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61

Multiple Access Capacity of UWB M-ary Impulse Radio Systems with Antenna Array S. S. Tan, B. Kannan, Member, IEEE, and A. Nallanathan, Senior Member, IEEE Abstract— In this paper, the multiple access capacity of an Mary pulse position modulation (PPM) impulse radio (IR) system with antenna array is analyzed in dense multipath environments. An antenna array with Rake receivers is used to capture the signal energy from multipaths. Multiple access performance of the system is evaluated in terms of number of supported users for a given bit error rate and bit transmission rate with different number of antenna elements and selected paths. Numerical results show that the multiple access capacity of an M-ary IR system can be improved significantly by increasing the number of antenna elements and/or by adding more paths coherently at the receiver. Index Terms— Antenna array, impulse radio, M-ary, multiple access.

I. I NTRODUCTION MPULSE radio (IR) is a form of Ultra-wideband (UWB) spread spectrum signaling, which has properties that make it a suitable candidate for short range communications in dense multipath environments. Due to the large bandwidth, an IR multiple access system can accommodate many users. In [1], multiple access capability of a binary PPM IR system is studied with the assumptions of free space propagation conditions and additive white Gaussian noise. The authors in [2] investigate the use of M-ary equally correlated (EC) block waveform encoded PPM signals to increase the number of users supported by the system in free space propagation conditions. In multiuser environments with fading channels, antenna array can be used to exploit the spatial diversity in conjunction with the path diversity provided by the Rake receiver to capture the energy from different multipath components that are separated in space and time in a way that improve the performance of the system. In this paper, we present the multiple access performance of M-ary IR systems with antenna arrays in the presence of multipaths.

I

II. S YSTEM M ODEL A typical time-hopping (TH) PPM signal for the uth user can be modeled as Y (u) (t) = =

∞ (i+1)N X Xs −1 i=0 ∞ X i=0

j=iNs u yi,d (u) (t),

(u)

w(t − jTf − cj Tc − δ j (u) ) di

(1)

i

Manuscript received October 8, 2003; revised November 24, 2004; accepted April 6, 2005. The associate editor coordinating the review of this letter and approving it for publication was A. Scaglione. S. S. Tan and A. Nallanathan are with the Department of Electrical & Computer Engineering, National University of Singapore, 119260 (email: {engp2444, elena}@nus.edu.sg). B. Kannan is with Institute for Infocomm Research, Heng Mui Keng Terrace, Singapore 119613 (email: [email protected]). Digital Object Identifier 10.1109/TWC.2006.01010

where w(t) represents the monopulse (monocycle) waveform, Tf is the frame interval, Tc is the chip duration and δ j (u) di

(1 ≤ δ j (u) ≤ M ) denotes the time shift for the data di n o (u) modulation. Each user is assigned a TH sequence, cj to th support the multiple access capabilities. The i data symbol (u) of the uth user, di is an M-ary symbol that conveys the information. The modulating data symbol changes only after every Ns hops, where Ns monopulses are transmitted per symbol. Hence, there are Ns frames in one symbol period, Ts = Ns Tf . The block waveform M-ary PPM signal set is {s1 (t), s2 (t), ...sM (t)}, where sm (t) can be written as sm (t) =

NX s −1

(u)

j w(t − jTf − cj Tc − δm ).

(2)

j=0

m = 1, 2, ...M. In this paper, an M-ary EC signal is considered. In an Mj ary EC signal, δm = ajm τδ ∈ {0, τδ }, ajm = {0, 1} [3], where τδ denotes the modulation index. of IR basic monopulse, Ew is denoted by Ew = R ∞Energy 2 w (t) dt and the corresponding normalized correlation −∞ function is defined as Z ∞ 1 γw (τδ ) = w(t)w(t − τδ ) dt. (3) Ew −∞ Normalized correlation function [2] of the signals is defined as Z ∞ 1 (u) (u) ρn,m = y (t)yi,m (t) dt ≈ [1 + γw (τδ )] /2, (4) Es −∞ i,n for Ns À 1 and n 6= m, where Es = Eb log2 M is the symbol energy and Eb is the bit energy. Note that for EC signals, ρn,m (= ρm,n ) = ρ when n 6= m and ρn,m (= ρm,n ) = 1 when n = m. III. A PPLICATION OF A NTENNA A RRAYS IN IR S YSTEMS In this paper, we consider uniform rectangular (A × B) and linear (A × 1) arrays. The delay and sum beamformer is used to process the received signals at the array elements. The beamformer output for an array steered towards an azimuth angle of φ and an elevation angle of θ on incidence of a plane wave from an azimuth angle of φ0 and an elevation angle of θ0 is given by [4] B(φ, θ, t) =

