# Method of Doppler Frequency Spread Estimation for High-Mobility OFDM Systems

Authors: Feng Zhao; Xihong Sun
Journal: International Journal of Distributed Sensor Networks
 (2015)

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## Abstract

A method of Doppler frequency spread (DFS) estimation algorithm is discussed which is suitable for high-mobility Orthogonal Frequency Division Multiplexing (OFDM) systems. DFS is a main factor which may mitigate the performance of system. The proposed method is based on autocorrelation function algorithm of received signal which utilizes cyclic prefix (CP) to estimate Doppler spread. The estimation algorithm uses least square (LS) method between the autocorrelation function of received signal and the zero-order Bessel function which is approximated by the expansion of power series. Then the estimation bias is analyzed and using polynomial fitting method can effectively correct estimation error at different DFS, respectively. Compared with the previously proposed method, the estimation performance of DFS in the proposed method has better performance. Simulation results show that when sufficient elements are used, the proposed scheme can effectively estimate DFS and can be applied into high-mobility OFDM systems.


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## Body

## 1. Introduction

The Orthogonal Frequency Division Multiplexing (OFDM) which is regarded as a key technology has been applied for the fourth generation wireless communication systems. Because of its advantages, such as antifrequency selective fading, resistance to strong narrowband interference, and high spectrum efficiency [1], OFDM has a wide application prospect in the Time Division Long Term Evolution (TD-LTE) mobile communications systems [2].The TD-LTE mobile communication systems need to support mobile communication systems with terminals moving at speed of up to about 350 km/h [3]. When we employ OFDM technology in TD-LTE mobile communication systems, the high speed between transmitters and receivers can cause some bad effects; one of the main drawbacks is DFS which can destroy the orthogonality between subcarriers. Besides, it also brings intercarriers interference (ICI) which will cause the degradation of BER performance for OFDM communication systems [4, 5]. If Doppler spread can be estimated at the receiver, we can improve the performance of the communication systems. Therefore, estimation of DFS is inevitable for the application of OFDM systems in high-mobility environment.In the existing technical literatures, researchers have proposed several Doppler frequency estimation methods for OFDM systems; the main DFS estimation methods can be classified as follows. In [6], a method based on the maximum-likelihood is employed in time-varying multipath channels. This method requires amounts of data and complexity computation. Another method based on the level crossing rate (LCR) method is presented in [7, 8] whose estimation accuracy is based on the complexity of estimator. Channel estimation is used to estimate DFS which is very perfect in lower Doppler frequency in [9]. But if the channel estimation is bad, it will reduce the accuracy of DFS. A conditional method of autocorrelation function of received signal is displayed in [10]; the estimation results show that the deviation between estimated value and real value is big; in addition, the method does not reflect high-mobility environment. In [11], this paper proposes a parametric spectrum estimation of DFS which considers two-ray arriving at the receiver. The DFS is estimated by angle of arrival which can be obtained from signal space decomposition. A method of DFS dependent on phase difference is used in [12]. In [13–15], the autocorrelation of channel response can also be used to estimate Doppler spread in both time domain and frequency domain. Although these methods can estimate DFS to some extent, all of these channel response methods need pilot information and complexity computation for accurate DFS.This paper proposes a method based on cyclic prefix for OFDM systems in time-varying fading channel to achieve the accurate Doppler frequency spread. The proposed method utilizes LS method between autocorrelation function of received signal and approximated zero-order Bessel function of first kind, and then method of quadratic polynomial fitting can reduce the error between the real Doppler spread and the estimated Doppler spread. Compared with the conditional method, the proposed scheme can improve the accuracy of DFS. The remainder of the paper is organized as follows. Section2 presents the OFDM system model. Section 3 discusses the proposed DFO estimation algorithm in detail. The results of computer simulations are described in Section 4. Finally, the paper concludes with a short summary in Section 5.


