# Doppler Spread Estimation Based on Machine Learning for an OFDM System

Authors: Eunchul Yoon; Soonbum Kwon; Unil Yun; Sun-Yong Kim
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

In this paper, we propose a Doppler spread estimation approach based on machine learning for an OFDM system. We present a carefully designed neural network architecture to achieve good performance in a mixed-channel scenario in which channel characteristic variables such as Rician K factor, azimuth angle of arrival (AOA) width, mean direction of azimuth AOA, and channel estimation errors are randomly generated. When preprocessing the channel state information (CSI) collected under the mixed-channel scenario, we propose averaged power spectral density (PSD) sequence as high-quality training data in machine learning for Doppler spread estimation. We detail intermediate mathematical derivatives of the machine learning process, making it easy to graft the derived results into other wireless communication technologies. Through simulation, we show that the machine learning approach using the averaged PSD sequence as training data outperforms the other machine learning approach using the channel frequency response (CFR) sequence as training data and two other existing Doppler estimation approaches.


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## Body

## 1. Introduction

Information about Doppler spread can be used for handoff, adaptive modulation, equalization, power control, etc., in wireless communication systems. Various Doppler spread estimation approaches have been studied for isotropic scattering and Rayleigh fading channels [1–6]. The Doppler spread estimation approaches of [5, 6] in single-antenna systems have been shown to outperform previous Doppler spread estimation approaches in isotropic scattering and Rayleigh fading channels. However, the performances of previous Doppler spread estimation approaches designed for isotropic scattering and Rayleigh fading channels degrade when the channels are shaped by nonisotropic scattering and line-of-sight (LOS) channel components. Therefore, it is still worthwhile to develop powerful new techniques for nonisotropic scattering and Rician fading channels for single-antenna systems. Meanwhile, machine learning has been the focus of extensive research in recent years because of its empirical success in various fields [7–24]. First of all, machine learning has been used for visual object recognition and speech recognition [7, 8]. Machine learning-based approaches have been successfully applied to wireless communication systems [9–18]. Recently, efforts have been made to apply machine learning to estimate the angle of arrival (AoA) [19–22] and apply it to indoor positioning [23, 24]. In [25], machine learning was used to improve the performance of human detection and activity classification based on the information of micro-Doppler signatures, i.e., Fourier transforms of the Doppler radar data measured under line-of-sight (LOS) conditions. In [26], machine learning was used to improve the performance of vehicle collision avoidance services based on information from the Doppler profiles, i.e., spectral representations of temporal non-line-of-sight (NLOS) Doppler energy. In [27], machine learning was used to estimate Doppler spread based on channel state information (CSI) by a high-speed train (HST) system. The channel considered in [27] differs from a typical mobile radio communication channel in that it features a LOS component formed based on the position of the HTS moving on the same track with respect to adjacent radio head units. To the best of our knowledge, machine learning has not been used for Doppler spread estimation in the context of general mobile radio communication channels. This led to the study of applying machine learning to Doppler spread estimation for mobile radio communication channels in this paper.Machine learning can work efficiently even when the system model is unknown, or the parameters cannot be accurately estimated. Doppler spread estimation techniques designed for specific channel conditions do not work well in real-world channel environments where channel characteristic variables such as Rician K factor, azimuth angle of arrival (AOA) width, average azimuth AOA, and channel estimation error are arbitrarily generated. Therefore, it is interesting to see if additional gains in performance can be obtained by applying a machine learning approach to Doppler spread estimation, especially when the channel characteristic variables are randomly generated. In [28], it was mentioned that machine learning requires a very large number of training data to effectively train the weights of a neural network for accurate classification. In other words, machine learning will not perform well if the amount of training data is not sufficient. However, it is often difficult to obtain a very large number of training data in real mobile radio communication systems. Given a limited number of training data, preprocessing the training data into high-quality training data by using feature selection and feature extraction can improve the performance of machine learning [29, 30]. This motivated our study to find out how to preprocess the collected CSI into high-quality training data to improve the machine learning performance for Doppler spread estimation.In this paper, we propose a Doppler spread estimation approach based on machine learning for an OFDM system. We present a carefully designed neural network architecture to achieve good performance in a mixed-channel scenario in which channel characteristic variables such as Rician K factor, azimuth AOA width, mean direction of azimuth AOA, and channel estimation errors are randomly generated. When preprocessing the CSI collected under the mixed-channel scenario, we propose averaged power spectral density (PSD) sequence as high-quality training data in machine learning for Doppler spread estimation. We detail intermediate mathematical derivatives of the machine learning process, making it easy to graft the derived results into other wireless communication technologies. Through simulation, we show that the machine learning approach using averaged PSD sequence as training data outperforms the other machine learning approach using channel frequency response (CFR) sequence as training data and two other existing Doppler estimation approaches. The main contributions of this paper are summarized as follows: (1) This paper applies machine learning to Doppler spread estimation in the context of mobile radio communication channels in a mixed-channel scenario. (2) The machine learning process detailed in this paper can be easily applied to other wireless communication technologies as well.The rest of this paper is organized as follows. Section2 describes the system model. Section 3 proposes two machine learning approaches and presents comprehensive algorithms for grafting machine learning into Doppler spread estimation. Section 4 presents the simulation results and compares the performance of the proposed and conventional approaches. Finally, Section 5 provides concluding remarks.


