Experimental evaluation of leaky LMS algorithms for active noise

Experimental evaluation of leaky LMS algorithms for active noise reduction in communication headsetsa

David A. Cartes, Laura R. Rayb, Robert D. Collier Thayer School of Engineering Dartmouth College 8000 Cummings Hall, Hanover, NH 03755 ph. (603) 646-1243 fax (603) 646-3856 email: [email protected]

Running Title: Experimental evaluation of leaky LMS algorithms

Received:

a

Portions of this work will be presented at the 2001 American Control Conference and in the IEEE Tran. on Signal Processing, D. Cartes, L.R. Ray, and R.D. Collier, Lyapunov Tuning of the Leaky LMS Algorithm for SingleSource, Single-Point Noise Cancellation. b Author to whom correspondence should be addressed, [email protected]

Cartes, Ray, and Collier Experimental Evaluation of Leaky LMS Algorithms J. Acoust. Soc. Am ________________________________________________________________________________________________________

ABSTRACT An adaptive leaky NLMS algorithm has been developed to optimize stability and performance of active noise cancellation systems. The research addresses performance issues related to insufficient excitation, nonstationary noise fields, and signal-to-noise ratio. The algorithm is based on a Lyapunov tuning approach in which three candidate algorithms, each of which is a function of the instantaneous measured reference input, measurement noise variance, and filter length, provide varying degrees of tradeoff between stability and performance. Each algorithm is evaluated experimentally for reduction of low frequency noise in communication headsets and performance is compared with that of traditional LMS algorithms. Acoustic measurements are made in a specially designed acoustic test cell which is based on the original work of Shaw et al. [J.G. Ryan, E.A.G. Shaw, A.J. Brammer, and G. Zhang. “ Enclosure for Low Frequency Assessment of Active Noise Reducing Circumaural Headsets and Hearing Protection,” Canadian Acoustics, 21(4), pp. 19-20, 1993] and which provides a highly controlled and uniform acoustic environment. The stability and performance of the ANR system, including a prototype communication headset, are investigated for a variety of noise sources ranging from stationary tonal noise to highly nonstationary measured F-16 aircraft noise over a 20 dB dynamic range. Results demonstrate significant improvements in stability of Lyapunovtuned LMS algorithms over traditional leaky or non-leaky normalized algorithms, while providing noise reduction performance equivalent to that of the NLMS algorithm for idealized noise fields.

Technical Areas: Acoustic Signal Processing, Active Noise Control PACS Subject Classification numbers:43.60.B; 43.60.Q; 43.50.K; 43.50.H

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INTRODUCTION Current research in active noise reduction (ANR) for communication headsets focuses on using digital feedforward technology based on least-mean-square (LMS) optimization. References 1-3 provide substantial experimental evidence that digital feedforward technology has the potential to improve performance of ANR headsets over state-of-the-art, commercially available, feedback systems. This is particularly true with respect to low frequency noise, which cannot be adequately attenuated passively by the headset structure -- typically 5 dB at 100 Hz, and for which feedback ANR systems exhibit limited performance, e.g., 15 dB between 100 and 250 Hz.

The goal of [1-3] was to document feasibility of feedforward ANR applied to

communication headsets using DSP technology that was available at that time. References [1-2] use the filtered-X NLMS (FXNLMS) algorithm and a dual sample-rate system to demonstrate active attenuation of 26 dB in response to band limited (80 - 750 Hz) white noise in flat plate testing of a prototype feedforward ANR communication headset. The filtered-X algorithm, first derived in [4-5], is used in systems where, like the communication headset, the transfer function of the path between the reference input and the LMS filter output must be precisely modeled. In communication headsets this path represents the combined transfer functions of several components such as microphones, amplifiers, and speakers, which may exhibit temporal variation, and whose frequency response functions must be accounted for in shaping the cancellation signal.

Using the FXNLMS algorithm, active feedforward

performance of 10 to 30 dB has also been demonstrated at frequencies up to 250 Hz for band limited (10-1000 Hz) noise from a helicopter at the location of the aircrew.3 For the helicopter noise experiments, ANR performance at the critical blade passage frequency (16 Hz) was reported to be as high as 26 dB.3

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The work presented in [1-3] fulfills the objective of proving feasibility of feedforward ANR systems for communication headsets. In practice, however, traditional feedforward LMS algorithms have significant stability and performance deficiencies caused by nonstationary reference inputs, finite precision arithmetic, and measurement noise.

