1. Introduction
As a potential solution for short range wireless communications, impulse radio ultra-wideband (UWB) is an attractive technology without acquiring additional spectrum resources. UWB offers some potentials, including low probability of interception, precise positioning capability, and coexistence with existing services [1]. These features have motivated interest in UWB technology for a variety of state-of-the-art wireless applications in recent years. The successful deployment of UWB systems, however, is encumbered by the harsh propagation environments. Each transmitted pulse arrives at the receiver over hundreds of delayed paths, with possibly severe shape distortion affected by diffraction and scattering effects. Moreover, due to the high frequency selectivity of UWB channels [2] , coherent receivers face many practical challenges, including intensive computational cost and extremely high sampling rate to estimate channel state information. These challenges make it difficult and costly to realize the optimal coherent receivers for UWB communications. Therefore, non-coherent UWB systems are preferred in view of their low implementation complexity [3]. In single antenna systems, the typical non-coherent detection is based on differential detection (DD), which employs the autocorrelation receivers (AcR) [4]. Although no explicit channel estimation is required for AcR, the non-coherent detection brings noise due to the use of noise template. Thus, DD results in severe bit error rate (BER) performance degradation as compared to the coherent receivers. To alleviate this issue, multiple symbol detection (MSD) is designed to suppress noise [5]. As an effective means to improve the BER performance of single antenna systems, multi-antennas non-coherent UWB systems are emerging recently. As a special form of DD, differential space-time block code (DSTBC) encoded AcR is investigated in UWB channels [6], where additional diversity gain has been achieved. In order to further enhance the detection performance, MSD is introduced in the DSTBC UWB systems. It has pointed out in [7] that MSD outperforms DD in terms of the BER performance. Furthermore, in order to fulfill a low-complexity MSD, decision-feedback is considered in [8]. However, all of these MSD are all about hard decision algorithm, which results in limited BER performance improvement. Therefore, soft information detection is inevitable, and needed to ensure detection accuracy in non-coherent UWB systems.
For many application scenarios, soft decision detection has been perceived as a promising approach to improve the systems performances [9]. In [10], a soft-input soft-output (SISO) MSD algorithm is proposed for the single antenna UWB systems. As an appealing benefit, trellis-based bidirectional information transmission is promising compared with that of the unidirectional transmission in [10]. An soft information detection has been proposed in [11] in Rayleigh fading channel, where DSTBC is concatenated with error correction code, e.g., convolutional code [12], and Turbo code [13]. The receiver in [11] employs a posteriori probabilities (APP) decoder, where linear prediction is presented to utilize the temporal correlation of fading for detection. Furthermore, a low-complexity iterative detection has been addressed in [14]. In order to avoid estimation for channel parameters, and further improve the detection performance, there is a need to investigate SISO MSD relying on the bidirectional message passing based on the DSTBC trellis [15].
In this paper, an effective SISO MSD algorithm is proposed in the DSTBC UWB systems, where the forward and bacward message passing mechanism [16] is implemented. The proposed algorithm aims to evaluate the APP of the information bits. In particular, a mathematical framework for MSD metric is analyzed for serving the APP model. To further exploit signals dependencies among DSTBC symbols, an AcR architecture is developed. For ease of explanation, in the following, signals sampling will be referred to as "observation window". Most of the MSD algorithms employ block-by-block sampling [7], where the observation window of size M + 1 symbols durations slides forwardsMsymbols when the current M symbols have been detected. We develop a different sampling architecture, where the observation window slides only one symbol duration each time. Benefitting from the proposed sampling and forward and backward message passing, the potential of the SISO MSD for improving the BER performance is achieved. In addition, the proposed SISO MSD can be exploited in conjunction with channel decoding to obtain an iterative detection approach. Simulations are presented to validate the effectiveness of the proposed appraoaches. To be more specific, the main contributions of this paper are summarized as follows.
The remainder of this paper is organized as follows. System description of a DSTBC UWB system is introduced in Section 2. In Section 3, an MSD metric is analyzed based on the DSTBC UWB system, and then an SISO MSD is proposed relying on the MSD metric. Simulation results are demonstrated in Section 4. Finally, conclusions are drawn in Section 5.
Notations: Lower-case (upper-case) boldface symbols denote vectors (matrices); IK represents the K × K identity matrix; for any matrix A, Tr(A) and AT denote the trace, and the transpose of A, respectively; E[⋅] and Var[⋅] denote expectation and variance, respectively; ⊗ stands for convolution; ∝ represents the proportion relationship.
