We next investigate the ANC performance of the proposed WD-FxLMS algorithm in multi-frequency noise field. The system is designed at the frequency of 1 kHz.
0 1 2 3 4 5 6 0 1 2 3 4 5 6 x (m) y (m) −20 −15 −10 −5 0 5 10 15 20 (a) 0 1 2 3 4 5 6 0 1 2 3 4 5 6 x (m) y (m) −30 −25 −20 −15 −10 −5 0 5 10 15 20 (b) 0 1 2 3 4 5 6 0 1 2 3 4 5 6 x (m) y (m) −30 −25 −20 −15 −10 −5 0 5 10 15 20 (c)
Figure 4.9: Results of ANC in reverberant environment. The inner and outer arrays are microphone array and loudspeaker array, respectively. (a) The energy of the initial noise field. The residual energy after 30 iterations of (b) WD-FxLMS, and
0 5 10 15 20 25 30 Number of iterations -80 -70 -60 -50 -40 -30 -20 -10 0 10 Enegy (dB) WD-FxLMS on the boundry MP-FxLMS on the boundry WD-FxLMS inside the region MP-FxLMS inside the region
Figure 4.10: Comparison of convergence performance for noise cancellation using WD-FxLMS and MP-FxLMS algorithm in reverberant environment.
The noise field within a radius of 1 m is due to 10 noise sources uniformly distributed on a circle 2.5 m from the origin and at angles [0 : 36 : 324]◦.
Figure 4.11 shows that for all frequencies below 1 kHz, using the WD-FxLMS algorithm, the noise can be reduced significantly within the desired quiet zone. The average noise reduction is around 80 dB and 50 dB in the free-field and reverberant environment, respectively. Notice that the performance is significantly degraded at 600 Hz and 900 Hz. This is due to the fact that at these particular frequencies, the Bessel functions are close to zero (i.e., Jm(kr)≈0) and the coefficient error is amplified. This problem can be avoided by placing two closely spaced microphone arrays [76].
4.6
Summary and Contributions
In this chapter, we presented adaptive wave-domain methods to achieve noise can- cellation over a spatial region using feedback control structure. We proposed a
100 200 300 400 500 600 700 800 900 1000 30 40 50 60 70 80 90 100 110 Noise Reduction (dB) Frequency (Hz) reduction in free−field
reduction in reverberant environment
Figure 4.11: Noise reduction using WD-FxLMS algorithm after 30 iterations in multi-frequency noise field.
feedback control system using a circular microphone array to measure the residual signals and a circular loudspeaker array to produce anti-noise signals. We rep- resented all the variables of the noise field in the wave domain and formulated the spatial ANC problem in terms of the wave-domain coefficients. We proposed the WD-FxLMS algorithm, and it was evaluated in both free-field and reverber- ant environments and compared with the conventional multi-point ANC algorithm. Simulation results showed that the proposed WD-FxLMS algorithm achieved sig- nificant noise reduction over the entire design region with fast convergence speed. We also introduced the sparse constraint in the WD-FxLMS which can reduce the total energy of the loudspeaker weights. There existed a trade off between noise reduction and the energy of the loudspeaker weights.
The major contributions made in this chapter are:
• We proposed a feedback control system using a circular microphone array
and a circular loudspeaker array, and formulated the spatial ANC problem in terms of the wave-domain coefficients.
• We proposed a wave-domain FxLMS algorithm (WD-FxLMS) for active noise control over spatial region in general noise field.
• Weproposed a `1-norm constrained wave-domain FxLMS algorithm (`1-WD-
FxLMS) for active noise control over a spatial region in a directional sparse noise field.
4.7
Related Publications
This chapter’s work has been published in the following conference proceedings: [96]: J. Zhang, W. Zhang, and T. D. Abhayapala, “Noise cancellation over
spatial regions using adaptive wave domain processing”, in IEEE Workshop on
Applications of Signal Processing to Audio and Acoustics (WASPAA) 2015, New Paltz, NY, USA, Oct. 2015, pp. 1-5
[113]: J. Zhang, T.D. Abhayapala, P. N. Samarasinghe, W. Zhang and S. Jiang,
“Sparse complex FxLMS for active noise cancellation over spatial regions”, inIEEE
International Conference on Acoustics, Speech and Signal Processing (ICASSP)
Wave Domain ANC: Different
Cost Functions and Adaptations
Overview: In this chapter, we investigate wave-domain spatial ac- tive noise control in terms of different cost functions and different update variables. We propose normalized wave-domain active noise control algorithms based on two minimization problems, (i) minimiz- ing the wave-domain residual signal coefficients and (ii) minimizing the acoustic potential energy over the region. We derive the update equations with respect to two variables, (a) the loudspeaker driving sig- nals and (b) wave-domain secondary source coefficients. Simulation results demonstrate the effectiveness of the four proposed algorithms, more specifically the convergence speed, energy of the loudspeaker driv- ing signals, and the noise cancellation performance in terms of the noise reduction level and acoustic potential energy reduction level over the entire spatial region. We also investigate the spatial active noise control performance with limited secondary sources in the simulations.
5.1
Introduction
As discussed in Chapter 3, the multi-point control systems minimize the sum of the squared pressures, which is equal to minimizing the potential energy density
at the microphone locations. Although these approaches lead to significant noise reduction at the target points, the consistency over a continuous spatial region is low.
In Chapter 4, we developed the wave domain ANC structure and proposed
FxLMS algorithm and the `1-norm constrained FxLMS algorithm in wave do-
main [96, 113]. We used cylindrical/spherical harmonics as basis functions and their respective coefficients to represent the noise field and secondary field over the desired spatial region. Instead of minimizing the sum of the squared error signals [64], wave-domain ANC tends to minimize the harmonic coefficients, which in turn control the entire spatial region directly. The simulation results [96, 113] showed that wave-domain ANC achieved significant noise cancellation over the entire region of interest with faster convergence speeds.
In this chapter, we further investigate the wave domain ANC strategy. As discussed in Chapter 2, ANC using multi-point structure can be solved by (i) min- imizing the squared residual signals and (ii) minimizing the energy densities. In this chapter, we investigate the spatial ANC problem by solving the minimization of (i) squared wave domain coefficients and (ii) acoustic potential energy. For each minimization problem, we update two different variables: (i) driving signals, and (ii) wave domain coefficients. We propose normalized version of the wave domain algorithms, and evaluate the proposed algorithms in terms of noise reduction over the region, acoustic potential energy reduction over the region and convergence performance. Simulations are conducted in both free field and reverberant envi- ronment, and in different numbers of loudspeaker setup.
The rest of the chapter is organized as follows. In Section 5.2, we refer back to Section 4.2, and formulate the spatial noise cancellation problem and the ANC system in the wave domain. The wave-domain multichannel ANC algorithms min- imizing the squared residual signal coeffients are proposed in Section 5.3, and the wave-domain multichannel ANC algorithms minimizing the acoustic potential en- ergy are proposed in Section 5.4. We demonstrate the simulation results to compare the ANC performance of the proposed wave-domain methods and the conventional multi-point method in Section 5.5, and draw some conclusions in Section 5.6.