A−1 X B−1 X

aa,b (φ, θ) × w(t − τa,b ),

(5)

a=0 b=0

dy dx (u − u0 ) + (b − bc ) (v − v0 )(6) c c where u = sin φ sin θ, u0 = sin φ0 sin θ0 , v = cos φ sin θ, v0 = cos φ0 sin θ0 and c is the speed of light. The spacing τa,b = (ac − a)

c 2006 IEEE 1536-1276/06$20.00 °

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between antenna elements in x and y directions are denoted by dx and dy respectively. The antenna elements are equally spaced, where dx = dy = 6”. The coordinate of the reference element is (ac , bc ). The pattern of the (a, b)th element is denoted by aa,b (φ, θ). Each element is assumed to have an isotropic pattern and thus, aa,b (φ, θ) is equal to 1/(4π) . Time delay of the received signal at the (a, b)th element is denoted by τa,b , which is measured with respect to the reference element. IV. C HANNEL M ODEL The channel model is based on Saleh-Valenzuela (S-V) model with slight modifications [5]. Typically, the cluster decay factor (Γ) is larger than the ray decay factor (γ). We consider constant cluster arrival rate (Λ) and ray arrival rate (λ), where λ > Λ. We denote the arrival time of the pth cluster by τp in which p = 0, 1, 2, ...P − 1 and the arrival time of the q th ray measured from the beginning of the pth cluster by τq,p , where q = 0, 1, 2...Q − 1. The total number of clusters is P and the total number of rays within each cluster is Q. The total number of multipaths is Ltotal = P × Q. The channel impulse response can be written as h(t)

=

A−1 −1 Q−1 X B−1 X PX X

αq,p,a,b δ (t − τa,b − τp − τq,p )

a=0 b=0 p=0 q=0

=

A−1 X B−1 X Ltotal X−1 a=0 b=0

αl,a,b δ (t − τl,a,b ) ,

(7)

l=0

where αq,p,a,b ∈ <, τl,a,b = τa,b + τp + τq,p and δ(t) is the Dirac delta function. τa,b is defined in (6) and {αq,p,a,b } is the gain coefficient of the q th ray in the pth cluster at the (a, b)th antenna element. Independent fading is assumed for each cluster, for each ray within the cluster and at each antenna element. The magnitudes of the channel gain follow a lognormal distribution where αq,p,a,b = χq,p,a,b ξp,a,b βq,p,a,b (8) ¡ ¢ 2 2 20 log10 (ξp,a,b βq,p,a,b ) ∝ N ormal µq,p,a,b , σ1 + σ2 . In (8), ξp,a,b is the fading associated with the pth cluster at the (a, b)th antenna element, βq,p,a,b reflects the fading associated with the q th ray within the pth cluster at the (a, b)th antenna element and χq,p,a,b is the +/- sign with equal probability. The log value of each channel gain follows a normal distribution with mean µq,p,a,b and variance σ12 + σ22 , where σ12 and σ22 are respectively, the variance of the cluster lognormal fading and ray lognormal fading in dB. The normalized mean square value of αq,p,a,b is defined as

where Ω0

2 E [αq,p,a,b ] = Ω0 e−τp,a,b /Γ e−τq,p,a,b /γ , (9) , PP −1 Q−1 P 2 2 = αq=0,p=0,a αq,p,ac ,bc is the avc ,bc

Σ

Σ

Glm= 0, a= 0, b = 0 ( i )

j = iN s

j 1 α l1= 0, a= 0,b = 0 Σ w (t − jTf −c (1) m = 1, 2,...,M j Tc − δm −τ l = 0, a = 0, b = 0 ) j

Σ

( i + 1) Ns − 1

Pulse Correlator

α l1= L

f

−1, a = 0, b= 0

Σ w (t − jT

f

Σ

Glm= Lf −1, a = 0, b= 0 ( i )

Σ

j = iN s

j 1 −c (1) j Tc − δm −τ l = L f −1, a = 0, b= 0

j

Fig. 1.

