# Sequentially Adapted Parallel Feedforward Active Noise Control of Noisy Sinusoidal Signals

Authors: Govind Kannan; Issa M. S. Panahi; Richard W. Briggs
Journal: Advances in Acoustics and Vibration
 (2009)

---

## Abstract

A large class of acoustic noise sources has an underlying periodic process that generates a periodic noise component, and thus their acoustic noise can in general be modeled as the sum of a periodic signal and a randomly fluctuating signal (usually a broadband background noise). Active control of periodic noise (i.e., for a mixture of sinusoids) is more effective than that of random noise. For mixtures of sinusoids in a background broadband random noise, conventional FXLMS-based single filter method does not reach the maximum achievable Noise Attenuation Level (NALmax). In this paper, an alternative approach is taken and the idea of a parallel active noise control (ANC) architecture for cancelling mixtures of periodic and random signals is presented. The proposed ANC system separates the noise into periodic and random components and generates corresponding antinoises via separate noise cancelling filters, and tends to reach NALmax consistently. The derivation of NALmax is presented. Both the separation and noise cancellation are based on adaptive filtering. Experimental results verify the analytical development by showing superior performance of the proposed method, over the single-filter approach, for several cases of sinusoids in white noise.


---

## Body

## 1. Introduction

Active noise control (ANC) is a technique of cancelling acoustic noise by generating an appropriate antinoise signal using loudspeakers, and directing it towards the region where noise cancellation is required. Rapid progress in digital signal processors (DSPs) and sensor technologies as well as new real-time adaptive control algorithm designs [1–3] has opened the door to new approaches in ANC. Active control of sinusoidal signals has been an important research and development topic [4] and can lead to different system architectures [5] and design simplifications [6]. The standard approach to realize an ANC system is to employ the FXLMS algorithm in feedforward configuration [4, 7] as portrayed in Figure 1. P(z) denotes the primary path and S(z) denotes the secondary path. From Figure 1, it follows that the exact solution to the noise cancelling filter (the control filter) W(z) is given by
(1)W0(z)=P(z)S(z).
Since the stable transfer function S(z) is in general nonminimum phase, W0(z) should in general be a stable but noncausal system, and hence unrealizable unless x(n) is perfectly predictable. The noncausality restriction no longer applies to a periodic signal since its past and future values can be determined from the knowledge of the period or alternatively from the frequencies, amplitudes, and phases of the constituent sinusoids. Hence a periodic noise signal can ideally be cancelled perfectly by designing an appropriate (predictive) cancelling filter in the ANC system. For linearly modeled ANC system with stationary signals, the causality restriction places an upper limit on the achievable noise attenuation level. The noise attenuation level (NAL) is defined as the ratio of noise power before cancellation to the noise power after cancellation. Computation of the maximum achievable NAL for some given stationary noise process is derived in this paper.  The maximum achievable NAL is denoted by NALmax. For an additive mixture of periodic and random signals, the NALmax can be calculated by assuming that the periodic component is cancelled completely. However experience with the FXLMS algorithm, using single cancelling filter, for additive mixtures of periodic and random signals shows that the NAL achieved is usually less than the NALmax. In this paper an ANC structure for such mixtures, that can achieve performances much closer to NALmax, is introduced.  The notion of separating the periodic and random noise components and generating corresponding anti-noise components is developed. Due to the simultaneous and additive way in which the two anti-noise components are generated and combined, the proposed method is termed as parallel feedforward ANC. It should be noted that in this paper, the adaptation scheme is sequential, that is, the two noise cancelling filters are computed by a two-step adaptation procedure.Figure 1
The feedforward FXLMS algorithm.W(z) generates an appropriate anti-noise so as to minimise e(n) in a mean square sense.This idea, which was initially introduced in [8] in an empirical fashion, is given a rigorous analytical flavor with supporting experimental results in this paper. The proposed separation-based parallel ANC system has three stages: the first stage estimates the sinusoidal parameters and separates the periodic part from the random part of the noise signal using adaptive noise cancelling technique. The second and third stages operate simultaneously, generating the corresponding anti-noise signals for periodic part and random part, respectively. Experimental results using different mixtures of noise signals show improved performance of the proposed ANC method.Active noise control of a mixture of sinusoids with or without harmonic relation is a well-studied topic. An indepth study of the convergence properties of the FXLMS algorithm when the noise to be controlled is a harmonically related mixture of sinusoids is presented in [9]. The idea of using a parallel bank of control filters, each handling a subset of the sinusoids is presented and analyzed. The more general case of active noise control of nonharmonically related sinusiods is analysed in a novel manner in [10] by employing a characteristic equation derived from an equivalent state-space model. A faster converging version of the FXLMS algorithm is also derived. A root-locus-based convergence analysis of an FXLMS-based ANC system for nonharmonically related sinusoids is done in [11].  A phase tuning approach based on an integrative internal model principle is developed in [12] for active noise control of sinusoidal disturbances. It should be noted that the methods developed in [9–12] are applicable to pure sinusoidal mixtures without any broadband random noise.The purpose of this paper is to introduce a new active noise control (ANC) architecture for effective active noise control of sinusoidal mixtures in a random background noise. The key idea is that for sinusoidal signals in random noise, it is possible to separate the sinusoidal and random parts and generate separate anti-noise signals in a parallel  fashion. Since periodic signals can be expressed as a mixture of sinusoids, the proposed method is also applicable to periodic signals with an additive random noise. The effectiveness of noise attenuation is expressed interms of the Noise Attenuation Level (NAL), which measures the relative level of attenuated noise with respect to the actual noise. The notion of, and an exact formula for, maximum achievable Noise Attenuation Level (NALmax) for sinusoidal signals in white noise is presented. The proposed ANC system design is shown to achieve performances close to NALmax.  The key ideas communicated are as follows:(1)
Given a feedforward ANC setup (one reference microphone and one error microphone setup) and a stationary acoustic noise input, there exists a maximum achievable noise attenuation level (NALmax) if linear estimation with an MMSE criterion is employed. The method to calculate this upper limit is developed for sinusoidal mixtures in random noise.(2)
In the active noise control of a mixture of sinusoids and a random noise with a single noise cancelling filter (or control filter), the adaptive algorithm finds an optimum filter that is neither equal to the optimum filter when sinusoids alone are present nor equal to the optimum filter when random noise alone is present. Alternatively, assuming that the sinusoidal and random parts can be separated, better cancellation can be achieved by formulating two mean square errors (MSEs) criteria-one for the sinusoidal component and the other for the random component.(3)
For the periodic signal part, perfect cancellation can be achieved by equalizing the nonminimum phase secondary path with a noncausal equalizer in a causal manner since for periodic signals, the future values can be determined from the knowledge of the period or alternatively from the frequencies, amplitudes, and phases of the constituent sinusoids. For the random part the best cancellation achieveable  is given byNALmax.(4)
For a mixture of sinusoids and random noise, it is possible to separate the periodic and random components and have a parallel ANC architecture. This architecture results in two separate control filters, each of which is optimum for the respective component and hence the overall system can reachNALmax  better than the conventional single-filter configuration. It seems that such an approach has not been considered in the literature so far.The paper is organized as follows. The notational framework and various definitions are introduced in “Notations and Definitions” section, towards the end of the paper. Section2 introduces the proposed parallel feed forward ANC architecture. Section 3 develops the methodology to calculate NALmax. Section 4 presents and discusses the experimental results. Concluding remarks are made in Section 5. The reader is advised to peruse the “Notations and Definitions” section before proceeding to read Sections 2 and 3.


## 2. The Parallel Feedforward ANC  Method (PFANC)

### 2.1. Motivation

The key idea of this paper is that for sinusoidal plus random noise mixtures, separating the sinusoids and random noise and generating antinoise signals separately improves the performance of the ANC system. The motivation for this approach lies in the observation that the optimum control (cancelling) filters for the periodic part and the random part are not the same. Assuming perfect separation ofx(n) into q(n) and ν(n), and separate generation of anti-noise signals for the periodic and random parts, the expression for the composite anti-noise signal generated by the ANC system is
(2)y′(n)=-(wq(n)*q(n)+wν(n)*ν(n))*s(n),
where wq(n) and wν(n) are the noise cancelling filters of q(n) and ν(n), respectively, and * denotes the convolution operation.The best noise cancellation is achieved whenWq(ω) and Wν(ω) are optimum in the MMSE sense. The superscript ° denotes the optimality of any solution in the rest of the manuscript.  From (1), it is obvious that for the tones at ωk, the optimum filter is any Wq∘(ω) such that(3)Wq∘(ωk)=P(ωk)S(ωk)=W0(ωk).
For the random part ν(n),  the optimum solution Wν∘(ω) is the causal Wiener filter (whose exact form is shown later in (51), approximating the noncausal function P(ω)/S(ω).Letepr(n) denote the residual noise for the proposed parallel anti-noise generation approach. Then assuming perfect separation of x(n) into q(n) and ν(n), epr(n) is given by(4)epr(n)=d(n)-wq(n)*q(n)*s(n)-wν(n)*ν(n)*s(n).
Since d(n) can be written as the sum of dq(n) and dν(n),  (4) can be written as
(5)epr(n)=(dq(n)-wq(n)*q(n)*s(n))+(dν(n)-wν(n)*ν(n)*s(n))=eq(n)+eν(n).
Thus epr(n) consists of two components:  eq(n) from the sinusoidal component and eν(n) from the random component. It should be noted that when Wq(ω)=Wq∘(ω), there is perfect cancellation of q(n), that is,  eq∘(n)=0. The best cancellation of ν(n) occurs when Wν(ω)=Wν∘(ω) with the corresponding minimum error eν∘(n). Since eq(n) and eν(n) are statistically independent, the MMSE is(6)E{epr∘2(n)}=E{eq∘2(n)}+E{eν∘2(n)}=E{eν∘2(n)},
where E{·} is the expectation operator.Letesgl(n) denote the residual noise for the single-filter approach. The antinoise is given by
(7)y′(n)=-wsgl(n)*(q(n)+ν(n))*s(n).
The expression for esgl(n) is then(8)esgl(n)=d(n)-wsgl(n)*(q(n)+ν(n))*s(n).
The best cancellation (in the MMSE sense) is achieved by minimising E{esgl2(n)} with respect to wsgl(n). Let the MMSE solution be wsgl∘(n) with the corresponding error esgl∘(n)(9)esgl∘(n)=d(n)-wsgl∘(n)*(q(n)+ν(n))*s(n)=(dq(n)-wsgl∘(n)*q(n)*s(n))+(dν(n)-wsgl∘(n)*ν(n)*s(n)).
In general, the MMSE solution, wsgl∘(n) will be different from both wq∘(n) and wv∘(n). Since only wq∘(n) will result in an error sequence esgl(n) with no tonal component, wsgl∘(n) will always result in a residual tonal component in esgl∘(n). The residual tonal component is denoted by eqres(n) and the residual random noise component is denoted by eνres(n). Thus (9) can be written as
(10)esgl∘(n)=eqres(n)+eνres(n),
where eqres(n)=dq(n)-wsgl∘(n)*q(n)*s(n) and   eνres(n)=dν(n)-wsgl∘(n)*ν(n)*s(n).The expression for MMSE owing to statistical independence is then(11)E{esgl∘2(n)}=E{eqres2(n)}+E{eνres2(n)}.
It should be recalled that the best attenuation of the random component is attained when wsgl∘(n) equals wv∘(n). Since it is not the case in general, it follows that
(12)E{eνres2(n)}≥E{eν∘2(n)}.
Since E{eνres2(n)}≥E{eν∘2(n)} and E{eqres2(n)} are positive quantities, it follows that
(13)E{esgl∘2(n)}=E{eqres2(n)}+E{eνres2(n)}≥E{eν∘2(n)}.
Since from (6), the MMSE for the proposed method E{epr∘2(n)}=E{eν∘2(n)}, it follows from (13) that
(14)E{epr∘2(n)}≤E{esgl∘2(n)}.
Thus the MMSE of the proposed method is always lesser than the MMSE of single-filter method.It should also be noted that when a single adaptive filter is used, the adaptive algorithm tends to cancel the dominating signal component, that is, the tones more effectively. As a result, the random component may be attenuated to a lesser degree, unaffected, or even reinforced. In fact, the adaptive filter in the single-filter case shows a non-Wiener behavior [13], the analysis of which is nontrivial. In contrast, the proposed ANC structure offers a simple means of enhancing performance.


