A Soft-Limiting Receiver Structure for Time-Hopping UWB in Multiple Access Interference

2006 IEEE Ninth International Symposium on Spread Spectrum Techniques and Applications A Soft-Limiting Receiver Structure for Time-Hopping UWB in Multiple Access Interference Norman C. Beaulieu, Fellow, IEEE, and Bo Hu, Member, IEEE Abstract A novel receiver structure is proposed for detecting a time-hopping ultra-wide bandwidth signal in the presence of multiple access interference. The proposed structure achieves better bit error rate performance than the conventional matched receiver when operating in multiple access interference. When operating in a multiple access interference-plus-gaussian-noise environment, the receiver structure outperforms the conventional matched filter receiver for moderate to large values of signal-tonoise ratio. Index Terms Demodulation, digital receivers, error rate, multiple access interference, ultra-wide bandwidth. I. INTRODUCTION Ultra-wide bandwidth technology (UWB) is currently being investigated as a promising solution for high capacity wireless multiple access systems. A time-hopping sequence is applied in UWB systems to eliminate catastrophic collisions in multiple access [1]. Studies of multiple access system performance for time-hopping systems were conducted in [1]- [7], in which a conventional single-user matched filter (correlation receiver) was used to detect the desired user signal. It has been shown that multiple access interference significantly degrades the bit error rate (BER). In [1] - [3], the BER was estimated by using a Gaussian approximation in which a central limit theorem (CLT) is employed to approximate the sum of the multiple access interference (MAI) as an additive Gaussian noise (AGN) process. As known, if a signal is corrupted by AGN, the matched filter is an optimum receiver in the sense that it maximizes the output signal-to-noise ratio (SNR) [8]. In the absence of intersymbol interference, it is also the minimum probability of error receiver [8]. However, the MAI in TH- UWB systems is not Gaussian-distributed interference. That this is so, is clear from the results in [4]- [7]. References [4] - [7] have showed that the Gaussian approximation significantly underestimates the BER of practical TH-UWB systems for medium and large SNR values, where the power of the MAI is large. In other words, multiple access interference in TH-UWB systems cannot be reliably modeled as AGN. Therefore, the conventional single-user matched filter or correlation receiver is not necessarily an optimal single-user receiver for UWB. Furthermore, in applications, where it is desired to achieve maximum user capacity, the performance of the system will be limited by MAI and the Gaussian noise may be negligible. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4. (e-mail: beaulieu@ece.ualberta.ca; bohu@ece.ualberta.ca). In this paper, we propose a new receiver structure for detecting a TH-UWB signal in MAI. It is shown that the new receiver structure outperforms the widely-used correlation receiver for the interference-limited case. The gain of the new receiver structure over the conventional matched filter UWB receiver varies from around 10 db at small values of signal-to-interference ratio (SIR) to 0 db as the SIR becomes infinitely large, provided that no Gaussian noise is present. In a practical mixed multiuser plus Gaussian noise environment, the proposed receiver underperforms the conventional receiver for small to moderate values of SNR, but can achieve more than 1 db performance gain for practical moderate to large values of SNR. II. TIME-HOPPING UWB SYSTEM MODELS In this letter, we consider a time-hopping binary phase shift keying (TH-BPSK) UWB system, but our analysis can also be used for time-hopping pulse position modulation (TH-PPM) systems. A typical TH-BPSK UWB signal has the form ( ) s (k) (t) = j/n p s t jt f c (k) j T c (1) j= where t is time, s (k) (t) is the kth user s signal conveying the ith data bit, and p(t) is the signal pulse with pulse width T p, normalized so that + p2 (t)dt =1. The structure of this TH-BPSK model is described as follows [7]: is the bit energy common to all signals. is the number of pulses required to transmit a single information bit, called the repetition code length in [5]. T f is the time duration of a frame, and thus, the bit duration T b = T f. T c is the hop width satisfying N h T c T f. { } c (k) j represents the TH code for the kth source; it is pseudorandom with each element taking an integer value in the range 0 c (k) j hops. j <N h, where N h is the number of represents the jth binary data bit transmitted by the kth source, taking values from {1, 1} with equal probability. Assuming N u users are transmitting asynchronously and the MAI dominates the ambient noise, the received signal is r(t) = A k s (k) (t τ k ) (2) 0-7803-9780-0/06/$20.00 2006 IEEE 417