In particular, the

FXNLMS algorithm, while assuring convergence when the secondary path transfer function is non-negligible, increases the output power and may cause distortion in the secondary path through nonlinear behavior of the cancellation speaker.6 In response to these stability and performance issues, the family of LMS algorithms developed over the past two decades includes leaky variants. As in the case of the well-known normalized LMS algorithm (NLMS), where adaptation of the step size of the traditional LMS algorithm addresses speed of convergence, the leakage factor addresses stability deficiencies that arise from nonstationary inputs, low signal-tonoise (SNR) ratio, and finite-precision arithmetic. The leakage factor can also be combined with the FXNLMS algorithm with a resultant increase in stability in the presence of finite precision and measurement noise, and a limit on output power to avoid distortion. However, in combining the leaky LMS algorithm with any of the family of LMS algorithms, tuning the leakage parameter is a highly empirical process, which, in order to retain stability during worst-case SNR conditions, results in significant performance reductions. In [7], the authors introduce a Lyapunov tuning method for choosing a combination of adaptive step size and leakage factor that addresses both stability and performance in the face of one of the factors that necessitates the use of a leaky LMS algorithm, namely measurement noise on the measured reference input. The Lyapunov tuning method results in a time-varying adaptive leakage factor and step size combination which maintains stability for low SNR on the measured reference input, while minimizing performance reduction for high SNR.7 This paper focuses on experimental evaluation of Lyapunov tuned LMS algorithms for active noise 4


reduction in a prototype communication headset. For experimental evaluation, a low frequency acoustic test chamber was designed and constructed, based on the original work of [8], to provide a precisely controlled acoustic environment with a flat frequency response from 0 to 200 Hz. The prototype was tested within this chamber for noise sources ranging from highly stationary tonal noise to highly nonstationary F-16 noise. Tonal noise was generated locally, while last three noise sources were taken from a compilation of noise files issued by the North Atlantic Treaty organization (NATO)9. These sound files provide a standard set of both military and nonmilitary noise sources for research in auditory acoustics. Testing was performed for source signals at two sound pressure levels - 80 dB and 100 dB, providing SNR of 35 dB and 55 dB, respectively, based on ambient noise levels measured under the headset by a calibrated B&K microphone. In addition, simulation analysis was performed to examine stability of Lyapunov tuned LMS algorithms at a SNR of 15 dB. This paper focuses on the experimental results, which demonstrate the superiority of Lyapunov tuned LMS algorithms over traditional NLMS algorithms or constant leakage parameter LNLMS algorithms. The contributions of this research are: 1) an experimentally verified tuning method for determining leakage and step size parameters that maintain stability of the LMS family of algorithms at minimal performance degradation, and 2) application of this method to enhance performance of adaptive feedforward noise cancellation in communication headsets. Though the focus of this paper is on tuning of the LNLMS algorithm, the resulting tuning parameters can also be used in combination with other members of the LMS family, such as the FXNLMS. The paper is organized as follows. Section I summarizes the Lyapunov tuned candidate LMS algorithms developed by the authors in [7]. Section II describes the acoustic test chamber, prototype communication headset, and experimental test procedure. Section III presents the

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results of experimental testing, and section IV presents additional simulation analysis performed to verify the Lyapunov tuning method.

I. THE LYAPUNOV TUNING METHOD A block diagram of a traditional feedforward LMS algorithm denoting signal definitions is shown in Figure 1. The LMS algorithm recursively selects a weight vector Wk ∈ Rn to minimize the squared error between dk and the adaptive filter output yk = WkT Xk , where Xk ∈ Rn is the sampled reference input and dk ∈ R1 is the output of the unknown continuous

acoustic process at time tk , which in the present application is the passive noise attenuation of a headset. The Wiener solution, or optimum weight vector is

[

Wo = E Xk X Tk

[

]

−1

E[ Xk dk ]

(1)

]

where E Xk X Tk is the autocorrelation of the input signal and E[ Xk dk ] is the cross correlation between the input vector and process output. The unbiased, recursive LMS weight vector filter is

Wk +1 = Wk + µek Xk

(2)

where ek = dk − WkT Xk . The weight update equation is guaranteed to converge to the Wiener solution in the ideal situation of stationary inputs, infinite signal-to-noise ratio, and infinite precision arithmetic, given an appropriate step size. In response to nonstationary inputs, finiteprecision arithmetic, non-persistent excitation, and low signal-to-noise ratio, the leaky LMS (LLMS) algorithm and step-size normalized versions of the leaky LMS algorithm “leak off” excess energy associated with weight drift. This results in a biased recursive weight update equation

Wk +1 = λWk + µek Xk

(3)

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λ is the leakage factor.10 The leakage factor prevents convergence of the weight vector to the Wiener solution and hence results in reduced performance over the unbiased LMS algorithm. The Lyapunov tuning method seeks time varying tuning parameters λk and µk that maintain stability and maximize performance in the presence of quantifiable measurement noise. With quantifiable noise Qk ∈ Rn corrupting the reference signal Xk , and with time varying leakage and step size parameters, λk and µk , the LNLMS weight update equation of eq. 3 becomes Wk +1 = λk Wk + µk (WoT Xk − WkT ( Xk + Qk ))( Xk + Qk )

(4)

For stability at maximal performance, time-varying parameters λk and µk that are functions of measurable quantities are determined, such that stability conditions on a candidate positivedefinite Lyapunov function Vk are satisfied for all k in the presence of quantifiable noise on reference input Xk . The conditions on Vk for uniform asymptotic stability are i ) Vk ≥ 0 , and

ii) Vk+1 − Vk ≤ 0 .11

If Vk+1 − Vk > 0 , the system may or may not be stable, thus no stability

claim or disproof can be made for such cases. Thus the objective of Lyapunov tuning is to define tuning parameters that guarantee that conditions i) and ii) are met for the largest possible space of the weight vector. Lyapunov tuning proceeds by first defining a positive definite Lyapunov function of a scalar projection w˜ k of the weight vector: Vk = w˜ kT w˜ k