### 2.2. Parallel Feedforward ANC (PFANC): Problem Formulation

The parallel feedforward ANC is formulated, as two subproblems, as follows.(1)
Given an acoustic noise signalx(n)=q(n)+ν(n), devise a method of separating x(n) into q(n) and ν(n), where q(n) is periodic, and ν(n) is random.(2)
Design an ANC system that has two parallel adaptive noise cancelling filtersWq(z) and Wν(z) for generating antinoise signals separately. It should be noted that, for analytical purposes, we consider a very short convergence time and hence drop the time indices that should accompany Wq(z) and Wν(z), owing to their time-varying (adaptive) nature. These two cancelling filters are parts of ANCq and ANCν, respectively, (see Figure 2). The overall system will monitor the error microphone output e(n) to adjust the weights of these cancelling filters.Figure 2
The proposed scheme: active noise control employing separation of sinusoidal and random parts of the reference signal.Figure 3
The optimum filterWν∘(ω) for the random noise can be way-off when compared to W0(ω), that is, Wν∘(ωk)≠W0(ωk). The dashed circled regions show frequencies where Wν∘(ω) deviates considerably from W0(ω). Presence of tones in these regions will affect the performance of single-filter ANC methods.In the proposed approach, the first sub-problem is addressed in three steps: a frequency analyzer block which estimates the sinusoidal frequencies, a synthesizer block which generates a reference signal based on the estimated frequencies, and an adaptive noise canceller block which separatesx(n) into q(n) and ν(n). The second sub-problem is solved by setting up two FXLMS-based ANC systems. The method is elaborated in Figure 2.  A more detailed explanation of the proposed approach is presented in subsequent sections.


### 2.3. PFANC Method

The proposed PFANC method can be divided into three major parts.(1)
Estimation of  sinusoidal parameters(2)
Separation of sinusoidal and random components of the noise signal.(3)
Generation of antinoise signals corresponding to the sinusoidal and random signal components.


#### 2.3.1. Estimation of Sinusoidal Parameters

This part estimates the sinusoidal parameters from the measured input noise signalx(n). Let M samples of x(n) be observed. The estimation problem can then be stated asGiven thatx(n)can be modeled as(15)x(n)=∑k=1L‍akcos(ωkn+ϕk)+ν(n)and givenMobserved samples ofx(n),estimateωk, ak, ϕk. The determination ofLis implicit in the problem.The above estimation problem is an important and a well-studied one [14].  Since there are many methods available, and since this is an important step in the proposed scheme, the tools and techniques that were found to be particularly effective are described. The first step is detecting the number L of sinusoids, which is essentially an order estimation problem. The next step is estimation of ωk, ak, ϕk.  An information-theoretic approach [15] for the first step and a subspace-based high resolution parametric approach (ESPRIT) [14] for the second step are prescribed.Detection ofL
The first step is the detection ofL, the number of sinusoids, which amounts to order estimation. The gist of all the order estimation methods is defining an information measure as a function of the number of sinusoids. The number of sinusoids that best explains the observations, minimizes the information measure. Out of the host of order estimation techniques available [16], the method described in [15] is particularly effective for the sinusoids in noise scenario. The information measure is based on the Minimum Description Length (MDL) principle introduced in [17].  The reasons for adopting this method are twofold: (1) the method has been shown to be consistent, that is, it asymptotically approaches the optimum MAP rule [15, 17] and (2) there is no need to choose a penalty function [15].The following quantities need to be defined, before introducing the technique.In general for any quantityH,  Ĥ denotes the estimated value of the quantity whereas the uncapped H denotes the true value. For example, if ω is the vector of frequencies. then ω̂ is the vector of estimated frequencies.  Let us define the vectors ω=[ω1⋯ωL]T, a=[a1⋯aL]T, φ=[ϕ1⋯ϕL]T, x(n)=[x(n)⋯x(n-m+1)]T, and matrix
(16)R=E{x(n)xH(n)}.R is the m×m, that is, order-m covariance matrix of the observations x(n).  Superscripts H and T denote Hermitian and transpose operations, respectively.However in practice, the covariance matrixR of order m is calculated from M observations (M>m) of x(n), that is, from {x(1),x(2),x(3),…,x(M)}.  The observations data length M should be greater than the covariance matrix order m so that an accurate estimate of R can be computed.   The expression for the sample covariance matrix R̂ is [14]
(17)R̂=1M∑t=mMxtxtHm×m,
where,
(18)xt=[x(t)⋯x(t-m+1)]T.
The eigen-structure of R̂ has all the information about the angular frequencies ωk. Additionally let λ̂1,λ̂2,…,λ̂2L,…,λ̂m be the eigenvalues of R̂ sorted in decreasing magnitude. The number of sinusoids is then determined from the l (for 2L complex exponentials) that minimizes [15]
(19)MDL(l)=M(m-l)log((1/(m-l))∑i=l+1mλ̂i∏i=l+1mλ̂i1/(m-l))+12l(2m-l)logM.
The number of sinusoids L can be found from  L=lmin/2, since for L real sinusoids there are 2L complex exponentials.The minimum is found by computing theMDL(l) for values of l ranging from 1,2,…,m-1. The eigenvalues are computed from an eigenvalue decomposition of R̂.  An example plot of MDL for l=1,2,…,m-1 is shown in Figure 4.Figure 4
Plot ofMDL as a function of l computed from (19). x(n)is a mixture of 10 (real) sinusoids in white noise where the SNR is η=0 dB.  It is apparent from the figure that the minimum occurs at l=2L=20.Estimation ofω,a,φ
The estimation ofω,a,φ  can be done using various ways like MUSIC, ESPRIT, FFT, and so forth, [14]. The method which has been shown to have very good statistical performance [14] is the nonlinear least squares (NLS) method in which the optimal values of ω,a,φ  minimizes the least squares measure,
(20)f(ω,α,φ)=∑n=1M‍|x(n)-∑k=1L‍akcos(ωkn+ϕk)|2.
In fact, for Gaussian noise scenarios the NLS is the best (in terms of minimum variance) estimator that achieves the Cramer-Rao bound. The estimation accuracy of NLS is given by
(21)var(ω̂k)=6M3ηk.
For ANC problems, M could be allowed to be reasonably large and as will be shown later, the PFANC method demonstrates superior performance for reasonable SNR values (η>10 dB). For example, for ηk=10 dB and M=10000,  var(ω̂k) will be of the order of 10-8.  Thus, very precise estimates can be expected from the NLS method. The drawback is that the minimization in (20) is a complex multidimensional search problem and could be computationally intractable. This drawback can be addressed by using the subspace methods, whose estimation performance is close to the best performance shown in (21) and is computationally efficient [14]. Out of the multitude of subspace methods (like MUSIC, ESPRIT, and Min-Norm), ESPRIT has been shown to have superior statistical performance [14, 18]. Moreover, ESPRIT does not give spurious frequencies once the correct order L is determined using (19).ESPRIT Calculations
For completeness of discussion, the steps in frequency estimation using the ESPRIT method are as follows.(1)
Calculate the sample covariance matrix using (17).(2)
Calculate the eigenvalue decomposition ofR̂(22)R̂=Ω̂Λ̂Ω̂H,
where, Λ̂=diag(λ̂1,λ̂2,…,λ̂m).(3)
LetΩ̂ be the matrix composed of the eigenvectors corresponding to the eigenvalues λ̂1,λ̂2,…,λ̂2L,
(23)Ω̂=[σ̂1,σ̂2,…,σ̂2L].
Define
(24)Ω̂1=[Im-10]Ω̂(m-1)×2L,Ω̂2=[0Im-1]Ω̂(m-1)×2L.(4)
The frequencies are estimated by solving the system of linear equations(25)Ω̂1Θ̂≈Ω̂2
whose least squares solution is given by
(26)Θ̂=(Ω̂1HΩ̂1)-1Ω̂1HΩ̂2.
The argument of the complex eigenvalues of Θ̂ gives the frequencies ωk. Let δk be the eigenvalue of Θ̂. Then the frequency estimates are
(27)ω̂k=-arg(δk).
The amplitudes and phases can be estimated as
(28)β̂=(GHG)-1GHx,
where β̂=[â1ejϕ̂1⋯âLejϕ̂L]T, x=[x(1)⋯x(M0)]T. The matrix G is calculated from the estimated frequencies ω̂k’s according to
(29)G=[eiω̂1⋯eiω̂L⋮⋮eiM0ω̂1⋯eiM0ω̂L],M0 indicates that the first M0 samples of the data frame x are used for ESPRIT calculations.The estimated frequencies are then used to generate a reference periodic signalr(n) shown in Figure 2 using
(30)r(n)=∑k=1L‍âkcos(ω̂kn+ϕ̂k).
The estimates âk and ϕ̂k can be further improved, by estimating q(n) from r(n), by using the adaptive cancellation technique as explained in the next section.