Decision circuit

)

m = 1, 2,...,M

Structure of SRake receiver with (A × B).

V. R ECEIVER P ROCESSING We use a Selective Rake (SRake) receiver with an antenna array and each antenna element has Lf Rake fingers as shown in Fig. 1, where the space-time correlators are used to process the received signals that are separated in space and time. Without lost of generality, we assume that the receiver is perfectly synchronized to the hopping code of the desired user (user 1) and the delays of the selected paths are known at the receiver. The receiver selects the L1f dominant paths of the user 1. The received signal can be written as r(t) =

L1f −1 A−1 X B−1 X X

1 αl,a,b y

(1) (1)

i,di

a=0 b=0 l=0

1 (t − τl,a,b )

1

+

A−1 X B−1 X Ltotal X−1 a=0 b=0

|

1 αl,a,b y

l=L1f

(1) (1)

i,di

1 (t − τl,a,b )

{z

(11)

}

nSI (t) u

+

Nu A−1 X X B−1 X−1 X Ltotal u=2 a=0 b=0

|

l=0

u u αl,a,b Y u (t − τl,a,b ) +na,b (t),

{z

}

nM AI (t)

where nSI (t) and nM AI (t) denote the self interference (SI) and the multiple access interference (MAI), respectively. The AWGN noise at the (a, b)th element, na,b (t) is with doubleu sided power spectral density of N0 /2. The notations τl,a,b and u αl,a,b are respectively, the time delay and the channel gain of the lth path at the (a, b)th antenna element for the uth user. An M-ary correlation receiver n o consists of M-filters matched to (1) 1 the signals yi,m (t − τl,a,b ) , m = 1, 2, ...M. Test statistics of the transmitted symbols depend on the sum of the correlator outputs of each selected path at each antenna element.

p=0 q=0

erage power gain of the first ray of the first cluster for the reference element (ac , bc ) which has been normalized by the total power gain of the reference element. The mean, µq,p,a,b is defined as µq,p,a,b

( i + 1) Ns − 1

Pulse Correlator

=

[(10 ln(Ω0 ) − 10τp,a,b − 10τq,p,a,b )/ln(10)] £ ± ¤ − (σ12 + σ22 ) ln(10) 20 . (10)

VI. BER A NALYSIS The decision variable that is used in a binary test to decide (1) (1) between the signals pair, y (1) and y (1) is given by i,di =n

G

n,m

(i) =

L1f −1 A−1 X B−1 X X a=0 b=0 l=0

i,di =m

Gn,m l,a,b (i),

(12)

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 1, JANUARY 2006

Z Gn,m l,a,b

=

1 (i+1)Ns Tf +τl,a,b

³

1 t=iNs Tf +τl,a,b

¡ ¢´ 1 ¡ ¢ 1 1 1 1 1 αl,a,b yi,d αl,a,b vi,n,m t − τl,a,b dt + NSI (t) + NM AI (t) + NAW GN (t), (1) t − τl,a,b | {z } i

NSI (t) =

1 (i+1)Ns Tf +τl,a,b



NM AI (t) =

X

 

1 t=iNs Tf +τl,a,b

NAW GN (t) =

1 (i+1)Ns Tf +τl,a,b

1 t=iNs Tf +τl,a,b

=

XX X

Θ=(

  0

A−1 P B−1 P

1 (αl,a,b )2 )Es (1 − ρn,m ). (18)

P

a=0 b=0 l=0

1 (αl,a,b )2 )Es (1 − ρ), n 6= m

otherwise.

Z

1

N0 ( 2 a=0

b=0 l=0

2 σM AI =

(21)

b=0 l=0

1 u αl,(a,b) αk,(a,b)

´2Z

∞ 2

R (x) dx,

−∞

k=0

R∞ where R (τ ) = −∞ w (t − τ ) [w(t) − w(t − τ )] dt. Similarly, the variance of decision variable caused by self interference, 2 σSI can be written as

1 (αl,a,b )2 )

(22)

1 A−1 B−1 Lf −1 Ltotal −1

Ns X X X 2Tf a=0

X

b=0 l=0 k=0,(k6=l)

