4.5.1 Repetition Codes
Figure 4.1 (a) shows the scenario of using repetition code with no diversity (Tx1-Rx1) in the case Np = 2. First, we define the processed data zp1,T1(f1) and zp1,T1(f2) in frames 1 and 2 as:
zp1,T1(f1) = sp1,T1+ rp1,T1+ n(f1)p1 + n(f1)p0 (4.30) and
zp1,T1(f2) = sp1,T1+ rp1,T1+ n(f2)p1 + n(f2)p0 , (4.31) where s, r, and n represent the signal part, the redundancy part, and the noise part of the processed data z, the superscript (fi) means the i-th frame, the subscript p1 means the message bit d(i) = 1, and the subscript Ti means the i-th transmit antenna. Denote the processed data for the no diversity scheme as zNDp1 . Then, we have
zp1ND = zp1,T1(f1) + zp1,T1(f2)
= 2sp1,T1 + 2rp1,T1+ n(f1)p1 + n(f1)p0 + n(f2)p1 + n(f2)p0 . (4.32)
From (4.17), (4.19), and (4.32), the mean and variance of the processed data zp1ND can be computed respectively by
E[zp1ND] = 2E[sp1,T1] + 2E[rp1,T1]
+E[n(f1)p1 ] +E[n(f1)p0 ] +E[n(f2)p1 ] +E[n(f2)p0 ]. (4.33) and
var[zp1ND] = var[2· sp1,T1] + var[2· rp1,T1] + var[n(f1)p1 ]
+ var[n(f1)p0 ] + var[n(f2)p1 ] + var[n(f2)p0 ]. (4.34) Represent Sp1ND and Np1ND as the signal energy and the noise energy of the processed data zp1ND, respectively. Then, we have
Sp1ND = 4E[sp1]2+ 4var[sp1], (4.35) where var[sp1] andE[sp1] can be obtained from (4.21) and (4.27), respectively.
Furthermore, the noise energy Np1ND can be derived as Np1ND = 2σn2
L−1 l=0
πH(l)E[al2]
+ 2
L−δ−1
l=0
πH(l)πH(1 + δ)E[a2l]E[a2l+δ]. (4.36) Thus, by substituting related channel information of aland πH(l) into (4.36), Np1ND can be also obtained analytically. From (4.35) and (4.36), we show how to calculate SNRNDp1 analytically.
4.5.2 Receive Diversity
Consider the receive diversity scheme (Tx1-Rx2) having repetition codes with Np = 2 as shown in Fig. 4.1 (b). We express the processed data zRDp1 for the receive diversity scheme as follows:
zp1RD = zp1,T1ND + zp1,T2ND , (4.37)
where the superscript RD means receive diversity. Clearly, we can use the same method of obtaining E[zp1ND] in (4.33) to compute the mean of the processed data zp1RD, which is defined as
E[zp1RD] =E[zp1,T1ND ] +E[zNDp1,T2]. (4.38)
Likewise, the variance of the processed data zp1RD can be calculated by
var[zp1RD] = var[zp1,T1ND ] + var[zNDp1,T2]. (4.39)
Denote Sp1RDand Np1RDas the signal energy and the noise energy of the pro-cessed data zp1RD, respectively and recall that the (Tx1-Rx2) receive diversity scheme and repetition length Np = 2 is considered. Then we have
Sp1RD = 16E[sp1]2+ 8var[sp1] (4.40)
and
Np1RD = 2Np1ND. (4.41)
4.5.3 Transmit Diversity
Now we consider a time-switched transmit diversity (TSTD) (Tx2-Rx1) scheme as shown in Fig. 4.1 (c). For the case with repetition length Np = 2, one can express the processed data zp1TD for the transmit diversity scheme as follows:
zp1TD= zp1,T1(f1) + zp1,T2(f2) . (4.42)
Since
E[zTDp1 ] = E[zp1,T1] +E[zp1,T2] = 2E[zp1], (4.43)
we can calculate E[zp1TD] by following the procedures of evaluating E[zp1] in (4.17). Define ρ as the correlation coefficient between the two transmit an-tennas. Then the variance of the processed data zp1TD is
var[zp1TD]
= var[zp1,T1] + var[zp1,T2] + 2cov[zp1,T1, zp1,T2]
= 2var[zp1] + 2ρ
var[sp1,T1]var[sp1,T2] +
var[rp1,T1]var[rp1,T2]
(4.44) where var[zp1] is defined in (4.19) of Proposition 2. Thus, we can compute SNRTD from
4.6.1 The UWB Channel Response
Figure 4.2 shows an example of the UWB channel response using the channel model described in Section 4.2 with parameters listed in Table 4.1. In the considered model, the channel response time is set to 225 nanoseconds as in [9], the average number of the resolvable paths is 80.72. Let N be the total time bin number during the channel response time, TA the first path arrival time, and tl is the arrival time of each resolvable path. Then, in our simulations, the mean excess delay Tm=
N
l=1(tl−TA)al2
N
l=1al2 = 34.61 nanoseconds,
and the root mean square delay spread TRM S =
Nl=1(tl−TNm−TA)2al2
l=1al2 = 37.98 nanoseconds.
