that the proposed method can effectively eliminate ISI and nonlinear distortion in the received signal and achieve accurate estimation of the original signal.

[0136] It should be noted that the above examples are only used to illustrate the technical solutions of the present application but not to limit the present application; although the present application has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the specific embodiments of the present application can be modified or equivalent replacements can be made to some technical features without departing from the spirit of the technical solutions of the present application, and all of them should be covered in the technical solution range of the present application claimed by the present application.

Claims

1. A method of joint channel estimation and inter-symbol interference and non-linear distortion cancellation, characterized by: The method comprises the following steps: Step 1: Set the number of transmission symbols N d , the number of signal carriers N, the length of the cyclic prefix υ, the pilot signal X 1 = [X 1 (0),...,X 1 (N-1)] T , the received signal where Y i = [Y i (0),...,Y i (N-1)] T , i = 1,...,N d , the number of iterations T, the nonlinear order D, the delay order Q, and 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 two: based on the nonlinear coefficient vector of power amplifier α, pilot signal X 1 and its corresponding received signal Y 1 , construct the measurement matrix Φ, and use the method of compressed sensing to estimate the ultra-wideband channel h = [h(0),..., h(L-1)] T ; The step two is specifically: The received signal Y corresponding to the pilot signal 1 The kth element of the vector is denoted as where τ q is the delay number of the qth delay, W 1 (k) is a Gaussian white noise, and the signal energy is calculated by the following equation At this time, the element Φ(k, l) in the kth row and the lth column of the measurement matrix Φ is constructed according to the following formula Φ(k, l) = RX 1 Wherein, the pth element R(p) in the vector R=[R(0),..., R(N-1)] is composed of the following formula Based on the measurement matrix Φ, the received signal Y 1 The relationship with the pilot signal X 1 is expressed as Y 1 = Φh+ W 1 where W 1 = [W 1 (0),...,W 1 (N-1)] T is a vector consisting of Gaussian white noise W 1 (k). Since the ultra-wideband channel is sparse, the channel estimation can be solved by using the compressive sampling matching pursuit algorithm wherein is the channel estimate result, ||h||1denotes the 1-norm of h; Step three: obtaining channel estimation result based on pilot signal The iterative algorithm is used to realize the elimination of the subsequent received signal symbol interference and non-linear distortion, and restore the original signal. The step three is specifically: Receiving signal Y i i = 2,..., N d is represented as Y i = A i X i-1 + B i X i + W i where X i-1 is the (i-1)th transmitted signal, X i = [X i (0),..., X i (N-1)] is the ith transmitted signal, W i = [W i (0),..., W i (N-1)] is the ith received signal with additive white Gaussian noise, the intersymbol interference matrix A i and the nonlinear distortion matrix B i are respectively And Matrix A i the element a in the kth row and pth column i (k,p) is represented as wherein is the channel estimate result is the lth element in the channel estimate result, and the signal energy is And the matrix B i The element b i (k,p) is expressed as wherein the signal energy is At this time, when the i-th original signal X i The estimation can be performed using the least square method, i.e. 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, and the algorithm specifically comprises the following steps: Input: received OFDM signal Pilot signal X 1 , cyclic prefix length υ, iteration number T, nonlinear order D, delay order 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: Restored original signal Step 1) initialization: symbol counter i=1, maximum iteration number T; Step 2) i=i+1; Step 3) According to X i-1 , calculate and construct matrix A i ; Step 4) Calculate the scaling factor and obtain the initial estimate of the ith original signal X i ​ Step 5) iteration counter t=1; Step 6) Based on the estimated value Computing the signal energy and constructing the matrix B i ; Step 7) updating 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 2. The method for canceling the inter-symbol interference and the nonlinear distortion of the joint channel estimation according to claim 1, characterized in that: The step one is specifically: The super wideband OFDM system with N carriers adopts the packet transmission mode, and each time the transmitting end transmits a packet composed of N d OFDM symbols, that is X 1 is a known pilot signal, is the transmitted data; the signal X i is subjected to inverse discrete Fourier transform and is added with a cyclic prefix (CP), and 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 the CP, and the signal The input signal to the PA is modeled using a memory polynomial model, and the output signal u(k) of the PA is expressed as wherein α 2d-1,q represents a coefficient on a nonlinear series 2d-1, a delay τ q 0<τ1<...<τ Q D is a nonlinear series, Q is a delay series, and the output signal of PA reaches a receiving end after passing through an ultra-wideband channel, at which time the signal received by the receiving end is where 0≤m≤N d (N+υ)-1, h(l) is the (l+1)th element of the ultra-wideband channel h = [h(0),...,h(L-1)], and h(l) = 0 when l < 0 or l ≥ L, w(m) is an additive white Gaussian noise with mean 0; without loss of generality, it is assumed that the system CP length υ < L + τ T , and L + τ Q ≤ N, the received signal y = [y(0),...,y(N Q (N+υ)-1)] d After discrete Fourier transform and removal of the CP, the transmitted signal X T = [X i (0),...,X i (N-1)] and the corresponding received signal Y i = [Y i (0),...,Y i (N-1)] are expressed as where W 1 (k) is a Gaussian white noise with signal energy and are respectively And

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