#### 2.3.2. Signal Separation

This block essentially estimatesq̂(n) from r(n) in (30) by trying to cancel q(n) from x(n)=q(n)+ν(n) (i.e., by estimating q̂(n) from r(n)). This step adaptively finetunes the parameters âk,ϕ̂k from (30) to their actual values and also separates x(n) into  q̂(n) and ν̂(n). It can be easily seen from Figure 2 that after convergence, the error signal is ν̂(n) and the cancelling signal becomes y(n)≈-q̂(n)⇒q̂(n)≈-y(n), thus  achieving separation. The update equation for the weight vector wsep(n+1) is given by
(31)wsep(n+1)=wsep(n)+μν̂(n)r(n)rT(n)r(n).
An initial portion of the reference signal x(n) is used to estimate the tones and achieve component separation. The separated signal components will be denoted by q̂(n) and ν̂(n) to indicate that they are estimates. For high η, the estimates are highly accurate. By incorporating the fact that the effect of an arbitrary transfer function on a sinusoid can be modeled by two taps [4, 9],  computationally efficient implementations of the above separation scheme can be derived.


#### 2.3.3. Generation of Antinoise Signal

This block generates the antinoise (i.e., the cancelling) signal in two steps via two parallel ANC systems,ANCq and ANCν. ANCq uses q̂(n) as the reference and generates an antinoise yq(n) which attempts to cancel dq(n). yq(n), after passing through S(z) becomes yq′(n).  The error signal eq(n) is given by
(32)eq(n)=d(n)-yq̂′(n)=dν(n)+(dq(n)-yq̂′(n)),
where d(n) denotes the desired noise to be cancelled and yq̂(n) denotes the anti-noise generated by ANCq as shown in Figure 2. The adaptive filter Wq(z) and S(z) linearly operate on the periodic signal q̂(n) to produce yq̂′(n) which lies in the same subspace as that of q̂(n). Since dν(n) is uncorrelated with dq(n), convergence in this case essentially means that (dq(n)-yq̂′(n)) is minimised in the mean square sense. Thus after convergence, from (32)
(33)eq(n)≈dν(n).
The update equation for the weight vector wq(n+1) is given by
(34)wq(n+1)=wq(n)+μeq(n)q̂′(n)q̂′T(n)q̂′(n),
where q̂′(n) is the signal vector obtained after filtering q̂(n) with Ŝ(z). After convergence is attained in ANCq, the adaptation is stopped and wq(n+1) is fixed to the steady-state value. The system ANCq continues to generate antinoise which cancels dq(n). The error microphone's output is dν(n). The next step is to switch on ANCν whose reference input is the separated random part ν̂(n) and error input is eν(n) which it tries to minimize by generating the antinoise signal yν̂(n).The update equation for the weight vectorwν(n+1) is given by
(35)wν(n+1)=wν(n)+μeν(n)ν̂′(n)ν̂′T(n)ν̂′(n),
where ν̂′(n) is the signal obtained after filtering ν̂(n) with Ŝ(z). After convergence the error is eν(n) and should not contain any component of the periodic noise as long as the acoustic system is stationary. Since we have access to only one error sequence, the procedure for finding the optimum Wq(z) and Wν(z) is sequential. However, this sequential adaptive tuning procedure can be repeated when the error sequence energy goes above a prespecified threshold, since the changes in the acoustic system will result in an increase in the energy of the error sequence.It should be noted that the adaptation methods presented here for the implementation ofANCq and ANCν are based on simple FXLMS for the sake of simplicity and quick experimentation. For the tonal active noise control part ANCq, the analysis and methods described in [9–12] can be incorporated. The proposed method can also be extended to accommodate multiple reference microphones in which case the reduced reference signal set procedure developed in [19] can lessen the controller dimension and computational complexity.


## 2.1. Motivation

The key idea of this paper is that for sinusoidal plus random noise mixtures, separating the sinusoids and random noise and generating antinoise signals separately improves the performance of the ANC system. The motivation for this approach lies in the observation that the optimum control (cancelling) filters for the periodic part and the random part are not the same. Assuming perfect separation ofx(n) into q(n) and ν(n), and separate generation of anti-noise signals for the periodic and random parts, the expression for the composite anti-noise signal generated by the ANC system is
(2)y′(n)=-(wq(n)*q(n)+wν(n)*ν(n))*s(n),
where wq(n) and wν(n) are the noise cancelling filters of q(n) and ν(n), respectively, and * denotes the convolution operation.The best noise cancellation is achieved whenWq(ω) and Wν(ω) are optimum in the MMSE sense. The superscript ° denotes the optimality of any solution in the rest of the manuscript.  From (1), it is obvious that for the tones at ωk, the optimum filter is any Wq∘(ω) such that(3)Wq∘(ωk)=P(ωk)S(ωk)=W0(ωk).
For the random part ν(n),  the optimum solution Wν∘(ω) is the causal Wiener filter (whose exact form is shown later in (51), approximating the noncausal function P(ω)/S(ω).Letepr(n) denote the residual noise for the proposed parallel anti-noise generation approach. Then assuming perfect separation of x(n) into q(n) and ν(n), epr(n) is given by(4)epr(n)=d(n)-wq(n)*q(n)*s(n)-wν(n)*ν(n)*s(n).
Since d(n) can be written as the sum of dq(n) and dν(n),  (4) can be written as
(5)epr(n)=(dq(n)-wq(n)*q(n)*s(n))+(dν(n)-wν(n)*ν(n)*s(n))=eq(n)+eν(n).
Thus epr(n) consists of two components:  eq(n) from the sinusoidal component and eν(n) from the random component. It should be noted that when Wq(ω)=Wq∘(ω), there is perfect cancellation of q(n), that is,  eq∘(n)=0. The best cancellation of ν(n) occurs when Wν(ω)=Wν∘(ω) with the corresponding minimum error eν∘(n). Since eq(n) and eν(n) are statistically independent, the MMSE is(6)E{epr∘2(n)}=E{eq∘2(n)}+E{eν∘2(n)}=E{eν∘2(n)},
where E{·} is the expectation operator.Letesgl(n) denote the residual noise for the single-filter approach. The antinoise is given by
(7)y′(n)=-wsgl(n)*(q(n)+ν(n))*s(n).
The expression for esgl(n) is then(8)esgl(n)=d(n)-wsgl(n)*(q(n)+ν(n))*s(n).
The best cancellation (in the MMSE sense) is achieved by minimising E{esgl2(n)} with respect to wsgl(n). Let the MMSE solution be wsgl∘(n) with the corresponding error esgl∘(n)(9)esgl∘(n)=d(n)-wsgl∘(n)*(q(n)+ν(n))*s(n)=(dq(n)-wsgl∘(n)*q(n)*s(n))+(dν(n)-wsgl∘(n)*ν(n)*s(n)).
In general, the MMSE solution, wsgl∘(n) will be different from both wq∘(n) and wv∘(n). Since only wq∘(n) will result in an error sequence esgl(n) with no tonal component, wsgl∘(n) will always result in a residual tonal component in esgl∘(n). The residual tonal component is denoted by eqres(n) and the residual random noise component is denoted by eνres(n). Thus (9) can be written as
(10)esgl∘(n)=eqres(n)+eνres(n),
where eqres(n)=dq(n)-wsgl∘(n)*q(n)*s(n) and   eνres(n)=dν(n)-wsgl∘(n)*ν(n)*s(n).The expression for MMSE owing to statistical independence is then(11)E{esgl∘2(n)}=E{eqres2(n)}+E{eνres2(n)}.
It should be recalled that the best attenuation of the random component is attained when wsgl∘(n) equals wv∘(n). Since it is not the case in general, it follows that
(12)E{eνres2(n)}≥E{eν∘2(n)}.
Since E{eνres2(n)}≥E{eν∘2(n)} and E{eqres2(n)} are positive quantities, it follows that
(13)E{esgl∘2(n)}=E{eqres2(n)}+E{eνres2(n)}≥E{eν∘2(n)}.
Since from (6), the MMSE for the proposed method E{epr∘2(n)}=E{eν∘2(n)}, it follows from (13) that
(14)E{epr∘2(n)}≤E{esgl∘2(n)}.
Thus the MMSE of the proposed method is always lesser than the MMSE of single-filter method.It should also be noted that when a single adaptive filter is used, the adaptive algorithm tends to cancel the dominating signal component, that is, the tones more effectively. As a result, the random component may be attenuated to a lesser degree, unaffected, or even reinforced. In fact, the adaptive filter in the single-filter case shows a non-Wiener behavior [13], the analysis of which is nontrivial. In contrast, the proposed ANC structure offers a simple means of enhancing performance.


## 2.2. Parallel Feedforward ANC (PFANC): Problem Formulation

The parallel feedforward ANC is formulated, as two subproblems, as follows.(1)
Given an acoustic noise signalx(n)=q(n)+ν(n), devise a method of separating x(n) into q(n) and ν(n), where q(n) is periodic, and ν(n) is random.(2)
Design an ANC system that has two parallel adaptive noise cancelling filtersWq(z) and Wν(z) for generating antinoise signals separately. It should be noted that, for analytical purposes, we consider a very short convergence time and hence drop the time indices that should accompany Wq(z) and Wν(z), owing to their time-varying (adaptive) nature. These two cancelling filters are parts of ANCq and ANCν, respectively, (see Figure 2). The overall system will monitor the error microphone output e(n) to adjust the weights of these cancelling filters.Figure 2
The proposed scheme: active noise control employing separation of sinusoidal and random parts of the reference signal.Figure 3
The optimum filterWν∘(ω) for the random noise can be way-off when compared to W0(ω), that is, Wν∘(ωk)≠W0(ωk). The dashed circled regions show frequencies where Wν∘(ω) deviates considerably from W0(ω). Presence of tones in these regions will affect the performance of single-filter ANC methods.In the proposed approach, the first sub-problem is addressed in three steps: a frequency analyzer block which estimates the sinusoidal frequencies, a synthesizer block which generates a reference signal based on the estimated frequencies, and an adaptive noise canceller block which separatesx(n) into q(n) and ν(n). The second sub-problem is solved by setting up two FXLMS-based ANC systems. The method is elaborated in Figure 2.  A more detailed explanation of the proposed approach is presented in subsequent sections.


