The implementation of delayless subband active noise control algorithms

Transcription

1 he implementation of delayless subband active noise control algorithms Xiaojun Qiu a and Ningrong i Key aboratory of Modern Acoustics Nanjing University, 10093, China Guoyue Chen b Dep. of Electronics & Information Systems Akita Prefectural University, Japan Colin H. Hansen c Active Noise & Vibration Control Group School of Mechanical Engineering University of Adelaide South Australia 5005 Australia ABSRAC Wideband active noise control systems usually have hundreds of taps for control filters and the cancellation path models, which results in high computational complexity and low convergence speed. Several active noise control algorithms based on subband adaptive filtering have been developed to reduce the computational complexity and to increase the convergence speed. he subband structure is similar to the frequency domain structure but differs in the time domain processing of the subband signals. his paper discusses several issues associated with implementing the delayless subband active noise control algorithms on a DSP platform, such as the modeling of the cancellation path in subbands and the partial update of different subbands. 1 INRODUCION Active noise control (ANC) has been successfully applied in some industrial applications such as the active control of sound in headsets, industrial air ducts and propeller aircraft cabins, as well as the active control of noise radiation from large transformers. However, one of the limitations of current active control systems is the limited bandwidth over which they operate. o increase the upper limiting frequency and to extend the attenuation zone of active noise control systems, a higher system sampling rate and multiple channel systems often have to be used. he number of control filter weights can be up to thousands and the number of control channels can be up to hundreds in some cases, and this brings a significant amount of computational load and sometimes even the fastest Digital Signal Processing (DSP) chips currently available cannot meet the need [1-3]. here are several different approaches that can be taken to solve the problem. One possible approach involves the use of a decentralized system, where the entire system is divided into a number of sub-systems, each of which independently adjusts a subset of the actuators signals to minimize a subset of the sensors signal [4]. A second possible approach is to apply the modal method on all error signals and all control signals to reorganize and decouple the inputs and outputs, so that the effective number of channels being processed can be reduced [5]. A third possible approach is to implement distributed adaptive algorithms with a network control structure, where a set of linearly connected computational modules are used, with each module having an input and output and transmitting data to and receiving data from its nearest neighbours [6]. None of the preceding approaches will be considered here; rather, this paper will focus on a address: xjqiu@nju.edu.cn; b address: chen@akita-pu.ac.jp c address: chansen@mecheng.adelaide.edu.au

2 using subband adaptive filtering techniques, on the most commonly used multiple channel FXMS algorithm, to reduce the computation requirement and to increase the convergence speed of the system [7-9]. he idea of using a filtering system coupled with multi-rate sampling was proposed many years ago, but as yet, this has not been implemented in a commercially available multi channel ANC controller. he common subband structures used in adaptive echo cancellation introduce a delay in the signal path, which limits their implementation in active noise control. he delayless subband adaptive architecture for the FXMS algorithm was proposed in 1995 [10], where the signal path delay was avoided by updating the adaptive weights in subbands while carrying out the signal filtering in fullband with additional computation load resulting from transforming the subband coefficients to fullband, using the frequency stacking method. Several new schemes have been proposed to improve the performance of the subband-fullband weight transformation [11-1]. Also, the subband cancellation path modeling and the subband filtered reference signal generation methods have also been proposed for the delayless subband adaptive architecture to further reduce the computation load [13]. However, none of the above delayless subband ANC algorithms have been extended to multi channel systems which are discussed in this paper along with implementation issues. SINGE CHANNE DEAYESS SUBBAND AGORIHM Figure 1 shows the structure of the single channel delayless subband ANC system. x(n) is the reference signal from the noise source and P(z) is the primary path transfer function (structural/acoustic system) between the primary noise p(n) and x(n). he actual control signal at the position of the error sensor results from filtering the output of the controller, y(n), with the physical cancellation path transfer function C(z). he error signal e(n) is the summation of the control signal at the error sensor, the modeling signal generated by r(n) and the primary noise. he cancellation path is modeled by injecting uncorrelated random noise into the system. Figure 1: Delayless subband ANC system using the FXMS algorithm

