Communications and Network, 2013, 5, 98102 http://dx.doi.org/10.4236/cn.2013.53B2019 Published Online September 2013 (http://www.scirp.org/journal/cn) Impulse Radio UWB Signal Detection Based on Compressed Sensing Xing Zhu, Youming Li, Xiaoqing Liu, Ting Zou, Bin Chen Institute of Communication Technology, Ningbo University, Ningbo, China Email: liyouming@nbu.edu.cn Received June, 2013 ABSTRACT The extremely high sampling rate is a challenge for ultrawideband (UWB) communication. In this paper, we study the compressed sensing (CS) based impulse radio UWB (IRUWB) signal detection and propose an IRUWB signal detec tion algorithm based on compressive sampling matching pursuit (CoSaMP). The proposed algorithm relies on the fact that UWB received signal is sparse in the time domain. The new algorithm can significantly reduce the sampling rate required by the detection and provides a better performance in case of the low signaltonoise ratio when comparing with the existing matching pu rsuit (MP) based detection algorithm. Simulation results demonstrate the effectiveness of the proposed algorithm. Keywords: IRUWB; Compressed Sensing; C oSaM P; MP; Si gnal Det ect i on 1. Introduction UWB is one of the key technologies in the shortrange broadband wireless communication. With the character istics of high data rates, low power, and low cost, UWB can be applied to many scenarios such as highspeed shortrange wireless personal area networks (WPAN), ranging, positioning, monitoring, and wireless sensor networks (WSN) [1]. In some of these applications, UWB signal detection is a very important component. Hence there is a need to study th e UWB signal detection to make it more practical. However, when using the traditional algorithm for UWB signal detection, a very high sampling rate is re quired according to ShannonNyquist sampling theorem for the UWB signal’s high bandwidth that is up to sev eral gigahertzes. This is difficult to implement with a practical analogtodigital converter (ADC) [2]. The emerging theory of CS enables the reconstruction of sparse or compressible signals from a small set of ran dom measurements. If adopted by the signal detection, the CS theory will make the sampling rate much lower than the Nyquist rate. The authors in [3,4] proved that it is effective to detect signal by processing the sampling values of compressed sensing directly. Reference [5] proposed a MP based signal detection algorithm. In [2], the authors proposed a CS based system for UWB echo signal detection at a sampling rate much lower than Ny quist rate. This study indicates that the lowdimensional random measurement method based on the CS theory can be used to sample UWB signal. A UWB signal detection method based on MP algorithm was developed in [6]. Whereas in a circumstance with low signaltonoise ratio (SNR), the performance of the MP based detection algo rithm is not good. Thus, it leaves room for improvement. In [7], the authors demonstrated that CS theory is par ticularly suitable to IRUWB signal detection, and a gen eralized likelihood ratio test (GLRT) detector was pro posed. However, the GLRT UWB receiver needs the pilot symbol assisted modulation, leading to high system complexity. Furthermore, in order to reach a high per formance, the number of mixerintegrators employed by the receiver is too large to be realized. In this paper, we propose a CoSaMP [8] based IR UWB signal detection method. Without other extra processes, the method is formed from extracting infor mation directly from sampling values acquired by CS. The complexity of the proposed detection method is re duced dramatically when comparing with the GLRT de tector. Moreover, computer simulation results are pro vided to verify the performance of the proposed method, which show that the performance of the new method is superior to that of the MP based detection algorithm, especially in low signaltonoise ratio. The remainder of the paper is organized as follows. In Section II, the background of CS is depicted. Section III provides the principle for generating IRUWB signal. Section IV presents a CS based IRUWB signal detection principle and proposes a CoSaMP based IRUWB signal detection algorithm. Simulation results are provided in Section V. Finally, conclusions are drawn in Section VI. C opyright © 2013 SciRes. CN
