Effect of the Audio Amplifier s Distortion on Feedforward Active Noise Control

Transcription

1 Effect of the Audio Amplifier s Distortion on Feedforward Active Noise Control Dongyuan Shi, Chuang Shi, and Woon-Seng Gan School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China dshi003@e.ntu.edu.sg shichuang@uestc.edu.cn ewsgan@ntu.edu.sg Abstract Active noise control (ANC) is an effective method for reducing low-frequency acoustic noise. An anti-noise wave is transmitted by the secondary source to destructively interfere with the noise wave. A quiet zone is thus formed around the error microphone, which provides the error signal in the adaptation process of an ANC controller. However, the real-world performance of an ANC system is often subjected to the distortion incurred in its electronic components. This distortion has been conventionally treated as a trivial part of the secondary path model. When the distortion is severe but the secondary path model is still forced to be linear, the nonlinearity of the true secondary path is no longer negligible and causes degradation in noise reduction performance or even divergence of the ANC controller. This paper revisits the causes of the amplitude distortion in the audio amplifier and how it influences the convergence of the filtered-x least mean square (FxLMS) algorithm in a feedforward ANC system for tonal noise cancellation. I. INTRODUCTION In modern urban living, traffic, construction and machinery noises are becoming more intrusive due to the proximity of residential buildings and noise sources. Exposure to excessive noise can lead to sleeping disturbance, hypertension and vascular diseases. Passive solutions are known effective in canceling the high-frequency noise due to their ability to absorb and reflect the noise wave, but become too expensive and impractical for mitigating the low-frequency noise, whose upper frequency bound is typically below 2 khz. Paul Leug patented the idea of active noise control (ANC) in 1936 to combat the low-frequency air-flow noise in a duct [1], [2], [3], [4]. Since then, many types of ANC systems have been developed with the open-loop, feedforward and feedback structures. The filtered-x least mean square (FxLMS) algorithm is the most effective and computationally efficient adaptive algorithm that is being used in many ANC systems [5]. The real-time digital processing platforms for ANC systems include microcontrollers [6], digital signal processors (DSP) [7] and field programmable gate arrays (FPGA) [8]. Over the past thirty years, ANC systems have been successfully deployed in applications such as canceling noise in ventilation ducts [9], [10], home windows [11], [12], [13], [14], [15], MRI equipment [16], [17], headsets and ear protectors [18], [19], [20]. Among those applications, the feedforward ANC structure is usually favorable when dealing with broadband noise, not to mention its robustness under different noise environments [21]. The real-time implementation of a feedforward ANC system encounters several practical constraints: (i) the electronic delay must be shorter than the acoustic delay from the reference microphone to the secondary source to ensure the overall system s causality [22]; (ii) imperfect modeling of the secondary path leads to non-optimized noise reduction performance; (iii) excessive analog amplification of the secondary source causes distortion that may break the controller s stability. However, the last constraint has not been well studied. There are only few papers about the effect of the nonlinear secondary path on the performance of an ANC system [23], [24]. Hence, there is a lack of explanation on how over-amplification can break down an ANC controller. When the output power of the audio amplifier exceeds its rated output power or greater than the rated power of the secondary source, termed as saturation and mismatch respectively, the anti-noise wave generated by the ANC system is distorted and results in additional highfrequency annoyance and sometimes divergence of the ANC