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In the communications literature, a number of different algorithms have been
proposed for channel estimation problems with the statistics of the channel noise
and observation noise exactly known. In practical systems, however, the channel
parameters are often estimated using training sequences which lead to the statistics
of the channel noise difficult to obtain. Moreover, the received signals are corrupted
not only by the ambient noises but also by multiple-access interferences, so the
statistics of observation noises is also difficult to obtain. In this paper, we will
investigate the

The estimation of rapidly changing parameters of the fast-fading channel is an important technology for cellular systems and has many applications, for example, multiuser detection under multipath fading channels. The detector performance mainly depends on the channel estimator tracking performance. In the communications literature, a number of different algorithms have been proposed for channel estimation problems with accurate models [

In this paper, we will investigate the

The remainder of this paper is organized as follows. In Section

In this paper, we will adopt a similar model as in [

We assume that the multipath channel is consisted of

The received signal component from the

On the other hand, according to the well-known Bello model [

The multipath fading channel of all users is given by

The problem investigated in this paper is stated as follows. Consider the state-space channel models (

For the forward channel, since every user’s spreading code is known by the base station, the coefficient matrices

It should be noted that an accurate representation of the channel coefficients with the autocorrelation function (

Note that the

In this section, we will demonstrate that the deterministic

Note that the denominator of the left side of (

Note that the received signal sequences

Now we introduce the following Krein space model associated with (

Note that the elements in (

Let

It is apparent that

To obtain the condition under which

The proof is similar to that in [

In this subsection, we will discuss the optimal estimation associated with the stochastic system (

We first define the following estimation error cross-covariance matrix:

From (

Due to the whiteness and uncorrelation of innovation sequence, we can obtain

Then, the channel estimation error cross-covariance matrix for filtering and smoothing can be calculated according to the following theorem.

The cross-covariance matrix

From the linear estimation theory [

From (

Therefore, we have

On the other hand, the one-step prediction

Then, we have the following boundary condition:

Furthermore, the auto correlation matrix of the one-step prediction error can be given by

Based on the solution

Consider the stochastic state space model (

In view of (

Based on the above results, the solution to the predictor

Based on the filtered and smoothed estimates

The predictor

The covariance matrices

In view of (

In the previous section, we have presented preliminary results on the innovation analysis associated the stochastic system (

Consider the channel model (

From

Using the LDU block triangular factorization of

By applying the above factorization, the minimum

Note that any choice of the channel estimator that renders

In this paper, the

This work is supported by the National Natural Science Foundation of China (nos. 61104050, and 61203029), the Natural Science Foundation of Shandong Province (no. ZR2011FQ020), and the Research Fund for the Doctoral Program of Higher Education of China (no. 20120131120058).