Comparisons of various pulse shapes for DS-UWB ... - Springer Link
Telecommun Syst (2009) 42: 213–222 DOI 10.1007/s11235-009-9181-x
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel Ertan Öztürk · Ergin Yılmaz
Published online: 24 July 2009 © Springer Science+Business Media, LLC 2009
Abstract In this paper, we first investigate the power spectral densities (PSD) of bipolar modulated Direct Sequence Ultra Wide Band (DS-UWB) signals using various pulse shapes under the FCC UWB emission mask. Considered pulse shapes are the first five derivatives of the Gauss pulse (p1 , p2 , p3 , p4 and p5 ), the first four orthogonal modified Hermite waveforms, and Daubechies wavelets (db-q). It is observed in the PSD results that p4 and p5 Gauss pulses, the Daubechies (db-q) for q > 4 comply with the FCC UWB rule by selecting proper values for the pulse duration. Then, we derive the pulse shape dependent probability of error expression for bipolar DS-UWB signals over the standard UWB channel. The five pulse shapes (p4 , p5 , db-5, db-6 and db-7) complying with the FCC emission mask are numerically compared by using the derived probability of error expression over the CM1 model of the Standard UWB channel. Results reveal that the Daubechies have better performance than those of the two Gauss pulses. Keywords DS-UWB · Pulse shape · Wavelets · CM1 channel model
1 Introduction Ultra Wide Band (UWB) has recently become very popular among researchers and industry. UWB technology is a wireless personal area network (WPAN) technology due to its E. Öztürk () · E. Yılmaz Department of Electrical and Electronics Engineering, Zonguldak Karaelmas University, 67100 Zonguldak, Turkey e-mail: [email protected] E. Yılmaz e-mail: [email protected]
very low output power and very high data rate. UWB technology utilizes Giga Hertz (GHz) level bandwidth within short distances. The Federal Communications Commission (FCC) in United States (US) specified emission limits in Power Spectral Density (PSD) for UWB applications in order to avoid interferences with the co-existed applications FCC [8]. The spectrum of UWB systems should have a maximum output power density of −41.3 dBm/MHz between 3.1 GHz and 10.6 GHz. The UWB technology has been standardized by IEEE 802.15.3a group for WPAN applications. The goals for this new standard are data rates up to 110 Mbps at 10 m distance, 200 Mbps at 4 m distance, and higher data rates at smaller distances Molisch, Foerster and Pendergrass [14]. There are various UWB techniques proposed by researchers, however these techniques can be roughly divided into two groups: Single and multi band UWB systems: In single band techniques, spreading sequences as in Code Division Multiple Access (CDMA) technology is used, whereas orthogonal frequency division multiplexing (OFDM) is used in multi-band UWB systems. In single band UWB systems, spreading sequences are pulse shaped by very narrow pulse signals to design direct sequence (DS) and time hopping (TH) UWB systems. These techniques are also called pulse based UWB systems. In DS-UWB systems, base-band modulation techniques; pulse amplitude modulation (PAM), pulse position modulation (PPM), on-off keying (OOK) and bipolar modulation (BPM) (also called antipodal signaling) are used Matthew and Welborn [12]. It is reported in Matthew and Welborn [12] that bipolar modulation is the most power efficient binary modulation. One of the distinguished characteristics of DS-UWB signals is that very narrow pulse signals with Giga Hertz bandwidth are used. Most known UWB pulses in literature are the derivatives of Gauss pulses Zuang, Shen and Bi [27]. Other
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proposed pulses for DS-UWB signals are orthogonal modified Hermite pulses Hu and Zeng [10] and wavelet packets Ciolina, Ghavami and Aghvami [5]. The channel model for UWB systems is proposed by IEEE 802.15.3a group based Saleh-Valenuela (S-V) channel model Saleh and Valenzuela [22], except that instead of a Rayleigh probability of density, it employs a log-normal probability density function (pdf) for the Rayleigh coefficients Molisch, Foerster and Pendergrass [14]. UWB measurement campaigns have been performed to obtain the parameter values for this channel model. Based on the results of four different channel measurement campaigns, four different sets of values for the channel parameters are given as CM1, CM2, CM3 and CM4 models: CM1 is based on line of sight (LOS) (0–4 m) channel measurements, CM2 is based on non line of sight (NLOS) (0–4 m) channel measurements, CM3 is based on NLOS (0–10 m) channel measurements, CM4 was generated to fit a 25 nano second (ns) RMS delay spread Molich, Foerster and Pandergrass [14]. In literature, there are several works about the performance evaluations of UWB systems. However, the performance evaluations are various with respect to considered multiple access and modulation techniques as well as considered channels. In this paper, we focus on the performance of DS-UWB systems. One of the challenges in the evaluation of the probability of error is the complexity of the analysis; hence some studies utilized simulations to obtain the performance results: Rajesvaran, Somayazulu and Foerster [21] investigated the performance of bipolar DS-UWB signals over the standard UWB channel by using MonteCarlo simulations. Tan, Nallanathan and Kannan [26] investigated the performance of DS-UWB systems utilizing multi antennas by using Monte-Carlo simulations. Liu and Elmirghani [11] obtained the performance results of bipolar DS-UWB with variable length spreading sequences by using simulation. Some of the works analysing DS-UWB signals can be summarized as followings: Hu and Beaulieu [9] analysed the performance of DS-UWB systems over additive white Gaussian noise (AWGN) channels. Boubaker and Letaief [1] compared antipodal DS-UWB and time-hopping UWB systems under both perfect and imperfect power control conditions over asynchronous AWGN channels. In Hu and Beaulieu [9], Boubaker and Letaief [1] the considered channels are non-fading channels. Ding, Zang and Wu [7] derived the probability of error expression for bipolar DSUWB signals over the standard UWB channel by using standard Gauss approximation and improved Gauss approximation. However, they didn’t utilize a Rake receiver in the designed system, which is vital for the channels like UWB channel with high number of multi-path resolutions. Chu and Murch [6] analysed the performance of Direct Sequence Multiple Access (DS-MA) UWB systems using biorthogonal pulse keying (BOPK) modulation over the CM3 channel model. In this work, the bit error rate (BER) expression
E. Öztürk, E. Yılmaz
conditioned on multipath fading experienced by each user is obtained, and then the unconditioned BER is obtained by averaging the conditional BERs over multiple channel realizations. Chen and Tsai [4] designed receivers based on Rake structure for DS-UWB systems and analysed the performances. Chen, Chiu and Yang [3] designed Rake receivers for DS-UWB systems over the UWB channel to mitigate inter symbol interference and obtained the performance results under minimum mean squared-error and maximum ratio combining. The works in Chen and Tsai [4], Chen, Chiu and Yang [3] neglected the effects of the pulse waveforms on the performance analysis. Nallanathan and Chai [15] studied multiple access performance of a prerake DS-UWB system over the CM1 and CM4 channel models. They used Gaussian second derivative as the pulse waveform in the numerical results. Chen et al. [2] proposed an approach to analyse pulse waveform dependent BER performance of a DS-CDMA UWB radio, which operates in a frequency selective fading channel. They introduced normalized mean squared auto-correlation function (ACF) of the pulse waveform, the duration of which is limited to one chip interval (Tc ). However, the pulse duration is determined by the bandwidth of the pulse obtained by utilizing the FCC mask Chu and Murch [6]. Besides, instead of calculating the values of the ACF for possible pulse shapes, some given values (0.1 and 0.3) of the ACF are used for the numerical BER calculations in Chen et al. [2]. The recent paper is the extended version of Öztürk and Yılmaz [19]. In aforementioned works and in most of other studies in literature relating to the performance evaluations of DSUWB systems expect Chu and Murch [6], the FCC UWB emission mask is not strictly applied. The study in Chu and Murch [6] presents that the fourth and fifth derivatives of Gauss pulse with 0.5 ns pulse duration (Tw ) satisfy the FCC regulation. Hence, there is a need to investigate the PSDs of other pulses like wavelets with respect to the FCC regulation. The durations of pulse shapes satisfying the FCC rule could be larger than one chip interval. Consequently, there is also a need to investigate the effects of the pulse shapes complying with the FCC rule on the performance of bipolar modulated DS-UWB signals over the standard UWB channel. Main contributions of our work are the followings: we first investigate proper values of pulse durations for the first five derivatives of Gauss pulse, the first four orthogonal modified Hermite pulses and Daubechies wavelet (db-q) pulses in order to comply with the FCC UWB mask. Then, we derive the pulse shape (not limited to chip interval) dependent probability of error expression of bipolar modulated DS-UWB signaling over the standard UWB channel. In the receiver, we use a Rake receiver in conjunction with maximum ratio combiner (MRC) to improve the output signal to noise ratio (SNR). We obtain an expression for the probability of error depending on the SNR at the output of the
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel
Rake receiver, by utilizing Hermite polynomials. The numerical values for the performance are calculated over the CM1 channel model for the pulse shapes complying with the FCC rule. The organization of the paper as the following: After this introduction, the bipolar DS-UWB signal is given in Sect. 2. Section 3 presents the PSDs of the pulse shapes under the FCC UWB emission mask. The standard UWB channel model is described in Sect. 4. Section 5 is about the probability of error derivation. Numerical results for the probability of errors over the CM1 channel model are given in Sect. 6. Section 7 concludes the paper.
