Full MIMO Channel Estimation Using A Simple Adaptive Partial Feedback Method
Abstract
Partial feedback in multipleinput multipleoutput (MIMO) communication systems provides tremendous capacity gain and enables the transmitter to exploit channel condition and to eliminate channel interference. In the case of severely limited feedback, constructing a quantized partial feedback is an important issue. To reduce the computational complexity of the feedback system, in this paper we introduce an adaptive partial method in which at the transmitter, an easy to implement least square adaptive algorithm is engaged to compute the channel state information. In this scheme at the receiver, the time varying stepsize is replied to the transmitter via a reliable feedback channel. The transmitter iteratively employs this feedback information to estimate the channel weights. This method is independent of the employed spacetime coding schemes and gives all channel components. Simulation examples are given to evaluate the performance of the proposed method.
1 Introduction
Multipleinput multipleoutput (MIMO) communication systems have
recently drawn considerable attention in the area of wireless
communications as they promise huge capacity increase and bandwidth
efficiency. In the derivation of MIMO fading channel capacity, it is
usually assumed that the channel state information (CSI) is known at
the receiver,
The proposed method is made and presented for multipleinput singleoutput (MISO) systems. However, the generalization of the algorithm for MIMO systems is easily possible. The rest of this paper is organized as follows: In section 3, the system model is introduced. Simulation examples are given in Section 3. Finally, the paper is concluded in Section 4.
2 System Model
The following theorem has an essential role in our proposed method.
Theorem 1
Assume the vector and the sequence are known. Define
(1) 
where is an arbitrary vector. Then, for all real
with
(2) 
is a decreasing sequence. Hence, the optimum value of to minimizes it is given by
(3) 
In this case we have
(4) 
Proof 1
See appendix 5.
In a MISO system with transmit antennas and receive antenna, the channel vector describes the CSI. The component of which is denoted by represents the channel fading coefficient from the transmit antenna to the receive antenna. In many real systems, the CSI (or ) is not fully provided to the transmitter. This is due to some basic limitations like limited capacity of feedback channel or rapid channel variations. Therefore, if the communication system is based on known CSI assumption at the transmitter, its estimated value or is alternatively used. Assume , where denotes the Euclidean vector norm, is a threshold level and its exact value depends on the system characteristics. Now assume that holding , the overall system has a satisfactory performance. Now suppose which may lead to a bad system performance. The proposed method in this paper starts to come into the role at this moment. It reduces the difference between and till the point that the corresponding difference comes under . The details of the proposed method are given based on the Theorem 1.
When , the receiver sends a signal to the transmitter indicating that the transmitter should stop data transmission and both parties run the procedure (1) in which:

, a pseudo white training vector sequence, is known at both transmitter and receiver.

the initial value is .

The stepsize is the quantized value of (3). Note that the right hand side of (3) is computed only at the receiver and its quantized value replied to the transmitter via a partial feedback channel. Our suggestion is the bits Lloyd quantization method [5] that we have used in our simulations to send the quantized value of .
If for , satisfies , the transmitter chooses as its new value for the channel estimate . This selection happens by receiving the procedure ending signal from the receiver and then, the communication system will go back to its normal operation. Note that when is a pseudo white sequence (as it is), in almost all iterations we have (see the recursion (4)). Therefore, is a decreasing sequence and for sufficiently large , we have
(5) 

The proposed method is independent of system coding and modulation schemes. Hence, it can be useful in all MISO systems equipped with partial feedback.

Generalization of the proposed method for MIMO systems with receiver antennas () is done by sending stepsize values back to the transmitter. In other words, there is one stepsize value for each corresponding receiver antenna.

Usually the feedback path is a narrow band channel. The required bandwidth for feedback is proportional to the CSI variations and the processing speed of both transmitter and receiver.

In order to have a good performance, it is reasonable to have interruption in symbol transmission by the transmitter during the channel estimation phase. However, as recursion (4) shows, in each time instance , the transmitter takes an access to a better channel estimate than that of the time instance . Hence, if the transmitter insists to continuously send its symbol and does not break it (even when the channel estimate error is over the threshold level ), it is reasonable that at each time instant , the old channel estimate is replaced by its new one at the transmitter. This is because the channel squared error is a decreasing sequence.
3 Simulations
In our simulation we consider a MISO system with flat Rayleigh fading channel in two cases. In case one, a binary phase shift keying (BPSK) modulation and in case two, a quadrature phase shift keying (QPSK) modulation are assumed. For each case we consider two values and which led to four different situations. Figures 1 and 2 compare the bit error rate (BER) versus the transmit power to noise ratio (TNR) of the OBS (see Appendix 6). Suboptimal beamforming scheme (SOBS) is used in which the unknown is replaced by its estimated value which has been given from the proposed method for .We assume no delay in feedback channel. Also it is assumed that along the channel estimation procedure, the symbol transmission is interrupted till the time the channel estimate error comes below . Figure 1 shows that with BPSK modulation and , both OBS and SOBS have the same performance. As it is expected and the histogram diagrams of the stepsizes for and (Figures 3a and 3b) show, the stepsize curve has a normal shape around zero.
4 Conclusion
In this paper, a full channel estimation procedure for MIMO systems is proposed. In the suggested method, an iterative adaptive algorithm is executed at both the receiver and the transmitter. The required stepsize value for the transmitter is sent back to the transmitter by means of the feedback channel.
5 The optimum value of
6 Optimal Beamforming Scheme
Assuming the known CSI, in optimal transmit beamforming scheme, the transmitted signal is multiplied by the normalized conjugate of the channel coefficient. In other words, the beamformer vector is applied at the transmitter, where denotes the conjugate transpose. It means that if is the transmit symbol, the received signal is
(8) 
or equivalently
(9) 
where is the additive Gaussian noise with zero mean and variance .
Footnotes
 The CSI can be estimated at the receiver by transmitting a training sequence from the transmitter.
 When , we must exchange its position with in relation (2).
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