3 he system consists of 5 parts, which are subband signal generation, subband cancellation path modeling, subband adaptive weight update, subband/fullband weight transformation and fullband control signal generation (control filtering). All these will be described below. Compared with the common fullband FXMS algorithm, the main difference is that, except for the control signal generation which is carried out in fullband to avoid delay, all other modules are carried out in subband at a decimated rate to reduce the computation load..1 Subband signal generation he m th subband signal xm ( l ) is calculated by bandpass filtering, frequency shifting, and down sampling the fullband signal x(n), K 1 k = 0 M 1 jπ mk M x () l = a e x( Dl k) m k K / M 1 j π mk M = e a x( Dl k nm) k= 0 n= 0 k+ nm (1) where l is the subband index, D is the down sampling rate, a k are the coefficients of a K point low pass prototype FIR filter and K usually is larger than the number of subbands M to avoid aliasing. he calculation complexity for all subband signal generation can be reduced by using the polyphase FF method as described in [10, 14]: [ x0() l x1() l xm 1() l ] = FF{ Fx()} l where the K point column vector l = [ x Dl x Dl x Dl K + ] () x() ( ) ( 1) 1( 1) K, the prototype filter matrix F is of size M K, and an example with M = 4 and K = 8 is shown below. a a a a5 0 0 F = (3) 0 0 a a a a7 he D new input samples are shifted into x () l and multiplied with the prototype filter matrix F. he M subband signals are obtained by applying a FF to the obtained M point product. In the algorithm shown in Figure 1, the reference signal, the error signal and the modeling signal are all decomposed into subband signals using this method. he generated subband signals are complex values, so complex valued adaptive filters are needed. However, as the fullband signal and the prototype filter matrix are real values, it is only necessary to do the calculation for the first M/+1 subbands.. Subband cancellation path modeling Figure 1 also illustrates the method used to obtain the subband cancellation path transfer functions. he modeling signal r(n) (random noise) is decomposed into M subband modeling signals {r m (l), m = 0, 1,, M-1}, which are used with the M

4 subband error signals (desire signals for modeling) {e m (l), m = 0, 1,, M-1} to directly obtain the subband cancellation path transfer functions. For example, for the m th subband, C ( l) = [ C,0( l) C,1( l)... C, 1( l)], m m m m S C C r, m = 0, 1,, M-1 (4) * m( l+ 1) = m() l + µ m() l em() l where µ is the convergence coefficient, s is the length of the complex valued FIR filter which is used to model the cancellation path for the m th subband with the vector r () [ () ( 1) ( 1) ] m l = rm l rm l rm l s + and the superscript * denotes the complex conjugate operator. he fullband cancellation path transfer function can be obtained by using the subband/fullband weight transformation method mentioned in section.4..3 Subband adaptive weight update he update equation of the fullband FXMS algorithm for the N tap control weight W( n) = w ( n) w ( n) w ( n) is, vector [ ] 0 1 N 1 W( n+ 1) = W( n) µ x ( n) e( n) (5) where x ( ) ( ) ( 1) ( 1) f n = xf n xf n xf n N + is the filtered reference signal column vector. he complex valued FXMS algorithm for the N = N / D tap subband control weight vector Wm() l = wm,0 () l wm,1 () l wm, N 1() l S is [13]: f s µ W l+ = W l x l e l * m( 1) m( ) ( ) ( ) * mf m xmf () l xmf () l, m = 0, 1,, M-1 (6) where x ( ) ( ) ( 1) ( 1) mf l = xmf l xmf l xmf l N s + is the filtered reference signal at the m th subband, s 1 mf m, k m k = 0 x () l = C () l x ( l k), m = 0, 1,, M-1 (7).4 Subband/fullband weight transformation he purpose of the subband/fullband filter weight transform is to transform a set of M subband filter weights Wm of length N s, into a corresponding fullband filter W of length N. Several methods have been developed, such as the DF stacking method, the DF- stacking method, the DF-FIR weight transform and the linear weight transform. he DF-FIR weight transform is used in this paper as it has almost the same computational complexity as the commonly used DF stacking method but with superior performance. With the DF-FIR weight transform, the fullband filter weights are obtained by using the subband filter weights as input subband signals to the synthesis filters [11-1]. Assuming that the subband signals are {x m (l), m = 0, 1,, M-1}, the fullband signal

