Generalized Regression Neural Network based Fast Fading Channel Tracking using Frequency-Domain CSI Smoothing

In the high mobility environment, the channel state information (CSI) in the last part of the packet is different from the beginning part’s actual channel. This phenomenon degrades channel estimation accuracy, and hence it is necessary to be compensated to realize reliable communications. Decision feedback channel estimation (DFCE) has been widely considered as the channel tracking approach. It still causes estimation errors due to the decision-making process in the presence of time and frequency selective fading environments. To address these issues, this paper newly proposes a generalized regression neural network (GRNN) based channel tracking scheme incorporated with frequency-domain CSI smoothing. The latter part is the key to improve the dependability of the training data sets. Computer simulation results confirm that the proposed scheme can achieve superior BER performance and the lower root mean square error (RMSE) value of estimated CSI better than the conventional ones.


I. INTRODUCTION
P Acket-based transmission has been the most common wireless communication method, in which the desired transmission bitstream is divided into packets and transmitted. In this case, the propagation channel needs to be estimated for each packet in order to accurately equalize and decode the signal. The pilot-aided channel estimation (PCE) is one of the basic methods for channel estimation. In PCE, known training symbols are generally inserted at the head of the packet. In a static environment, it can estimate the frequency selectivity of channel state information (CSI). However, there is a growing demand for reliable and high-capacity communications even in an environment where transmitters and receivers move at high speed. Under such a fast moving environment, the channel state rapidly changes in the time-domain. Accordingly, estimated CSI by PCE is largely dissimilar to the actual channel state, particularly in the last part of the packet.
Many challenges have been made so far to trace the channel state transition [1]- [5]. In [1], a data-aided decision feedback channel estimation (DFCE) was proposed. DFCE directly estimates channel variation at any specific data symbol using the difference between the received symbol and the replica signal which is the multiplication of remodulated signal by CSI obtained by PCE. The authors in [2] proposed a computationally efficient channel estimation for an OFDM system with space-time coding in time-varying dispersive multipath fading channels. In [3], the low-complexity windowed DFT-based minimum mean square error (MMSE) channel estimator was proposed. These methods reduced the computation complexity without degrading the channel estimation performance. In [4], the authors proposed the pilot-based channel estimation scheme on a rapidly-varying channel environment in OFDM systems. It used channel interpolation to handle rapid variation within a transmission block. Moreover, subspace-based channel estimation techniques for OFDM systems over fast-fading channels have been proposed [5]. Although these precedent channel esti-mation approaches are tolerant of fast fading environments, their computation complexity have not been evaluated. The recent advancement of machine learning technologies has enabled its application to a variety of fields. In wireless communications, several applications are being actively investigated. Automatic modulation recognition is the most widely known application of machine learning to signal processing technology [6], [7]. Emerging approaches have been proposed to estimate communication environment, mainly for SNR [8], K-factor [9], and Doppler shift estimation [10]. There are also many methods for channel estimation using machine learning [11]- [13]. Literature in [11] proposed the data-aided MIMO channel estimation using the support vector machine (SVM) based on regression of multiple variables. This method shows better estimation accuracy in nonlinear channels. In [12], the authors proposed to employ deep neural networks for channel estimation and symbol detection in an OFDM system. The model is trained offline using simulation data where the wireless channel was considered as black boxes. The simulation results indicated that the deep learning method is beneficial when the wireless channel is complicated by severe distortions and interference. Authors in [13] proposed deep learning-based super-resolution channel and direction of arrival (DOA) estimation in massive MIMO systems. While machine learning is an effective method for channel estimation, there are still issues in considering its effectiveness in high-speed mobile environments.
In view of the above background, we previously proposed a multilayer feedforward neural network (MLFNN) based channel tracking scheme which is effective even in fast-moving environments [15]. It employs MLFNN to estimate the whole channel state transition and compensates the channel variation using the generalization capability derived from the relationship between inputs and outputs. MLFNN is trained by partially obtained CSI by DFCE in the beginning part of the packet. It showed satisfactory tracking performance against channel variations. Meanwhile, there was a need for improvement in terms of estimation accuracy and computational complexity. We then conceived its extension by using a generalized regression neural network (GRNN) [16]. A one-pass learning process of GRNN can eliminate the iterative training process without impairing the generalization capability. While these methods show superior performance in fast-moving environments, these channel tracking performance depend on the accuracy of CSI used for the supervised data. Especially, DFCE sometimes produces incorrect CSI due to decision errors when the channel state transition has a sharp fluctuation even in the beginning part of the packet. For that reason, estimator has to reduce the difference between data-aided CSI and real channel state to achieve further improvement of BER performance.
In order to overcome this problem, this paper newly proposes a frequency-domain CSI smoothing scheme for GRNN based channel tracking method. Improving the accuracy of the network and BER performance by smoothing the scattered CSI samples used for supervised training. Our proposal applies Savitzky-Golay (SG) filtering which can provide outstanding smoothing capability of data-aided CSI and thus it can contribute to more precise GRNN training. In many cases, the values of CSIs in neighboring subcarriers are similar; behave as if it had continuity. If DFCE produces inaccurate CSI in some subcarriers, the estimated frequency selectivity is dispersed in various places. SG smoothing method can eliminate or mitigate such influence of dispersed data in consideration with the importance of not only specific but also peripheral information. Accordingly, the frequency selectivity of data-aided CSI becomes more accurate than the conventional method by dispersed data correction capability of SG smoothing method. Finally, the estimated whole channel state transition by GRNN can be more accurate in time domain after the smoothing process. To summarize, key contributions of this paper are: 1) To rectify the impact of inaccurate CSI and estimate the whole channel state transition accurately even if dataaided CSI is in error. 2) To achieve high-precision channel estimation with limited CSI information even under a high-speed mobile environment. The rest of this paper is structured as follows. Section II defines the channel model and OFDM system. Section III describes the conventional approach. Section IV describe smoothing methods and present the proposed approach concretely. The simulation results are disclosed in Section V. Finally, Section VI concludes the paper.