X. ZHU ET AL. 99 2. Compressed Sensing Background CS is a technology that can recover the highdimensional signals from the lowdimensional and subNyquist sam pling data with the prior information that the signals are sparse or compressible [9]. The mathematical model of CS can be described as yAx (1) where is a signal which can be sparsely rep resented in a basis matrix N1 xRNN 12 {,, ,} N φR TN 12 [, ,,] N , that is, , the vector xφθ 1 θR MN AR θ x consists of K (K<<N) nonzero elements (we often say that is Ksparse). Th e vector (K<M<<N) is a random measurement matrix that is uncorrelated with . denotes the M samples obtained by CS. Our purpose is to recover the sparse coefficient vector from the M samples, and then multiply it by the basis matrix , thus recovering the original signal . θ φM1 yR φ In order to figure out the sparse coefficient vector , we need to find the solution to the following norm optim i zat i on p r ob l em [10] θ 0 l 0 argmin. .st θθyAφθ (2) Unluckily, solving the optimization problem (2) is prohibitively complex for it is an NPhard nonconvex optimization problem. A modified problem is to replace the restrict with the restrict 0 l1 l 1 argmin. .st θθyAφθ (3) This optimization problem transforms (2) into a con vex optimization problem which can be easily solved by linear programmi ng. 3. Impulse Radio UWB Theory The US Federal Communications Commission (FCC) provided a definition that a signal can be classified as an UWB signal if its fractional bandwidth is greater than 0.2 or its bandwidth is 500MHz or more [1]. According to this definition, there are several ways to generate UWB signals, among which impulse radio is the most common method. In this paper, we focus on the impulse radio UWB (IRUWB) signal. IRUWB communication is based on transmitting ul trashort (typically on the order of a nanosecond) dura tion pulses that are subsequently modulated by the binary information symbols. The two most popular modulation schemes are pulse amplitude modulation (PAM) and pulse posit i o n modulat i o n ( PPM). Owing to different approaches are employed to gener ate the pulse train, the UWB systems can be divided into two main categories: time hopping UWB (THUWB) and direct sequence UWB (DSUWB). To take a specific case, we will discuss the PAMDSUWB signal and its detection in the following. A block diagram of the PAMDSUWB transmitter [1] is shown in Figure 1. In Figure 1, 01 1 ,,,,, , kk bb bb b is a binary sequence to be sent and generated at a rate of 1/ bb RT (bits/s). After passing through the repeat encoder, every bit of the sequence is repeated b N times, therefore we get a new s e q u ence 000111 (,,, ,,,, ,, ,,, ,,) kk k bbbbb b bbb b* where has a rate of (bits/s). Then the second system of the block diagram converts into a sequence 01 1 b* /1/ cbs bs RNT T (,,,,,mmm b* , ,) jj m 11 * 21( jj mb m )j m that contains two kinds of elements,and . The con version equation of this is . When the sequence enters the transmission encoder, a binary zero correlation duration (ZC D ) co de 01 1 (,,,,,,) jj aa aa a composed of 1 ’s is applied to it [11] and the output of the transmission encoder is a new sequence , which can be expressed as *m 00 010 10 111 01 01 *(,, ,,, ,,,,, ,,,,, (,,,,,) j j jj jj j ma mama ma mama ma mama mmm ) m (4) The period of the ZCD code 01 1 (,,,,,,) jj aa aa a is N, we often assume that s (a more general hypothesis is that NN N is an integer multiple of N). The rate of sequence is bs (bits/s). Next, the sequence goes into the PAM modulator, and a sequence of unit pulses (Dirac pulses *m *m /1/T T cs RN ()t ) lo cated at times jT 1 are generated by the PAM modulator [1,11]. The rate of the sequence of unit pulses is / / sb s RNT T T (pulses/s). At last, the output of the PAM modulator passes through the pulse shaper, whose impulse response is. The duration of is , and ms ()pt ()pt m T T . Thus, we get the final output signal () t, which is given by * ()( ) s j tmptj T (5) where often has the following form ()pt 2 2 2 2 2 ()(1 4) t t pt e (6) Figure 1. Block diagram of PAMDSUWB transmitter. Copyright © 2013 SciRes. CN
X. ZHU ET AL. 100 where 2 42 is the pulse shap er facto r, and 2 is the variance. In practice, a PAMDSUWB transmitter’s parameters set by the user are: the average transmit power , the number of bits generated by the binary source 0, the sampling frequency 0 P n c , N, T, N, , and m T2 [1]. Figure 2 shows an example of the PAMDSUWB signal. From this figure, we can see that PAMDSUWB signal presents an intuitive sparse characteristic in the time domain. That is, the signal has only a few nonzero values. Thus, according to the Section II, the basis matrix of the PAMDSUWB signal can be an identity matrix. Based on the above, we can apply the CS theory into the PAMDSUWB signal detection. 