Adaptive blind equalization of fast time-varying channel with frequency estimation in impulsive noise environment

In this paper, a novel source signal recovery method for fast time-varying channels described by complex exponential-basis expansion model (CE-BEM) in the impulsive noise environment is proposed. This method consists of two phases. The ﬁrst phase is the equalization of fast time-varying channels, in this phase, a novel algorithm FSE-FLOS-CMA is proposed. The convergence performance of this newly proposed algorithm is much better than that of existing fractionally spaced equalizer-constant modulus algorithm (FSE-CMA) in impulsive noise environment. In the second phase, a novel frequency estimation method is proposed. The estimated frequency in impulsive noise environment is obtained by calculating a speciﬁc p-order fractional low-order cyclic moment of the equal-ized signal. Simulation results show that the proposed FSE-FLOS-CMA and frequency estimation method can effectively estimate the source signals transmitted through the fast time-varying channels in impulsive noise environment.


INTRODUCTION
In wireless communications, inter-symbol interference (ISI), as a form of signal distortion, is caused by multipath propagation and band limited propagation [1]. In real systems, the truncation of filter and the deviation of sampling time will both cause the overlap of the tail before and after the symbol, which will cause ISI. So, equalizers need to be designed to eliminate the influence of ISI. Including training sequence in the transmitted data will increase the transmission cost. In the fast time-varying (hereinafter referred to as TV) channel communication system, the time used to transmit the training sequence even takes up 50% of the total transmission capacity overhead [2,3]. And in the measuring instrument, the transmitted signal is likely to be unknown. Therefore, the adaptive blind equalization is necessary to study. Communication signals often have the characteristic of constant envelope, that is, constant modulus. According to this characteristic, Godard [4]. This algorithm is one of the best known and the simplest adaptive blind equalization algorithms. CE-BEM model is widely used in the TV FIR channel [5][6][7]. The TV taps are expressed as a superposition of TV bases (e.g. complex exponentials when modelling Doppler effects) with time-invariant (hereinafter referred to as TI) coefficients. Timevariation offers diversity. Since the same input is modulated by different complex exponentials, some redundancy is introduced at the output which we call "channel diversity" [8,9]. In order to deal with faster TV channels, a number of channel models have been developed to describe the TV SIMO-FIR channels [10,8,11,12]. Just like the TI case, sampling faster than the symbol rate creates diversity that enables the problem to be cast into an SIMO framework. Multichannel diversity can also be achieved by using multiple antennas at the receiver [13][14][15]. The traditional adaptive blind equalization algorithms can only track slowly TV channels and many of them suffer from local minima. They fail when the speed of channel variations exceeds the convergence speed of the algorithms [10]. In practice, the CM algorithm has often been implemented using fractionally spaced equalizer (FSE) [16].
In many practical problems, the noise distribution encountered is more impulsive than Gaussian distribution. For examples, underwater noise, low frequency atmospheric noise, noise of semiconductor devices and many types of man-made noise are impulsive [17]. There exists a class of distributions called alpha-stable distributions that can be used to model impulsive noise [18]. In references [19,20], a novel FLOS-CMA algorithm based on fractional low order constant modulus property of the measured signal is proposed. This algorithm can deal with impulsive noise and ISI in the data robustly.
The fourth-order statistics based estimation method [8] and higher-order statistics (HOS) based method both need to use a very large number of samples to achieve reasonable statistical performance and are expensive in computation [21]. Because the variance of second-order statistics (SOS) is usually lower than that of fourth-order and higher-order statistics, Tsatsanis et al. proposed an effective frequency estimation method combining second-order and fourth-order statistics [10], which has been widely used to solve the blind identification and equalization problems of TV channels [10,8,15,22]. There is a special kind of non-stationary signal among communication, astronomy, ocean and other signals, their non-stationarity is characterized by cyclostationarity, that is the statistical characteristics change periodically or in many cycles (each cycle is incommensurable). The second-order cyclic moment is a powerful tool for utilizing cyclostationarity of signals [23].
In this paper, we propose a novel source signal recovery method which realize adaptive blind equalization and frequency estimation of fast TV channel in impulsive noise environment. The main contributions of this paper are summarized as follows. Firstly, we propose FSE-FLOS-CMA algorithm for the fast TV channel based on CE-BEM model. This algorithm can converge signals effectively in impulsive noise environment. Moreover, we propose a method to estimate the frequency of fast TV channel in impulsive noise environment, we calculate the fractional lower-order cyclic moment of the signal to obtain the frequency estimation.
The rest of the paper is organized as follows. In Section 2, the alpha-stable distribution is introduced as the model of impulsive noise. CE-BEM model applied to the fast TV channel and TV SIMO channel blind equalization architecture are both introduced in Section 3. In Section 4, we proposed a new channel equalization algorithm and a new frequency estimation method for the fast TV channel in impulsive noise environment. In Section 5, the simulation results are presented. Finally, the conclusion is shown in Section 6.

