# Phasor Representation for Narrowband Active Noise Control Systems

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## Abstract

The phasor representation is introduced to identify the characteristic of the active noise control (ANC) systems. The conventional representation, transfer function, cannot explain the fact that the performance will be degraded at some frequency for the narrowband ANC systems. This paper uses the relationship of signal phasors to illustrate geometrically the operation and the behavior of two-tap adaptive filters. In addition, the best signal basis is therefore suggested to achieve a better performance from the viewpoint of phasor synthesis. Simulation results show that the well-selected signal basis not only achieves a better convergence performance but also speeds up the convergence for narrowband ANC systems.

## Keywords

Convergence Speed Adaptive Filter Notch Filter Convergence Performance Phase Compensation## 1. Introduction

The problems of acoustic noise have received much attention during the past several decades. Traditionally, acoustic noise control uses passive techniques such as enclosures, barriers, and silencers to attenuate the undesired noise [1, 2]. These passive techniques are highly valued for their high attenuation over a broad range of frequency. However, they are relatively large in volume, expensive at cost, and ineffective at low frequencies. It has been shown that the active noise control (ANC) system [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] can efficiently achieve a good performance for attenuating low-frequency noise as compared to passive methods. Based on the principle of superposition, ANC system can cancel the primary (undesired) noise by generating an antinoise of equal amplitude and opposite phase.

The design concept of acoustic ANC system utilizing a microphone and of a loudspeaker to generate a canceling sound was first proposed by Leug [3]. Since the characteristics of noise source and environment are nonstationary, an ANC system should be designed adaptively to cope with these variations. A duct-type noise cancellation system based on adaptive filter theory was developed by Burgess [4] and Warnaka et al. [5]. The most commonly used adaptive approach for ANC system is the transversal filter using the least mean square (LMS) algorithm [6]. In addition, the feedforward control architecture [6, 7, 8] is usually applied to ANC systems for practical implementations. In the feedforward system, a reference microphone, which is located upstream from the secondary source, detects the incident noise waves and supplies the controller with an input signal. Alternatively, a transducer is suggested to sense the frequency of primary noise, if to place the reference microphone is difficult. The controller sends a signal, which is in antiphase with the disturbance, to the secondary source (i.e., loudspeaker) for canceling the primary noise. In addition, an error microphone-located downstream picks up the residual and supplies the controller with an error signal. The controller must accommodate itself to the variation of environment.

The single-frequency adaptive notch filter, which uses two adaptive weights and a 90^{°} phase shift unit, was developed by Widrow and Stearns [9] for interference cancellation. Subsequently, Ziegler [10] first applied this technique to ANC systems and patented it. In addition, Kuo et al. [11] proposed a simplified single-frequency ANC system with delayed-X LMS (DXLMS) algorithm to improve the performance for the fixed-point implementation. In addition, the fact that convergence performance depends on the normalized frequency is pointed. Generally, a periodic noise contains tones at the fundamental frequency and at several harmonic frequencies of the primary noise. This type of noise can be attenuated by a filter with multiple notches [12]. If the undesired primary noise contains *M* sinusoids, then *M* two-weight adaptive filters can be connected in parallel. This parallel configuration extended to multiple-frequency ANC has also been illustrated in [6]. In practical applications, this multiple narrowband ANC controller/filter has been applied to electronic mufflers on automobiles in which the primary noise components are harmonics of the basic firing rate. Furthermore, the convergence analysis of the parallel multiple-frequency ANC system has been proposed in [12]. It is found by Kuo et al. [12] that the convergence of this direct-form ANC system is dependent on the frequency separation between two adjacent sinusoids in the reference signal. In addition, the subband scheme and phase compensation have been combined with notch filter in the recent researches [13, 14, 15].

Using the representation of transfer function [6, 7, 8, 9, 10, 11, 12, 13], the steady state of weight vector for the ANC systems can be determined and the convergence speed can be analyzed by eigenvalue spread. However, it can not explain the fact that the performance will be degraded at some frequencies. Based on the concepts of phasor representation [16], this paper discusses the selection of reference signals in narrowband ANC systems to illustrate the effect of phase compensation in delayed-X LMS approach [11]. The different selections of signal phasor to the reference signal are considered to describe the operation of narrowband ANC systems. In addition, this paper intends to modify the structure of Kuo's FIR-type ANC filter in order to achieve a better performance. This paper is organized as follows. Section 2 briefly reviews the basic two-weight adaptive filter and the delayed two-tap adaptive filter in the single-frequency ANC systems. Besides, the solution of weight vectors will be solved by using the phasor concept. In Section 3, the signal basis is discussed and illustrated for the above-mentioned adaptive filters based on the phasor concept. In Section 4, the eigenvalue spread is discussed to compare the convergence speed for different signal basis selections. The simulations will reflect the facts and discussions. Finally, the conclusions are addressed in Section 5.