## 2.3. PFANC Method

The proposed PFANC method can be divided into three major parts.(1)
Estimation of  sinusoidal parameters(2)
Separation of sinusoidal and random components of the noise signal.(3)
Generation of antinoise signals corresponding to the sinusoidal and random signal components.


### 2.3.1. Estimation of Sinusoidal Parameters

This part estimates the sinusoidal parameters from the measured input noise signalx(n). Let M samples of x(n) be observed. The estimation problem can then be stated asGiven thatx(n)can be modeled as(15)x(n)=∑k=1L‍akcos(ωkn+ϕk)+ν(n)and givenMobserved samples ofx(n),estimateωk, ak, ϕk. The determination ofLis implicit in the problem.The above estimation problem is an important and a well-studied one [14].  Since there are many methods available, and since this is an important step in the proposed scheme, the tools and techniques that were found to be particularly effective are described. The first step is detecting the number L of sinusoids, which is essentially an order estimation problem. The next step is estimation of ωk, ak, ϕk.  An information-theoretic approach [15] for the first step and a subspace-based high resolution parametric approach (ESPRIT) [14] for the second step are prescribed.Detection ofL
The first step is the detection ofL, the number of sinusoids, which amounts to order estimation. The gist of all the order estimation methods is defining an information measure as a function of the number of sinusoids. The number of sinusoids that best explains the observations, minimizes the information measure. Out of the host of order estimation techniques available [16], the method described in [15] is particularly effective for the sinusoids in noise scenario. The information measure is based on the Minimum Description Length (MDL) principle introduced in [17].  The reasons for adopting this method are twofold: (1) the method has been shown to be consistent, that is, it asymptotically approaches the optimum MAP rule [15, 17] and (2) there is no need to choose a penalty function [15].The following quantities need to be defined, before introducing the technique.In general for any quantityH,  Ĥ denotes the estimated value of the quantity whereas the uncapped H denotes the true value. For example, if ω is the vector of frequencies. then ω̂ is the vector of estimated frequencies.  Let us define the vectors ω=[ω1⋯ωL]T, a=[a1⋯aL]T, φ=[ϕ1⋯ϕL]T, x(n)=[x(n)⋯x(n-m+1)]T, and matrix
(16)R=E{x(n)xH(n)}.R is the m×m, that is, order-m covariance matrix of the observations x(n).  Superscripts H and T denote Hermitian and transpose operations, respectively.However in practice, the covariance matrixR of order m is calculated from M observations (M>m) of x(n), that is, from {x(1),x(2),x(3),…,x(M)}.  The observations data length M should be greater than the covariance matrix order m so that an accurate estimate of R can be computed.   The expression for the sample covariance matrix R̂ is [14]
(17)R̂=1M∑t=mMxtxtHm×m,
where,
(18)xt=[x(t)⋯x(t-m+1)]T.
The eigen-structure of R̂ has all the information about the angular frequencies ωk. Additionally let λ̂1,λ̂2,…,λ̂2L,…,λ̂m be the eigenvalues of R̂ sorted in decreasing magnitude. The number of sinusoids is then determined from the l (for 2L complex exponentials) that minimizes [15]
(19)MDL(l)=M(m-l)log((1/(m-l))∑i=l+1mλ̂i∏i=l+1mλ̂i1/(m-l))+12l(2m-l)logM.
The number of sinusoids L can be found from  L=lmin/2, since for L real sinusoids there are 2L complex exponentials.The minimum is found by computing theMDL(l) for values of l ranging from 1,2,…,m-1. The eigenvalues are computed from an eigenvalue decomposition of R̂.  An example plot of MDL for l=1,2,…,m-1 is shown in Figure 4.Figure 4
Plot ofMDL as a function of l computed from (19). x(n)is a mixture of 10 (real) sinusoids in white noise where the SNR is η=0 dB.  It is apparent from the figure that the minimum occurs at l=2L=20.Estimation ofω,a,φ
The estimation ofω,a,φ  can be done using various ways like MUSIC, ESPRIT, FFT, and so forth, [14]. The method which has been shown to have very good statistical performance [14] is the nonlinear least squares (NLS) method in which the optimal values of ω,a,φ  minimizes the least squares measure,
(20)f(ω,α,φ)=∑n=1M‍|x(n)-∑k=1L‍akcos(ωkn+ϕk)|2.
In fact, for Gaussian noise scenarios the NLS is the best (in terms of minimum variance) estimator that achieves the Cramer-Rao bound. The estimation accuracy of NLS is given by
(21)var(ω̂k)=6M3ηk.
For ANC problems, M could be allowed to be reasonably large and as will be shown later, the PFANC method demonstrates superior performance for reasonable SNR values (η>10 dB). For example, for ηk=10 dB and M=10000,  var(ω̂k) will be of the order of 10-8.  Thus, very precise estimates can be expected from the NLS method. The drawback is that the minimization in (20) is a complex multidimensional search problem and could be computationally intractable. This drawback can be addressed by using the subspace methods, whose estimation performance is close to the best performance shown in (21) and is computationally efficient [14]. Out of the multitude of subspace methods (like MUSIC, ESPRIT, and Min-Norm), ESPRIT has been shown to have superior statistical performance [14, 18]. Moreover, ESPRIT does not give spurious frequencies once the correct order L is determined using (19).ESPRIT Calculations
For completeness of discussion, the steps in frequency estimation using the ESPRIT method are as follows.(1)
Calculate the sample covariance matrix using (17).(2)
Calculate the eigenvalue decomposition ofR̂(22)R̂=Ω̂Λ̂Ω̂H,
where, Λ̂=diag(λ̂1,λ̂2,…,λ̂m).(3)
LetΩ̂ be the matrix composed of the eigenvectors corresponding to the eigenvalues λ̂1,λ̂2,…,λ̂2L,
(23)Ω̂=[σ̂1,σ̂2,…,σ̂2L].
Define
(24)Ω̂1=[Im-10]Ω̂(m-1)×2L,Ω̂2=[0Im-1]Ω̂(m-1)×2L.(4)
The frequencies are estimated by solving the system of linear equations(25)Ω̂1Θ̂≈Ω̂2
whose least squares solution is given by
(26)Θ̂=(Ω̂1HΩ̂1)-1Ω̂1HΩ̂2.
The argument of the complex eigenvalues of Θ̂ gives the frequencies ωk. Let δk be the eigenvalue of Θ̂. Then the frequency estimates are
(27)ω̂k=-arg(δk).
The amplitudes and phases can be estimated as
(28)β̂=(GHG)-1GHx,
where β̂=[â1ejϕ̂1⋯âLejϕ̂L]T, x=[x(1)⋯x(M0)]T. The matrix G is calculated from the estimated frequencies ω̂k’s according to
(29)G=[eiω̂1⋯eiω̂L⋮⋮eiM0ω̂1⋯eiM0ω̂L],M0 indicates that the first M0 samples of the data frame x are used for ESPRIT calculations.The estimated frequencies are then used to generate a reference periodic signalr(n) shown in Figure 2 using
(30)r(n)=∑k=1L‍âkcos(ω̂kn+ϕ̂k).
The estimates âk and ϕ̂k can be further improved, by estimating q(n) from r(n), by using the adaptive cancellation technique as explained in the next section.


### 2.3.2. Signal Separation

This block essentially estimatesq̂(n) from r(n) in (30) by trying to cancel q(n) from x(n)=q(n)+ν(n) (i.e., by estimating q̂(n) from r(n)). This step adaptively finetunes the parameters âk,ϕ̂k from (30) to their actual values and also separates x(n) into  q̂(n) and ν̂(n). It can be easily seen from Figure 2 that after convergence, the error signal is ν̂(n) and the cancelling signal becomes y(n)≈-q̂(n)⇒q̂(n)≈-y(n), thus  achieving separation. The update equation for the weight vector wsep(n+1) is given by
(31)wsep(n+1)=wsep(n)+μν̂(n)r(n)rT(n)r(n).
An initial portion of the reference signal x(n) is used to estimate the tones and achieve component separation. The separated signal components will be denoted by q̂(n) and ν̂(n) to indicate that they are estimates. For high η, the estimates are highly accurate. By incorporating the fact that the effect of an arbitrary transfer function on a sinusoid can be modeled by two taps [4, 9],  computationally efficient implementations of the above separation scheme can be derived.


### 2.3.3. Generation of Antinoise Signal

This block generates the antinoise (i.e., the cancelling) signal in two steps via two parallel ANC systems,ANCq and ANCν. ANCq uses q̂(n) as the reference and generates an antinoise yq(n) which attempts to cancel dq(n). yq(n), after passing through S(z) becomes yq′(n).  The error signal eq(n) is given by
(32)eq(n)=d(n)-yq̂′(n)=dν(n)+(dq(n)-yq̂′(n)),
where d(n) denotes the desired noise to be cancelled and yq̂(n) denotes the anti-noise generated by ANCq as shown in Figure 2. The adaptive filter Wq(z) and S(z) linearly operate on the periodic signal q̂(n) to produce yq̂′(n) which lies in the same subspace as that of q̂(n). Since dν(n) is uncorrelated with dq(n), convergence in this case essentially means that (dq(n)-yq̂′(n)) is minimised in the mean square sense. Thus after convergence, from (32)
(33)eq(n)≈dν(n).
The update equation for the weight vector wq(n+1) is given by
(34)wq(n+1)=wq(n)+μeq(n)q̂′(n)q̂′T(n)q̂′(n),
where q̂′(n) is the signal vector obtained after filtering q̂(n) with Ŝ(z). After convergence is attained in ANCq, the adaptation is stopped and wq(n+1) is fixed to the steady-state value. The system ANCq continues to generate antinoise which cancels dq(n). The error microphone's output is dν(n). The next step is to switch on ANCν whose reference input is the separated random part ν̂(n) and error input is eν(n) which it tries to minimize by generating the antinoise signal yν̂(n).The update equation for the weight vectorwν(n+1) is given by
(35)wν(n+1)=wν(n)+μeν(n)ν̂′(n)ν̂′T(n)ν̂′(n),
where ν̂′(n) is the signal obtained after filtering ν̂(n) with Ŝ(z). After convergence the error is eν(n) and should not contain any component of the periodic noise as long as the acoustic system is stationary. Since we have access to only one error sequence, the procedure for finding the optimum Wq(z) and Wν(z) is sequential. However, this sequential adaptive tuning procedure can be repeated when the error sequence energy goes above a prespecified threshold, since the changes in the acoustic system will result in an increase in the energy of the error sequence.It should be noted that the adaptation methods presented here for the implementation ofANCq and ANCν are based on simple FXLMS for the sake of simplicity and quick experimentation. For the tonal active noise control part ANCq, the analysis and methods described in [9–12] can be incorporated. The proposed method can also be extended to accommodate multiple reference microphones in which case the reduced reference signal set procedure developed in [19] can lessen the controller dimension and computational complexity.


