Multi-Tone Active Noise Equalizer With Spatially Distributed User-Selected Profiles

In work we propose a multi-channel active noise equalizer (ANE) that can deal with multi-frequency noise signals and assigns simultaneously different equalization gains to each frequency component at each monitoring sensor. For this purpose, we state a pseudo-error noise signal for each sensor of the ANE, which has to be cancelled out in order to get the desired equalization profiles. Firstly, the optimal analytic solution for the ANE filters in the case of single-frequency noise is provided, and an adaptive algorithm based on the Least Mean Squared (LMS) is proposed for the same case. We also show that this adaptive strategy reaches the theoretical solution in steady state. Secondly, we state an equivalent approach for the case of multi-frequency noise based on two alternatives: a common pseudo-error signal at each sensor for all frequencies, and a different pseudo-error signal at each sensor for each frequency. The analytic and adaptive solutions for the ANE control filters have been developed for both pseudo-error alternatives. Finally, the ability of the proposed ANE to achieve simultaneously different user-selected noise profiles in different locations has been validated by their transfer functions and simulations.

Some real life situations require simultaneously different noise profiles at different locations within an enclosure, which gives rise to the need of a personalized active equalization of the multifrequency noise at each position. In this sense, Fig. 1 shows a system with K control points or sensors where a particular gain is required at each sensor position. The primary noise signal generated by the noise source, x(n) = A ref cos(ω 0 n + φ ref ), arrives at each kth sensor filtered through the acoustic path between this noise signal and the sensor, p k (n), which is usually modelled as a Finite Impulse Response (FIR) filter and it is called primary acoustic path. We call d k (n) the noise signal that is recorded at each sensor, d k (n) = x(n) * p k (n), where * This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ stands for the convolution operation. Without loss of generality, we assume at this point that the original noise contains only the frequency f 0 with ω 0 = 2πf 0 . However, in Section III we will extend the analysis to noise sources whose spectral content includes multiple frequencies. As said before, the system should provide a particular noise profile at each sensor. For this purpose, J secondary sources or loudspeakers are used, whose output signals y j (n), j = 1, . . . , J, result from filtering the noise signal x(n) through a filter h j , as shown in Fig. 1. It should be noted that the reference signal of the system and the noise that is emitted by the source are assumed to be the same signal x(n) in Fig. 1. In case that the reference signal differs in amplitude or phase from the original noise, the proposed model would still be valid by simply including the phase and amplitude differences within the phase and amplitude of their respective primary transfer functions, P k (ω 0 ) 1 , k = 1, . . . , K. In this sense, a reference signal can easily be synthesized just knowing the frequency of the noise signal without knowing its exact phase and amplitude.
Taking into account that the acoustic transfer function between the secondary source j and the sensor k at ω 0 is denoted by C jk (ω 0 ), and that P k (ω 0 ) is the transfer function between the original noise source and the sensor kth at the same frequency, then, using linear superposition (see Fig. 1), we can express the error signal as where Y j (ω 0 ) is the Fourier transform (FT) of the output signal y j (n) of loudspeaker j at ω 0 . E k (ω 0 ) and D k (ω 0 ) are the FTs of the error signal and the desired error signal at the kth sensor respectively, both particularized for ω 0 . We also assume that a particular residual error noise signal e k (n) is desired at each sensor location. Let us express this requirement on the signal picked up by the kth sensor, e k (n), such that where E k (ω 0 ) is the FT of the desired error signal at the kth sensor, and β k (β k ≥ 0 and β k = 1), is a constant that sets the desired equalization profile for ω 0 at this sensor location. Substituting (2) into (1), the following expression must be fulfilled for every sensor: The equality stated in (3) can be expressed in matrix form to include all the sensors as 1 Throughout the paper the transfer function of a system is denoted by the z-transform of the respective time impulse response particularized at z = e iω . where . . . . . . . . . . . .
is a matrix containing the acoustic transfer functions of the secondary paths, is a vector that contains the transfer function of the filters that feed each loudspeaker and is a vector with the acoustic transfer functions of the primary paths weighted by the corresponding gain (β k − 1). The feasibility of the scheme shown in Fig. 1 to fulfill (4) depends mostly on the acoustic system characteristics, in particular on the matrix C. If the system has more loudspeakers than control points, J > K, then (4) is underdetermined and its minimum norm solution is given by as long as C is full rank, which becomes h opt = C −1 p β , when K = J. If the system in (4) has more control points than loudspeakers, K > J, there are not enough degrees of freedom to reach the desired solution.