controller. This paper revisits the distortion incurred in the audio amplifier and elaborates how this distortion degrades the noise reduction performance and breaks the stability of a feedforward ANC system. II. AMPLIFIER DISTORTION The audio amplifier, which drives the secondary source to generate the anti-noise wave, is a very critical but often overlooked electronic component in an ANC system. Class A and Class AB amplifiers own low distortion and short latency but also low power efficiency. In contrast, Class D amplifiers have high power efficiency but long latency. Despite of types of audio amplifiers, they may have the same three types of distortion, which are the amplitude distortion, frequency distortion and phase distortion. Among them, the amplitude distortion plays the most important role in an ANC system. There are two causes of the amplitude distortion. Firstly, the inappropriate biasing level applied to the audio amplifier causes only a portion of the input being amplified and the rest being clipped. This usually happens to the discrete component amplifier. Using the integrated audio amplifier can avoid this problem. Secondly, the output level of the audio amplifier

2 Proceedings of APSIPA Annual Summit and Conference 2017 Fig. 1. The curve of output amplitude vs. input amplitude of an audio amplifier. Fig. 2. The clipped output signal of an audio amplifier. is limited by the supply voltage. Therefore, large input to the audio amplifier cannot be faithfully amplified. Figure 1 shows the input-output curve of an audio amplifier, where Vthr indicates the supply voltage and the voltage gain of this audio amplifier is set to 1. The input signal is a sinusoidal signal, given by y(t) = C sin(ω0 t), (1) where ω0 and C denote the frequency and amplitude respectively. When C is greater than Vthr, the output of the audio amplifier is clipped at the top and bottom as shown in Fig. 2. This clipped waveform can be mathematically expressed as ω0 t + 2k [ 2 t, 2 + t] Vthr, (2) y(t) = C sin(ω0 t) others Vthr ω0 t + 2k [ 2 t, 2 + t] where t = can be written as 2 arcsin( Vthr C ) and k Z. Its Fourier series Fig. 3. Single-frequency adaptive noise canceler. thus given by x0 (n) = sin(ω0 n + θr ) x1 (n) = cos(ω0 n + θr ), (5) (6) 2 t + + sin(2 t) C sin(ω0 t). (3) 4 cos( t) sin3 ( t) + C sin(3ω0 t) Equation (3) tells that the clipped waveform consists of the odd-order harmonics. Furthermore, when an audio amplifier exhibits the amplitude distortion, it can be regarded as a memoryless system, because the output y(n) relies only on the current input y(n), where y(n) and y(n) represent y(t) and y(t) in the digital domain. where θr is the initial phase of the reference signals. Based on the FxLMS algorithm, the coefficient updating equation of the single-frequency feedforward ANC controller is written as III. I NFLUENCE OF THE AMPLITUDE DISTORTION ON ANC where y(n) is the output signal of the control filter. y(t) = PERFORMANCE Figure 3 displays the block diagram of a single-frequency feedforward ANC controller [25]. H s (z) is the estimate of the secondary path Hs (z). To simplify the analysis, it is assumed that H s (z) = Hs (z) = As e jθs. The primary disturbance signal d(n) is defined as d(n) = D sin(ω0 n), (4) where D is the amplitude of the disturbance signal and ω0 is the noise frequency. The two orthogonal reference signals are @2017 APSIPA w0 (n + 1) = w0 (n) + µe(n)[x0 (n) hs (n)] (7) w1 (n + 1) = w1 (n) + µe(n)[x1 (n) hs (n)], (8) where µ is the step size and e(n) is the error signal [25]. When the coefficients of the controller reach the optimal solution, the anti-noise y (n) should equal to the disturbance signal d(n), i.e. y (n) = y(n) hs (n) = d(n), (9) A. Case A: Moderate Overdriving When the amplitude of the disturbance signal exceeds the threshold of the audio amplifier, the output signal y(n) is distorted and expressed by (3). As the over-driving is moderate, the fundamental and third-order harmonics are dominating in the distorted output, which are written as y f irst (n) = AC(n) sin(ω0 n + θ0 ) (10) y third (n) = BC(n) sin(3ω0 n + θ1 ), (11) APSIPA ASC 2017