2. System Model
We consider a DSTBC system, which is equipped with Q (Q ≥ 1) receive antennas. Note that each DSTBC code applied has to be an orthogonal unitary matrix. When the orthogonal unitary matrices are designed according to the DSTBC scheme [17], the proposed algorithm can be extended to the systems with more than two transmit antennas. In order to facilitate the encoding scheme, we consider two transmit antennas at the transmitter. The basic system structure is shown in Fig. 1.
Fig. 1.DSTBC UWB system, where (a) and (b) denote the transmitter and the receiver, respectively
The information bits are encoded with a forward error correction code, e.g., convolutional code. Then, the coded bits are interleaved and fed into the DSTBC encoder. As the encoding strategy mentioned in [6], the encoded information are divided into blocks of two bits. Finally, every block is mapped onto a information symbol from the code book Ω = {U0,U1,U2,U3}, where the code Ux ∈Ω, (x = 0,1,2,3) is an orthogonal unitary matrix and has the form . Then, the code selector determines the information symbol according to the rule, 00 → U0; 01 →U1; 10 → U2; 11 → U3. In the differential encoded system, the transmitted code for the k -th information symbolUk ∈ Ω is obtained by Dk = Dk-1Uk where k = 1,2,L , K , and K represents the number of transmitted symbols; the 2×2 matrix Dk is sent over two antennas during two frame durations.The reference matrix is set as . Let dp,2k+n-1 (p = 1,2;n = 1,2) denotes the p -th row and n -th column of Dk. Then, the transmitted signal from the p -th antenna can be expressed by
where ω(t) denotes the monocycle pulse waveform; Tf is frame duration; one transmitted symbol duration is Ts = 2Tf; the single time index j = 2k+n-1 is introduced to replace the double index (k,n), and dp,2k+n-1 is rewritten by dp,j, correspondingly. Considering a quasi-static dense multipath fading environment [18], UWB channel impulse response between the p -th transmit antenna and the q -th (1 ≤ q ≤ Q) receive antenna is given by
where Lp,q denotes the number of propagation paths; are the gain coefficient and the delay associated with the l -th path, respectively; δ(t) is Delta function. The overall channel response gp,q(t) between the p -th transmit antenna and the q -th receive antenna can be described as
Then, the signal at the q -th receive antenna is obtained as
where nq(t) denotes additive white Gaussian noise (AWGN) with zero mean, and two-sided power spectral density N0 / 2. Based on (4), several MSD algorithms have been developed [6-8]. However, all these investigations result in limited performances improvement due to their hard decision detection. Therefore, in order to further improve the BER performance of the existing MSD, it is necessary to investigate an efficient algorithm relying on soft decision. In the next section, we will attempt to analyze an SISO MSD algorithm.
3. SISO MSD for DSTBC UWB Systems
In this section, an SISO MSD mathematical model will be derived for the DSTBC UWB systems. Firstly, by employing an AcR structure sampling, an MSD metric will be analyzed. Furthermore, an SISO MSD will be proposed with the aids of the MSD metric and BCJR approach in message passing mechanism.
3.1 MSD Metric for SISO MSD
In the following, an MSD metric will be presented for the DSTBC UWB systems. Given the transmitted symbols matrices, the conditional probabilities metric of the received signals matrices will be referred to as MSD metric. An AcR architecture is developed to further exploit signals dependencies among information symbols. For explanation convenience, the observation window size is set to M + 1 symbol durations, during which we assume that the channel remains invariant [5]. The observation window in [7] slides M symbols each time. As a result, the AcR ignores information dependencies among different symbols blocks. Different from the previous approaches, in our sampling, the observation window slides forwards only one symbol duration each time. To this end, the proposed sampling contributes to exploiting dependencies among different symbols blocks.
Fig. 2 illustrates the details of the proposed sampling for two adjacent observation windows. In the k -th observation window (k - M)Ts ≤ t ≤ kTs, the receivr samplings are obtained as Yk = [Yk,1,Yk,2,L ,Yk,M] where 2×2 matrix Yk,m(m = 1,L ,M) is the correlation function given by
Fig. 2.The proposed AcR sampling mechanism for the k -th and (k + 1)-th observation window
where Ti ≤ Ts is integration interval, is the received signals waveforms at the q -th antenna. The (u,v)-entry of Yk,m is given by
with u,v = 1,2.