ALPHA-STABLE DISTRIBUTION
The impulsive noise can be described by many models, such as Middleton Class A [24], Bernoulli-Gaussian [25][26][27][28][29][30] and symmetric alpha-stable distribution [18]. The parameters in model "Middleton Class A" are directly related to the physical channel, but not easy to deal with; models "Bernoulli-Gaussian" and "symmetric alpha-stable distribution" can both generate Gaussian and non-Gaussian noise. The alpha-stable distribution is the only distribution that satisfies the generalized central limit theorem, so we use alpha-stable distribution in this paper. Alpha-stable distribution does not have a unified probability density function (pdf) expression, but it has a unified characteristic function expression.
is the characteristic exponent, which controls the level of impulse in the stable distribution process, the smaller the , the stronger the impulse; is the symmetry parameter, when = 0, the distribution is symmetric, such distribution is calledS S (symmetry -stable) distribution; is the dispersion parameter, which is similar to the variance of Gaussian distribution; a is the location parameter, which corresponds to the mean value or mid-value of the stable distribution. When = 2, = 0, alpha-stable distribution corresponds to the Gaussian distribution, and it has explicit probability density function expression.
If 0 < < 2, there is no statistic higher than for the alphastable distribution. However, any order statistic exists for the Gaussian distribution ( = 2).
For any random variables X and Y which submit to theS S distribution, their p-order fractional lower-order moment is defined as where < ⋅ > represents: z <a> = z <a−1> z * Whenp = 2, the p-order fractional moment is the general second-order moment.
Due to the characteristic of alpha-stable distribution, the conventional SNR becomes meaningless. Therefore, generalized SNR (GSNR) [31] is defined in impulsive noise environment, the expression is: where 2 s is the variance of the input signal, is the dispersion parameter of alpha-stable distribution.
The Figure 1 below shows the impulse noise with GSNR is 25. Because we do not know the channel characteristics before processing the signal, we cannot process the channel noise in advance.

COMPLEX EXPONENTIAL-BASIS EXPANSION MODEL (CE-BEM) AND TV SIMO CHANNEL BLIND EQUALIZATION ARCHITECTURE
The basis expansion model is the most ideal model for linear TV channels so far. Complex exponential-basis expansion model (CE-BEM) is the most commonly used basis expansion model in the research of blind equalization technology. The model uses the linear combination of a group of complex exponential basis to represent TV channel, which is simple to implement and has clear physical meaning for each parameter.
CE-BEM is a deterministic model, and it approximates the well-known random coefficient models as the number of paths increases (chosen to be ten) [32,33]. Therefore, the multipath is usually caused by a few strong reflectors so that it can be considered deterministic. As shown in Figure 2, Q is the number of paths and is set to less than ten [8]. Moreover, because the time-varying multipath is too rapid, the number of paths is usually between 2-5. h q (l )is the time-invariant coefficient, wherel is the delay. b q (n)is the complex exponential basis of a channel, b q (n) = e j q n , where q represents the basis frequency. We can get the impulse response of a channel: Set the basis frequencies0 = 1 < 2 < … < Q , the difference between two different basis frequencies is different, i.e. if SupposeLis the order number of the channel, s(n)is the input source signal, v(n)is the channel noise. Therefore, as shown in Figure 2, the signal received by a channel is: Because the channel variations of the TV systems are too rapid, CE-BEM model uses spatial diversity, which is achieved by using multiple antennas at the receiver, to improve the quality of received signals. Therefore, TV channels usually use singleinput multiple-output (SIMO) architecture [15].
The figure below shows the architecture of TV SIMO channel blind equalization.
As shown in Figure 3 above, there are M receiving antennas, which are expressed as x 1 According to the analysis of the signals in CE-BEM model and TV SIMO channel blind equalization architecture, we can get: The expression of matrix H l (l = 0, 1, … , L) is as follows. Its dimension isM × Q.
The dimension of matrix H is M (K + 1) × Q(P + 1),P = K + L. The expression of the matrix H is as follows.
The expressions of the matrix C (n) is as follows, and its dimension is Q(P + 1) × Q(P + 1).