A. CHANNEL MODEL
We assume Jakes' time-varying multipath fading channel as the channel model of the wireless communications system. It is composed of L discrete paths with different time delays. Its impulse response is expressed as, where h l and τ l denote the complex channel coefficient and the time delay of the l-th propagation path, respectively. g l , α q and φ q are the l-th path gain, angle of arrival of the qth wave and its initial phase, respectively. ω D represents the maximum Doppler radian frequency shift, i.e. ω D = 2π f D , where f D presents the maximum Doppler frequency. Here assumes normalized path gain, i.e. L−1 p=0 E |h l | 2 = 1, where E [·] indicates the ensemble average (expectation) operation. It is well-known that the probability density function of |h l | has Rayleigh distribution in non-line-of-sight communication. The multipath fading which has this feature is called Rayleigh fading. From Eq. (2), we can confirm that the channel coefficient has the time-varying component and highspeed mobile environment makes the variation more drasti-  cally. The frequency response H( f, t) via Fourier transform of the impulse response can be obtained as, where f denotes the frequency component. The frequency response is generally not flat in a mobile communication environment. Frequency selective fading channel is provided by L > 1 where the discreet expression of |H( f, t)| fluctuates. Because of this fading, the received signal level and phase component significantly vary in the transmission frequency bandwidth as in Fig. 1, especially in case of broadband transmission [17], [18]. From these equations, it can be observed that the multipath fading channel varies with the Doppler shift, which is proportional to the transceiver's velocity of movement. Therefore, general channel estimation is greatly affected by the Doppler shift, causing degradation in decoding performance and throughput, especially in fast-moving environments. The objective of this paper is to compensate for these influences and to improve the BER performance.

B. OFDM SYSTEM
Orthogonal Frequency Division Multiplexing (OFDM) is adopted for many commercial wireless standards which include IEEE 802.11 wireless local area networks, IEEE 802.16 wireless metropolitan area networks, and 3GPP long term evolution cellular networks [27]. The system uses sub-carriers which are orthogonal to each other. OFDM system converts a high-speed transmission signal into a low-speed and narrow-band signal for a parallel transmission on a frequency characteristic. OFDM can improve the utilization efficiency of a frequency band. In addition, each subcarrier can make the length of signal longer. This can prevent intersymbol interferences by using a guard interval. From these advantages, OFDM system is suitable for mobile communication. Fig. 2 shows the original system model of OFDM transmitter and receiver.