5 x(n) can be obtained by summation of all the subband signals after up sampling, bandpass filtering and frequency shifting, M 1 K 1 xm ( n/ D), n/ D Z xn ( ) = gm( kx ) m( n k), x m( n) = m= 0 k= 0 0, n/ D Z (8) where gm( k ) is the modulated low pass prototype FIR filter. By using the polyphase FF method and the sum of the auxiliary filter state vectors [14], the following equation can be obtained, u l s () l D 1 s ( l 1) ( K D) 1 = + l G s () l 0 ( K D) 1 K 1 D 1 K 1 i IFF{ x ( l)} (8) K M m where the subband signal vector x () [ 0() 1() 1()] m l = x l x l xm l, and the prototype filter matrix G is of size M. An example of G with M = 4 and K = 8 is shown K below (assume that the same low pass prototype FIR filter as used in the subband signal generation is used here), a a a a 0 0 G (9) 1 5 = 0 0 a a a a 3 7 D fullband signals are obtained with every subband signal vector input, u [ x( ld D + 1) x( ld 1) x( ld)] =s ( l ) D 1 (10) For the subband/fullband filter weight transform, the subband input signal vectors are, x x m m x (0) = [ w () l w () l w ()] l 0,0 1,0 M 1,0 (1) = [ w () l w () l w ()] l 0,1 1,1 M 1,1 ( N 1) = [ w () l w () l w ()] l m s 0, Ns 1 1, Ns 1 M 1, Ns 1 (11) A total of N = NsD fullband filter weights can be obtained. It should be noted the magnitude of all the synthesis signals (vector) may need to be multiplied by a certain constant to have the same value as the true one..5 Fullband control filtering he fullband control signal generation is standard FIR filtering. Although some techniques exist that can be used to reduce its computation load while still maintaining its delayless property, these will not be used nor discussed here.

6 3 MUI CHANNE DEAYESS SUBBAND ANC AGORIHM 3.1 Fullband multi channel FXMS algorithm For an ANC system with one reference signal, J control sources and K error sensors, the control signals for the (n +1) th sample are obtained using, N 1 j, l (1) l= 0 y ( n + 1) = w ( n + 1) x( n l), j = 1,,..., J j where the length of the FIR control filter is N, x(n) is the reference signal at sample n, w j,l (n+1) is the l th weight of the j th control filter at sample (n+1), which is estimated by the multiple error filtered reference MS (MFXMS) algorithm outlined below, K w ( n+ 1) = w ( n) µ f ( n 1 l) e ( n) (13) jl, jl, jk k k = 1 1 f jk ( n l) = Cjk, mx( n l m). (14) m= 0 where e k (n) is the k th error signal and C jk is the tap cancellation path transfer function from the j th control source to the k th error sensor. 3. Subband multi channel FXMS algorithm For the subband MFXMS algorithm, the system consists of the same 5 parts as for the single channel case. he subband signal generation function is applied to the reference signal, K error signals and the modeling signal (usually only one actuator is fed the signal at any particular time). he cancellation path modeling, adaptive weight update and subband/fullband weight transformation are all carried out in each subband at the decimated sampling rate to reduce the computation load, and only J control signals are generated by fullband FIR filters to avoid delay. Compared with the single channel subband FXMS algorithm, which has a maximum computation complexity reduction of about 30%, the computation complexity reduction provided by the subband MFXMS algorithm can be much more. his will be shown in the following section. 3.3 Computational complexity For the fullband MFXMS algorithm described above, the computation requirement for the control filter update per sample is JK -order FIR filtering and J N-order FIR filter MS update with K error signals. he number of real multiplications is JK+JKN. he control signal generation takes JN multiplications. he memory requirement is words for the reference signal, JK words for the cancellation path coefficients, JKN words for the states of the control filters in the MS update, and there are also JN words for the coefficients of the control filters and JN words for the states of the control filters, in total of about JK(+N)+JN+ words. he computation complexity and the memory requirement for the cancellation path modeling can be estimated in the same way; however, these estimates are not included here. For the subband MFXMS algorithm, assume that the length of the prototype filter is K, the down sampling rate is D, and the number of subband is M. For each subband signal generation, ( K M)/ D real multiplications are needed per input sample, and for 1 reference and K errors, the number of multiplications is ( K M)( K + 1)/ D. For each subband, the complex filtered reference signal generation needs 4 = 4 / D real multiplications, and the complex MS update needs s