## 2. System Model

We consider an OFDM system with FFT sizeNF. We assume that the number of OFDM symbols required to compute the PSD function is Ndata, and the number of test user equipments (TUEs) required to collect training data for machine learning is NTUE. The channel impulse response of the m-th TUE for m=1,2,⋯,NTUE is modeled as a tapped delay line with L taps,
(1)hmn=h0mnh1mn⋯hL−1mn,for n=0,1,⋯,Ndata−1, where n denotes the OFDM symbol index and hlmn denotes the coefficient of the l-th channel path. To consider the effects of nonisotropic scattering and LOS channel components on Doppler spread estimation, we model the channel coefficients as in [1] by
(2)hlmn=Ωh,lKrm+1hl,amn+KrmΩh,lKrm+1hl,bmn,where hl,amn and hl,bmn denote non-LOS and LOS components, respectively, Krm denotes the Rician K-factor defined on a linear scale meaning the ratio of LOS component power to non-LOS (or fluctuating) component power, and Ωh,l denotes the power delay profile. For Rician fading, most estimated K-factor values are less than 3 dB in urban areas [31]. For the simulation, we assume an exponential power delay profile where Ωh,l for l=0,⋯,L−1 is given by
(3)Ωh,l=ρl∑l=0L−1ρl,with ρ=0.8. The autocorrelation function of hl,amn was presented in [1, 5, 32] as
(4)rhl,amn=I0κm2−2πfDmTSn2+j4πκmfDmTSncosαmI0κm,where fDm denotes the Doppler spread, κm denotes the azimuth AOA width factor which controls the width of the azimuth AOA, αm denotes the mean direction of the azimuth AOA, TS denotes the OFDM symbol period, and I0· denotes the zero-th order modified Bessel function of the first kind. The l-th Rician channel component, hl,bn, can be written as
(5)hl,bmn=e−j2πfDmTSncosα0m+jϕ0m,where α0m is the parameter controlling the azimuth AOA direction of the LOS component and ϕ0m is the parameter representing the phase of the LOS component with a uniform distribution between 0 and 2π. The autocorrelation function of hlmn was presented in [5, 32] as
(6)rhlmn=Ωh,lKrm+1rhl,amn+KrmΩh,lKrm+1e−j2πfDmTSncosα0m.The CFR coefficient over the k-th subcarrier is given by the discrete Fourier transform (DFT) of hmn,
(7)Hkmn=∑l=0L−1hlmne−j2π/NFkl,for k=0,1,⋯,NF−1. The OFDM demodulated signal over the k-th subcarrier of the n-th OFDM symbol for the m-th TUE can be written as
(8)Ykmn=HkmnXkmn+Wkmn,where Xkmn denotes the transmitted symbol and Wkmn denotes a zero-mean circularly symmetric complex Gaussian noise with variance σ2. SNR is defined as 1/σ2 assuming that the transmitted symbol has unit average power. If the number of pilot symbols in the OFDM block is NP, the CFR coefficient over the subcarrier of the p-th pilot symbol for p=0,1,⋯,NP−1 can be estimated by the least square detection method in [33] as
(9)H^nPmn=YnPmnXnPmn,where nP denotes the subcarrier index of the p-th pilot symbol. We use the minimum mean square error-based channel interpolation method in [34] to estimate the CFR coefficients, H^kmn0≤n≤Ndata−1,0≤k≤NF−1. By using this channel estimation method, the channel estimation error increases as the SNR decreases. The ideal PSD function of the channel can be obtained as in [35] by taking the Fourier transform of rhlmn,
(10)Sideal,lmu=∑n=−∞∞rhlmne−j2π/Ndataun,for u=0,1,⋯,Ndata/2−1. The Doppler spread index is defined by
(11)uDm=fDmΔf=fDmT,where T=NdataTS denotes the period of collecting information of Ndata consecutive OFDM symbols and fDm=1/T denotes the frequency resolution required to determine the Doppler spread index with the Doppler spread. Since base stations (BSs) in macrocells generally suffer from severe nonisotropic scattering, Sideal,lmu has its maximum quantity at umax,lm=uDmcosαm [5]. For severe nonisotropic scattering, it is reasonable to assume α0m=αm for all hlmnl=0L−1 [36]. The Doppler spread is found by transforming umax,lm to fD,lm based on fD,lm=umax,lm/T/cosαm for l=0,1,⋯,L−1 and finding fDm=maxl∈0,⋯,L−1fD,lm [5]. For severe nonisotropic scattering, the angle spread θm is mostly less than 30°, and for very severe nonisotropic scattering, it is less than 10° [5]. If κm is not too small, κm can be approximated as κm=360°/θm/π2 [37]. We assume that the transmitted signal undergoes severe nonisotropic scattering and that θm has a uniform distribution between 10° and 30°. Although there are several ways to estimate the PSD function [38], periodogram-based nonparametric techniques are often used because of its simplicity. Using the periodogram-based nonparametric technique, multiple PSD functions can be computed with multiple series of CFR coefficients; the k-th PSD function computed using a series of the k-th estimated CFR coefficient, H^kmn0≤n≤Ndata−1, can be written as
