An Introduction to Ultra Wideband Communication Systems

6.4. Case Study: Improving Range of UWB Using RAKE Receivers

The fine time resolution of multipaths induced by the channel (typically in the order of 100300 picoseconds) can be exploited using a RAKE receiver to capture a significant amount of energy found in the multipath components, and to benefit from multipath diversity gain. Energy capture is important in UWB receiver design, and the number of detectable multipath components could be as large as a few hundred since UWB signals occupy bandwidth greater than 500 MHz or have a fractional bandwidth greater than 0.25. Clearly, classical RAKE receiver design solutions, such as MRC or selection combining, are no longer appropriate from the viewpoint of complexity and performance trade-offs. The realistic trade-off between the energy capture and RAKE diversity level in dense multipath for UWB systems have been quasi-analytically/experimentally presented in [54]. The experimental results are shown in Figure 6.56.

In this section, we examine the throughput range trade-off of UWB communication systems that employ two variants of RAKE receiver structures in multipath fading channels: (a) the generalized selection combining receiver (hereafter, referred to as GSC(N,L)) that combines a subset of N resolvable multipaths with the highest instantaneous SNR out of L resolvable multipaths in a similar fashion to MRC; and (b) MRC with finite tap statistics or partial MRC (hereafter, referred to as PMRC(N, L)) that combines N multipaths with the largest mean SNR. Figures 6.57 a and b show the schematic for these receiver structures respectively.

Clearly, PMRC is simpler to implement compared to GSC owing to the simpler selection circuitry complexity (i.e., the need to select N paths with the largest instantaneous SNR in the latter). The motivation for performance analyses of the previous two suboptimal receiver structures is based on the propagation measurements of UWB signals at Virginia Tech [84] that indicates the significant amount of energy that is found in a few dominant multipaths, particularly for the LOS case. For example, approximately 80% of the receiver energy could be captured by combining only 12 dominant multipaths. However, for the NLOS situation, approximately 75% of energy is captured by combining 30 dominant multipaths.

In [85], the performance comparisons between GSC and PMRC over Nakagami-m channels have been studied in the context of spread spectrum communications but with only a few multipaths. In [86], the authors resorted to a semianalytical approach for analyzing the average bit error rate (ABER) performance of GSC(N,L) and PMRC(N, L) at a fixed distance. In contrast, we now examine the throughput range performance of UWB for two different reduced-complexity receiver structures over generalized fading channels using exact and approximate ABER expressions (rather than the semianalytical approach discussed in [86]).

We assume a UWB system operating in a Rayleigh or a Nakagami-m fading environment and present analytical expressions for the throughput as a function of the distance for the GSC(N,L) and PMRC(N, L) receiver structures. The bit error probability for an M-ary PAM system is given by

Equation 6.153

where M is the alphabet size and c = log ₂M denotes the number of information bits carried by each symbol. The instantaneous SNR/bit is g = (E_b/N₀)a² where E_b denotes received energy per bit, N_o is noise spectral density, and a denotes random fading amplitude. The average SNR per bit at the receiver input can be computed as , where P_r is the average received signal strength and T_b denotes the bit duration (time between transmitted pulses). The received power at a distance d from the transmitter can be modeled as

Equation 6.154

where n denotes the path loss exponent, L_s is the system loss factor not related to propagation (L_s 1), l is the carrier frequency used for computing distance loss, and {G_t, G_r} denotes the transmitter and receiver antenna gains, respectively. Using those expressions, it can be shown that the maximum achievable throughput, R_b, at distance d, is given by

Equation 6.155

The noise spectral density is N₀ = kTF where k is Boltzman's constant, T is system temperature, and F denotes the noise figure. From (6.155) it is apparent that for fixed transmit power, throughput is a function of dⁿ. The average SNR per bit required to evaluate (6.155) for a given bit error rate performance can be found by averaging (6.153) over the probability distribution function (PDF) of GSC(N,L) or PMRC(N, L).

6.4.1. GSC(N,L) with Independent but Nonidentically Distributed Fading Statistics

Exact Analysis of GSC(N,L)

Using the moment-generating function (MGF) approach, we obtain

Equation 6.156

where f_g(.) denotes the MGF of the GSC(N,L) output SNR, g_gsc. The MGF expressions for the GSC(N,L) output SNR in a myriad of fading environments has been developed in [87,88], namely

Equation 6.157

where

Equation 6.158

while S_L is the set of all permutations of integers [1, 2, . . ., L], and ss = [s (1), s (2), . . ., (L)] which permutes the integers [1, 2, . . ., L]. In (6.157), f_s(k) (.), F_s(k) (.), and f_s(k) (.) denote the PDF, cumulative distribution function (CDF), and the MGF of SNR of the sk^th multipath, respectively.