4. CS Based IRUWB Signal Detection 4.1. The Signal Detection Model We implement the detection by distinguishing between the following two hypotheses 0 1 : :( H H yAn ) Axn (7) where denotes the PAMDSUWB signal to be detected, and N1 xR 2 ~(0, ) N n M1 I is the independent and identically distributed additive white Gaussian noise. (M<<N) is a known random measurement matrix, and is the sample obtained by the de tector. Next, we let MN yR AR 11 10 Pr( chosen when true )and Pr( chosen when true) d f PH H PH H denote the probability of detection and the probability of false alarm respectively [3]. Figure 3 illustrates the principle block diagram of the IRUWB signal detection based on CS. 4.2. The Proposed IRUWB Signal Detection Algorithm In this section, we propose an IRUWB signal detection algorithm based on CoSaMP [8]. The procedure of the algorithm is as follows: 00.5 11. 5 x 10 8 5 0 5 x 10 3 Time[s ] Am plitude[V ] PAMDSUW B Transmitter Sparse Signal Reconstruction Decision Output Detector Random Samplin g Figure 3. IRUWB signal detection principle block diagram. Let MN ΦR ometr denote the measurement matrix with restricted isy constant 2sc (c is a constant), u denote the noisy sample . Furthermore, the arsity level (the number of the nonzero values) is s, and the s pa rse vector sp approximation of the target signal is a. a)ze the approximation 00a and resid Initiali ual vu. Initialize the iteration cou nter. Find a proxy * yΦv for the c 1k b) urrent residual and locate the 2 largens of the prox y () st colum 2 supp Ω 2 () supp means the index set of the 2 where larg est columns of . c) Merge the index set of the newly identified compo nents with that of the largest components of the current approximation 1 (a ) k Tsupp Ω d) Solve a least squares problem to make an estimation of 0 the signal † ;  c TT T bΦub †1 () TTTT ΦΦΦΦ.where For an arbitrary N1 bR, assume {1,2,,N}, we define that T is a subset of , 0,otherwise i TbiT b. e) Reserve the largest components in the approxima tio h f) Update the residual n obtained by te step d) to produ ce a new approxima tion ks ab k a uΦ v , if 2 v, Figure 2. PAMDSUWB signal. g) 1kk where sis a known con stant, the b); or en go to steplse , go totep h). h) If a , where is a threshold value, then detect 1 ; otherwise, choose 0 . 5. Simulation Results In this section, the performance of the proposed detection algorithm and the MP based detection algorithm are compared. First, we set the parameters of the PAMDS Copyright © 2013 SciRes. CN
X. ZHU ET AL. 101 UWB transmitter as follows: 030 (dBm)P , 02n , 509 (Hz) c fe, 10 (s) p N, 39(s) s Te ,5 s N , ), lef Btected is 01500 scs NnN fT [12 0.59(s m Te the PAMDSU 20.25e signal to be 9. Hence the de ngth o W]. We set the sparsity level of the signal. The the measurement matrix as s250 signal is shown in Figure 4. In simulation, we let MN R be an independent and identically distributed andom matrix with zeromean and unit vari ance. Further, the mean and variance of the additive white Gaussian noise are 0 and 1, respectively. For the proposed detection algorithm, we let constant 5 10 A Gaussian r . For the MP based detection algorithm [5], we number of iterations as 10. Suppose that the prior prob abilities of the two hypotheses are equal, that is, r0 r1 P()P()0.5HH. The probability of detection is 000 trials. In order to demonstrate the effectiveness of the proposed algorithm, we have implemented the following three simulations. Figure 5 illustrates d P as a function of M set the the statistic result of 10 which is the number of measurents. We set the S as 2dB, and 0.01 f P. M ranges [150, 750].The threshold value me NR and theeshold value of MP based detection algorithm are both chosen by Monte Carlo simulations [5]. The number of Monte Carlo simulations is 2000. If we use the traditional detection algorithm, the number of measurements should be 01500 scs NnN fT according to the ShannonNyq we can see from this figure, the proposed algorithm can acquire a very high probability of detection at about 20% of the sampling rate required by the ShannonNyquist theorem. What’s more, in this condition, the proposed algorithm is superior to the MP based detection algo rithm. Figu thr uist samporem. Asling the re 6 shows as a function of the SNR which ra d P e cnges [10, 5]. Wonsider M 300 and M 500 respectively. Let 0.01 f P. The threld valu two algorithms unent SNRs are also acquired by the same means as in Figure 5. According to Figure 6, we can see that the perf ormance of the proposed algo rithm is better than that of the MP based detection algo rithm when the SNR is less than 1 dB. shoes of the der differ 00.5 11.5 22.5 3 x 10 8 5 0 5 x 10 3 Time[s] Amplit ude[V] 200 300 400500 600 700 0. 5 0. 55 0. 6 0. 65 0. 7 0. 75 0. 8 0. 85 0. 9 0. 95 1 Number of measurements (M) P robabilit y of det ec t ion MPbased proposed a l gori thm Figure 5. Probability of detection comparison of the proposed algorithm and the MP based detection algorithm under different number of measurements, SNR=2 dB and Pf = 0.01. 