Remark 1: The proposed AcR-based MSD is specifically tailored to the distinct signal structure of UWB impulse radio. It requires correlation operations between different segments of the received signals. The computed correlations will be fed into a digital processor. This can be implemented by using analog delay line, multiplier, and integrator; which avoids analog to digital converter with ultra high sampling rate. With Q = 2 as an example, Fig. 3 illustrates the AcR structure for k -th and (k − m)-th matrixes.
Fig. 3.AcR structure of the k -th and (k − m)-th received matrixes in UWB system with two receive antennas
In order to facilitate analysis for MSD metric, substituting (4) into (6), we have
where
and
denote the time-shifted noises. For convenience of description, we divide Yk,m(u,v) into signal component Sk,m(u,v)and noise component Nk,m(u,v). Then, (7) is reformulated as
The specific expression of each term in (10) is given by Appendix A. In order to simplify the analysis, we assume that nq(t) in (4) is a wideband AWGN process with bandwidth W > 1/Ti [18]. Thus, Nk,m(u,v)can be considered as a noise with zero mean, and conditional variance σ2. A detailed derivation of σ2 can be found in Appendix A. Now, Yk,m in (5) can be rewritten as a matrix form
where
and
represent the signal component and noise component, respectively. Based on the differential modulation , we have
Let . Combining (11) and (14), Yk,m is given by
Based on (15), the MSD metric can be expressed by
Now, MSD metric has been evaluated [18].
Remark 2: The mathematical model for MSD metric has been analyzed in DSTBC UWB systems. From the theoretical analysis, it can be seen that the proposed metric is capable of exploiting the transmitted symbols information from the received signals accurately. With the aid of MSD metric, the SISO MSD algorithm will be formulated in the next subsection.
Remark 3: From the MSD metric in (16), we can explore the correlations among multiple symbols to get information assisting the detection of the current symbol. By increasing the observation window size M, detection performance can be enhanced. Note that there is a trade-off between detection performance and practical implementations. Thus, from the perspective of detection performance, the BER performance is becoming better and better at the cost of a higher and higher computational complexity when M increases.
3.2 BCJR-Based SISO MSD Algorithm
The target of the SISO MSD is to calculate APP of information bits as Λ(dk(i)) = p{dk(i) = z|Y} , where dk(i) is the i -th bit corresponding to the information symbol Uk with i = 1,2 and z = 0,1. In order to obtain soft information by implementing message passing with trellis diagram, the information source can be modeled as a Markov finite-state process [19]. Correspondingly, the state transitions of the information source are characterized by the transition probabilities, from which we can obtain the APP. In the following, we will develop a trellis representation and apply it to the BCJR-based message passing approach. The differential modulation can be represented by a DSTBC trellis structure [10], which is shown in Fig. 4.
Fig. 4.DSTBC trellis diagram, where the nodes denote states, and the branches represent states transitions
The trellis diagram represents the time progression of the state sequences, where the nodes denote the states, and the branches represent the transitions having non-zeros probabilities. For every state sequence, there is a unique path through the trellis diagram, and vice versa. In order to integrate the trellis diagram with MSD effectively, we define the trellis as follows. When the source state group is Sk-1 = [Uk-1,L ,Uk-M]T inputting information symbol Uk, then the output is Xk = [Uk,Uk-1,L ,Uk-M+1]T. Set Adk(i)=z ={(Sk-1→Sk):dk(i) = z} denotes all the state transitions Sk-1 → Sk associated with input Uk containing dk(i) = z. Let Y = {Yk} be the total received signals matrixes. Consequently, the associated trellis nodes correspond to the probabilities p(Sk|Y) and the associated trellis branches correspond to the probabilities p(Sk-1,Sk|Y). Given Y, p(Y) is a constant, then we can derive the joint probabilities to obtain the probabilities p(Sk|Y) and p(Sk-1,Sk|Y), respectively. Therefore, the APP of information bits can be rewritten in terms of the trellis diagram as
Note that the received signals matrixes Y can be split into three sections, namely, the present received matrix Yk corresponding to the present state, the matrixes Ym