(5)

( X + Qk ) where W˜ k = Wk − Wo , w˜ k = W˜ kT uk and uk = k . λk and µk are to be selected such that Xk + Qk

Lyapunov difference  (φk2 − 1) A2 + γ 12 + γ 22 B2 + 2φkγ 1 A  T T k k k Vk +1 − Vk =   Wo uk uk Wo  + 2φkγ 2 k AB + 2γ 1k γ 2 k B 

(6)

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is negative for largest possible space of W˜ k , where

φk = λk − µk ( Xk + Qk )T ( Xk + Qk )

(7)

γ 1k = λk − 1

(8)

γ 2 k = − µk ( Xk + Qk )T ( Xk + Qk )

(9)

W˜ kT uk A= T Wo uk

(10)

WoT α k WoT uk

(11)

B=

αk =

Qk Xk + Qk

(12)

The expression for the Lyapunov function difference Vk +1 − Vk in eq. 6 takes the n-dimensional vector space of Wk and projects it to a two-dimensional space that is a function of scalars A and B.

The constant A represents the output error ratio between the actual output

yk = WkT ( Xk + Qk ) of a noise corrupted system and the ideal output WoT ( Xk + Qk ) of a noise

corrupted system converged to the Wiener solution. B represents the output noise ratio, or portion of the ideal output that is due to noise vector Qk .

Physically, these constants are

inherently bounded based on i) the maximum output that a real system is capable of producing, and ii) signal-to-noise ratio. Though bounds are difficult to determine in practice, at high SNR, B approaches zero, and when the weight vector converges to the Wiener solution, A = 0. Thus, high SNR implies that both A and B approach zero. With low excitation or low signal-to-noise ratio, larger instantaneous magnitudes of A and B are possible, but it is improbable that these constants are persistently much greater than unity in practice.

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Since several terms are positive in eq. 6 independent of λk and µk , each individual term of eq. 6cannot be guaranteed to be negative. Thus, the approach taken by the Lyapunov tuning method is to define the region of stability around the Wiener solution in terms of A and B. Hence, these constants provide a convenient two-dimensional parameterization of the multidimensional weight vector stability analysis problem that enables visualization of the region of stability of the weight update equation in two dimensions. In [7], the structure of the Lyapunov function difference of eq. 6 is used to propose parameters λk and µk that simplify the analysis of eq. 6 and subsequently determine conditions on any remaining scalar parameters such that Vk +1 − Vk < 0 for the largest region possible around the Wiener solution. Such a region is now defined by parameters A and B, providing a means to graphically display the stable region and to visualize performance/stability tradeoffs introduced for candidate leakage and step size parameters. Three candidate leakage parameter and adaptive step size combinations are proposed in [7] for such parameterization. The first candidate uses a traditional choice for leakage parameter as presented in [12] in combination with a traditional choice for adaptive step size from [13]:

λk = 1 − µkσ q2 µk =

µo ( Xk + Qk )T ( Xk + Qk )

(13) (14)

σ q2 is the variance of quantifiable noise corrupting each component of vector Xk . To determine the remaining scalar parameter, µo , we perform a scalar optimization of Vk +1 − Vk with respect to µo and evaluate the result for worst case values of A and B. This results in a value of µo that makes Vk +1 − Vk most negative for worst case deviations of weight vector Wk from the Wiener solution and for worst case effects of measurement noise Qk . For the candidate adaptive leakage

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parameter and step size combination of eq. 13 and 14, this procedure results in µo = 1/3, which is consistent with the choice for µo from [13]. The second candidate also retains the traditional leakage factor of eq. 13, and finds an expression for µk as a function of the measured reference input and noise covariance directly by performing a scalar optimization of Vk +1 − Vk with respect to µk . Again, the results are evaluated for worst-case conditions on A and B, as described above. This results in

µk =

(

2( Xk + Qk )T ( Xk + Qk ) + 4σ q2

2 ( Xk + Qk )T ( Xk + Qk )

)

2

+ 8σ q2 ( Xk + Qk )T ( Xk + Qk ) + 8σ q4

(15)

The final candidate appeals to the structure of eq. 6 to determine an alternate parameterization as a function of µo as follows:

µo λk ( Xk + Qk )T ( Xk + Qk )

(16)

XkT Xk − QkT Qk λk = ( Xk + Qk )T ( Xk + Qk )

(17)

µk =

The expression for λk in eq. 17 cannot be measured, but it can be approximated as ( Xk + Qk )T ( Xk + Qk ) − 2 Lσ q2 λk = ( Xk + Qk )T ( Xk + Qk )

(18)

for Xk > Qk . L is the filter length. The optimum µo for this candidate, which is again found by scalar optimization subject to worst case conditions on A and B is µo = 1/2. Further details regarding the Lyapunov tuning method are provided in [7]. Evaluation of the Lyapunov difference of eq. 6 as a function of A and B reveals insights on the stability and performance of each candidate tuning law. The Vk +1 − Vk vs. A and B surface is shown in Figure 2 for SNR = 35 dB and for each candidate, along with the plane