2 Transmitted signal In bipolar DS-UWB systems, the transmitted signal is similar to that of BPSK/DS-CDMA systems described in Öztürk and Sahin ¸ [17], Öztürk [18] except there is no carrier signal, and for a total of U users it is given by S(t) =
U
s u (t)
(1)
u=1
where the signal for the uth user is s u (t) =
∞
bu [n/N ]cu [n]ψ(t/Tc − n)
(2)
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where τm determines the pulse width. The nth nderivative p0 (t) of the Gauss pulse is obtained by pn (t) = d dt and n n its Fourier Transform is Pn (f ) = (2πf ) P0 (f ) Chu and Murch [6]. By selecting proper values for τm and n, a proper pulse shape can be obtained with a PSD contained in the FCC UWB emission mask. Other proposed UWB pulse shapes are orthogonal modified Hermite functions given by Hu and Zeng [10] (t/τm )2 (t/τm )2 d n−1 hn (t) = (−1)n−1 exp exp 4 2 dt n−1 (4) where integer n denotes the pulse order and τm determines the pulse width. Finally, orthogonal wavelets are also used as pulse shapes of DS-UWB signals. Most known orthogonal wavelets are Daubechies wavelets, represented by db-q where 2q is the number of finite impulse response (FIR) filter coefficients. Daubechies wavelets don’t have mathematical expressions; instead they are generated from FIR coefficients by using two equations, called dilation equation and wavelet equation Strang and Nguyen [25]. Daubechies wavelets are used widely due to the simplicity of their generations Öztürk [20].
3 Power spectral density
n=−∞
where the parameters are defined similarly to those in Öztürk and Sahin ¸ [17], Öztürk [18]: {bu [n] ∈ {−1, 1}}∞ n=−∞ , u ∈ {1, 2, . . . , U } is the data sequence of uth user. The elements of bu [n] for all u and n, are independent and identically distributed (i.i.d.) equally probable binary bits {+1, −1}. {cu [n] ∈ {−1, 1}}∞ n=−∞ is assumed to be the aperiodic random spreading sequences. The elements of cu [n] for all u and n are also i.i.d. equally probable binary real chips {+1, −1}. Spreading codes and data sequences are individually and mutually independent. y denotes the largest integer less than or equal to y. In (2), the chip and bit rates are Rc = 1/Tc and Rb = 1/Tb (Tb = NTc ), respectively. The processing gain is denoted by N and it is the same for all users. Therefore the energy per bit (Eb ) is Tb . The pulse shaping ψ(t) is a continuous real time function with duration of Tw . Most common pulse shapes used in the literature are the derivatives of Gauss pulse given in time and frequency domains Chu and Murch [6] 2 t ↔ P0 (f ) p0 (t) = exp −2π τm √ π 2τm 2 (3) exp − (τm f ) = 2 2
In bipolar direct sequence spread spectrum (DSSS) systems, the frequency spectrum of pulse shapes define the PSD of the transmitted signal. The difference between the DS-UWB signal and the conventional DSSS signals is that DS-UWB signals has no carrier and pulse shaping waveforms are band-pass signals. The below equation gives the PSD of a bipolar DS-UWB signal for a single user using random bits and chips Mo and Gelman [13], SS (f ) =
1 |ψ(f )|2 Tc
(5)
where ψ(f ) is the Fourier Transform of ψ(t) pulse. In this section, we present the PSD of the pulses defined in the previous section in order to investigate whether they comply with FCC UWB emission mask or not. First, we consider Gaussian pulses. The most proper value of τm for each Gauss pulse is investigated in order to obtain a PSD falling into the FCC UWB mask. Only p4 and p5 comply with the mask by using 0.170 ns and 0.182 ns of τm ns resulting 0.45 and 0.5 ns pulse durations, respectively. The PSDs of the other Gauss pulses can not be contained into the FCC mask for any value τm . Figure 1 shows the PSDs of the first five derivatives of the Gauss pulse. Second, we consider Daubechies wavelets. It is known that higher order of Daubechies (i.e., higher value of q in
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Fig. 1 PSDs of the first five derivatives of Gauss pulse under the FCC UWB mask
Fig. 2 PSDs of the Daubechies pulses (db-5, db6 and db7) under the FCC UWB mask
db-q) results into higher interference Öztürk [16], consequently we consider first several Daubechies which are db-q (q = 1, 2, 3, 4, 5, 6, 7). By investigating a proper value of the pulse width for each wavelet, it is concluded that only db-5,
db-6, and db7 wavelets comply with FCC mask for different pulse widths (1.08 ns, 1.3 ns and 1.52 ns, respectively) as presented in Fig. 2. The PSD of db-q (q = 1, 2, 3, 4) can not be contained into the FCC mask for any pulse width.
