16 fingers. In the case of CM4, on the other hand, non-negligible SINR loss occurs for small to moderately large number of fingers F < 50. This is due to the long delay spread associated with CM4. A rather large number of fingers F > 64 would be required to approach the optimum performance. Fig. 3.7 - 3.9 present the curves for the DS-UWB for finite number of fingers selected by the G A for three scenarios (CM1, N=24), (CM4, N=24) and (CM4, N=6), respectively. The length of the processing windows is the same as that was used in previous plots of corresponding scenario. In general, the B E R performance improves as the number of selected finger increases. For CM1, 35 6 8 10 12 14 16 18 10log l 0(E b/N 0) [dB] Figure 3.8: B E R versus 101og10(f76/^b) using the GA for CM4, N = 24. 32 fingers selected by the G A can approach within 0.2 dB of the optimum (all fingers), while more fingers are needed for CM4 to achieve good performance. For the same number of selected fingers, low data rate achieves a performance better than high data rate for CM4, with respect to the all fingers case. For example, for L E built from the selected 24 fingers in CM4 channels, the gap in power efficiency to all taps is 2.3 dB for N=24, and 3.2 dB for N=6. W L processing provides performance better than linear processing in all scenarios. A large gap in power efficiency is discovered for CM4 and N = 6, with this gap becoming narrower as more fingers are selected. We also consider the performance of the combination of equalization schemes and the GA. Fig. 3.10 shows the B E R curves as function of 10\\ogw(Eb/N0) for the equalizers for CM4 N — 6. These equalizers are built from the fingers selected by the GA. It is observed that W L DFE significantly outperforms 36 4 6 8 10 12 14 16 18 10log,0(Eb/N0) [dB] Figure 3.9: B E R versus 101og10(£6/./Vo) using the G A for CM4, N — 6. L E for the same number of selected fingers, but the gap in power efficiency decreases as more fingers are chosen. For a fixed number of fingers, the perfor-mance for W L MMSE is very close to that of linear DFE. We further observe that the performance for W L DFE with 16 fingers approaches within 0.3 dB of that of L E with 24 fingers at low BER. Fig. 3.11 shows a B E R performance comparison between SP and GA for DS-UWB of CM4, N = 6. It is observed that the G A outperforms the SP for a fixed number of selected fingers, at the expense of additional computational cost. The GA algorithm has a performance significantly better than the SP for a small umber of fingers, but the gap in efficiency between the GA and the SP drops as more fingers are chosen because the GA and the SP might end up selecting same fingers if the number of fingers to be used for detection is large. This observation is supported by Fig. 3.12, which shows the SINR loss 37 BER performance of CM4 N=6 based on Greedy Algorithm - — r — • 1—• 1 r l O l o g ^ / t y IriB] Figure 3.10: B E R performance comparison for 4 equalization schemes with Greedy Algorithm (GA) for CM4, N=6. due to a finite number of selected fingers. A similar conclusion can be drawn for other scenarios (CM1, N = 6) and (CM4, N = 24). Simulation Results for Simulated Annealing By comparing the SINR obtained by the SA with an initial state, to the SINR of that initial state, we can determine the SINR gain of the SA. It is informa-tive to compare the performance between the SA and the GA. Fig. 3.13 shows the average SINR gain, which is obtained by comparing the SINR of the SA with different initial states to a common initial state obtained by SP, for various number of iterations as a function of search space for 100 channel realizations. The search space M is defined as the M strongest pathes. The number of selected fingers is fixed as 16. The initial state is obtained by 38 10log, 0(E b/N 0) [dB] Figure 3.11: B E R performance comparison between Strongest Paths (SP) method and Greedy Algorithm (GA) for CM4, N = 6. GA and SP, respectively. We found that both the geometry cooling and con-stant temperature cooling schedules achieve the same performance, so only the constant temperature cooling schedule results are shown here. It is observed that, regardless of initial state, for a large search space, SA converges in only a few iterations. For example, for search space of all taps, the SINR gain of SA over initial state by SP is very close to that of initial state by GA, after only 8 iterations. The SINR gain is found to increase as the search space expands, for a fixed number of iterations. For the initial state obtained by SP and search space of 32 taps, an SINR gain of 0.95 dB is achieved after only 4 iterations. If the search space is increased to the 48 strongest taps, the resulting SINR gain increases to 1.03 dB. However, if the search space is enlarged to all taps, only 39 0| -2 -3 1 /• * / t I J / i / ' / f / 1 / ' / ' / ' /...I / ' I 1 / ' / ' / ' l 1 / ' / 1 I ' I ' / 1 1 I 1 1 •'• • Greedy Algorithm — - — - Strongest Pathes method i i i 0 50 100 150 200 250 300 Number of fingers Figure 3.12: Average SINR loss as function of number of fingers for the SP and the G A for CM4, A' = 6. another 0.1 dB gain can be achieved, and this small gain comes at the expense of additional computations. Furthermore, a notable SINR improvement is achieved with SA compared to the SP method; on the other hand, no significant SINR improvement is found compared to the GA. More specifically, gains of up to 1.35 dB and 0.3 dB are achieved compared to the SP method and the GA, respectively. We also ob-serve that SA with search spaces of 100 strongest paths results in performance very close to that with full space search. Finally, it is observed that 8 iterations are adequate for SA because only little gain is obtained by performing more iterations. In general, GA is preferred for finger selection because it can achieve per-40 • Initial state obtained by GA, 16 iterations Initial state obtained by GA, 8 iterations Initial state obtained by GA, 4 iterations Initial state obtained by SP, 16 iterations 0 Initial state obtained by SP, 8 iterations - •©- - Initial state obtained by SP, 4 iterations * Gain of GA over SP, 0 Iteration 0 50 100 150 200 250 300 Search space (fingers) used by Simulated Annealing Figure 3.13: Average SINR gain as function of search space used by Simulated Annealing for selecting 16 fingers with initial state obtained by SP method, for CM4, N = 6 at SNR=14dB. formance very close to the optimum while incurring moderate computational complexity. We can also conclude that a search space of 64 is adequate to get a good performance since little gain is obtained if the number of taps increased. Comparison between the GA and the RAKE combining followed by equalization from [13] It is proposed in [13] that equalization be employed at the output of the R A K E combiner to mitigate ISI is . The structure of these receivers is simple because equalization is processed at the symbol level. In this section, we compare the performance and complexity of the receiver between the receiver in [13] and GA. For a fair comparison, we use the same set of channel realizations in both cases. 