**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 = νΦΦ^{H}. 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 Φ_{dir} since this matrix
directly uses the first m measurements. Note that Φ_{dir} = [I_{m} 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:

(Φ_{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 Φ_{des}. 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, N_{s} = 10 and performing
300, 000 different ω = [ω_{1} ω_{2}]^{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 Φ_{dir} 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 DD^{H} = (L/M )IM holds for the presented scenario which converts (4.18) to
Φ^{∗}= arg min_{Φ}kΦ^{H}Φ − ˜cIMk^{2}_{F} =√

˜

cΦdir, 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 Φ_{dir}, Φ_{rg}, Φ_{ber}, and Φ_{des} for different
sector indices by fixing M = 50, m = 10, K = 2, N_{s} = 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 Φ_{dir} is
used, the decreasing CRLB is clearly observed since Φ_{dir}only receives information
from the first m sensors in the array. In other words, for Φ_{dir}, increasing m is
equivalent to the use of a larger sensor array. When Φ_{des} 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 Φ_{dir}, Φ_{rg}, Φ_{ber}, and Φ_{des} 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 N_{s} increases since increasing
N_{s} provides more information. If the DOA estimation performance provided by
a sensor array is needed to be increased, this result suggests that increasing N_{s} 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
N_{s} 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}
10^{0}
10^{2}

CRLB () dir

rg ber des

Figure 4.4: The CRLB comparison among Φ_{dir}, Φ_{rg}, Φ_{ber}, and Φ_{des} for different
K values by fixing M = 50, m = 10, N_{s} = 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 Φ_{dir}, Φ_{rg}, Φ_{ber}, and Φ_{des} for different
M values by fixing m = 10, N_{s}= 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, Φ_{des} achieves the lowest
CRLB values. Furthermore, Φ_{ber} and Φ_{rg} 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.