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel
217
Fig. 3 PSDs of the first four Hermite pulses under the FCC UWB mask
Daubechies db-4 for 0.84 ns pulse duration which gives the best form approaching to the FCC mask is also shown in Fig. 2. For higher values of q (q > 7), the PSD of db-q also falls into the FCC mask, but the interference derived in the next section (the inter path interference in (11)) for these pulses become higher than those of the Gauss pulses, hence we don’t consider them in this study. Finally, we consider Hermite pulses and search for proper τm values to comply with the FCC regulation. Hermite pulses always occupy bands not allowed by the FCC regulation and their PSDs are shown in Fig. 3 for τm = 0.1 ns. However, they can be shifted to the permitted band by using modulation Chu and Murch [6]. Actually, some of the aforementioned pulses also can be shifted to the allowed band. But, this increases the complexity of the receiver that requires very sensitive timing recovery. Consequently, we don’t modulate ultra wide band signals in our study.
4 Channel model The channel model includes clusters of rays and it is described as the followings Molisch, Foerster and Pendergrass [14]: The channel impulse response for the uth user is hu (t) =
K−1 M−1 k=0 m=0
u u αk,m δ(t − Tku − τk,m )=
L−1
αlu δ(t − τlu )
l=0
(6)
u is fading gain coefficient for mth ray of kth cluswhere αk,m u u is the delay of the ter, Tk is the delay of the kth cluster, τk,m mth ray of kth cluster, for the uth user. There are a total of K clusters and M rays per cluster in the channel model. The rays in the channel arrive with a rate of λ (1/ns) and the clusters arrive with a rate of (1/ns). For simplicity in the channel model, we define l = M ×k +m 0 ≤ k ≤ K −1, 0 ≤ m ≤ M − 1, thus the two summations reduces to one summation. u u ξ uβ u The fading coefficients are given as: αk,m = εk,m k k,m u where εk,m is the random equiprobable ±1 to account for signal inversion due to reflections. The coefficient ξku relates u relates the fading for the fading for the kth cluster and βk,m mth path of the kth cluster. The product of the fading coefu is a random variable having log-normal disficients ξku βk,m u ) ∝ N ormal(μu , σ 2 + σ 2 ) tribution i.e., 20 log10 (ξku βk,m k,m 1 2 u | = 10(μk,m +n1 +n2 )/20 where n ∝ Normal(0, σ 2 ) or |ξku βk,m 1 1 2 and n2 ∝ Normal(0, σ2 ) are independent and corresponded to the fading on each cluster and ray, respectively. The mean value μuk,m is given by u
μuk,m =
u /γ 10ln(0 ) − 10Tku / − 10τk,m
ln(10) −
(σ12 + σ22 )ln(10) 20
(7)
where and γ are the cluster decay factor and ray decay factor, respectively and 0 is the mean energy of the first path of the first cluster. The second moment of the fading
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E. Öztürk, E. Yılmaz
Table 1 The values of the channel parameters for the CM1 model Molisch, Foerster and Pendergrass [14] Parameters
(1/ns)
λ (1/ns)
γ
σ1 (dB)
σ2 (dB)
Values
0.0233
2.5
7.1
4.3
3.4
3.4
γr =
coefficients is given by u u E{(ξku βk,m )2 } = 0 exp(−Tku / ) exp(−τk,m /γ )
(8)
As mentioned earlier, this study considers the CM1 model which is based on LOS (0–4 m) measurement campaign, the values of the channel parameters are given in Table 1. Then, the received signal becomes r(t) =
U L−1
The SNR for the reference path in (10) is independent from the user, i.e., it is the same for any user, and can be written as
− τlu ) + n(t)
(9)
where n(t) is a Gaussian noise with a power spectral density of N0 /2.
where (αr ) is the random variable with log normal distribution, and all other terms are deterministic. Hence, the SNR of a path γr is a random variable with log normal pdf given by Simon and Alouini [24] 10/ ln 10 (10 log10 γr − μr )2 , pγr (γr ) = √ exp − 2σr2 2πσ 2 γr
In this study, the considered receiver is a match filters based Rake receiver similar to that in Öztürk and Sahin ¸ [17]. Since the bandwidth of the transmitted signal is UWB, the number of resolvable paths is quite large. Hence, the considered rake receiver selects, combines and processes only a subset of the resolvable paths. We assume that the Rake receiver selects the first Lf nonzero paths. The outputs of branches of the rake receiver are combined by a MRC. In the MRC, the branches of rake receiver are weighted for maximum SNR, and decision is made based on the combined signal. Based on the similar assumptions for the Rake receiver in Öztürk and Sahin ¸ [17], the SNR γr1 at the output of the reference finger r {r ∈ 0, 1, 2, . . . , Lf − 1} of the desired user (u = 1) is derived for the assumption of unit energy u 2 constraint on the fading process (i.e., L−1 l=0 (αl ) = 1) and for no carrier, then it is obtained as (αr1 )2 Eb N0 + 2(U − 1/L)Iψ
(14)
where μr (dB) and σr (dB) are the mean and standard deviation of 10 log10 γr , respectively, and obtained as μr =
5 Probability of error analysis
(σ 2 + σ22 ) ln(10) 10 ln(0 ) − 10Tr / − 10τr /γ − 1 ln 10 20 Eb 10 (15) ln + ln 10 N0 + 2(U − 1/L)Iψ
σr2 = σ12 + σ22
(16)
The total SNR for any desired user at the output of the MRC is Lf −1
γ=
(17)
γr
r=0
The summation of log normal random variables yields to a log normal variable, i.e., γ is also a log normal random variable. The probability of error using bipolar modulation Lf −1 can be expressed as conditioned on a {γr }r=0
L f −1 Lf −1 Pd ({γr }r=0 ) = Q 2 γr
(18)
r=0
(10)
where Iψ is the Inter Path Interference (IPI) and given by 1 ∞ ˆ )|2 dτ Iψ = |ψ(τ (11) Tc −∞ and it depends on the time correlation of the pulse shaping that is ∞ ψ(t) = ψ(s)ψ(s − t)ds (12) −∞
(13)
γr ≥ 0 αlu slu (t
u=1 l=0
γr1 =
(αr )2 Eb N0 + 2(U − 1/L)Iψ
where Q(x) is the Gaussian Q-function which can be represented in the following definite integral form Simon and Alouini [24]: Q(x) =
1 π
π/2 0
exp −
x2 2 sin2 θ
dθ,
x≥0
(19)
the average Pb with the PDF of γ becomes Pb = 0
∞
Q( 2γ )pγ (γ )dγ
(20)
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel L −1
f where pγ (γ ) is the joint pdf of {pγr (γr )}r=0 , and written as
Lf −1
pγ0 ,γ1,...,γL
f
(γ0 , γ1 , . . . , γLf −1 ) = −1
pγr (γr )
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Table 2 IPI Values for the pulse shapes Pulse
p5
p4
db-7
db-6
db-5
IPI value
0.4607
0.4316
0.4145
0.4089
0.3958
(21)
r=0
Following the work in Simon and Alouini [24], the average probability of error is expressed as
∞ ∞ ∞ L f −1 Pb = ··· Q 2γr 0 0 0 r=0
6 Numerical calculations for the probability of error
n √ 1 1 (xi 2σr +μr )/10 Mr (x, θ ) = √ wi exp − 2 10 π sin θ
7 Conclusions