where {A k } Nu represent the channel gains for all transmitted signals, and {τ k } Nu represent time shifts which account for user asynchronisms. Without loss of generality, we assume τ 1 =0[5]. Following a widely-adopted assumption on τ k, we further assume {τ k } Nu are uniformly distributed on a bit duration [0,T b ) as in [9]. III. RECEIVER STRUCTURES Consider using a conventional single-user matched filter or correlation receiver to coherently demodulate the desired user signal in an asynchronous system. Assume s (1) (t) to be the reference signal and d (1) 0 to be the transmitted symbol. Without loss of generality, we will set c (1) j =0, for all j [5]. Assuming perfect synchronization with the reference signal, the decision statistic of the conventional single-user correlation receiver is obtained as [7] (m+1)tf r = r(t)p(t τ 1 mt f )dt mt f = S + I (3) where S = A 1 d (1) 0 depends on user 1 s target signal bit d (1) 0, and I is the total MAI from the N u 1 active users in the TH-BPSK UWB system, given by where I = (m+1)tf A k I (k) (4) I (k) = s (k) (t τ k )p(t τ 1 mt f )dt. (5) mt f We model the difference of time shifts for user asynchronism as [1, eqn. (55)] τ k τ 1 = m k T f + α k, T f /2 α k <T f /2 (6) where m k is the value of the time difference τ k τ 1 rounded to the nearest frame time, and α k is uniformly distributed on [ T f /2,T f /2). Based on the assumption [1, eqn. (57)], which is rewritten as N h T c < T f 2 2T p (7) We can rewrite (5) in the form [1, eqn. (76)] I (k) = (m+m k )/ p ( x α k c (k) m T c ) p(x)dx and the MAI, I, can be expressed as I = A k (8) ( ) (m+m k )/N R s α k + c (k) m T c. (9) Then, the desired data symbol can be detected based on the output of the conventional single-user correlation receiver. It is seen from (9) that the decision statistic is obtained with a summation of integrals over the number of frames required to transmit one information bit,. Each integration is a partial correlation for the corresponding frame. One can rewrite the decision statistic r as r = r m = S m + I m (10) where S m is the desired signal component on the mth frame, E given by S m = A b 1 d (1) 0, and I m is the MAI on the mth frame, given by I m = ( ) A k (m+m k )/N R s α k + c (k) m T c. (11) The output of the conventional correlation receiver, r, is the sum of the partial correlations on each frame. Based on this observation, we propose a novel UWB receiver structure in this paper. Unlike the conventional correlation receiver which makes its decision based on r = N r m, the decision variable r is calculated as r = r m where S m, if S m r m r m = r m, if S m <r m < S m S m, if r m S m. (12a) (12b) The transmitted information bit d (1) 0 is then decided according to the rule r >0 d (1) 0 =1 r 0 d (1) 0 = 1. The novel receiver structure is plotted in Fig. 1. The implementation of the receiver (12) requires more information than the implementation of the receiver (3) since the former requires knowledge of S m whereas the latter does not. Receiver (12) is an optimal structure for a signal embedded in additive Laplace noise [10]. Our proposal to use it for TH-UWB is new. Our use of it here for UWB is motivated by the fact that the Laplace noise model is used for impulsive noise and the short bursts of multiple access interference are impulse-like. The superiority of receiver (12) over the matched filter is intuitive. An interference burst in one frame has limited influence on the bit recovery owing to the limiting inherent in each frame in the receiver structure of (12). IV. RESULTS AND DISCUSSION We evaluate the BER performance of the proposed receiver structure and compare it to the conventional correlation receiver (matched filter). Monte-Carlo simulation is used for predicting the BER. We restrict our simulation to a UWB system using the Gaussian monocycle given in [1]. The parameters for the example UWB system are listed in Table I following [1] and [7]. 418

TABLE I EXAMPLE TH-BPSK SYSTEM PARAMETERS Parameter Notation Typical Value Time Normalization Factor τ p 0.2877 ns Frame Width T f 20 ns Chip Width T c 0.9 ns Number of Users N u 4, 16 Number of Chips per Frame N h 8 Repetition Code Length 4, 8 First, BER curves of the TH-BPSK system in the presence of MAI are plotted as a function of SIR for different values of in Fig. 2. The SIR is given by SIR = A2 1 var[i] A 2 = 1 σ 2 a Nu A2 k (13) (14) where σa 2 is defined as [1, eq. (79)] σa 2 = 1 [ 2 p(x s)p(x)dx] ds. (15) T f We observe that when the transmission is interferencelimited, the proposed receiver outperforms the matched filter receiver for all SIR values. For example, when the BER is 10 2, the performance gain achieved by using the new receiver with =8is around 1.7 db, and the gain achieved with =4is 0.6 db. We note from Fig. 2 that the performance gain achieved by using the proposed structure is strongly dependent on the value of SIR and decreases as the value of SIR increases. For example, when the BER is 5 10 2, the gains are 8.8 db and 1.8 db for =8and =4, respectively, whereas the gains