Low-Density Parity-Check Codes for Noncoherent UWB Communication Systems

Zhonghua Liang , Junshan Zang , Xiaojun Yang , Xiaodai Dong 3,, Huansheng Song

1 School of Information Engineering, Chang’an University, Xi’an 710064, China

2 Jushri Technologies Inc, Shanghai, China

3 Department of Electrical and Computer Engineering, University of Victoria, BC 8W 3P6, Canada

* The corresponding author, email： lzhxjd@hotmail.com

I. INTRODUCTION

Noncoherent UWB communication systems have received considerable attention from both academia and industry due to their low-complexity and low-power consumption without the need for channel estimation and accurate synchronization [1]. Correspondingly, there is an impellent need for simple noncoherent UWB receivers, such as energy detectors(EDs) and autocorrelation receivers (AcRs)[1]. The ED collects the energy of the received signal over a given time and frequency window. Typically, ED receivers are used in conjunction with noncoherent pulse position modulation (NC-PPM) to avoid complicated decision threshold computation. The AcR collects energy from all multipath components and selectively accumulates signal energy via the delay ofTdseconds. Accordingly transmitted reference (TR) signaling with AcR offers low-complexity reception [2-4]. TR signaling consists of a reference pulse and a data pulse with a large delay interval which can avoid inter-pulse interference (IPI) [5]. NC-PPM and TR systems have been widely developed as the two popular noncoherent UWB systems[1-8]. However, a long wideband delay line is a challenge for the implementation of TR systems [6-7]. Hence transmitted reference pulse cluster (TRPC) was proposed as an improved TR signaling to address this problem with the compact and uniform spacing between reference and data pulses [9]. Therefore, in this paper, NC-PPM, TR, and TRPC systems are considered as the three investigated noncoherent UWB systems.

In order to guarantee reliable data transmission, several forward correction (FEC)codes, such as Reed-Solomon and convolutional codes were employed in noncoherent UWB systems [10-12]. Results reported in the previous work [12] show that significant performance gains can be obtained using the FEC codes. However, to make full use of the benefits of channel coding for applications with low cost and low power consumption, it is still interesting to develop more powerful FEC codes for noncoherent UWB systems.

Based on the above discussion, Low-density parity-check (LDPC) codes are considered in this paper. Actually, LDPC codes have been used in several radio communication systems[13-18] due to their excellent property of approaching Shannon limits [19]. Therefore, in order to evaluate the performance of LDPC codes in low-rate UWB systems, in this paper LDPC codes are introduced to noncoherent UWB communication systems. Moreover, a comprehensive performance comparison between the LDPC codes and other exiting FEC codes is presented for the three investigated noncoherent UWB communication systems.

The reminder of this paper is organized as follows. Section II describes the system model for the three investigated noncoherent UWB communication systems. Section III introduces two LDPC coding schemes with different parity-check matrix, and section IV gives the LDPC decoding procedure for the three investigated noncoherent UWB communication systems. In section V, some simulation results and discussion are presented. Finally, section VI provides the concluding remarks.

Two LDPC codes and the corresponding decoding procedures are presented in this paper for noncoherent UWB systems.

II. SYSTEM MODEL

In this section, we introduce the signal models for the three investigated noncoherent UWB communication systems, and the UWB channel models specified in the IEEE 802.15.4a standard.

2.1 Noncoherent pulse position modulation signaling

2.2 Transmitted reference signaling

2.3 Transmitted reference pulse cluster signaling

2.4 IEEE 802.15.4a channel model

The UWB channel models described in the IEEE 802.15.4a standard can be written as[21]

III. CODING SCHEMES

Several FEC codes, such as Reed-Solomon and convolutional codes, have been used in noncoherent UWB communication systems[10], [12]. In this section, two speci fic LDPC codes with different parity-check matrices are considered for the three investigated noncoherent UWB communication systems to obtain higher coding gains.

3.1 Encoding of LDPC codes

LDPC codes are a series of linear block codes.To encode a LDPC code, a generator matrix converted by the parity-check matrix, is required. Using the generator matrixG, we can encode the original information sequencesas

wherecis the LDPC codeword. For an LDPC code, the circle of Tanner graph [19] determines its BER performance. The shorter the circle is, the worse the BER performance will be. Therefore, in quasi-cyclic (QC-) LDPC codes, the circulant permutation matrix is used to construct the parity-check matrix to avoid the short circle [22]. The key point of designing a good LDPC code lies in searching for the parity-check matrix without length-4 circle and length-6 circle as far as possible. In this paper, two approaches presented in [23] were employed to the coded noncoherent UWB communication systems to construct QC-LDPC codes.