Vk +1 − Vk = 0 that identifies the region of guaranteed stability. From Figure 2, this stability

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region, which is defined by the extent of the surface that lies above Vk +1 − Vk = 0 is largest for candidate 3, followed closely by candidate 1, while candidate 2 shows the smallest stability region overall. Performance is indicated both by the magnitude and gradient of the Vk +1 − Vk vs. A and B surface, with a larger gradient denoting more aggressive performance. For example, in the limiting case of Vk +1 − Vk = 0 for every value of A and B, as would be the case for λk = 1 and µk = 0 , the system is guaranteed to be stable, and the gradient Vk +1 − Vk is zero. However, no convergence results, as Wk +1 = Wk . Based on the gradient of the Vk +1 − Vk surface rank order, candidate 2 is expected to have the most aggressive performance, followed by candidates 3 and 2. These stability-performance tradeoffs point to candidate 3 as the best overall choice of the three Lyapunov tuned candidates.

II. EXPERIMENTAL APPARATUS AND TEST PROCEDURE The three Lyapunov tuned candidate ANR algorithms are analyzed and compared experimentally with three traditional fixed parameter NLMS and LNLMS algorithms at two sound pressure levels - 80 dB and 100 dB - representing signal-to-noise ratio environments of 35 dB and 55 dB, respectively. Comparisons are made for four acoustic noise sources ranging from stationary tonal noise to highly nonstationary F-16 aircraft noise. Results for the two extreme noise sources are presented here. Comparisons are made within a low frequency test cell that provides a controlled acoustic environment. This section documents the test cell and prototype construction, and the experimental procedure. A. Low Frequency Test Cell The low frequency acoustic test cell (LFATC) is designed to provide a flat frequency response from 0 to 200 Hz for sound pressure levels of up to 140 dB. The design and experimental verification of the test chamber, which is based on the original work of [8], is described in detail in [14]. Figure 3 provides photographs of the assembled and disassembled 11


test chamber, with a prototype headset mounted in its base. A 15.24 cm diameter Rockford Fosgate model FNQ1406, 100 Watt speaker provides the sound source. A B&K 4190 precision microphone is mounted axially in the base plate near where the ear would be located inside of the headset if worn by a human. A second precision microphone is mounted radially through the cylinder wall to measure the reference sound pressure level in the test cell. With a prototype headset in the test cell, these two microphones allow simultaneous, calibrated measurement of the sound pressure level outside of the prototype headset and inside the headset, and they also serve to calibrate the reference and error microphones of the prototype headset. The test chamber is mounted on two layers of packing material separated by a 35 kg brass plate to minimize noise due to structural vibration.

The predicted and measured frequency

response of the test cell at the B&K microphone mounted in the flat plate at its base is shown in Figure 4. B. Prototype Headset The prototype headset is depicted in Figure 5, along with pertinent transfer functions. H1( s) represents the transfer function of the test cell, reference microphone, and its amplifier. H2 ( s) represents the transfer function of the cancellation path, which includes the cancellation

speaker and its amplifier, the acoustic path between the speaker and the error microphone, and the error microphone and its amplifier. H3 ( s) is the unknown transfer function of the passive headset, which is to be found using the LMS filter, and H4 ( s) is the acoustic feedback path from the cancellation speaker to the reference microphone. The reference and error microphones are Panasonic WM-34B electret microphones. The error microphone is located to the side of and slightly recessed from the speaker, and the reference microphone is located on the outside periphery of the headset dome. When mounted

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on the flat plate in the test cell, the prototype's reference microphone is located 0.6 cm from the precision microphone in the side of the test cell, and the error microphone is located 2.5 cm above the precision microphone in the base of the test cell. A Sennheiser HD570 cancellation speaker is used due to its rated low frequency characteristics and relatively low distortion. Figure 6 shows the experimentally determined speaker transfer function, which is the major contributor to H2 ( s) in Figure 5. Figure 6 shows a relatively flat speaker transfer function, with a low frequency roll off at approximately 50 Hz, and a notch at 100 Hz. The cancellation speaker distortion was evaluated in the test cell for active noise reduction of single tones at 50 Hz, 100 Hz, 150 Hz, and 200 Hz, at sound pressure levels of 80 and 100 dB. These tests were performed at a sample frequency of 10 kHz, weight update rate of 5 kHz, and a filter length of 200, using the NLMS algorithm. At 100 dB SPL, total harmonic distortion was measured to be 0.7%, 1.8%, 0.6%, and 0.3%, respectively for the four input frequencies. At 80 dB, total harmonic distortion is less than 0.3% at all frequencies. In addition, the feedback path transfer function ( H4 ( s) in Figure 4) was measured during these experiments. In all cases, the magnitude response of the feedback path presents a minimum of 35 dB attenuation and is therefore considered an insignificant contribution to the reference signal

Xk . C. Noise Sources Four noise sources with increasing degrees of stationarity were selected for evaluation of candidate algorithms: 1) a sum of pure tones at the 1/3 octave center band frequencies between 50 and 200 Hz, 2) Lynx helicopter noise, 3) Pink noise, and 4) F-16 aircraft noise. Results for the two extremes - tonal noise and F-16 noise - are presented in this paper.

