TH-CDMA-PPM with Noncoherent Detection for Low ... - IEEE Xplore

446

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 2, FEBRUARY 2008

TH-CDMA-PPM with Noncoherent Detection for Low Rate WPAN S. H. Song, Member, IEEE, and Q. T. Zhang, Senior Member, IEEE

Abstract— Ultra wideband (UWB) impulse radio (IR) is a prospective transmission technology for low-rate indoor communications, as described in the physical-layer proposals for IEEE 802.15.4a wireless personal area networks (WPAN). Time hopping (TH) code division multiple access (CDMA) is considered as an access scheme for multiuser UWB-IR systems. The THCDMA system widely addressed in the literature adopts binary phase-shift keying (BPSK) with coherent detection, which requires accurate channel estimation and thus increasing the implementation complexity. In this correspondence, we suggest using TH-CDMA-PPM (pulse position modulation) with non-coherent detection to simplify the receiver structure. The influence of different combinations of TH and CDMA processing gains on error performance of the new scheme is analyzed and numerical results are presented for illustration.

scheme is called TH-CDMA-PPM. Studies on TH-CDMABPSK reveal that there exists a tradeoff between the TH and CDMA processing gains [9], and this tradeoff is further analytically investigated in [3] by using the improved Gaussian approximation method. In what follows, we will first introduce the new multiple access scheme, then investigate the way the two constituent sequences influence the system overall performance, aiming to provide a basis for the design of THCDMA-PPM systems.

Index Terms— Ultra wideband, wireless personal area network, pulse position modulation, non-coherent detection.

as

II. N EW SCHEME FOR MULTIPLE ACCESS AND DETECTION A. System description A typical TH-CDMA-PPM signal for user u can be written u

s (t) = I. I NTRODUCTION

U

LTRA Wideband (UWB) impulse radio (IR) is a proposed air interface technology for IEEE 802.15.4a wireless personal area networks (WPAN) targeting at low-rate indoor communications and ranging applications. In these networks, multiple users share the same IR channel usually through time hopping (TH). To improve the access performance of TH, one can introduce another spreading sequence to further randomize each user’s information bit sequence, resulting in the joint TH-CDMA multiple access system. In a TH-CDMA system, CDMA is often utilized alongside binary phase-shift keying (BPSK) for ease of implementation. This can be done by directly using a spreading sequence to randomize the polarity of the users’ bit information sequence [1], [2], [3], leading to what is called TH-CDMA-BPSK. The cost, however, is that coherent detection required in THCDMA-BPSK implies the need for channel estimation or transmit-reference signalling [4] thereby increasing the system implementation complexity [5]. In this correspondence, we propose the use of pulse position modulation (PPM) [6], [7], [8] for TH-CDMA, instead, which enables the position randomization of the user’s information bits on each TH frame on one hand, while allowing for noncoherent detection at the receiver on the other. The resulting

Manuscript received August 10, 2006; revised November 29, 2006 and April 2, 2007; accepted May 25, 2007. The associate editor coordinating the review of this paper and approving it for publication was Z. Tian. This work was fully supported by a strategic research grant from the City University of Hong Kong, Hong Kong, China (Project No. 7001856). This paper was presented in part at the IEEE Wireless Communications and Networking Conference 2007- Phy/MAC, WCNC07, Mar. 2007. The authors are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/TWC.2008.060572.

Ns −1 Eb w(t − iTf − cui Tc − aui ⊕ bu Δ) Ns i=0

(1)

where Tf and Tc denote the TH frame length and chip width, respectively, and w(t) is the transmitted pulse with unit energy. In each frame, the transmitted pulse is allowed to hop over Nh = Tf /Tc possible locations. The user information bu and the spread sequence aui take the value of 0 or 1, and cui assumes an integer value from [0, Nh − 1]. The hopping pattern of each user is generated under the control of both the time hopping sequence {cui } and the spreading sequence {aui }. The former sequence randomizes, with a relatively large scale, the pulse positions (time hopping chips) over different frames whereas the latter randomizes the binary pulse location of the information symbol just inside each hopping chip through the operation {aui ⊕ bu }. Here, ⊕ denotes modula-2 addition, and Δ is the pulse position modulation depth. The THCDMA-PPM system is different from the conventional THPPM system [7] by the second spectral spreading operation. The idea of TH-CDMA-PPM is illustrated in part (b) of Fig. 1. It differs from the TH-CDMA-BPSK, shown in part (a) of Fig. 1, mainly in that the post-spreading user information is carried by the pulse position inside each hopping chip, thereby enabling the non-coherent detection. Suppose a TH-CDMA-PPM system is used to support Nu + 1 synchronous users with each TH cycle consisting of Ns frames. An information symbol bu from user u undergoes time hopping and spreading over Ns consecutive frames before it is used to pulse-position modulate a basic transmitted pulse w(t). The resulting signal is transmitted over a frequency-selective fading channel. The channel model we use is the one adopted by IEEE 802.15 4a [10], which consists of N independent paths as shown by

c 2008 IEEE 1536-1276/08$25.00

h(t) =

N −1 j=0

hj (t)δ(t − jTw )