III. CONVENTIONAL NN BASED CHANNEL TRACKING A. DECISION FEEDBACK CHANNEL ESTIMATION (DFCE)
Here we assume that single-input single-output based orthogonal frequency division multiplexing (SISO-OFDM) transmission and the insertion of a pilot symbol at the beginning of the packet. The DFCE can estimate the CSI of a particular data symbol using the demodulated signal and the CSI obtained by the PCE. Fig. 3 shows the process of DFCE for the i-th data symbol and the characteristic of time-varying channel. The received signal of the m-th subcarrier and the n-th data symbol, Y(m, n), is represented as follows: where H(m, n), X(m, n), and N(m, n) indicate the channel coefficient, the transmitted symbol, and Additive white Gaussian noise (AWGN), respectively. Given the pilot symbol X(m), CSIH(m) is estimated as It is used for subcarrier equalization and the decision result of the n-th transmitted data symbol is then obtained.
where F (m)[.] stands for the decision function. Dada-aided CSI at the n-th symbolȞ(m, n) can be derived as, At this point, CSI is contaminated due to the additive noise effect and decision failure in (??), which may cause degradation of demodulation accuracy. The noise reduction is possible by using adjacent symbols. Here CSI is averaged over adjacent three samples [14]: For the purpose of tracking the whole channel state transition, DFCE based channel estimation is performed for all VOLUME 4, 2016 symbol instants. However, this method sometimes estimates inaccurate CSI due to decision errors. Thus, the channel compensation performance cannot be increased even though CSI given DFCE is applied under the fast fading environment. Specifically, decision errors have often happened in the last part of the packet. This is because the channel variation from the pilot-aided CSI to the channel state is quite huge especially at the last part of the packet.

B. NEURAL NETWORK (NN)
Because of the low reliability of DFCE in the last part of the packet, additional approach is needed to stabilize the whole CSI transition using partially obtained CSI at the beginning part of the packet. To resolve this problem, we previously proposed the NN-aided approach [15], [16]. This method applies a NN for channel tracking using partially obtained CSI by PCE and DFCE. The NN can construct relationships between input signals and output signals because of a non-linear statistical modeling capability [20]. Accordingly, the generalization capability of the NN can achieve high tracing accuracy of whole CSI transition even by training only a few estimated CSI. It is essential to find an optimal structure of hidden layers because of following two perspectives: an unsatisfactory performance produced by few neurons, and the considerable system complexity caused by the oversized hidden layers.

1) Multilayer Feedforward Neural Network (MLFNN)
Previous work [15] employed a fully connected 2-layer feedforward neural network. MLFNN has a layered structure constructed by neurons, and every neuron passes signals to neurons of the next layer. Every MLFNN has an input layer, one or more hidden layers, and an output layer. The input signals of MLFNN are passed directly to the hidden layer in every input neuron. Thus an input layer is not generally counted as layer of MLFNN. The size of input and output layers depend on the size of signals of interest. On the other hand, the system size of hidden layers is variable. One hidden layer normally has sufficient approximation capability to present any continuous function. In general, the sigmoid function is applied for the activation function of hidden layer in regression analysis. The derivative type of this function can be expressed simply using the function itself. Accordingly, using the sigmoid function is suitable for backpropagation (BP) training which is the representative method of the MLFNN training [20]- [22]. However, MLFNN has to renew parameters iteratively in BP training until any training parameter reaches the target value. From this reason, the iterative process imposes huge computation complexity and causes processing delay.

2) Generalized Regression Neural Network (GRNN)
In order to eliminate the iterative training process without impairing the generalization capability, we previously employed GRNN for the NN part [16]. GRNN was originally developed by Donald Specht in 1991 [23]. It is represented as an improved version of the radial basis function neural network (RBFNN), which is based on nonparametric regression. This network can smooth the data transition using sparse training samples using probability distribution from each signal. GRNN sets the weights from the expected value of response directly. Therefore, GRNN does not require any iterative learning process. As shown in Fig. 4, GRNN is constructed by 3 layers except for an input layer: a pattern layer, a summation layer, and an output layer [24]. Following describes how to calculate output values of the 1st output neuron. Let x = [x 1 , x 2 , . . . , x j ] and c i denote the input vector and the i-th training input vector (1 ≤ i ≤ N t ), respectively. N t indicates the number of training sets. The i-th pattern neuron outputs where r denotes the radius of the radial basis function (RBF) applied for the pattern layer. This parameter can adjust the probability distribution and controls a smoothing level of regression results. When r is large, GRNN produces smoother transitions of dispersed training samples and they are not forced to desired outputs. In contrast, small value of r generates rapidly changing regression curves and desired responses are more faithfully tracked by regression results [23]. The output of the i-th pattern neuron is multiplied by the i-th desired response, and inputs to the numerator neuron of the summation layer. Accordingly, the output of the numerator neuron, a 1 (x), is calculated by The output of the denominator neuron, b(x), is generated by From (10) and (11), the output at the 1st output neuron, y 1 (x), is represented as FIGURE 5: Block diagram of the GRNN based channel estimation.
As mentioned above, GRNN output presents a weighted average of the desired signals when the outputs of the pattern layer are regarded as weights [25], [26]. By replacing w 1 (i) with w k (i) and a 1 with a k , we can also calculate the output at the k-th output neuron (k = 1, 2, . . . , N o ) using (9)- (12), where N o denotes the number of GRNN outputs.