C(n) = diag
( e j 1 n , … , e j Q n , … , e j 1 (n−P ) , … , e j Q (n−P ) ) (9) Based on the above definitions of matrices and vectors, we can get the following equation in the noiseless case.
x(n) = HC(n)s(n) (10) wheres It is known that g 1 , g 2 , … , g M are the equalizers of the corresponding receivers. The length of each equalizer is K + 1.

SOURCE SIGNAL RECOVERY METHOD FOR TIME-VARYING CHANNELS BASED ON CE-BEM IN IMPULSIVE NOISE ENVIRONMENT
For a fast TV channel based on CE-BEM, due to the existence of the complex exponential basis, the source signals cannot be recovered by equalization only, the estimation of the basis frequencies is also needed.
When the received signal is in impulsive noise environment, a novel method including channel equalization and frequency estimation is proposed for recovering the source signal.
As shown in Figure 3 above, after equalization, we can get: where g andx(n) have the same meanings as above. The noise contained in the received signalx(n) is impulsive noise. After frequency estimation, the recovery value of the source signal can be obtained by the following formula [9]: wherēq represents the estimated frequency.

FSE-FLOS-CM algorithm of channel equalization
Constant modulus algorithm (CMA) is a common adaptive blind equalization algorithm for constant modulus signals [4]. The cost function of CMA is: where is a constant determined by the statistical properties of the source signal.
Obviously, the cost function shown in the above formula contains the fourth-order moment. If the received signal is in impulsive noise environment, the second or higher order statistics do not exist, CMA is no longer applicable. Therefore, CMA is applicable only when the received signal is in Gaussian noise environment.
According to the characteristics of alpha-stable distribution, when the received signal is in the impulsive noise environment, only the statistics with order less than alpha are finite. We call this kind of statistics "fractional lower-order statistics (FLOS)". The FLOS-CMA algorithm proposed in [19,20] can be used in this environment. The cost function of FLOS-CMA shown below can suppress impulsive noise by using the fractional lower-order statistics. where Using the stochastic gradient descent method, we can get: Then the iterative solution equation of weight vector is obtained: The above iterative solution equation of weight vector can achieve the signal convergence in impulsive noise environment. It can be seen, if p = 2, FLOS-CMA degenerates into CMA algorithm.
Enough diversity (at least M > Q) is required for zeroforcing FIR equalizers in the fast TV channels described by CE-BEM model. According to [15], the quadruplet (M, L, Q, K) obeys In Figure 3, there are M channels to transmit signal s(n). In this situation, the existing CMA and FLOS-CMA are not applicable, because they are all used for symbol rate equalizers. Therefore, the existing FSE-CMA algorithm based on the idea of fractionally spaced equalization can equalize the fast TV channels in Gaussian noise environment.
Learned from FSE-CMA, we apply the idea of fractionally spaced blind equalization to FLOS-CMA and the fast TV channels in impulsive noise environment can be equalized effectively. We call this algorithm FSE-FLOS-CMA. For one channel, the order of the corresponding equalizer is K + 1, so the order The pseudo-code of the novel source signal recovery method Input: Signalx(n), which is received by M receivers. Output: Signalŝ(n), which is obtained by the equalization and the estimation of basis frequencies.
First step:Calculate the dispersion constan tR p , which is only related to the statistical characteristics of the transmitted signal.