In this section, the numerical performance results over the CM1 model are obtained by using the probability of error expression derived in the previous section. We consider that the delay resolution (τ ) is 0.167 ns which is based on measurements in the 2 GHz–8 GHz band Molisch, FoLf fold erster and Pendergrass [14]. The typical CM1 channel imLf −1 pulse response contains 200 ∼ 300 bins with a time reso × pγr (γr )dγ0 · · · dγLf −1 (22) lution of 167 ps and it is typically taken as 250 Ding, Zang r=0 and Wu [7]. Hence, the delay spread is 250×167 ps = 42 ns. Since the cluster arrival rate ( ) of the CM1 is 0.0233 (1/ns), By using the definite integral form of Q(x) function in (19), there is approximately one cluster within delay spread of 42 the Pb in (22) is expressed as ns. On the other hand, the ray arrival rate (λ) of the CM1 is Lf −1 ∞ ∞ ∞ π/2 2.5 (1/ns) implies that there are approximately 105 non zero γr 1 dθ Pb = ··· exp − r=02 rays within a delay spread of 42 ns. The chip interval Tc is π 0 sin θ 0 0 0 taken as 0.2 ns, and we consider two data rates in the per Lf fold formance calculations: 50 Mbps and 100 Mbps. Thus, the processing gain N becomes 100 and 50, respectively. The Lf −1 number of users U is 10. We assume that the Rake receiver × pγr (γr )dγ0 · · · dγLf −1 (23) process 20 paths because they capture 85% of the channel r=0 energy. In the numerical calculations, the values of the IPI Lf −1 1 π/2 in (11) are calculated for the pulses comply with the FCC Pb = Mr (γr , θ )dθ (24) regulation. The obtained values are presented in Table 2. π 0 r=0 Then, the values of the probability of error versus the SNR (0–20 db) are calculated for N = 50 and N = 100 rewhere lating to Rb = 100 Mbps and Rb = 50 Mbps for Tc = 0.2 ns. ∞ γr Results are shown in Fig. 4 and Fig. 5, respectively. It is obpγr (γr )dγr Mr (γr , θ ) = exp − 2 (25) sin θ 0 served in the figures that the performance results from the √ best to the worst are db-5, db-6, db-7, p4 and p5. This order By changing the variable as x = (10 log γr − μr )/ 2σr , is due to the IPI results in Table 2. Because, lower values of (25) becomes the IPI lead to better probability of error performance. It is ∞ seen in Fig. 5 that db-5 has approximately 1.8 dB and 4 dB √ 1 1 2 Mr (x, θ ) = √ exp − 2 10(x 2σr +μr )/10 e−x dx better results than those of p4 and p5 at a BER of 10−4 , reπ 0 sin θ spectively. Besides, it is observed by comparing both figures (26) that the performance increases with a decrement of the data rate (or an increment of the code length). Mr (xr , θ ) can be written by using Hermite polynomial Simon and Alouini [24].
i=1
(27) where {xi }, i = 1, 2, . . . , n are the zeros of the nth order Hermite polynomial and {wi } i = 1, 2, . . . , n are the weight factors tabulated in Salzer, Zucker and Capuano [23].
In this work we consider three different sets of pulses for bipolar DS-UWB signals: the first five derivatives of Gauss pulse, Daubechies wavelets and the first four orthogonal Hermite pulses. The PSDs of these pulses are investigated under the FCC UWB emission regulation. The Gauss pulses p4 and p5 and db-q (q > 4) wavelets coincides with the FCC
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Fig. 4 Probability of error versus the SNR for N = 50, Tc = 0.2 ns, Rb = 100 Mbps, U = 10
Fig. 5 Probability of error versus the SNR for N = 100, Tc = 0.2 ns, Rb = 50 Mbps, U = 10
regulation, whereas the PSD of the other pulses (Gauss pulse for n < 4, and the Hermite pulses) are not contained in the FCC mask. Then, we compare the pulses complying with the
FCC rule in terms of the probability of error over the CM1 channel model. Results reveal that the Daubechies (q = 5, 6 and 7) have better performance than those of p4 and p5 .
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel Acknowledgements This work was supported by Zonguldak Karaelmas University Research Fund Project number: 2007-YDP-45-19.
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18. Öztürk, E. (2007). Performance analysis of asynchronous DSCDMA for general chip waveforms over multi-path Rayleigh fading channels. IET Communications, 1(4), 570–576. 19. Öztürk, E., & Yılmaz, E. (2007). Performance of DS-UWB signals over the CM1 channel model. In Proceedings of the 5th ACM international workshop on mobility management and wireless access (MobiWac 2007) (pp. 107–111). Chania, Crete, Greece, October 2007. 20. Öztürk, E. (2008). Performance of wavelet based multi-chip rate DS-CDMA signals over multi-path Nakagami-m fading channels. Wireless Communications and Mobile Computing, 8(6), 745– 757. 21. Rajeswaran, A., Somayazulu, V. S., & Foerster, J. R. (2003). Rake performance for a pulse based UWB system in a realistic UWB indoor channel. In IEEE international conference on communications (Vol. 4, pp. 2879–2883) Alaska, USA, May 2003. 22. Saleh, A. R., & Valenzuela, R. (1987). A statistical model for indoor multipath propagation. IEEE Journal on Selected Areas in Communications, 5, 128–137. 23. Salzer, E. H., Zucker, Z., & Capuano, R. (1952). Table of the zeros and weight factors of the first twenty Hermite polynomials. Journal of Research of the National Bureau of Standards, 48(2). 24. Simon, M. K., & Alouini, S. M. (2000). Digital communications over fading channels. New York: Wiley. 25. Strang, G., & Nguyen, T. (1996). Wavelet and filter banks. Cambridge, Wellesley: Wellesley-Cambridge Press. 26. Tan, S. A., Nallanathan, A., & Kannan, B. (2006). Performance of DS-UWB multiple access with diversity reception in dense multipath environment. IEEE Transactions on Vehicular Technology, 55, 1269–1280. 27. Zuang, W., Shen, X., & Bi, Q. (2003). Ultra wideband wireless communications. Wireless Communications and Mobile Computing, 3, 663–685.
˙ Ertan Öztürk was born in Iznik, Bursa, Turkey on September 10, 1971. He received the B.S. degree in Electrical and Electronics Engineering from Gazi University, Ankara, Turkey in 1992, and the M.S. and the Ph.D. degrees both in Electrical Engineering from Illinois Institute of Technology, Chicago, in 1995 and 2001, respectively. He worked as a senior system engineer from 2000 to 2002 at Motorola, Illinois where he worked on UMTS system simulations and planning. Dr. Öztürk has been working as a faculty member in the Department of Electrical and Electronics Engineering at Zonguldak Karaelmas University, Turkey since July 2002. His research interests are in the area of wireless communication systems including code division multiple access systems and ultra wide band wireless access.
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E. Öztürk, E. Yılmaz Ergin Yılmaz was born in Karabük, Turkey on 27 November 1979. He received his B.S. degree in Electronics and Telecommunications Engineering from Kocaeli University, Kocaeli, Turkey in 2003. He worked as a Service Engineer from 2005 to 2006 at Agfa Film, Turkey. He has been working as a research assistant at Zonguldak Karaelmas University, Turkey since January 2006. He is currently pursuing the master degree in Electrical and Electronics Engineering at Zonguldak Karaelmas University. His research inter-
ests are in the area of digital communications including ultra-wideband communication systems.
