Chapter 4 Experimental Results
4.1 Narrow-Band Simulations of the Proposed Robust Beamformer
4.1.1 Analyses of the Norm Constraint Value for Half-Wavelength Spacing . 53
In this sub-section, we show that an appropriate selection of norm constraint value
() = 0.035 is less sensitive to the input signal powers, central angle mismatches, and number of sensors than the selection of the diagonal loading level () with a standard linear array. For all the analyses, the theoretical covariance matrices of the signals were used. The equivalent () of a chosen () can be derived from the relationship in
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0.005 0.01 0.015 0.02 0.025 0.03 0.035
Angle (Degree)
Normalized Angular Power Density
Presumed Actual
Figure 4-1 Presumed and actual angular power density functions for 5 and 5
central angle mismatch
(2-20).
For the Simulation 1, the sensitivity of () and () to the input signal powers are studied. A ULA of M = 10 sensors with half-wavelength spacing was used. The multi-rank signal model given in (4-1) was utilized with presumed central angle equal to 0 and angular spread = 5. The scattered desired source and interference with angular spread = 5 impinged into the array from the central angles 5 and 45, respectively.
Thus, 5 central angle mismatch was considered. The presumed and actual normalized angular power density functions are shown in Figure 4-1. The sensor noise power was set to 1, and the interference-to-noise ratio (INR) was 30 dB. The signal-to-noise ratio (SNR) varies from -20 dB to 30 dB. Figure 4-2 shows the output SINRs versus input SNRs for different selections of () and (). It can be seen that () = 102 (note that the sensor noise power = 1) and () = 0.035 (i.e., T() = 1/M + 0.035 = 0.135) are good choices considering all input SNR conditions. These values will be used as the best choices for the rest simulations. For the selection of (), high values give more penalties on the spatially white noise (or incoherent noise), which leads to the matched
-20 -15 -10 -5 0 5 10 15 20 25 30 interferences, hence it may have poor output SINR performance. As () 0, it turns into the MVDR solution without norm constraint, which is sensitive to the array mismatches and has severe self-cancellation at high input SNRs. For the selection of
(), it can be expected that the variation of output SINR with different () is relatively smaller than that with different (). This is because in (2-16) and (2-20), the
10-1 100 101 102 103 104 105 106
factor of averaged input power has been removed by the division. When the norm constraint value () 0, the weight vector turns into the matched filter. When () approaches to the upper bound given in (2-23), it gives the MVDR solution without norm constraint. From Figure 4-3 and Figure 4-4, the optimal selections of () and
() for different input SNR conditions are illustrated. In the figures, the stars point out the optimal selections for each case. It is obvious that the optimal selection of () is
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related to the input signal power, while the optimal selection of () is less sensitive to the input signal power. Since the powers of speech signals are unknown a priori and even time varying, the proposed robust beamformer provide consistent SINR performances due to the insensitivity of () to the signal powers. It is interesting to note that there is a jump at () 0.027 in Figure 4-4. A possible explanation of this phenomenon is that the feasible set that satisfies the norm constraint and the distortionless constraint is too small in this case. Thus the solution fails to reject the interference in order to satisfy the constraints, which gives the solution of the matched filter. Figure 4-5 gives the comparison between the best selected () and () for different input SNR conditions. Since the optimal choice of () dependents on the desired signal powers, the proposed norm-constrained robust beamformer with the optimal () gives better output SINRs for most SNR conditions.
For the Simulation 2, the comparison between the best selected () and () for different central angle mismatches is analyzed. It can be seen from Figure 4-6 that for small angular mismatches, the proposed norm-constrained robust beamformer with the
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Central Angle of the Desired Source (Degree)
Output SINR (dB)
Figure 4-6 Comparison between the best selected () and () for different central angle mismatches
4 5 6 7 8 9 10 11 12 13 14 15 for different number of sensors
best selected () performs better than using the best selected diagonal loading.
For the Simulation 3, the comparison between the best selected () and () for different number of sensors is investigated. Again, the superiority of using the norm-constrained robust beamformer is demonstrated in Figure 4-7.
In this section, the superiorities of using the norm constraint value () over the diagonal loading level () were demonstrated in the narrow-band simulations.
However, for wide-band applications, the properties differ from frequency bands according to different directivities [1]. The parameter selection strategy for wide-band applications will be discussed in Section 4.3.1.
4.1.2 Narrow-Band Comparisons
In this section, the comparisons of the proposed beamformers and other adaptive beamformers are studied. The sensor noise power was set to 1. The simulation condition is the same as Simulation 1, except the generated simulated data is used. For each scenario, the average of 100 simulation runs is used to obtain each simulated point. The detailed parameter settings and abbreviations of the algorithms are listed below:
1) MRSMI: Multi-rank sample matrix inverse [20]. The algorithm was implemented by (2-8).
2) MRLSMI: Multi-rank loaded sample matrix inverse [20]. The algorithm was implemented by (2-10), where the diagonal loading level was chosen as ()
= 102 (note that the sensor noise power = 1).