7 4Ns = 4 N / D. Altogether, there are JKM subband cancellation paths and JM subband complex MS updates. As the input signals are real, only (M/+1) complex subbands need to be processed per D samples. hus, the total number of real multiplications per fullband input sample for the filtered reference signal generation and control filter update are 4 sjkm / Di( M /+ 1)/ M JKM / D and JKMN / D respectively. For the subband to fullband weight transformation, for each control filter, it needs K M multiplications per D samples (can be per N samples), and there are J control filters, so that the actual number of multiplications per fullband sample is ( K M) J ( K M)( J + K + 1) JKM ( + N). he total is + + JN. D D D he memory requirement for the subband FXMS algorithm consists of the following parts. For subband signal generation, K words are needed for the prototype filter, and K ( K + 1) words for the reference and error signal states. For each subband, JK s words are needed for the complex cancellation path filter coefficients, and for M/ subbands (due to the symmetry), the memory requirement is JKMs words. For the subband filtered reference signal generation, the corresponding complex filter states for M/ subbands need s M words. For each subband, JKN s words are needed for the states of the control filters and JN s words for the filter coefficients in the multi channel complex MS algorithm, and for M/ subbands, the total number of words needed is JNs ( K + 1) M. For fullband control filtering, JN words are needed for the coefficients of the control filters and JN words for the states of the control filters, a total of about K ( K + 1) + JK M + M + JN ( K + 1) M + JN words. s s s able 1: Complexity Comparison of the Multichannel FXMS Algorithms Operation Fullband MFXMS Subband MFXMS Subband signal generation None ( K M)( K + 1)/ D Filtered x signal generation JK JKM / D Control filter update JKN JKMN / D Subband/fullband transform None ( K M) J / D Control signal generation JN JN otal JK(+N)+JN ( K M)( J + K + 1)/ D + JKM ( + N)/ D + JN able 1 summarizes the computation complexity for both the fullband and subband MFXMS algorithms. able shows an example of the computation complexity comparison between the fullband and subband MFXMS algorithms for a system with 16 control outputs and 16 error inputs, length of the control filter and cancellation path filter, N = = 4096, number of subbands, M, down sampling rate, D = M/, and length of the prototype filter, K = 4M. It can be seen that with the increase of subband number, the computation load of the subband MFXMS algorithm can be reduced to 6% of that of the fullband. It can also be seen that when M is larger than a particular value, further increasing M cannot reduce the computation load of the subband algorithm significantly, as the main contributor becomes the control signal generation

