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Effect of the Audio Amplifier's Distortion on Feedforward Proceedings of APSIPA Annual Summit and Conference 2017 12 - 15 December 2017, Malaysia Effect of the Audio Amplifier’s Distortion on Feedforward Active Noise Control Dongyuan Shi∗, Chuang Shiy, and Woon-Seng Ganz ∗z School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore y School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China ∗ E-mail: [email protected] y E-mail: [email protected] z E-mail: [email protected] Abstract—Active noise control (ANC) is an effective method with broadband noise, not to mention its robustness under for reducing low-frequency acoustic noise. An anti-noise wave is different noise environments [21]. transmitted by the secondary source to destructively interfere The real-time implementation of a feedforward ANC system with the noise wave. A quiet zone is thus formed around the error microphone, which provides the error signal in the encounters several practical constraints: (i) the electronic delay adaptation process of an ANC controller. However, the real-world must be shorter than the acoustic delay from the reference performance of an ANC system is often subjected to the distortion microphone to the secondary source to ensure the overall sys- incurred in its electronic components. This distortion has been tem’s causality [22]; (ii) imperfect modeling of the secondary conventionally treated as a trivial part of the secondary path path leads to non-optimized noise reduction performance; (iii) model. When the distortion is severe but the secondary path model is still forced to be linear, the nonlinearity of the true excessive analog amplification of the secondary source causes secondary path is no longer negligible and causes degradation in distortion that may break the controller’s stability. However, noise reduction performance or even divergence of the ANC con- the last constraint has not been well studied. There are only troller. This paper revisits the causes of the amplitude distortion few papers about the effect of the nonlinear secondary path on in the audio amplifier and how it influences the convergence of the the performance of an ANC system [23], [24]. Hence, there filtered-x least mean square (FxLMS) algorithm in a feedforward ANC system for tonal noise cancellation. is a lack of explanation on how over-amplification can break down an ANC controller. When the output power of the audio I. INTRODUCTION amplifier exceeds its rated output power or greater than the rated power of the secondary source, termed as saturation In modern urban living, traffic, construction and machinery and mismatch respectively, the anti-noise wave generated by noises are becoming more intrusive due to the proximity of the ANC system is distorted and results in additional high- residential buildings and noise sources. Exposure to excessive frequency annoyance and sometimes divergence of the ANC noise can lead to sleeping disturbance, hypertension and controller. This paper revisits the distortion incurred in the vascular diseases. Passive solutions are known effective in audio amplifier and elaborates how this distortion degrades canceling the high-frequency noise due to their ability to the noise reduction performance and breaks the stability of a absorb and reflect the noise wave, but become too expensive feedforward ANC system. and impractical for mitigating the low-frequency noise, whose upper frequency bound is typically below 2 kHz. Paul Leug II. AMPLIFIER DISTORTION patented the idea of active noise control (ANC) in 1936 to The audio amplifier, which drives the secondary source combat the low-frequency air-flow noise in a duct [1], [2], [3], to generate the anti-noise wave, is a very critical but often [4]. Since then, many types of ANC systems have been devel- overlooked electronic component in an ANC system. Class A oped with the open-loop, feedforward and feedback structures. and Class AB amplifiers own low distortion and short latency The filtered-x least mean square (FxLMS) algorithm is the but also low power efficiency. In contrast, Class D amplifiers most effective and computationally efficient adaptive algorithm have high power efficiency but long latency. Despite of types that is being used in many ANC systems [5]. The real-time of audio amplifiers, they may have the same three types digital processing platforms for ANC systems include micro- of distortion, which are the