3.2 Two methods for constructing the parity-check matrix of LDPC codes

whereAis a square submatrix of dimentor matrixGcan be obtained as

In the first method, to ensure the uniqueness of encoding, the index matrixPcan be modi fied as

Similarly, the modi fied parity-check matrix is given as

In the second method, the design ofHcan avoid length-6 circle. First, exchange the columns ofHin (9) to getAwhose diagonal elements are all 1’s. When exchanging the columns ofH, the code weight and circle property remain unchanged. Then the following steps are performed to ensureAis non-singular.

Step 1： Replace the elements 1 with 0 below the diagonal from row 1 to rowjMand judge whetherAis non-singular. IfAis singular, repeat the deleting operation untilAis nonsingular.

Step 2： GetGfromA. We can ensure that length-6 circle is nonexistent as we just exchange the columns ofHand delete elements.

The second method can get better performance than the first one due to the avoidance of length-6 circle.

IV. DECODING PROCEDURES

After encoding and transmission through the UWB channel, the received signal is given as[9]

4.1 Decision variable for NC-PPM systems

Then the decision variable can be defined as

4.2 Decision variable for tr and trpc systems

4.3 Decoding algorithm

LDPC codes are constructed by very sparse factor graphs and have excellent error correction performance via several improved decoding algorithms [26-28]. However, considering the complexity and efficiency, the belief-propagation (BP) algorithm [19] is only considered in this paper.

The key steps of BP decoding procedure are presented in Algorithm 1.

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Table I Parameters for LDPC codes

Fig. 1 Relative error between the estimated training length

Fig. 2 BER performance of RS-std, and LDPC coded NC-PPM systems in CM1 channels

whereLrepresents the length of training varies with the training lengthL.

V. PERFORMANCE EVALUATION AND COMPARISON

5.1 BER evaluation of NC-PPM systems

5.2 BER Evaluation of coded TR Systems

5.3 BER Evaluation of coded TRPC Systems

From the simulation results above, we also see that the performance of H1-LDPC and H2-LDPC are similar at low and medium Eb/N0values. H2-LDPC only tends to be better than H1-LDPC at high Eb/N0values. Therefore, only the BER curves of H1-LDPC are used to compare with other codes in the following.

5.4 Comprehensive analysis

In order to make an intuitive comparison between the three investigated systems, we evaluate their uncoded and H1-LDPC coded BER performance, respectively.

Fig. 3 BER performance of RS-std and LDPC coded NC-PPM systems in CM8 channels

Fig. 4 BER performance of LDPC coded TR systems in CM1 and CM8 channels

According to Figs. 8 and 9, we see that uncoded TRPC systems outperform uncoded TR and NC-PPM systems by more than 2 On the other side, the performance gain of TR and NC-PPM systems reaches 4 dB in CM1 channels. The TRPC systems have not only better BER performance, but also greater performance gains than TR and NC-PPM systems when using LDPC codes.

5.5 Complexity analysis

Table III presents the computational complexity required for the investigated codes. The computational complexity is measured by consuming time with temporal resolution 0.4836 ms (i.e., 0.4836 ms is the smallest time unit).The results were obtained by averaging 1000 encoding and decoding loops in a practical simulation system. We see that compared to convolutional codes, the LDPC codes possess similar encoding complexity, however, their decoding delays are much larger. The performance improvement of LDPC codes is at the cost of complexity for both EDs and AcRs.Therefore, a trade-o ff between the performance and complexity should be considered to implement a desired system. For example,in the TRPC system, when both complexity and real-time capability are required, CC-L-nonsys code is more suitable. In the case of higher requirement on BER performance but more relaxed constrains on decoding latency, a LDPC code, such as H1-LDPC or H2-LDPC,can be considered.

Fig. 5 BER performance of LDPC coded TRPC systems in CM1 and CM8 channels

Fig. 6 BER performance of RS-std, CC-L-sys, CC-L-nonsys, and LDPC coded TRPC systems in CM1 channels

Table II Abbreviations of the investigated codes

Table III Computational Complexity Comparison

VI. CONCLUSIONS

In this paper, the performance of LDPC codes has been evaluated for noncoherent UWB communication systems. According to the simulation results, we show that with limited increased computational complexity, LDPC codes have better BER performance than the existing FEC codes specified in the IEEE 802.15.4a standard and those used in [11].Therefore, they can be considered as good alternatives to the FEC codes for noncoherent UWB applications with low cost and low power consumption.

Fig. 7 BER performance of RS-std, CC-L-sys, CC-L-nonsys, and LDPC coded TRPC systems in CM8 channels

Fig. 8 BER performance of unocded and H1-LDPC coded TR, NC-PPM and TRPC systems in CM1 channels

ACKNOWLEDGEMENTS

This work was supported in part by the National Natural Science Foundation of China under Grant 61271262, 61473047 and 61572083, in part by Shaanxi Provincial Natural Science Foundation under Grant 2015JM6310, and in part by the Special Fund for Basic Scientific Research of Central Colleges, Chang’an University under Grant 310824152010 and 0009-2014G1241043.

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