3) CKF: Constrained Kalman filter [67]. The CKF uses the rank-1 model without the norm constraint and formulates the state space using the SCPO method. The parameter matrices Q() was set to a zero matrix, and the diagonal terms of R() were estimated as ˆ12
and 105 ˆ12
, where
2
ˆ1
was estimated using (2-41) with the forgetting factor = 0.9.
4) RLSVL: Recursive least square with variable loading [71]. The RLSVL uses the rank-1 model and the GSC structure. A norm constraint was imposed on the nulled vector for improving the robustness. The forgetting factor and the
norm constraint were set as 0.999 and 0.2 as suggested in the paper [71].
5) MRNCKF-FOE: Multi-rank norm-constrained first-order extended Kalman filter. The proposed beamformer was implemented using (2-35) by letting both (k) and (k) be zero. The parameter matrix Q() was set to a zero matrix, and R() was set as in (2-40) with = 10-5 for all the proposed Kalman filters, where ˆ12
was estimated using (2-41) with the forgetting factor = 0.9. The norm constraint value () = 0.035 is chosen, which corresponds to the constraint () = 1/M + () = 0.135.6) MRNCKF-SOE: Multi-rank norm-constrained second-order extended Kalman filter. The proposed beamformer was implemented using (2-35) with the second-order terms (k) and (k).
7) MRNCKF-U: Multi-rank norm-constrained unscented Kalman filter. The proposed beamformer was implemented using (2-36) and (2-37).
For the first case, the convergences and beam patterns at SNR = 0 dB are studied.
Figure 4-8a shows the output SINR performance versus the training size. The presence of the desired signal deteriorates the SINR performance due to the self-cancellation phenomenon, which can be observed in Figure 4-8b around the central angle of the desired source 5. Considering the performance of the pairs (MRSMI, MRLSMI) and (CKF, RLSVL), it can be observed that the norm constraint improves the SINR performance. It is also worth to note that the Kalman filter solutions seem to be more robust to the steering mismatches than the beamformers using the estimation of sample matrix. The Kalman filter is a close-loop system who constrains the weight vector to the desired array response at each iterationon; on the other hand, the sample matrix inverse method is an open-loop system who constrains the weight vector after the sample matrix is estimated. Therefore, the latter one can be easily affected by the contaminated sample
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(a) Output SINR vs. training size
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Figure 4-8 Comparisons of the beamformers at 0 dB input SNR.
matrix since the importance of the training data is the same when estimating the sample matrix. From the beam pattern, it is shown that the proposed beamformer, MRNCKF-FOE, gives the best output SINR since it has the smallest signal distortion at
5 while keeping the same order of noise rejection at 45.
For the second case, the convergences and beam patterns at SNR = 20 dB are studied. In Figure 4-9a, the large signal power slows down the convergence of the
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(a) Output SINR vs. training size
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Figure 4-9 Comparisons of the beamformers at 20 dB input SNR.
algorithms. The strong signal power leads to larger self-cancellation for the MRSMI and MRLSMI beamformers. Despite of the difference between using the Kalman filter and the sample matrix, this case also reveals that the chosen diagonal loading level of the MRLSMI beamformer is not appropriate under this SNR condition (see Figure 4-3).
This demonstrates the advantage of using the norm constraint with a more robust selection of the norm constraint value.
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Figure 4-10 Output SINR vs. input SNR for different beamformers
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Figure 4-11 Output SINR vs. input SNR for proposed Kalman filters
For the third case, the output SINR of the beamformers versus the input SNR is illustrated in Figure 4-10. The training size of this simulation is N = 500. When the input SNR is small, all the beamformers converges to the optimal MVDR solution. As the SNR increases, the differences between algorithms become obvious. It is shown that the proposed beamformer has the best performance through different SNR conditions.
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Angle (Degree)
Beam Pattern (dB)
MRNCKF-FOE MRNCKF-SOE MRNCKF-U
-5 45
Figure 4-12 Beam patterns of the proposed Kalman filters at 20 dB input SNR.
For the last case, the output SINR of proposed Kalman filter solutions versus the input SNR is illustrated in Figure 4-11. It can be seen that the performances of the first- and second-order extended Kalman filters are almost the same. This indicates that the first-order approximation is good enough for our problem. Compared to the extended solutions, the unscented Kalman filter (UKF) has worse SINR performance. The UKF implicitly estimates the first- and second-order approximation terms of the Taylor expansion using sigma points. The sigma points were spread based on M times eigenvectors of the error covariance P(k). An issue for the spreading of sigma points is invoked when some error dominants the covariance P(k). In this case, some sigma points are spread far away from the constraint sets and the neighborhood of the current state estimate, which can induce improper nonlinear transformations that degrade the performance of the UKF. In Figure 4-12, it can be seen that the large error of interference rejection enforces the noise reduction at both 45 and 5, which results in the self-cancellation.
4.2 Speech Enhancement Results of the Proposed SCPF and