Numerical Evaluation and Comparison

In the evaluation of the estimators and CRBs discussed in the previous sections, we

will be considering two different antenna models. On the one hand, we assume an ESA

with M = 6 sectors and, on the other hand, we use the LWA model from [P2]. This

model was obtained by approximating the measured radiation patterns of the LWA [81]

according to the procedure described later in Section 4.4. The resulting model consists

of M = 12 sectors each approximated by a Gaussian radiation pattern (4.1) with the

parameters listed in [P2, Tab. III]. The ESA can be used in combination with any of

the discussed estimators, whereas the LWA consists of sectors where the main beam is

different in all sectors meaning that it can be used with TSLS only (see [P2]). For the

ESA, we use TSLS with L = 3 and variance weighting, which results in the overall best

performance [P2]. The influence of L and the weighting scheme on the performance of

TSLS is then studied using the LWA.

All results presented in the following were obtained assuming a DoA uniformly

distributed over the whole angular coverage area of the respective antennas. This

means that the DoA is distributed as Ï

≥ U(≠180

; 180

) for the ESA and as Ï

≥

U(≠60

_{; 60}

_{) for the LWA. An overview of the expressions used in the figures can be}

found in Tables 4.2 and 4.3. For a detailed description of the simulation setup please

refer to the references in those tables.

Figures 4.6 and 4.7 depict the performance of DoA estimation and RSS estimation

with the ESA as a function of the SSP a

, respectively. We notice that, in certain

conditions, both SLS and TSLS result in an RMSE that is lower than the respective

4.2 DoA/RSS Estimation Using Sectorized Antennas Tab. 4.3. Expressions used in the evaluation of RSS estimation performance.

Expression Alg./CRB Type References

RRMSERSS CRB numerical average over DoA (4.19), [P1]

RRMSEa

RSS CRB approximation, numerical average over DoA (4.21), [P1]

RRMSESNRRSS CRB asymptotical SNR æ Œ, analytical (4.25), [P1]

RRMSEN

RSS,m MaxE asymptotical N æ Œ, analytical (4.45), [P4]

SLS SLS numerical algorithm evaluation Sec. 4.2.2.2, [P5]

Side-sector suppression, as 0 0.2 0.4 0.6 0.8 1 RM S E ˆϕ [d eg] 10-1 100 101 102 103 RMSEDoA RMSEa_DoA RMSESNRDoA SLS RMSESLS RMSEN_DoA,m TSLS

Fig. 4.6. DoA estimation performance as a function of the side-sector suppression. Parameters: ESA

with M = 6, N = 100, and SNR = 5 dB.

CRB. Clearly, this indicates that both algorithms are not entirely unbiased. As shown

and discussed in [P2] and [P5], a significant non-zero bias in SLS and TSLS occurs in

adverse operation conditions, such as for a low SNR, for strong multipath or for antennas

with disadvantageous values for the SSP. Now the choice of the SSP is a compromise.

As can be seen from the asymptotic CRB in Figure 4.6, the SSP should be as small

as possible for SNR æ Œ. However, for moderate to large SNR, the lowest CRB is

attained for a SSP around a

œ [0.2; 0.4]. This is in line with the SSP interval where the

DoA estimators SLS and TSLS result in the lowest RMSE. However, as evident from

Figure 4.6 TSLS is much less susceptible to the choice of a

than SLS. For RSS estimation

and SNR æ Œ, the asymptotic CRB is entirely independent of a

. However, for finite

SNR, the trend in the RSS estimation CRB is opposite to the trend in RMSE

as an

omnidirectional antenna (a

= 1) results in the lowest CRB. Overall, a SSP a

œ [0.2; 0.4]

is thus a good compromise. Consequently, the approximations RMSE

and RMSE

are very accurate for well-tuned antennas since the respective curves in Figures 4.6 and

4.7 match perfectly with the CRBs for a

>0.1.

Figures 4.6 and 4.7 also include the asymptotic RMSEs N æ Œ of MaxE DoA and

RSS estimation. Even in the asymptotic case, the performance of MaxE is far from the

Side-sector suppression, as 0 0.2 0.4 0.6 0.8 1 RRM S E ˆγ 10-3 10-2 10-1 100 101 RRMSERSS RRMSEa_RSS RRMSESNRRSS SLS RRMSEN_RSS,m

Fig. 4.7. RSS estimation performance as a function of the side-sector suppression. Parameters: ESA

with M = 6, N = 100, and SNR = 5 dB.

CRBs and the performance of SLS and TSLS with finite N. Note that for the asymptotic

case N æ Œ, the other estimators as well as the CRB result in an RMSE = 0. Due to

this significant difference in performance, we will, in the following, not discuss MaxE

anymore. For a detailed performance discussion of MaxE refer to [P4] and [P5] instead.

Figure 4.8 depicts the performance of DoA estimation with an ESA as a function of

the SNR. As can be seen, RMSE

DoA

is a perfect approximation for the CRB RMSE

and the analytical expression for the RMSE of SLS is also very accurate for SNR > 2 dB.

Moreover, for moderate SNRs around 5 dB both SLS and TSLS perform quite close to

the CRB. For increasing SNRs, TSLS approaches the CRB even closer. The performance

of SLS, in contrast, saturates at a much higher level (RMSE

) than the CRB for

SNR > 15 dB. As discussed in more detail in [P1], this is due to SLS excluding all but

two sectors from the DoA estimation. For increasing SNRs, our assumption that all

but two sectors are too noisy to exploit the TN signal component becomes increasingly

worse. For low SNRs, on the other hand, the performance of TSLS and SLS is in fact

equal. This is due to the weighting in TSLS, which practically excludes all other but

the two sectors that are also used in SLS.

In TSLS the number of sectors L that results in the best performance depends on

the type of antenna, as well as on the SNR. This is illustrated in Figure 4.9 using the

LWA as an example. For large SNRs, the best performance is achieved with L = 11,

i.e., with significantly more sectors compared to the ESA. While the ESA has M = 6

sectors with a beam-width — ¥ 0.22 rad distributed over 360°, the LWA has M = 12

sectors with beam-widths between 0.5 rad and 0.7 rad distributed over 120° [P1]. This

means that the number of sectors that receive the TN signal with a significant signal

strength is much larger in the LWA than in the ESA. Consequently, L should also be

4.3 Localization Using Sectorized Antennas