Considering now (4)- (7), whose solution are the filter coefficients (6) for each loudspeaker j, it is very unlikely to know or even estimate the transfer functions P k (ω 0 ) in practice. However, adaptive systems do not usually know P k (ω 0 ). Furthermore, adaptive strategies can evolve with changes of the initial system setup. In this work we present an adaptive strategy for the solution of (4)-(7) that provides the required attenuation β k at each sensor position, as described in (2). We will also show that, on average, the steady-state solution of the adaptive algorithm is equal to the solution of (4). Additionally, we will extend the proposed adaptive strategy to signals composed of multiple frequencies.
The rest of the paper is organized as follows: in Section II the Minimum Mean Square pseudo Error solution for the filters h in (4) is obtained in closed-form, provided that the transfer functions of the primary paths P k (ω 0 ) are known. Since P k (ω 0 ) are usually unknown, an adaptive algorithm based on the Multiple Error Filtered-x LMS (MEFxLMS) scheme [31] is developed, together with the operating conditions to assure stability. Section III presents two strategies for the extension of the proposed algorithm to deal with multiple frequency noise profiles, while Section IV presents some relevant simulation results. Finally, Section V summarizes the conclusions. Furthermore, Appendices A and B show the development of the adaptive filtering expressions for the multi-frequency controllers, and Appendix C illustrates the steady-state analysis of both multifrequency equalization strategies. Throughout this paper the following notation will be used: scalars and signals are written in italic lowercase letters, a, values of the transfer functions in italic uppercase, A, vectors in lowercase and bold, a, while matrices are shown in uppercase and bold, A. The following symbols and operators used are: i = √ −1, Re{·} and Im{·} are respectively the real and imaginary part of a complex number, E{·} denotes the statistical mean operator and tr{·} stands for the trace of a matrix. Finally, diag{·} denotes a diagonal matrix whose diagonal is formed by the elements of its argument.

II. MULTI-CHANNEL ANE FOR SINGLE-FREQUENCY NOISE
A multi-channel adaptive equalizer that weights a multifrequency noise signal by an arbitrary equalization factor β lk for each ω l was presented in [30], [32]. However, the gains were assumed to be the same for all the sensors at one particular frequency, that is, β l1 = β l2 = · · · = β lK . This equalizer was based on notch adaptive filters as proposed by Kuo in [15], [16], [33]. Thus it used the adaptive strategy of filtered-x LMS [31], [34] and adapted the filter coefficients by considering each complex coefficient as two real ones, which were fed by the in-phase and quadrature components of the reference signal. Fig. 2 shows a generic scheme of this type of equalizer where the instantaneous power of the pseudo-error signal, defined as can be minimized in order to design the control filters. Here c jk (n) represents the estimated value of the true secondary acoustic path c jk (n). As said before, the equalization factor β k corresponds to ω 0 and exhibits the same value for every sensor, [30]. Although this scheme can achieve e k (n) = β k d k (n) at steady-state for the case of perfect estimation of the secondary paths, c jk (n) = c jk (n), it cannot be straightforwardly extended to the case of different equalization factors, β k , for each sensor. Therefore, let us assume in the following that a different equalization factor, β k , is desired for each sensor in Fig. 2, where the "LMS" block performs the MEFxLMS algorithm. This algorithm selection is not exclusive, so different, and generally more complex, alternative adaptive strategies could be explored, as the recently proposed Recursive Least Squares based Model Predictive Control (RLSMPC) [35] or noise-robust low complexity adaptive schemes like described in [36], [37]. The system needs to know all the pseudo-error noise signals and all the signals that result from filtering the in-phase and quadrature components of the reference signal through their corresponding estimated frequency response of the secondary paths, C jk . From Fig. 2, the kth pseudo-error signal can be expressed for ω 0 as where f (·) is a function that will be specified later, and exact knowledge of the secondary paths, C jk (ω 0 ) = C jk (ω 0 ), has been assumed. Function f (·) in (10) should fulfil E k (ω 0 ) → β k D k (ω 0 ) when E k (ω 0 ) = 0. On the other hand, we can consider a more general case where the filter outputs y j (n) are weighted by a parameter (1 − γ j ) in order to control the dynamic range of the adaptive filter's output signal, which models the need in practice to deal with finite precision effects and adjust the input working range of the D/A converter. Thus the residual error signal for each sensor can be expressed by Note that the above expression (11) is similar to (1) except for the weight (1 − γ j ) that affects to the output signals Y j (ω 0 ).