## 2.3.1. Estimation of Sinusoidal Parameters

This part estimates the sinusoidal parameters from the measured input noise signalx(n). Let M samples of x(n) be observed. The estimation problem can then be stated asGiven thatx(n)can be modeled as(15)x(n)=∑k=1L‍akcos(ωkn+ϕk)+ν(n)and givenMobserved samples ofx(n),estimateωk, ak, ϕk. The determination ofLis implicit in the problem.The above estimation problem is an important and a well-studied one [14].  Since there are many methods available, and since this is an important step in the proposed scheme, the tools and techniques that were found to be particularly effective are described. The first step is detecting the number L of sinusoids, which is essentially an order estimation problem. The next step is estimation of ωk, ak, ϕk.  An information-theoretic approach [15] for the first step and a subspace-based high resolution parametric approach (ESPRIT) [14] for the second step are prescribed.Detection ofL
The first step is the detection ofL, the number of sinusoids, which amounts to order estimation. The gist of all the order estimation methods is defining an information measure as a function of the number of sinusoids. The number of sinusoids that best explains the observations, minimizes the information measure. Out of the host of order estimation techniques available [16], the method described in [15] is particularly effective for the sinusoids in noise scenario. The information measure is based on the Minimum Description Length (MDL) principle introduced in [17].  The reasons for adopting this method are twofold: (1) the method has been shown to be consistent, that is, it asymptotically approaches the optimum MAP rule [15, 17] and (2) there is no need to choose a penalty function [15].The following quantities need to be defined, before introducing the technique.In general for any quantityH,  Ĥ denotes the estimated value of the quantity whereas the uncapped H denotes the true value. For example, if ω is the vector of frequencies. then ω̂ is the vector of estimated frequencies.  Let us define the vectors ω=[ω1⋯ωL]T, a=[a1⋯aL]T, φ=[ϕ1⋯ϕL]T, x(n)=[x(n)⋯x(n-m+1)]T, and matrix
(16)R=E{x(n)xH(n)}.R is the m×m, that is, order-m covariance matrix of the observations x(n).  Superscripts H and T denote Hermitian and transpose operations, respectively.However in practice, the covariance matrixR of order m is calculated from M observations (M>m) of x(n), that is, from {x(1),x(2),x(3),…,x(M)}.  The observations data length M should be greater than the covariance matrix order m so that an accurate estimate of R can be computed.   The expression for the sample covariance matrix R̂ is [14]
(17)R̂=1M∑t=mMxtxtHm×m,
where,
(18)xt=[x(t)⋯x(t-m+1)]T.
The eigen-structure of R̂ has all the information about the angular frequencies ωk. Additionally let λ̂1,λ̂2,…,λ̂2L,…,λ̂m be the eigenvalues of R̂ sorted in decreasing magnitude. The number of sinusoids is then determined from the l (for 2L complex exponentials) that minimizes [15]
(19)MDL(l)=M(m-l)log((1/(m-l))∑i=l+1mλ̂i∏i=l+1mλ̂i1/(m-l))+12l(2m-l)logM.
The number of sinusoids L can be found from  L=lmin/2, since for L real sinusoids there are 2L complex exponentials.The minimum is found by computing theMDL(l) for values of l ranging from 1,2,…,m-1. The eigenvalues are computed from an eigenvalue decomposition of R̂.  An example plot of MDL for l=1,2,…,m-1 is shown in Figure 4.Figure 4
Plot ofMDL as a function of l computed from (19). x(n)is a mixture of 10 (real) sinusoids in white noise where the SNR is η=0 dB.  It is apparent from the figure that the minimum occurs at l=2L=20.Estimation ofω,a,φ
The estimation ofω,a,φ  can be done using various ways like MUSIC, ESPRIT, FFT, and so forth, [14]. The method which has been shown to have very good statistical performance [14] is the nonlinear least squares (NLS) method in which the optimal values of ω,a,φ  minimizes the least squares measure,
(20)f(ω,α,φ)=∑n=1M‍|x(n)-∑k=1L‍akcos(ωkn+ϕk)|2.
In fact, for Gaussian noise scenarios the NLS is the best (in terms of minimum variance) estimator that achieves the Cramer-Rao bound. The estimation accuracy of NLS is given by
(21)var(ω̂k)=6M3ηk.
For ANC problems, M could be allowed to be reasonably large and as will be shown later, the PFANC method demonstrates superior performance for reasonable SNR values (η>10 dB). For example, for ηk=10 dB and M=10000,  var(ω̂k) will be of the order of 10-8.  Thus, very precise estimates can be expected from the NLS method. The drawback is that the minimization in (20) is a complex multidimensional search problem and could be computationally intractable. This drawback can be addressed by using the subspace methods, whose estimation performance is close to the best performance shown in (21) and is computationally efficient [14]. Out of the multitude of subspace methods (like MUSIC, ESPRIT, and Min-Norm), ESPRIT has been shown to have superior statistical performance [14, 18]. Moreover, ESPRIT does not give spurious frequencies once the correct order L is determined using (19).ESPRIT Calculations
For completeness of discussion, the steps in frequency estimation using the ESPRIT method are as follows.(1)
Calculate the sample covariance matrix using (17).(2)
Calculate the eigenvalue decomposition ofR̂(22)R̂=Ω̂Λ̂Ω̂H,
where, Λ̂=diag(λ̂1,λ̂2,…,λ̂m).(3)
LetΩ̂ be the matrix composed of the eigenvectors corresponding to the eigenvalues λ̂1,λ̂2,…,λ̂2L,
(23)Ω̂=[σ̂1,σ̂2,…,σ̂2L].
Define
(24)Ω̂1=[Im-10]Ω̂(m-1)×2L,Ω̂2=[0Im-1]Ω̂(m-1)×2L.(4)
The frequencies are estimated by solving the system of linear equations(25)Ω̂1Θ̂≈Ω̂2
whose least squares solution is given by
(26)Θ̂=(Ω̂1HΩ̂1)-1Ω̂1HΩ̂2.
The argument of the complex eigenvalues of Θ̂ gives the frequencies ωk. Let δk be the eigenvalue of Θ̂. Then the frequency estimates are
(27)ω̂k=-arg(δk).
The amplitudes and phases can be estimated as
(28)β̂=(GHG)-1GHx,
where β̂=[â1ejϕ̂1⋯âLejϕ̂L]T, x=[x(1)⋯x(M0)]T. The matrix G is calculated from the estimated frequencies ω̂k’s according to
(29)G=[eiω̂1⋯eiω̂L⋮⋮eiM0ω̂1⋯eiM0ω̂L],M0 indicates that the first M0 samples of the data frame x are used for ESPRIT calculations.The estimated frequencies are then used to generate a reference periodic signalr(n) shown in Figure 2 using
(30)r(n)=∑k=1L‍âkcos(ω̂kn+ϕ̂k).
The estimates âk and ϕ̂k can be further improved, by estimating q(n) from r(n), by using the adaptive cancellation technique as explained in the next section.


## 2.3.2. Signal Separation

This block essentially estimatesq̂(n) from r(n) in (30) by trying to cancel q(n) from x(n)=q(n)+ν(n) (i.e., by estimating q̂(n) from r(n)). This step adaptively finetunes the parameters âk,ϕ̂k from (30) to their actual values and also separates x(n) into  q̂(n) and ν̂(n). It can be easily seen from Figure 2 that after convergence, the error signal is ν̂(n) and the cancelling signal becomes y(n)≈-q̂(n)⇒q̂(n)≈-y(n), thus  achieving separation. The update equation for the weight vector wsep(n+1) is given by
(31)wsep(n+1)=wsep(n)+μν̂(n)r(n)rT(n)r(n).
An initial portion of the reference signal x(n) is used to estimate the tones and achieve component separation. The separated signal components will be denoted by q̂(n) and ν̂(n) to indicate that they are estimates. For high η, the estimates are highly accurate. By incorporating the fact that the effect of an arbitrary transfer function on a sinusoid can be modeled by two taps [4, 9],  computationally efficient implementations of the above separation scheme can be derived.


## 2.3.3. Generation of Antinoise Signal

This block generates the antinoise (i.e., the cancelling) signal in two steps via two parallel ANC systems,ANCq and ANCν. ANCq uses q̂(n) as the reference and generates an antinoise yq(n) which attempts to cancel dq(n). yq(n), after passing through S(z) becomes yq′(n).  The error signal eq(n) is given by
(32)eq(n)=d(n)-yq̂′(n)=dν(n)+(dq(n)-yq̂′(n)),
where d(n) denotes the desired noise to be cancelled and yq̂(n) denotes the anti-noise generated by ANCq as shown in Figure 2. The adaptive filter Wq(z) and S(z) linearly operate on the periodic signal q̂(n) to produce yq̂′(n) which lies in the same subspace as that of q̂(n). Since dν(n) is uncorrelated with dq(n), convergence in this case essentially means that (dq(n)-yq̂′(n)) is minimised in the mean square sense. Thus after convergence, from (32)
(33)eq(n)≈dν(n).
The update equation for the weight vector wq(n+1) is given by
(34)wq(n+1)=wq(n)+μeq(n)q̂′(n)q̂′T(n)q̂′(n),
where q̂′(n) is the signal vector obtained after filtering q̂(n) with Ŝ(z). After convergence is attained in ANCq, the adaptation is stopped and wq(n+1) is fixed to the steady-state value. The system ANCq continues to generate antinoise which cancels dq(n). The error microphone's output is dν(n). The next step is to switch on ANCν whose reference input is the separated random part ν̂(n) and error input is eν(n) which it tries to minimize by generating the antinoise signal yν̂(n).The update equation for the weight vectorwν(n+1) is given by
(35)wν(n+1)=wν(n)+μeν(n)ν̂′(n)ν̂′T(n)ν̂′(n),
where ν̂′(n) is the signal obtained after filtering ν̂(n) with Ŝ(z). After convergence the error is eν(n) and should not contain any component of the periodic noise as long as the acoustic system is stationary. Since we have access to only one error sequence, the procedure for finding the optimum Wq(z) and Wν(z) is sequential. However, this sequential adaptive tuning procedure can be repeated when the error sequence energy goes above a prespecified threshold, since the changes in the acoustic system will result in an increase in the energy of the error sequence.It should be noted that the adaptation methods presented here for the implementation ofANCq and ANCν are based on simple FXLMS for the sake of simplicity and quick experimentation. For the tonal active noise control part ANCq, the analysis and methods described in [9–12] can be incorporated. The proposed method can also be extended to accommodate multiple reference microphones in which case the reduced reference signal set procedure developed in [19] can lessen the controller dimension and computational complexity.