(12)Skmu=1Ndata∑n=0Ndata−1H∧kmne−j2π/Ndataun2,for k=0,1,⋯,NF−1 and u=0,1,⋯,Ndata/2−1. Because of channel estimation error and the lack of channel coefficients used to compute the PSD function, the accuracy of the PSD function decreases. With an inaccurate PSD function, the maximum value of Skmu given in (12) does not always occur at the ideal Doppler spread index u=uDmcosαm. Since fDm=ΔfmuDmcosαm, a slight deviation in the value of uDm results in a Δfmcosαm times greater deviation in the fDm value. Therefore, it is important to set Δfm to a small value to improve the performance of the nonparametric Doppler spread estimation approach. In order for Δfm to be small with TS fixed, Ndata should be larger because Δfm=1/NdataTS. The sampling theorem states that in the time dimension, the channel sampling rate must be at least twice the Doppler spread, i.e., 1/TS≥2fDm. A good rule of thumb for finding the maximum of fDm that can be estimated by a Doppler spread estimation approach is to assume that the maximum of fDmTS lies between 1/12 and 1/6 ([35], pp. 845–846). We set the maximum value of fDmTS to 0.1. Therefore, we assume that 0≤fDm≤0.1/TS, or equivalently, 0≤uDm≤Ndata/10.


## 3. Proposed Doppler Spread Estimation

Figure1 shows the block diagram of the proposed neural network structure to realize forward signal propagation in machine learning. The proposed neural network structure consists of an input layer with Nin nodes (or neurons), an output layer with Nout nodes, and three hidden layers. Since 0≤uDm≤Ndata/10, we set Nout to Ndata/10 and let the output layer act as a multiclass classifier generating a one-hot encoded binary sequence of multiple zeros and single ones. The position of a single 1 in the one-hot encoded binary sequence represents uDm. The output signals of the three hidden layers are processed with the batch normalization function [39], the dropout function [40], and the rectified linear unit (ReLU) activation function [41]. The output signal of the output layer is processed with a softmax regression (SoftMax) activation function [42]. Since a bias factor of 1 is added to every sequence entering each hidden layer, the dimensions of the four weight matrices, W1, W2, W3, and W4, are chosen as Nmid−1×Nin+1, Nmid−1×Nmid, Nmid−1×Nmid, and Nout×Nmid, respectively, where Nmid is selected as 1000. We randomly initialize all components of W1, W2, W3, and W4 using a Gaussian distribution with mean 0 and variance 2/Ncol as in [43], where Ncol denotes the column number of the matrix to initialize. The CFR can be said to be a kind of raw CSI, and the averaged PSD sequence can be said to be a kind of processed CSI. In this paper, as two representative machine learning approaches, we consider a machine learning approach using the CFR sequence (i.e., raw CSI) as training data and a machine learning approach using an averaged PSD sequence (i.e., preprocessed CSI) as training data. We name the two machine learning approaches ML1 and ML2, respectively. In ML1, the training data are selected from a CFR sequences of length NF×Ndata (i.e., Nin=NF×Ndata). The m-th training data for m=1,2,⋯,NTUE can be written as
(13)sm=HFm0THFm1T⋯HFmNdata−1TT,where
(14)HFmn=H0mnH1mn⋯HNF−1mnT.Figure 1
A block diagram of the proposed neural network structure to realize the forward signal propagation of machine learning.In ML2, the training data are selected from the averaged PSD sequences of lengthNdata/2 (i.e., Nin=Ndata/2). The m-th training data for m=1,2,⋯,NTUE can be written as
