BER analysis in a generalized UWB frequency selective fading channel with randomly arriving clusters and rays

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BER Analysis in A Generalized UWB Frequency

Selective Fading Channel With Randomly Arriving

Clusters and Rays

Wei-Cheng Liu and Li-Chun Wang

Department of Communication Engineering

National Chiao Tung University, Hsinchu, Taiwan

lichun@cc.nctu.edu.tw, Tel: +886-3-5712121 ext 54511

Abstract— In this paper, we present an analytical method to evaluate the bit error rate (BER) of the ultra-wideband (UWB) system in the IEEE 802.15.4a standardized channel model. The IEEE 802.15.4a UWB channel model is more general and based on more measurements than the earlier IEEE 802.15.3a model. It also poses new challenge in analyzing UWB performance in such a channel. First, the power delay profile become a function of randomly arriving cluster and ray arrival time. Second, the signal amplitude in the IEEE 802.15.4a channel is modeled by a Nakagami random variable of which the Nakagami fading parameter is log-normally distributed. Thus, the signal amplitude is a nonlinear function of a log-normally distributed random variable. By means of counting integral of Lebesgue measure theory, the analytical expression for the BER performance in the IEEE 802.15.4a UWB channel is presented. We apply this analytical model to investigate the impacts of various UWB channel parameters on the system performance and provide some useful insights into the design of UWB transceiver.

Index Terms— Ultra-wideband (UWB), IEEE 802.15.4a chan-nel model, bit error rate (BER).

I. INTRODUCTION

T

HE ultra-wideband (UWB) communications is a

promis-ing technique to achieve the objectives of higher data rates and better quality for the modern wireless systems. The complexity of the UWB channel has posed new challenges on performance analysis. Thus, for a more sophisticated channel model such as the IEEE 802.15.4a [1], the performance eval-uation of UWB system is mainly performed by simulations.

Compared to traditional narrow-band channel, the UWB channel has two important properties. First, the bandwidth of the UWB signals is much wider than the coherence bandwidth of the channel. Thus, the severely frequency selective fading occurs in the frequency domain. Second, The large bandwidth of a UWB signal results in a arrival time with high resolution. Thus, the UWB waves arrive in many clusters and contain some non-Rayleigh multipath components.

A. Motivation and Challenges

Based on measurement results, the IEEE 802.15.4a has captured these important characteristics, but also pose new

1_{This work is supported by the National Science Council, Taiwan, under}

the contract NSC95-2221-E-009-147.

challenges for performance analysis, which will be discussed next.

• Unlike the fixed number of impulse responses in a

narrow-band channel, the UWB signals may arrive with random number of clusters and rays. In the IEEE 802.15.4a, the number of clusters is modeled as the Poisson random variable, and the inter-arrival time of the rays within a cluster is modeled by a hyperexponential random variable. Thus, computing the collected signal energy in a UWB channel with random number of both clusters and rays is complicated since one may need to find a joint distribution of infinite number of random variables.

• When the number of arrival rays in a very narrow time bin (or chip duration) is not large enough, the central limit theorem is no longer applicable to model the distribution of fading signal. Unlike a traditional Rayleigh random variable as in the narrow-band case, the multipath fading signal in the IEEE 802.15.4a UWB channel is charac-terized by a Nakagami m random variable of which the parameter m is a log-normal random variable related to the ray arrival time. Thus, the UWB signal amplitude become a multidimensional random variable and is a nonlinear function of another variable. The analysis of such a signal is rarely seen in the currently literature Based on the IEEE 802.15.4a UWB channel model, a UWB signal is mathematically characterized by a joint continuous Nakagami m random variable for fading amplitude, a discrete Poisson random variable for cluster’s number, and a discrete counting random variable with a hyperexponential distributed inter-arrival time. In [2], the author compared the IEEE 802.15.3a [3] and 802.15.4a channel model and concluded that the 4a model is more general and based on more measurements than the earlier 3a channel model. Thus, we are motivated to develop an analytical model to evaluate the impact of different UWB channel parameters on the system performance. There are nine sets of UWB channel parameters specified in the IEEE 802.15.4a for various environments. These channel parameters include the inter-cluster arrival rate, ray arrival rates (mixed Poisson model parameters), inter-cluster decay

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constant, intra-cluster decay time constant parameters, the mean and variance of Nakagami m parameter, Nakagami m factor for the strongest components, and the alternative power delay profile (PDP) shape.