## 2. System Model

We consider the scenarios of high mobile communication, such as communications between high-speed train and the base stations in the mountainous environment. The high-speed train receives the downlink signals from the base station (BS). As shown in Figure1, without loss of generality, let N be the number of orthogonal subcarriers; the original signal is modulated by N-point inverse Fast Fourier Transform (IFFT). In order to avoid intersymbol interference (ISI), a copy of the last samples which is longer than the length of the maximum channel delay is inserted at the front of the useful symbol’s time. We make it as the guard interval between two adjacent symbols. Figure 2 is the cyclic prefix structure. An integrated OFDM symbol is composed of useful signal and cyclic prefix.Figure 1
Rayleigh channel model for high-mobility communication.Figure 2
Cyclic prefix structure.Frequency domain signal is denoted byX=[X0,X1,…,X(N-1)], and then the time-domain transmitted signal can be written as [15](1)xmn=1N∑k=0N-1Xmkexp⁡j2πknN,-M≤k≤N-1,where Xmk denotes the frequency domain at the kth subcarrier of the mth OFDM symbol. As shown in Figure 2, M is the cyclic prefix length and N is the number of useful symbols. So for a complete signal, its length is L=M+N.Figure3 is the block diagram of DFS estimation process [16]. At first, at the transmitter, random input data is modulated by quadrature phase shift modulation (QPSK), and then the serial information sequence undergoes the serial-to-parallel (S/P) conversion. After IFFT, the output signals of the OFDM transmitter will be orthogonal with each other. A cyclic prefix is appended in order to mitigate the multipath effects and is regarded as a guard interval between two adjacent symbols. Then the data is converted by parallel-to-serial conversion, after transmitting over Rayleigh fading channels. AWGN and Doppler frequency spread are superimposed on the data. Then the autocorrelation function of cyclic prefix signals can be calculated and DFS can be estimated by LS estimation. Polynomial fitting can mitigate the DFS estimation error.Figure 3
The block diagram of Doppler spread estimation process for OFDM systems.In this paper, the environment we consider is high-mobility train in the mountainous environment; the obvious features of wireless channel are frequency selective fading and time selective fading. We consider a communication system employing a single transmit and receive antenna. In the mountainous environment, we assume the channel is Rayleigh channel and the signal has propagated throughp different limited paths with each having independent phase shift, attenuation, and delay. So in the paper the channel transfer function which is based on time-varying multipath can be defined as [17](2)hn,τ=∑p=0p-1σpexp⁡⁡j2πnTcfdNδτ-τp,where σp is the pth channel amplitude determined by path loss, τp denotes the time delay of pth delay path, fd obeys uniform distribution from -fmax⁡ to fmax⁡, and fmax⁡ denotes the maximum value of DFS. We assume that Doppler frequency spread fd does not change for each path. Doppler frequency spread is related to the velocity υ and the carrier frequency fc, and the formula of fmax⁡ can be written as(3)fmax⁡=υfcc∗cos⁡θ,where υ is the relative velocity between transmitter and receiver and c≈3∗108 m/s is the speed of light. θ is the angle between the direction of movement of the mobile station and the direction of the incident wave.After passing through multipath channels, the received signal is the superposition of all multipath components at the output of the received antennal. So themth received time-domain OFDM signal ym(n) is the linear convolution of the channel input signal and the discrete-time channel response, and the received signal corrupted by additive white noise can be written as(4)yn=∑p=0p-1hpnx(n-τp)+wn,where w(n) is the zero-mean complex Additive White Gaussian Noise (AWGN) and it is independent of the transmitted signal. Based on (4), the corresponding received frequency domain signal Ym(k) can be expressed as(5)Ymk=1N∑n=0N-1ymnexp⁡-j2πnkN.


