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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 307342, 8 pages
Channel Estimation for DS-CDMA Systems: A Partial Difference Equation Approach
1School of Control Science and Engineering, Shandong University, Jingshi Road 73, Jinan 250061, China
2School of Control Science and Engineering, The University of Jinan, Jiwei Road 106, Jinan 250022, China
Received 23 January 2013; Accepted 25 February 2013
Academic Editor: Xiaojie Su
Copyright © 2013 Wei Wang and Chunyan Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 channel estimation problem for direct-sequence code-division multiple-access (DS-CDMA) communication systems with time-varying multipath fading channels. The channel estimator is designed by applying a partial difference equation approach together with the innovation analysis theory. This method can give a sufficient and necessary condition for the existence of an channel estimator.
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 [1–6]. In , a subspace-based estimation algorithm is developed. The algorithms in [2, 3] are based on the maximum likelihood estimation method. Due to the performance benefits of the Kalman algorithms, many works have focused on the Kalman filter-based channel estimation algorithms. These algorithms require a state-space model for the random process to be estimated. It is thus necessary to employ an autoregression (AR) or autoregression moving average (ARMA) model to account for the behavior of the actual process [4, 7]. In , the multipath fading channels are modeled as a first-order AR model, and a robust Kalman filer is employed to estimate the Rayleigh fading channels. A Kalman channel estimator based on a higher AR model has been proposed in . In , a reduced Kalman/LMS algorithm was proposed. In , a linear-trend tracking approach was developed, which uses the self-tuning scheme to track the time-varying fading channels. The aforementioned channel estimators implicitly assume that the ambient noises are additive white Gaussian noises and have accurate statistics. However, in many applications, 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 are also difficult to obtain. In , the channel estimation problem was investigated for second-order channel model, where a polynomial approach was proposed. concepts have also been employed in communication systems for robust equalization .
In this paper, we will investigate the channel estimation problem for direct-sequence code-division multiple-access (DS-CDMA) communication systems with time-varying multipath fading channels. A th-order AR channel model is used to present the actual channel process. Note that the system model for fading channel is time varying, so an approximated time-invariant system model has to be found in order to apply the polynomial approach. Different from , the channel estimator is designed by applying a partial difference equation approach together with the innovation analysis theory in this paper. This method can give a sufficient and necessary condition for the existence of an channel estimator, and do not need to approximate a time-invariant system before estimator design.
The remainder of this paper is organized as follows. In Section 2, the state space system model is introduced firstly, and the channel estimation is formulated. In Section 3, the channel estimation algorithm was introduced. Finally, some conclusion remarks are drawn in Section 4.
2. System Model and Problem Statement
In this paper, we will adopt a similar model as in [8, 12, 14]. Consider a binary DS-CDMA communication system with -multiple access users, the transmitted baseband signal of the th user is given by  as follows: where is the transmitted bit energy, is symbol duration, is the modulated information symbol of the th user and is chosen randomly from the set , and represents the transmitted waveform and has the form where is the spreading gain, is the spreading code of the th user with period , and is the real transmitted monocycle waveform shape in the time interval , that is, if , and has energy .
We assume that the multipath channel is consisted of resolvable propagation path and the channel coefficients are time invariant; then, the channel impulse response for the th user can be described by 
The received signal component from the th user can be represented as where denotes the convolution operator. The total received signal at the receiver is the superposition of the signal of the users, given by where is a white Gaussian noise with zero mean. The discrete-time signal is generated by sampling the output of the chip-matched filter at the chip rate. By collecting successive samples, the channel output from the th user at the th symbol can be expressed as where is the parameter collection of all multipath components and and are the signature matrices with dimension and have the form where For a synchronous CDMA forward channel, the total received discrete-time signal of all users is given by where
On the other hand, according to the well-known Bello model , different path’s channel components are independent. Thus, and are uncorrelated if . The channel coefficients of a path have the autocorrelation function as  where denotes the Hermitian transpose, denotes the zero-order Bessel function of the first kind, and is the Doppler frequency. We assume that the fading channel coefficients are invariant at a chip duration, then the fading coefficient can be approximated by a -order model as follows: where denotes the state transition coefficients of the th user in the th channel component and can be obtained by solving the Yule-Walker equations with an autocorrelated property .
The multipath fading channel of all users is given by where
The problem investigated in this paper is stated as follows. Consider the state-space channel models (11) and (15), find a causal estimator such that where is a given scalar and is a positive integer. Note that is the filtering, is the smoothing, and is the prediction.
Remark 1. For the forward channel, since every user’s spreading code is known by the base station, the coefficient matrices and in (11) are known. The symbols of all users can be obtained via the training sequence in the training mode or decision feedback in the training model. The transition coefficient matrices are often estimated using training sequences which lead to the item in (15) not to be a white noise process with unknown statistics. Moreover, the signal is corrupted not only by the ambient noise but also by the additive interference from other users, so the statistics of noise is also difficult to obtain. Therefore, the channel estimation has an important value.
Remark 2. It should be noted that an accurate representation of the channel coefficients with the autocorrelation function (13) would require an AR model with infinite order. In most of the related references [8, 12], however, the first- and second-order AR models are used because of their simplicity. In this paper, we will considered a general case, where the AR model may be a higher one.