According to the properties of Markov process [20], if Sk is known, then the events after k do not depend on Ym and Substituting (19) and (20) into (18), we have For brevity, we define the probabilities functions as follows. Based on (22), (21) is simplified as where α(Sk−1) is the forward state transition probability, β(Sk) denotes the backward state transition probability, and γ(k) represents the branch transition probability. α(Sk−1) and β(Sk) can be calculated through the forward and backward recursions based on the DSTBC trellis structure. In order to implement the bidirectional message passing mechanism with MSD, we need to reformulate an expression for the branch transition probability. Note that, γ(k) can be further reformulated as where p(Sk|Sk-1) represents the state transition probability, which can be considered as the local check function for the state transition Sk-1 → Sk given by and p(Yk|Sk-1,Sk) corresponds to the conditional probability with output Xk expressed as From (24)-(26), we have where p(Yk|Uk,Uk-1,L ,Uk-M+1) is given by (16). Assume that channel is memoryless [16], α(Sk) in (23) can be obtained by the forward message passing as Similarly, β(Sk) is given by the backward message passing as A detailed derivation of (28) and (29) can be found in Appendix B. Combining (17) and (23), APP is obtained as where the summation is over all possible output associated with Uk containing dk(i)z=; γ(k) is given by (27), α(Sk) and β(Sk) are obtained from (28) and (29), respectively. In order to reveal the intrinsic message passing pattern and visualize the proposded SISO MSD algorithm, we employ a delivery representation to explain the implications. The message passing model between the adjacent states is demonstrated by Fig. 5, where circles with ring are state variables, the circles represent variables nodes, and squares denote the transition function nodes. The resultant message for APP consists of forward and backward message. During the specific implementation, firstly, we calculate γ(k) from the received signals. Then, the forward passing message α(Sk), and the backward passing message β(Sk) are recursively calculated, respectively. Fig. 5.A forward and backward message passing representation of the proposed algorithm. Circles with ring are state variables; the circles represent variables nodes; squares denote transition function node Finally, APP of information symbols will be obtained from (30). For clarity, the calculation details are summarized in Table 1. Table 1.Algorithm: Calculating (30) Based on the Trellis Diagram. Remark 4: When the proposed SISO MSD is combined with channel decoding, an iterative detection and decoding approach is obtained. In particular, the messages exchanged between the detector and the decoders are known as extrinsic information [21]. During the iterative decoding process, the inputs to DSTBC detector are a priori or the extrinsic information of the input symbols, which will be delivered to channel decoder; the decoder computes the APP from (30), and then sends back to the detector as a priori information for the next iteration [22-24]. In this section, simulations are conducted to validate the BER performance of the proposed SISO MSD algorithm. A DSTBC UWB system is considered for the case of receiver antennas Q = 1, 2, and 4, respectively. The channel is generated according to the IEEE 802.15.3a CM1 model [5], and the monocycle waveform is the normalized second derivative of a Gaussian function ω(t)=[1-4π(t/Tm)2]exp[-2π(t/Tm)2], where the pulse duration Tm = 0.287 ns. In order to eliminate the inter-symbol interference, the frame duration is chosen as Tf = 100 ns, such that Tf > Tn [5], where Tn = 50 ns denotes the maximum excess delay of the channel. Firstly, we investigate the BER performance of the proposed algorithm without considering channel coding. Fig. 6 illustrates the performance comparisons of the BCJR-based MSD and MSD for different observation window size M with Q = 1. It is clear that the proposed algorithm has a better BER performance than the MSD in [6]. This is attributed to more accurate detection information provided by the bidirectional message passing of the SISO MSD when compared to MSD in [6]. Furthermore, performance gain becomes larger when M increases. Then, we consider the BER performance for different Q with M = 2. It is shown in Fig. 7 that a performance improvement can be achieved with the proposed MSD when compared to MSD in [6]. Fig. 6.BER performance comparisons of the BCJR-based SISO MSD and MSD with Q = 1 and different M Fig. 7.BER performance comparisons of the BCJR-based SISO MSD and MSD with M = 2 and different Q Next, an iterative SISO MSD algorithm is further investigated. The generator for the rate 1/2, (5, 7) octal convolution code with a constraint length of 3 is employed [14]. For the iterative SISO MSD, 4 iterations between the detector and the decoder are performed. Fig. 8 compares the performance of the proposed SISO MSD, and the existing MSD in [6]. It can be seen that gain can be achieved at the cost of extra iterations between the SISO detector and the SISO channel decoder as compared to MSD in [6]. These results clearly highlight the great benefits of iterative SISO MSD in conjunction with channel decoding. Fig. 8.BER performance comparisons of iterative SISO MSD and MSD with different M and Q for 4 iterations An SISO MSD algorithm has been proposed for the DSTBC UWB systems. Specifically, we have constructed an MSD metric to serve the SISO MSD, where the forward and backward message passing mechanism is implemented. In addition, an iterative detection and decoding approach is developed. Simulations illustrate that the proposed SISO MSD can offer significant gains as compared with the existing MSD, and iterative detection approach further improves the BER performance.
4. Simulations
5. Conclusion