(19) For an IR system in a multiuser environment, the transmission time difference between the lth path of the user 1 at th the (a, b) element and the other paths from the same user u 1 or the other active users is defined as τk,(a,b) − τl,(a,b) = u u u jk,l,(a,b) Tf + qk,l,(a,b) , where jk,l,(a,b) is the time uncertainty u rounded to the nearest integer, and qk,l,(a,b) is the error in this rounding process, which is assumed to be uniformly distributed over [−Tf /2, Tf /2]. In this paper, we assume that the number of users and the number of multipaths are sufficiently large. Hence, NM AI and NSI in (13) can be modeled as Gaussian random variables using the central limit theorem (CLT). Furthermore, NM AI and NSI can be considered as independent random variables as the user signals are all assumed to be independently generated. Therefore, Ntotal can be modeled by a Gaussian random variable with zero mean 2 2 2 2 2 = σSI + σM variance, where σtotal and σtotal AI + σAW GN . 2 2 2 The notations, σSI , σM AI and σAW GN are the variances of the decision variable caused by SI, MAI and AWGN 2 respectively. The AWGN variance, σAW GN can be written as 2 σAW GN =

2 The variance, σM AI can be written as

2 = σSI

L1f −1

Lf −1 A−1 X B−1 X X

(17)

Ns 2Tfu=2 a=0

Furthermore, for EC signals,

Θn,m =

(15)

(16)

u Nu A−1 f −1 Ltotal −1 ³ X X B−1 X LX X

a=0 b=0 l=0

  

¡ ¢ u 1 1 1  αl,a,b t − τl,a,b vi,n,m t − τl,a,b dt,

h i (1) (1) 1 ∈ {m, n} and vi,n,m (t) = yi,n (t) − yi,m (t) .

n o (1) E Gn,m (i) /di = n (

¢

¡ ¢ 1 1 1 vi,n,m t − τl,a,b dt, na,b (t) αl,a,b

1 A−1 B−1 Lf −1

=

u αl,a,b Y

¡ u

l=0

where Gn,m l,a,b (i) is defined in (13). The conditional mean, Θn,m of the Gn,m (i) is given by Θn,m



u

Nu Ltotal X X−1

u=2

(14)

i

l=0,l6=lth

1 (i+1)Ns Tf +τl,a,b

Z

 ¡ ¢ ¡ ¢ 1 1 1 1 1 1  αl,a,b αl,a,b yi,d vi,n,m t − τl,a,b dt, (1) t − τl,a,b

L1total −1



1 t=iNs Tf +τl,a,b

Z

(1)

(13)

Ntotal

Z

di

63

∞

−∞

1 (vi,n,m )2 (t) dt.

(20)

³

1 1 αl,(a,b) αk,(a,b)

´Z2

∞ 2

R (x) dx.

−∞

The union bound of the symbol error probability (SER) conditioned on a particular SNR can be written as M M ´ 1 X X ³p SN Rn,m (Nu ) , m 6= n. Q M m=1 n=1 (23) From (18) and (19), the signal-to-noise ratio is defined by

Ps/γ (Nu ) =

=

. ± 2 2 2 SN Rn,m (Nu ) = (Θn,m ) σtotal = Θ2 σtotal h¡ ±£ ¤¢−1 ¡ 2 ± 2 ¢−1 i−1 2 2 Θ2 σAW + Θ σM AI GN + σSI

=

SN R(Nu ).

Substituting (24) in (23), we obtain ³p ´ Ps/γ (Nu ) = (M − 1) Q SN R (Nu ) .

(24)

(25)

Corresponding upper bound of BER [3] can be written as ³p ´ Pb/γ (Nu ) = (M/2) Q log2 (M ) SN Rb(Nu ) . (26) SN Rb(Nu ) in (26) is the output bit SNR and is defined by (27). h i R∞ 2 In (27), β = 2 [Ew ((1 − γw (τδ ))/2)] / (1/Tf −∞ R2 (x)dx , Rb = log2 (M )/Ns Tf is the bit transmission rate and

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

−1 −1  1    (αl,a,b )2 β/Tf       a=0 b=0 l=0  −1  Ã ! SN Rb(Nu ) = (SN Rb(1)) +    u  ³ ´ L −1 L −1   Nu A−1 B−1 f 2 total P   (Rb ) P P P P   u   αl,(a,b)1 αk,(a,b) 