E[|s(n)| p ]
, p < ∕2 Second step:Set the initial value of equalizer tap, and the value of the middle two taps of M × (K + 1) taps is set to 1.  For adaptive fractionally spaced blind equalization algorithm FSE-FLOS-CMA, the number of the basis frequencies (i.e. the value of Q) of the fast time-varying channels described by CE-BEM model can be up to 5. As mentioned in Section 3, Q is set to 2-5 due to the high-speed time-varying multipath. Therefore, five basis frequencies can be applied to most channel cases using CE-BEM model. Ref. [15] proposes a blind equalization (nonadaptive) algorithm based on CE-BEM model, on which we can further study the adaptive blind equalization algorithm with Q from 6 to 9.

Frequency estimation method
The fast time-varying channel model proposed in this paper is based on SIMO architecture, therefore, the received signals have cyclic frequencies. In the real system, the inconsistency of the frequencies and phases of the received and transmitted signals lead to the cyclic frequency offset [34,35]. In this paper, it is assumed that the transmitted and received signals are completely synchronized, so there is no cyclic frequency offset.
In the Gaussian noise environment, the second-order statistics of signals can effectively suppress the stochastic noise, then can extract the useful signal information. The frequencies of complex exponentials in CE-BEM are calculated from the socalled cyclic moments, the Fourier series of the TV moments [36,37].
However, when there is severe impulsive noise in the environment, the performance of the second-order statistics is degraded significantly. So, in the impulsive noise environment, the p-order fractional lower-order moment, which has been defined above, is needed to calculate the frequencies.
If signals x(n) and y(n) have at least one non-zero cyclic frequency, we expand their p-order fractional lower-order moment into Fourier series in order to get the p-order fractional lowerorder cyclic moment. Therefore, the fractional lower-order cyclic moment function of signals x(n) and y(n) is: where = k∕N is the cyclic frequency. It can be seen, whenp = 2,R (p) xy ( ) = ⟨xy * e − j 2 n ⟩ n , that is, fractional lower-order cyclic moment degenerates to secondorder cyclic moment.
As can be seen above, the output after equalization isy(n), the received signal of one channel isx m (n). From Equation (12), the p-order fractional lower-order cyclic moment is: From Equations (4), (5) and the definition of< ⋅ >, the above formula can be further transformed into: Because the source signal is temporally white, the above formula can be further transformed into:

4.3
The detailed process of the novel source signal recovery algorithm Based on Sections 4.1 and 4.2, the following summarizes the steps of recovering the source signal from the received signal through a fast TV channel in impulsive noise environment.
Step 1: Use FSE-FLOS-CMA to equalize data through fast TV channels based on the architecture of linear TV SIMO channel blind equalization.
Step 2: Calculate p-order fractional lower-order cyclic moment to find the peak values of the waveform. The peak values represent the differences between̄q and q ,q = 1, 2, … , Q. Since 1 is equal to 0, the maximum frequency value corresponding to the peak values repre-sents̄q.
Step 3: The equalized signal y(n) is obtained in step 1; the desired basis frequencȳq is obtained in step 2. According to Equation (12):ŝ(n) = y(n) ⋅ e − j̄qk , the estimated value of the source signal can be calculated by y(n) and q . Table 1 shows how to use the above three steps to realize the novel source signal recovery method in MATLAB.

SIMULATIONS
When the transmission system is full response system, the signal source used in the simulation is 4 quadrature-amplitude modulation (4-QAM) signal, which is a constant modulus signal. If the transmission system is partial response system, the modulation mode of 4-QAM signal is changed to 9 quadrature-partial response (9-QPR) signal, which is not a constant modulus signal [36]. In the partial response system, the inter-symbol interference (ISI) is introduced at the sampling time of some symbols, which can increase the utilization of frequency band and reduce the requirement of timing accuracy. However, the output of partial response system is changed from two-electrical level form to three-electrical level form, which means the anti-noise ability is reduced, so 4-QAM signal is still used in this paper.
Simulation results have illustrated the effective recovery performance of the proposed novel source signal recovery method in the impulsive noise environment.
The novel proposed algorithm is mainly used for the fast TV channels, but it can also be used for the TI channels. When used for the TI channels, Q in CE-BEM model is equal to 1. According to Equation (18), M min = 2. The following simulation cases focus on the TV channels, therefore, we set Q ≥ 2 and M = 4.