## 2. Two-Weight Notch Filtering for ANC System

^{°}phase shifter or another cosine wave generator [17, 18] is required to produce the quadrature reference signal Open image in new window . As illustrated in Figure 1, Open image in new window is the residual error signal measured by the error microphone, and Open image in new window is the primary noise to be reduced. The transfer function Open image in new window represents the primary path from the reference microphone to the error microphone, and Open image in new window is the secondary-path transfer function between the output of adaptive filter and the output of error microphone. The secondary signal Open image in new window is generated by filtering the reference signal Open image in new window with the adaptive filter Open image in new window and can be expressed as

*T*denotes the transpose of a vector, and Open image in new window is the weight vector of the adaptive filter Open image in new window . By using the filtered-X LMS (FXLMS) algorithm [6, 7, 8], the reference signals, Open image in new window and Open image in new window , are filtered by secondary-path estimation filter Open image in new window expressed as

*A*and phase Open image in new window . And, assume that the phase and amplitude responses of the secondary-path Open image in new window at frequency Open image in new window is Open image in new window and

*A*, respectively. Since the filtering of secondary-path estimate Open image in new window is linear, the frequencies of the output signal Open image in new window and the input signal Open image in new window will be the same. To perfectly cancel the primary noise, the antinoise from the output of the adaptive filter should be set as Open image in new window Therefore, the relationship Open image in new window holds. In the following, the concept of phasor [16] is used for representing the system to solve the optimal weight solution instead of using the transfer function and control theory [6, 7, 8]. The output phasor of adaptive filter Open image in new window would be the linear combination of signal phasors Open image in new window and Open image in new window , that is,

which depends on the system parameter Open image in new window .

^{°}phase shift unit and the two individual weights by a second-order FIR filter. As shown in Figure 2, the structure does not need two quadratic reference inputs and the filter-x process is reduced. Especially, Kuo et al. inserted a delay unit located in the front of the second-order FIR filter to improve the convergence performance for considering the implementation over the finite word-length machine. This inserted delay can be called the phase compensation to the system parameter Open image in new window . For Kuo's approach, the output phasor of adaptive filter would be the linear combination of Open image in new window and Open image in new window , where

*D*is the inserted delay. That is,

*D*, Open image in new window and Open image in new window shown as

## 3. Signal Basis Selection

In practical applications, adaptive notch filter is usually implemented on the fixed-point hardware. Therefore, the finite precision effects play an important role on the convergence performance and speed for the adaptive filter. It is difficult to maintain the accuracy of the small coefficient and to prevent the order of magnitude of weights from overflowing simultaneously, as the ratio of two weights in the steady state is very large. When the ratio of two weights in the steady state, Open image in new window , is close to one, the dynamic range of weight value in adaptive processing is fairly small [11]. Thus, the filter can be implemented on the fixed-point hardware with shorter word length, or the coefficients will have higher precision (less coefficient quantization noise) for given a word length.

*S*to Open image in new window ( Open image in new window ) along the Open image in new window -axis, as shown in Figure 5. When the system phase has been compensated, the operation point in Open image in new window -plane can locate in the desired region which the ratio of two weights is close to one. Using the signal bases Open image in new window and Open image in new window , the ratio of two weights satisfies

The solution to (9) is Open image in new window , where *k* is any integer. The optimal delay *D* can be expressed as Open image in new window samples, where the operation Open image in new window denotes to take the nearest integer. These solutions confirm the results in [11] in which the solution is derived by transfer-function representation. Besides, since the relationship Open image in new window holds, there are four solutions for delay *D* ; these solutions are the possible operation points, Open image in new window , and Open image in new window , as shown in Figure 5. From the phasor point of view, the operation points Open image in new window and Open image in new window mean that the synthesis phasor *y* (*n*) is located in the acute angle formed by basis phasors Open image in new window and Open image in new window , as shown in Figure 3(c). Therefore, the range of weights value can be efficiently used. In addition, observing Figure 5, it can be found that the area of the desired regions varies with the normalized frequencies. It means that the performance will vary with the normalized frequency. This fact also confirms the experimental results in [11]. To solve the problem that the performance depends on the normalized frequency, another signal bases should be found for the two-tap adaptive filters.

## 4. Discussion and Simulations

Since the eigenvalue spread Open image in new window is larger than one, the convergence speed will be slower than the conventional two-weight notch filter. It can be found that the convergence speed will depend on the normalized frequency Open image in new window .

and the corresponding eigenvalue spread is Open image in new window . Since the eigenvalue spread has been reduced from Open image in new window to Open image in new window 1, the proposed two-tap adaptive filter will have higher convergence speed.

## 5. Conclusion

In this paper, the phasor representation instead of transfer function is introduced and discussed for the narrowband ANC systems. Based on the concepts of signal basis and phasor rotation, the reference signal/phasor for two-tap adaptive filters has been modeled and well-selected. Using the representation of phasor can explain the reason why the performance of the narrowband ANC systems is degraded for some normalized frequency. In addition, to achieve a better performance, the proposed two-tap adaptive filter can choose the near-orthogonal phasors for the fixed-point hardware implementation. With the same complexity, the inserted delay in Kuo's two-tap adaptive filter can be moved back to construct the proposed approach, which would achieve a better performance.

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