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel Ertan Öztürk · Ergin Yılmaz
Published online: 24 July 2009 © Springer Science+Business Media, LLC 2009
Abstract In this paper, we first investigate the power spectral densities (PSD) of bipolar modulated Direct Sequence Ultra Wide Band (DS-UWB) signals using various pulse shapes under the FCC UWB emission mask. Considered pulse shapes are the first five derivatives of the Gauss pulse (p1 , p2 , p3 , p4 and p5 ), the first four orthogonal modified Hermite waveforms, and Daubechies wavelets (db-q). It is observed in the PSD results that p4 and p5 Gauss pulses, the Daubechies (db-q) for q > 4 comply with the FCC UWB rule by selecting proper values for the pulse duration. Then, we derive the pulse shape dependent probability of error expression for bipolar DS-UWB signals over the standard UWB channel. The five pulse shapes (p4 , p5 , db-5, db-6 and db-7) complying with the FCC emission mask are numerically compared by using the derived probability of error expression over the CM1 model of the Standard UWB channel. Results reveal that the Daubechies have better performance than those of the two Gauss pulses. Keywords DS-UWB · Pulse shape · Wavelets · CM1 channel model
1 Introduction Ultra Wide Band (UWB) has recently become very popular among researchers and industry. UWB technology is a wireless personal area network (WPAN) technology due to its E. Öztürk () · E. Yılmaz Department of Electrical and Electronics Engineering, Zonguldak Karaelmas University, 67100 Zonguldak, Turkey e-mail: [email protected] E. Yılmaz e-mail: [email protected]
very low output power and very high data rate. UWB technology utilizes Giga Hertz (GHz) level bandwidth within short distances. The Federal Communications Commission (FCC) in United States (US) specified emission limits in Power Spectral Density (PSD) for UWB applications in order to avoid interferences with the co-existed applications FCC [8]. The spectrum of UWB systems should have a maximum output power density of −41.3 dBm/MHz between 3.1 GHz and 10.6 GHz. The UWB technology has been standardized by IEEE 802.15.3a group for WPAN applications. The goals for this new standard are data rates up to 110 Mbps at 10 m distance, 200 Mbps at 4 m distance, and higher data rates at smaller distances Molisch, Foerster and Pendergrass [14]. There are various UWB techniques proposed by researchers, however these techniques can be roughly divided into two groups: Single and multi band UWB systems: In single band techniques, spreading sequences as in Code Division Multiple Access (CDMA) technology is used, whereas orthogonal frequency division multiplexing (OFDM) is used in multi-band UWB systems. In single band UWB systems, spreading sequences are pulse shaped by very narrow pulse signals to design direct sequence (DS) and time hopping (TH) UWB systems. These techniques are also called pulse based UWB systems. In DS-UWB systems, base-band modulation techniques; pulse amplitude modulation (PAM), pulse position modulation (PPM), on-off keying (OOK) and bipolar modulation (BPM) (also called antipodal signaling) are used Matthew and Welborn [12]. It is reported in Matthew and Welborn [12] that bipolar modulation is the most power efficient binary modulation. One of the distinguished characteristics of DS-UWB signals is that very narrow pulse signals with Giga Hertz bandwidth are used. Most known UWB pulses in literature are the derivatives of Gauss pulses Zuang, Shen and Bi [27]. Other
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proposed pulses for DS-UWB signals are orthogonal modified Hermite pulses Hu and Zeng [10] and wavelet packets Ciolina, Ghavami and Aghvami [5]. The channel model for UWB systems is proposed by IEEE 802.15.3a group based Saleh-Valenuela (S-V) channel model Saleh and Valenzuela [22], except that instead of a Rayleigh probability of density, it employs a log-normal probability density function (pdf) for the Rayleigh coefficients Molisch, Foerster and Pendergrass [14]. UWB measurement campaigns have been performed to obtain the parameter values for this channel model. Based on the results of four different channel measurement campaigns, four different sets of values for the channel parameters are given as CM1, CM2, CM3 and CM4 models: CM1 is based on line of sight (LOS) (0–4 m) channel measurements, CM2 is based on non line of sight (NLOS) (0–4 m) channel measurements, CM3 is based on NLOS (0–10 m) channel measurements, CM4 was generated to fit a 25 nano second (ns) RMS delay spread Molich, Foerster and Pandergrass [14]. In literature, there are several works about the performance evaluations of UWB systems. However, the performance evaluations are various with respect to considered multiple access and modulation techniques as well as considered channels. In this paper, we focus on the performance of DS-UWB systems. One of the challenges in the evaluation of the probability of error is the complexity of the analysis; hence some studies utilized simulations to obtain the performance results: Rajesvaran, Somayazulu and Foerster [21] investigated the performance of bipolar DS-UWB signals over the standard UWB channel by using MonteCarlo simulations. Tan, Nallanathan and Kannan [26] investigated the performance of DS-UWB systems utilizing multi antennas by using Monte-Carlo simulations. Liu and Elmirghani [11] obtained the performance results of bipolar DS-UWB with variable length spreading sequences by using simulation. Some of the works analysing DS-UWB signals can be summarized as followings: Hu and Beaulieu [9] analysed the performance of DS-UWB systems over additive white Gaussian noise (AWGN) channels. Boubaker and Letaief [1] compared antipodal DS-UWB and time-hopping UWB systems under both perfect and imperfect power control conditions over asynchronous AWGN channels. In Hu and Beaulieu [9], Boubaker and Letaief [1] the considered channels are non-fading channels. Ding, Zang and Wu [7] derived the probability of error expression for bipolar DSUWB signals over the standard UWB channel by using standard Gauss approximation and improved Gauss approximation. However, they didn’t utilize a Rake receiver in the designed system, which is vital for the channels like UWB channel with high number of multi-path resolutions. Chu and Murch [6] analysed the performance of Direct Sequence Multiple Access (DS-MA) UWB systems using biorthogonal pulse keying (BOPK) modulation over the CM3 channel model. In this work, the bit error rate (BER) expression
E. Öztürk, E. Yılmaz
conditioned on multipath fading experienced by each user is obtained, and then the unconditioned BER is obtained by averaging the conditional BERs over multiple channel realizations. Chen and Tsai [4] designed receivers based on Rake structure for DS-UWB systems and analysed the performances. Chen, Chiu and Yang [3] designed Rake receivers for DS-UWB systems over the UWB channel to mitigate inter symbol interference and obtained the performance results under minimum mean squared-error and maximum ratio combining. The works in Chen and Tsai [4], Chen, Chiu and Yang [3] neglected the effects of the pulse waveforms on the performance analysis. Nallanathan and Chai [15] studied multiple access performance of a prerake DS-UWB system over the CM1 and CM4 channel models. They used Gaussian second derivative as the pulse waveform in the numerical results. Chen et al. [2] proposed an approach to analyse pulse waveform dependent BER performance of a DS-CDMA UWB radio, which operates in a frequency selective fading channel. They introduced normalized mean squared auto-correlation function (ACF) of the pulse waveform, the duration of which is limited to one chip interval (Tc ). However, the pulse duration is determined by the bandwidth of the pulse obtained by utilizing the FCC mask Chu and Murch [6]. Besides, instead of calculating the values of the ACF for possible pulse shapes, some given values (0.1 and 0.3) of the ACF are used for the numerical BER calculations in Chen et al. [2]. The recent paper is the extended version of Öztürk and Yılmaz [19]. In aforementioned works and in most of other studies in literature relating to the performance evaluations of DSUWB systems expect Chu and Murch [6], the FCC UWB emission mask is not strictly applied. The study in Chu and Murch [6] presents that the fourth and fifth derivatives of Gauss pulse with 0.5 ns pulse duration (Tw ) satisfy the FCC regulation. Hence, there is a need to investigate the PSDs of other pulses like wavelets with respect to the FCC regulation. The durations of pulse shapes satisfying the FCC rule could be larger than one chip interval. Consequently, there is also a need to investigate the effects of the pulse shapes complying with the FCC rule on the performance of bipolar modulated DS-UWB signals over the standard UWB channel. Main contributions of our work are the followings: we first investigate proper values of pulse durations for the first five derivatives of Gauss pulse, the first four orthogonal modified Hermite pulses and Daubechies wavelet (db-q) pulses in order to comply with the FCC UWB mask. Then, we derive the pulse shape (not limited to chip interval) dependent probability of error expression of bipolar modulated DS-UWB signaling over the standard UWB channel. In the receiver, we use a Rake receiver in conjunction with maximum ratio combiner (MRC) to improve the output signal to noise ratio (SNR). We obtain an expression for the probability of error depending on the SNR at the output of the
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel
Rake receiver, by utilizing Hermite polynomials. The numerical values for the performance are calculated over the CM1 channel model for the pulse shapes complying with the FCC rule. The organization of the paper as the following: After this introduction, the bipolar DS-UWB signal is given in Sect. 2. Section 3 presents the PSDs of the pulse shapes under the FCC UWB emission mask. The standard UWB channel model is described in Sect. 4. Section 5 is about the probability of error derivation. Numerical results for the probability of errors over the CM1 channel model are given in Sect. 6. Section 7 concludes the paper.