Tonal noise

represents the easiest of the four noise sources to cancel using ANR, and F-16 noise represents

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an extremely challenging source for which to achieve high performance ANR while maintaining stability of the weight update equation. Performance for the remaining two noise sources falls between that for the two noise sources presented here. All sources are band limited at 50 Hz to maintain a low level of low frequency distortion due to the cancellation speaker roll off and 200 Hz, the upper limit for a uniform sound field in the low frequency test cell. Figure 7 shows the power spectral density for the F-16 noise used for testing. D. Experimental procedure The three candidate Lyapunov tuned leaky LMS algorithm are evaluated and compared to i) empirically tuned, fixed leakage parameter leaky, normalized LMS algorithms (LNLMS), and ii) an empirically tuned normalized LMS algorithm with no leakage parameter (NLMS). These algorithms assume that the cancellation path transfer function is constant, and that the gain of the cancellation path is well known. In Figure 5, the cancellation path is represented by H2 ( s) and the portion of H1( s) associated with the reference microphone and its amplifier. In practice, the transfer function of this path is generally slowly time varying due to factors such as amplifier drift, temperature variations, and variations in mounting of the prototype headset within the test cell. In order to accommodate for slowly time varying transfer functions in the secondary path, the frequency response amplitude of the microphone and cancellation speaker and their amplifiers are adjusted prior to each experiment such that the frequency response magnitude of the cancellation path is unity and is reasonably flat. To do so, the calibrated microphones in the base and side of the test cell are used to adjust the amplifier gains for the reference and error microphones of the prototype such that the transfer function magnitude of each microphone/amplifier path is unity. The speaker amplifier is adjusted such that the speaker

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transfer function, as shown in Figure 6 is approximately unity. By band limiting the noise source, as explained above, and by making these adjustments prior to each set of experiments, the need for compensation of the cancellation path is avoided. Once adjustments are make, the cancellation path transfer function remains relatively constant for the duration of the experiments, which is a period of two to three hours. All experimental data are collected during this time period, to ensure accurate comparisons between traditional and Lyapunov tuned leaky LMS algorithm performance. All LMS algorithms are implemented in Simulink and run on a dSPACE 1103 DSP board, which contains 16-bit A/D and 12-bit D/A converter channels for input and output, as well as processing capabilities for algorithm implementation. The amplitude of the noise source is established to evaluate algorithm performance over a 20 dB dynamic range, i.e., sound pressure levels of 80 dB and 100 dB. These sound pressure levels are measured at the B&K microphone mounted in the base of the test cell and thus reflect the SPL inside the headset after passive attenuation of the reference noise and before active attenuation. All SPL measurements use a C weighting. The 20 dB dynamic range tests the ability of the traditional and Lyapunov tuned leaky LMS algorithms to adapt to different signalto-noise ratios, from 35 dB for the 80 dB SPL to 55 dB for 100 dB SPL. The quantization noise magnitude is 610e-6 V, based on a 16-bit round-off A/D converter with a +10 V range and one sign bit, and the noise floor of the test cell is 49 dB. A filter length of 250 and weight update frequency of 5 kHz are used. Each noise cancellation test is repeated four times for each SPL and noise source. Ensemble average results of the four trials are reported here. All active noise reduction results are presented as measured at the error microphone of the prototype headset. In the first part of the comparative study, empirically tuned NLMS and LNLMS filters with constant leakage parameter and the traditional adaptive step size of eq. 14 are used with the F-16 noise source. These algorithms are tuned to maximize performance at 100 dB SPL and 15


subsequently applied without change to all other noise sources at both 100 dB SPL and 80 dB SPL. On the other hand, the constant leakage parameter LNLMS filter is empirically tuned for the F-16 noise at 80 dB and subsequently applied to all other test conditions.

These two

empirically tuned algorithms are denoted LNLMS(100) and LNLMS(80), respectively. For both filters, µo = 1/3, and the respective leakage parameters are given in Table I. At 100 dB SPL, SNR is high enough to minimize to possibility of instability due to measurement noise on the reference input; hence, the LNLMS(100) filter has a leakage factor that is closer to unity than that of the LNLMS(80) filter. However, in a real ANR system, the reference noise SPL is not known a priori, thus the leakage factor must be chosen based on worst case SNR in order to prevent instability. Thus, in the test procedure, application of the algorithm tuned for a specific SPL to cancellation of noise not matching the tuning conditions demonstrates the loss of performance that results for constant tuning parameters. The Lyapunov based tuning approach aims to retains stability and performance in the presence of any noise source, ranging from stationary tones to highly nonstationary F-16 noise over the 20 dB dynamic range, i.e., at both 80 and 100 dB SPL. A successful Lyapunov tuned candidate should provide better performance than the LNLMS(80) filter at low SNR, while maintaining performance equal to or exceeding that of the LNLMS(100) filter at high SNR.