(2)


Tc

Δ

Fig. 1.

447

Tf

Tc

Tf

Signal structures for TH-CDMA-BPSK and TH-CDMA-PPM systems.

where Tw is the length of the resolution bin, and hj (t) signifies the channel gain of path j. We assume, for convenience, that the transmitted pulse has the same width Tw as the resolution bin and each resolvable delay bin contains significant energy [10]. By collecting the first L significant path components of each user, we can write the received signal corresponding to one TH cycle as Nu +1 N s −1 L−1 Eb hu w(t − iTf − cui Tc r(t) = Ns u=1 i=0 j=0 ij −aui ⊕ bu Δ − jTw ) + n(t)

(3)

where huij denotes the channel gain of path j for user u over frame i and n(t) is the additive white Gaussian noise (AWGN). The use of PPM enables both coherent and noncoherent detections. To meet the low complexity requirement, we consider the non-coherent square-law detection for which no information about the channel path gain {huij } is needed at the receiver. B. Decision variable for noncoherent detection An impulse radio system usually transmits a pulse of width Tw at the order of a few nanoseconds, and the received signal is normally spread to a much wider waveform with the received signal length Tg Tw . For applications with low-rate transmission between 1 kbit/s and several Mbit/s as described in IEEE 802.15.4a [10], the inter-symbol interference caused by multi-path spread can be fully removed by applying orthogonal PPM with Tc /2 ≥ Δ ≥ Tg . We therefore neglect the effect of inter-symbol interference in this study. As indicated in (3), each bit information is spilled over Ns L pulses, which therefore form the basis for a bit decision. For binary PPM, the information bit 1 and bit 0 are carried by their positions. We can therefore collect signal components at their respective locations producing two groups of variables. One group corresponding to bit 0 is denoted by {xij } with xij signifying the correlator output from jth path in the ith frame. Another corresponds to bit 1 and is denoted by {yij }. More specifically, assume user 1 is the desired user, we can write

xij yij

Ns Tf

= =

0 Ns Tf 0

r(t)w(t − iTf − c1i Tc − a1i Δ − jTw )dt r(t)w(t − iTf − c1i Tc − (1 − a1i )Δ − jTw )dt. (4)

The receiver will accumulate, in an appropriate manner, all the signal energy for bits 1 and 0, respectively, and use their difference for a symbol decision. In some WPAN applications, the channel remains nearly constant during an entire data burst, while there are other circumstances in which channel variations occur due to the relative movement of the transceiver [11]. Thus, we consider two kinds of channel conditions: fast fading and slow fading. The former implies that the channel gains remain unchanged within a frame duration while varying independently from one frame to another [12] and the latter corresponds to channel gains maintaining constant over an entire TH symbol. 1) Fast fading channel: For a fast fading channel, we can non-coherently combine the Ns L components. We accumulate signal energy on the expected positions of all paths for bit zero and bit one, and calculate their difference leading to the decision variable rf =

N s −1 L−1

2 rij , rij = x2ij − yij .

(5)

i=0 j=0

2) Slow fading channel: In a slow fading channel, the channel gain of each path remains unchanged over Ns frames; namely, hu0j = hu1j = · · · = hu(Ns −1)j = huj for j = 0, · · · , L − 1. This slow fading enables a coherent combining. In particular, we add the correlation output of path j from all frames to obtain xj =

N s −1 i=0

xij ,

yj =

N s −1

yij .

(6)

i=0

If we regard xij as the recovered symbol information over the ith chip in the conventional CDMA system, the above operation is equivalent to coherently collect the same symbol information over different chips. The symbol information so

448


obtained is further accumulated over L independent paths, in a non-coherent manner, to form the decision statistic =

rs

L−1

x2j − yj2 .