decrease to 0.6 db and 0.15 db, respectively, when the BER is 10 3. The reduction of gain as SIR increases can be explained as follows. Both receivers add r m to the receiver decision variable if S m <r m < S m. The two receivers differ when r m > S m ; the conventional receiver adds r m to the receiver decision statistic whereas the new receiver adds signum(r m )S m. Now consider S m fixed and reduce the variance of the interference. As the variance decreases, r m S m + if r m > S m and r m S m if r m < S m. Thus, the conventional receiver decision statistic approaches the decision statistic of the soft-limiting receiver, and the BER performance of the former approaches that of the latter. We remark that the gain of the soft-limiting receiver over the conventional matched filter receiver for small values of SIR is very large. As a check, we have performed a mathematical analysis and proved the following. In the case that the MAI has precisely the Laplace distribution, the gain is G =8=9.03 db in the limit of vanishingly small SIR. Hence, although the exact distribution of the MAI is not known, this check gives confidence in the validity of the small SIR results, i.e, the large gain. Note that this gain requires the absence of any Gaussian receiver noise component, which is unrealistic. We will consider this point further in the sequel. The choice of system parameters in Fig. 2 implies that multiuser collisions will be frequent. It is of interest to assess the performances of the two receiver structures when collisions are less frequent. Fig. 3 is similar to Fig. 2 except that the number of interferers has been reduced from 15 to 3. We observe that the benefit of the soft-limiting receiver is greater at small values of SIR, but similar for usable values of BER around 10 2. In order to investigate the sensitivity of the BER performance of the soft-limiting receiver to estimation error in the threshold value S m, we also provide simulation BER results for the receiver with different threshold values, shown in Fig. 4. Note that when the threshold goes to infinity, the soft-limiting receiver structure is equivalent to the matched filter; when the threshold goes to 0, the receiver structure becomes a hard-limiting or hard decision receiver. Two other cases between these two extreme cases are also considered for comparisons, with the threshold taking values 2S m and 1 2 S m. The BER performances of the receiver structure with different threshold values are plotted in Fig. 4 for N u =16and =8. We observe that when the SIR values are less than 6 db, the receivers with thresholds S m, 1 2 S m and 0 achieve similar bit error rates, all outperforming the matched filter and the receiver with threshold 2S m.however,asthesir value increases, the performance of the hard decision receiver significantly deteriorates, and the receiver with threshold S m outperforms the receivers using other threshold values. In addition, it is seen that the receiver with threshold 2S m outperforms the matched filter receiver for all values of SIR, although the performance difference diminishes to nothing as the SIR increases. While the soft-limiting receiver offers very large gains for small values of SIR, these gains cannot be realized in a practical receiver because of thermal receiver noise. As an example for the proposed receiver operating in MAI and background noise, BER curves of the TH-BPSK system are presented as a function of signal-to-noise power ratio (SNR) with 15 interfering users for a value of SIR = 10 db in Fig. 5. The SNR is defined as /N 0.InFig.5,thecurvewith triangular markers represents the BER performance achieved by the matched filter, and the curve with diamond markers represents the BER s obtained using the soft-limiting receiver structure with threshold S m. Observe that the conventional correlation receiver achieves better performance than the new receiver for small SNR values. However, the soft-limiting receiver structure outperforms the matched filter for medium and large SNR values, when the SNR is greater than 14 db, lowering the error rate floor caused by the MAI. These observations can be explained as follows. For small values of SNR, the background Gaussian noise N is dominant in the term I +N, and then I +N can be approximated as a Gaussian distributed random variable. In this case, the conventional correlation receiver works almost as an optimal receiver, and thus, it outperforms the soft-limiting structure as observed. On the other hand, when the SNR is large, the interference I is dominant in the sum I +N and I +N cannot be approximated as a Gaussian random variable. Then, the correlation receiver is not optimal. Receiver (12) effectively suppresses (limits) 419