C. CONVENTIONAL SCHEME
This paper defines the GRNN-based channel tracking using partially obtained CSI via PCE and DFCE as the conventional scheme [16]. It can obtain accurate estimation results of whole channel state transition thanks to the generalization capability of GRNN trained by estimated CSIs at the beginning and intermediate parts of the packet. It also unnecessitates independent NN training process. Fig. 5 where h out (m, n) (n = 1, 2, ..., N d ) is the trained NN based estimated CSI applied for the m-th subcarrier and the n-th data symbol. Decision errors can happen even in the beginning part of the packet when the channel state transition in the beginning part is quite rapid. Namely, data-aided CSI has some risk of inaccurate information even if DFCE is conducted in the beginning part of the packet. Because of the problem mentioned above, impact of inaccurate CSI caused by the decision error should be alleviated to further improve the performance of channel compensation.

IV. PROPOSAL: FREQUENCY-DOMAIN CSI SMOOTHING
This paper newly proposes to incorporate CSI smoothing method before the GRNN estimation part in the frequencydomain. When DFCE contains some error component, the frequency selectivity of CSI is dispersed; continuity among adjacent subcarriers is broken. By using a smoothing method, such an unnatural channel fluctuation of data-aided CSI can be mitigated by refining the frequency selectivity. The following subsections introduce the representative smoothing methods and derive the suitable one for our proposed approach concretely.

A. SMOOTHING METHOD
Smoothing methods can compensate discrete or scattered outliers and estimate smoother transition than the original information. These techniques are applied as preprocessing in order to correct data originally having continuity or mutuality. These preprocessing for data sets are especially important to conduct machine training accurately. The wellknown smoothing methods are the moving average (MA) and Savitzky-Golay (SG) filtering. Each smoothing method has respective features of calculating a smoothing value, and there is a possibility that the optimal smoothing scheme is different depending on the problem and target. On that account, the SG filter is suitable for the proposed approach. It is elaborated by the simulation results of performance comparison in Section V.
Here, α (0 < α < 1) denotes a constant smoothing factor. In EWMA, the weights fluctuation does not reach anywhere zero. The whole raw data is concerned for the calculation of smoothing value using (16). As shown in (15) and (16), the former diminishes a weight value linearly; in contrast, a weight value of the latter is lessened non-linearly. Therefore, LWMA is suitable for weighted average with restricting smoothing window, and EWMA can make much account of adjusting values and cannot abandon distant data.