B. Related Work

Here we provide a literature survey on the related works in performance analysis of the UWB system. In [4], the authors derived the analytical BER formula for the binary and M-ary UWB systems with Walsh codes under the AWGN channel with multiple access interference (MAI). In [5], the authors studied the impacts of the interference from the universal mobile telecommunications system (UMTS)/wideband code division multiple access (WCDMA) system on the UWB systems. In [6], the BER formula of the UWB system under the flat and dispersive Rayleigh fading channels with timing jitters was derived. In [7], the authors analyzed the performance of a transmit-reference (TR) UWB system with an autocorrelation receiver under a slowly fading channel of which the amplitude is characterized by an appropriate moment generating function approach.

The BER formula in the IEEE 802.15.3a UWB channel model [3] was shown as a function of finite window size without considering the RAKE receiver in [8]. The statistics of the output signal-to-noise ratio (SNR) with the RAKE receiver under the IEEE 802.15.3a UWB channel were also obtained [9]. However, the explicit BER formula was unavailable and the shadowing effects are ignored. In [10], we derived the BER analytical formula with a coherent RAKE receiver in the complete IEEE 802.15.3a UWB channel model where shad-owing is taken into account. In [11], an analytical expression for the SNR of the pulse position modulated (PPM) signal in a multi-input multi-output (MIMO) system was provided in a UWB channel characterized by Gamma distributed signal power with Poisson distributed arrival clusters and rays. In [12], the error performance of a zero-forcing (ZF) RAKE receiver in a MIMO system was analyzed in the frequency-selective UWB lognormal fading channels with a fixed number of clusters and rays.

C. Objective and Outline of This Paper

The objective of this paper is to evaluate BER performance with various UWB channel parameters specified in the IEEE 802.15.4a model. The rest of this paper is organized as follows. In Section II, we provide the required mathematical back-ground about counting integrals to analyze the IEEE 802.15.4a channel. In Section III, we derive the analytical expression for the BER performance of the antipodal and orthogonal binary signals under the IEEE 802.15.4a UWB channel. We show the numerical results in Section IV and give our concluding remarks in Section V.

II. MATHEMATICALBACKGROUND

The IEEE 802.15.4a channel model can be viewed as a random process. The random variables associated with the ran-dom process can be categorized into two kinds. First, the time

domain random variables, which contains the arrival time of all the clusters and rays, i.e., the variables{Tl} and {τk,l}, where

the indices k and l can be any nonnegative integers. Second, the amplitude domain random variables, which contains the amplitudes of all the channel impulse, i.e., the variables a_k,l. Thus, in order to investigate the characteristics of the channel, we must consider infinite number of random variables. This makes the analysis very complicated.

Let us treat the problem in another way. Let us rewrite the channel impulse response h(t) to (15) in [1] in a simpler form:

h(t) =

k

G_kδ(t − T_k), (1) where T_k is the arrival time of the k-th multipath component, regardless it is a cluster or ray. We arrange {T_k} such that it is a nondecreasing sequence. G_k is the gain of the k-th multipath component. In [14], the author focused on finding the characteristic function of the random variableΦ, which is the sum of path gains that arrive in the time window [a, b]. The mathematical expression of the random variableΦ can be written as Φ = k G_kI_[a,b](T_k), (2) where I_[a,b](x) = 1, if x ∈ [a, b], 0, if x /∈ [a, b]. (3)

The purpose of defining I_[a,b](x) here is to consider the multipath components arriving within the time interval[a, b]. The key idea to compute the statistics ofΦ is the use of the counting integral [14] Φ = k ϕ(T_k, G_k) = _∞ −∞ _∞ 0 ϕ(s, g)N(ds × dg), (4) where N(·) is the counting measure [15], ϕ(s, g) = gI[a,b](s).

The counting integral is the Lebesgue integral [16] defined based on the counting measure. That is, we first consider the value of the function ϕ within a small time interval ds and a small gain interval dg. Second, we integrate over all possible

values of s and g, i.e., s ∈ [0, ∞) and g ∈ (−∞, ∞). It

is equivalent to sum up the value of the function ϕ(Tk, Gk)

for all k, which is the summation term in (4). By exploiting the counting integral, the original probability problem with infinitely many random variables has been transformed into a integral with only two dummy variables s and g. The remained problem is to find the characteristic function ofΦ. The detail derivation can be found in [14].