41 Figure 3.14: B E R versus 10 log10(jEVJV0) of the G A and R A K E combining [13] for CM4, N = 24. Fig. 3.14 and 3.15 shows the BER versus 10logw(Eb/NQ) of the G A and R A K E combiner [13] for CM4, AT = 24 and CM4, N = 6, respectively. A R A K E com-biner with 16 fingers is employed to capture multipath components. We can observe that, for CM4 and N = 24, the R A K E combiner followed by linear MMSE equalization results in a gain of 0.6 dB over the G A with 16 fingers, and 0.8 dB for the widely linear MMSE equalization scheme. For CM4 and N = 6, these gains are 1.1 dB and 1.0 dB, respectively. It seems that R A K E combiner followed by a equalizer can result in performance gain over the GA, however, we need to compare the complexity of these two algorithms before making any conclusions. At the R A K E combiner, in our comparison, each of the employed 16 fingers 42 Figure 3.15: B E R versus 10logw(Eb/N0) of the GA and R A K E combining [13] for CM4, N = 6. has to sample N consecutive received chip-rate signals. Equalization schemes are processed at the symbol rate at the output of the R A K E combiner. The computational cost of the equalization is dominated by the matrix inversion to decide the filter coefficients for LE . The matrix inversion is done once, and the cost is 0((Lchannei + l ) 3 ) , with Lchananei denotes the length of overall chip rate IR. At least (16 + 24 — 1 = 39) signals are sampled at R A K E to detect one bit (in the extreme case, 16 x 24 = 384 signals are needed), and this will result in high complexity for hardware implementation. While for the GA, each of the employed 16 fingers needs to capture one chip-rate signal, and the equalization schemes are also processed at chip rate. 16 received signals are used for symbol detections. The overall computational cost is mostly caused by the matrix inversion, which is 0(163). Therefore, the 43 overall computational cost of R A K E combiner followed by equalizer is higher than that of the GA. In terms of hardware implementation, R A K E combiner followed by equalizer is more complex than the GA. By comparing Fig. 3.8 and Fig. 3.14, we observe that the G A with 20 fingers achieves performance slightly better than that of R A K E combiner with 16 fingers followed by an equalizer. We conclude that the GA is preferred over the R A K E combiner for implementation due to its simple structure and low complexity. 44 Chapter 4 Channel Estimation for a Single In a UWB system, an MMSE receiver (or other receivers discussed in Chap-ter 3) is typically employed to detect information symbols. To fully capture the signal energy spread over multiple paths, channel parameters values are necessary to construct an MMSE receiver. However,channel information is not known a priori. Recently, maximum-likelihood (ML) channel estimation methods have been proposed in [37], which provide a theoretical guidance for evaluation of other channel estimators. However, ML-based channel estima-tion methods are computationally complex. In this chapter, we will investigate several low-complexity channel estimation schemes for the DS-UWB systems. A DS-UWB system with a single user was considered in the previous chapter. We also consider in this chapter channel estimations for a single user. We study various schemes for channel estimation, depending on the availability of training sequences. In this section, we consider the problem of estimating the physical channel of a given user from the received signal, based on the knowledge of the spreading DS-UWB 4.1 Blind Channel Estimation 45 code for that user. The transmission model used here is the one described in Chapter 2. In practical applications, the spreading code of the desired user is usually known to the receiver. The equivalent overall spreading sequence for the user is h = Cg (4.1) where C is the (N + Lf) x (L/ +1) spreading matrix formed from the spreading code of the user, g is the UWB channel vector, and Lf is the order of the channel. c[N-l] 0 ... 0 c[N - 2] c[N - 1] C = . c[N-2] ... . (4.2) 0 0 c[l] c[0] g = g[o] (4.3) [g[Lf] g[qf-l] Assume a symbol with decision delay ko unit is desired. Recall from (2.19)that the effective spreading sequence for the desired symbol is the k0th column of H, denoted as Hfc0 which can be expressed as H f c 0 = Cg (4.4) where 0 C 0 (4.5) is an N(qf + 1) x (qj + 1) matrix and A 2 > • • • > Ajv( g / + 1) be the eigenvalues of 46 Cr. Since the matrix H has full column rank, the signal component of the covariance matrix C r (i.e., HH H ) has rank (qf+L+1). Therefore, we have [16] A; > for i = 1,..., qf+L+1, Xi = ol for i = qf+L+2,..., N(qf+1), Because the ambient channel noise is white, by performing an eigendecompo-sition of the matrix C r , we can obtain [16] C r = U 8 A s U ? + <#J n U? (4.7) where A s = diag(Ai,...,Ajv(9/+i)) contains the (qf+L+1) largest eigenvalues of C r in descending order, U s = [u\\- • -uqf+L+i] contains the corresponding or-thogonal eigenvectors, and U n = [uqf+i+2- • •ix./v(g/+i)] contains the (N(qf+1)-(qf+L+1)) orthogonal eigenvectors that correspond to the eigenvalues o\\. The column space Us is called the signal subspace and its orthogonal complement, the noise subspace, is spanned by the columns of U n . The focus in this section is on the performance analysis of the subspace-based blind detection in a multipath environment, assuming the signal subspace com-ponents are known. If a training sequence is not available, the multipath channel g can be estimated blindly by exploiting the orthogonality between the signal and noise subspaces. Specifically, since U n is orthogonal to the column space of H, and Hfc0 is in the column space of H, we have U ^ H f c o = U ^ C g = 0. (4.8) Some constraint on g needs to be imposed in order to avoid the trivial solution g = 0. In particular, it is desirable to constrain the energy of the user of interest to a constant. It is proposed in [38] that, g, an estimate of the channel response g can be obtained by computing the minimum eigenvector of the matrix ( C T U n U ^ C ) . The resulting linear MMSE detector is determined by / = U ^ U f C g . (4.9) 47 Note that the estimated channel response is constrained to have unit norm, but this constraint will not affect the result of symbol detection because the output of the detector is different only by a real (nonzero) multiplicative term. 4.2 Training-Based Channel Estimation In this section, we investigate the estimation of the UWB channels with train-ing sequences. Commonly used channel estimation techniques are the least squares (LS) [39] and maximum likelihood (ML) estimation. Because the UWB channel delay spread is very large, the application of these methods is computationally complex and is not practical. We need to