(15)sm=sm,0sm,1⋯sm,Nin−1T,where
(16)sm,u=1NF∑k=0NF−1Skmu.The following is a summary of the notations used in machine learning algorithms:(i)
xn represents the n-th component of a vector or sequence, x(ii)
x1ex2 represents the component-wise product of the two vectors, x1 and x2(iii)
Ak,: represents the k-th row of the matrix A(iv)
lenx represents the length of the x vector(v)
xk:l represents a column vector whose components are the components of the x vector from the k-th position to the l-th position(vi)
Given two vectorsx and y of equal length, x>y represents a vector whose n-th component is 1 if xn>yn and 0 if xn≤yn for n=1,2,⋯,lenx(vii)
randnN1,N2 represents a randomly generated N1×N2 matrix whose components follow a Gaussian distribution with zero mean and unit variance(viii)
·T represents the transpose operator(ix)
For input vectorx, fSumx represents a function whose output is given as ∑n=1lenxxn(x)
diagx represents a square matrix whose diagonal components are given by the components of the vector x(xi)
For two column vectors,x1 and x2, x1T;x2T represents a matrix created by stacking two row vectors x1T and x2TA full overview of the proposed machine learning algorithm is given in Algorithm1. In both ML1 and ML2, the forward signal propagation algorithm, the backward error propagation algorithm, and the parameter update algorithm are performed Nepoch×NTUE times in two “for” loops. In the forward signal propagation algorithm, the input vector xin is initialized with sm. The output vector yout resulting from xin is used to yield the final loss (or final cost) ξ. To compute ξ, the sequence of d=fSoftMaxtm that satisfies ∑n=1Ndata‍dn=1 is prepared in advance, where tm denotes the m-th target Doppler spread index sequence written as a one-hot encoded binary sequence consisting of multiple 0’s and a single 1. One example of tm can be written as
(17)tm=00⋯010⋯0T,where the index of a single 1 in tm represents the Doppler spread index defined in (11). The whole data that needs to be prepared in advance to run the proposed machine learning algorithm for Doppler spread estimation can be arranged in two matrices; first, the target Doppler spread index sequence matrix,
(18)T=t1T;t2T;⋯;tNTUET,and second, the training data matrix,
(19)S=s1T;s2T;⋯;sNTUET.To gather information aboutT, the BS can have a TUE move with a constant velocity v and have that TUE report its Doppler spread index, uD=T·fc·v/c, where c=3×108(m/sec) and fc is the carrier frequency. To gather information about S, the BS needs to estimate the CSI from that TUE and use the CSI to compute the training data. This method of collecting information by the BS entails system overhead. However, unless there are significant changes in Doppler spread statistics, reusing trained machine learning weights for Doppler spread estimation can reduce the overhead. Since it is difficult to actually collect the measured CSI at various times and places that can generate different Doppler spreads, in this paper, the collection of CSI necessary for machine learning or performance evaluation is limited to work through simulation.Algorithm 1: Overall machine learning algorithm.
1 Input:2S which denotes the data of the averaged PSD sequences3T which denotes the data of the target Doppler4 shift index sequences56 Fixed Parameters:7αA=0.001,αB=0.01,RD=0.0289 Weight Matrix Initialization:10W1=2/Nmid−1·randnNmid−1,Nin+111W2=2/Nmid−1·randnNmid−1,Nmid12W3=2/Nmid−1·randnNmid−1,Nmid13W4=2/Nout·randnNout,Nmid14γ1=1,β1=0,γ2=1,β2=0,γ3=1,β3=01516 fori∈1,2,⋯,Nepoch do17 form∈1,2,⋯,NTUE do18xin=Sm,:T19  t=Tm,:T20  Forward Signal Propagation (Algorithm2)21  Backward Error Propagation (Algorithm4)22  Parameter Update (Algorithm6)23 end24 endAlgorithm 2: Forward signal propagation.
1x~1=1;xin2x˘1=W1·x~13x¯1=fBatchNormx˘1,γ1,β14x^1=fReLUx¯15x2=fDropx^1,RD6x~2=1;x27x˘2=W2·x~28x¯2=fBatchNormx˘2,γ2,β29x^2=fReLUx¯210x3=fDropx^2,RD11x~3=1;x312x˘3=W3·x~313x¯3=fBatchNormx˘3,γ3,β314x^3=fReLUx¯315x4=fDropx^3,RD16x~4=1;x417x˘4=W4·x~418yout=fSoftMaxx˘4Algorithm 3: Batch normalization function.
