#### Abstract

The UWB channel estimation and multiuser detection problem are investigated. The information symbol and channel parameter are considered as unknown variables. The multiuser detector and UWB channel estimator are designed jointly. For symbol detection, the one-step predictor of channel parameter is used and the estimation error is treated as a multiplicative noise; then a Riccati equation and a Lyapunov equation will be needed. If the transmitted symbols are uncorrelated and identically distributed random variables with zero mean and unit variance, only a Riccati equation needs to be solved. For UWB channel estimation, the one-step predictor of information symbol is used and the estimation error is also considered as a multiplicative noise. The solutions to the above two problems are obtained by solving a couple of Riccati equations together with two Lyapunov equations.

#### 1. Introduction

In the communications literature, a number of different algorithms have been proposed for channel estimation problems with accurate models [1–19]. In [1], 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 [13–15]. These algorithms require a state-space model for the random process to be estimated.

As for the UWB channel, many different types of channel models have been proposed. In general, the UWB propagation channel models are characterized by a dense multipath propagation and the clustering phenomenon and can be classified as deterministic and statistical [17, 18]. Deterministic models apply an electromagnetic simulation tool to obtain exact propagation characteristics for a specified geometry. Statistical models are normally less complex than the deterministic models and can provide sufficiently accurate channel information. In [20], three channel models were considered, namely, the Rayleigh tap delay line model, the -K model, and the Saleh-Valenzuela (S-V) model. The comparisons showed that the S-V model gives the best fit to the measured channel characteristics. This double exponential channel model is commonly used for UWB realistic indoor channel, and the channel impulse response is given by where(i) are the multipath gain coefficients, refers to the cluster, and refers to the rays in one cluster;(ii) is the delay of the th cluster which is defined as the TOA of the first arriving multipath component within the th cluster;(iii) is the delay of the th multipath component relative to the th cluster arrival time .

The clustering channel model relies on two classes of the parameters, namely,* intercluster* and* intracluster* parameters, which characterize the cluster and multipath component, respectively. In the above model, and are classified as the intercluster and intracluster parameters, respectively. The distributions of the cluster arrive time and the ray arrive time can be given by two Poisson processes:
where and are mean cluster arrival rate and mean ray arrival rate, respectively. The channel coefficients are defined as follows:
where is equiprobable to to account for signal inversion due to reflection; correspond to the fading associated with the th ray of the th cluster, which can be modeled as a log-normal distribution:
where is given by
where is the mean power of the first path of the first cluster. The behavior of the averaged power delay profile is
which reflect the exponential decay of each cluster.

With mapping, the above two-dimensional channel model can be reduced to a one-dimensional channel model: where and are the number of the resolvable multipath components; is the delay of the th path relative to the first path; is the fading coefficient of path .

After mapping, the one-dimensional model can be dealt with by using some conventional channel estimation algorithm that is used for narrowband systems, such as maximum likelihood approach and least mean square approach. In this paper, we will pursue a Kalman-filter-based approach with information symbols unknown.

#### 2. UWB System Model

In this paper, we consider a binary DS-CDMA UWB communication system with multiple access users. The transmitted baseband signal of the th user is given by [15, 16] 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 .

Note that the channel coefficient in (7) is fading with respect to time ; the channel impulse response for the th user can be described by where is the time delay for the th path of user . Then the received signal component from the th user can be represented as where denotes the convolution, and The total received signal at the receiver is the superposition of the signals 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 a pulse-matched filter (PMF) at the monocycle rate (as shown in Figure 1) and given by [13] In this paper the multipath delay of the UWB channel is assumed to be an integral multiple of the monocycle length ; then the above equation can be rewritten as where Further assuming that is invariant during a symbol interval and using to denote the channel parameter in the th symbol, then we have By collecting successive samples, the channel output from the th user at the th symbol can be expressed as where and are the signature sequence matrices with dimension and have the formsand is the parameter collection of all multipath components The total received discrete-time signal of all users can be given by where Considering that the channel parameters and the user information symbols are unknown, in this paper, we pursue a joint design method for user detection and channel parameter estimation.