When the target has been reached, which means E k (ω 0 ) = β k D k (ω 0 ), the following expression is fulfilled: Then multiplying by β k both sides of (12) and considering expression (10) when E k (ω 0 ) = 0, the following definition for the function f (·) of (10) arises: and, consequently, the pseudo-error signal can be expressed as The pseudo-error expression in (14) is valid for any equalization factor, except for β k = 1, and for any k = 1, . . . , K. Therefore, the algorithm exhibits no solution for those sensors whose desired equalization means to keep the noise at its original level, E k (ω 0 ) → D k (ω 0 ).
In the following subsection we will find the optimal solution for the problem of minimizing the mean power of the pseudoerror noise expressed in (14). Then, we will further state the corresponding adaptive strategy that minimizes (14) without the need of explicitly knowing the value of the noise at every sensor, D k (ω 0 ), or, alternatively, their corresponding primary acoustic paths P k (ω 0 ).

A. The Minimum Mean Square Pseudoerror Solution
We define the following vector formed by the components of the reference signal, , filtered by the secondary paths c jk (n) as: where A jk and φ jk are the magnitude and the phase of C jk (ω 0 ) respectively. Let us assume that the whole system is linear, thus the acoustic, c jk , and electric adaptive systems can be exchanged. Then we substitute the value of E k (ω 0 ) from (11) into (14) and perform the inverse FT to achieve the following expression of the pseudo-error noise signal in time domain: where y j (n) is generated as in Fig. 2 by filtering the reference signal through the adaptive filter given by The mean square of the pseudo-error signal in (16) can be expressed as E{e (n) T e (n)}, where e (n) = e 1 (n) e 2 (n) . . . e K (n) T is composed by the samples of the K pseudo-error signals at time n. It can be shown [38] that the values of w j that minimize the above mean square are given by where w is defined as the 2J × 1 vector The 2J × 2J correlation matrix R and the 2J × 1 crosscorrelation vector p are respectively defined as and where the 2J × 1 vector x k (n) is defined as The correlation matrix R can be partitioned in J 2 matrices such that where the expression of each 2 × 2 matrix R fc located at the f th row and cth column of the partition (22) is given by Taking into account the expression of x jk (n) in (15), the above matrix R fc can be explicitly expressed as where we have considered that the amplitudes A ref , A fk , A ck , and phases φ fk , φ ck are known.
In a similar way, the cross-correlation vector p can be partitioned in J vectors such that where each 2 × 1 vector p j is expressed as and where A k and φ k are the magnitude and the phase of the acoustic transfer function of the primary path P k (ω 0 ) respectively. Therefore the noise signal d k (n) could be written as It can be shown that the solution given by (17) is equivalent to (8) when J = K and A ref = 1. That is, where we define a new matrix C γ based on matrix C from (5) Notice that the expressions for R expressed as (22)-(24) and p defined as (25) are equivalent to the previously proposed ones of Section I.

B. Adaptive Solution Based on the Least Mean Squared (LMS) Algorithm
From the expression of each cross-correlation vector p j in (25), the closed-form solution to (17) can only be found if the acoustic system is completely characterized, which means knowing its primary paths, P k (ω 0 ), and its secondary ones, C jk (ω 0 ). Regarding the secondary paths, their estimate can be usually obtained through channel measurements [39], [40] (or even applying on-line estimation methods [41]). However, primary paths P k (ω 0 ) can rarely be measured or known. Therefore, we propose in the following an stochastic gradient algorithm that does not need to know P k (ω 0 ) to look for the optimum control filters. In particular we will use the widely known LMS algorithm [38], whose updating equation at time n for the particular system of Fig. 2 can be expressed as where w(n) is the same vector defined in (18) but now with an explicit dependence on the sample time n, μ is the step size to assure stability [38], e (n) was defined in (II-A), and the 2J × 1 vector ∇ w represents the gradient operator 2 .