The performance of FSE-FLOS-CMA
The signal is received through TV channel described by CE-BEM model and it is added by the impulsive noise. GSNR of the signal is 30 dB. The values of variables in CE-BEM model are as follows: L = 1, M = 4, Q = 2, K = 4. Moreover, the characteristic exponent of the impulse noise is equal to 1.1. The figure below shows the signals after equalization according to the above settings.
As can be seen from Figure 4(a), FSE-CMA cannot effectively equalize the fast TV signals with obvious impulse noise; FLOS-CMA shown in Figure 4(c) cannot equalize the fast TV signals described by CE-BEM. However, as can be seen from Figure 4(b), the new proposed FSE-FLOS-CMA can effectively equalize this kind of signal.
When the parameters in CE-BEM model are assigned as L = 1, M = 4, Q = 2, and the characteristic exponent of the impulse noise is equal to 1.1, it can be seen from the Figure 5(a,b) that the more the number of equalizer taps, the better the signal convergence.

The performance of ISI index
Based on BE-CEM model, the signal after channel equalization is expressed as: where q(n) = (HC (n)) H g is the system impulse response. C (n) is the matrix of complex exponential bases of a TV channel.
Because of the channel diversity of BEM model, the matrix is related to the time point n of the input signal s(n).
Usually, ISI is used to measure the convergence performance of blind equalization. The definition is as follows: where q(n) is defined in Equation (22). The parameters in CE-BEM model are assigned as L = 1, M = 4, Q = 2, K = 2, and the characteristic exponent of the impulse noise is equal to 1.1.
As can be seen from Figure 6, if the received signal is equalized by FSE-CMA, the residual ISI does not converge; however, if the received signal is equalized by the newly proposed FSE-FLOS-CMA, the residual ISI can converge.

5.3
The performance of fractional lower-order cyclic moment The parameters in CE-BEM model are assigned as L = 1, M = 4, Q = 3, K = 2, and the characteristic exponent of the impulse noise is equal to 1.1. The three basis frequencies in the TV channel are 1 = 0, 2 = 2 ∕60, 3 = 2 ∕40.
When the received signal is equalized by FSE-CMA, the waveform of the second-order cyclic moment is shown in the lower plot of Figure 7  As can be seen from Figure 6(b), the largest frequency of the peaks of the waveform is 2 ∕40, sōq =̄q − 1 = 2 ∕40.
If the received signals are all equalized by FSE-FLOS-CMA, the waveform of the second-order cyclic moment is shown in the upper plot of Figure 8. The lower plot of Figure 8 shows the waveform of the p-order fractional lower-order cyclic moment. From these two plots, we can see that if the signal contains impulse noise, the noise of the second-order cyclic moment is too large to submerge useful signal; while the p-order fractional low-order cyclic moment can output all useful information of the signal.
FSE-FLOS-CMA algorithm can equalize the fast timevarying channel described by CE-BEM model with up to 5 basis frequencies in impulsive noise environment. Figure 9 below shows the equalization algorithm and frequency estimation algorithm for fast time-varying signals with 5 basis frequencies.

CONCLUSION
In this paper, we propose a novel source signal recovery method for TV channels described by CE-BEM in impulsive noise environment. In channel equalization phase of the source signal estimation, we proposed FSE-FLOS-CM algorithm. Simulation results show that the convergence performance of FSE-FLOS-CMA is much better than that of FSE-CMA in impulsive noise environment. In frequency estimation phase of the source signal estimation, we obtain the estimated frequency in impulsive noise environment by calculating the p-order fractional loworder cyclic moments. Simulation results show that the correct estimated frequency can be obtained by calculating the fractional lower-order cyclic moment using equalized signal, which is obtained by FSE-FLOS-CMA. Otherwise, the correct estimated frequency can't be obtained in the cases of equalizing signals by CMA or calculating the second-order cyclic moment of the equalized signals. Moreover, we use the ISI index to show that FSE-FLOS-CMA is effective for the signal convergence in impulsive noise environment. The ISI index is calculated based on CE-BEM and SIMO architecture. The newly proposed adaptive equalization algorithms can equalize the fast time-varying channels described by CE-BEM with up to 5 basis frequencies in impulsive noise environment. For this deterministic model, Q is the number of basis frequencies which is less than 10. So future work is required for extending the adaptive equalization algorithm in impulsive noise environment.

ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (No. 61801092) and Fundamental Research Funds for the Central Universities (No. ZYGX2020J012).

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.