3 when the controller s output y(n) is actually given by y(n) = C(n) sin(ω 0 n + θ 0 ). (12) The corresponding harmonics in the anti-noise are written as y first(n) = A s AC(n) sin(ω 0 n + θ 0 θ s ) (13) y third(n) = A s BC(n) sin(3ω 0 n + θ 1 θ s ). (14) It is noteworthy that A changes slowly when the amplitude of y(n) is only slightly greater than the amplifier s threshold V thr. The error signal is further written as e(n) = d(n) y first(n) y third(n)... (15) where the... denotes the higher-order harmonics in y (n). The modified coefficient updating equation is thus derived as w 0(n + 1) = w 0(n) + µe(n)x 0(n) (16) w 1(n + 1) = w 1(n) + µe(n)x 1(n), (17) where w 0(n) and w 1(n) are the coefficients of the controller when the audio amplifier generates amplitude distortion. Because x 0(n) and x 1(n) are orthogonal to the harmonics in the anti-noise, after calculating the expectation of (16) and (17), the coefficient updating equation becomes w 0(n + 1) = w 0(n) + µe{[d(n) A C(n) y (n)]x 0(n)} (18) w 1(n + 1) = w 1(n) + µe{[d(n) A C(n) y (n)]x 1(n)} (19) where C(n) is independent with the reference signal x 0(n) and x 1(n). Therefore, the optimal coefficients of the controller with the amplitude distortion are derived as w o 0 = 1 A wo 0 (20) w o 1 = 1 A wo 1, (21) where w o 0 and w o 1 are the optimal coefficients of the controller without the amplitude distortion. It is hence confirmed that the FxLMS algorithm has an optimal solution under this situation. The disturbance d(n) is finally canceled by the fundamental harmonic y first (n) but the third- and high-order harmonic are introduced as the high-frequency annoyance. The error transfer function of the single-frequency ANC controller can be written as G(z) = Y (z) E(z) = 2µ[zA s cos(ω 0 + θ r θ s ) 1] z 2 2zA s cos(ω 0 + θ r θ s ) + 1, (22) where Y (z) and E(z) are the z-domain output and error signals respectively [25]. Based on the simplified structure shown in Fig. 4, the transfer function of the third-order harmonic y third (n) is written as H e (z) = 1 1 G(z)H s (z), (23) Fig. 4. The equivalent diagram of the single-frequency adaptive noise canceler for the third-order harmonic. which is a positive feedback loop and hence easily becomes unstable. The amplitude of y third (n) will increase until the gradient of the FxLMS algorithm decreases to 0. In the moderate over-driving case, the amplitude of the disturbance is only slightly greater than the threshold V thr. Thus, A = D C opt, where C opt is the amplitude of y(n) when the control filter reaches its optimal solution. From (3), the third-order harmonic is finally derived as y third (n) 2A sv thr (1 ξ 2 ) 3/2 3(ξ 1 ξ 2 + arcsin ξ) sin(3ω 0n), (24) where ξ = V thr D. Using to the same procedure, the other harmonics can also be found to be constant when the control filter reaches the optimal solution. B. Case B: Severe Overdriving When the amplitude of the disturbance is much greater than threshold of the audio amplifier, the distorted output approximates a bipolar rectangular wave. In this case, we assume that the y(n) has the initial phase of θ s and can be expressed as { V thr ω 0 n + 2k ( θ s, θ s ] y(n) (25) V thr ω 0 n + 2k ( θ s, 2 θ s ], where k Z. The Fourier series of the anti-noise is written as y (n) 4A sv thr sin(ω 0 n) + 4A sv thr 3 sin(3ω 0 n).... (26) If the amplitude of the disturbance is greater than the amplitude of the fundamental harmonic, i.e. D > 4A sv thr, (27) the ANC controller diverges and fails to cancel the disturbance. This is because that the power of the error signal leads to negative gradient in the FxLMS algorithm, i.e. E{ e 2 } = 1 2 (4A sv thr D) < 0. (28) In summary, the single-frequency ANC controller is stable and does not generate high-frequency annoyance, when the disturbance s amplitude D [0, V thr ]. The high-frequency annoyance is incurred, when there is moderate overdriving, i.e. D (V thr, 4AsV thr ]. The ANC controller becomes unstable, when D ( 4A sv thr, + ).