Taking into account the generic expression of the pseudo-error signals given by (16), the updating equation for the jth filter, j = 1, . . . , J, is 2 The gradient operator is defined as: where x jk is defined as in (15) but the true secondary paths c jk have been substituted by the estimated secondary paths c jk .
In this sense, it is assumed that these estimates can be obtained with enough accuracy. It is shown in Appendix C that the filter coefficients given by (29) tend, in steady-state, to the optimum coefficients provided by (17) under the assumption of perfect estimation of these secondary paths.

C. Stability Condition of the Multi-Channel ANE LMS Algorithm
It is well known that the stability of iterative gradient algorithms depends on the step size μ. It can be shown that μ must be less than the inverse of the largest eigenvalue of the correlation matrix R given by (22). In the particular case of a system with one loudspeaker and one sensor, R = R 11 , we can express the bound for μ as Note that the convergence speed will reduce as β → 1. The convergence behaviour can be explored by means of zero-pole analysis as in [32]. A lower bound for the step size that does not require the eigenvalue decomposition of matrix R is given for the general multi-channel case by where tr{R} is the trace of R, which can be easily computed as the sum of the elements of its main diagonal as

III. MULTI-CHANNEL ANE FOR MULTI-FREQUENCY NOISE
In this section we will extend the adaptive solutions proposed for the multi-channel single-frequency ANE of Section II to the case of noises whose spectral content is formed by multiple frequencies. These noises are called periodic when their multiple tones are harmonic among them, but, in general, they are called multi-tone or multi-frequency noises.
In the case of a multi-channel multi-frequency ANE, the equalization factor or gain, β lk , will be defined for each frequency, l, and each sensor, k; and a weighting parameter, γ lj , will be defined for the output signals of each frequency, l, and each loudspeaker, j. Thus, we will define LK equalization gains and LJ control filters in an equalizer system with K sensors and J loudspeakers where L frequency components have to be controlled. According to the theoretical model, the system is composed by the linear superposition of L single-frequency systems, equivalent to the one presented in Section II, working in parallel. Thus the optimal control filter for each system is given by (27), where ω 0 is replaced by ω l . In order to obtain the optimal control filters for multi-frequency noises, we will follow [32], where an adaptive algorithm was proposed to design a multi-channel multi-frequency ANE with the same equalization profile for all sensors.
At this point, we will extend the notation used in Section II to deal with multi-frequency signals: r In-phase and quadrature components of the reference noise signal are now given by r β lk is the equalization gain for the kth sensor at ω l . r w lj (n) is the vector formed by the two coefficients of the adaptive filter that controls the jth loudspeaker output signal at ω l . Note that we have kept the dependence with sample time n in case the proposed solution is adaptive in time: r γ lj is the weighting factor at the output of the jth adaptive filter for the lth frequency component.
r A lk and φ lk are respectively the magnitude and phase of the transfer function of the primary path, P k (ω l ).
r A ljk and φ ljk are respectively the magnitude and phase of the transfer function of the secondary path C jk (ω l ). r x ljk (n) is the 2 × 1 vector formed by the in-phase and quadrature signals of the lth frequency component of the reference signal filtered by the estimate of the secondary path, c ljk , between loudspeaker j and sensor k at ω l . In case that the true secondary paths c ljk are available, then x ljk (n) = x ljk (n). Once the new multi-frequency notation has been stated, two different strategies based on [32] are developed in the following.