Ã

!2

1

f −1 A−1 P B−1 P LP

u=2 a=0 b=0 l=0

(27)

k=0

  2 u L1f −1 Nu A−1 f −1 Ltotal −1 ³ A−1 B−1 ´2 X X B−1 X LX X X X X   1 1 u αl,(a,b) αk,(a,b) D= (αl,a,b )2  β/Tf  / u=2 a=0 b=0 l=0

a=0 b=0 l=0

SN Rb(1) is the bit SNR of a single user system and is defined as £ ±¡ 2 ¢¤ 2 SN Rb(1) = log2 (M ) Θ2 σAW . (28) GN + σSI

(31)

k=0

0

10

Nu=150 M=4

theoretical simulation

−1

The average BER, Pb = E [Pb/γ (Nu )] is evaluated using Monte Carlo method. BER

10

VII. D EGRADATION FACTOR AND M ULTIPLE ACCESS C APACITY

10

Degradation factor is a measure of degradation in performance as the number of users with Lutotal paths increases. As in [3], SN Rbs denotes the specified bit SNR to achieve the desired probability of error and SN Rbr denotes the required value of SN Rb(1) to meet SN Rb(Nu ) = SN Rbs , where

10

SN Rbr (Nu ) = SN Rbs =

SN Rbs

−1 ,

(29)

−1 ,

(30)

1 − SN Rbs [(1/Rb )D] SN Rbr (Nu )

1 + SN Rbr (Nu ) [(1/Rb )D]

and D is given by (31). The degradation factor, DF is defined as DF

=

SN Rbr (Nu )/SN Rbs (1) .h i −1 = 1 1 − SN Rbs [(1/Rb )D] .

Rb (DF ) = (1/SN Rbs ) [1 − (1/DF )] D.

lim

SN Rbr (Nu )→∞

2

4

6

8

10

12

14

Eb/No (dB)

Fig. 2. The numerical bound and simulation results of IR system with different number of antenna elements and selected paths for a uniform linear array.

By substituting SN Rbs (Nu ) in (34) with SN Rblim (Nu ) and expanding (34) using the power series, the maximum value of C(B) for a given B can be written as ∞ X (−1)k+1 k=1

k

k

[D/B] .

(36)

By setting B → ∞, the multiple access capacity per user can be written as (33)

(34)

where B = 1/Tw is the bandwidth of the signal with Tw as the duration of the monopulse. For a given Nu , the SN Rbs takes the maximum value when SN Rbr (Nu ) → ∞ in (30); i.e., ∆

0

(32)

By using the Shannon capacity formula, we can derive the multiple-access channel capacity per user, Ca (Nu ) with antenna array at the receiver. The capacity formula is defined as C(B) = B log2 (1 + (1/B) Rb SN Rbs (Nu )) ,

−3

C(B) = [B/loge 2]

From (32), we can get the bit transmission rate of Rb as

SN Rblim (Nu ) =

R =(2x1), L =1 x f Rx=(3x1), L f=1 R =(4x1), L =1 X f Rx=(5x1), L f=1 R =(6x1), L =1 x f Rx=(4x1), L f=3

−2

SN Rbs = (1/Rb )D. (35)

∆

Ca (Nu ) = lim C(B) ≈ D/loge (2). B→∞

(37)

VIII. N UMERICAL R ESULTS We consider a Gaussian monopulse, which is defined by w(t) = ³h i h i´ 2 2 1 − 4π ((t − td )/τm ) exp −2π ((t − td )/τm ) ,(38) where td = 0.35 ns is the pulse center, and τm = 0.2877 ns is the pulse width. In this paper, Tf = 40 ns, τδ = 0.156 ns, and the channel parameters, Γ = 24, γ = 12, σ1 = σ2 = 3.3941 dB, λ = 0.5/ns and Λ = 0.1/ns. The signals received at all antenna elements are assumed to be in phase, where φ = 150 , θ = 900 , φ0 = 450 and θ0 = 900 . We assume

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 1, JANUARY 2006

600

65

350

M=4

−3

Pb(1)=10

−3

P (1)=10 b

R =6.25Mbps

300

Rx=(25x1), L =1 f Rx=(9x1), L f=1 Rx=(3x1), L =5 f Rx=(3x1), L f=2 Rx=(3x1), L =1 f Rx=(1x1), L =1

b

400

Rx=(5x1) L =2 f

250

Number of users, Nu

Number of users, Nu

500

Rb=6.25Mbps

f

300

200

200

150

100

M=32 M=16 M=8 M=4 M=2

100

50 1

0

5

10

15

20

25

30

35

40

1

Degradation factor (dB)

0

5

10

15

20

25

30

35

40

Degradation factor (dB)

Fig. 3. Number of active users, Nu as a function of degradation factor with different number of antenna elements and selected paths for a uniform linear array.