## 3. DFO Estimation

### 3.1. Algorithm Principle

Every received signal can be described as a sequence of complex and the form of discrete sampling points can be expressed as(6)yk=yi(k)+jyq(k)=uk+wik+jvk+wqk,where u(k) and v(k) are the real component and image component of received signal (excluding noise) and, respectively, E[v2(k)]=E[u2(k)]=σ2/2, E[wi2(k)]=E[wq2(k)]=N0B/2, E[·] is the average operation, yi(k) is the real component of yk, and yq(k) is the image component of y(k).After joining cyclic prefix, integrated OFDM symbol contains two same parts, so Doppler frequency spread can be estimated based on these two same parts. Assuming that OFDM’s synchronization has been reached and the received signal is transmitted over a correlator, the autocorrelation function of thekth cyclic prefix symbol can be described as(7)CNTs,k=12∑k=0M-1yikyik+Lyi2k+yqkyqk+Lyq2k.Based on Central Limited Theorem, the autocorrelation function of received signal can also be calculated and displayed as [10](8)Rii=Rqq=Eyikyik+τ=σ22J02πfdτ+N0B2δτ,where J0(·) is the zeroth order Bessel function of the first kind and δ(τ) is the impulse response function with a fixed noise bandwidth. Signal to noise (SNR) is available in this paper, so from (7) and (8), DFS can be obtained from the following relationship:(9)J⌢02πf⌢dNTs=1+1SNR1K∑k=0K-1CNTs,k.Form the analysis of theory, normalized autocorrelation function has a relationship with zero-order Bessel function of the first kind. By calculating the autocorrelation function, Doppler frequency shift can be estimated by least square:(10)Lf⌢d=argminJ02πfdNTs-1+1SNR1K∑k=0K-1CNTs,kssssssshss·∑k=0K-1CNTs,k2.


### 3.2. Correct Estimation Error

Although Doppler frequency spreadf⌢d can be got from (10), the deviation between estimated Doppler frequency spread and the actual Doppler frequency spread value is large. That is because the estimation error when computing the inverse Bessel function is difficult to avoid, so we need to correct estimation errors. Polynomial fitting method can effectively correct estimation error; the modified formula can be displayed as follows:(11)f^d=a0+a1fd+a2fd2+⋯+ap-1fdp-1,where fd is the real Doppler frequency spread, and the error of estimated Doppler frequency spread is large and it can be modified by polynomial fitting. The function in (11) we constructed is used to mitigate this difference and then the modified Doppler frequency spread f^d is closer to the real Doppler frequency. So we exploit the method of polynomial fitting to mitigate the difference [8]. In MATLAB software, based on mathematical function, the command of an=polyfit(fd,f⌢d,n) can be realized, where fd and f⌢d are real DFS and estimated DFS. n is the order of the polynomial fitting. When n is chosen as an appropriate value, the coefficients an can be obtained. Here we just give an example; set n=3 and SNR = 6; by this statement, we can get some coefficients which are a3, a2, a1, and a0. The relative prediction formula can be displayed as(12)f^d=-0.1654fd3+1.0628fd2-0.1134fd+0.3916.By (12), the estimated error is mitigated and the corrected DFS is more accurate.


### 3.3. The Feature of Bessel Function

Based on the characteristic of Bessel function, it can be seen that Bessel function is not a monotonic curve and is a nonlinear curve. Therefore, these two aspects can affect the accuracy of the simulation. Because inverse Bessel function is also not a specific formula, we can use look-up table (LUT) method to obtain the Doppler frequency spread. Bessel function and its inverse function graphs are shown in Figure4; we can see from the figure that zeroth Bessel function of the first kind is not a monotonic function. So choose an appropriate N to ensure that J0(x) is monotonic within a certain interval range.Figure 4
J
0
(
x
) and its inverse function.
(a)
(b)Based on the explanation above, the property of Bessel function is nonlinear. So the independent variable of zeroth Bessel function of the first kind in (10) should be located in the estimated monotonic range; that is, it must be limited between 0 and x=min⁡(J-10(x)). In order to estimate the nonlinear property, J0(x) can be approximated as f(x) by the expansion of power series which is shown in Figure 5.Figure 5
Approximation of Bessel function.


