DEVELOPMENT OF ORTHOGONALIZED ACTIVE NOISE CONTROL SYSTEMS ROGELIO BUSTAMANTE B.1,2, SERGIO BARRETO A.2, EDGAR BERMEJO A. 2, VICENTE MADRIGAL T. 2, ROCÍO VALLE R2, HÉCTOR M. PÉREZ M.1, BOHUMIL PSENICKA3 (1) Engineering Graduate Studies Section. ESIME Culhuacán, IPN. 1000 Santa Ana Ave. San Francisco Culhuacán, 04430, Ciudad de México, México.
[email protected] (2) Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Ciudad de México. 222 Del Puente St. , Ejidos de Huipulco, 14380, Ciudad de México, México.
[email protected] (3) Universidad Autónoma Nacional de México, Facultad de Ingeniería, P. O. Box 70-389, Delegación Coyoacán, 04510 Ciudad de México, México.
[email protected]
ABSTRACT In the present paper, several algorithms are analyzed and implemented for active noise cancellation To test their performance we used multiple sound patterns which we simulate separately to specify the best algorithm performance for each sound. In order to analyze and control all parameters from each algorithm, Simulink basic blocks (delays, algebraic operators) were used for simulation development. Simulations results show how some algorithms are better than other depends on the noise to cancel in the system.
KEY WORDS ANC, active noise cancellation, DCT, Lattice, orthogonal samples, normalizing. I. INTRODUCTION TO ANC SYSTEMS The ANC involves electroacoustic or electromechanic systems that cancel primary noise based on the superposition principle; in fact, a “pseudo-noise” is generated with same amplitude but with contrary phase. The amount of cancelled noise depends on the amplitude and phase of the signal generated [1]. ANC attenuates low frequency noise where passive systems result no efficient at all. In the digital signal processing field there are some adaptive systems which implements ANC systems. In these systems a digital filter coefficient is adjusted to minimize an error signal which is stated as the desired signal minus the control signal. These adaptive filters can be designed as transversals or FIR (finite impulse response), which are the most used; recursive or IIR (infinite impulse response); Lattice and frequency domain filters. In the block diagram of figure 1 we can see the noise cancellation system. From the block diagram the system P(z) has an output signal d(n) which are both unknown because depend on uncontrollable variables.
Adaptive filter W(n)
Physical system
Noise d(n) Physical plant Output signal y(n)
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-
Σ
Signal reference x(n) = estimated noise d´(n)
Error signal e(n)
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Σ
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LMS algorithm: coefficients estimation of the W(n) filter
Figure 1: ANC system without reference
We do know the W(z) system producing the signal y(n) which estimates the d(n) signal. For closing the control loop we use the LMS algorithm that modifies the filter’s coefficients and reduces the error signal. For this job we intend to modify this part with our algorithm proposals.
II. ACTIVE NOISE CANCELATION (ANC) ALGORITHMS There are many algorithms that gobern adaptive filters for ANC systems. In the following proposal we present the LMS (Least Mean Square) algorithm. This algorithm is one of the simplest regarding its implementation, and in its simpler version, we have the stocastic gradient LMS algorithm. In this algorithm we have a reference signal, which for our purposes is the estimated signal x’(n). The LMS algorithm uses expression (1) in order to calculate the optimum coefficients. From these equations it is possible to see the algorithm’s recursivity which means that the present value of the coefficient depends on the former one. The LMS is essentialy a descendent gradient search method. The behavior of this algorithm has been studied extensively in the ralated literature and it has been determined that m, a constant corresponding to the step
size used by the algorithm to search the optimum solution, is chosen by (2). Where λmax is the maximum eigenvalue of the autocorrelation matirx of the signal to be canceled. It is important to notice that m should not be very large in orther to avoid the method’s divergence, but it also should not be very small so that the convergence time turns out to be too long and by consequence useless for ANC purposes. Speaking of convergence time, other factors that can modify this behavior are the time constants where the algorithm can converge, and are determined by (3). Where λi are the n autocorrelation matrix eigenvalues of the input signal to the LMS. As we can see, the speed of convergence depends on the election of the autocorrelation matrix eigenvalues of the input signal. It is important to mention that the eigenvalues of the input signal are related directly with the power of the signal. From the literature it can be derived form this equation the slower time constant that is given by [4]. From which it can be deduced that for those cases in which there is a great dispersion index of the input signal’s eigenvalues, the algorithm will have large convergence times and as a result it will not be useful for pratical implementation.
ω (n) = ω (n − 1) + µ (∇j ) ∇j = −2e(n) x(n) 1 0