x¯=fBatchNormx˘,γ,β1M=lenx˘2μ=1/MfSumx˘3σ2=1/MfSumx˘−μ⊙x˘−μ4ε=10−75x=x˘−μ/σ2+ε6x¯=γ·x+β7 returnx¯By the double “for” loops of Algorithm1, three subalgorithms, namely, “Forward Signal Propagation” (Algorithm 2), “Backward Error Propagation” (Algorithm 4), and “Parameter Update” (Algorithm 6) are executed iteratively. In Algorithm 2, the training data are propagated through the proposed neural network structure. Firstly, the input sequence xin is incremented by a bias factor 1 to produce the increased input sequence x~1. Then, x~1 is multiplied by W1 to produce x˘1. Then, x˘1 is batch-normalized to yield x¯1. The batch normalization function [39] is written in Algorithm 3. Then, x¯1 is activated by the ReLU function [41] to generate x^1. The ReLU function can be written as
(20)fReLUx¯=fReLUx¯1⋯fReLUx¯lenx¯T,where
(21)fReLUx¯=x¯ifx¯>0,0otherwise.Then,x^1 is applied to the dropout function [40] resulting in x2. Dropout is a technique to solve the overfitting problem by omitting hidden nodes with predefined probabilities. The dropout function can be written as
(22)x2=fDropx^1,RD=x^1⊙11−RD·zD,1,where RD denotes the dropout rate given by 0.02. zD,i for i=1,2,3 denotes a randomly generated vector of zeros and ones that satisfy the condition that the number of zeros is equal to the product of the value of RD and the number of components in x^1. In (22), for the purpose of maintaining the same power of x2 as in x^1, zRD,1 is multiplied by 1/RD−1. Then, x~2 is obtained by inserting a bias factor of 1 into x2. Similar to the process above, x~3 and x~4 will also be obtained. Then, W4 is multiplied by x~4 to produce x˘4. Finally, yout is created by activating x˘4 with SoftMax function ([42]), i.e., yout=fSoftMaxx˘4, where
(23)youtn=ex˘4n∑i=1Noutex˘4i,for n=1,⋯,Nout. In Algorithm 4, the back-propagation error is derived using the backward error propagation technique [44, 45]. To determine the final cost (or final loss) ξ, cross-entropy function [42] is used with two input vectors yout and d as
(24)ξ=−∑n=1Ndatadn·logyoutn.When an error of amplitude 1 is back-propagated from the output position of the cross-entropy function to then-th input position of the SoftMax function, a back-propagation error is induced as
(25)∂ξ∂x˘4n=youtn−dn.Therefore, if an error of amplitude 1 propagates back from the output position of the cross-entropy function to the input vector position of the SoftMax function, the resulting back-propagation error can be written as(26)e˘4=∂ξ∂x˘4=yout−d.Sincex˘4=W4·x~4, we derive
(27)dx˘4dx~4=W4T.Algorithm 4: Backward error propagation.
1d=fSoftMaxt2ξ=−∑n=1Ndatadn·logyoutn3e˘4=∂ξ/∂x˘4=yout−d4e~4=∂ξ/∂x~4=W4T·e˘45e^3=∂ξ/∂x^3=e~42:lene~4⊙zD,3/1−RD6e¯3=∂ξ/∂x¯3=x^3>0⊙e^37e˘3,dγ3,dβ3=fBackwardBatchNormx˘3,γ3,e¯38e~3=∂ξ/∂x~3=W3T·e˘39e^2=∂ξ/∂x^2=e~32:lene~3⊙zD,2/1−RD10e¯2=∂ξ/∂x¯2=x^2>0⊙e^211e˘2,dγ2,dβ2=fBackwardBatchNormx˘2,γ2,e¯212e~2=∂ξ/∂x~2=W2T·e˘213e^1=∂ξ/∂x^1=e~22:lene~2⊙zD,1/1−RD14e¯1=∂ξ/∂x¯1=x^1>0⊙e^115e˘1,dγ1,dβ1=fBackwardBatchNormx˘1,γ1,e¯1Algorithm 5: Backward error propagation of batch normalization function e˘,dγ,dβ=fBackwardBatchNormx˘,γ,e¯.
1M=lenx˘2μ=1/MfSumx˘3σ2=1/MfSumx˘−μ⊙x˘−μ4ε=10−75x=x˘−μ/σ2+ε6dβ=fSume¯7dγ=fSumx⊙e¯8e˘=γ/Mσ2+εM·e¯−fSume¯·1M−fSumx˘−μ⊙e¯/σ2+εx˘−μ9 returne˘,dγ,dβBy the chain rule [46] accompanying (26) and (27), back-propagation error generated at the input position of W4 can be written as
(28)e~4=∂ξ∂x~4=dx˘4dx~4·∂ξ∂x˘4=WT·e˘4.Taking into account the bias factor contained inx~4, we derive
(29)dx~4dx^3=0Nmid−111−RD·diagzD,3,where 0Nmid−1 denotes a zero-column vector of length Nmid−1. Therefore, the back-propagation error at the input position of the dropout function can be written as
(30)e^3=∂ξ∂x^3=∂x~4∂x^3·∂ξ∂x~4=e~42:lene~4⊙zD,31−RD.Sincex^3=fReLUx¯3, we can write
(31)∂x^3n∂x¯3m=x^3n>0ifn=m,0otherwise,for n,m=1,2,⋯,Ndata−1. By expanding (31), we derive
(32)∂x^3∂x¯3=diagx^3>0.From the result of (30) and (32), the back-propagation error at the input position of the ReLU function can be written as
(33)e¯3=∂ξ∂x¯3=∂x^3∂x¯3·∂ξ∂x^3=x^3>0⊙e^3.Algorithm 6: Parameter update.