2) Savitzky-Golay (SG) filtering
The Savitzky-Golay (SG) filtering was initially proposed in 1964 [30]. Even now, the SG filter is mainly used for noisy data in spectral analysis, and various improvements of the SG filter have been studied. Also, it was shown that the SG filter is adequate for smoothing time-series data in the literature [31]- [33]. SG filtering conducts an arbitrary order polynomial approximation using the least square method in an object of a window section. A smoothed value obtained by the SG method is defined as an output value of the approximated polynomial at the center of the window section. We give the precise method of calculation to extract the smoothed value at the i-th explanatory variable, which is assumed to be p below.
where d n is the regression coefficient of the n-th degree of q. The definition vector of regression coefficients is, The vectors of explanatory variables and objective variables are defined as where f q presents the objective value, i.e. CSI at the qth subcarrier obtained by PCE and DFCE in the proposed method. This is the same as u(p − i). The normal equation of least square method is given by, By variation of Eq. (22), we can obtain the values of d using the following equation: where Z = (Q T Q) −1 Q T . Thus, the regression coefficients are obtained and the SG filter can express the approximate equation of arbitrary order. From (17), the smoothed value at p = i is represented as the polynomial approximation of the SG filtering result at q = 0. The constant term, d 0 , is only remained when 0 is substituted for q in Eq. (18). Accordingly, the smoothed value at p = i is expressed as, where Z(a, b) denotes the matrix element of Z at the a-th row (1 ≤ a ≤ N + 1) and b-th column (1 ≤ b ≤ 2M S G + 1). Z can usually be calculated in advance by defined the number of N and M S G . For that reason, SG filtering is practically conducted with (21) and (24) only. This smoothed scheme is basically applied for all data areas in an explanatory variable (but except for circumferential parts of both ends). Incidentally, transitions of obtained expressions in p = M S G + 1 and p = L d − M S G is applied in 1 ≤ p ≤ M S G and L d − M S G + 1 ≤ p ≤ L d using (18) and (23). Here, L d denotes the all length of data area processed smoothing. We can eventually obtain a smoothed transition of all data and remove the influence of noisy or scattered information by these processes. VOLUME 4, 2016

B. INCORPORATION WITH GRNN TRAINING AND CHANNEL TRACKING
The proposed scheme employs the smoothing approach in the frequency-domain after the DFCE process. In this method, the SG filtering process is inserted between the DFCE block and desired responses block of Fig. 5. It can compensate for the influence of inaccurate CSI before GRNN training. Due to the preprocessing of desired symbols, GRNN estimates whole channel state transitions more precisely than the conventional approach. Fig. 7 shows the process of the proposed scheme. Using the pilot symbol, the first symbol's CSI is estimated by PCE in (12), and the 10th symobol's CSI estimated by DFCE in (8) is treated as the training input data for GRNN. After CSI estimation, the smoothing method is applied to DFCEbased CSIs in the frequency domain. SG filtering enables to optimize the value of CSI, which improves the channel compensation performance by GRNN. Here, the channelH for p = i smoothed by the SG filtering is represented as follows, On the other hand, This smoothing can remove inaccurate frequency responses, e.g. outliers, without degradation of frequency selectivity and obtain the refined CSI,ĥ smooth = (ĥ smooth (1),ĥ smooth (2), · · · ,ĥ smooth (N c )). Then PCE-based CSIs and smoothed DFCE-based CSIs are set as the desired symbols. Symbol indices (1, 10) are defined as the training inputs. Lastly, same as the conventional scheme, the index vector for all data symbols is entered into the GRNN. Thanks to improved accuracy of the desired responses, GRNN can produce H out more accurately, and this information can be utilized for channel tracking.