(7)

j=0

The proposed receiver requires the pulse waveform w(t) for correlation operation as described in (4). In fact, the correlation can be removed if a decision is simply based on energy comparison [13], certainly at the cost of performance degradation. We do not pursue this issue further for brevity. III. P ERFORMANCE ANALYSIS The important issue is to understand how the overall processing gain is dictated by TH and CDMA sequences. In this section, we investigate such mechanism aiming to provide some guideline for system design.

B. Performance evaluation Without loss of generality, assume that user 1 is the desired user and bit zero is transmitted by user 1. The probability of error is given by Pe = Pr (r < 0)

which will serve as a means for studying the interaction between TH and CDMA. 1) Fast fading channel: Given the presence of multiuser interference and multipath fading with Nakagami distribution, it is extremely difficult to directly evaluate the exact Pe by averaging over all possible interfering patterns and fading channel gains. We note, however, that the decision variable rf in (5) contains Ns × L independent terms, which is usually large, thus justifying the use of the central limit theorem to assert that the variable rf approaches Gaussian distribution. Hence, we can write rf = μrf + erf with

A. Channel model A 1ns-long monocycle, when applied to a UWB channel of N paths, produces a channel output typically of 200 ns or even longer [14]. It is usually assumed [10] that the power delay profile (PDP) attenuates with time delay j, as shown by Ωj = K exp(−jβ) where β stands for the exponentially decaying factor. For we choose K such that the convenience, L−1 total channel gain j=0 Ωj = 1. To specify the distribution functions for path gains, we adopt the two-sided Nakagami model as suggested in IEEE 802.15.4a [10]. The path gain hj can take on positive and negative values with equal probability to account for signal inversion due to reflection [11]. Its magnitude |hj | follows the Nakagami distribution such that we can write [10] m m 2 P|hj | (x) = x2m−1 exp(−mx2 /Ωj ) (8) Γ(m) Ωj where Ωj = E[h2j ] and the m parameter is modeled as a lognormally distributed random variable, whose logarithm has mean μm and standard deviation σm . Thus, m has the distribution m = exp(z),

2 ) z ∼ N (μm , σm

(9)

2 ) N (μm , σm

means that z is Gaussian distributed where z ∼ 2 2 . The parameters (μm , σm ) with mean μm and variance σm for different channel conditions are listed in [10]. Given the symmetric probability density function (PDF) of hj about the vertical axis and the Nakagami distribution of |hj |, we can determine the nth moments of hj as [15] ⎧ n/2 ⎨ Γ(m+n/2) Ωj E , n is even; m Γ(m) m E[hnj ] = (10) ⎩ 0, n is odd. where E[·] and Em [·] denote expectation and the expectation operator with respect to m, respectively. Further averaging this result over the lognormally distributed parameter m, we obtain [16] E[h2j ]

=

E[h4j ]

=

Ωj ; 2 σm 2 − μm Ωj 1 + exp . 2

(11)

(12)

μrf

= E[rf ],

erf

2 ∼ N (0, σrf ).

From (12) and (13), it follows that μ2rf Pef = Q 2 σrf

(13)

(14)

2 . The intended which relies on the parameters μrf and σrf signal is collided by other users with different hit patterns. Therefore, we need to average rij over all possible hit patterns 2 . to obtain μrf and σrf Let Hca denote the event that c out of Nu users collide with user 1 in the ith TH frame and a out of the c colliding users transmit the same information bit as user 1, and let P (Hca ) denote its probability. When the event Hca occurs, there are a users colliding the desired user’s pulse (xij ) and other (c − a) users arrives at the same time hopping chip but different PPM depth (yij ). Directly calculating (4) with these observations enables us to write a+1 Eb u xij | Hca = h + nxij ; Ns u=1 ij c+1 Eb u h + nyij , (15) yij | Hca = Ns u=a+2 ij

where the AWGN components nxij and nyij result from the integration defined in (4) both having the same mean and variance as n(t). The probability of Hca can be determined to be P (Hca ) = Pc Pa where c (Nu −c) Nu Nh − 1 1 c Nh Nh c c 1 = (16) a 2

Pc

= P1/Nh (c | Nu ) =

Pa

= P1/2 (a | c)

are the probabilities for two binomial distributions. Averaging rf over all possible hit patterns and fading realization, we can obtain E[rf ] and Var[rf ]. It then follows


from (11) that the total mean and variance is expressible as μrf 2 σrf

= Eb L−1 Eb2

(17)

(Nu + Nh ) 2 1 + exp(σm /2 − μm ) Ns j=0 Nh Nu Nu (Nu − 1) + − 1 + Nh Nh2 E 2 Nu 4Eb (Nu + Nh ) 2 + b + σn + 4Ns Lσn4 . Ns Nh Nh