find a compu-tationally simple algorithm that can achieve good performance. A matching pursuit (MP) algorithm [41] is investigated in [42] for estimation of the sparse channels. In [18], an MP followed by cancelation (MPC) algorithm is investi-gated to estimate C D M A channels with large delay spreads. Performing better than LS algorithm, M P C algorithm is discovered to have significantly low com-plexity compared with LS algorithm. Since the application of the M P C on the DS-UWB has not been previously studied, we will investigate the performance of this algorithm for the DS-UWB and the resulting complexity. The UWB systems are utilized for short-range high data rate transmission, and the channels stay unchanged duration the transmission a data packets. We fo-cus only on time-invariant channels.The M P C algorithm is now discussed. Starting from the UWB system model (2.20) for a single user, an alternative representation of the signal model as r = Th + n, (4.10) where r and n are as defined in (2.20), and h denotes the chip-rate generalized UWB channel impulse response. T is a Toeplitz matrix consisting of chip-rate 48 sampling training symbols t[n] 0 ... 0 t[n-l] 0 0 t[n] 0 ... 0 t[n - 1] 0 0 . ... 0 0 ... 0 t[n + l] 0 ... 0 t[n] 0 0 t[n + l] 0 ... 0 t[n] t[n + N t r a i n i n g - 1] 0 ... 0 t[n + Ntraining - 2] 0 T = (4 11) where Ntraining denotes the length of training symbols and T is of dimension NNtraining x (L/ +1)- N is the spreading factor, and Lf is the order of the chip rate channel impulse response. The problem now is to approximate the vector r in terms of a linear combination of columns from the matrix T, i.e., from (4.10), we must find h such that Th « r. The M P C algorithm described next gives an efficient method for obtaining a suboptimal solution to this problem. The M P C algorithm consists of two steps: (1) The matching pursuit (MP) step and (2) the cancellation (C) step. • The matching pursuit (MP) step: The MP step approximates the value of each channel coefficient. We first find the column in the matrix T = [t1? t2, ...,tjr,/+1], which is best aligned with the signal vector r 0 = r and this is denoted [41]. Then the projection of r along this direction is removed from r 0 and the residual r i is obtained. Now the next column from T, tfc2, which is best aligned with r i and a new residual, r2, is found. The algorithm proceeds by sequentially choosing the column which best matches the residual until the number of channel coefficients has been se-lected (we assume the number of channel coefficients is known, however, the value of each coefficient needs to be determined). The pth iteration is described in the follows. 49 The vector from T most closely aligned with the residual r p _i is chosen, where the alignment is measured as the projection of the residual onto the vector, i.e., I t^r _ i I2 kp = arg max (4.12) ' I w lr The new residual vector is then computed as (tgrp_i)tfc rkp = rkp-i j|-7—p— (4.13) and the channel coefficient at position kp is hkp — (tj^rp_i)/ || tkp || 2. The iteration is repeated until the number of coefficients has been selected. • The cancelation (C) step: The performance of MP deteriorates signif-icantly due to column correlations in the matrix T [43]. To overcome this performance degradation, the cancelation step uses the already estimated channel coefficients to remove interference from the received vector in or-der to identify the channel coefficients more accurately. The interference cancelation is as follows. Lf+l £fcp = t £ ( r - £ tkphki)/ || tfcp || 2 . (4.14) The updating is stopped when the change in the coefficients falls below a certain threshold, but it is observed in our simulation results that the cancelation step needs to be performed only once to achieve good performance. 4.3 Complexity Comparison between the MPC and other Channel Estimation Algorithms In this section, we compare the computational complexity of the M P C al-gorithm with other channel estimation algorithms. Commonly used channel estimation techniques include the LS and the M L estimation. Since the com-plexity of the M L is higher than that of the LS, we will compare here the 50 Table 4.1: Complexity comparison for channel estimation algorithms. M P C 0(2(L, + 1)) LS 0((Lf + l ) 3 + (L; + l)NNtraining + (Lf + l)NNtraining) complexity of the M P C and the LS. Table 4.1 shows complexity comparison for channel estimation algorithm. LS based estimations [46] can be obtained as h = (T T T)- 1 T T r (4.15) The complexity of LS is dominated by inversion of an ((L/+1) x(L/+l ) ) matrix (TTT), and it is not practical for long spread delays UWB channels. The complexity of M P C mainly comes from the M P step, and the computational cost of this step is the sum of (4.12) (0(Lf + 1)) and (4.13) (0(Lf + 1)). The comparison clearly shows the complexity of the M P C is much lower than that of the LS algorithm. 4.4 Results and Discussions In this section, simulation results for the blind channel estimation and channel estimation with training sequences for the BPSK DS-UWB systems are pre-sented. In Section 4.3.1, we present results for the blind channel estimation. The results for channel the estimation with training sequences are presented in Section 4.3.2. For all simulations, qf = 15 is for (CM4, N = 24) and qf = 50 for (CM4, N = 6). 4.4 .1 Simulation Results for the Blind Channel Estima-tion Fig. 4.1 shows the B E R performance of the DS-UWB system with N = 24 and CM4 as function of 101og10(£^f,/A/o). The simulation shows the performance of the subspace-based blind detection in multipath, assuming the signal subspace components U s of (4.7) are perfectly known. This is a lower performance 51 0 Blind Linear MMSE equalizer ; Exact Linear MMSE equalizer ... 10lo» 1 0 (E l /N 0 ) [dB] Figure 4.1: B E R versus 101og10(E,6/A''o) for Blind Channel Estimation for CM4 and N=24. bound for the subspace-based blind channel estimation, because in practical applications, perfect knowledge of C r is not known at the receiver. An L E blind detector with (N + Lf) taps is built from the estimated channel. No finger selection schemes are performed in this simulation since all taps are used. We compare the performance of the blind detector with the performance of the all-tap L E assuming perfect channel information. It is observed that the blind L E suffers a performance loss of approximate 8 dB -compared to the exact L E for at a B E R of 10 - 3 . Therefore, the blind channel estimation cannot produce a good performance for the DS-UWB systems. This result is consistent with the result shown in figure 2.14 of [40]. We can conclude that the blind channel estimation methods are not suitable for the DS-UWB due to the long delay spreads of the UWB channels. 