FxLMS
In accordance with the method described in the previous section, we use a real system of P (z) and S(z) to verify the proposed algorithm. The impulse response and frequency response of P (z) are shown as Fig.5.11 and Fig.5.12:
And the impulse response of S(z) is shown as Fig.5.13. The impulse response sample length of P (z) and S(z) are 145 and 51 taps. Based on
10 7 4 2.2 1 0 −3 −6 −7.8 −9 −10
−24.5
−24
−23.5
−23
−22.5
−22
−21.5
−21
SNR 10log(σd2/σ2ε)
Steady−state MSD (dB)
Proposed, variable step size
Figure 5.10: The steady-state MSD of different variances of auxiliary noisesε(n).
The simulation is to compare noise reduction ratio NRR (dB) of pro-posed algorithm with different on online secondary path modeling FxLMS algorithms shown in Fig.5.14, Fig.5.15 and Fig.5.16. The NRR defini-tion of the on online secondary path modeling FxLMS is different from traditional FxLMS, which is given by
NRR(dB) = 10log
(E[d2(n)]
E[e′2(n)]
)
. (5.3)
About the proposed algorithm setting, the generality optimal coefficients parameters M′ andN′ are set to 64 and 316, theτ1′ andτ2′ are set to 0.0925
and0.0626. The terminating thresholdL′ andR′are set to10−1 and5×10−4. About the normalized LMS(NLMS) algorithm [2] setting, the smoothing
0 50 100 150
Figure 5.11: The impulse response of real P(z).
parameter is set to 0.99, the normalized step size α is set 0.4 and the tap length L is set to 95. We use two 95-taps FxLMS algorithms one with a large step size and the other with the small step size, where the large step size is set as µmax = (95+2)1.2×σ2
x′
and the small step size µmin = (95+2)0.4×σ2 x′
. A 95-taps FxLMS with variable step size is similar to [8], whose the param-etersρ,Pe,Pe,min andPe,max are the same as the previous traditional FxLMS simulation, and the maximum and minimum step sizes are between the other two 95-taps FxLMS with large and small step size.
In Fig.5.14, we adjust the final performance of proposed algorithm, NLMS [2], variable step size [8] and 95-taps FxLMS with small step size are close, and then compare the convergence speeds. The step size of the 95-taps FxLMS with large step size is adjusted so that its convergence
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−8000
−6000
−4000
−2000 0 2000
Normalized Frequency (×π rad/sample)
Phase (degrees)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−40
−30
−20
−10 0 10
Normalized Frequency (×π rad/sample)
Magnitude (dB)
Figure 5.12: The frequency response of real P(z).
final performance.
In Fig.5.15, the convergence speed of proposed algorithm is better than the NLMS [2], variable step size [8] and 95-taps FxLMS with small step size. The final performance of proposed algorithm is better than the 95-taps FxLMS with large step size.
In Fig.5.16(a), the tap length of proposed algorithm is terminated at 92 taps, which is close the tap length of optimal coefficients 95 taps. Shown in Fig.5.16(b), the step size of proposed algorithm gradually decreases slower than the others.
0 10 20 30 40 50 60
Figure 5.13: The impulse response of real S(z).
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0
Proposed, variable step size with limit NLMS[2]
Variable step size LMS[8]
95−tap FxLMS, small step size 95−tap FxLMS, large step size
Figure 5.14: Comparison of NRR (dB) performance for different online secondary path modeling FxLMS algorithm.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Proposed, variable step size with limit NLMS[2]
Variable step size LMS[8]
95−tap FxLMS, small step size 95−tap FxLMS, large step size
Figure 5.15: The result of the comparison of NRR (dB) performance for different online secondary path modeling FxLMS algorithm before 2000 iterations.
0 2000 4000 6000 8000 0
Tap Length L(n) and R(n)
(a)
L(n) Proposed, variable step size R(n) Proposed, variable step size
0 2000 4000 6000 8000 10−4
Proposed, variable step size with limit NLMS[2]
Variable step size LMS[8]
Figure 5.16: Convergence comparison of tap-length and step size for pro-posed algorithm and other online secondary path modeling FxLMS algo-rithms: (a) Tap lengthM (n); (b) Step sizeµ(n).
Chapter 6 Conclusion
We propose a new ANC system using a variable tap length and step size FxLMS algorithm where a simple recursive form is obtained as well to estimate the tap length. Here, the new FxLMS algorithm is developed based on the assumption that the impulse response of the control filter in the ANC secondary path has an unsymmetric and exponential decay-ing enve- lope in order to deal with the lowpass filter in the loudspeaker system. The proposed FxLMS algorithm has a much faster convergence rate than the conventional and variable step size FxLMS algorithms with-out intensively computational cost of implementing complicated DFT or subband filters.
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Vita
Fei-Tao Chu was born in Taiwan R.O.C., in 1978. He received the B.S. degree in Computer Science and Information Engineering from Chung Hua University.
From June 2000 to January 2012, serving in AVID Electronics Corp, ELAN Microelectronics Corp. and WYS SoC Corp., he was engaged in the researches of the audio signal processing.
In 2013, he got the M.S. degree in the Institute of Communications Engineering in National Chiao Tung University, Hsin-Chu, Taiwan. His research interests include active noise cancellation and adaptive filtering.