A. Multi-Channel Multi-Frequency ANE Based on Common Pseudo-Error Signal
In this case we will define a pseudo-error signal per sensor but common to all the frequency components. Since the different frequency components of the signals involved are considered uncorrelated, we can follow the mathematical development described in Section II. If we apply superposition to include all the frequencies, l = 1, . . . , L, the pseudo-error signal for sensor k can be expressed as As detailed in Appendix A, the updating equation for the adaptive filter that feeds loudspeaker j at lth frequency can be expressed as A bound of the step size μ l , l = 1, . . . , L, that assures stability of the above algorithm can be calculated as in (31): where tr{R(ω l )} is now given by

B. Multi-Channel Multi-Frequency ANE Based on Multiple Pseudo-Error Signals
This strategy of multiple pseudo-error signals is very similar to the previous one, where a common pseudo-error signal was used. The benefit of proposing multiple pseudo-error signals, which means a different pseudo-error signal for each frequency, is the possibility of running L adaptive algorithms in parallel. Consequently, the 2J filter coefficients for every frequency can be computed independently for l = 1, . . . , L. The update equation of the filter coefficients, derived in appendix B, is given by where μ l for l = 1, . . . , L, is given by (38)-(39) and the pseudoerror signal for sensor k and frequency l is now defined as We assume that both the noise signal and the error signal, e k (n), depend only on the single frequency component ω l . We show in Appendix C how this assumption does not prevent the adaptive algorithm to reach, on average, the same steady state solution than that of Section III-A.
It can be seen from (40) that the adaptive algorithm based on multiple pseudo-error signals can be run as L algorithms in parallel. Each one would independently achieve the optimal solution of the single frequency system as described in Section II-A. However each of them also contributes to reach an optimal global solution for all coefficients, even for those that control other frequency components, since the pseudo-error signal in (41) has content in all frequencies. Taking into account that x ljk (n) and x mjk (n) are uncorrelated, the solution reached by (40) tends on average to the solution reached by (37), as shown in Appendix C.

C. Transfer Function Analysis
It is known that ANE systems can be considered linear systems under usual conditions [2]. Thus, their behavior can be characterized by the knowledge of their transfer functions over the whole working frequency range. These transfer functions, H k (z), relate the corresponding error and desired signals at each sensor as and they should fulfill that H k (e iω l ) = β lk for D k (e iω ) = 0, ∀l, k. These transfer functions allow to validate the equalizer performance according to the designed specifications at the control frequencies, although they can also provide information about the behavior of the equalizer outside the frequencies of interest.
It should be noted that there exists a different transfer function for each error sensor. Unfortunately, these transfer functions only present a closed expression for single-channel systems due to the cross dependence between the transfer functions corresponding to different error sensors in a general multi-channel system, as it is shown in [32]. However, they can be numerically computed and then used to predict the behaviour of the adaptive system, both at the steady state by estimating their corresponding frequency responses, and at transient state by computing their pole moduli [42]. The transfer functions can be described by a quotient of polynomials, N k (z) and M k (z), where the denominator can be factored using the transfer function poles as where we have considered the usual assumptions that these functions have 2L poles (p k1 , p k2 , . . . , p k2L ), being L the number of control frequencies, and the denominator order is equal to the numerator order [32].
The error signal at each sensor can be estimated for a given desired (input) signal using its corresponding transfer function and a deterministic approach. Let the input noise be a complex sinusoidal signal starting at n = 0, denoted by: d k (n) = e iω 0 n u(n), where u(n) is the unit step function, then the k-th error signal is given by which can be alternatively expressed by its partial fraction expansion as Finally, the error signal at the k-th error sensor can be estimated as e k (n) = H k (e iω 0 )e iω 0 n u(n) + A k1 p n k1 u(n) Since the system can be considered causal and stable for the convergence parameter fulfilling (38), all the poles remain within the unit circle and then e k (n) → H k (e iω 0 )e iω 0 n = β lk e iω l n , when n → ∞ and ω 0 = ω l . The duration of the transient stage depends on the values of the amplitudes given by (46) and the poles of (43), which can be calculated from the transfer functions.
The poles of (43) depend on the adaptive system parameters through the transfer function. The pole moduli can be calculated from the maximum values of the transfer function absolute value. These poles, which appear as conjugated pairs, are associated with the control frequencies. Consequently the poles' phases are very close to the corresponding control frequencies' angles, which eases their search through numerical methods [42].