Remark 3. Note that the -order AR model is a linear system with multiple delays. For such system, one can pursue an LMI approach. However, such approach usually can only give a sufficient condition for the existence of an estimator. An augmentation approach can also be used to solve such problem, but this will be deduced to solve higher-dimension Riccati equations. In this paper, we will investigate the channel estimation problem by applying a partial difference equation approach together with the innovation analysis theory. This method can give a sufficient and necessary condition for the existence of an channel estimator.
3. Channel Estimator Design
3.1. Existence of an Channel Estimator
In this section, we will demonstrate that the deterministic channel estimation problem can be converted to an innovation analysis for an associated stochastic system in Krein space.
Note that the received signal sequences are fixed, the only variable in is the disturbance ; thus, the channel estimation can be equivalently restated as  (i) has a minimum over ; (ii) can be chosen such that the value of at its minimum is positive.
Now we introduce the following Krein space model associated with (11), (15), and (18): where , and are the same as in (15) and (11), , and are mutually uncorrelated white noises with zero means and known covariance matrices as , , and .
Remark 4. Note that the elements in (11), (15)–(18), which are denoted in normal letters, are from the Euclidean space, while the elements in (20)–(22), denoted by bold face letters, are from Krein space. They satisfy the same constraints.
It is apparent that where with covariance matrix .
To obtain the condition under which has a minimum over , we introduce the innovations associated with observation sequence . Like in the Kalman filtering, the innovation is defined as the one-step prediction error of , that is, where is the projection of onto the linear space . In view of (24), we have where is the projection of onto the linear space . We denote the covariance matrix of as . From the linear estimation theory , we know that the linear space spanned by the innovation sequence contains the same information as the one spanned by the observation sequence, that is,
Theorem 5. An channel estimator that achieves (17) exists if and only if and have the same inertia. The minimum of , if exist, can be given in terms of as
Proof. The proof is similar to that in  and omitted here.
3.2. Optimal Estimation in Krein Space
In this subsection, we will discuss the optimal estimation associated with the stochastic system (20)–(22) in Krein space. For the convenience of discussion, we will use to denote the filtered and smoothed estimate of in Krein space and to denote the predicted estimate of .
We first define the following estimation error cross-covariance matrix: where denotes the inner product, and where is the projection of onto the linear space . In view of (30), it is obvious that .
From (31), we have
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.
Theorem 6. The cross-covariance matrix defined in (30) is the solution to the following Riccati type difference equation: with the following boundary conditions where and the initial value and .
Proof. From the linear estimation theory , we know that is the projection of onto the linear space , that is,
where , the parameter of the projection of onto , yields the stationary point of the following error Gramian,
from which we can obtain
From (20) and (38), it is obvious that
Therefore, we have which is (34).
On the other hand, the one-step prediction can be given by
Then, we have the following boundary condition: which is (34).
Furthermore, the auto correlation matrix of the one-step prediction error can be given by which is (35).
Based on the above results, the solution to the predictor of the stochastic system (20) and (24) can be obtained in terms of filter and smoother. For prediction, the innovation covariance matrix and estimation gain are different from filtering and smoothing. The estimation gain in prediction have the same form as in (37), only with “” instead by “,” that is, where is given by where and can be obtained according to the following theorem.
Theorem 8. Based on the filtered and smoothed estimates and the covariance matrices , for a given , one has the following: (i)The predictor is given by where and is defined as with , . (ii)The covariance matrices and are given by where
Proof. In view of (20) and by using the mathematical instruction method, we have that for all , where is as defined in (51). From (54), (50) can be obtained directly. In view of (50) and (54), the proof of (52) is straightforward and omitted here.
3.3. Channel Estimator
In the previous section, we have presented preliminary results on the innovation analysis associated the stochastic system (20) and (24) in Krein space. In this section, we will give the main results on the channel estimator.
Theorem 9. Consider the channel model (11) and (15) and the associated performance criterion (17). For given scalar and integer , an channel estimator that achieves (17) exists if and only if and have the same inertia. A suitable channel estimator is given by where is defined as the Krein space projection of onto the linear space and obtained from Theorem 7 for filtering and smoothing and Theorem 8 for prediction only with and instead by and , respectively.
Proof. From Theorem 5, has a minimum if and only if and have the same inertia, which is the condition for the existence of an channel estimator that achieves (17). If a minimum exists, then we can find a channel estimator such that the value of at its minimum is positive. Furthermore, from Theorem 5, the minimum of can be given by
where and are obtained from the Krein space projections of and onto the linear space , respectively.
Using the LDU block triangular factorization of , we can easily obtain that where
By applying the above factorization, the minimum can be equivalently rewritten as where we have defined
Note that any choice of the channel estimator that renders is an acceptable one, the simplest is which is (55).
In this paper, the channel estimation problem for DS-CDMA communication systems with time-varying multipath fading channels was investigated. The channel estimator is designed by applying a partial difference equation approach together with the innovation analysis theory in this paper. This method can give a sufficient and necessary condition for the existence of an channel estimator and do not need to approximate a time-invariant system before estimator design.
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).
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