2 Transmitted signal In bipolar DS-UWB systems, the transmitted signal is similar to that of BPSK/DS-CDMA systems described in Öztürk and Sahin ¸ [17], Öztürk [18] except there is no carrier signal, and for a total of U users it is given by S(t) =
U
s u (t)
(1)
u=1
where the signal for the uth user is s u (t) =
∞
bu [n/N ]cu [n]ψ(t/Tc − n)
(2)
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where τm determines the pulse width. The nth nderivative p0 (t) of the Gauss pulse is obtained by pn (t) = d dt and n n its Fourier Transform is Pn (f ) = (2πf ) P0 (f ) Chu and Murch [6]. By selecting proper values for τm and n, a proper pulse shape can be obtained with a PSD contained in the FCC UWB emission mask. Other proposed UWB pulse shapes are orthogonal modified Hermite functions given by Hu and Zeng [10] (t/τm )2 (t/τm )2 d n−1 hn (t) = (−1)n−1 exp exp 4 2 dt n−1 (4) where integer n denotes the pulse order and τm determines the pulse width. Finally, orthogonal wavelets are also used as pulse shapes of DS-UWB signals. Most known orthogonal wavelets are Daubechies wavelets, represented by db-q where 2q is the number of finite impulse response (FIR) filter coefficients. Daubechies wavelets don’t have mathematical expressions; instead they are generated from FIR coefficients by using two equations, called dilation equation and wavelet equation Strang and Nguyen [25]. Daubechies wavelets are used widely due to the simplicity of their generations Öztürk [20].
3 Power spectral density
n=−∞
where the parameters are defined similarly to those in Öztürk and Sahin ¸ [17], Öztürk [18]: {bu [n] ∈ {−1, 1}}∞ n=−∞ , u ∈ {1, 2, . . . , U } is the data sequence of uth user. The elements of bu [n] for all u and n, are independent and identically distributed (i.i.d.) equally probable binary bits {+1, −1}. {cu [n] ∈ {−1, 1}}∞ n=−∞ is assumed to be the aperiodic random spreading sequences. The elements of cu [n] for all u and n are also i.i.d. equally probable binary real chips {+1, −1}. Spreading codes and data sequences are individually and mutually independent. y denotes the largest integer less than or equal to y. In (2), the chip and bit rates are Rc = 1/Tc and Rb = 1/Tb (Tb = NTc ), respectively. The processing gain is denoted by N and it is the same for all users. Therefore the energy per bit (Eb ) is Tb . The pulse shaping ψ(t) is a continuous real time function with duration of Tw . Most common pulse shapes used in the literature are the derivatives of Gauss pulse given in time and frequency domains Chu and Murch [6] 2 t ↔ P0 (f ) p0 (t) = exp −2π τm √ π 2τm 2 (3) exp − (τm f ) = 2 2
In bipolar direct sequence spread spectrum (DSSS) systems, the frequency spectrum of pulse shapes define the PSD of the transmitted signal. The difference between the DS-UWB signal and the conventional DSSS signals is that DS-UWB signals has no carrier and pulse shaping waveforms are band-pass signals. The below equation gives the PSD of a bipolar DS-UWB signal for a single user using random bits and chips Mo and Gelman [13], SS (f ) =
1 |ψ(f )|2 Tc
(5)
where ψ(f ) is the Fourier Transform of ψ(t) pulse. In this section, we present the PSD of the pulses defined in the previous section in order to investigate whether they comply with FCC UWB emission mask or not. First, we consider Gaussian pulses. The most proper value of τm for each Gauss pulse is investigated in order to obtain a PSD falling into the FCC UWB mask. Only p4 and p5 comply with the mask by using 0.170 ns and 0.182 ns of τm ns resulting 0.45 and 0.5 ns pulse durations, respectively. The PSDs of the other Gauss pulses can not be contained into the FCC mask for any value τm . Figure 1 shows the PSDs of the first five derivatives of the Gauss pulse. Second, we consider Daubechies wavelets. It is known that higher order of Daubechies (i.e., higher value of q in
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E. Öztürk, E. Yılmaz
Fig. 1 PSDs of the first five derivatives of Gauss pulse under the FCC UWB mask
Fig. 2 PSDs of the Daubechies pulses (db-5, db6 and db7) under the FCC UWB mask
db-q) results into higher interference Öztürk [16], consequently we consider first several Daubechies which are db-q (q = 1, 2, 3, 4, 5, 6, 7). By investigating a proper value of the pulse width for each wavelet, it is concluded that only db-5,
db-6, and db7 wavelets comply with FCC mask for different pulse widths (1.08 ns, 1.3 ns and 1.52 ns, respectively) as presented in Fig. 2. The PSD of db-q (q = 1, 2, 3, 4) can not be contained into the FCC mask for any pulse width.
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel
217
Fig. 3 PSDs of the first four Hermite pulses under the FCC UWB mask
Daubechies db-4 for 0.84 ns pulse duration which gives the best form approaching to the FCC mask is also shown in Fig. 2. For higher values of q (q > 7), the PSD of db-q also falls into the FCC mask, but the interference derived in the next section (the inter path interference in (11)) for these pulses become higher than those of the Gauss pulses, hence we don’t consider them in this study. Finally, we consider Hermite pulses and search for proper τm values to comply with the FCC regulation. Hermite pulses always occupy bands not allowed by the FCC regulation and their PSDs are shown in Fig. 3 for τm = 0.1 ns. However, they can be shifted to the permitted band by using modulation Chu and Murch [6]. Actually, some of the aforementioned pulses also can be shifted to the allowed band. But, this increases the complexity of the receiver that requires very sensitive timing recovery. Consequently, we don’t modulate ultra wide band signals in our study.