## 3.1. Algorithm Principle

Every received signal can be described as a sequence of complex and the form of discrete sampling points can be expressed as(6)yk=yi(k)+jyq(k)=uk+wik+jvk+wqk,where u(k) and v(k) are the real component and image component of received signal (excluding noise) and, respectively, E[v2(k)]=E[u2(k)]=σ2/2, E[wi2(k)]=E[wq2(k)]=N0B/2, E[·] is the average operation, yi(k) is the real component of yk, and yq(k) is the image component of y(k).After joining cyclic prefix, integrated OFDM symbol contains two same parts, so Doppler frequency spread can be estimated based on these two same parts. Assuming that OFDM’s synchronization has been reached and the received signal is transmitted over a correlator, the autocorrelation function of thekth cyclic prefix symbol can be described as(7)CNTs,k=12∑k=0M-1yikyik+Lyi2k+yqkyqk+Lyq2k.Based on Central Limited Theorem, the autocorrelation function of received signal can also be calculated and displayed as [10](8)Rii=Rqq=Eyikyik+τ=σ22J02πfdτ+N0B2δτ,where J0(·) is the zeroth order Bessel function of the first kind and δ(τ) is the impulse response function with a fixed noise bandwidth. Signal to noise (SNR) is available in this paper, so from (7) and (8), DFS can be obtained from the following relationship:(9)J⌢02πf⌢dNTs=1+1SNR1K∑k=0K-1CNTs,k.Form the analysis of theory, normalized autocorrelation function has a relationship with zero-order Bessel function of the first kind. By calculating the autocorrelation function, Doppler frequency shift can be estimated by least square:(10)Lf⌢d=argminJ02πfdNTs-1+1SNR1K∑k=0K-1CNTs,kssssssshss·∑k=0K-1CNTs,k2.


## 3.2. Correct Estimation Error

Although Doppler frequency spreadf⌢d can be got from (10), the deviation between estimated Doppler frequency spread and the actual Doppler frequency spread value is large. That is because the estimation error when computing the inverse Bessel function is difficult to avoid, so we need to correct estimation errors. Polynomial fitting method can effectively correct estimation error; the modified formula can be displayed as follows:(11)f^d=a0+a1fd+a2fd2+⋯+ap-1fdp-1,where fd is the real Doppler frequency spread, and the error of estimated Doppler frequency spread is large and it can be modified by polynomial fitting. The function in (11) we constructed is used to mitigate this difference and then the modified Doppler frequency spread f^d is closer to the real Doppler frequency. So we exploit the method of polynomial fitting to mitigate the difference [8]. In MATLAB software, based on mathematical function, the command of an=polyfit(fd,f⌢d,n) can be realized, where fd and f⌢d are real DFS and estimated DFS. n is the order of the polynomial fitting. When n is chosen as an appropriate value, the coefficients an can be obtained. Here we just give an example; set n=3 and SNR = 6; by this statement, we can get some coefficients which are a3, a2, a1, and a0. The relative prediction formula can be displayed as(12)f^d=-0.1654fd3+1.0628fd2-0.1134fd+0.3916.By (12), the estimated error is mitigated and the corrected DFS is more accurate.


## 3.3. The Feature of Bessel Function

Based on the characteristic of Bessel function, it can be seen that Bessel function is not a monotonic curve and is a nonlinear curve. Therefore, these two aspects can affect the accuracy of the simulation. Because inverse Bessel function is also not a specific formula, we can use look-up table (LUT) method to obtain the Doppler frequency spread. Bessel function and its inverse function graphs are shown in Figure4; we can see from the figure that zeroth Bessel function of the first kind is not a monotonic function. So choose an appropriate N to ensure that J0(x) is monotonic within a certain interval range.Figure 4
J
0
(
x
) and its inverse function.
(a)
(b)Based on the explanation above, the property of Bessel function is nonlinear. So the independent variable of zeroth Bessel function of the first kind in (10) should be located in the estimated monotonic range; that is, it must be limited between 0 and x=min⁡(J-10(x)). In order to estimate the nonlinear property, J0(x) can be approximated as f(x) by the expansion of power series which is shown in Figure 5.Figure 5
Approximation of Bessel function.