=

Ω2j

2 We observe that serious fading with small μm and large σm will clearly increase the interference. We next show how the TH and CDMA sequences influence the system performance. The error performance is uniquely determined by the signal to interference plus noise ratio, SINR 2 , or equivalently by its reciprocal 1/SINR. For a = μ2rf /σrf fair comparison of different settings for TH-CDMA-PPM, we confine the total number of chips in a TH cycle to a fixed number, say Nr . Then, we have Ns = Nr /Nh . Defining γ = Eb /σn2 , we can use (17) to represent SINR as a function of Nr and Nh : L−1 2 2

1 SINR

=

exp(σm /2 − μm ) 4 + Nh γ Nr

1 + Nh +

(Nu2 − Nu )

L−1 j=0

j=0

Ω2j

Nr

Nu (2 + exp(

2 σm 2

− μm ))

Ωj

4Nu γ + 4Nr L + γ2

L−1 j=0

Ω2j + 1

,

Nr

(18)

which reveals the impact of TH and CDMA on the error performance. Clearly, there is an optimal choice of Nh or Ns . For a given γ and Nr , we differentiate (18) with respect to Nh to obtain the optimum value of Nh , for which the SINR is maximized. The resulting relation is given by L−1 2 2 2 2 Nho =

(Nu − Nu ) + 4(Nr Nu γ + Nr L)/(γ

j=0

Ωj )

2 exp(σm /2 − μm )

.

449

Since coherent combining is performed, we cannot consider the hit pattern independently for each frame. Instead, we u , u = handle the whole symbol hit pattern user by user. Let Hca 1 denote the event that user u collides with user 1 in cu out of total Ns frames, and among all cu colliding frames, user 1 and user u transmit the same information in au frames. Clearly, u v and event Hca are independent for u = v. Further event Hca u denote {Hca }, u = 1, ..., Nu as the event when each interu fering user takes its specific hit pattern Nu +1Hca , independently. u au paths colliding Thus, for event {Hca } there are u=2 Nu +1 the desired user’s pulse (xj ) and other u=2 (cu −au ) paths arrives at the same time hopping chip but different PPM depth (yj ). Calculating (4) and (6) with these observations enables us to write xj |

u {Hca }

=

Ns

γ. This suggests a strategy for allocating processing gains between TH and CDMA. Namely, a larger Ns is allocated in a high signal to noise power ratio (SNR) environment whereas a larger Nh is allocated in a low SNR environment. Besides, we observe that Nho is an increasing function of number of users. 2) Slow fading channel: The bit error rate (BER) performance of the receiver (7) for slow fading channel is ⎛ ⎞ L−1 Pes = Pr ⎝rs = x2j − yj2 < 0⎠ ⎛ = Pr ⎝

j=0

−1 L−1 s N j=0

i=0

2 xij

−

N −1 s

2 yij

⎞ < 0⎠ .

i=0

(20)

u=2

u } yj | {Hca

Nu +1

=

(cu − au )

u=2

Ns −1 Eb u hj + nxij ; Ns i=0

Eb u hj + Ns

Ns −1

nyij .

(21)

i=0

Since we need to average both the channel fading and hit pattern effects, the evaluation of Pes by (20) is extremely complicated. In fact, the visit of all the hit patterns itself is too complicated to handle. For example, {cu } in (21) has total NsNu possible realizations, not saying the further combination with {au }. As mentioned in the channel model section, the number of resolvable paths L is very large. This fact, together with many independent users, justifies the use of Gaussian approximation for rs . Following the same procedure as that for the fast fading case, we can obtain the mean and variance of the decision variable μrs

= Ns Eb

2 σrs

2 = Ns2 Eb2 exp(σm /2 − μm )

(19)

Once Nho is obtained, the optimal Ns is determined by Nso = Nr /Nho . It is clear from (19) that Nso increases monotonically with γ while Nho being a monotone decreasing function of

Nu +1 Eb 1 hj + au Ns

⎛

L−1

Ω2j

j=0

2 +2Eb2 Nu ⎝1 + exp(σm /2 − μm )

L−1

⎞ Ω2j ⎠

j=0

Nh2

+ 3(Ns − 1)Nh + (Ns − 1)(Ns − 2) Ns Nh3 L−1 2Ns Ns (Ns − 1) +Eb2 Nu Ω2j + N Nh2 h j=0 (Nu − 1)(Ns + Nh − 1) + + 4LNs2 σn4 Nh3 4Nu 2Nu (Ns − 1) +Eb Ns σn2 4Ns + + , Nh Nh2 (22) ×

which can be utilized for performance evaluation. The optimum Nho that maximizes SINR can be determined by numerical methods, and we do not pursue it further due to space limitation.

