3.4 Numerical Results
3.4.3 Numerical Results for BER in IEEE 802.15.4a Channel 50
Comparison of Simulation and Analytical BER
Figure 3.8 shows the BER v.s. Eb/N0 for the CM1 model in the IEEE 802.15.4a standard with/without the shadowing by simulation and analysis.
CM1 represents the residential line-of-sight (LOS) environment. The related channel parameters for CM1 can be found in Table 3.1. For the analytical curves, the orthogonal binary signal, i.e., ρr = 0, is considered and δ = 1 for the PPM signal. For each Eb/N0, we simulate 100,000 bits to obtain the BER. As seen from the figure, the analytical results match the simulation results quite well. Also, the BERs for CM1 with shadowing standard devia-tion σx = 3 and 6 dB are worse than the BER for CM1 without shadowing.
Moreover, the BER for σx = 6 dB is worse than the BER for σx = 3 dB.
This result agrees with our common sense that more severe shadowing fading causes worse BER performance.
Figure 3.9 shows the BER v.s. L, the number of fingers of the RAKE receiver, for the CM1 model in the IEEE 802.15.4a standard by simulation and analysis. The simulation parameters are the same as that of the last figure. From the figure we see that the simulation and analytical BER are very close. Hence we can conclude that the approximation of ˜E in (3.36) is good for a large range of L.
Impact of the Inter-Cluster Arrival Rate Λ
Figure 3.10 shows the BER v.s. Eb/N0 for various inter-cluster arrival rates Λ = 0.01, 0.1, 0.5, and 1. When the cluster arrival rate increases, we find that the BER decreases and the slope of BER v.s. Eb/N0 becomes steeper.
This is because when the inter-cluster arrival rate increases, more clusters arrive in the time interval [0, LTc] and the RAKE receiver can collect more energy of the channel impulse, thereby improving the BER performance and increasing the diversity order.
Impact of the Ray Arrival Parameters λ1 and λ2
Figure 3.11 shows the effects of different values of ray-arrival parameter λ1on the BER v.s. Eb/N0, when λ2 = 0.15 and β = 0.095 according to the IEEE 802.15.4a model. The figure shows that in the considered case, a larger λ1 results in lower BER. When λ1 increases, the RAKE receiver can obtain higher energy from more rays. Compared with the impact of the cluster arrival rate, changing λ1 yields smaller variations on the BER performance.
This is because the cluster arrival rate is a macroscopic parameter that influ-ences all the clusters and λ1 is a microscopic parameter that influences only rays within each cluster.
Figure 3.12 shows the impacts of various values of λ2 on the BER for
the case λ1 = 1.54 and β = 0.095. As shown in the figure, a larger value of λ2 leads to a lower BER. Compared to Fig. 3.11, λ2 affects the BER more significantly than λ1. Referring to [4, (18)], the condition of β = 0.095 implies that ray interarrival time is decided by the parameter λ1 with a probability of 0.095 and is decided by the parameter λ2 with a probability of 0.905. This explains why λ2 becomes a more dominant parameter than λ1 in the case of β = 0.095.
Impact of the Ray Arrival Parameter β
Figure 3.13 shows the effect of various values of β on the Eb/N0 in the case λ1 = 1.54 and λ2 = 0.15. From [4, (18)], we expect that when β increases, the ray process is more likely to choose the arrival rate λ1 than λ2. According to CM1 model of the IEEE 802.15.4a channel, λ1 = 1.54 and λ2 = 0.15.
Thus, a larger value of β indicates a higher ray arrival rate. Thus, the BER is improved for a larger β in the considered case.
Impact of the Inter-Cluster Decay Constant Γ
Figure 3.14 shows the effect of various inter-cluster decay constants Γ. One can see that a larger value of Γ yields a better BER performance. From [4, (21)] one can see that the total energy Ωl is proportional to exp(−Tl/Γ).
Thus, when Γ increases, the total energy Ωl of the l-th cluster also increases.
With more signal energy captured by the RAKE receiver, the BER perfor-mance is therefore improved.