8 part at fullband. For this ANC system, the memory requirement for the subband MFXMS algorithm is about twice that of the fullband for M from 4 to 56. able : Complexity Comparison of the Multichannel FXMS Algorithms for a 16 channel controller with 4096 tap control filters and cancellation path filers (Unit: K multiplications) M = Operation Full Sub Sub Sub Sub Sub Sub Sub band band band band band band band band Subband signal generation Filtered x signal generation Control filter update Subband/fullband transform Control signal generation otal (K multiplications) Partial subband update algorithm In some ANC applications, the disturbance is narrow band noise. For the subband MFXMS algorithm under these circumstances, only some subbands which contain the noise need to be processed. his is called the partial subband update algorithm. For the above example, if the fullband sampling rate is 16KHz with 18 subbands being used, the subband band width is about 15Hz. If the bandwidth of the narrow band disturbance is less than 15Hz and is located in one subband, then only the control filter weights in this subband need to be updated, and this can significantly reduce the computation complexity and memory requirement. For this example, referring to able, for M = 18, the number of multiplications for the filtered reference signal generation and control filter update can be reduce to about 1/64 of the values listed in able, which means that the total computation load of the partial subband MFXMS algorithm can be reduced to about 70/16 = 3.% of that of fullband with 18 subbands. he memory requirement is also reduced to 8% of that of fullband. 3.5 Discussions and implementation Applying subband techniques in active noise control has two advantages: faster convergence is possible because the dynamic range is greatly reduced in each subband and each subband can have its own convergence coefficients; also, the computation complexity for the control filter update is reduced by approximately the number of the subbands, since both the number of taps and weight update rate can be decimated in each subband. he first advantage has been confirmed by other researchers [10, 13], and is not discussed in the paper. For the second advantage, as has been shown in Section 3.4, if the disturbance is a narrow band noise, both the computation complexity and the memory requirement can be reduced. he disadvantage of the subband MFXMS algorithm is that although there is no delay in the control signal generation path, there is a delay at the control filter update path. Unlike the fullband MFXMS algorithm which updates every sample, the subband MFXMS algorithm updates every D samples. In some applications where the primary or secondary path changes rapidly, the subband MFXMS may exhibit inferior performance. One problem with the subband FXMS algorithm is that although numerical simulations have shown that substituting the filtered reference signal generation from fullband real value filtering to subband complex value filtering can still achieve good

9 noise attenuation, a formal derivation has not been given [13]. In the original delayless subband FXMS algorithm proposed by Morgan and hi, the filtered reference signal is generated in the fullband by real value cancellation path filtering [10]. here are also some other algorithms that can be used to reduce the computation complexity of the MFXMS algorithm [3]; for example, the periodic FXMS (FPMS) algorithm and the periodic block FXMS (FPBMS). By comparing these with the subband MFXMS, it is found that the latter algorithm with a large number of subbands has more computation complexity reduction and processing flexibility in addition to the potential faster convergence speed. he subband MFXMS algorithm is especially suited to the third generation controller architecture proposed in [1], where each I/O module contains a low cost DSP and a small delay A/D and D/A converter sampling at a high frequency. he DSP on each I/O board provides ample processing power for the I/O management tasks and multi-rate filtering as well as transducer failure and signal overload management. he central processor board contains a cluster multiprocessing system, which has multiple DSPs connected to the cluster bus via the processors external port and supports interprocessor access of on-chip memory-mapped registers and shared global memory. As an example, the DSP used in each I/O module can be ADSP-16 (one of third generation of SHARC Processors), which runs at 00MHz and has Mbits on-chip SRAM. If the system sampling rate is 16KHz, the interval between two continuous samples is 1.5K clock cycles or 6.5µs. he processing time for a 4096 tap FIR filter for an ADSP-16 is about 10.4 µs, which needs to be completed within the interval. he processing time for the subband signal generation and subband/fullband transform is less than 0µs, and this only needs to be carried out every 64 samples. At the decimated sampling rate (50Hz), all the subband signals are transferred to the central processor board via dual port RAM, which contains a cluster multiprocessing system and processes the signals at the decimated rate. In a typical cluster of SHARC processors, up to six processors and a host can arbitrate for the bus with the on-chip bus arbitration logic, which allows the DSPs to have a very fast node-to-node data transfer rate, allowing a simple, efficient software communication model to be implemented, especially when sharing a large amount of cancellation path model data. However, a bottleneck exists within the cluster because only two DSPs can communicate over the shared bus during each cycle, and other DSPs are held off until the bus is released. hus, using more DSPs on one board does not significantly improve the overall speed of the system due to bottlenecks associated with using external SDRAM. he above subband MFXMS algorithm and third generation controller architecture can be easily expanded to hundreds of channels or can be configured to a specific number of channels as needed. 4 CONCUSIONS Single channel ANC systems often use subband techniques to overcome the difficulties of high computational complexity and low convergence speed associated with a wideband control filter containing thousands of taps. his paper has extended the use of subband techniques to multi channel systems. For a single channel delayless subband FXMS algorithm, the maximum computation complexity reduction is about 30%, but the computation complexity reduction provided by the multi channel algorithm is much more. For a system with 16 control outputs and 16 error inputs and 4096 tap control filters, the computation load of the subband MFXMS algorithm can