4 Channel model The channel model includes clusters of rays and it is described as the followings Molisch, Foerster and Pendergrass [14]: The channel impulse response for the uth user is hu (t) =
K−1 M−1 k=0 m=0
u u αk,m δ(t − Tku − τk,m )=
L−1
αlu δ(t − τlu )
l=0
(6)
u is fading gain coefficient for mth ray of kth cluswhere αk,m u u is the delay of the ter, Tk is the delay of the kth cluster, τk,m mth ray of kth cluster, for the uth user. There are a total of K clusters and M rays per cluster in the channel model. The rays in the channel arrive with a rate of λ (1/ns) and the clusters arrive with a rate of (1/ns). For simplicity in the channel model, we define l = M ×k +m 0 ≤ k ≤ K −1, 0 ≤ m ≤ M − 1, thus the two summations reduces to one summation. u u ξ uβ u The fading coefficients are given as: αk,m = εk,m k k,m u where εk,m is the random equiprobable ±1 to account for signal inversion due to reflections. The coefficient ξku relates u relates the fading for the fading for the kth cluster and βk,m mth path of the kth cluster. The product of the fading coefu is a random variable having log-normal disficients ξku βk,m u ) ∝ N ormal(μu , σ 2 + σ 2 ) tribution i.e., 20 log10 (ξku βk,m k,m 1 2 u | = 10(μk,m +n1 +n2 )/20 where n ∝ Normal(0, σ 2 ) or |ξku βk,m 1 1 2 and n2 ∝ Normal(0, σ2 ) are independent and corresponded to the fading on each cluster and ray, respectively. The mean value μuk,m is given by u
μuk,m =
u /γ 10ln(0 ) − 10Tku / − 10τk,m
ln(10) −
(σ12 + σ22 )ln(10) 20
(7)
where and γ are the cluster decay factor and ray decay factor, respectively and 0 is the mean energy of the first path of the first cluster. The second moment of the fading
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E. Öztürk, E. Yılmaz
Table 1 The values of the channel parameters for the CM1 model Molisch, Foerster and Pendergrass [14] Parameters
(1/ns)
λ (1/ns)
γ
σ1 (dB)
σ2 (dB)
Values
0.0233
2.5
7.1
4.3
3.4
3.4
γr =
coefficients is given by u u E{(ξku βk,m )2 } = 0 exp(−Tku / ) exp(−τk,m /γ )
(8)
As mentioned earlier, this study considers the CM1 model which is based on LOS (0–4 m) measurement campaign, the values of the channel parameters are given in Table 1. Then, the received signal becomes r(t) =
U L−1
The SNR for the reference path in (10) is independent from the user, i.e., it is the same for any user, and can be written as
− τlu ) + n(t)
(9)
where n(t) is a Gaussian noise with a power spectral density of N0 /2.
where (αr ) is the random variable with log normal distribution, and all other terms are deterministic. Hence, the SNR of a path γr is a random variable with log normal pdf given by Simon and Alouini [24] 10/ ln 10 (10 log10 γr − μr )2 , pγr (γr ) = √ exp − 2σr2 2πσ 2 γr
In this study, the considered receiver is a match filters based Rake receiver similar to that in Öztürk and Sahin ¸ [17]. Since the bandwidth of the transmitted signal is UWB, the number of resolvable paths is quite large. Hence, the considered rake receiver selects, combines and processes only a subset of the resolvable paths. We assume that the Rake receiver selects the first Lf nonzero paths. The outputs of branches of the rake receiver are combined by a MRC. In the MRC, the branches of rake receiver are weighted for maximum SNR, and decision is made based on the combined signal. Based on the similar assumptions for the Rake receiver in Öztürk and Sahin ¸ [17], the SNR γr1 at the output of the reference finger r {r ∈ 0, 1, 2, . . . , Lf − 1} of the desired user (u = 1) is derived for the assumption of unit energy u 2 constraint on the fading process (i.e., L−1 l=0 (αl ) = 1) and for no carrier, then it is obtained as (αr1 )2 Eb N0 + 2(U − 1/L)Iψ
(14)
where μr (dB) and σr (dB) are the mean and standard deviation of 10 log10 γr , respectively, and obtained as μr =
5 Probability of error analysis
(σ 2 + σ22 ) ln(10) 10 ln(0 ) − 10Tr / − 10τr /γ − 1 ln 10 20 Eb 10 (15) ln + ln 10 N0 + 2(U − 1/L)Iψ
σr2 = σ12 + σ22
(16)
The total SNR for any desired user at the output of the MRC is Lf −1
γ=
(17)
γr
r=0
The summation of log normal random variables yields to a log normal variable, i.e., γ is also a log normal random variable. The probability of error using bipolar modulation Lf −1 can be expressed as conditioned on a {γr }r=0
L f −1 Lf −1 Pd ({γr }r=0 ) = Q 2 γr
(18)
r=0
(10)
where Iψ is the Inter Path Interference (IPI) and given by 1 ∞ ˆ )|2 dτ Iψ = |ψ(τ (11) Tc −∞ and it depends on the time correlation of the pulse shaping that is ∞ ψ(t) = ψ(s)ψ(s − t)ds (12) −∞
(13)
γr ≥ 0 αlu slu (t
u=1 l=0
γr1 =
(αr )2 Eb N0 + 2(U − 1/L)Iψ
where Q(x) is the Gaussian Q-function which can be represented in the following definite integral form Simon and Alouini [24]: Q(x) =
1 π
π/2 0
exp −
x2 2 sin2 θ
dθ,
x≥0
(19)
the average Pb with the PDF of γ becomes Pb = 0
∞
Q( 2γ )pγ (γ )dγ
(20)
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel L −1
f where pγ (γ ) is the joint pdf of {pγr (γr )}r=0 , and written as
Lf −1
pγ0 ,γ1,...,γL
f
(γ0 , γ1 , . . . , γLf −1 ) = −1
pγr (γr )
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Table 2 IPI Values for the pulse shapes Pulse
p5
p4
db-7
db-6
db-5
IPI value
0.4607
0.4316
0.4145
0.4089
0.3958
(21)
r=0
Following the work in Simon and Alouini [24], the average probability of error is expressed as
∞ ∞ ∞ L f −1 Pb = ··· Q 2γr 0 0 0 r=0
6 Numerical calculations for the probability of error
n √ 1 1 (xi 2σr +μr )/10 Mr (x, θ ) = √ wi exp − 2 10 π sin θ
7 Conclusions