## 4. Simulating Results

Computer simulation is performed to evaluate the DFS. In this section, we will validate the performance of modified scheme in high-mobility OFDM communication systems. The main parameters of the investigated OFDM systems are set. We assume an OFDM system withN=2048 subcarriers, cp = 256 is the length of cyclic prefix, the carrier frequency of the OFDM system is fc=3 GHz, and bandwidth is B=10 MHz. The maximum relative speed between transmitter and receiver is about v=350 km/h which means the maximum Doppler frequency spread is approximately equal to 1 kHz and there are p=6 multipaths. In this simulation, we set n=5 and the model of multipath channel is shown in Table 1.Table 1
The reference model of multipath channel.
Path number
Relative time (us)
Relative power (dB)
1
0
0
2
0.31
−1
3
0.71
−9
4
1.09
−10
5
1.73
−15
6
2.57
−20Figure6 shows the performance of the proposed Doppler frequency spread estimation method under different SNR as 6 dB, 15 dB, and 20 dB when the order n=5. The real Doppler spread is changed from 0 kHz to 1 kHz and, respectively, the speed is from 0 km/h to 350 km/h. From the figure we can see that the estimated Doppler frequency spread is close to the real Doppler frequency spread. Even when the real DFS is up to 1 KHz which means the speed is 350 km/h, the simulation result is also very close to the ideal value.Figure 6
Estimatedf⌢d versus real Doppler spread at different SNR when n=5.The normalized mean square error (NMSE) is employed in the evaluation of accuracy for the modified estimation method to display the performance of the proposed method:(13)NMSE=1Q∑n=0QFdn-fdfd2,where Q is the number of samples to be used in the evaluation of NMSE and Q=1000.Figure7 shows the comparison about NMSE of Doppler frequency spread between the conventional cyclic prefix algorithm in [10] and the modified method. As we can see from this figure, when the relative Doppler frequency spread is small, the conventional cyclic prefix method is not suitable and the NMSE is higher than the proposed method. Although at some point the performance of proposed method is poor, from the entire graphics view, the NMSE of the method in this paper is lower than conditional cyclic prefix method. The proposed MSE is below 1% when DFS is 1 kHz.Figure 7
NMSE comparison between modified and conventional method.Figure8 shows the comparison of estimation accuracy between the conventional cyclic prefix algorithm and the modified method with changing the real Doppler frequency shift from 0.1 kHz to 1 kHz. In this figure, it can be observed that the deviation value of the modified method is smaller than conditional cyclic prefix algorithm. Although simulation results are not stable, from the entire graphics view, the modified DFS estimation method has better performance.Figure 8
Comparison between conventional and modified method based on different SNR.


## 5. Conclusion

A new Doppler frequency spread estimation method used in high-mobility OFDM communication systems is displayed in this paper which has a large estimation range in high-mobility situation. The proposed method is based on autocorrelation algorithm of received signal which utilizes cyclic prefix to estimate Doppler spread. By comparing the ideal autocorrelation function value with the estimated autocorrelation function of the received signal, we can use LS method to estimate Doppler frequency spread. However, error exists in estimated DFS; in order to mitigate the estimation bias, polynomial fitting method we used can effectively correct estimation error. The modified DFS is used for minimizing the impact of DFS on the BER performance. The estimation error analysis and the simulations results show that the proposed scheme has better performances and can be applied into high-mobility OFDM communication systems.



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