V. COMPUTER SIMULATION A. SIMULATION RESULTS
The simulation parameters are shown in Table 1, here N p denotes the number of pilot symbols. The maximum Doppler frequency is set as 700 Hz to represent a fast fading environment, and its normalized value is 2.8×10 −3 per OFDM symbol duration. Radius of RBF r is 4.5 in the proposed method. At first, the best parameters for each smoothing method are examined by comparing the bit error rate (BER) value of the CSI estimate in this simulation condition. Fig. 8 shows the BER performance of various unilateral window size M for the revised method when MA is introduced in the smoothing process. The best BER performance is shown for M = 3. The optimal window size M could depend on the coherence bandwidth. This parameter setting is based on Wi-Fi systems with relatively wider subcarrier spacing than   cellular-based specifications such as LTE or WiMAX. It also implies that the MA-based approaches do not apply to the channel environment having dynamic frequency selectivity. Fig. 9 shows the BER performance which adopts LWMA. In smoothing with LWMA, the best BER performance is observed when M is 2 and 3. compared to MA, the range of performance degradation due to M is wide. From (15), since LWMA performs smoothing with a larger amount of information than MA when M becomes too large, even harmful effects are captured and smoothed, resulting in significant degradation of BER.   FIGURE 10: BER performance of various constant smoothing factor α for the proposed method using EWMA. Fig. 10 presents the BER performance of the various smoothing factor α for the proposed method when EWMA is introduced in the smoothing process. As shown from this graph, setting α to 0.3 achieves the best BER performance. When α is 0.7, the BER performance is the most degraded. In EWMA, as α increases, the influence of distant data becomes stronger, resulting in significant performance degradation.    Smoothing methods are then compared using the above parameters. Fig. 12 shows the comparison results of the smoothing methods. Each smoothing method is employed after DFCE part. All channel tracking performance using smoothing method can be improved in both metrics against the conventional GRNN approach. MA and LWMA should always contain adjacent data values, and it may limit the effectiveness of CSI smoothing in a wide subcarrier spacing setting. Meanwhile, the SG filtering based scheme exhibit the lowest BER value. From this result, SG filtering is introduced for the smoothing part of the proposed method hereafter. Fig. 13 presents the BER performances of the conventional PCE, MLFNN [15], GRNN-based method [16], and the proposed method that employs SG filtering. The BER performance of the PCE is seriously deteriorated because decision errors are frequently happened in the last part of the packet; it shows an error floor. MLFNN-based method shows superior performance compared to the conventional PCE, but the BER never reaches 10 −4 and an error floor occurs. This is due to the fact that the generalization capability by MLFNN is not able to track the channel tracking sufficiently. The conventional GRNN-based method presents the better BER performance than the PCE and MLFNN methods, however, the error floor still remains around 10 −4 . On the other hand, the proposed method shows the best BER performance, achieving a value close to 10 −5 . It can be confirmed that the proposed method obtains excellent performance even in a fast-moving environment because the smoothing on the CSI, which is the supervised data, can make full use of the superior generalization capability of GRNN. Based on this result, the proposed method can compensate for inaccurate CSI estimates derived from DFCE and improved channel tracking performance by fully leveraging GRNN capability for all data symbols. From these results, our proposed GRNN-based channel tracking with the smoothing filter is quite effective to realize stable wireless communication in a high mobility environment. FIGURE 13: BER performance compared to various conventional methods.

B. COMPUTATION COMPLEXITY
method [16], and the proposed method in terms of computation complexity. All methods use DFCE for channel estimation, therefore, its complexity is not considered. At first, the computation complexity of MLFNN can be expressed as follows, Υ MLFNN = N hidden (N input + N output ), (27) where N input is the number of the network input, N hidden is the number of hidden layers, and N output is the number of outputs. Similarly, for GRNN, it can be represented as, Υ GRNN = N hidden (N input + N output + 1) + N output .
In the proposed method, smoothing by SG is performed on the estimated CSI. Its computation complexity is expressed as follows, Table 2 summarizes the CPU processing time consumed for channel estimation and BER at SNR = 30 dB of each method. For the computation environment, Intel(R) Core(TM) i9-10900K CPU @ 3.70 GHz and 64 GB memory are used. Here, the processing time is measured and averaged over 10000 trials. As can be seen from the table, the processing speed of GRNN is much faster than that of MLFNN. MLFNN requires only a small amount of computation to implement once training is completed, but the running time required for convergence is enormous. On the other hand, because the computational cost for training GRNN is minimal, it shows a high processing time for solving problems that require online phase processing, such as those considered in the proposed method. Since the computation cost of the SG smoothing is heavy, it increases overall processing time for the proposed method. Meanwhile, its BER performance can be improved significantly; such additional processing can be allowable considering practical hardware implementation.

VI. DISCUSSION
The most significant factor affecting the channel tracking and compensation is the Doppler shift caused by the moving velocity. The proposed method performs accurate channel compensation from a small number of CSI data and is completed only in the online phase. Hence, the proposed method can achieve robust channel compensation to changes in moving speed. In this paper, we have shown the effectiveness of the proposed method by using only simulated data. The proposed method is expected to be still effective even when using actual data, when CSI as the supervised data can be estimated correctly to a certain extent. It should be further investigated as our future work.

VII. CONCLUSION
In this paper, we proposed the GRNN based channel tracking method that applied the smoothing preprocessing in order to refine incorrect desired responses produced by DFCE. From the comparison of smoothing candidates, SG filtering is the most effective for refining inaccurate CSI outliers without disruption of original channel frequency selectivity. It can reduce the impact of erroneous CSI estimation by DFCE and improve the whole channel tracking capability of GRNN. Therefore the proposed method has provided the best BER performance compared with the conventional PCE and our previous GRNN-based method even at a maximum Doppler frequency of 700 Hz. Accordingly, we can conclude that the proposed approach can be the most suitable channel tracking method even in high mobility communication environment.