Fig. 5. Number of active users, Nu as a function of degradation factor for a uniform linear array of Rx = (5 × 1)with Lf = 2 for different values of M.

1200

M=4 −3 Pb(1)=10 1000

7

10

6

10

f

800

5

10

Ca(Nu)(kbps)

Number of users, Nu

8

10

Rx=(7x7), L f=1 Rx=(5x5), L =1 f Rx=(3x3), L f=3 Rx=(3x3), L =2 f Rx=(3x3), L =1

Rb=6.25Mbps

600

400

4

10

3

10

2

10

200

1

10 1

0

5

10

15

20

25

30

35

Rx=(25x1), L f=1 Rx=(9x1), Lf=1

40

Degradation factor (dB)

0

10 0 10

1

10

2

10

3

10

4

10

5

10

6

10

Number of users, Nu

Fig. 4. Number of active users, Nu as a function of degradation factor with different number of antenna elements and selected paths for a rectangular planar array.

Lutotal = Ltotal = 20. The number of pulses in one symbol, Ns = log2 (M ) Nsb where Nsb is the number of pulses used in binary communications and is set to 4 in this paper. Fig. 2 shows the theoretical and simulation results of the system’s BER performance for different number of antenna elements and selected paths. Theoretical and simulation results match reasonably well. The figure also shows that the BER of an M-ary system improves with increasing number of antenna elements and/or with increasing number of selected paths. Fig. 3 and Fig. 4 show the effect of number of selected paths and antenna elements on the multiple access performance of the system with uniform linear and rectangular arrays, respectively. One can observe from these two figures that the number of simultaneous users supported by the system increases with increasing number of antenna elements and/or with increasing number of selected paths. It allows one to capture more multipath signal energy and hence, to enhance the multiple access capability of the system.

Fig. 6. The multiple access capacity per user Ca (Nu ) as a function of number of active users Nu for Rx = (9 × 1), (25 × 1) elements of a uniform linear array with Lf = 1.

Fig. 5 shows the effect of M on the multiple access performance of the system with a uniform linear array. This figure shows that it is possible to increase the number of supported users, without increasing each user’s signal power, by increasing the value of M . Fig. 6 shows the multiple access capacity per user Ca (Nu ) as a function of number of users. This figure suggests that the multiple access capacity per user of an IR system can be improved by increasing the antenna elements. IX. C ONCLUSIONS In this paper, the multiple access performances of M-ary IR systems with antenna arrays in multipath environments are presented. Antenna array can be used in conjunction with the Rake receiver to increase the multiple access capability. The results indicate that the multiple access capacity of an

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 1, JANUARY 2006

IR system improves significantly when the number of the antenna elements and/or selected paths at the receiver is increased. However, increasing the antenna elements and/or selecting more paths will increase the complexity of a receiver. Therefore, a system designer has to decide a suitable number of antenna elements and/or the multipaths in order to achieve the required multiple access capacity. R EFERENCES [1] R. A. Scholtz, “Multiple access with time hopping modulation,” in Proc. IEEE MILCOM ’93, pp. 447-450, Oct. 1993.

[2] F. Ramirez-Mireles and R. A. Scholtz, “Multiple-access with time hopping and block waveform PPM modulation,” in Proc. IEEE ICC ’98, vol. 3, pp. 775-779, June 1998. [3] F. Ramirez-Mireles, “Performance of ultra wideband SSMA using time hopping and M-ary PPM,” IEEE J. Select. Areas Commun., vol. 19, pp. 1186-1196, June 2001. [4] R. J. -M. Cramer, M. Z. Win, and R. A. Scholtz, “Impulse radio multipath characteristics and diversity reception,” in Proc. IEEE ICC ’98, vol. 3, pp. 1650-1654, June 1998. [5] A. F. Molisch, J. R. Foerster, and M. Pendergrass, ”Channel models for ultrawideband personal area networks,” IEEE Pers. Commun., vol. 10, pp. 14-21, Dec. 2003.