Impact of the Intra-Cluster Decay Constant γ0
Figure 3.15 illustrates the effect of the intra-cluster decay constant γ0 of [4, (20)]. As γ0 increases, the BER first increases and then remains the
same or even decreases. This phenomenon can be explained by (19) and (20) in [4]. From [4, (20)], γ0 is proportional to γl. According to [4, (19)], E[|ak,l|2] = Ωl
γl exp(−τk,l/γl). For a small value of γl, the term 1/γl dominates the value of E[|ak,l|2]. Thus, as γ0 increases, a smaller E[|ak,l|2] will yield higher BER. On the other hand, for a larger γl, exp(−τk,l/γl) will affect E[|ak,l|2] more significantly. Thus, a larger γ0 leads to a larger E[|ak,l|2], and lower BER. The maximal BER occurs at γ0 = 30, 40, and 90 for Eb/N0 = 5, 10, and 15 dB, respectively.
3.5 Conclusions
First, we have derived the computable BER formula for a RAKE receiver in the complete IEEE 802.15.3a UWB channel models. In particular, we find that deriving a BER formula taking account of RAKE finger numbers and shadowing is quite challenging for the IEEE 802.15.3a UWB channel. This is mainly because the jointly two-dimension lognormal and doubly-stochastic Poisson random variables yield infinite number of rays. We propose an ap-proximation technique for the collected energy at an L-finger RAKE receiver.
We find that the proposed BER computation method can save a significant amount of computer simulation time. Furthermore, we propose a character-istic function based BER formula to overcome the convergence problem of MGF-based BER formula when shadowing is included. The accuracy of the proposed technique is verified by simulations. Our results quantitatively indi-cate the effect of shadowing and RAKE finger numbers on BER performance in the IEEE 802.15.3a UWB channel.
Second, we have derived the BER analytical formula for a coherent RAKE receiver under the IEEE 802.15.4a UWB channel model. Our proposed
an-alytical method can accurately and quickly compute the BER values for the sophisticated IEEE 802.15.4a UWB channel, and evaluate the impact of var-ious channel parameters. We find that of all the parameters in the IEEE 802.15.4a channel, the inter-cluster arrival rate Λ has the most significant impact on the BER performance. We also observe that the BER can be low-ered due to the increase on the inter-cluster arrival rate Λ, the inter-cluster decay constant Γ, the ray arrival parameters λ1, λ2, and β. We also find that increasing the intra-cluster decay constant γ0 causes the BER to first increase and then remain the same or even decrease.
In general, the time-domain parameters in the IEEE 802.15.4a UWB channel affect BER performance quite significantly. This reflects the com-ment in [94], that time-of-arrival characteristics are more important than am-plitude characteristics for MIMO-UWB systems. In the future, it would be worth extending the suggested analytical method to other multipath chan-nel models with any given fading distribution, PDP, and cluster and ray inter-arrival time distributions.
0 0.5 1 1.5 2 2.5 3 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
fE(x) simulation fE˜(x) analysis
(a)
0 1 2 3 4 5 6 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
CDF
CDF ofE (simulation) CDF of ˜E (analysis)
(b)
Figure 3.1: The distributions of E by simulation and ˜E by analysis for a RAKE receiver with 10 fingers in the IEEE 802.15.3a UWB channels CM1, where the standard deviations of lognormal fading and shadowing are σ = 4.8 dB and σx = 3 dB, respectively. (a) PDFs. (b) CDFs.
0 0.5 1 1.5 2 2.5 3 0
0.5 1 1.5 2 2.5 3 3.5
x f˜E(x)
CM1 CM2 CM3 CM4
Figure 3.2: The PDF fE˜(x) of the received energy ˜E for a RAKE receiver with 10 fingers in the IEEE 802.15.3a UWB channels CM1, CM2, CM3, and CM4, where the standard deviations of lognormal fading and shadowing are σ = 4.8 dB and σx = 3 dB, respectively.
