**4.5 Numerical Results**

**4.5.3 Summary of the Comparison Study**

sector. The results are presented in Fig. 4.11. The FDOA outperforms all other
techniques which can achieve RMSE = 3.5^{o} while the second best technique, the
A-MMD, can achieve RMSE = 6^{o}. Compared to the previous scenario, since L^{(i)}r

is increased, the general performance of the FDOA, the A-MMD, and the EA-MMD degrades. For the A-EA-MMD and the EA-EA-MMD, the reason is the increasing uncertainty in the prior information. For the FDOA, focusing on larger sectors is expected to be more difficult which explains the performance degradation.

0 10 20 30 40

SNR (dB) 0

5 10 15 20 25 30

RMSE (o )

FDOA IRG A-MMD EA-MMD RGD

Figure 4.11: DOA estimation performance comparison among the FDOA, the
IRG, the A-MMD, the EA-MMD and the RGD for K = 4 where the first two
targets emerge at the 1^{st} sector and the remaining two targets emerge at the 3^{rd}
sector with L = 120 and L^{(i)}r = 40

achieves the lowest CRLBs in all scenarios. After the CRLB comparison, the
DOA estimation performance of different algorithms is investigated. The FDOA
can achieve RMSE ≈ 0^{o} in various scenarios and outperforms the alternatives
including the A-MMD and the EA-MMD that were proposed in Chapter 3.
Fur-thermore, even at the low SNR regime, the FDOA achieves high performance
even in the most challenging scenarios despite the highly noisy environment.

## Chapter 5

## Conclusions and Future Research Directions

In this thesis, two essential essential questions are addressed: why and how the CS based techniques should be used in DOA estimation. In Chapter 2, it is shown that the CS based DOA estimation techniques outperform the classical methods under various scenarios. Furthermore, unlike the well-known classical techniques the MUSIC and the Capon’s beamformer, the CS based techniques can typically work with a single snapshot and they are not affected by the multipath effects.

That is esentially “why” the CS based DOA estimation should be preferred to the classical methods.

To improve the performance of a CS based DOA estimation system, the natu-ral attempt would be to improve the encoding and the decoding strategies. The encoding of data in a CS based system is determined by measurement matri-ces; therefore, measurement matrix design is among the most important aspects of the CS based sensor processing applications. Various measurement matrix de-sign algorithms are proposed in this thesis by using the mutual coherence and the CRLB criteria. The mutual coherence based design, namely the A-MMD adapts to the prior information on the target scene. Hence, a computationally feasible,

closed-form expression for the measurement matrix is derived enabling online up-dates of the measurement matrix at a high compression rate for each snapshot of the sensor data. Further improvements on the software and the hardware complexity of the proposed measurement matrix methodology are done which allows for more efficient hardware and software implementation of the proposed system, namely, the more software-efficient EA-MMD and the more hardware-efficient A-MMD-BD algorithms are presented. The CRLB based measurement matrix design methodology requires solving a more complicated non-convex op-timization compared to the A-MMD algorithm and does not have a closed-form solution. However, the CRLB based design does not need a prior information on the target scene, while it is able to adapt to the prior information depending on the application. To sum up, the measurement matrix design is extensively studied in this thesis. For the decoding part, an iterative reconstruction algo-rithm that can work with the proposed CRLB based measurement matrix design methodology is proposed leading to the FDOA algorithm. The FDOA finds the DOAs of the sources by focusing on particular sectors and iteratively canceling out the interferences with the received signal leading to the successive estimations on the varying residual signal. The proposed methodologies in the thesis provide alternative answers to the question of “how” the CS based techniques should be used in DOA estimation.

The proposed techniques are eligible for the joint optimizization over both the measurement matrix and the signal dictionary. While a detailed investigation of the dictionary design is not performed in the presented thesis, a simple adaptive dictionary design algorithm is proposed and shown to improve the performance under some scenarios when it is used with the A-MMD algorithm. As a future study, dictionary design in the CS based DOA estimation can be studied in more detail and the resulting designs can be integrated with the techniques proposed in the thesis. Furthermore, while the hardware complexity constraints are taken into account, the other restrictions caused by the hardware implementation of the presented system such as mutual coupling effects, quantization limits and the calibration errors in the antenna displacement are not investigated. Another research direction can be the wideband extensions of the proposed results since

a narrowband DOA estimation signal model is assumed throughout the thesis.

Finally, although the DOA estimation has been the main focus, in the future, the proposed methodologies can be applied to other areas of sensor processing including detection of sparse signals in video and audio streams.

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## Appendix A

To achieve the result given in (3.23), we first write the eigendecomposition of
Z = QΛQ^{H} for QQ^{H} = Q^{H}Q = I_{M} and Λ ≡ diag(λ). Since the Frobenius
norm is a unitarily invariant norm, we can write

kGΨ− Zk^{2}_{F} = kQ^{H}GΨQ − Λk^{2}_{F}, (A.1)
and optimize over GΨQ ≡ Q^{H}GΨQ since GΨQ satisfies the constraints if and
only if G_{Ψ} satisfies them. Let

Λ ≡¯

λ_{1} . . . 0
... . .. ...

0 . . . λm

0

0 0

, ˜Λ ≡

λ_{1} . . . 0
... . .. ...

0 . . . λz

0

0 0

. (A.2)

The proof consists of two parts for m < z and m ≥ z.

I) For m < z, let G_{ΨQ} be any feasible matrix. Our aim is to show that
k ¯Λ − Λk^{2}_{F} ≤ kG_{ΨQ} − Λk^{2}_{F} for any G_{ΨQ} to prove that ¯Λ is the solution. We
write G_{ΨQ} and Λ as:

G_{ΨQ} =

"

G^{zz}_{ΨQ} G˘^{zz}_{ΨQ}
G˘^{rr}_{ΨQ} G^{rr}_{ΨQ}

# , Λ =

"

Λ^{zz} 0
0 Λ^{rr}

#

, (A.3)

where G^{zz}_{ΨQ} and Λ^{zz} consist of the first z rows and the first z columns of the
matrices G_{ΨQ} and Λ while G^{rr}_{ΨQ} and Λ^{rr} include rest of the entries that are
on the block diagonal of G_{ΨQ} and Λ respectively. G˘^{zz}_{ΨQ} and ˘G^{rr}_{ΨQ} denote the