Approximate Analysis of GSC(N,L)

The throughput range performance of UWB systems employing GSC receivers can be analyzed using the previous equations. However, as the number of resolvable multipath components increases, it becomes computationally intensive to evaluate (6.156) along with (6.157), and turns out to be impractical for the case L 15. As such, an approximate ABER formula is needed to predict the performance of GSC(, m (K + 1) and in the analytical expression for independent and identically distributed multipath components. This hypothesis is further validated by the numerical results presented in the following subsection.

6.4.2. PMRC(N, L) with Independent but Nonidentically Distributed Fading Statistics

It is not difficult to show that the MGF of PRMC (N, L) output SNR may be computed as

Equation 6.159

where f_gk (.) denotes the MGF of the k^th strongest (mean SNR) diversity path. Hence, the ABER performance of PMRC may be analyzed using (6.156) and (6.159), and the maximum achievable throughput can be easily computed as before.

6.4.3. GSC(N,L) with Equally Correlated Nakagami-m Fading Statistics

In order to gain significantly from the use of diversity, there must be a sufficient level of statistical independence in the fading of the received signal in different diversity paths. In UWB communication systems with closely received resolvable multipath components, an assumption of independent multipath signals may lead to grossly overestimated diversity gains. As such, a study of the effect of multipath correlation on the performance of a UWB communication system is important. In this section, we will provide analytical expressions to evaluate the ABER performance of the GSC(N,L) receiver structure in equally correlated Nakagami-m multipath fading channels.

Let g₍₁₎, . . ., g(_L) denote the instantaneous SNRs in ascending order, namely g₍₁₎< g₍₂₎< . . . < g_(L). Using the property of exchangeability, the joint PDF of the ordered set of instantaneous SNR can be expressed as

Equation 6.160

The joint PDF of the ordered instantaneous SNRs in a Nakagami-m channel can be obtained using Equation 21 of [90] and (6.160) as

Equation 6.161

where m 1/2 denotes the fading severity index, W is the average received SNR per bit per branch, r denotes the power correlation coefficient , v_k = l_k + m, , and g_k(., ., .) is defined as

Equation 6.162

It is apparent from (6.161) that we can compute the MGF f_ggsc (.) in a similar fashion to GSC with i.n.d fading statistics [88], namely

Equation 6.163

where I(., .) is defined as

Equation 6.164

where , Using the identity in Equation 3.381.3 of [91], functions G_k (.) and f_k (., .) may be computed in closed form as

Equation 6.165

Equation 6.166

where denotes the complementary incomplete Gamma function. It may be noted that (6.163) is much more concise, yet more general and numerically efficient, than Equation 36a of [90]. Now the throughput performance can now be easily analyzed using (6.155), (6.156), and (6.163).

The results discussed in this section use the assumptions in Table 6.2. Figure 6.58 compares the accuracy of "approximate ABER analysis" against the "exact ABER analysis" using (6.157) for a GSC(N, 12) receiver with i.n.d diversity branches in Rayleigh fading channels. It is evident that the throughput curves corresponding to approximate ABER match closely with the exact throughput curves. Readers are referred to [92] for further details.

Figure 6.59 draws a comparison between the throughput performance of GSC(N, 30) and PMRC(N, 30) receiver structures in a Rayleigh environment with an exponentially decaying multipath intensity profile (d = 0.2). The average SNR of the n^th diversity path is , where denotes the average SNR/bit, and the parameter Q is chosen such that the constraint is satisfied.

Solving for C yields

Equation 6.167

As expected, the GSC receiver outperforms PMRC for a fixed number of combined paths, N. For instance, GSC(10,30) and PMRC(20,30) have identical throughput performances. From Figure 6.59 it is also apparent that as the number of combined paths N are increased, the throughput performance of PMRC becomes virtually identical to GSC. Based on these observations, we can conclude that deployment of a GSC(N,L) receiver is desirable for the cases in which N << L, whereas for N N,L). Figure 6.60 depicts the performance of the 5-tap PMRC(5, 20) RAKE receiver in a Nakagami-m fading channel for varying fade distributions. It is evident that improving channel conditions translates into a considerable improvement in the receiver throughput. Figure 6.61 presents the throughput analysis for a GSC(N, 20) receiver for path loss exponent n = 3.5. A comparative study of Figures 6.60 and 6.61 suggests a major fallback in throughput with even the slightest increase in the path loss exponent n. The rate at which the throughput falls declines gradually as the range increases. Finally, Figure 6.62 illustrates the effect of multipath correlations on the throughput performance of the GSC receiver for different numbers of combined paths, N. The throughput performance degrades with increasing multipath correlation, as expected. Throughput falls by almost 50% for r = 0.8.