In this paper, a low complexity time domain semi-blind algorithm is proposed to estimate and track the time varying MIMO OFDM channels. First, the proposed least mean squares (LMS) based algorithm is developed for the training mode and then is extended for the blind mode of the operation by combining with the decision direction (DD) or adaptive Bussgang algorithm (ABA) techniques. In the blind mode, because of decision errors, a smaller step size is considered for the LMS algorithm and the channel estimation is run a few times to improve its precision. In each round of the estimation in the blind mode, the step size is decreased to form some kind of annealing. Both DD LMS and ABA LMS techniques are simulated and compared to the full training case and MSE of channel estimation error is considered as comparison criterion. It is shown for 2x4 DD LMS and for 4x4 ABA LMS algorithms present near full training case estimation error. Of course in some scenarios the former proposed technique performs better and in other scenarios the latter is better and therefore combine of it can be very interesting in all channel conditions.
In recent years, MIMO channels are introduced to achieve high data rates required by the next generation wireless communication systems [1]. Using multiple-input multiple output (MIMO) channels, when bandwidth is limited, provides much higher spectral efficiency compared to Single-Input Single-Output (SISO), Single-Input Multiple-Output (SIMO), and Multiple-Input Single-Output (MISO) channels. Moreover, when channel is full rank or in other words all propagation coefficients are independent, the diversity gain of MIMO channels is product of transmitter and receiver elements numbers. Therefore, employing MIMO channels not only increases the throughput, but also increases their robustness against fading; thus, makes it efficient for the requirements of the next generation wireless services. Using orthogonal frequency division multiplexing (OFDM) combats the frequency selective fading by converting a wide band channel to a couple of narrowband flat fading channels [2]. To detect the received signal in a time varying wireless channel, use of either equalization or channel estimation is obligatory [3][4][5][6][7][8][9][10]. While use of the channel estimation, designer has more flexibility to use more kinds of the different detection and decoding algorithms than equalization and therefore channel estimation is a more common option.
Estimation of the MIMO OFDM channels can be done in the time domain or frequency domain [11]. The most researches done in the MIMO OFDM channel estimation topic, is in frequency domain where because of especial features of the OFDM signals, implementation of the algorithms is more feasible. But while using the frequency domain channel estimation, information related to the delay spread of the channel paths is ignored and this simplification decreases the performance. Usually to compensate this degradation, a mapping to the time domain and then channel shortening is used. But this process needs, whole of an OFDM symbol is assigned to the training, or in other word all sub-carriers must be available for designer. This requirement not only decreases the effective throughput but also in some scenarios is not feasible because all sub-carriers are not available. Therefore, time domain channel estimation provides more flexibility for the designer and better performance compared to the frequency domain algorithms. On the other hand, when channel undergoes time variations, the algorithm should follow channel variations in the blind mode and the extension off the time domain algorithms for the blind mode is straitforward. Two conventional techniques to extend training based channel estimation and equalization algorithms to the semi-blind mode are decision directed [12][13][14] and modulation-statistics-based algorithms [15][16][17][18] which respectively use hard and soft decision detected data as virtual training. Bussgang technique which uses a non-linear function of soft detected data is conventionally used in equalization and its application as a channel estimator is novel [19]. The optimum non-linear function used in Bussgang algorithms is dependent to the SNR and other channel parameters, therefore its adaptation provide improvement in the performance [11]. In this paper, LMS algorithm is developed as a semi-blind time domain estimation algorithm for time varying MIMO channel and to adapt non-linear function used by Bussgang algorithm, two parameters in it are considered to be adapted.
The rest of this paper is organized as follows. In Section II, models used for signal transmission and channel are introduced. In Section III, the proposed algorithm is derived. In Section IV simulation results of the proposed receiver are presented. Concluding remarks are presented in Section V.
The block diagram of the transmitter in a spatial multiplexed MIMO-OFDM system with M antennas is shown in Fig. 1 [2].
The main input block is converted to a couple of parallel sub-streams using a serial to parallel converter. Then each sub-stream is converted to M OFDM symbols. Finally, after inserting cyclic prefix with minimum length equal to the delay spread of the channel, all M sub-blocks are transmitted separately via transmitters.
In the receiver side, linear combinations of all transmitted sub-blocks are distorted by time-varying Rayleigh or Ricean fading, and the inter symbol interference (ISI) are observed under the additive white Gaussian noise.
Frequency selective fading MIMO channel with time varying Rayleigh distribution and exponentially decaying paths is assumed. The input/output equations can be summarized as
where cp L and p L are the length of inserted cyclic prefix and the number of resolvable paths, and k r is the received vector,
is the channel matrix, and k s is the transmitted symbol all in time index k, and k w is the vector with i.i.d. AWGN elements with variance 2 w . Of course usually channel matrices have slow variation on an OFDM block and therefore sub-script k can be omit
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