## 3. Calculation ofNALmax

The goal of this section is to sketch the method for calculatingNALmax.  NALmax is obtained, assuming that the adaptive ANC system has converged and that the adaptive ANC system is linear time-invariant (stationary), as shown in Figure 5(a). First the NALmax for wide sense stationary random noise is derived and then expanded to accommodate sinusoidal mixtures in a wide sense stationary random noise. This derivation is important since it gives an insight into how well the proposed method performs when compared to conventional ANC structure with a single cancelling filter.  NALmax calculation is also very useful during actual real-time implementation of an ANC system. Based on a few basic measurements on an actual real-time ANC setup,  the computation of NALmax gives an indication of the expected performance. This comes in handy when experimenting with real-time ANC systems like the acoustic laboratory shown in Figure 9. Once the maximum limit is known, the sources of error in an underperforming system can be analyzed.(a) Feedforward ANC system  (b) interchanging theW(z) and S(z) blocks owing to their linear time-invariant nature.
(a)(b)In the three subsections that follow, the expressions forNALmax are derived for three cases: when the acoustic noise is (i) wide-sense stationary, (ii) a sum of sinusoidal signals, and (iii) a mixture of sinusoids and a wide-sense stationary random noise, that is, combination of (i) and (ii).


### 3.1.NALmax for Wide-Sense Stationary Signals

In linear estimation problems involving stationary signals, Wiener filtering provides the best estimate of a linear time-invariant (LTI) system based on (minimum mean square error) MMSE criterion [20, 21].  In a linear stationary ANC system as shown in Figure 5(a), the transfer function W(z)=P(z)/S(z) often has poles outside the unit circle, that is,  when S(z) is nonminimum-phase. In such cases, a stable noncausal Wiener filter is the best estimate of W(z)=P(z)/S(z). An expression for MMSE using noncausal Wiener filter is given in [4, 19]. However if x(n) is not perfectly predictable, a  noncausal Wiener filter  is not realizable. The realizeable MMSE estimate is then the causal and stable Wiener filter that approximates P(z)/S(z).In order to derive a measurable expression forNALmax, let Wco(z) be the causal stable IIR Wiener filter that best approximates P(z)/S(z):
(36)Wco(z)≈P(z)S(z),
with the superscript and subscript denoting the optimality and causality, respectively, of the solution.The expression forWco(z) is obtained as follows:LetΨuv(z) be the cross-power spectral density of two jointly stationary processes u(n) and v(n). The expression for Ψuv(z) is
(37)Ψuv(z)=∑n=-∞n=+∞Ruv(n)z-n,
where Ruv(n) denotes the cross-covariance sequence. Ψuu(z) denotes the power spectral density of a stationary process u(n) and can be expressed using (37) after appropriate substitution.Assuming the signals in the system have real positive (power spectrum density) PSD satisfying Paley-Wiener condition [21], the stable causal Wiener solution is given in [20, 22]. Referring to Figure 5(b) the causal Wiener solution is
(38)Wc∘(z)=1Ψx′(z){Ψdx′(z)Ψx′(z-1)}+,
where {·}+ denotes the causal part of the quantity inside. The subscript c stands for causality. Ψx′(z) is the minimum-phase and causal spectral factor of Ψx′x′(z), that is,
(39)Ψx′x′(z)=Ψx′(z)Ψx′(z-1).
The next few steps are devoted to the computation of Ψdx′(z) and Ψx′(z) in terms of P(z), S(z) and the input PSD Ψxx(z). The transfer function that maps x(n) to x′(n) is S(z) and the transfer function that maps x′(n) to d(n) is  P(z)/S(z). Thus using the theory of linear filtering of random processes [21, 23] Ψdx′(z) is related to Ψxx(z) as follows:
(40)Ψdx′(z)=P(z)S(z)Ψx′x′(z)=P(z)S(z)(S(z)S(z-1)Ψxx(z))=S(z-1)P(z)Ψxx(z).
It should be noted that(41)Ψx′x′(z)=S(z)S(z-1)Ψxx(z)=S(z)S(z-1)Ψx(z)Ψx(z-1),
where, the spectral factorization of Ψxx(z) is
(42)Ψxx(z)=Ψx(z)Ψx(z-1),Ψx(z) is minimum-phase and causal. S(z) can be decomposed into Smin(z) and Smax(z) as follows:
(43)S(z)=Smin(z)Smax(z).Smax(z) is a polynomial that contains only the zeros outside the unit circle and is responsible for the noncausality of P(z)/S(z). Hence it follows that Smin(z) is a minimum phase, stable and rational function. Thus (41) becomes(44)Ψx′x′(z)=Smin(z)Smax(z)Ψx(z)︸causalSmin(z-1)Smax(z-1)Ψx(z-1)︸noncausal,Smax(z-1) is a polynomial of positive powers of z and hence is noncausal. The causal Smax,c(z) can be expressed as
(45)Smax,c(z)=z-rSmax(z-1),
where r is the order of Smax(z-1).It should be noted that any stable transfer function can also be written as a product of a stable minimum-phase and a stable all-pass system. In this case(46)S(z)=Smin(z)z-rSmax(z-1)Smax(z)z-rSmax(z-1)=Smin(z)Smax,c(z-1)Smax(z)Smax,c(z).By definition,Smin(z)Smax,c(z-1) is a minimum-phase system and will be denoted by SMIN(z).Smax(z)/Smax,c(z) is an all-pass transfer function and will be denoted as SAP(z). It should be noted that the inverse of SAP(z) is SAP(z-1). The above calculations lead to the familiar minimum-phase, all-pass decomposition [22](47)S(z)=SMIN(z)SAP(z).Equation  (44) can now be written as
(48)Ψx′(z)Ψx′(z-1)=[Smin(z)Smax,c(z)Ψx(z)]︸min-phase,causal[Smin(z-1)Smax,c(z-1)Ψx(z-1)]︸nonmin-phase,noncausal.
The minimum-phase and causal Ψx′(z) is computed as
(49)Ψx′(z)=Smin(z)Smax,c(z)Ψx(z).
Substituting (49) and (40) in (38)
(50)Wc∘(z)=1Smin(z)Smax,c(z)Ψx(z)×{P(z)·Ψx(z)·Smax,c(z)Smax(z)}+
which can be written as
(51)Wc∘(z)=1SMIN(z)Ψx(z)×{P(z)·Ψx(z)·SAP(z-1)}+
and the corresponding output error of the ANC system is computed by
(52)ec∘(n)=d(n)-y′(n)=p(n)*x(n)-wc∘(n)*s(n)*x(n),
where * denotes the convolution operation.When using numerical software, it should be noted that even though the form ofWc∘(z) in (51) is IIR, it should be considered as a very long FIR filter. The calculation of (51) can be done entirely in the frequency domain by using numerical software like MATLAB. The minimum-phase, maximum-phase decomposition can be done employing cepstral techniques [22]. The causal part can be found by taking the (Inverse Fast Fourier Transform) IFFT of the quantity inside the bracket and discarding the negative index terms. The maximum possible NAL, that is, the upper bound, can then be computed using
(53)NALmax,xrandom=20log10(∥(d(m))∥2∥(ec∘(m))∥2)
for
(54)n-Lf2<m≤n+Lf2,
where Lf is the frame length and n the frame position and ∥·∥2 denotes the 2-norm operation.It should be noted thatNALmax,xrandom given in (53) can be computed as long as the spectral factorization of (42) is valid.


### 3.2.NALmax for a Mixture of Sinusoids with No Random Noise

Whenx(n) contains sinusoidal components with no random noise, the power spectral density Ψxx(z) of x(n) becomes singular. Hence the technique developed in Section 3.1 cannot be applied. The way out of this difficulty is by recognising that for a pure mixture of sinusoids or a purely periodic signal without any background random noise, the effect of nonminimum phase secondary path can be perfectly equalised. Hence perfect cancellation can be acheived. The nonminimum phase nature essentially translates to a noncausal equaliser which can be realized for purely periodic signals because the future values of the signal are known by indexing the past, that is, by knowing one past period of the signal. For example placing a (noncausal) equalizer Heq(z)=SAP(z-1) as shown in Figure 6 can perfectly equalize a nonminimum phase secondary path. As a result, the effective secondary path becomes the minimum-phase SMIN(z), which can be exactly equalized since it is perfectly invertible. Factorizing S(z) as SMIN(z)SAP(z), the equalizer Heq(z) is obtained as
(55)Heq(z)=∑k=0LS-1‍sAP(-n)z-k≈SAP(z-1),
where LS is the order of SAP(z) (after sufficient truncation) when expressed in FIR form. Thus heq(n)=sAP(-n). The equalization operations are summarized as
(56)xeq(n)=∑k=1LS-1‍heq(k)x(n+k)=∑k=1LS-1‍heq(k)x(n+k-mNp),
where Np is the period and m=⌊k/Np⌋+1.  Here ⌊·⌋  denotes the integer floor. For cases for which Np≥LS,
(57)xeq(n)=∑k=0LS-1‍heq(k)x(n+k-Np).
Thus in theory, a pure mixture of sinusoids can be perfectly cancelled. Thus arbitrarily high noise attenuation level can be reached, that is, NALmaxsinusoids→∞.Figure 6
Perfect nonminimum phase equalization for periodic signals.Heq(z) can be noncausal since the input x(n) is periodic.