amplitude distortion, frequency controllers [6], digital signal processors (DSP) [7] and field distortion and phase distortion. Among them, the amplitude programmable gate arrays (FPGA) [8]. distortion plays the most important role in an ANC system. Over the past thirty years, ANC systems have been suc- There are two causes of the amplitude distortion. Firstly, cessfully deployed in applications such as canceling noise the inappropriate biasing level applied to the audio amplifier in ventilation ducts [9], [10], home windows [11], [12], causes only a portion of the input being amplified and the rest [13], [14], [15], MRI equipment [16], [17], headsets and ear being clipped. This usually happens to the discrete component protectors [18], [19], [20]. Among those applications, the amplifier. Using the integrated audio amplifier can avoid this feedforward ANC structure is usually favorable when dealing problem. Secondly, the output level of the audio amplifier 978-1-5386-1542-3@2017 APSIPA APSIPA ASC 2017 Proceedings of APSIPA Annual Summit and Conference 2017 12 - 15 December 2017, Malaysia 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(!0t); (1) where !0 and C denote the frequency and amplitude respec- tively. 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 8 Fig. 3. Single-frequency adaptive noise canceler. >− 2 π − π < Vthr !0t + 2kπ [ 2 ∆t; 2 + ∆t] y(t) = C sin(!0t) others ; (2) thus given by :> 2 π − π Vthr !0t + 2kπ [ 2 ∆t; 2 + ∆t] x0(n) = sin(!0n + θr) (5) π − Vthr 2 Z where ∆t = 2 arcsin( C ) and k . Its Fourier series x1(n) = cos(!0n + θr); (6) can be written as where θr is the initial phase of the reference signals. Based −2∆t + π + sin(2∆t) y(t) = C sin(!0t) on the FxLMS algorithm, the coefficient updating equation of π : (3) the single-frequency feedforward ANC controller is written as 4 cos(∆t) sin3(∆t) + C sin(3!0t) + : :: ∗ 3π w0(n + 1) = w0(n) + µe(n)[x0(n) hs(n)] (7) ∗ Equation (3) tells that the clipped waveform consists of the w1(n + 1) = w1(n) + µe(n)[x1(n) hs(n)]; (8) odd-order harmonics. Furthermore, when an audio amplifier where µ is the step size and e(n) is the error signal [25]. When exhibits the amplitude distortion, it can be regarded as a the coefficients of the controller reach the optimal solution, the memoryless system, because the output y(n) relies only on 0 anti-noise y (n) should equal to the disturbance signal d(n), the current input y(n), where y(n) and y(n) represent y(t) i.e. and y(t) in the digital domain. 0 y (n) = y(n) ∗ hs(n) = d(n); (9) III. INFLUENCE OF THE AMPLITUDE DISTORTION ON ANC where y(n) is the output signal of the control filter. PERFORMANCE Figure 3 displays the block diagram of a single-frequency A. Case A: Moderate Overdriving feedforward ANC controller [25]. H^s(z) is the estimate of the When the amplitude of the disturbance signal exceeds the secondary path Hs(z). To simplify the analysis, it is assumed threshold of the audio amplifier, the output signal y(n) is dis- −jθs that H^s(z) = Hs(z) = Ase . The primary disturbance torted and expressed by (3). As the over-driving is moderate, signal d(n) is defined as the fundamental and third-order harmonics are dominating in the distorted output, which are written as d(n) = D sin(!0n); (4) yfirst(n) = AC(n) sin(!0n + θ0) (10) where D is the amplitude of the disturbance signal and !0 is the noise frequency. The two orthogonal reference signals are ythird(n) = BC(n) sin(3!0n + θ1); (11) 978-1-5386-1542-3@2017 APSIPA APSIPA ASC 2017 Proceedings of APSIPA Annual Summit and Conference 2017 12 - 15 December 2017, Malaysia when the controller’s output y(n) is actually given by y(n) = C(n) sin(!0n + θ0): (12) The corresponding harmonics in the anti-noise are written as 0 − yfirst(n) = AsAC(n) sin(!0n + θ0 θs) (13) 0 − ythird(n) = AsBC(n) sin(3!0n + θ1 θs): (14) Fig. 4. The equivalent diagram of the single-frequency adaptive noise canceler for the third-order harmonic. It is noteworthy that A changes slowly when the amplitude of y(n) is only slightly greater than the amplifier’s threshold which is a positive feedback loop and hence easily becomes Vthr. 0 The error signal is further written as unstable. The amplitude of ythird(n) will increase until the gradient of the FxLMS algorithm decreases to 0. − 0 − 0 e(n) = d(n) yfirst(n) ythird(n) ::: (15) In the moderate over-driving case, the amplitude of the disturbance is only slightly greater than the threshold Vthr.
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