In this section, the numerical performance results over the CM1 model are obtained by using the probability of error expression derived in the previous section. We consider that the delay resolution (τ ) is 0.167 ns which is based on measurements in the 2 GHz–8 GHz band Molisch, FoLf fold erster and Pendergrass [14]. The typical CM1 channel imLf −1 pulse response contains 200 ∼ 300 bins with a time reso × pγr (γr )dγ0 · · · dγLf −1 (22) lution of 167 ps and it is typically taken as 250 Ding, Zang r=0 and Wu [7]. Hence, the delay spread is 250×167 ps = 42 ns. Since the cluster arrival rate ( ) of the CM1 is 0.0233 (1/ns), By using the definite integral form of Q(x) function in (19), there is approximately one cluster within delay spread of 42 the Pb in (22) is expressed as ns. On the other hand, the ray arrival rate (λ) of the CM1 is Lf −1 ∞ ∞ ∞ π/2 2.5 (1/ns) implies that there are approximately 105 non zero γr 1 dθ Pb = ··· exp − r=02 rays within a delay spread of 42 ns. The chip interval Tc is π 0 sin θ 0 0 0 taken as 0.2 ns, and we consider two data rates in the per Lf fold formance calculations: 50 Mbps and 100 Mbps. Thus, the processing gain N becomes 100 and 50, respectively. The Lf −1 number of users U is 10. We assume that the Rake receiver × pγr (γr )dγ0 · · · dγLf −1 (23) process 20 paths because they capture 85% of the channel r=0 energy. In the numerical calculations, the values of the IPI Lf −1 1 π/2 in (11) are calculated for the pulses comply with the FCC Pb = Mr (γr , θ )dθ (24) regulation. The obtained values are presented in Table 2. π 0 r=0 Then, the values of the probability of error versus the SNR (0–20 db) are calculated for N = 50 and N = 100 rewhere lating to Rb = 100 Mbps and Rb = 50 Mbps for Tc = 0.2 ns. ∞ γr Results are shown in Fig. 4 and Fig. 5, respectively. It is obpγr (γr )dγr Mr (γr , θ ) = exp − 2 (25) sin θ 0 served in the figures that the performance results from the √ best to the worst are db-5, db-6, db-7, p4 and p5. This order By changing the variable as x = (10 log γr − μr )/ 2σr , is due to the IPI results in Table 2. Because, lower values of (25) becomes the IPI lead to better probability of error performance. It is ∞ seen in Fig. 5 that db-5 has approximately 1.8 dB and 4 dB √ 1 1 2 Mr (x, θ ) = √ exp − 2 10(x 2σr +μr )/10 e−x dx better results than those of p4 and p5 at a BER of 10−4 , reπ 0 sin θ spectively. Besides, it is observed by comparing both figures (26) that the performance increases with a decrement of the data rate (or an increment of the code length). Mr (xr , θ ) can be written by using Hermite polynomial Simon and Alouini [24].
i=1
(27) where {xi }, i = 1, 2, . . . , n are the zeros of the nth order Hermite polynomial and {wi } i = 1, 2, . . . , n are the weight factors tabulated in Salzer, Zucker and Capuano [23].
In this work we consider three different sets of pulses for bipolar DS-UWB signals: the first five derivatives of Gauss pulse, Daubechies wavelets and the first four orthogonal Hermite pulses. The PSDs of these pulses are investigated under the FCC UWB emission regulation. The Gauss pulses p4 and p5 and db-q (q > 4) wavelets coincides with the FCC
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E. Öztürk, E. Yılmaz
Fig. 4 Probability of error versus the SNR for N = 50, Tc = 0.2 ns, Rb = 100 Mbps, U = 10
Fig. 5 Probability of error versus the SNR for N = 100, Tc = 0.2 ns, Rb = 50 Mbps, U = 10
regulation, whereas the PSD of the other pulses (Gauss pulse for n < 4, and the Hermite pulses) are not contained in the FCC mask. Then, we compare the pulses complying with the
FCC rule in terms of the probability of error over the CM1 channel model. Results reveal that the Daubechies (q = 5, 6 and 7) have better performance than those of p4 and p5 .
Comparisons of various pulse shapes for DS-UWB signals over the UWB channel Acknowledgements This work was supported by Zonguldak Karaelmas University Research Fund Project number: 2007-YDP-45-19.
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18. Öztürk, E. (2007). Performance analysis of asynchronous DSCDMA for general chip waveforms over multi-path Rayleigh fading channels. IET Communications, 1(4), 570–576. 19. Öztürk, E., & Yılmaz, E. (2007). Performance of DS-UWB signals over the CM1 channel model. In Proceedings of the 5th ACM international workshop on mobility management and wireless access (MobiWac 2007) (pp. 107–111). Chania, Crete, Greece, October 2007. 20. Öztürk, E. (2008). Performance of wavelet based multi-chip rate DS-CDMA signals over multi-path Nakagami-m fading channels. Wireless Communications and Mobile Computing, 8(6), 745– 757. 21. Rajeswaran, A., Somayazulu, V. S., & Foerster, J. R. (2003). Rake performance for a pulse based UWB system in a realistic UWB indoor channel. In IEEE international conference on communications (Vol. 4, pp. 2879–2883) Alaska, USA, May 2003. 22. Saleh, A. R., & Valenzuela, R. (1987). A statistical model for indoor multipath propagation. IEEE Journal on Selected Areas in Communications, 5, 128–137. 23. Salzer, E. H., Zucker, Z., & Capuano, R. (1952). Table of the zeros and weight factors of the first twenty Hermite polynomials. Journal of Research of the National Bureau of Standards, 48(2). 24. Simon, M. K., & Alouini, S. M. (2000). Digital communications over fading channels. New York: Wiley. 25. Strang, G., & Nguyen, T. (1996). Wavelet and filter banks. Cambridge, Wellesley: Wellesley-Cambridge Press. 26. Tan, S. A., Nallanathan, A., & Kannan, B. (2006). Performance of DS-UWB multiple access with diversity reception in dense multipath environment. IEEE Transactions on Vehicular Technology, 55, 1269–1280. 27. Zuang, W., Shen, X., & Bi, Q. (2003). Ultra wideband wireless communications. Wireless Communications and Mobile Computing, 3, 663–685.
˙ Ertan Öztürk was born in Iznik, Bursa, Turkey on September 10, 1971. He received the B.S. degree in Electrical and Electronics Engineering from Gazi University, Ankara, Turkey in 1992, and the M.S. and the Ph.D. degrees both in Electrical Engineering from Illinois Institute of Technology, Chicago, in 1995 and 2001, respectively. He worked as a senior system engineer from 2000 to 2002 at Motorola, Illinois where he worked on UMTS system simulations and planning. Dr. Öztürk has been working as a faculty member in the Department of Electrical and Electronics Engineering at Zonguldak Karaelmas University, Turkey since July 2002. His research interests are in the area of wireless communication systems including code division multiple access systems and ultra wide band wireless access.
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E. Öztürk, E. Yılmaz Ergin Yılmaz was born in Karabük, Turkey on 27 November 1979. He received his B.S. degree in Electronics and Telecommunications Engineering from Kocaeli University, Kocaeli, Turkey in 2003. He worked as a Service Engineer from 2005 to 2006 at Agfa Film, Turkey. He has been working as a research assistant at Zonguldak Karaelmas University, Turkey since January 2006. He is currently pursuing the master degree in Electrical and Electronics Engineering at Zonguldak Karaelmas University. His research inter-
ests are in the area of digital communications including ultra-wideband communication systems.