1γ3=γ3−αA·dγ32β3=β3−αA·dβ33γ2=γ2−αA·dγ24β2=β2−αA·dβ25γ1=γ1−αA·dγ16β1=β1−αA·dβ17W4=W4−αB·e˘4·x~4T8W3=W3−αB·e˘3·x~3T9W2=W2−αB·e˘2·x~2T10W1=W1−αB·e˘1·x~1TThe backward error propagation function for batch normalization is defined in Algorithm5, whose detailed derivation is provided in the Appendix. Applying e¯3 to that function will result in the back-propagation error, e˘3, at the input vector position of the batch normalization function. Similar to the process above, e˘2 and e˘1 can also be obtained. In Algorithm 6, batch normalization parameters, γ1, β1, γ2, β2, γ3, and β3, and weight matrices, W1, W2, W3, and W4, are updated according to stochastic gradient descent [47]. The batch normalization parameters, γi and βi for i=1,2,3, are updated according to
(34)γi=γi−αA·dγi,βi=βi−αA·dβi,where αA denotes the learning rate chosen as 0.001. The values of dγi and dβi are generated by the backward batch normalization function, the detailed derivation of which is presented in the Appendix. W4 is updated according to the stochastic steepest descent method as
(35)W4=W4−αB·∂ξ∂W4,where αB denotes the learning rate chosen as 0.01. To find ∂ξ/∂W4, we first derive
(36)∂ξ∂W4i,j=∂x˘4∂W4i,j·∂ξ∂x˘4=∂x˘4∂W4i,j·e˘4.Sincex˘4=W4·x~4, we derive
(37)∂x˘4∂W4i,j=x~4j·uiT.ui denotes a vector of length lenx~4, where the i-th component is given as 1 and the other components are given as 0’s. By substituting the result of (37) into (36), we derive
(38)∂ξ∂W4i,j=x~4j·uiT·e˘4=e˘4i·x~4j.By stacking the results of (38) for all indices of i and j, we can write
(39)∂ξ∂W4=e˘4·x~4T.By substituting the result of (39) into (35), we derive
(40)W4=W4−αB·e˘4·x~4T.Similar to the process above, the formula for updatingWi for i=1,2,3 can also be derived as
(41)Wi=Wi−αB·e˘i·x~iT.Note that the BS can train the weight matrices offline using Algorithm1. The trained weight matrix can be used by the BS to estimate the Doppler spread of the target user equipment (UE) online. When the weight matrices are trained by ML1, the BS can find the Doppler spread of the target UE by computing the CFR sequence based on the CSI of the target UE and using the CFR sequence as xin in Algorithm 2 to generate x˘4. When the weight matrices are trained by ML2, the BS can find the Doppler spread of the target UE by calculating the averaged PSD sequence based on the CSI of the target UE for xin and applying xin to Algorithm 2 to find x˘4. Since the maximum component index of x˘4 represents the Doppler spread index uD, the Doppler spread of the target UE is given by fD=uD/T based on (11).


## 4. Simulation Results

In the simulation, OFDM system parameters are chosen asNF=128, L=5, and TS=0.001 sec. The number of pilot subcarriers is chosen to be NP=8, which means that the subcarrier spacing between two neighboring pilots is given by NF/NP=16. The machine learning settings for two machine learning approaches, ML1 and ML2, are summarized in Table 1.Table 1Machine learning settingsParameterValueParameterValueNTUE (training sequence #)2000Ndata=T/Ts1000Nin (input node #) in ML1128000Nout (output node #) in ML1100Nin (input node #) in ML2500Nout (output node #) in ML2100Hidden and output layer#4Epoch#2000Neuron # inW1, W2, and W31000Neuron# in W4100Initial values of weightsRandomDropout ratio2%Activation functionSoftMax (23)Cost functionCross entropy (24)To evaluate the Doppler spread estimation performance, we consider the root mean square error (RMSE)(42)RMSE=1NM∑m=1NMf∧Dm−fDm21/2,and the normalized RMSE (NRMSE)
(43)NRMSE=1NM∑m=1NMf∧Dm−fDmfDm21/2,where f^Dm denotes the estimated Doppler spread at the m-th TUE and NM denotes the Monte Carlo simulation number, which is set to 105. For comparison, the two previous Doppler estimation approaches introduced in [5, 6] are also considered and named PREV1 and PREV2, respectively. PREV1 estimates Doppler spread by referring to the maximum position of the maxima of the multiple PSD functions derived from channel multipaths [5]. PREV2 estimates Doppler spread by referencing the maximum position of the averaged PSD values greater than 10% of the maximum averaged PSD value [6].In Figure2, we evaluated the RMSEs of ML1, ML2, PREV1, and PREV2 in terms of SNR in a Rayleigh and isotropic channel scenario, where fD, Kr, κ, and α were assumed to be 40 Hz, 0, 0, and 0°, respectively. The same channel scenario was used to generate the CSI used for training ML1 and ML2 in the simulation. In the figure, the curve for ML2 is not shown because all RMSEs for ML2 are given as zero. This means that ML2 perfectly estimated the Doppler spread at all SNRs, unlike other approaches. PREV2 outperformed PREV1 and ML1 at SNRs above -0.8 dB, while PREV2 degraded as SNR decreased, performing worse than PREV1 and ML1 at SNRs below -2.5 dB. The reason for the poor performance of PREV2 at low SNR is that the maximum position of the averaged PSD values greater than 10% of the maximum averaged PSD value becomes more uncertain due to the PSD values adversely affected by small SNRs or large channel estimation errors. Although ML1 had higher implementation complexity than ML2, it performed much worse than ML2 because it could not effectively train the weight matrices using 2000(=NTUE) CFR sequences. This indicates the effectiveness of using averaged PSD sequences as training data in machine learning for Doppler spread estimation compared to using CFR sequences when the number of the training data is limited.Figure 2