The transient stage duration tends to be dominated by the pole with largest modulus as shown in (47), which will be the closest one to the unit circle. Let ρ max k be the largest modulus of the poles of H k (z). We can define the effective time constant for the error signal at sensor k (47), n , as the value of n when ρ max k has decreased to , then(ρ max k ) n = ⇒ n = ln( ) ln(ρ max k ) (i.e. for a signal decreasing of 90%, ≈ 0.1). This effective time constant can be used to estimate the system transient stage duration at each error sensor.
The summation given by (47) is leaded by only two terms when the poles' moduli are very similar (and close to the unit circle) and the control frequency angles are separated enough. This fact is due to the value of 1 1−e iω 0 z −1 | z=p kl in (46), which will be large when ω 0 = ω l for a given l and notably smaller for the rest of control frequencies in this case. This fact allows the estimation of the transient duration of each control frequency at each error sensor by using the modulus of its corresponding pole within the k-th transfer response.
On the other hand, there exist K transfer functions (K sensors) and L control frequencies in the general case. Thus there will be (K × 2L) different poles, whose moduli can be denoted by ρ kl . Each frequency component commands the convergence of two coefficients of each secondary source, therefore it can be assumed that the convergence behaviour of each pair of coefficients are mostly conditioned by the maximum pole modulus for this control frequency, which can be defined as ρ l = argmax{ρ kl }, ∀k. Then the convergence behaviour of each adaptive coefficient pair can be modeled by where w jl and w jl are the coefficient values given by (17) for the j-th secondary source.

IV. SIMULATION RESULTS
The performance of the proposed adaptive strategies is illustrated in the following by numerical simulations using real data. The actual acoustic paths used in the experiments are available online at [43], but decimated to a sampling frequency of 2 kHz. The multi-channel ANE system is composed by two loudspeakers (J = 2, number 20 and 22 in the set-up of [43]) and two sensors (K = 2, number 11 and 13 in the set-up of [43]), and the noise to be equalized has content at five frequencies, {100, 300, 500, 700, 900} Hz, thus L = 5. The convergence parameter used in all simulations was derived according to (38). It should be noted that the common pseudo-error strategy adjusts the transfer functions perfectly at the equalized frequencies, whereas the multiple pseudo-error strategy exhibits slight deviations due to the mutual influence of the filters working at different frequencies, especially those in the middle of the band, i.e., with two adjacent control frequencies. Fig. 4 shows the spectrum of the original noise signal, d k (n), versus the spectrum of the error signal, e k (n), after algorithm convergence. The equalizer gains are the same as those used for Figure 3. The black stars indicate the theoretical values of the original noise magnitudes, |D k (ω l )|, whereas the red circles indicate those of the theoretical error signal according to (42), |E k (ω l )| = |β lk D k (ω l )|. All the components of the noise reference signal have an amplitude of A ref,l = 1, ∀l, although the magnitudes |D k (ω l )| are different at each sensor and frequency due to the influence of their respective primary acoustic paths. It can be observed in Figure 4 that the performance of both common and multiple pseudo-error strategies is very similar, both achieving the desired equalization profile β lk D k (ω l ) at each sensor.
A Gaussian white noise was added to the multi-frequency noise signal in order to test the robustness of both algorithms against the presence of broadband noise of non-tonal nature. The  power ratio between the multi-frequency noise signal and the Gaussian noise was 15 dB. In this case, the equalization profile for Sensor #1 is set to β l1 = 0, ∀l, meaning that the system acts as an active noise canceller for that sensor, whereas β l2 = 0.5, ∀l, for Sensor #2. Fig. 5 shows the spectrum of the original noise signal d k (n) versus the spectrum of the error signal e k (n) for the common pseudo-error strategy, whereas Fig. 6 illustrates the temporal evolution of the adaptive filters where each filter is composed by 2 coefficients. Since both strategies behave similarly, only the results of the common error strategy have been shown in Figs. 5 and 6. It can be observed in Fig. 5 that the algorithm equalizes quite well the frequency components to be controlled, while keeping the rest of the spectral components unaltered.
As it can be seen from Fig. 6, the coefficients move around their theoretical values with a variance proportional to the power of the added broadband noise and their corresponding convergence step. As expected, none of the proposed strategies was able to provide the desired equalization profiles at the controlled frequencies when the additive broadband noise exhibited a power higher than the multi-frequency noise signal.
