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4.5 Numerical Results

4.5.1 The CRLB Comparison

In this section, SNR is fixed to Ps/ν = 10 which is not an important parameter since the linear effect of the noise variance on the CRLB is clear from (4.7) by writing R = νΦΦH. L = 100 and L(i)r = 20 are fixed, and since L/L(i)r = 5 in that case, 5 different measurement matrices corresponding to these 5 different

“sectors” are designed. The sectors are formed by partitioning ω domain where ω intervals corresponding to each sector is presented on TABLE 4.1.

Table 4.1: ω Intervals for Each Sector when L = 100 and L(i)r = 20 Sector 1 ω ∈ [0, 0.4π)

Sector 2 ω ∈ [0.4π, 0.8π) Sector 3 ω ∈ [0.8π, 1.2π) Sector 4 ω ∈ [1.2π, 1.6π) Sector 5 ω ∈ [1.6π, 2π)

The CRLB comparison among four matrices is performed. First matrix uses the first m measurements out of M which is called as Φdir since this matrix directly uses the first m measurements. Note that Φdir = [Im 0] and this is also a solution of the common design criterion given in (4.18) after an unimportant scaling constant for the investigated scenario 2. The second matrix Φrg is a random Gaussian matrix, i.e., its real and imaginary entries are independently drawn from a zero-mean normal distribution with 1/√

2m standard deviation.

The entries of the third matrix are independently drawn from the ± Bernoulli distribution:

ber)i,j =

 +1/√

m, with probability 1/2

−1/√

m, with probability 1/2

. (4.22)

The last matrix is designed based on the design procedure described in Algorithm 2, which is called as Φdes. Since the optimization problem proposed in (4.19) is non-convex, different CRLBs can be achieved with different initializations of Φ.

However, it requires an exhaustive search over a one dimensional grid of different Φ initializations. Hence, the demonstrated results are achieved by using only a single random initialization of Φ. In other words, the presented results do not actually reflect the minimum achievable CRLBs by the proposed technique.

To compare the CRLBs achieved by Φdir, Φrg, Φber, and Φdes; all K sources are assumed to locate on the grid points correspondig to a given sector. Otherwise, the proposed methodology cannot be used since it is designed to focus on a single sector only. Assuming M = 50, m = 10, K = 2, Ns = 10 and performing 300, 000 different ω = [ω1 ω2]T realizations for each sector and with independent noise realizations, the results given in Fig. 4.1 are achieved. It is observed that Φdes achieves the lowest CRLBs at all sector indices. Furthermore, the dramatic difference between Φdir and the other matrices demonstrates the importance of using the linear combination of all M measurements instead of directly using the first m ones. In other words, wisely designed measurement matrices can provide drastic performance improvements.

2The equality DDH = (L/M )IM holds for the presented scenario which converts (4.18) to Φ= arg minΦHΦ − ˜cIMk2F =

˜

dir, see [32] for the details.

1 2 3 4 5 Sector Index

10-7 10-6 10-5 10-4

CRLB () dir

rg ber des

Figure 4.1: The CRLB Comparison among Φdir, Φrg, Φber, and Φdes for different sector indices by fixing M = 50, m = 10, K = 2, Ns = 10, L = 100 and L(i)r = 20

In Fig. 4.2, dependency of the CRLB on m is demonstrated. The CRLB decreases in general as m increases since the amount of information increases.

This is also compatible with our previous finding presented in (4.6). When Φdir is used, the decreasing CRLB is clearly observed since Φdironly receives information from the first m sensors in the array. In other words, for Φdir, increasing m is equivalent to the use of a larger sensor array. When Φdes is used, an interesting observation is that after m = 12, the CRLB does not decrease much. Even though the measurement matrix design technique was different, this is also compatible with our previous finding shown in Fig. 3.9, where the increasing m did not result in drastic performance improvements. These two results suggest that increasing m may result in a complicated hardware which cannot provide the expected performance gain. Hence, if a hardware with a measurement matrix is to be

implemented, it is crucial to perform a CRLB analysis to decide on the number of digital channels.

10 12 14 16 18

m 10-7

10-6 10-5 10-4 10-3

CRLB ()

dir rg ber des

Figure 4.2: The CRLB Comparison among Φdir, Φrg, Φber, and Φdes for different m values by fixing M = 50, K = 2, Ns = 10, L = 100 and L(i)r = 20

In Fig. 4.3, dependency of the CRLB on the number of snapshots Nsis demon-strated. For all techniques, the CRLB decreases as Ns increases since increasing Ns provides more information. If the DOA estimation performance provided by a sensor array is needed to be increased, this result suggests that increasing Ns is a hardware-efficient alternative since it does not change the hardware structure.

However, processing more snapshots requires more computation as the dimensions of data increase.

5 10 15 20 25 Ns

10-8 10-7 10-6 10-5 10-4 10-3

CRLB () dir

rg ber des

Figure 4.3: The CRLB comparison among Φdir, Φrg, Φber, and Φdes for different Ns values by fixing M = 50, m = 10, K = 2, L = 100 and L(i)r = 20

Dependency of the CRLB on K is demonstrated in Fig. 4.4. It is observed that the increasing K results in the increasing CRLB for all measurement matrices.

This observation is compatible with our expectations since K is the number of unknowns.

1 2 3 4 5 K

10-8 10-6 10-4 10-2 100 102

CRLB () dir

rg ber des

Figure 4.4: The CRLB comparison among Φdir, Φrg, Φber, and Φdes for different K values by fixing M = 50, m = 10, Ns = 10, L = 100 and L(i)r = 20

In Fig. 4.5, the effect of M on the CRLB is investigated. The first observation is the unchanging CRLB when Φdir is used since Φdir does not exploit the infor-mation coming from the remaining M −m sensors. For other matrices, the CRLB decreases with the increasing M . This is an important observation since it shows the importance of the amount of information even if it is not digitized. This result further implies the possibility of achieving drastic performance improvements by using measurement matrices.

25 30 35 40 45 50 M

10-7 10-6 10-5 10-4

CRLB () dir

rg ber des

Figure 4.5: The CRLB comparison among Φdir, Φrg, Φber, and Φdes for different M values by fixing m = 10, Ns= 10, K = 2, L = 100 and L(i)r = 20

In all of these simulations, sector index is set to 3 and the corresponding CRLBs achieved by the designed measurement matrix Φdes = Φ(3) is demonstrated for the sake of visual clarity. Otherwise, we would need to report 4 more sets of CRLBs achieved by Φ(i)’s for i ∈ {1, 2, 4, 5}, which are almost the same as previously demonstrated in Fig. 4.1. Finally, in all simulations, Φdes achieves the lowest CRLB values. Furthermore, Φber and Φrg perform very similar to each other. The similarity between the reconstruction performance achieved by random Gaussian and random Bernoulli matrices is studied well in the CS literature [11]. Our findings in this study support them by using a more traditional technique, i.e., computation of the CRLB.

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