450


0

0

10

10 Simulation: Slow fading Simulation: Fast fading Theoretical Results

Ns=20, Nh=5 Ns=10, Nh=10 Ns=5, Nh=20

−1

10

−1

10

BER

BER

20 Users −2

10

−2

10 −3

10

10 Users

−4

10

−3

0

5

10

15 Eb/N0

20

25

10

30

Fig. 2. Performance of TH-CDMA-PPM systems over fast and slow fading channels using square law detection: theoretical approximation .vs. simulation.

0

5

10

15

20

25

Fig. 3. Performance comparison of TH-CDMA-PPM systems with different system parameters using square law detection: fast fading. Ns=20, Nh=5 Ns=10, Nh=10 Ns=5, Nh=20

IV. N UMERICAL R ESULTS A. Synchronous systems

−1

10

−2

10 BER

Consider a TH-CDMA-PPM system with synchronous users working on L-path independent fast and slow fading channels. The Nakagami distributed channel gain huij is generated according to the NLOS indoor case [10] with the following para2 = 1.15 and β = 0.0845. We use (14) meters μm = 0.3dB, σm and (17) to determine the theoretical error performance for the fast fading channel, and the theoretical error performance for the slow fading channel is obtained using (22). The true error performance is obtained through Monte Carlo simulations. The comparison results of systems with 10 users are shown in Fig. 2 where we set Nh = 5, Ns = 20 and L = 20. Good agreements between the theoretical approximation and simulation results are observed. It should be mentioned that the Gaussian approximation method utilized in this paper is valid with relative large number of users and paths. For the performance evaluation of a system with small number of users and paths, which hardly occurs for a typical UWB system, we can adopt the improved Gaussian approximation method [17] by considering each interfering pattern separately. We next compare the performance of three system configurations with Nh = 5, 10, and 20, respectively, by simulation. Again, we set Nr = 100 and L = 20. The simulation results for the fast fading channel are depicted in Fig. 3 where two sets of curves are shown for 10 and 20 users, respectively. As predicted by (19), for the case of 20 users or at low SNR, systems with larger Nh demonstrate a better performance. For the case of 10 users, the system with Nh = 5, 10 outperforms its counterpart with Nh = 20 at high SNR environment. We can conclude from these observations that at low SNR or with large number of users more TH processing gain (Nh ) is beneficial, but if a small number of users are active at high SNR environment, it is better to have more CDMA processing gain (Ns ). Similar simulation results are obtained for slow fading channels, as shown in Fig. 4, although the improvement gained from increasing the CDMA processing gain at high

30

Eb/N0

30 Users

−3

10

10 Users

−4

10

0

5

10

15 Eb/N0 (dB)

20

25

Fig. 4. Performance comparison of TH-CDMA-PPM systems with different system parameters using square law detection: slow fading.

SNR is not obvious. To choose the (Ns , Nh ) combination in a practical system, we should consider other factors besides SINR. For example, the spectral shaping, which is critical for system coexistence, also depends much on the tradeoff between TH and CDMA. B. Asynchronous systems The analysis in this paper is performed for system with TH-chip level synchronization, which is equivalent to frame level synchronization since the random TH sequence is utilized. In practical systems, asynchronous transmission may occur. Fortunately, our focus in this paper is on the tradeoff between two kinds of processing gain, while the decreasing of probability of collision caused by asynchronism should have similar influence on different combinations of TH and CDMA processing gains, thus will not greatly change the concerned tradeoff. Shown in Fig. 5 is the performance comparison for


Ns=20, Nh=5 Ns=10, Nh=10 Ns=5, Nh=20

451

provide a basis for the design of TH-CDMA-PPM system for low rate WPAN.