10 50 5 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 SNR(dB) Probability of detec tion MPbase d, M =300 MPbase d, M =500 propos ed al gorithm, M = 300 propos ed al gorithm, M = 500 Figure 6. Probability of detection comparison of the Figure 7 illustrates as a function of proposed algorithm and the MP based detection algorithm under different SNRs, M = 300 or M = 500, Pf = 0.01. d P P which ranges [0, 0.2]. Let M150 and SNR= 2B. The threshold values of thlgorithms under different probabilities of false alarm are chosen by using the method in Figure 5 and Figure 6. As we can see from Figure 7, the proposed algorithm is superior to the MP based detection algorithm in this situation. According to Figure 5, Figure 6 and Fig d e two a ure 7, we can speculate that the performance of the proposed IRUWB signal detection algorithm is better than that of the MP based detection algorithm in case of the low SNR. Meanwhile, the proposed algorithm needs a much lower sampling rate than the Nyquist rate. Figure 4. The PAMDSUWB signal of interest. Copyright © 2013 SciRes. CN
X. ZHU ET AL. Copyright © 2013 SciRes. CN 102 [1] M. D. Benedetto and G. Giancola, “Understanding Ultra Wide Band Radio Fundamentals,” Upper Saddle River, NJ: Prentice Hall, 2004. 00.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180.2 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 [2] G. Shi, J. Lin, X. Chen, et al., “UWB Echo Signal Detec tion with Ultralow Rate Sampling based on Compressed Sensing,” IEEE Transactions on Circuits and Systems, Vol. 55, No. 4, 2008, pp. 379383. doi:10.1109/TCSII.2008.918988 P robabi l i ty of false al arm P robability of detec tion [3] M. A. Davenport, M. B. Wakin and R. G. Baraniuk, “De tection a nd Estimat ion with Compre ssive Me asurements, ” Elect. Comput. Eng. Dept., Rice University, Houston, Tx, 2006, Tech. Rep. TREE0610. [4] J. Haupt and R. Nowak, “Compressive Sampling for Sig nal Detection,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), Honolulu, HI, April 2007, pp. 15091512. MPbased proposed algori thm [5] M. F. Duarte, M. A. Davenport, M. B. Wakin and R. G. Baraniuk, “Sparse Signal Detection from Incoherent Pro Jections,” in Proc. IEEE Int. Conf. Acoust., Speech, Sig nal Process. (ICASSP), Toulouse, France, May 2006, pp. 305308. Figure 7. Probability of detection comparison of the 6. Conclusions esent a CS based IRUWB signal 7. Acknowledgements art by the National Science REFERENCES proposed algorithm and the MP based detection algorithm under different probabilities of false alarm, M = 150, SNR = 2 dB. [6] J. L. Paredes, G. R. Arce and Z. Wang, “Ultrawideband Compressed Sensing: Channel Estimation,” IEEE J. Se lect. Topics Signal Process., Vol. 1, No. 3, pp. 383395, 2007. [7] Z. Wang, G. R. Arce, J. L. Paredes and B. M. Sadler, “Compressed Detection for Ultrawideband Impulse Ra dio,” in Proc. of 8th IEEE Int. Workshop SPAWC, June 2007, pp. 15. In this paper, we pr de tection model and an IRUWB signal detection algorithm based on CoSaMP. Compared with the GLRT based model, the proposed model is easier to realize. The new detection algorithm solves the detection problem by di rectly processing the sampling values obtained by CS. It is proved that the proposed algorithm can detect the IRUWB signal at a sampling rate that is much lower than the Nyquist rate. Numeric simulations show that the new algorithm yields performance gains over the MP based detection algorithm in case of the low SNR. [8] D. Needell and J. A. Tropp, “CoSaMP: Iterative Signal Recovery from Incomplete and Inaccurate Samples,” Ap plications and Computational Harmonic Analysis, Vol. 26, No. 3, 2009, pp. 301321. doi:10.1016/j.acha.2008.07.002 [9] L. Jiao, S. Yang, F. Liu, and B. Hou, “Development and prospect of compressive sensing,” Acta Electronia Sinica, Vol. 39, No. 7, 2011, pp.16511662. [10] P. Zhang, Z. Hu, R. C. Qiu and B. M. Sadler, “A Com pressed Sensing Based Ultrawideband Communication System,” in Proc. IEEE Int. Conf. Communications (ICC), Dresden, Germany, June 2009, pp. 15. This work was supported in p [11] E. C. Kim, S. Park, J. S. Cha and J. Y. Kim, “Improved Performance of UWB System for Wireless Body Area Networks,” IEEE Transactions on Consumer Electronics, Vol. 56, No. 3, 2010, pp. 13731379. doi:10.1109/TCE.2010.5606272 Foundation of China (61071119), Scientific and Tech nological Innovations Teams of Zhejiang Province (2010R50009), Ningbo Natural Science Foundation (2012A610017), and Innovation Team of Ningbo (No. 2011B81002). [12] L. Shi, “Applied Research on Compressed Sensing for Ultra Wideband Channel Estimation,” Ph.D. Thesis, Bei jing University of Posts and Telecommunications, 2011.