0 2 4 6 8 10 12 14 16 10−3
10−2 10−1 100
Eb/N0 (dB)
BER
σx = 3 dB σx = 6 dB
Figure 3.3: Effect of various shadow standard deviations (σx = 3 dB and 6 dB) on the BER performance of a 10-finger RAKE receiver in the IEEE 802.15.3a UWB channels CM3.
0 2 4 6 8 10 12 14 16 10−3
10−2 10−1 100
Eb/N0 (dB)
BER
CM2 (analysis) CM2 (simulation) CM3 (analysis) CM3 (simulation) CM4 (analysis) CM4 (simulation)
Figure 3.4: BER v.s. Eb/N0 for the 10-finger RAKE receiver in the IEEE 802.15.3a UWB channels CM2, CM3, and CM4 with shadowing standard deviation σx = 6 dB, where the analytical BER is obtained from the charac-teristic function based approach, i.e. (3.28).
0 2 4 6 8 10 12 14 16 10−3
10−2 10−1 100
Eb/N0 (dB)
BER
CM2 (analysis) CM2 (simulation) CM3 (analysis) CM3 (simulation) CM4 (analysis) CM4 (simulation)
Figure 3.5: BER v.s. Eb/N0 for the 10-finger RAKE receiver in the IEEE 802.15.3a UWB channels CM2, CM3, and CM4 with shadowing standard deviation σx = 6 dB, where the analytical BER is obtained from the MGF-based approach, i.e. (3.29).
0 0.5 1 1.5 2 2.5 3 3.5 4 10−2
10−1 100
x
f˜E(x) L=10 → ← L=20~50
L=10 L=20 L=30 L=40 L=50
Figure 3.6: The PDF fE˜(x) of the received energy ˜E of a RAKE receiver with number of fingers L = 10, 20, 30, 40, and 50 in the IEEE 802.15.3a UWB channel CM1.
0 10 20 30 40 50 60 70 80 10−2
10−1 100
L
BER
CM1 (analysis) CM2 (analysis) CM3 (analysis) CM4 (analysis)
Figure 3.7: BER v.s. the number of fingers of the RAKE receiver (L) for CM1, CM2, CM3, and CM4, where Eb/N0= 5 dB.
0 2 4 6 8 10 12 14 16 18 20 10−3
10−2 10−1 100
Eb/N0 (dB)
BER
CM1 (no shadowing, analysis) CM1 (no shadowing, simulation) CM1 with σx=3dB (analysis) CM1 with σx=3dB (simulation) CM1 with σx=6dB (analysis) CM1 with σx=6dB (simulation)
Figure 3.8: The BER v.s. Eb/N0 for the CM1 model without shadowing and CM1 model with shadowing standard deviation σx = 3 and 6 dB in the IEEE 802.15.4a standard by simulation and analysis. In CM1, the default value of σx is 2.22 dB.
10 20 30 40 50 60 70 80 90 100 10−1
100
M
BER
CM1 (analysis) CM1 (simulation)
Figure 3.9: The BER v.s. L (number of fingers of the RAKE receiver) for the CM1 model in the IEEE 802.15.4a standard by simulation and analysis.
The SNR is Eb/N0 = 0 dB. The shadowing standard deviation σx is 2.22 dB.
0 2 4 6 8 10 12 14 16 18 20 10−6
10−5 10−4 10−3 10−2 10−1 100
Eb/N0 (dB)
BER
Λ = 0.01 Λ = 0.1 Λ = 0.5 Λ = 1
Figure 3.10: The BER v.s. Eb/N0 for various inter-cluster arrival rates Λ = 0.01, 0.1, 0.5, and 1 under the CM1 model of the IEEE 802.15.4a UWB channel. In CM1, the default value of Λ is 0.047. The shadowing standard deviation σx is 2.22 dB.