52 BER performance of UWB CM4 N=24 applvng MPC algorithm using 100 training bits and two-stage adaptive filter —fa— MMSE equalizer based on MPC algorithm with 100 training symbols ; -- 8 — MMSE equalizer based on MPC algorithm with 150 training symbol! .. . • \" D \" MMSE equalizer based on MPC algorithm with 200 training symbol! • -Exact linear MMSE equalizer 10log t 0(Eb/N0) |dB] Figure 4.2: B E R versus 10log10(Eb/N0) for channel estimation with training symbols of different length for CM4 and N=24. 4.4 .2 Simulation Results for Channel Estimation with Training Sequences Fig. 4.2 shows the B E R performance of the DS-UWB systems with N = 24 and CM4 as function of 101og10(£b/./Vo), and the B E R performance assuming exact channel information is available. The receiver first estimates the UWB channel using the M P C algorithm with training sequences, an L E is then built from the estimated channel. We assume the UWB channels are time-invariant during the channel estimation and the data transmission. We compare the B E R performance of the M P C algorithm with training se-quences of 100, 150, and 200 symbols, respectively. As expected, the B E R performance improves as the length of training sequence increases. At the B E R of 10 - 5 , the M P C algorithm a training sequence of 200 symbols achieves 53 Figure 4.3: B E R versus 10\\og10(Eb/N0) for channel estimation with training symbols of different length for CM4 and A=6. a performance gain of 0.6 dB over that with a training sequence of 100 sym-bols. We further observe that the M P C algorithm with 200 training symbols suffers a performance loss by approximate 1 dB compared to L E with exact channel information. Fig. 4.3 shows the B E R performance of the DS-UWB system for N = 6 and CM4 as function of 10\\ogw(Eb/N0), and the B E R performance assuming exact channel information is available. At a B E R of 10 - 5 , the M P C algorithm with 500 training symbols achieves a performance gain of 1.2 dB over that with 200 training symbols. It is observed that the M P C algorithm with 500 training symbols suffers a performance loss by approximate 0.6 dB compared to the L E with exact channel information. 54 SINR of UWB CM4 N=24 applying MPC algorithm at SNR = 14dB 15 r 1 i 1 1 1 1 1 r 8I i I I I l I I l 1 20 40 60 80 100 120 140 160 180 200 Number of training symbols Figure 4.4: SINR of M P C algorithm versus number of training symbols of different length for CM4 and iV=24 at SNR = 14dB. The results in Figs. 4.2 and 4.3 indicate that the channel estimation by the M P C algorithm with a training sequence is suitable for the DS-UWB systems because with only a few hundreds training symbols, the performance is rela-tively good. We also investigate the SINR performance of the M P C algorithm with a train-ing sequence. Fig. 4.4 shows, at SNR = 14 dB, the SINR obtained by the M P C algorithm with different number of training symbols for A=24 and CM4. As expected, the SINR improves as number of training symbols increases. The SINR improves rapidly with increasing training symbols when the number of training symbols is less than 80, but the SINR improvement is small as more training symbols are transmitted; for example, difference of only 0.5 dB is ob-served between training sequences of 200 symbols and 100 symbols. We can 55 20 40 60 80 100 120 140 160 180 200 Number of training bits Figure 4.5: Channel estimation error of M P C algorithm versus number of training symbols of different length for CM4 and JV=24 at SNR = 14dB. conclude that, for (N=2A and CM4), 200 training symbols are adequate for the channel estimation with the M P C algorithm. This conclusion is further supported by the simulation results shown in Fig. 4.5. Fig. 4.5 shows the channel estimation error versus number of training symbols for N=24 and CM4, at SNR = 14dB. qf = 15. The channel estimation error is defined as the Euclidean distance between the estimated channel and the exact channel averaged over the number of channel coefficients [18]. Assuming h is the estimated of exact channel h, the Euclidean distance de can be expressed as dE =|| h - h || 2 . (4.16) Recall that we assume the number of channel coefficients is known. The chan-nel estimation error is observed to decrease as the number of training symbols 56 Figure 4.6: SINR of M P C algorithm versus number of training symbols of different length for CM4 and N = 6 at SNR = 14dB. increases, but after the number of training symbols reach a certain value (i.e., 80), the channel estimation error does not improve much even if more training symbols are used. Fig. 4.6 shows the SINR versus the number of training symbols for N=6 and CM4, at SNR = 14 dB. It is observed that the SINR at the output of the L E from the estimated channel improves as more training symbols are used for channel estimation. The channel estimation error curve for CM4 and N = 6 is shown in Fig. 4.7, and it is observed to undergo small changes when the number of training symbols exceeds 300. In practical applications, it is desired to select a limited number of taps for symbol detection, based on the estimated channels, for the sake of simple 57 t i i i r 1 0- ' l i i : i I i 1 i i 1 50 100 150 200 250 300 350 400 450 500 Number of training symbols Figure 4.7: Channel estimation error of the M P C algorithm versus number of training symbols of different length for CM4 and N = 6 at SNR = 14dB. hardware implementation. Therefore, the performance of the combination of channel estimation schemes and finger selection schemes is desired. Fig. 4.8 presents the B E R performance versus 10\\og10(Eb/N0) for channel estimation with the M P C algorithm with 100 training symbols for CM4 and iV = 24. Different number of fingers are selected with the GA, and an L E equalizer is built correspondingly. Parameters for L E are identical to those used in chapter 3. It can be observed that the B E R performance improves as the number of selected fingers increases. The L E with 48 fingers by the G A approaches the L E with all taps within 0.6 dB. Fig. 4.9 presents B E R versus 101og 1 0(£ ,t/A 0) for channel estimation with the M P C algorithm with 200 training symbols for CM4 and N = 6. Different number of fingers are selected with the GA, and an L E is built. It can be 58 T Figure 4.8: B E R versus 10\\ogw(Eb/N0) for channel estimation with the M P C algorithm with 100 training symbols, finger selection by the GA, for CM4 and N = 24. observed that the B E R performance improves as the number of selected fingers increases. The L E with 48 fingers chosen by the GA is within 0.8 dB of the L E with all fingers. Training symbol length on the order of hundreds can provide an acceptable channel estimate for the DS-UWB systems. Because DS-UWB systems oper-ate at very high data rates, the transmission of the overhead is small. We also investigate the performance of the different combinations of channel estimation schemes and other equalization strategies. Fig. 4.10 shows the performance comparison between L E and W L E for CM4 and N — 24. The channels are first estimated with the M P C algorithm with 100 training sym-bols, and 16 fingers are selected with the GA from the estimated channel. 