Finally, we can shed some light to the transient behaviour of the proposed algorithms using the transfer functions. It has been shown that it depends on the moduli of the poles of these transfer functions. Because these multi-channel transfer functions do not have analytical expressions, they have been numerically estimated.
For = 0.1, the effective time constant, n e , for the system whose transfer functions are shown in Fig. 3 is 2302 samples for both common and multiple pseudoerror (the highest modulus of the poles found numerically was ρ max k = 0.999 with a search resolution of 0.001 [42]). Fig. 7 illustrates the wave-forms of the noise signal before and after equalization. It is remarked within a square the theoretical transient stage using n e . It can be noted that the effective time constant properly bounds the transient duration of the algorithm. Furthermore, the transient duration of each tonal noise component could be estimated by means of the maximum modulus of the poles corresponding to this component frequency due to the fact that the poles had similar modulus and the five control frequencies' angles are equispaced. As example, Fig. 8 shows the transient behaviour of the fastest and the slowest noise components (for sensor 2 and the multiple pseudoerror strategy) and their estimated transient durations, using only the magnitude of the corresponding poles for these components, which was estimated as explained in [42].
The temporal evolution of the filter coefficients could be approximated using this pole information by (48). Fig. 9 shows an example of this theoretical model compared with the simulation result for the filter coefficients that correspond to the third harmonic of the noise signal. It can be seen that this simple

V. CONCLUSION
A multi-channel active equalizer for multi-frequency noise signals, which allows different equalization gains for each sensor and for each frequency component of the noise, has been proposed. To this end, new pseudo-error signals have been defined that depend on the desired equalization profiles at each sensor, and on the weighting parameters of the output signals generated by the loudspeakers. We have obtained the expression of the optimal filter coefficients associated to every loudspeaker of the system in the case of single-frequency noise. However, taking into account that this solution requires the knowledge of the primary acoustic paths, we have alternatively proposed an adaptive strategy based on the LMS algorithm. Furthermore, we have extended the analysis performed for single-frequency noises to the case of multi-frequency noises. In this sense, two adaptive strategies have been derived: the first uses a common pseudo-error signal at each sensor, whereas the second proposes a different pseudo-error signal for each frequency at each sensor, which we have called multiple pseudo-error signal. Simulations have been carried out with real acoustic paths and they have shown that both strategies provide a similar behavior and manage to reach the desired equalization profiles at each sensor. However, it should be noted that the strategy based on the multiple pseudo-error signal may show uncontrolled influences in nearby frequencies, since it actually reaches its solution by solving an individual parallel problem for each frequency component. On the other hand the strategy based on a common pseudo-error signal achieves a global solution for all frequency components. Furthermore, it has been shown that the steady state behaviour of these algorithms and the duration of their transient stages can be estimated from the transfer function analysis. Together with the simulations, it has been also shown that the active controllers provided by the iterative strategies tend in practice to their corresponding theoretical values.