### 3.3.NALmax for Mixture of Sinusoids  in Wide-Sense Stationary Random Noise

Whenx(n) contains sinusoidal components in addition to the wide-sense stationary random noise, the upper bound becomes a function of η and the NALmax associated with the random noise part. The derivation below is fundamentally based on the fact that every stationary sequence can be expressed as a sum of a perfectly predictable sequence and a stochastic sequence which are orthogonal to each other (Wold’s Decomposition) [24].  In this case the sinusoidal components constitute the perfectly predictable part and the background broadband noise component constitutes the stochastic part.The noise attenuation levelΓx by definition is given by
(58)Γx=ξdξe=ξdξeq+ξeν,
where ξd is the power of d(n), ξe is the power of e(n), ξeq is the power of the tonal component (after convergence) of  e(n), and ξeν is the power of the random component (after convergence) of e(n).Due to linearity,d(n)=p(n)*(x(n))=p(n)*(q(n)+ν(n))=dq(n)+dν(n). Since  the deterministic component q(n) and the random component ν(n) are independent,
(59)ξdx=ξdq+ξdν,(60)Γx=ξdq+ξdνξeq+ξeν.
The performance gain due to the proposed separation-based parallel feedforward method can be understood from (60). By using a dedicated cancelling filter, the periodic part can be almost perfectly canceled, that is, ξeq≈0. Perfect cancellation of the periodic part is not possible with conventional single filter  ANC. Hence the denominator of (60) is always higher for the conventional method. On the assumption that the periodic part can be perfectly cancelled, (60) becomes
(61)Γx≈ξdq+ξdνξeν,Γx in (61) defines an upper bound for the ANC system since it is the maximum NAL possible under the assumptions of perfect separation and perfect cancellation of the periodic part q(n). Thus for a mixture of periodic tones in random noise, the maximum achievable NAL, that is, the NALmax,xmixture is given by Γx.  Denoting the NAL of an ANC system to a random noise input by Γν=ξdν/ξeν, the expression in (61) can be written as
(62)NALmax,xmixture=Γx=(ξdνξeν+ξdνξeνξdqξdν)=Γν+Γνξdqξdν=Γν(1+ξdqξdν).
The above expression generalizes the form of maximum achievable NAL derived for stationary random noise in Section 3.1 to stationary random noise along with sinusoids. It should be noted that Γν of (62) is calculated from (51)–(53) after setting Ψxx(z)=Ψνν(z), that is, Γν=NALmax,vrandom. To see the effect of the primary path P(z) it should be noticed that
(63)Γx=Γν(1+∑k=1L‍(Ak2/2)|P(ejωk)|2ξν((1/2π)∫-π+π|P(ejω)|2dω)),
where ωk’s are the frequencies of the sinusoids in q(n)=∑k=1L‍akcos(ωkn+ϕk).In case of a single tone (L=1),
(64)Γx=Γν(1+η1|P(ejω1)|2((1/2π)∫-π+π|P(ejω)|2dω)),
and for
(65)α1=|P(ejω1)|2((1/2π)∫-π+π|P(ejω)|2dω).
The expression becomes
(66)Γx=Γν(1+η1α1).
For the parallel ANC system, (66) can be written as(67)NALmax,xmixture=NALmax,νrandom(1+η1α1).
That is, the improvement in NAL of a mixture of periodic and random noise signals is greater than that of the performance of a purely random noise by a factor of (1+η1α1). If we assume η1α1≫1, then Γx≈Γνη1α1. It seems that this way of characterizing maximum achievable performances of ANC system is new. Furthermore, α1 is low when there are deep notches in P(ejω) at the frequency ω1, in which case the relative position of reference microphone should be changed.As a summary, the procedure described in the paper is as follows.Step 1.
Given a feedforward ANC set up as shown in Figure9, record x(n) and estimate P(z) and S(z). The same ANC setup can be used for both measurements. The primary path P(z) can be measured switching off the anti-noise generation (i.e., y(n)=0) and by recording the measurements x(n) from the reference microphone and d(n) from error microphone, and estimating the transfer function that maps x(n) to d(n). For measurement of S(z), methods given in [4] can be employed.Step 2.
Ifx(n) does not have tones use (51) through (53) to calculate NALmax.  The conventional single-filter approach is sufficient for this case. Additionally the optimum Wiener filter can also be calculated from (51) and can be used to initialize the adaptive filter W(z).Step 3.
Ifx(n) has tones along with random noise, use (67) to calculate the NALmax.  This step needs calculation of the power spectral density of ν(n) which can be generated from x(n) using the adaptive noise cancellation-based separation that was developed in Sections 2.3.1 and 2.3.2.Step 4.
Implement the PFANC scheme.ANCq requires the calculation of the period Np from the autocorrelation sequence of x(n) and the calculation of Heq(z) from (55). Additionally the optimum Wiener filter can also be calculated from (51) and can be used to initialize the adaptive filter Wν(z).


## 3.1.NALmax for Wide-Sense Stationary Signals

In linear estimation problems involving stationary signals, Wiener filtering provides the best estimate of a linear time-invariant (LTI) system based on (minimum mean square error) MMSE criterion [20, 21].  In a linear stationary ANC system as shown in Figure 5(a), the transfer function W(z)=P(z)/S(z) often has poles outside the unit circle, that is,  when S(z) is nonminimum-phase. In such cases, a stable noncausal Wiener filter is the best estimate of W(z)=P(z)/S(z). An expression for MMSE using noncausal Wiener filter is given in [4, 19]. However if x(n) is not perfectly predictable, a  noncausal Wiener filter  is not realizable. The realizeable MMSE estimate is then the causal and stable Wiener filter that approximates P(z)/S(z).In order to derive a measurable expression forNALmax, let Wco(z) be the causal stable IIR Wiener filter that best approximates P(z)/S(z):
(36)Wco(z)≈P(z)S(z),
with the superscript and subscript denoting the optimality and causality, respectively, of the solution.The expression forWco(z) is obtained as follows:LetΨuv(z) be the cross-power spectral density of two jointly stationary processes u(n) and v(n). The expression for Ψuv(z) is
(37)Ψuv(z)=∑n=-∞n=+∞Ruv(n)z-n,
where Ruv(n) denotes the cross-covariance sequence. Ψuu(z) denotes the power spectral density of a stationary process u(n) and can be expressed using (37) after appropriate substitution.Assuming the signals in the system have real positive (power spectrum density) PSD satisfying Paley-Wiener condition [21], the stable causal Wiener solution is given in [20, 22]. Referring to Figure 5(b) the causal Wiener solution is
(38)Wc∘(z)=1Ψx′(z){Ψdx′(z)Ψx′(z-1)}+,
where {·}+ denotes the causal part of the quantity inside. The subscript c stands for causality. Ψx′(z) is the minimum-phase and causal spectral factor of Ψx′x′(z), that is,
(39)Ψx′x′(z)=Ψx′(z)Ψx′(z-1).
The next few steps are devoted to the computation of Ψdx′(z) and Ψx′(z) in terms of P(z), S(z) and the input PSD Ψxx(z). The transfer function that maps x(n) to x′(n) is S(z) and the transfer function that maps x′(n) to d(n) is  P(z)/S(z). Thus using the theory of linear filtering of random processes [21, 23] Ψdx′(z) is related to Ψxx(z) as follows:
(40)Ψdx′(z)=P(z)S(z)Ψx′x′(z)=P(z)S(z)(S(z)S(z-1)Ψxx(z))=S(z-1)P(z)Ψxx(z).
It should be noted that(41)Ψx′x′(z)=S(z)S(z-1)Ψxx(z)=S(z)S(z-1)Ψx(z)Ψx(z-1),
where, the spectral factorization of Ψxx(z) is
(42)Ψxx(z)=Ψx(z)Ψx(z-1),Ψx(z) is minimum-phase and causal. S(z) can be decomposed into Smin(z) and Smax(z) as follows:
(43)S(z)=Smin(z)Smax(z).Smax(z) is a polynomial that contains only the zeros outside the unit circle and is responsible for the noncausality of P(z)/S(z). Hence it follows that Smin(z) is a minimum phase, stable and rational function. Thus (41) becomes(44)Ψx′x′(z)=Smin(z)Smax(z)Ψx(z)︸causalSmin(z-1)Smax(z-1)Ψx(z-1)︸noncausal,Smax(z-1) is a polynomial of positive powers of z and hence is noncausal. The causal Smax,c(z) can be expressed as
(45)Smax,c(z)=z-rSmax(z-1),
where r is the order of Smax(z-1).It should be noted that any stable transfer function can also be written as a product of a stable minimum-phase and a stable all-pass system. In this case(46)S(z)=Smin(z)z-rSmax(z-1)Smax(z)z-rSmax(z-1)=Smin(z)Smax,c(z-1)Smax(z)Smax,c(z).By definition,Smin(z)Smax,c(z-1) is a minimum-phase system and will be denoted by SMIN(z).Smax(z)/Smax,c(z) is an all-pass transfer function and will be denoted as SAP(z). It should be noted that the inverse of SAP(z) is SAP(z-1). The above calculations lead to the familiar minimum-phase, all-pass decomposition [22](47)S(z)=SMIN(z)SAP(z).Equation  (44) can now be written as
(48)Ψx′(z)Ψx′(z-1)=[Smin(z)Smax,c(z)Ψx(z)]︸min-phase,causal[Smin(z-1)Smax,c(z-1)Ψx(z-1)]︸nonmin-phase,noncausal.
The minimum-phase and causal Ψx′(z) is computed as
(49)Ψx′(z)=Smin(z)Smax,c(z)Ψx(z).
Substituting (49) and (40) in (38)
(50)Wc∘(z)=1Smin(z)Smax,c(z)Ψx(z)×{P(z)·Ψx(z)·Smax,c(z)Smax(z)}+
which can be written as
(51)Wc∘(z)=1SMIN(z)Ψx(z)×{P(z)·Ψx(z)·SAP(z-1)}+
and the corresponding output error of the ANC system is computed by
(52)ec∘(n)=d(n)-y′(n)=p(n)*x(n)-wc∘(n)*s(n)*x(n),
where * denotes the convolution operation.When using numerical software, it should be noted that even though the form ofWc∘(z) in (51) is IIR, it should be considered as a very long FIR filter. The calculation of (51) can be done entirely in the frequency domain by using numerical software like MATLAB. The minimum-phase, maximum-phase decomposition can be done employing cepstral techniques [22]. The causal part can be found by taking the (Inverse Fast Fourier Transform) IFFT of the quantity inside the bracket and discarding the negative index terms. The maximum possible NAL, that is, the upper bound, can then be computed using
(53)NALmax,xrandom=20log10(∥(d(m))∥2∥(ec∘(m))∥2)
for
(54)n-Lf2<m≤n+Lf2,
where Lf is the frame length and n the frame position and ∥·∥2 denotes the 2-norm operation.It should be noted thatNALmax,xrandom given in (53) can be computed as long as the spectral factorization of (42) is valid.