59 A 6 8 10 12 14 16 18 IOIob^ Ej/N, IdSJ Figure 4.9: B E R versus 101og10(Eb/A70) for channel estimation with the M P C algorithm with 200 training symbols, finger selection by the GA, for CM4 and N = 6. Finally, L E / W L E is built for symbol detections. The B E R performance of L E and W L E with exact channel information are also shown. W L E is found to achieve a small performance gain of approximate by 0.2 dB over L E at B E R of 10~5. A performance loss by approximate 1.0 dB is observed between the L E with M P C algorithm and that with exact channel information. As for CM4 and N = 6The performance gap between L E and W L E is observed to be about 0.5 dB at low SNR, as from Fig. 4.11. The channels are first estimated with the M P C algorithm with 200 training symbols, and 16 fingers are selected by the G A from the estimated channel. Finally, L E / W L E is built for symbol detections. The 16-finger L E from the estimated channels suffers a performance loss of about 2.0 dB to that with exact channel information. 60 1 1 T 1 I I Figure 4.10: B E R versus 101og10(£ ,fc/A /0) comparison for channel estimation with the M P C algorithm with 100 training symbols for CM4 and N = 24, comparison is made between L E and W L E , finger selection with the GA. Also shown: B E R performance of L E and W L E of exact channels. 61 Figure 4.11: B E R versus 101og10(i?(,/iVo) comparison for channel estimation with the M P C algorithm with 200 training symbols for CM4 and N = 6, comparison is made between L E and W L E , finger selection with the GA. Also shown: B E R performance of L E and W L E of exact channels. 62 Chapter 5 Interference Suppression We have investigated various equalization schemes for single-user DS-UWB systems in Chapter 3, assuming no other interferences. But in practical ap-plications, UWB systems operate as multiple-access systems, in which mul-tiple users share the same radio resources. Similar to CDMA, a DS-UWB system assigns channels in a way that allows all users to use all frequency resources simultaneously, through the assignment of a spreading code to each user. A DS-UWB user of interest is therefore interfered with by other possi-ble DS-UWB users in the same network, and this kind of interference is called Multiple-access Interference(MAI). Also, a DS-UWB user might suffer from in-terference arising from other devices operating in the same band, such as IEEE 802.11a that operats in the same 5.0 GHz band. Interference mitigation is a major factor in the design of DS-UWB receivers. In this chapter, we discuss al-gorithms used for joint equalization and interference suppression for DS-UWB. It is well known that multiuser detection [7] can achieve good performance in multiple access systems. In addition, the coefficients of the receiver filters can be conventionally adapted to the channel and interference situation by using data-aided [19], [44] or blind [7], [45] adaptive algorithms. In practical UWB applications, it is reasonable to assume the receivers are decentralized receivers, i.e., the receivers exploit the knowledge of the spreading code and propagation channel of the user of interest only. In this chapter, we are concerned with 63 decentralized adaptive receivers, i.e., receivers formed by the concatenation of an adaptive filter with a suitable detection operation acting on the filter output. Data-aided recursive least square (RLS) and least mean square (LMS) algo-rithms have been investigated in [46]. The rate of convergence of the RLS algorithm is typically an order of magnitude faster than that of the LMS algo-rithm, at the expense of more computational cost. This is because the step-size parameter in the LMS algorithm is replaced by the inverse of the correlation matrix of the input vector [46]. The computational cost of RLS algorithm is impractical in UWB channels with large delay spreads. For this reason, this chapter focuses on the interference suppression with the LMS algorithm. Fur-thermore, it has been shown that under certain conditions, the performance of linear receivers can be significantly improved if not only the received signal, but also its complex conjugate, is precessed [7], [45] - [49]. It is shown in [21] that W L algorithms require a slightly lower computational complexity than their linear counterparts. In the next section, we briefly talk about the widely linear LMS (WL-LMS) algorithm and the widely linear minimum output energy (WL-MOE) algo-rithm. We assume that the CSI of the desired user is known at the receiver. The desired user is interfered with by other unknown asynchronous DS-UWB users (for MAI suppression) or NBI (for NBI suppression). The WL-LMS (WL-MOE) algorithm starts from the filter that is built with linear MMSE equalization scheme (3.2) based on the CSI of the desired user. 5.1 WL-LMS Algorithm The WL-LMS algorithm can be applied to recursively update the MMSE filter if the desired user transmits a training sequence or if reliable decisions on the desired user's transmitted data symbols are available. 64 The WL-LMS algorithm minimizes the instantaneous squared error signal e^sik] = t[k] - Re{^s[k]Hr[k]} ( 5 . 1 } where t[k] denotes the training symbol at time index k. f^Msik] a n d r[^] a r e the adaptive filter and the received signal vector at time k, respectively. The WL-LMS update equation is [21] i^Msik + 1] = & W + l*Y£8[k]T[k] (5.2) where is the step size. The only difference between the linear LMS algo-rithm [46] and the WL-LMS algorithm is the different definition of the error signal. For the LMS algorithm, eLMS{k] = t[k] - fLMs[k]\"r[k]. The WL-LMS algorithm is obtained from the linear LMS algorithm simply by replacing the LMS error signal by its real part. Hence, the computational complexity of the WL-LMS is slightly lower than that of the conventional LMS algorithm [21]. The convergence and SINR analysis for the WL-LMS algorithm is shown in [21]. The steady-state SINR of the WL-LMS algorithm is found to be 1 T HLMS + SINRivt where SINRivt denotes the SINR of the widely linear MMSE receiver from equation (5) of [21]. For small step size ji where EK is the signal energy of «th user and E\\ the energy of desired user. LT0W x K is the dimension of transmission matrix defined similarly to that of (2.19). 5.2 WL-MOE Algorithm The adaptive filter can be updated blindly using the W L M O E criterion if no training sequence is available. The only requirement for the W L - M O E algo-rithm is the knowledge of the desired user's spreading sequence. 