The problems investigated in this paper can be stated as follows.

*Problem I*. Given the received signal sequence , with not exactly known, find an optimal symbol detector by using* a priori* estimate of channel parameter which is recursively calculated in Problem-II.

*Problem II*: Given the received signal sequence , with the information symbol not exactly known, find a channel estimator by using* a priori* estimate which is recursively calculated in Problem-I.

*Remark 1. *In this section, we have adopted a signal model for DS-CDMA communication systems similar to that in [15, 16]. Different from [15], in (7) the information symbol matrix is unknown which will be detected together with the channel parameter. In most relevant works, the information symbol is considered known for channel estimation [14], or channel parameter is known for user detection [12] and only few works investigate a joint estimation scheme considering both of the aforementioned unknown variables [13]. This paper will propose a Kalman-filter-based joint design method for multiuser detection and channel estimation.

*Remark 2. *The above two problems cannot be solved separately, because the channel parameters and information symbols are not exactly known. Different from [13], in this paper the symbol detector and channel estimator are designed simultaneously. The solutions of the detector and channel estimator will be obtained via solving coupled Riccati equations together with two Lyapunov equations.

#### 3. Multiuser Detector

In this section, a first-order state-space model is applied to symbol detection for the proposed UWB system, where the channel parameter is not exactly known. Then we can employ the Kalman filter to estimate all users’ symbols simultaneously. In view of (7), for multiuser detection the total received discrete-time signal can be expressed as whereNote the symbol vector defined in (23); the first-order non-Gaussian Markov transition model is defined as where where denotes the all-zero matrix, and is the identity matrix; the noise vector is white with zero mean and covariance matrix For the convenience of discussion, we first give the following definitions.

*Definition 3. *For a given symbol , let denote the optimal estimation of , which is the projection of onto the linear space

*Definition 4. *For multiuser detection with unknown UWB channel parameters, define
For UWB channel estimation with unknown information symbols, define
where is defined as in Definition 3.

As in the standard Kalman filtering, we define the one-step prediction error covariance matrix of the information symbol and channel parameter as where where is the stack of channel parameters of all users and and are defined as in Definition 3.

Note that the elements of UWB channel parameter matrix are unknown. In this section, we will use the one-step prediction instead of and consider the estimation error as a multiplicative noise for symbol detection. The optimal detector is given according to the following theorem.

Theorem 5. *Consider the discrete-time state-space signal model (23) and (25); when the channel parameter matrix is unknown, the information symbol detector is given by
**
where is the one-step prediction of UWB channel parameter obtained from the next section, and is the detector gain matrix
**
where is the covariance matrix of innovation **
where denotes the Hadamard product and is the Kronecker product. satisfies the following Lyapunov equation:
**
where is the symbol estimation error covariance matrix and satisfies the following Riccati equation:
**
where is the UWB channel parameter estimation error covariance matrix which will be calculated recursively in the next section. The one-step prediction of the information symbol is given by
**
which will be used for channel estimator design.*

*Proof. *From Definition 3 we know that the* a priori* estimate is the projection of onto the linear space and consider the channel parameter matrix as an unknown variable; then we have
In view of (7) and Definition 4, we obtain
It is apparent that is with zero mean and if . The stochastic process is termed as the innovation sequence associated with the received signal sequence. The covariance matrix of , denoted as , is calculated as follows:
where denotes the inner product, and
where denotes the Hadamard product and is the Kronecker product; denotes the inner product of symbol vector , given by
and satisfies the following Lyapunov equation:
where is the parameter estimation error covariance matrix which will be calculated recursively in the next section. In the third step are as follows:where, for the convenience, have been replaced by without confusion. Then substituting (43) into (42), we can get (36).