0 1 2 3 4 5 6 7 8 9 10 10−0.9
10−0.8 10−0.7 10−0.6 10−0.5
Eb/N0 (dB)
BER
λ1 = 0.01 λ1 = 0.1 λ1 = 1 λ1 = 10
Figure 3.11: The effects of different values of ray-arrival parameter λ1 = 0.01, 0.1, 1, and 10 on the BER v.s. Eb/N0, where λ2 = 0.15 and β = 0.095 according to the CM1 model of the IEEE 802.15.4a channel. In CM1, the default value of λ1 is 1.54. The shadowing standard deviation σx is 2.22 dB.
0 1 2 3 4 5 6 7 8 9 10 10−3
10−2 10−1 100
Eb/N0 (dB)
BER
λ2 = 0.01 λ2 = 0.1 λ2 = 1 λ2 = 10
Figure 3.12: The impacts of various values of λ2 on the BER v.s. Eb/N0 for the case λ1 = 1.54 and β = 0.095 in the CM1 model of the IEEE 802.15.4a UWB channel. In CM1, the default value of λ2 is 0.15. The shadowing standard deviation σx is 2.22 dB.
1 2 3 4 5 6 7 8 9 10 10−3
10−2 10−1 100
Eb/N0 (dB)
BER
β = 0 β = 1/3 β = 2/3 β = 1
Figure 3.13: The effect of various β on the BER v.s. Eb/N0 for λ1 = 1.54 and λ2 = 0.15 in the CM1 model of the IEEE 802.15.4a UWB channel. In CM1, the default value of β if 0.095. The shadowing standard deviation σx is 2.22 dB.
1 2 3 4 5 6 7 8 9 10 0.09
0.1 0.2 0.3 0.4
Eb/N0 (dB)
BER
Γ = 0.1 Γ = 1 Γ = 10 Γ = 100
Figure 3.14: The effect of the inter-cluster decay constant Γ = 0.1, 1, 10, and 100 in the IEEE 802.15.4a UWB channel for various of Eb/N0, where a 10-finger RAKE receiver is adopted in the CM1 model. In CM1, the default value of Γ is 22.61. The shadowing standard deviation σx is 2.22 dB.
0 20 40 60 80 100 120 140 160 180 200 10−2
10−1 100
γ0
BER
Eb/N0 = 5 dB Eb/N0 = 10 dB Eb/N0 = 15 dB
Figure 3.15: The effect of intra-cluster decay constant γ0 in the IEEE 802.15.4a UWB channel for various values of Eb/N0, where a 10-finger RAKE receiver is adopted in the CM1 model. The shadowing standard deviation σx is 2.22 dB.
Chapter 4
On the Performance of Using Multiple Transmit and Receive Antennas in Pulse-Based
Ultrawideband Systems
This chapter presents an analytical expression for the signal-to-noise ratio (SNR) of the pulse position modulated (PPM) signal in an ultra-wideband (UWB) channel with multiple transmit and receive antennas. We consider a generalized fading channel model that can capture the cluster property and the highly dense multipath effect of the UWB channel. Through sim-ulations, we demonstrate that the derived analytical model can accurately estimate the mean and variance properties of the pulse based UWB signals in a frequency selective fading channel. Furthermore, we investigate to what extent the performance of the PPM based UWB system can be further en-hanced by exploiting the advantage of multiple transmit antennas or receive antennas. Our numerical results show that using multiple transmit antennas
in the UWB channel can improve the system performance in the manner of reducing signal variations. However, because of already possessing rich diversity inherently in the UWB channel, using multiple transmit antennas does not provide diversity gain in the strict sense (i.e., improving the slope of bit error rate (BER) v.s. SNR), but can possibly reduce the required fingers of the Rake receiver for the UWB channel. Furthermore, because multiple receive antennas can provide higher antenna array combining gain, multiple receive antennas technique can be used to improve the coverage performance for the UWB system, which is crucial for a UWB system due to the low transmission power operation.
4.1 Motivation
Wireless systems continue to pursue even higher data rates and better quality.
The ultra-wideband (UWB) technique and space time processing techniques are two promising techniques to achieve this objective. However, how to merge these two techniques together to further increase the data rates is not an easy task. This chapter investigates how multiple transmit/receive antennas and the UWB system can function together to exploit the synergy of marrying these two advanced techniques.