65 The derivation of the W L - M O E algorithm can be found in [21]. The canonical representation for the W L - M O E filter is introduced as [21] ^Most^] — P i + X M O E [ ^ ] (5-5) where f^o£[fc] denotes the updated W L - M O E filter at time index k, and p x denotes the desired user's spreading sequence. x^ost^] is the component orthogonal to the desired user's spreading sequence, and it fulfills the following equation [21] < Pi,WL= R e { p f x ^ [ A ; ] } = 0. (5.6) At each updating iteration, xj^^f/c], the component orthogonal to the desired user's spreading sequence, needs to be determined such that the variance of the output energy M O E H / L = R e l f ^ ^ / c ^ r ^ ] } is minimized. The widely lin-ear minimum-output-energy (WL-MOE) algorithm is obtained using gradient descent algorithm and is given by [21] i[fc] as training symbol. The output at the decision device of the second stage also follows the detection rule: b2[fc] = Re{z2[k}} = Re{(m^s[k])Hv[k]}. (5.11) Since the outputs of both stages of the two-stage receiver are always available, their SINRs are easily measured and the better one can be selected. While for a dual-mode receiver with a single adaptive filter switching between adaption modes, the output SINR must be compared with an empirical threshold in order to determine which adaption mode is to be used. A bad choice of this threshold could make the dual-mode receiver work in the wrong adaption mode most of the time [22]. The advantage of the two-stage receiver is obtained at the cost of an increased computational complexity because two adaptive algorithms must be run in parallel. 5.4 Narrow Band Interference DS-UWB systems unavoidably suffer from NBI (this kind of interference is called narrow band because its bandwidth is relatively small compared to UWB systems, but they usually transmit at much higher PSD), because they operate 68 ISM Band P C S IEEE 802.11b IEEE 602.11a 1.6 1.9 2.4 3.1 4 5 10.6 Frequency (GHz) Figure 5.2: Spectrum crossover of the NBI in UWB systems. in the same frequency band as UWB systems. This narrowband interference can sometimes be so serious that the UWB communication is totally prevented. Therefore, NBI suppression is of primary importance for UWB systems [50]. When dealing with NBI in UWB systems, it should be considered that there are some potential interferers with predetermined center frequencies and band-widths. The most serious interferer among all the potential interferers is the IEEE 802.11a W L A N (see Fig. 5.2). In this section, we will focus on IEEE 802.11a W L A N suppression in UWB communications. The IEEE 802.11a W L A N system uses 52 sub-carrier orthogonal frequency division multiplexing (OFDM). This system has a fixed bandwidth of 20 MHz. The operating bands of IEEE 802.11a are three 100 MHz wide frequency bands: 5.15-5.25, 5.25-5.35, and 5.725-5.825 GHz. Based on modeling an NBI as a single carrier BPSK modulated waveform, different types of NBI suppression have been studied for UWB systems [24] and [51]. A maximum ratio com-bining (MRC) rake receiver is employed in [51] to suppress NBI for DS-UWB systems by considering NBI as a sinusoidal tone. With the bandwidth as large as hundreds of MHz, IEEE 802.11a interference can not be modeled as a BPSK modulated signal or as a sinusoidal tone any more. Hence, many proposed methods are ineffective at suppressing IEEE 69 . Barid-sto{> filter Figure 5.3: Receiver with Bandstop filter for NBI suppression. 802.11a interference. Recently, an IEEE 802.11a signal was demonstrated to be modeled as a bandlimited additive white Gaussian noise [56]. In [23], the interference is first approximated using a singular value decomposition (SVD) algorithm, and it is later subtracted from the received signals, but this suppres-sion involves matrix decomposition and thus impose a heavy computational burden. The bandwidth of an IEEE 802.1 l a system is known to be 20MHz but its center frequency can be anywhere in the operating band. If the center frequency of this kind of single narrowband interferer is unknown, the two-stage adaptive filter structure discussed early can be applied to mitigate the NBI. If the center frequency is known by the receiver, a bandstop filter is proposed for interference removal. The resulting receiver is shown in Fig. 5.3. In the proposed receiver, the front-end band-stop filter will eliminate the narrow band interference IEEE 802.11a out of UWB signal band. The filtered signal then is fed to any of the equalizers discussed in Chapter 3. The advantage of this receiver structure includes complete elimination of narrow band interference, regardless of the PSD of the interferer signal. This advantage is crucial because UWB signals are restricted to very low PSD (-41.3 dBm/MHz) while IEEE 802.11a signals have a PSD that is tens of dB higher than that of the UWB signals [23]. 70 4 6 8 10 12 14 16 18 lO log^ lE^ , ) |dS| Figure 5.4: B E R versus 101og10(£b/A/V0) for MAI suppression for CM4, N = 24. 5.5 Simulation Results and Discussions In this section, we present the simulation results for multiple user interference and narrow band interference with the two-stage widely linear adaptive filter. 5.5 .1 M A I Suppression The CSI of the desired user is assumed known at the receiver, and the desired user's communication is interfered with by one BPSK DS-UWB user. The spreading code for the interferer is as shown in Table 3.2. Fig. 5.4 shows B E R versus 10 logw(Eb/N0) for multiple access interference for CM4, N = 24, with the interferer and the desired user having equal power. qf = 15 is used. The two-stage W L adaptive filters start with an L E with all taps. The L E is built based on the CSI of desired user. While for finite 71 T — ^ — Interferer's CSI unknown, F=32 taps by GA, Linear MMSE filter without interference suppressior ;;; O Interferer's CSI unknown, F=32 taps by GA, suppression with two-stage adaptive filter 4 6 fl 10 12 14 16 18 lO-log^/H,, ) |dB| Figure 5.5: B E R versus 101og10(E6/iVo) for MAI suppression for CM4, N = 6. number of taps, the two-stage adaptive filers are initialized by the linear MMSE equalizer formed with fingers being selected by the GA. The step sizes used in the first stage and the second stage of the adaptive filter are 10 - 4 and 3 x 10~3, respectively. It is observed that, for both all-tap and finite-tap cases, the two-stage adaptive filter can effectively suppress multiple access interference and provide a performance very close to the case where the CSI of the interferer is known, at the cost of the extra adaptive filter structure. Comparing the performance of the case with one unknown interferer suppressed by the adaptive filter to that with the interferer known at the receiver, at low BER, we observe that the gap in power efficiency is 0.1 dB for all-tap filter and 0.2 dB for 24-tap filter. The B E R versus 101og10(^b/Ao) for multiple access interference for CM4, N = 6 is shown in Fig. 5.5. The two-stage adaptive filter, is again observed to be 72 SINR of two-stage adaptive filter for CM4 N=24 * SINR at the 2nd stage of two-stage WL adaptive Alter • - * - - SINR at the 2nd stage of two-stage linear adaptive filter SINR at the 1 st stage of two-stage WL adaptive filter . - - - SINR at the 1 st stage of two-stage linear adaptive filter SINR at the receiver without MAI suppression