In terms of the definition of projection, we know that
From the linear estimation theory, the linear space spanned by the innovation sequence contains the same information as the one spanned by the received signal sequence ; that is,
Then the projection in (48) can be rewritten as
where is the parameter of the projection of onto , which yields the stationary point of the following error Gramian matrix:
and satisfies
Thus we have
In view of (52) and (53), it is apparent that we have
which is (38).

In view of (49), the projection in (48) can be further given by
where we have defined detector as
which is (34).

*Remark 6. *In Theorem 5, the UWB channel estimation error is considered as a multiplicative noise which is in matrix form, and the transmitted symbols may be colored and cross-correlated for different users. Their statistic characteristics are represented in the Lyapunov equation in (37). If the transmitted symbols are uncorrelated and identically distributed random variables with zero mean and unit variance, then the above result is equivalent to that proposed in [13] where the channel estimation error is considered as an additive noise.

Corollary 7. *If one assumes the transmitted symbols are uncorrelated and identically distributed random variables with zero mean and unit variance, that is, , then the information symbol detector is given by
**
where
**
with being the all-one matrix with dimension and .*

*Proof. *The proof is straightforward. Note that the transmitted symbols are uncorrelated and identically distributed random variables with zero mean and unit variance; then we can easily obtain that and . It is apparent that .

#### 4. Channel Estimator

For UWB channel estimation, the received discrete-time signal in (23) can be reexpressed as where is the information symbol matrix and is defined as and is as shown in (33), which can be modeled by using a first-order autoregressive (AR) model as where is a white random variable with zero mean and covariance matrix , and is the channel correlation matrix, given by where the scalar factor denotes the state transition coefficient of the th user in the th path. The above AR model for the channel parameter is only an approximation to the actual statistics of these random processes.

Similar to multiuser detection, for the UWB channel estimation the symbol matrix is treated as an unknown variable and uses the one-step prediction instead of and considers the estimation error as a multiplicative noise for channel estimation. The optimal estimation is given according to the following theorem.

Theorem 8. *Consider the discrete-time state-space signal model (60) and (62); when the information symbol matrix is unknown, the channel estimator is given by
**
where is the one-step prediction of information symbol obtained from the previous section, and is the estimator gain matrix:
**
where is the covariance matrix of innovation :
**
where is the information symbol estimation error covariance matrix which is obtained in the previous section, and satisfies the following Lyapunov equation:
**
where is the channel estimation error covariance matrix and satisfies the following Riccati equation:
**
The one-step prediction of the information symbol is given by
**
which will be used for the design of the user detector.*

*Proof. *Consider the information symbol matrix as an unknown variable; then we have
In view of (7) and Definition 4, we obtain
It is apparent that is with zero mean and if . The covariance matrix of is denoted as . The following proof of this theorem is very similar to that of Theorem 5, so we omit it here.

*Remark 9. *Different from [7, 10], for UWB channel estimation, the users’ symbols are also considered as unknown variables in this paper. The one-step prediction of symbol matrix is used, and the estimation error is treated as a multiplicative noise in matrix form. The detector and channel estimator are designed jointly and cannot be solved separately. The algorithm structure is as shown in Figure 2.

#### 5. Conclusions

The information symbol and channel parameter are considered as unknown variables in this paper. The multiuser detector and UWB channel estimator are designed jointly. For symbol detection, the one-step predictor of channel parameter is used and the estimation error is treated as a multiplicative noise; then a Riccati equation and a Lyapunov equation will be needed. If the transmitted symbols are uncorrelated and identically distributed random variables with zero mean and unit variance, only a Riccati equation needs to be solved. For UWB channel estimation, the one-step predictor of information symbol is used and the estimation error is also considered as a multiplicative noise. The solutions to the above two problems are obtained by solving a couple of Riccati equations together with two Lyapunov equations.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding to the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61203029, 61104050, and 61273124), the Natural Science Foundation of Shandong Province (Grant no. ZR2011FQ020), the Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20120131120058), and the Scientific Research Foundation for Outstanding Young Scientists of Shandong Province (Grant no. BS2013DX008).