Comparison of RMSE in terms of SNR whenfD was 40 Hz, Kr was 0, κ was 0, α0 was 0°, and α was 0°.Depending on the location and velocity of the UE and the scattering environment between the UE and the BS, channel characteristic variables such as SNR,fD, θ, Kr, and α can be randomly generated. For the purpose of making ML1 and ML2 work well anytime and anywhere, in the following simulations, ML1 and ML2 were trained based on the information of the channels generated assuming a mixed-channel scenario in which SNR, fD, θ, Kr, and α=α0 have a uniform distribution between -6 dB and 30 dB, between 0 Hz and 100 Hz, between 10° and 30°, between 0 and 2, and between -180° and 180°, respectively.In Figure3, we evaluated the RMSEs of ML1, ML2, PREV1, and PREV2 in terms of SNR for the channels whose characteristic variables, fD, Kr, θ, and α, have a uniform distribution between 0 Hz and 100 Hz, between 0 and 2, between 10° and 30°, and between -180° and 180°, respectively. α0 was chosen equal to α as in [36]. It was observed that ML2 outperformed the other approaches at all SNRs because ML2 effectively trained the weight matrices in the mixed-channel scenario. ML2 yielded RMSE values of less than 21 for SNR above 0 dB, which means that when the carrier frequency is 2.4 GHz and the SNR is above 0 dB, the amount of error in user speed estimation is less than 9.45 km/hr. On the contrary, ML1 performed worst for SNRs higher than -1 dB because the 2000 CFR sequences were not sufficient to properly train ML1’s weight matrices. Since ML1 used unprocessed CSI (i.e., CFR sequences) for Doppler spread estimation, it yielded a constant RMSE curve. PREV2 outperformed PREV1 at SNRs above 6 dB, while PREV2 degraded as SNR decreased, performing worse than PREV1 at SNRs below 6 dB. The reason for the poor performance of PREV2 at low SNR is that the maximum position of the averaged PSD values greater than 10% of the maximum averaged PSD value becomes more uncertain due to the PSD values adversely affected by small SNRs or large channel estimation errors.Figure 3
Comparison of RMSE in terms of SNR whenfD, Kr, θ, and α=α0 have a uniform distribution between 0 Hz and 100 Hz, between 0 and 2, between 10° and 30°, and between -180° and 180°, respectively.In Figure4, we evaluated the NRMSEs of ML1, ML2, PREV1, and PREV2 in terms of fD for the channels whose characteristic variables, SNR, Kr, θ, and α=α0, have a uniform distribution between -6 dB and 30 dB, between 0 and 2, between 10° and 30°, and between -180° and 180°, respectively. It was observed that ML2 outperformed the other approaches for all values of fD because ML2 effectively trained the weight matrices in the mixed-channel scenario. ML1 yielded a constant NRMSE curve because it used unprocessed CSI (i.e., CFR sequences) for Doppler spread estimation. ML1 performed worst when fD was greater than 53 Hz. PREV2 performed worse than PREV1 at all fD values because PREV2 tended to yield very large NRMSEs at low SNRs, which led the overall performance of PREV2 to be worse than PREV1 when the SNR was uniformly generated between -6 dB and 30 dB.Figure 4
Comparison of RMSE in terms offD when SNR, Kr, θ, and α=α0 have a uniform distribution between -6 dB and 30 dB, between 0 and 2, between 10° and 30°, and between -180° and 180°, respectively.In Figure5, we evaluated the RMSEs of ML1, ML2, PREV1, and PREV2 in terms of θ for the channels whose characteristic variables, SNR, fD, Kr, and α=α0, have a uniform distribution between −6 dB and 30 dB, between 0 Hz and 100 Hz, between 0 and 2, and between -180° and 180°, respectively. Note that κ representing the azimuth AOA width can be expressed as θ through κ=360°/θ/π2 [37]. It was observed that ML2 outperformed the other approaches at all values of θ because ML2 effectively trained the weight matrices in the mixed-channel scenario. PREV1 outperformed ML1 and PREV2 for all θ values. As θ increased, the performance of PREV2 decreased, which can be explained as follows. As θ increases, the channel autocorrelation takes a sharper shape according to (5). With sharper shaped channel autocorrelation, the PSD function forms a smoother shape according to (10), resulting in a smaller maximum PSD value. Therefore, the maximum position of the averaged PSD values greater than 10% of the maximum averaged PSD value becomes more uncertain by a larger θ, especially if the PSD values are adversely affected by a small SNR or large channel estimation error.Figure 5