Number of Iterations Figure 5.6: SINR versus number of iterations for MAI suppression for CM4, N = 24 at SNR = 14dB. able to effectively suppress interference and result in good performance, for both all-tap and finite-tap. Fig. 5.6 shows SINR versus number of iterations for multiple access interfer-ence for CM4, N = 24 at SNR = 14dB. The SINR at different stages of the W L adaptive filter and the linear adaptive filter are compared in the figure. The theoretical steady-state SINRs is calculated using equation (5.3), which calculates the steady-state SINR obtained by using training sequences. We ob-serve that the theoretical and simulated steady-state SINRs coincide for this scenario. The simulated SINRs are observed to increase as the number of itera-tions increases, and only negligible changes are discovered after 500 iterations. The simulated SINR with 1000 iterations is 0.1 dB lower than the theoretical value. The theoretical SINR value is higher than the simulated SINR value because the theoretical SINR is the result of using training symbols, while the 73 SINR of two-stage adaptive filter for CM4 N=6 . . » . i . . . . . . . . . . . . 0 200 400 600 600 1000 1200 1400 1600 1600 2000 Number of Iterations Figure 5.7: SINR versus number of iterations for MAI suppression for CM4, N = 6 at SNR = 14dB. two-stage W L LMS adaptive filters do not require any training symbols. It can be observed that the W L two-stage adaptive filter achieves a higher SINR than its linear counterpart, with less computational cost. The SINR at the output of 1st stage of W L adaptive filter is very close to that of the 1st stage of the linear adaptive filter, but the SINR at the output of the 2nd stage of the W L two-stage adaptive filters is 0.13 dB higher than that of the output of the 2nd stage of the linear two-stage filters. Suppressing MAI by the W L two-stage filters leads to an SINR gain of 2.3 dB over that without MAI suppression. Similar results can be observed in Fig. 5.7, which presents SINR versus number of iterations for multiple access interference suppression for CM4, N = 6 at SNR = 14dB, qf = 50. We can observe that the employ-ment of the W L two-stage filter for MAI suppression achieves an SINR gain of 4.5 dB over that without MAI suppression. Also, the simulated steady-state 74 Figure 5.8: B E R versus 10log10(Eb/N0) for single NBI suppression for CM4, N = 24. SINR gain of the W L two-stage filter over its linear counterpart after 2000 iterations is about 0.7 dB. The above observations indicate that the W L two-stage filter structure is a good choice for MAI suppression for DS-UWB systems because: 1) it can achieve significant SINR gains at the output over equalization without MAI suppression; 2) it has a structure simpler than its linear counterpart while resulting in better performance. 5.5.2 N B I Suppression It has been shown MAI can be effectively suppressed by the two-stage adap-tive filter. We now investigate the performance of NBI interference suppression with the two-stage adaptive filter. Single narrow-band interference suppres-\\ I r r Single NBI, SIR = -20dB, Linear MMSE - • - Single NBI, SIR = -20dB, iwo-atage adatpive filtei Single NBI. SIR = -10dB, Unear MMSE • Single NBI. SIR = -1 OdB, two-stage adatpive filtei Figure 5.9: BER versus 101og10(Efe/A^o) for single NBI suppression for CM4, N = 6. sion with the two-stage adaptive filter is first considered. Recall that the single narrow-band interference has a fixed bandwidth of 20 MHz. We will look at different scenarios where the center frequency is known and unknown at the receiver, respectively. We also study the worst situation narrow-band interfer-ence, with IEEE 802.11a signals occupying the entire 300 MHz bandwidth. Fig. 5.8 plots B E R versus 101og10(Eb/A /o) for a single narrow band inter-ference for CM4, iV = 24. qf = 15 is used. It is observed that random NBI (its center frequency is unknown to the receiver) has significant impact on the UWB systems and results in a poor performance at high signal-to-interference-ratio (SIR), if no interference suppression is employed. A UWB receiver with two-stage adaptive filter, on the other hand, can work well under strong narrow-band interference, even at SIR = -20dB. A similar conclusion can be drawn for scenario CM4 and N = 6, as shown in Fig. 5.9. We observe 76 10-1 10\"* - 1 1 1 1 1 — 9 — worst case NBI, bandstop titer followed by MMSE filter - if — worst case NBI, bandstop filer followed by two-stage adaptive filte single NBI. SIR = -20dB, with interference suppression —B— single NBI, SIR = -10dB, with interference suppression - - - single NBI, with band-stop Alter and MMSE filter . _ . - , Linear MMSE filter, without Interference ^ N J 5 * ^ ^ ^ - . » v > ^ . . . . x ^ . . « > s s . ^ s * * ^ - : ' : ; : : : : : ; : : : : : : : ; : ; : ; : , * » X \\ -~ -i i i i \\ 4 6 8 10 12 14 16 lOloo^lE,/^) [dB] Figure 5.10: B E R versus 10 log10(Eb/A^o) for NBI suppression with band-stop filter for CM4, N=24. that at low SNR, two-stage adaptive filter gives a performance slightly worse than the linear MMSE equalizer, and this can be explained as follows. At low SNR, the detection errors in the first stage (nondata-aided stage) of the two-stage filter affects the detection of the data symbols. But at high SNR, the two-stage adaptive filter greatly outperforms the linear MMSE without interference suppression. Fig. 5.10 shows B E R versus 101og10(£0/-Wo) for single narrow band interfer-ence using band-stop filter for CM4, N = 24. If the center frequency of the single narrow-band interferer is known at the receiver, a band-stop filter can be employed at the front end of the receiver to completely remove the NBI, regardless of the power of the NBI. In our simulations, an order 10 Chebyshev Type I bandstop filter with 0.5 dB of peak-to-peak ripple in the passband is 77 Figure 5.11: B E R versus 10 logw(Eb/N0) for NBI suppression with band-stop filter for CM4, N = 6. used for NBI removal, and it is followed by a linear M M S E equalizer. It is observed that only a small performance degradation between this structure and the optimal case without the presence of interference. The small perfor-mance loss is due to the fact that some useful UWB signals are also removed by the bandstop filter. We also notice that employing the two-stage adaptive filter after the bandstop filter does not lead to performance improvement. As for the worst case, we observe that pre-filtering NBI using a smiple bandstop filter is an effective way to reduce the effect of NBI. At low BER, the receiver structure with a bandstop filter approach the case without NBI, within 0.8 dB for CM4 and N = 24, and 1.2 dB for CM4 and AT = 6 (Fig. 5.11). We conclude that if the center frequency of the single NBI is known, or if multiple homogeneous narrowband interferers are expected, a simple receiver structure 78 consisting of a bandstop filter followed by a linear MMSE equalizer is able to result in good performance. On the other hand, if the center frequency of a single NBI is not available at the receiver, a two-stage W L LMS adaptive filter is proposed for the NBI suppression. 