III. EXPERIMENTAL COMPARISON OF EMPIRICALLY TUNED AND LYAPUNOV TUNED ALGORITHMS Figures 8 and 9 show the results of the NLMS, LNLMS, and the three candidate Lyapunov tuned LMS algorithms acting tones and F-16 noise, respectively. The results presented in Figures 8 and 9 represent an ensemble average of performance of four trials of each

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algorithm on five-second noise samples, for tonal noise and F-16 noise, respectively for both sound pressure levels. Beginning with results for pure tones in Figure 8, the NLMS is unstable when applied to the 80 dB noise source. This is indicated by the drift in SPL after convergence, which is caused by weight drift associated with measurement noise. The LNLMS(100) also is unstable, albeit the weight drift is slower than for the NLMS algorithm.

Instability of the NLMS algorithm

indicates that a leaky LNLMS algorithm is required, and instability of the LNLMS(100) indicates that a fixed leakage parameter providing optimal performance for 100 dB F-16 noise simply is not large enough to retain stability for the 80 dB tonal noise source. The LNLMS(80) retains stability at a performance loss of approximately 12 dB. Of the Lyapunov tuned LMS algorithms (Figure 8c), candidates 2 and 3 are also unstable at 80 dB SPL, while candidate 3 retains stability and provides a steady-state noise reduction exceeding that of the LNLMS(80). For 100 dB SPL, all algorithms are stable, though it is clear that a reduction in performance from that of the NLMS algorithm results for the empirically tuned LNLMS(100) and LNLMS(80) algorithms, as indicated by the steady-state SPL. The Lyapunov tuned algorithms each retain performance comparable to that of the NLMS algorithm, indicating that for the higher SPL, the time varying leakage parameter is closer to unity, on average. In ranking the stability and performance characteristics of each candidate, as predicted, candidate 2 provides the most aggressive performance; it exhibits the fastest transition to steadystate of the Lyapunov tuned algorithms, as noted from Figure 8d. Candidate 2 also exhibits the poorest stability of the three candidates as indicated by the response to the 80 dB source in Figure 8c.

Here, the weight drift is fastest for candidate 2, followed by candidate 1, while

candidate 3 remains stable. These experimental results confirm the analytic results derived from

17


Figure 2 that rank candidate 3 as the algorithm of choice for maintaining both stability and performance. For F-16 noise, stability and performance results, which are shown in Figure 9, are similar. In all cases, the NLMS algorithm, LNLMS(100), and first two candidate Lyapunov tuned LMS algorithms are unstable for each 80 dB noise source. LNLMS(80) and candidate 3 are stable for each 80 dB noise source, and candidate 3 outperforms the fixed leakage parameter LNLMS(80). For each 100 dB noise source, all algorithms are stable. However, both the LNLMS(80) and LNLMS(100) exhibit significant performance reduction over the NLMS algorithm, while the candidate Lyapunov tuned algorithms retain performance comparable to the NLMS algorithm. Performance gains of Lyapunov tuned candidates over the fixed leakage parameter LMS algorithms is confirmed by the experimentally determined mean and variance of the leakage factor for each candidate, as shown in Table I. The Lyapunov tuned LMS algorithms are more aggressively tuned and operate closer to the Wiener solution (λ = 1), providing better performance than constant leakage factor algorithms. The experimental results provide evidence that stability and performance gains are achieved in the reduction of both stationary and highly nonstationary noise for an optimized combination of both adaptive step size and adaptive leakage factor without requiring empirical tuning. We further investigate the stability of Lyapunov tuned LMS algorithms for the stationary tones and nonstationary F-16 noise sources by examining steady-state noise attenuation and filter weights for a single100-sec noise sample. For the 80 dB noise source, for which candidate 3 is stable for five-second noise samples, stability is also evident for a 100 second sample, as indicated in Figure 10. Figures 10a and 10c show that steady-state performance of at least 15 dB

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attenuation in SPL is maintained, and Figures 10b and 10d show that no weight drift is evident for the 100-second noise samples. For the 100 dB SPL, Figures 11a and 11c shows that all three candidates reach a steady-state performance of at least 30 dB attenuation in SPL. A slow weight drift is apparent for candidates 1 and 2, while candidate 3 asymptotically approaches steadystate. Overall noise reduction performance of 30 dB for the more difficult nonstationary F-16 source is only 3 dB lower than the 33 dB noise reduction for pure tones. Thus, minimal performance degradation in incurred as the LMS filter attempts to cancel a source whose statistics can vary rapidly with time.