−1

10

R EFERENCES 30 Users BER

−2

10

−3

10

10 Users

−4

10

0

5

10

15 Eb/N0

20

25

30

Fig. 5. Performance comparison of asynchronous TH-CDMA-PPM systems with different system parameters: fast fading.

asynchronous systems over fast fading channels in which we assume the time delays between different users are integer times of path length Tw and uniformly distributed over [0, Tc ]. The results show that the pulse collision probability is indeed reduced by asynchronism and the asynchronous system obtains better performance than the synchronized case. However, this asynchronism has not change the tradeoff between TH and CDMA processing gains such that the tradeoff exists in a similar way as the synchronous system shown in Fig. 3. V. C ONCLUSION A new joint TH-CDMA-PPM scheme with non-coherent detection is proposed for low rate WPAN application. We have investigated the mechanism that governs the performance of the TH-CDMA-PPM system with non-coherent square-law detection over fast and slow fading channels. Good agreement between the theoretical analysis and simulation results is observed. The results reveal that there exists an optimal strategy for allocating processing gain between TH and CDMA. In the case of low SNR environment, a longer TH frame should be used. In a high SNR environment, however, a longer CDMA sequence is preferred. If a large number of users is expected, we should select a larger TH processing gain. These results

[1] K. A. Hamdi, “On the multiple-access capability of the bi-phase modulated TH-CDMA impulse radio networks,” in Proc. IEEE Vehicular Technology Conference 2004-Fall VTC04, Sept. 2004. [2] S. Gezici, H. Kobayashi, H. V. Poor, and A. F. Molisch, “Performance evaluation of impulse radio UWB systems with pulse-based polarity randomization,” IEEE Trans. signal processing., vol. 53, no. 7, pp.25372548, July 2005. [3] M. A. Rahman, S. Sasaki, J. Zhou, and H. Kikuchi, “Simple-to-evaluate error probabilities for impulse radio UWB multiple access systems with pulse-based polarity randomization,” IEEE Commun. Lett., vol. 9, no. 9, pp. 772-774, Sept. 2005. [4] F. Tufvesson, S. Gezici, A. F. Molisch, “Ultra-wideband communications using hybrid matched filter correlation receivers,” IEEE Trans. Wireless Commun., vol. 5, no. 11, Nov. 2006. [5] Q. T. Zhang and S. H. Song, “Eigen-based receivers for the detection of random UWB signals,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1184-1189, July 2006. [6] P. Withington II and L. W. Fullerton, “An impulse radio communications system,” in Proc. Int. Conf. on Ultra-Wide Band, Short-Pulse Electromagnetics, pp. 113-120, Oct. 1992. [7] M. Z. Win and R. A. Scholtz, “Impulse radio: How it works,” IEEE Commun. Lett., vol. 2, no. 2, pp. 36-38, Feb. 1998. [8] B. Hu and N. C. Beaulieu, “Exact bit error rate analusis of TH-PPM UWB systems in the presence of multiple-access interference,” IEEE Commun. Lett., vol. 7, no. 12, pp. 572-574, Dec. 2003. [9] E. Fishler and H. V. Poor, “On the tradeoff between two types of processing gains,” IEEE Trans. Commun., vol. 53, no. 9, pp. 1744-1753, Sept. 2005. [10] A.F., Molisch, D. Cassioli, C-C. Chong, S. Emami, D. Fort, B. Kennan, J. Karedal, J. Kunisch, H.f. Schantz, K. Siwiak, M.Z. Win, “A comprehensive standardized model for ultrawideband propagation channels,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3151-3166, Nov. 2006. [11] A. F. Molisch, J. F. Foerster, and M. Pendergrass, “Channel models for ultrawideband personal area networks,” IEEE Wireless Commun. Mag., vol. 10, pp. 14-21, Dec. 2003. [12] L. Liu, J. Tong and Li Ping, “Analysis and optimization of CDMA systems with chip-level interleavers,” IEEE J. Select. Areas Commun., vol. 24, no. 1, pp. 141-150, Jan. 2006. [13] I. Guvenc, Z. Sahinoglu, A. F. Molisch and P. Orlik, “Non-coherent TOA estimation in IR-UWB systems with different signal waveforms,” in Proc. IEEE International Conference on Broadband Networks, pp. 245-251, Oct. 2005. [14] Q. T. Zhang and S. H. Song, “Parsimonious correlated non-stationary models for real baseband UWB data,” IEEE Trans. Veh. Technol., vol. 54, no. 2, pp.447-455, Mar. 2005. [15] J. G. Proakis, Digital Communications, Third Ed. McGraw-Hill, Inc., 1995. [16] A. Stuart, J. Keith Ord, Kendall’s Advanced Theory of Statistics. London: Charles Griffin Co. Ltd., 1987. [17] R. K. Morrow, Jr. and J. S. Lehnert, “Bit-to-bit error dependence in slotted DS/SSMA packet systems with random signature sequences,” IEEE Trans. Commun., vol. 37, no. 10, pp. 1052-1061, Oct. 1989.