III. BER ANALYSIS

A. Receiver Structure

Consider a coherent RAKE receiver with LRAKE fingers.

The received SNR γb is γb= Eb N₀ L k=1 c2_k , (5)

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where Eb/N0is the bit SNR, ck is the channel amplitude that

appears at the k-th finger of the RAKE receiver. From [17] we know that the conditional error probability for binary signals for the coherent RAKE receiver is

P₂(γb) = Q

γb(1 − ρr)

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where ρr = −1 for antipodal signals and ρr = 0 for

orthogonal signals. Next we will derive the characteristic function of the received energy E L_k=1c2_k in the IEEE 802.15.4a UWB channel.

B. Characteristic Function of the Received Energy (E) In the following theorem, we give the formula of the characteristic function of E. We modify and extend the result in [10] to the case of the IEEE 802.15.4a UWB channel.

Lemma 1: Let L_T,t(ν) be the characteristic function of the squared gain of each path in the IEEE 802.15.4a UWB channel, where T and t= T + τ denotes the cluster arrival time and the ray arrival time, respectively. Also, let e−ψν(T ) and e−ΛJ(ν) be the characteristic functions of a shot-noise random variables related to the ray arrival process with parameter λ and the cluster arrival process with parameter Λ, respectively. Then, it can be proved that the characteristic function of the received energy (E) in the IEEE 802.15.4a UWB channel can be computed by

Ψ(ν) = L0,0(ν)e−ψν(0)−ΛJ(ν). (7)

Proof: In the IEEE 802.15.4a UWB channel, the ray arrival rate λ(τ) is a function of τ. Thus, different from the 3a case, we have to redefine the function ψ_ν(T ) by putting λ(τ) into the integral:

ψ_ν(T ) = _b

max(a,T )[1 − LT,t(ν)]λ(t − T )dt, T ≤ b,

0, T > b, (8)

The term λ(τ) can be found in Theorem 2.

Theorem 1: Consider a RAKE receiver with LRAKE fingers

in the IEEE 802.15.4a UWB channel. The characteristic function LT,t(ν) can be computed by

LT,t(ν) = (1 − jνΩ/m)−m (9) where Ω = 1 γ_lexp −T_Γ −t− T γ_l (10) and m= expm₀+ m2₀/2 . (11)

The parameterγ_l is defined in (20) in [1]. Proof: See Appendix I.

Theorem 2: The parameter λ in Lemma 1 is a function of τ and can be expressed as

λ(τ) = _βλ

1e−λ1τ+(1−β)λ2e−λ2τ

βe−λ1τ_+(1−β)e−λ2τ , τ ≥ 0,

0, τ <0. (12)

Proof: See Appendix II.

The equations for calculating ψν(T ) and J(ν) can be found

in Theorem 2 in [10].

With characteristic function of E, i.e. Ψ(ν) in (7), the probability density function (PDF) ofE can be computed by the Gauss-Hermite formula as follows:

f_E(x) = 1 2π _∞ −∞Ψ(ν)e −jxν_dν ≈ _2π1 N(H) k=1 w(H)_k Ψ(ν)e−jxνeν2 ν=x(H) k . (13) Combining (9), (13), and (16) and (17) in [10], the BER of the RAKE receiver in the IEEE 802.15.4a UWB channel can be computed as P2 = EE Q (1 − ρr)Eb N₀E = _∞ 0 Q (1 − ρr)_NEb 0x f_E(x)dx = 1 2π 1 − j ν γ₀exp (m₀+ m2₀/2) _{− exp}₍_m₀₊_m2 0/2) N(H) k=1 w(H)_k exp  −1₂λ(L − 1)Tc N(L) p=1 w(L)_p [1 − L0,t(ν)]|_t=1 2(L−1)Tc(x(L)p+1) exp  −1₂Λ(L − 1)Tc N(L) i=1 w_i(L) 1 − LT,T(ν)e−λψν(T ) T =1 2(L−1)Tc(x(L)_i +1) _∞ 0 Q (1 − ρr)Eb N0x exp(−jxν)dx exp(ν2₎ ν=x(H) k . (14) C. Discussion