IV. SIMULATION OF LYAPUNOV TUNED CANDIDATES FOR INDUCED, LOW SNR The Lyapunov tuning method presents stability bounds as a function of scalar constants A and B defined in eq. 11 and 12. These scalar constants, along with a Lyapunov function of the projected the weight vector difference w˜ k collapse the n-dimensional weight vector space to a two-dimensional space, such that stability and performance tradeoffs of candidate tuning laws can be visualized in a three-dimensional plot of Vk +1 − Vk vs. A and B. To evaluate the use of scalar parameters in the analysis of stability of LMS algorithms, a simulation was performed in which A and B can be determined for each time step, and in which stability and performance for very low SNR was measured. The simulation was required because actual measurement noise sequences are known during simulation, allowing for computation of A and B at each time step, but noise sequences are not known during an actual experiment. In the simulation, the unknown acoustic transfer function of the headset ( H3 ( s) in Figure 5) is modeled by a finite impulse response filter which represents the passive attenuation of the headset. The acoustic noise sources are input into this simulated system to provide the signal dk

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which is to be cancelled. The candidate LMS algorithm acts on a noisy measured source sequence Xk + Qk , as in the actual experimental evaluation. However, in simulation, the noise sequence Qk is known a priori, as is the Wiener solution, which is the finite impulse response filter representing the passive system. The noise sequence Qk is generated by recording a time history of the actual test cell noise floor, and scaling this measured sequence to represent a desired SNR for the simulation. The scaled noise sequence is added to the acoustic noise source to generate Xk + Qk . Again, tonal noise and band limited F-16 noise are used as the noise sources. Figure 12 shows the results of simulation analysis, presented as a three-dimensional plot of Vk +1 − Vk vs. A and B, showing 5000 samples of the actual system operating points for a 35 dB SNR. Points satisfying Vk +1 − Vk < 0 represent stable operating conditions, while for

Vk +1 − Vk > 0, the system may or may not be stable. From the simulation analysis, it is evident that a majority of the instantaneous operating points are guaranteed to be stable for both pure tones and F-16 noise, with fewer excursions to larger values of A and B, or away from the Wiener solution, for stationary tonal noise than for nonstationary F-16 noise. Additional insight is gained by viewing Figure 12 as a two-dimensional plot, as shown in Figure 13. Here, the plane Vk +1 − Vk = 0 is defined by solid lines, and the stable region is denoted. Figure 14 shows the same results for a lower SNR of 15 dB. Figures 13 and 14 show a 'cluster' of operating points near the Wiener solution, with occasional excursions to larger values of A and B caused by measurement noise. For 35 dB SNR, the cluster is nearer to the Wiener solution than for 15 dB SNR as expected, as the bias in the weight vector update equation due to the leakage factor increases with decreased SNR. In addition, for the F-16 noise source, variation or scatter in the operating points is greater than for the tonal noise source, as would be expected for the

20


nonstationary noise. Analysis such as that presented in Figures 13 and 14 aids in visualizing the effects of nonstationarity and measurement noise on stability and performance of Lyapunov tuned algorithms. To further aid in the analysis, we define P, the probability of operating outside of the guaranteed stability boundary, i.e., the probability that Vk +1 − Vk > 0. For pure tones P = 0.1162 and 0.1704 for 35 dB and 15 dB, respectively, indicating the expected reduction in stability as SNR decreases, and for F-16 noise, P = 0.037 and 0.2556, respectively for 35 dB and 15 dB SNR. For F-16 noise, a significant bias away from the Wiener solution is indicated by a center of the cluster of points far from A = B = 0, as compared to tonal noise. As a consequence, the F16 simulation at 35 dB SNR has a lower overall P, though excursions in operating points away from A = B = 0 are more frequent, as would be expected for this nonstationary noise source. Confirming the experimental evaluation, simulation results of candidate 3 compared with other candidate algorithms indicate that it retains the best stability properties, while candidate 2 retains more aggressive performance at a sacrifice to stability. Table II provides summary of the probabilities of exceeding Vk +1 − Vk = 0 for candidates 2 and 3. Here, with the exception of pure tones at 35 dB, the probability of exceeding the guaranteed stability boundary is larger for candidate 2, indicating this stability sacrifice. The steepness of the Vk +1 − Vk vs. A and B surface of candidate 2 compared with candidate 3, as shown in Figure 2, contributes to a lower P for the 35 dB SNR and pure tones. However, the same phenomenon generally contributes to the reduction in stability for this algorithm. That is, equal excursions away from A = B = 0 for each candidate result in a larger value of Vk +1 − Vk for candidate 2 than for candidate 3, increasing the likelihood of instability for candidate 2. In all simulation results, the weight update was stable,

21


even though the probability of exceeding the stability boundary was non-zero. Thus, Lyapunov analysis errs on the conservative side when predicting stability.

CONCLUSION Experimental evaluation of Lyapunov tuned leaky LMS algorithms in a prototype ANR communication headset demonstrates the Lyapunov tuning method as a means for a priori evaluation of stability and performance tradeoffs in time-varying leakage and step size LMS algorithms. The comparative study shows that LNLMS filters with adaptive leakage parameter and step size can provide improved stability at low SNR, while retaining low frequency performance comparable to a non-leaky NLMS algorithm at high SNR. Candidate algorithm 3, which results from a non-traditional step-size and leakage parameter combination obtained through appealing to the structure of the Lyapunov function difference, demonstrated the best overall experimental performance. Candidate 3 retains stability for all noise sources at 35 dB SNR, while demonstrating steady-state performance of 33 and 30 dB, respectively in the reduction of pure tones and band limited F-16 noise at 100 dB SPL.