4 Proceedings of APSIPA Annual Summit and Conference 2017 whereby the disturbance is sufficiently canceled but high-order harmonics are observed in the attenuated noise. Subsequently, D is adjusted from 0.74 to The curves of the coefficient W 1(n) are drawn in Fig. 6 for different values of D. When D is slightly less than the threshold, the coefficient W 1 is close the optimal solution; when the D [0.75, ), the controller converges to a different optimal solution; when D > , the FxLMS algorithm fails to converge. Therefore, the simulation results obey well with our analysis in the previous section. V. C ONCLUSIONS Fig. 5. Spectrum of the disturbance and attenuated noise at 500 Hz in the moderate over-driving case. The amplitude distortion is a common issue of the audio amplifier, but its effect is seldom studied in ANC systems. This paper takes the single-frequency feedforward ANC controller as an example and elaborates the generation of the additional high-frequency annoyance due to the moderate overdriving and the slightly distorted anti-noise signal. As long as the amplitude of the disturbance is less than the maximum amplitude of the fundamental harmonic in the distorted output of the secondary source, the FxLMS algorithm can still converge. Otherwise, the mild overdriving can surely lead to the divergence of the FxLMS algorithm. VI. ACKNOWLEDGMENT This material is based on research work jointly supported by the Haliburton Singapore-NTU Fund (Project Code M ) and the Fundamental Research Funds for the Central Universities (Project No. A ). Fig. 6. The curves of weight w1 (n) for disturbances with different power. IV. S IMULATION In this simulation, the output function of the audio amplifier is configured as 0.75 y(n) < 0.75 y(n) = y(n), (29) others 0.75 y(n) > 0.75 where y(n) and y(n) are the input and output signal of the audio amplifier. The primary noise is given by x(n) = D sin(0.1n), (30) where D is the amplitude of the d(n). The disturbance signal is generated by a simple primary path as d(n) = 0.01x(n 1) x(n 2) + v(n), (31) where v(n) is a white Gaussian noise whose variance is The secondary path is a single delay which has a unit gain (As = 1). Moreover, the sampling frequency is set to 10 khz. Using (29) and Vthr = 0.75, the threshold of the disturbance is calculated as (4As Vthr )/ = When D equals to 0.94, the spectrum of the attenuated noise is shown in Fig. 5, @2017 APSIPA R EFERENCES [1] S. M. Kuo and D. R. Morgan, Active noise control: a tutorial review, Proceedings of the IEEE, vol. 87, no. 6, pp , [2] P. A. Nelson and S. J. Elliott, Active control of sound. Academic press, [3] C. N. Hansen, Understanding active noise cancellation. CRC Press, [4] Y. Kajikawa, W. S. Gan, and S. M. Kuo, Recent applications and challenges on active noise control, in th International Symposium on Image and Signal Processing and Analysis (ISPA), Sept 2013, pp [5] D. R. Morgan, History, applications, and subsequent development of the fxlms algorithm [dsp history], IEEE Signal Processing Magazine, vol. 30, no. 3, pp , May [6] K. K. Shyu, C. Y. Ho, and C. Y. Chang, A study on using microcontroller to design active noise control systems, in 2014 IEEE Asia Pacific Conference on Circuits and Systems (APCCAS), Nov 2014, pp [7] S. M. Kuo and D. Morgan, Active noise control systems: algorithms and DSP implementations. John Wiley & Sons, Inc., [8] D. Shi, C. Shi, and W. S. Gan, A systolic fxlms structure for implementation of feedforward active noise control on fpga, in 2016 AsiaPacific Signal and Information Processing Association Annual Summit and Conference (APSIPA), Dec 2016, pp [9] E. Esmailzadeh, A. Alasty, and A. Ohadi, Hybrid active noise control of a one-dimensional acoustic duct, Journal of Vibration and Acoustics, vol. 124, no. 1, pp , [10] P. Minogue, N. Rankin, and J. Ryan, Adaptively canceling server fan noise, Analog Dialogue, vol. 2, pp , [11] B. Lam, S. Elliott, J. Cheer, and W. S. Gan, The physical limits of active noise control of open windows, in 12th Western Pacific Acoustics Conference, December [12] T. Murao and M. Nishimura, Basic study on active acoustic shielding, Journal of Environment and Engineering, vol. 7, no. 1, pp , APSIPA ASC 2017

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