It is possible to apply the above analytical method to the generalized UWB frequency selective fading channel with any given fading distribution, PDP, and the distribution of cluster and ray inter-arrival time. For a give PDP, we only need to express it as a function of T and t and place it in (10) as the procedure in Appendix I. For a given PDF of the fading amplitude, we can first find the PDF of the squared amplitude as in (17) and use (18) to find L_T,t(ν). For any given PDF of ray inter-arrival time, we can use the same method as in Appendix II to find the intensity function and λ(τ) and applied it in (7). For any given PDF of cluster inter-arrival time, we can also find its intensity functionΛ(T ). Since it is a function of T we need to redefine the function J(ν) as

J(ν) = _b

0 Λ(T )[1 − LT,T(ν)e

−ψν(T )_{]dT .} ₍₁₅₎

That is, we put the function Λ(T ) into the integrand and integrate it with respect to T . On the other hand, the function

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Ψ(ν) should be rewritten as

Ψ(ν) = L0,0(ν)e−ψν(0)−J(ν) . (16)

IV. NUMERICALRESULTS

We will apply the developed analytical model to evaluate the effect of different UWB channel parameters on the BER performance of the system, including cluster arrival rate Λ, inter-cluster decay constantΓ, intra-cluster decay constant γ₀, and ray arrival parameters (λ₁, λ₂, and β). In our recently work [13], we have validated the correctness of the analytical model by simulations in a simple case. In the following, we consider a RAKE receiver with 10 fingers and the CM1 model in the IEEE 802.15.4a standard.

A. Impact of the Inter-Cluster Arrival Rate Λ on BER Figure 1 shows the BER v.s. Eb/N0 for various

inter-cluster arrival rate Λ = 0.01, 0.1, 0.5, and 1. We can see that when the cluster arrival rate increases, the BER decreases. This is because when the inter-cluster arrival rate increases, more clusters arrive in the time interval [0, (L − 1)Tc] and

the RAKE receiver can collect more energy of the channel impulse, thereby improving the BER.

B. Impact of the Inter-Cluster Decay ConstantΓ on BER

Figure 5 shows the BER v.s. Eb/N0 for CM1 by

anal-ysis. The number of the RAKE receiver fingers is 10. We show four curves with the inter-cluster decay constant Γ = 0.1, 1, 10, and 100. We can see that when Γ increases, the BER decreases. This is because according to (21) in [1], when Γ increases, the Ω_l, which is the integrated energy of the lth cluster, will increases, and the RAKE receiver can collect more energy of the channel impulse, thus the BER will decrease. C. Impact of the Ray Arrival Parameterλ₁ on BER

Figure 2 shows the BER v.s. Eb/N0 for different value of

ray-arrival parameter λ₁ = 0.01, 0.1, 1, and 10. One can see that when λ₁ increases, the BER decreases. This is because when λ₁increases, more rays in each cluster arrive in the time interval[0, (L − 1)Tc] and the RAKE receiver can collect more

energy of the channel impulse. Compared with the impact of the cluster arrival rate, changing λ₁ leads to less variation on the BER than the cluster arrival rate does. It is reasonable because the cluster arrival rate is a macroscopic parameter that influences every clusters and λ₁ is a microscopic parameter that influences only rays within each cluster.

D. Impact of the Ray Arrival Parameter λ₂ on BER

Figure 3 shows the BER v.s. Eb/N0 parameter λ2 =

0.01, 0.1, 1, and 10. When λ2 increases, the BER decreases.

This is because when λ₂ increases, more rays in each

cluster arrive in the time interval[0, (L − 1)Tc] and the RAKE

receiver can collect more energy of the channel impulse. Thus the BER with larger λ₂will improve BER performance. Furthermore, Compared with Figure 2, changing λ₂influences the BER more significantly than λ₁the BER. It is reasonable because for the CM1 in IEEE 802.15.4a standard, β= 0.095.

According to (18) in [1], we can see that ray interarrival time has the probability 0.095 decided by the parameter λ1and has the probability of 0.905 decided by the parameter λ₂. Thus λ₂ has more significant impact on BER than λ₁ does.

E. Impact of the Ray Arrival Parameterβ on BER

Figure 4 shows the BER v.s. Eb/N0 for different values

of β= 0, 1/3, 2/3, and 1. As β increases, the BER decreases. According to (18) in [1], when β increases, the ray process has the higher probability to choose the arrival rate λ1rather than λ2. For CM1, λ1= 1.54 and λ2= 0.15. From the discussion

in the last two subsection, we know that the higher the ray arrival rate is, the lower the BER. This phenomenon is similar to the cluster. As the cluster arrival rate increases, the BER decreases, as we have seen in Subsection IV-A.

