Multiple Error Filtered-X LMS algorithm - iBrarian

design of the filters. ... generated by the secondary loudspeakers S1 and S2. ... microphone and antinoise loudspeakers are represented by the Noise filter 1 and ...
311KB taille 12 téléchargements 409 vues
Multiple Error Filtered-X LMS algorithm Miroslav Sedlák, Branislav Bača Department of Radioelectronics, FEEIT SUT Bratislava, Slovak Republic [email protected], [email protected] This contribution bring new modified algorithm in Active Noise Control. MEFX LMS algorithm is adaptation to need for ANC and simulated in program Matlab - simulink. Further, this method is modified and used for multichannel ANC. One of methods to obtaining the optimum LMS solution, is to use an iterative algorithm such as the Multiple Error Least Mean Square (MELMS) algorithm. Such an adaptive approach is preferred over the direct calculation of wopt(n) since it offers in-situ design of the filters. It also enables a convenient method to readjust the filters whenever a change occurs in the electro-acoustic transfer functions. The MELMS algorithm employs the steepest descent approach to search for the minimum of the performance index [1]. This is achieved by successively updating the filters‘ coefficients by an amount proportional to the negative of the gradient ∇(n) w (n + 1) = w (n) + µ(−∇(n)) , (1) where µ is the step size that controls the convergence speed and the final misadjustment . An approximation often used in such iterative LMS algorithms is to update the vector w using ~ ~ the instantaneous value of the gradient ∇(n) instead of its expected value ∇(n) = E{∇(n)} , leading to the well known LMS algorithm. After rewriting gradient into ∇(n) = 2E{−x Tf (n)e(n)} , the update equation for the MELMS algorithm is given by replacing ∇(n) in (1) by its instantaneous value w (n + 1) = w (n) + 2 µx Tf (n)e(n) ,

(2)

Fig. 1 Block diagram of the multiple error filtered-x LMS algorithm This update algorithm is often referred to as the Multiple Error Filtered-X (MEFX) algorithm. Implementation of (2) requires calculating the matrix xf(n), which implies measuring all the electro-acoustical transfer functions cml and filtering each input signal through all ML transfer functions to construct the KML elements of xf(n). This is shown in Fig. 1, where the measured matrix of electro-acoustic transfer functions is represented by the ˆ to distinguish it from the physical one represented by the block C. block C The detailed implementation of the MEFX algorithm is further explained by unpacking the composite vector w(n) in (2) into its individual filters {wlk(n): l = 1,2, … L; k = 1,2, … K} M

giving w lk (n + 1) = w lk (n) + 2 µ ∑ c m (n)x f lmk (n) , m =1

(3)

where xflmk(n) is the result of filtering the kth input through the measured electro-acoustic th ˆ microphone and lth loudspeaker. The detailed impulse response C ml between the m implementation of (3) is visually illustrated in Fig. 2. for a [K x M x L = 2 x 2 x 2] active

noise control system. In this system, it is desired to reduce the sound field due to the primary sources P1 and P2 at two receiver points (microphones) R1 and R2 using an anti-sound field generated by the secondary loudspeakers S1 and S2. Using frequency domain representations, the inputs to the secondary sources Y1(ω) and Y2(ω) are controlled by four adaptive filters {Wlk(ω) : l = 1, 2; k = 1, 2} to achieve the above cancellation task. The desired response is the sound field generated by the primary sources P1 and P2 at microphones’ positions: D(ω) = [D1(ω)D2(ω)] = H(ω)X(ω), where H(ω) is the matrix of electro-acoustic transfer functions between the two microphones and primary sources and X(ω) = [X1(ω)X2(ω)]. The microphones’ outputs are, therefore, the sum of the sound fields ˆ (ω) , due to the primary and secondary sources E(ω) = [E1(ω)E2(ω)] = D(ω) + D ˆ (ω) = [D ˆ (ω)D ˆ (ω)] = C(ω)Y(ω) is the (unmeasurable) sound field due to the where D 1

2

secondary sources alone at the microphones. At each time sample, the update algorithm adjusts the coefficients of the adaptive filters to minimise the error signals measured by the microphones using the MEFX algorithm (3). Four update blocks are needed, one for each adaptive filter. Those are indicated by the MEFX boxes in the Fig. 2. According to (3), each of the update blocks requires two filtered input signals, in total KML = 8 filtered input signals are needed. The filtered input signals are calculated by filtering each of the two input signals through the four measured impulse ˆ (ω) : m = 1, 2; l = 1, 2}, that are estimates and the microphones represented in responses { C ml Fig. 2. By the matrix C.

Fig. 2 Block diagram of the MEFX ANC system

Fig. 3 Secondary path modelling scheme

After successful convergence, the sound waves generated by S1 and S2 in R1 and R2 equal in magnitude and opposite in phase to that generated by P1 and P2, and reduction in the net sound field at R1 and R2 results. This may be expressed mathematically as H(ω)X(ω) = - C(ω)W(ω)X(ω), and the solution to the matrix of control filters is given by (4) W(ω) = - C-1(ω)H(ω) . Described algorithm was simulated in the Matlab – Simulink program. Simulation consist of two parts. First one is secondary path modelling, illustrated in Fig. 3, where are electro-acoustic transfer functions, represented by Noise Filter 11 – 22 blocks, measured by adaptive identification technique. Modelling process of the secondary paths is shown in Fig. 4, where are illustrated curves of the difference between signals from unknown secondary paths and adaptive filters, that secondary paths identify. Coefficients of individual filters after identification are illustrated in Fig. 5.

Fig. 4 Secondary path modelling error signal curves

Fig. 5 Multichannel ANC system scheme

Second part is simulation of the multichannel ANC system. Realised scheme of Fig. 2 in Matlab – Simulink program is displayed in Fig. 5. Transfer functions between input microphone and antinoise loudspeakers are represented by the Noise filter 1 and 2 blocks, secondary paths are represented by the Noise filter 11 – 22 blocks. Each of Adaptive filter blocks consist of the adaptive filter with LMS algorithm and two filters for input signals filtering to eliminate secondary path influence, that have same coefficients as filters from secondary path modelling. Signal curves are shown in Fig. 6. First one is signal generated by noise source, next two are antinoise signals. Last two are error signals.

Fig. 6 Signal curves of ANC system adaptation

References [1] [2] [3]

Garas, J.: Adaptive 3D sound systems, Technische Universiteit Eindhoven, 1999 Scott, C. D.: Fast Implementations of the Filtered-X LMS Algorithms for Multichannel Active Noise Control, Salt Lake City, 1999 Scott, C. D.: Reducing the Computational and Memory Requirements of Multichannel Filtered-X LMS Adaptive Controller, 2000