Comparison of RMSE in terms ofθ when SNR, fD, Kr, and α=α0 have a uniform distribution between -6 dB and 30 dB, between 0 Hz and 100 Hz, between 0 and 2, and between -180° and 180°, respectively.In Figure6, we evaluated the RMSEs of ML1, ML2, PREV1, and PREV2 in terms of Rician K-factor Kr for the channels whose characteristic variables, SNR, fD, θ, and α=α0, have a uniform distribution between -6 dB and 30 dB, between 0 Hz and 100 Hz, between 10° and 30°, and between -180° and 180°, respectively. It was observed that ML2 outperformed the other approaches at all Kr values because ML2 effectively trained the weight matrices in the mixed-channel scenario. PREV2 degraded as Kr increased, which can be explained as follows. As Kr increases, the effect of the fixed channel component becomes more significant than the fading (or fluctuating) channel component. The greater the effect of the fixed channel component, the more imprecise it is to find the Doppler spread by referencing the maximum position of the averaged PSD values greater than 10% of the maximum averaged PSD value. Therefore, the performance of PREV2 degrades as Kr increases. At high SNR, PREV1, which estimates the Doppler spread by referring to the maximum position of the maxima of the multiple PSD functions derived from channel multipaths, outperformed ML1 and PREV2 because at least one channel multipath can have a strong fading component with high probability [5] and PREV1 could benefit from that multipath. Very small values of Kr cause very large RMSEs in PREV1 and PREV1. This is because for very small Kr values, a low SNR or a large channel estimation error greatly deteriorates their Doppler spread estimation performance and thus causes a large overall Doppler spread estimation error even if the SNR is uniformly generated.Figure 6
Comparison of RMSE in terms ofKr when SNR, fD, θ, and α=α0 have a uniform distribution between -6 dB and 30 dB, between 0 Hz and 100 Hz, between 10° and 30°, and between -180° and 180°, respectively.In Figure7, we evaluated the RMSEs of ML1, ML2, PREV1, and PREV2 in terms of mean direction of the azimuth AOA α for the channels whose characteristic variables, SNR, fD, θ, Kr, have a uniform distribution between -6 dB and 30 dB, between 0 Hz and 100 Hz, between 10° and 30°, and between 0 and 2, respectively. α0 was chosen equal to α as in [36]. It was observed that ML2 outperformed the other approaches for most α values. However, for α values with 165°<∣α∣<180°, ML2 performed worse than PREV1 and PREV2. The reason for this phenomenon is that ML2 trained the weight matrices with the aim of improving the overall Doppler spread estimation performance for all α values while effectively avoiding the overfitting problem. It was observed that PREV1, PREV2, and ML2 produced RMSE curves that fluctuate in terms of α because PREV1, PREV2, and ML2 used PSD sequences for Doppler spread estimation. Notice the fact that the maximum position of the PSD function fluctuates due to the cosα0 term as shown in (6). However, ML1 yielded a constant RMSE curve because it used unprocessed CSI (i.e., CFR sequences) for Doppler spread estimation. Because PREV1 and PREV2 assumed α0=0° and estimated the Doppler spread, the performance of PREV1 and PREV2 deteriorated as α0=α value changed from 0° to 90°.Figure 7
Comparison of RMSE in terms ofα=α0 when SNR, fD, θ, and Kr have a uniform distribution between -6 dB and 30 dB, between 0 Hz and 100 Hz, between 10° and 30°, and between 0 and 2, respectively.In Figures8(a) and 8(b), we investigated the impact of NTUE on the performance of ML2. In Figure 8(a), we evaluated the RMSE in terms of SNR using the same simulation parameters as given in Figure 3. In Figure 8(b), we evaluated the NRMSE in terms of fD using the same simulation parameters as given in Figure 4. It can be seen from Figures 8(a) and 8(b) that the RMSE and NRMSE performance improved with increasing NTUE. However, increasing NTUE to a value greater than 2000 produced only minor gains. From the above simulation results, we conclude that 2000 is a good choice for the NTUE value in ML2.Figure 8
(a) Comparison of RMSE of ML2 in terms of SNR for various values ofNTUE by using the same simulation parameters as given in Figure 3. (b) Comparison of NRMSE of ML2 in terms of fD for various values of NTUE by using the same simulation parameters as given in Figure 4.
(a)(b)


## 5. Conclusion

A neural network structure in machine learning for Doppler spread estimation was proposed for an OFDM system. The weight matrix update algorithm was derived with the help of the backward error propagation technique and the stochastic steepest descent method. Numerical simulations have shown that averaged PSD sequences can be used to effectively train machine learning weights for Doppler spread estimation. The proposed machine learning approach using averaged PSD sequence as training data outperformed other Doppler spread estimation approaches under various channel conditions.



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*Source: 5586029-2021-09-22.xml*