F. Impact of the Intra-Cluster Decay Constantγ0 on BER Figure 6 shows the BER v.s. the intra-cluster decay constant γ₀ for Eb/N0 = 5, 10, and 15 dB. When γ0 increases, the

BER first increases and then decreases. This phenomenon can be explained as follows. Based on (20) in [1], a larger value of γ0 also leads to a larger value of γl. According to (19) in [1], for a small value of γl, the term1/γldominates the BER performance, which cause the decrease ofΩ_l and the increase of BER. When γ_l is large, the exponential term dominates and thereforeΩ_lincreases and BER decreases. The maximum of the BER occurs at γ₀ = 30, 40, and 100 for Eb/N0 =

5, 10, and 15 dB, respectively.

V. CONCLUSIONS

In this paper, we have derived the BER analytical formula for a coherent RAKE receiver under the IEEE 802.15.4a UWB channel model. Our proposed analytical BER formula can obtain the BER values quickly, compared to the computer simulation. Further, we also discuss the impact of various parameters of the IEEE 802.15.4a UWB channel model on BER. Furthermore, it is worthwhile to emphasize that the suggested analytical method can be applied to other multipath channel models with any given fading distribution, PDP, and cluster and ray inter-arrival time distributions.

ACKNOWLEDGEMENT

The authors would like to express their thanks to Prof. A. Molisch for introducing them the IEEE 802.15.4a UWB channel model and providing many insightful discussions on the characteristics of the UWB channel model.

APPENDIXI PROOF OFTHEOREM1

From (24) in [1], we can easily find the PDF of x= a2 by exploiting the resulting of Example 7b in [18]. That is,

f_x(x) = 1 2√x f_a(√x) + f_a(−√x) = _exp₍₋mx Ω )(mxΩ )m xΓ(m) , x≥ 0, 0, x <0. (17)

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The characteristic function of x is LT,t(ν) = _∞ −∞ f_x(x)ejνxdx = (1 − jνΩ/m)−m_. ₍₁₈₎

The termΩ = E{x} is defined in (19) in [1]. To fit it into our formula, we substitute T_lby T and M_clusterby its mean, zero, in (21) in [1] and τ_k,l by(t − T ) in (19) in [1]. Then we can get (10).

Finally, we set m to its mean and get (11). The mean is given by (4) in [19].

APPENDIXII PROOF OFTHEOREM2

According to (18) in [1], the PDF of interarrival time of ray is

f(τ) =

βλ₁e−λ1τ+ (1 − β)λ₂_e−λ2τ_, _τ≥ 0,

0, τ <0. (19)

The cumulative distribution function (CDF) is

F(τ) = _τ −∞f(x)dx = 1 − βe−λ1τ− (1 − β)e−λ2τ_, _τ≥ 0, 0, τ <0.(20)

The parameter λ in Lemma 1 is the intensity function [20] of the interarrival time of rays:

λ(τ) = f(τ) 1 − F (τ) = _βλ 1e−λ1τ+(1−β)λ2e−λ2τ βe−λ1τ_+(1−β)e−λ2τ , τ≥ 0, 0, τ <0. (21)

The ray arrival rate is a function of time τ , which means the ray arrival is a nonhomogeneous Poisson process. Note that when β= 1, the PDF in (19) reduces to the exponential PDF and the intensity function reduces to the constant λ₁. This case corresponds to a homogenous Poisson process.

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Fig. 1. The BER v.s.Eb/N0for the RAKE receiver with 10 fingers in the IEEE 802.15.4a UWB channel CM1. The inter-cluster arrival rateΛ is 0.01, 0.1, 0.5, and 1.

Fig. 2. The BER v.s.Eb/N0for the RAKE receiver with 10 fingers in the

IEEE 802.15.4a UWB channel CM1. The parameter λ₁ is 0.01, 0.1, 1, and 10.

IEEE 802.15.4a UWB channel CM1. The parameter λ2 is 0.01, 0.1, 1, and 10.

IEEE 802.15.4a UWB channel CM1. The parameterβ is 0, 1/3, 2/3, and 1.

IEEE 802.15.4a UWB channel CM1. The inter-cluster decay constant Γ is 0.1, 1, 10, and 100.

Fig. 6. The BER v.s.γ₀for the RAKE receiver with 10 fingers in the IEEE 802.15.4a UWB channel CM1.Eb/N0is 5 dB, 10 dB, and 15 dB.