APPENDIX A MULTI-CHANNEL ANE ADAPTIVE FILTERING STRATEGY BASED ON COMMON PSEUDO-ERROR SIGNAL
The optimal coefficients that minimize the mean square of the pseudo-error signal can be obtained as in Section II-A, but considering the new pseudo-error signal defined in (36). The signal recorded at each sensor is given by Assuming perfect estimation of the secondary paths x T ljk (n) = x T ljk (n), then (36) can be expressed as where the 2LJ × 1 vector shares the notation with (21) but now it is formed by the extension of (21) to the case of L frequencies, and is a 2LJ × 1 vector with the same notation as in (18) but now formed by stacking the filter coefficients for the L frequencies ordered according to the expression of x k (n) in (51). Therefore, the optimal coefficients that minimize E{(e (n)) T e (n)} can be obtained as where the 2LJ × 2LJ augmented correlation matrix R A is defined as in (19) and (20) respectively, with dimensions 2LJ × 2LJ for R A and 2LJ × 1 for p A . Taking into account that components belonging to different frequencies are uncorrelated, i.e. E{x ljk x mjk } = 0 if l = m, then the augmented correlation matrix R A is a block diagonal matrix: where every submatrix R(ω l ) is defined in the same way as (22) but for ω l , l = 1, . . . , L and 0 2J×2J is a square matrix with zero elements. For the sake of clarity we state here the expression of the 2 × 2 matrix R fc (ω l ) located at the f -th row and c-th column of the partition of R(ω l ) as in (22): where At this point the augmented cross-correlation vector p A follows the same composition of above, but for L frequencies. Thus it can be expressed as a partition of LJ subvectors of 2 × 1 elements: where It can be shown that the solution given by (53) for a particular ω l is the same as the one obtained in Section II-A particularizing w 0 = w l in (27). Considering now an adaptive strategy similar to that proposed for single-frequency signals in (28), we can express where is a 2JL × 2JL diagonal matrix formed by L different values (μ 1 , μ 2 , . . . , μ L ), with μ l the step size for the frequency component ω l . Dependence on l must be included in μ l because different frequency components can exhibit different convergence conditions. Taking into account the generic expression of the pseudo-error signals given in (36), the updating (58) for the adaptive filter of loudspeaker j and l-th frequency can be expressed as where we have split the different frequency components of (49) into two parts: one term containing the frequency component of interest, ω l , which is affected by the equalization gain β lk , and the other term including the rest of frequency components such that m = l. The pseudo-error signal in (61) can be expressed in compact form as where vector w(n) is given by (52) and x T k (n) is newly defined as The above expression is similar to the expression of the common pseudo-error signal given by (50) except for the fact that (62) depends only on the equalization gain β lk for the l-th frequency, as it can be seen from the definition of x k (n) in (63). In this sense, the solution that gives the optimal coefficients that minimize E{(e (n)) T e (n)} is provided by (53), but the correlation matrix R A and cross-correlation vector p A have to be written according to the new definition of x k (n) in (63). Therefore, the expression of the correlation matrix R A is given by (54), but the 2J × 2J submatrices R(ω m ) are now partitioned into the following 2 × 2 submatrices: where and The expression of the cross-correlation vector p A is given by (57), but the 2 × 1 subvectors of the partition are now expressed as where β mk is given by (66). As said before, the advantage of this strategy based on multiple pseudo-error signals relies on the possibility of updating the filter coefficients independently for every frequency component. In order to state the adaptive algorithm that converge to the above solution, let us define the 2J × 1 partition of w(n) in (52) that contains the filter coefficients affecting only to the frequency component ω l as The gradient algorithm is then expressed by where e l = e l1 . . . e lK T . In a similar way to (37), the updating equation can be stated as for ω l with l = 1, . . . , L. The above expression can be rewritten in compact form for the whole set of filter coefficients for the given frequency component ω l as

APPENDIX C CONVERGENCE IN MEAN OF THE ADAPTIVE MULTIPLE PSEUDO-ERROR EQUALIZER ALGORITHM
Assuming perfect acoustic path estimation, the filter updating equation, (71), can be rewritten in matrix form as where vector e l (n) can be written, following (61), as when n → ∞, then Since tonal signals are uncorrelated for different frequencies, that is, E{x Miguel Ferrer Contreras (Member, IEEE) received the Graduation degree in telecommunications engineering from Universidad Politécnica de Valencia (UPV), Valencia, Spain, in 2000. Since a year before, he has been collaborating with the Audio and Communications Signal Processing Group (GTAC) performing a six months research stay with Instituto de Investigación Aplicada al Automóvil, Tarragona, Spain (Automobile Applied Research Institute). Subsequently, he was the recipient of several grants offered both by the Communications Department of the UPV and by the Research, Development and Innovation Vice-Chancellery of the same university, enabling him to start his Doctorate studies and to collaborate in different Research Projects within the GTAC. During this period he has authored or co-authored more than seventy papers related with signal processing in renowned journals and conferences.
Since 2005, he has been an Assistant Lecturer with the Communications Department, Polytechnic University of Valencia (UPV) teaching different subjects related with Signal Processing and Communications. His research interests include the study of adaptive algorithms and its application to audio digital processing and noise active control, a subject about which he developed his doctoral thesis, and he continues to develop his research work with the GTAC, now belonging to the Institute of Telecommunications and Multimedia Applications (iTEAM) of the UPV.