79 Chapter 6 Conclusions Equalizations at the output of R A K E receivers have been employed in many papers for DS-UWB systems. The combining schemes of R A K E receivers, however, are not optimal due to the long delay spread nature of UWB chan-nels. Therefore, we have proposed developing chip rate equalization. We have also studied the channel estimation and interference suppression for DS-UWB systems. In Chapter 3, we investigated equalizations for DS-UWB systems with BPSK modulation. We also developed two finger selection schemes to capture a re-duced number of multipaths, resulting in reduced computational complexity while at the same time producing good performance. The results of our inves-tigation can be summarized as follows. • L E is suitable for low data-rate modes of DS-UWB, Non-linear equaliza-tion (DFE) provides performance slightly better than LE. • The W L processing was found to provide gain of up to 0.4 dB over LE , and these gains come at added complexity. W L E is also found to achieve a performance very close to DFE, for both low data-rate and high data-rate. W L E is recommended for implementation. • The proposed finger selection schemes can effectively capture energy with a reduced number of paths. The SA achieves performance slightly better 80 than the GA, but has a higher computational cost. Compared with R A K E receivers, the G A has the\"advantage of simple structure and low computational complexity. The GA is recommended for implementation. The number of selected fingers to achieve acceptable performance was found to be 16 for low date-rate and 50 for high data-rate. The equalization schemes and finger selection schemes are performed based on the knowledge of the CSI of the desired user. In a practical implementation, the CSI is not always available and may need to be estimated by the receiver. Furthermore, the DS-UWB systems are subject to interference by other co-existing narrow-band wireless communications systems with relatively high PSD, or other DS-UWB systems. In Chapters 4 and 5, channel estimation and interference suppression for DS-UWB systems were studied. To this end, we first investigated blind channel estimation for DS-UWB systems, and the performance was compared to that obtained by channel estimation with train-ing sequences. Adaptive filters were then proposed to suppress the MAI and NBI for DS-UWB systems. The results of our investigations can be summa-rized as: • Blind detection scheme for DS-UWB yields a poor performance, due to the long delay spread nature of the UWB channels. Therefore, channel estimation with training symbols is necessary. • By utilizing training symbols, the proposed channel estimation schemes provide a good performance, while at the same time causing low imple-mentation complexity. At low data rates, the proposed channel estima-tion scheme can approach the optimal (with perfect CSI) within 1.0 dB, with 200 training symbols. Channel estimation with training symbols are appropriate because the transmission of training symbols results in negligible bandwidth wastage for the high data rates DS-UWB systems. Compared with other channel estimation methods (LS and ML) , the proposed scheme is simple to implement. • The proposed two-stage W L LMS adaptive filters can effectively suppress 81 unknown MAI and NBI and provide a negligible performance loss to the optimal case with CSI of interference being known. The proposed filters converges rapidly to the steady-state than linear adaptive filters, while yielding a higher SINR. • If the center frequency of NBI is available, we proposed a simple inter-ference suppression receiver to largely eliminate the negative effects of NBI, regardless of the PSD of the interference. Simulation results show that the proposed receiver almost perfectly with a single NBI. Even for the worst case scenario with NBI occupying the entire possible operating band, the proposed receiver can still result in a performance within 1.0 dB of the optimal case without interference. Compared with other inter-ference suppression methods for DS-UWB systems, the proposed scheme is preferred because of its simplicity. An interesting topic for further research would be to implement the two-stage adaptive filters in hardware and study the implementation-specific is-sues. Moreover, throughout this work, time-invariant channels were assumed. It would be interesting to investigate the performance of channel estimation schemes with time-varying channels. 82 Appendix A Derivation of SINRs for Equalization Schemes In this appendix, we derive the SINRs for the equalizers in Chapter 3. We first start with the derivation of SINR for linear MMSE receiver. Since the trans-mitted data signals are real, only the real part of linear MMSE filter output is of interest for the performance of the receiver. The error signal e[k — k0] is defined as e[k — k0] = a[k — ko] — Rej /^r} . The variance of the error signal is [21] at = E{(e[k - k0])2} = E{(a[k - k0] -\\\\fHr + r\"/])2} (A.l) = a\\- 2Re{fHE{a[k - k0]r} + E{(Re{fHv})2} Also, we have E{a[k-ko]r} = a2aHko, (A.2) and £{(Re{/\"r}) 2 } = ia 2Re{/\"HHT } + V ( H H ^ + a2I)/a2 (A3) + \\fH(n^al + all)raj Define R = H H H ^ + aft. From (A.l) , (A.2) and (A.3), we can obtain °l = °l- lK^-^l + ^Re{Hj^R_1HHr(R_1)*Hfco}(T^, (A.4) 83 where a\\ is the variance of the transmitted signals. Useful signal power is ( H ? R ^ f c j V 2 , then we can have S I N R - P ° W e r interference • and • noise a\\ - [1 - H J R - 1 ^ ] ^ Notice that a 2 = 1 for BPSK signals. After some manipulation, the above equation can be simplified as u 1 S I N R = - = — - , (A.6) 1 — u - — 1 u where u = H f o ( H H \" + ^ I ) \" 1 ^ . The SINRs for other equalization schemes can be derived with similar proce-dure. After some manipulation, the SINR for D F E is derived as „ e - r - x f ^ - ' w . I2 with ( H ^ R - 1 ^ ) 2 al [1 - H R _ 1 H f e o j S I N R D F E = v \"*° (A.7) a\\ = l - ^ H f o R - 1 H f e o + ^ R e { H f o R - 1 ( H H r - H 1 : g B H ^ B ) ( R - 1 ) * H ^ o } . (A.8) The SINR for widely linear MMSE is derived as [21] S I N R ™ = Hk°„ „ _, f c: (A.9) with 1 _ H f c 0 ^ H f c o M al R = H H + f I 2 N w . (A. 10) and H and Hfc0 are defined in equation (3.11). 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Jayant, \"Analysis of IEEE 802.11a interference on UWB systems,\" IEEE Conference on Ultra Wideband Sys-tems and Technologies, pp. 297-301, Nov. 2002. 91 "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0065640"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Electrical and Computer Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Detection, channel estimation and interference suppression for DS-UWB"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/31889"@en .