10 be reduced to 6% of that of fullband. For a narrow band ANC system with a sampling rate of 16KHz, the total computation load of the partial subband MFXMS algorithm can be reduced to about 3.% of that of the fullband with 18 subbands, and the memory requirement is also reduced to 8% of that of fullband. he delayless subband MFXMS algorithm is especially suited to the third generation controller architecture, and the implementation of the algorithm in such a control architecture was also discussed. 5 ACKNOWEDGEMENS Projects and supported by NSFC, and the projects are also partially sponsored by SRF for ROCS, SEM and SRFDP. 6 REFERENCES [1] C. H Hansen, Current and future industrial applications of active noise control, Proceeding of Active 004, Williamsburg, Virginia, USA (004). [] S. D. Snyder, Microprocessors for active control: Bigger is not always enough, Noise Control Engineering Journal, 49(1), 1-9 (001). [3] X. Qiu, and C. H. Hansen, A comparison of adaptive feedforward control algorithms for multi-channel active noise control, Proceedings of 8th Western Pacific Acoustics Conference, Melbourne, Australia (003). [4] W. P. Engels, O. N. Baumann, S. J. Elliott and R. Fraanje, Centralized and decentralized control of structural vibration and sound radiation, Journal of the Acoustical Society of America 119(3), , (006). [5] S. D. Snyder, N. Burgan and N. anaka, Acoustic based modal filtering approach, Mechanical Systems and Signal Processing, 16(1), (00). [6] D. V. V. Barry,. Olivier, P. M. Vijay and S. J. Daniel, Distributed adaptive algorithms for large dimensional MIMO systems, IEEE rans. on Signal Processing, 48(4), (000). [7] S. M. Kuo and D. R. Morgan, Active Noise Control Systems - Algorithms and DSP Implementations (John Wiley & Son Inc., 1996). [8] C. H. Hansen and S. D. Snyder, Active Control of Noise and Vibration (E&FN SPON, 1997). [9] S. J. Elliott, Signal Processing for Active Control (Academic Press, ondon, 001). [10] D. R. Morgan and J. C. hi, A delayless subband adaptive filter architecture, IEEE rans. Signal Processing, 43, (1995). [11] J. Huo, S. Nordholm, and Z. Zang, New weight transform schemes for delayless subband adaptive filtering, in Proceedings of IEEE Global elecommunications Conference, pp , (001). [1]. arson, J. M. de Haan and I. Claesson, A new subband weight transform for delayless subband adaptive filtering structures, in IEEE Digital Signal Processing Workshop 00, pp (00). [13] S. J. Park, J. H. Yun and Y. C. Park, A delayless subband active noise control system for wideband noise control, IEEE rans. Speech and Audio Processing, 9, (001). [14] S.. Gay and J. Benesty, Acoustic signal processing for telecommunication, (Kluwer Academic Publishers, 000).