ACKNOWLEDGEMENT This research is supported in part by the United States Air Force contract number F41624-99-C6006 though a subcontract with Creare, Inc. The authors are grateful to the Air Force Research Laboratory Human Effectiveness Directorate, Wright Patterson Air Force Base, Dayton Ohio, for the support.

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REFERENCES 1. G.J. Pan, A.J. Brammer, R. Goubran, J.G. Ryan, and J. Zera, "Broad-Band Active Noise Reduction in Communication Headsets," Canadian Acoustics, 22(3), pp. 113-114, March 1994. 2. A.J. Brammer and G.J. Pan, "Opportunities for Active Noise Control in Communication Headsets," Canadian Acoustics 26(3), pp. 32-33, September 1998. 3. A.J. Brammer, G.J. Pan, and R.B. Crabtree, "Adaptive Feedforward Active Noise Reduction Headset for Low-Frequency Noise," Proceedings of Active '97 - Symposium on Active Control of Sound and Vibration, pp. 365-372, 1997. 4. J.C. Burgess, "Active Adaptive Sound Control in a Duct: A Computer Simulation," Journal of the Acoustical Society of America, 70, pp. 715-726, Sept 1981. 5. B. Widrow, D. Shur, and S. Shaffer, "On adaptive inverse control," Proc. 15th Asilomar Conf., pp. 185-189, 1981. 6. S.M. Kuo and D.R. Morgan Active Noise Control Systems, John Wiley and Sons, New York, 1996. 7. D. Cartes, L.R. Ray, and R.D. Collier, "Lyapunov Tuning of the Leaky LMS Algorithm for Single-Source, Single-Point Noise Cancellation," submitted to IEEE Transactions on Signal Processing, May 2000. 8. J.G. Ryan, E.A.G. Shaw, A.J. Brammer, and G. Zhang. “ Enclosure for Low Frequency Assessment of Active Noise Reducing Circumaural Headsets and Hearing Protection,” Canadian Acoustics, 21(4), pp. 19-20, 1993. 9. H. J. Steeneken and F. W. Geirtsen, "Description of the RSG-10 Noise Database," TNO Human Factors Institute, Soesterberg, The Netherlands, 1988. 23


10. R. P. Gitlin, H.C. Meadows, and S.B. Weinstein, “The Tap-Leakage Algorithm: An Algorithm for the Stable Operation of a Digitally Implemented Fractionally Spaced Equilizer.” Bell Syst. Tech. J., 61(8), pp. 1817-1839, Oct 1982. 11. J.E.Slotine, and W. Li, Applied Nonlinear Control, Prentice Hall, New Jersey, 1991. 12. S.Haykin, Adaptive Filtering Theory, Prentice Hall, NJ, 1996. 13. S.B. Gelfand, Y. Wei, and J.V. Krogmeier, “The Stability of Variable Step-Size LMS Algorithms,” IEEE Transactions of Signal Processing, 47(12), pp. 3277-3288, Dec. 1999. 14. D. Cartes, L.R. Ray, and R.D. Collier, "Low Frequency Acoustic test Cell for the Evaluation of Circumaural Headsets and Hearing Protection," submitted to Canadian Journal of Acoustics, July 2000.

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List of Tables Table I

Mean tuning parameters for three candidate adaptive LNLMS algorithms

Table II

Probability of Vk +1 − Vk > 0 for simulation of candidate 2 and candidate 3 in response to pure tones and F-16 noise and archived measurement noise on the reference input.

25


Table I Mean tuning parameters for three candidate adaptive LNLMS algorithms LMS algorithm and sound pressure level (SPL)

Mean leakage factor (tones)

LNLMS(80) 80 dB LNLMS(100) 100 dB Candidate 1 80 dB 100 dB Candidate 2 80 dB 100 dB Candidate 3 80 dB 100 dB

0.999 930 6 0.999 988 305 0.999 999 8130 0.999 999 9983 0.999 999 4077 0.999 999 9947 0.999 732 4048 0.999 997 3676

Standard deviation of leakage factor x 10000 (tones) 0 (constant) 0 (constant) 0.000 035 40 0.000 000 071 0.000 054 41 0.000 000 241 0.123 289 85 0.000 000 24

Mean leakage factor (F-16)

0.999 930 6 0.999 988 305 0.999 999 8749 0.999 999 9986 0.999 999 5584 0.999 999 9972 0.999 999 5584 0.999 997 8927

Standard deviation of leakage factor x 10000 (F-16) 0 (constant) 0 (constant) 0.000 156 3 0.000 004 67 0.001 308 12 0.000 005 60 0.528 805 03 0.004 826 22

26


Table II Probability of Vk +1 − Vk > 0 for simulation of candidate 2 and candidate 3 in response to pure tones and F-16 noise and archived measurement noise on the reference input. Noise source Pure tones • 35 dB SNR • 15 dB SNR F-16 noise • 35 dB SNR • 15 dB SNR

P, Candidate 2

P, Candidate 3

0.0328 0.2018

0.1162 0.1704

0.0502 0.3274

0.0379 0.2556

27

Experimental evaluation of leaky LMS algorithms for active noise

Experimental evaluation of leaky LMS algorithms for active noise

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