## 3.2.NALmax for a Mixture of Sinusoids with No Random Noise

Whenx(n) contains sinusoidal components with no random noise, the power spectral density Ψxx(z) of x(n) becomes singular. Hence the technique developed in Section 3.1 cannot be applied. The way out of this difficulty is by recognising that for a pure mixture of sinusoids or a purely periodic signal without any background random noise, the effect of nonminimum phase secondary path can be perfectly equalised. Hence perfect cancellation can be acheived. The nonminimum phase nature essentially translates to a noncausal equaliser which can be realized for purely periodic signals because the future values of the signal are known by indexing the past, that is, by knowing one past period of the signal. For example placing a (noncausal) equalizer Heq(z)=SAP(z-1) as shown in Figure 6 can perfectly equalize a nonminimum phase secondary path. As a result, the effective secondary path becomes the minimum-phase SMIN(z), which can be exactly equalized since it is perfectly invertible. Factorizing S(z) as SMIN(z)SAP(z), the equalizer Heq(z) is obtained as
(55)Heq(z)=∑k=0LS-1‍sAP(-n)z-k≈SAP(z-1),
where LS is the order of SAP(z) (after sufficient truncation) when expressed in FIR form. Thus heq(n)=sAP(-n). The equalization operations are summarized as
(56)xeq(n)=∑k=1LS-1‍heq(k)x(n+k)=∑k=1LS-1‍heq(k)x(n+k-mNp),
where Np is the period and m=⌊k/Np⌋+1.  Here ⌊·⌋  denotes the integer floor. For cases for which Np≥LS,
(57)xeq(n)=∑k=0LS-1‍heq(k)x(n+k-Np).
Thus in theory, a pure mixture of sinusoids can be perfectly cancelled. Thus arbitrarily high noise attenuation level can be reached, that is, NALmaxsinusoids→∞.Figure 6
Perfect nonminimum phase equalization for periodic signals.Heq(z) can be noncausal since the input x(n) is periodic.


## 3.3.NALmax for Mixture of Sinusoids  in Wide-Sense Stationary Random Noise

Whenx(n) contains sinusoidal components in addition to the wide-sense stationary random noise, the upper bound becomes a function of η and the NALmax associated with the random noise part. The derivation below is fundamentally based on the fact that every stationary sequence can be expressed as a sum of a perfectly predictable sequence and a stochastic sequence which are orthogonal to each other (Wold’s Decomposition) [24].  In this case the sinusoidal components constitute the perfectly predictable part and the background broadband noise component constitutes the stochastic part.The noise attenuation levelΓx by definition is given by
(58)Γx=ξdξe=ξdξeq+ξeν,
where ξd is the power of d(n), ξe is the power of e(n), ξeq is the power of the tonal component (after convergence) of  e(n), and ξeν is the power of the random component (after convergence) of e(n).Due to linearity,d(n)=p(n)*(x(n))=p(n)*(q(n)+ν(n))=dq(n)+dν(n). Since  the deterministic component q(n) and the random component ν(n) are independent,
(59)ξdx=ξdq+ξdν,(60)Γx=ξdq+ξdνξeq+ξeν.
The performance gain due to the proposed separation-based parallel feedforward method can be understood from (60). By using a dedicated cancelling filter, the periodic part can be almost perfectly canceled, that is, ξeq≈0. Perfect cancellation of the periodic part is not possible with conventional single filter  ANC. Hence the denominator of (60) is always higher for the conventional method. On the assumption that the periodic part can be perfectly cancelled, (60) becomes
(61)Γx≈ξdq+ξdνξeν,Γx in (61) defines an upper bound for the ANC system since it is the maximum NAL possible under the assumptions of perfect separation and perfect cancellation of the periodic part q(n). Thus for a mixture of periodic tones in random noise, the maximum achievable NAL, that is, the NALmax,xmixture is given by Γx.  Denoting the NAL of an ANC system to a random noise input by Γν=ξdν/ξeν, the expression in (61) can be written as
(62)NALmax,xmixture=Γx=(ξdνξeν+ξdνξeνξdqξdν)=Γν+Γνξdqξdν=Γν(1+ξdqξdν).
The above expression generalizes the form of maximum achievable NAL derived for stationary random noise in Section 3.1 to stationary random noise along with sinusoids. It should be noted that Γν of (62) is calculated from (51)–(53) after setting Ψxx(z)=Ψνν(z), that is, Γν=NALmax,vrandom. To see the effect of the primary path P(z) it should be noticed that
(63)Γx=Γν(1+∑k=1L‍(Ak2/2)|P(ejωk)|2ξν((1/2π)∫-π+π|P(ejω)|2dω)),
where ωk’s are the frequencies of the sinusoids in q(n)=∑k=1L‍akcos(ωkn+ϕk).In case of a single tone (L=1),
(64)Γx=Γν(1+η1|P(ejω1)|2((1/2π)∫-π+π|P(ejω)|2dω)),
and for
(65)α1=|P(ejω1)|2((1/2π)∫-π+π|P(ejω)|2dω).
The expression becomes
(66)Γx=Γν(1+η1α1).
For the parallel ANC system, (66) can be written as(67)NALmax,xmixture=NALmax,νrandom(1+η1α1).
That is, the improvement in NAL of a mixture of periodic and random noise signals is greater than that of the performance of a purely random noise by a factor of (1+η1α1). If we assume η1α1≫1, then Γx≈Γνη1α1. It seems that this way of characterizing maximum achievable performances of ANC system is new. Furthermore, α1 is low when there are deep notches in P(ejω) at the frequency ω1, in which case the relative position of reference microphone should be changed.As a summary, the procedure described in the paper is as follows.Step 1.
Given a feedforward ANC set up as shown in Figure9, record x(n) and estimate P(z) and S(z). The same ANC setup can be used for both measurements. The primary path P(z) can be measured switching off the anti-noise generation (i.e., y(n)=0) and by recording the measurements x(n) from the reference microphone and d(n) from error microphone, and estimating the transfer function that maps x(n) to d(n). For measurement of S(z), methods given in [4] can be employed.Step 2.
Ifx(n) does not have tones use (51) through (53) to calculate NALmax.  The conventional single-filter approach is sufficient for this case. Additionally the optimum Wiener filter can also be calculated from (51) and can be used to initialize the adaptive filter W(z).Step 3.
Ifx(n) has tones along with random noise, use (67) to calculate the NALmax.  This step needs calculation of the power spectral density of ν(n) which can be generated from x(n) using the adaptive noise cancellation-based separation that was developed in Sections 2.3.1 and 2.3.2.Step 4.
Implement the PFANC scheme.ANCq requires the calculation of the period Np from the autocorrelation sequence of x(n) and the calculation of Heq(z) from (55). Additionally the optimum Wiener filter can also be calculated from (51) and can be used to initialize the adaptive filter Wν(z).


## 4. Experimental Results

Simulation of the proposed method, using FXLMS for adaptation, was implemented by synthesizingx(n)=q(n)+ν(n) for different SNR values η and different linear combinations of multiple sinusoids. Single and multitone signals of various frequencies were synthesized and ν(n) of varying power was added. The P(z) and S(z) used in these experiments were obtained from [4] and characterize real life acoustic transfer functions (Figure 7).Figure 7
Magnitude responses of the primary and secondary paths used in the simulation.Each experiment was done on a 100000 sample longx(n) generated at a sampling rate of 16 kHz. The first 10000 samples were used for the frequency estimation part, that is, determination of L, ωk, ak, ϕk.The next 20000 samples were used to train the adaptive noise canceller that is, to separate  q(n) and ν(n) as discussed in Section 2.3.2.  The rest of the samples were used to generate antinoises via ANCq and ANCν. The performance of the proposed method is compared with the conventional FXLMS-based ANC [4]. The results are shown in Table 1. The last column shows the maximum achievable noise attenuation level NALmax,xmixture using (62). NALmax,νrandom was found to be 19.58 dB for the P(z) and S(z) used.   It can be seen that the NAL obtained by the proposed method is closer to NALmax,xmixture than that obtained by the conventional single-filter method. For low SNR values (like 10 dB) the performance improvement is less noticeable when compared to higher SNR values. For higher SNR values excellent performance of the proposed method can be observed and thus the proposed method targets high SNR applications. It should also be noticed that perfect separation and cancellation of the periodic part may not be achieved in practice resulting in maximum NAL being less than that given by (62).Table 1
Performance comparison of the proposed separation-based parallel ANC and conventional single-filter ANC for different tones and SNR combination.Γx (dB) is the maximum possible NAL.
NumberNumber of sinusoids,Lη(dB)Proposed PFANC NAL (dB)Single-Filter ANC NAL (dB)NALmax,xmixture from (62)111025.5724.6126.38212033.6920.1636.33313044.5625.9746.24414053.0335.6056.24521031.8333.2134.20622042.8832.2044.09723052.3435.9154.05824058.0044.6464.00931031.9631.8633.001032041.9930.6943.001133050.0335.3453.801234055.8644.4364.001341031.5431.4333.001442040.3630.542.671543048.3235.4953.641644058.4044.3563.002171031.4530.0531.722272041.4841.8742.052373049.0344.6554.022474057.4047.6562.951781031.4530.6430.641882044.4042.1443.251983042.1429.5246.792084055.4047.0663.00Experiments 21 through 24 were carried out using a signal derived from functional magnetic resonance imaging (fMRI) acoustic noise [25, 26]. fMRI acoustic noise has SNR’s > 20 dB and hence amenable to the proposed approach. Seven significant tones from a recorded fMRI acoustic noise were estimated and the sinusoidal component q(n) was created from the seven tones. White noise was then added with different SNR values. Figures 8(a) and 8(b)  present a comparison of noise cancellation performance between the conventional single-filter method and the proposed PFANC for an 8 tones case and a 10 tones case, respectively.  As it can be seen, PFANC considerably reduces the background noise in addition to sinusoidal noise.(a) Comparison of the conventional FXLMS with single filter and the proposed PFANC for 8 tones in white noise. (b) Comparison of the conventional FXLMS with single filter and the proposed PFANC for 10 tones in white noise.
(a)(b)Figure 9
The real-time Active Noise Control setup. The transparent semicircular structure mimics the MRI cavity. Recorded noise from actual MRI room is used to drive the speakers to recreate MRI environment. The area around the manikin’s ears is where cancellation is desired.


## 5. Conclusion

In this paper the focus has been on active noise control of acoustic signals that  have a periodic/harmonic structure along with a random noise. The key idea stems from the observation that the periodic component and the random component may need different noise cancelling filters for the best performance. This leads to the idea ofseparation  of the periodic and random components and the proposed parallel form of ANC, in which the reference input is separated into periodic and random components and separate anti-noise signals are parallely generated. A novel adaptive filtering-based implementation of signal separation and anti-noise generation is developed and simulated. A simple but an insightful formula is derived to quantify performance. The method is shown to work very well with single and multi-tone signals in white noise.Some important and recurring signals, blocks, and notation that are used in the paper are listed in the Notations and Definitions section.  Other definitions are made as they are introduced.



---

*Source: 694290-2009-06-24.xml*