4.6.2 Average SNR and Variance of the Pulse Based UWB Signals
Figure 4.3 compares the SNR of PPM signals for no diversity, receive diver-sity, and transmit diversity schemes. Through simulations, we validate the analytical results obtained by (4.35), (4.36), (4.40), (4.41), (4.45), and (4.46) in Section 4.4. From Fig. 4.3, one can find that the SNR of the receive diver-sity is the highest, while the no diverdiver-sity scheme and the transmit diverdiver-sity have the similar SNR.
Figure 4.4 shows the variance of PPM signals with no diversity, receive diversity, and transmit diversity schemes in the UWB channel by analysis and simulations. From the viewpoint of the signal variance, transmit diversity is the best, no diversity ranks second, and receive diversity is the worst. Here, we assume that the antennas of both receive diversity and transmit diversity are mutually independent.
Figure 4.5 shows the effect of spatial correlation ρ of transmit diversity on the variance of the PPM signals over the UWB channel. As shown in the figure, the variance of the PPM signals increases as the correlation of transmit diversity increases. From the results, it is implied that the diversity gain of transmit diversity may not be significant in the UWB channel. In the following, we will quantify the performance difference between no diversity and having antenna diversity in terms of BER performance.
4.6.3 Comparison for Different Diversity Schemes for the PPM UWB System
Figure 4.6 shows the BER performances of different diversity schemes for the binary PPM signals in the UWB channel. In the figure, the numbers adjacent to Tx and Rx represent the numbers of the transmit and receive antennas; L represents the finger number in the RAKE receiver; f represents the frame number; and δ represents the modulation index associated with the message bit which is an integer multiple of the chip time Tc. From Fig. 4.6, we have the following observations:
• Comparing the no diversity (Tx1-Rx1) scheme to the time switched transmit diversity (Tx2-Rx1) scheme, one can find that the TSTD scheme can improve BER performance by about 2 dB at BER = 10−4. As shown in Fig. 4.4, the signal of the transmit diversity scheme is more stable than that of the no diversity scheme, which can explain the BER performance improvement of the transmit diversity scheme over the no diversity scheme even though the SNRs of these two diversity schemes are about the same in Fig. 4.3.
• Recall that he diversity order can be roughly viewed as the slope of BER v.s. SNR in the region with high SNRs where the slope does not increase any more. The higher the diversity order, the steeper will be the slope of the performance curve for BER v.s. SNR. As shown in the figure, the (Tx2-Rx1) TSTD scheme indeed achieves the same diversity order as the (Tx1-Rx2) receive diversity scheme. Furthermore, comparing the Tx2-Rx2 and the Tx1-Rx4 schemes, we find that the Tx2-Rx2 scheme can achieve about the same diversity order as the Tx1-Rx4 scheme but at the cost of about 3 dB Eb/N0 loss. In this figure, it is demonstrated
that employing multiple time switched transmit diversity or multiple receive antennas can improve the UWB performance even though the UWB channel possesses inherently rich diversity.
Note that because the UWB MIMO channel may perform differently from the narrowband MIMO channel. For example, severe correlation between channel paths may exist in a UWB channel. Thus, the above results should be used cautiously as an upper bound that quantifies the extent to which transmit or receive antenna combining techniques can improve the perfor-mance for the PPM based UWB system. In the following, we will examine how to exploit transmit diversity in the UWB channel from a different per-spective – reducing the complexity of Rake receiver.
4.6.4 Effect of RAKE Finger Numbers
Figure 4.7 shows the BER performance of the PPM UWB system with dif-ferent RAKE finger numbers. Two major remarks are given below:
• The transmit diversity scheme (Tx2-Rx1) with L = 30 (with the squared legend) has the similar performance to the scheme (Tx1-Rx1) with L = 50 (with the triangle legend). It is implied that the complexity of Rake receiver can be alleviated at the cost of increasing the transmit antennas by using time-switched transmit diversity.
• Because of inherit large path diversity, adding more transmit antennas in the UWB system cannot increase the diversity order significantly. In the figure, the slope of BER v.s. SNR for the cases of L > 50 with single antenna (with the triangle legend) and that of L > 30 with two transmit antennas (with the squared legend) are about the same. Nevertheless, transmit diversity can slightly improve the BER performance for the
PPM UWB system from the signal variance perspective as explained in Fig. 4.4.