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3.1 Channel estimation
The channel estimation technique for MIMO-OFDM systems is crucial issue as a good channel estimation technique results in less estimation errors and consequently good signal detection. Least Square (LS), Least Mean Square (LMS), and Minimum Mean Square Error (MMSE) algorithms are of the most known training based channel estimation technique. In this work we adopt the LS algorithm, because it is the simplest and it is performance approaches MMSE for higher SNR values. Another reason to use LS is because our focus in this paper is to design effective prediction technique able to overcome the dynamics of the channel, and mitigate the feedback delay effect in MIMO-OFDM systems, not on designing a high performance channel estimation technique. Comb-type pilot structure channel estimation is adopted in this work as it is shown in [Ref_A1] - [Ref_A2] that comb-type pilot structure performs better than block type structure for fast fading channels.pilot symbols with known data are uniformly inserted into each OFDM symbol at each transmit antenna. The pilot sequences from each antenna are assumed to be orthogonal with other sequences from other antennas.
Then the frequency domain signal at thetransmit antenna can be expressed as where and is the pilot carrier value on the transmit antenna. At the receiver the pilot signals are firstly extracted from the received signal, and the transfer function at the pilot subcarriers is then estimated using the received signal and the predetermined pilot values. Letbe the channel frequency response of the pilot subcarriers between the transmit antenna and the receive antenna. Then the LS estimation of the channel at pilot subcarriers between the transmit and the receive antennas can be expressed as
Where is the received signal at the pilot subcarrier of received from the transmit antenna, and is the signal transmitted from the transmit antenna at the pilot subcarrier.
3.2 Channel Prediction
As we mention before, in this paper we consider a time-varying environment where the channel changes from one OFDM symbol to another. Hence the precoding matrix chosen at the receiver from the current OFDM symbol becomes outdated due to signal processing and feedback delay. Consequently the outdated information result in system performance degradation. As a solution to this problem a channel prediction scheme based on Kalman filter is proposed to overcome the performance degradation in the system performance due to delay in the feedback channel. Kalman filter is used to predict the channel state for each subcarrier utilizing the collection of the past estimated channel values. The predicted channels state is used to design the precoding matrices for the next OFDM symbols, and the indices of the precoding matrices are fed back to the transmitter through the limited feedback channel.
It is well known that a dynamic system can be modeled as an autoregressive (AR) process of order [ ]. An order AR model for is presented as:
where are the coefficients of the AR process, and is a white Gaussian process vector. The AR process coefficients can be found by solving the Yule-Walker equation , however our aim in this paper is not to find these coefficients, and therefore they assumed to be known. The Yule-Walker equation is given by
Where is the autocorrelation matrix which assumed to be non-singular. The choice of is a trade off between the accuracy of the model (e3) and the parameters estimation complexity. For simplicity in this paper we model the channel as a first order AR process, furthermore a first order AR model provides an adequate model for time varying channels
According to Jakes model:
Where represents zeroth-order Bessel function of the first kind, and is the maximum Doppler frequency, and is the OFDM symbol duration.
The input output relationship at the pilot subcarrier can be written as where is the received signal vector at the pilot subcarriers of the received from the transmit antenna, is a diagonal matrix with the transmit pilot signal vector being its diagonal, and is the white Gaussian noise vector.
In the preceding section we estimate the channel at the pilot subcarriers between each transmitter and each receiver. In this section Kalman filter is employed to predict the future state of the channel at the pilot subcarriers based on the collection of the estimated channels. In order to employ Kalman filter the state space equations are needed. Combining of and give the state space model for the channel between the transmitterand the receiver as
Where the first equation represents the process equation and the second represents the measurement equation, denotes the time varying transition matrix, and is known measurement matrix.and are the process and the measurement noise vectors respectively. The noise vectors and are mutually uncorrelated with noise sequences with covariance matrices and .
Kalman filter is well described in [Monson Hayes] and [Simon Haykin]. Using the estimated channel values given by and the measurement data from, the channel at the pilot subcarriers of the next OFDM symbol can be obtained using the following recursive computation:
Where is the Kalman gain, is the error correlation matrix, and is the innovation vector. The predicted channel states can be found as
3.3 Interpolation previous
In the last two sections we estimate the channels at the pilot subcarrier for the current OFDM symbol using the LS estimation based on Comb-type pilot distribution. The future state of the channel at the pilot subcarriers were also obtained using one step Kalman predictor. To estimated and predicted the next states of the channels at the data subcarriers an efficient interpolation technique is needed. Different interpolation techniques have been investigated in [Ref_A1], [Ref_A2]; however in this work we used the Time Domain Interpolation (TDI) technique because it outperforms linear interpolation in terms of BER. [Ref_A2].
Using the estimated channel frequency response vectorof the pilots between the transmit antenna and the receive antenna is given by (e2), and the predicted channel at the pilot subcarriersis given by (e17). The channel frequency response at the data subcarriers can now be found using time-domain interpolation by converting and to time domain vectors and using Inverse Discrete Fourier Transform (IDFT), Zero pads each of and to point, and finally transform the zero padded time domain vectors back to frequency domain using Discrete Fourier Transform (DFT).