Two-Stage Channel Frequency Response Estimation in OFDM Systems

© 2020 The Authors. This article is licensed under a Creative Commons Attribution 4.0 License Abstract. This paper proposes two-stage channel frequency response estimation algorithm in communication systems with OFDM technology. Algorithm is based on Kalman filter. Pilots from current and previous OFDM symbols are used for channel estimation. At the first stage data is processed in time and frequency directions. Pilots from the current OFDM symbol are filtered and at the position, where the pilots from the previous OFDM symbols should be placed, predictions are made. Predictions are based on the pilots and channel correlation characteristics. The data processing carried out on both sides relative to the array of processed data in frequency direction and on one side at processing in time direction. The results of processing are optimally combined at the second stage. The autoregressive process was used as a channel model. The analysis of the developed algorithm carried out on a model example by statistical modeling. Modeling showed that application of designed algorithm allows reducing the standard deviation of the estimation error of channel frequency response. The efficiency of designed algorithm studied using Rayleigh channel with Doppler spectrum described by Jakes model. The autocorrelation characteristics of the channel were considered as known. Modeling showed a decrease in the probability of a bit error during reception using the proposed algorithm. It is also shown that an increase in the order of the autoregressive model reduces the error in estimating the frequency response of the communication channel.


INTRODUCTION
One of the widely used technologies for transmitting information in channels with frequency selective fading is orthogonal frequency division multiplexing (OFDM). The main advantages of OFDM are high spectrum efficiency and ability to cope with severe channel conditions without complex equalization filters [1]. One of the problems that must be solved when demodulating OFDM signals is the estimation of communication channel parameters. The probability of error in reception depends on the accuracy of the channel estimate: the more accurate the estimate, the less error probability can be achieved during data transmission. This problem also exists in narrow-band communication channels, for which optimal compensation algorithms for frequency-selective distortion are proposed in [2]. Pilot signals are widely used in channel estimation. They are located in positions known to the receiver [3]. The simplest methods for estimating channel characteristics by pilots are the least square (LS) method and the method of minimum mean square error (MMSE). As established previously [4], the MMSE method has better accuracy characteristics, but it requires knowledge of the channel correlation matrix and is more laborious. A comparison of these two methods on the accuracy of estimating the channel characteristics is considered in [5]. The pilot layout in the form of a parallelogram grid is widely used in practice [6]. In case of static channels for frequency response estimation can be used the pilots from current and previous symbols. However, if channel parameters are variable in time, using the pilots from previous symbols is impossible because they contain outdated information. In [7] an algorithm for estimating the channel frequency response based on the Kalman filter with using pilots only from the current OFDM symbol was proposed. The essence of the algo-Section "Technics" 1002 rithm is that the pilots from the current symbol filtered and at the position, where the pilots from the previous OFDM symbols should be placed, predictions are made. All predictions are based on the pilots from the current symbol and the channel correlation characteristic. The data processing carried out in the frequency direction on both sides relative to the array of processed data with the subsequent optimal combination of the results. We propose to additionally take into account the predictions based on the pilots from previous OFDM symbols. Thus, the data will be processed in both frequency and time directions at the first stage with the subsequent optimal combination of the results at the second stage ( Figure 1).

Figure 1 -The essence of designed algorithm
It is proposed to use the autoregressive (AR) process as a channel model. Studies in this area were considered in [8,9,10]. In particular, a method for estimating the frequency response based on the Kalman filter for a channel model in the form of a first-order AR process was propose in [8].
Problem statement. After execution the analog to digital (ADC) conversion and discrete Fourier transformation (DFT) with length N the received signal takes the next form (1): is a vector of received signal; Denote the coordinates of all necessary points for channel frequency response estimation by plu- Thus the task is to estimate channel frequency response in e  positions according to available Development of the algorithm. Any stationary random process can be represented as an infinite order AR process [8]. This property allows representing the elements of channel frequency response vector as the following autoregressive (AR) process (3): For filtering and extrapolation of the n-th OFDM symbol along the data array from left to right (forward), we introduce the system state vector for the j-th subcarrier of the n-th OFDM symbol F is the matrix of system evolution which has the form (5): The observation equation according to (2) takes next form (6): According to the model (4), (6), algorithm of filtration and extrapolation describes by the next equations (7-10): * , , 1   Following the technique given in [11] the algorithm for optimal combining the processing of the results in forward and backward directions at point j is described by the next equations (12-15):  F is the matrix of system evolution which has the next form (18): The observation equation takes next form (19): where  is a Gaussian white noise.
Model parameters:    As can be seen from the plots, theoretical and experimental results agree, which confirms the validity of the algorithm. Moreover, the algorithm (12)-(15) allows to decrease the standard deviation of the estimation error by 50% in comparison with algorithm (7)- (10). Application of algorithm (24)-(27) allows to further reduce the standard deviation of the estimation error by 10% compared with algorithm (12)-(15). The efficiency of designed algorithm is studied using Rayleigh channel with Doppler spectrum that describes by Jakes model [13]. Coefficients of AR process are obtained by Yule-Walker equa-Section "Technics" 1006 tions are obtained from channel autocorrelation characteristic. The autocorrelation characteristic of the channel is considered known. Figure 5 shows dependence of the mean square error (MSE) of the channel estimate on variance of the noise for communication channel with 20 rays, 10 e-6 s maximum delay, 10 Hz Doppler shift. The autoregressive models of the first and seventh order in frequency and time directions were used.  As can be seen from the above plots, an increase in the order of the autoregressive model reduces the error in estimating the frequency response of the communication channel. Figure 7 shows dependence of the bit error rate (BER) on signal to noise ratio (SNR) for channel with 10 Hz Doppler shift when using QPSK modulation on OFDM symbol subcarriers.

CONCLUSIONS
Thus, the proposed algorithm makes the optimal combination of the three results of estimation the frequency response of the communication channel, which is represented by the autoregressive model. The estimation is based on the Kalman filtering theory using pilot signals and the known correlation characteristics of the communication Section "Technics" 1007 channel. Two estimates for the point, at which the combining is made, are obtained using the pilot signals of the current symbol by filtration and extrapolation in both sides in frequency direction. The third estimate is obtained using the measurements obtained in the previous symbols, which corresponds to the coordinate of the given point. Modeling showed a decrease in the probability of a bit error during reception using the proposed algorithm. A further area of the research is the development of an algorithm for jointly estimating the frequency response of a communication channel and decision making on transmitted symbol.