T 0 f m m Fig. 1. The block diagram of the proposed receiver. SIR 10 4 Threshold 2S m /2 10 4 BER of the matched filter with Ns = 4 BER of the novel receiver with Ns = 4 BER of the matched filter with Ns = 8 BER of the novel receiver with Ns = 8 SIR Fig. 2. The average BER versus SIR of soft-limiting and conventional TH- BPSK UWB receivers assuming 15 asynchronous interferers. BER of the matched filter with Nu = 4 and Ns = 4 BER of the novel receiver with Nu = 4 and Ns = 4 SIR Fig. 3. The average BER versus SIR of soft-limiting and conventional TH- BPSK UWB receivers assuming 3 asynchronous interferers. Fig. 4. The average BER versus SIR of soft-limiting and conventional TH- BPSK UWB receivers assuming 15 asynchronous interferers for =8. Three values of threshold are shown for the soft-limiting receiver. BER of the matched filter with Ns = 8 BER of the novel receiver with Ns = 8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 E / N b 0 Fig. 5. The average BER versus SNR of soft-limiting and conventional TH- BPSK UWB receivers assuming 15 asynchronous interferers with =8. part of the interference, and achieves better performance. Note that the region where the conventional UWB receiver outperforms the soft-limiting receiver is an impractical region of operation as the BER is too large. Fig. 6 shows the case where there are 3 interferers rather than 15. Observe that the benefits of the soft-limiting receiver are greater for a smaller number of interferers, i.e., the reduction in the error floor is greater. Also, the differences in performance between the conventional receiver and the soft-limiting receiver for small values of SNR are smaller. The effects of estimation error in the threshold on the BER performance are investigated in Figs. 7 and 8. It is seen that the receiver with threshold 1 2 S m achieves comparable performance to the receiver with perfect estimation of the threshold S m, and the performance of the receiver with 420

BER of the matched filter with Nu = 4 and Ns = 4 BER of the novel receiver with Nu = 4 and Ns = 4 Threshold 2 S m /2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 14 16 18 20 22 24 26 Fig. 6. The average BER versus SNR of soft-limiting and conventional TH-BPSK UWB receivers assuming 3 asynchronous interferers. Fig. 8. The average BER versus SNR of soft-limiting and conventional TH- BPSK UWB receivers assuming 15 asynchronous interferers with =8for large values of SNR. Three values of threshold are shown for the soft-limiting receiver. threshold 2S m is close to the performance of the matched filter. These observations coincide with the results obtained for the interference-limited case shown in Fig. 4. V. CONCLUSION A novel receiver structure for single-user TH-UWB has been proposed. It was shown to achieve better performance than the matched filter receiver for all SIR values when operating in multiple access interference without Gaussian noise. The proposed receiver also outperforms the matched filter receiver in MAI-plus-Gaussian-noise environments for Threshold 2S m /2 0 2 4 6 8 10 12 14 medium and large values of SNR, where the MAI dominates the Gaussian noise. REFERENCES [1] M. Z. Win and R. A. Scholtz, Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications, IEEE Trans. Commun., vol. 48, pp. 679 691, Apr. 2000. [2] A. Taha and K. M. Chugg, A theoretical study on the effects of interference on UWB multiple access impulse radio, in Proc. Asilomar Conference on Signals, Systems and Computers, Nov. 3-6, 2002, pp. 728 732. [3] V. S. Somayazulu, Multiple access performance in UWB systems using time hopping vs. direct sequence spreading, in Technical Papers in Intel Lab. [4] G. Durisi and G. Romano, On the validity of Gaussian approximation to characterize the multiuser capacity of UWB TH-PPM, in Proc. IEEE Conf. on Ultra Wideband Systems and Technologies, Baltimore, USA, May 20-23, 2002. [5] G. Durisi and S. Benedetto, Performance evaluation of TH-PPM UWB systems in the presence of multiuser interference, IEEE Commun. Lett., vol. 7, pp. 224 226, May 2003. [6] B. Hu and N. C. Beaulieu, Exact bit error rate of TH-PPM UWB systems in the presence of multiple access interference, IEEE Communications Letters, vol. 7, 572 574, December 2003. [7] B. Hu and N. C. Beaulieu, Accurate performance evaluation of time- Hopping and direct-sequence UWB systems in multi-user interference, IEEE Trans. Commun., vol. 53, pp. 1053 1062, Jun. 2005. [8] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [9] J. S. Lehnert and M. B. Pursley, Error probabilities for binary directsequence spread-spectrum communications with random signature sequences, IEEE Trans. Commun., vol. 35, pp. 87 98, Jan. 1987. [10] M. Schwartz and L. Shaw, Signal Processing: Discrete Spectral Analysis, Detection and Estimation. New York: McGraw-Hill, 1975. [11] B. Hu and N. C. Beaulieu, Accurate evaluation of multiple access performance in TH-PPM and TH-BPSK UWB systems, IEEE Trans. Commun., vol. 52, pp. 1758 1766, Oct. 2004. Fig. 7. The average BER versus SNR of soft-limiting and conventional TH- BPSK UWB receivers assuming 15 asynchronous interferers with =8for small values of SNR. Three values of threshold are shown for the soft-limiting receiver. 421