Luigi Piroddi Active Noise Control course notes (January 2015)

Transcription

1 Active Noise Control course notes (January 2015) 9. On-line secondary path modeling techniques Luigi Piroddi

2 Introduction In the feedforward ANC scheme the primary noise is canceled by a secondary noise generated by an adaptive filter through a secondary path. For the adaptive filter to converge to the desired solution it is necessary to compensate for the secondary path transfer function S(z). The secondary path transfer function S(z) can be estimated: off-line, prior to the ANC system operation on-line, when the ANC system is in operation On-line estimation may be necessary to ensure stability and convergence of the adaptive filter, if the secondary path is known to be time varying. 9. On-line secondary path modeling techniques 2

3 Why direct on-line secondary path modeling is not viable x(n) S^(z) (copy) P(z) W(z) y(n) S(z) S^(z) d(n) y (n) y^ (n) e(n) An adaptive filter S^(z) is inserted in parallel with the secondary path S(z). The secondary signal y(n) is used also as excitation signal for secondary path modeling. x (n) e(n) f(n) The adaptive filter S^(z) is adapted with the algorithm to minimize f(n). F(z) = E(z) S^(z)Y(z) = [P(z) S(z)W(z) S^(z)W(z)] X(z) 9. On-line secondary path modeling techniques 3

4 Assuming that: S^(z) is of sufficient order x(n) is a persistently exciting signal P(z) and W(z) are time-invariant systems the algorithm will make f(n) converge to 0. The resulting secondary path transfer function estimate would be: S^ (z) = S(z) P(z) W(z) which is biased by an unknown factor P(z)/W(z). correct identification is possible only if d(n) is 0, i.e., if the primary noise is canceled with other techniques the input y(n) to the S^(z) block must be persistently exciting S^ (z) depends on W(z), which in turn depends on S(z) again, since the optimum solution for W(z) is W (z) = P(z)/S(z)! Indeed, if W(z) were optimal, S^ (z) would be 0, which is obviously an undesired solution 9. On-line secondary path modeling techniques 4

5 Time-difference signal algorithm (van Overbeek, 1991) If the primary noise is periodic and the primary and secondary path transfer functions are slowly changing, the influence of d(n) can be reduced by subtracting two successive periods of the error signal e(n). y(n) S(z) d(n) y (n) e(n) z N ~ ~ y (n) y^ (n) e (n) S^(z) f(n) z N This time-differencing process is equivalent to comb filtering the output and error signals in order to prevent contamination by the periodic noise. 9. On-line secondary path modeling techniques 5

6 ~ y (n) = y(n) y(nn) ~ e (n) = e(n) e(nn) = [d(n)s(n)*y(n)] [d(nn)s(nn)*y(nn)] Remarks: the periodicity of the primary noise implies that d(n) d(nn) furthermore, assuming that S(z) is slowly changing, the system impulse response is almost time-invariant, i.e. s(n) s(nn). Therefore, the differenced error becomes: ~ e (n) s(n)*[y(n) y(nn)] = s(n)*y ~ (n) and the z-transform of the modeling error signal can be expressed as: F(z) = E ~ (z) S^(z)Y ~ (z) = [S(z) S^(z)]Y ~ (z) 9. On-line secondary path modeling techniques 6

7 Assuming that S^(z) is of sufficient order f(n) will converge to 0 and S^(z) will tend to S(z), at least at frequencies for which Y ~ (z) 0. Remarks: apparently, the result is not so good since we need accurate estimation mostly at the harmonics of the primary signal, i.e., exactly at the frequencies which have been filtered away however, if we assume that the frequency response S(e jω ) cannot vary too rapidly, and that the model order is not too high (M << N), the modeling filter will interpolate the frequency response near these frequencies obtaining a reasonable approximation 9. On-line secondary path modeling techniques 7

8 Additive random noise technique (Eriksson and Allie, 1989) x(n) S^(z) (copy) x (n) P(z) W(z) y(n) e(n) v(n) Random noise generator S(z) S^(z) d(n) y (n)v (n) v^ (n) f(n) e(n) A random noise, uncorrelated with the primary noise, or a chirp signal is added to the secondary signal y(n). Only the noise component is fed to the secondary path model. 9. On-line secondary path modeling techniques 8

9 The residual error is expressed as: e(n) = d(n) s(n)*y(n) s(n)*v(n) = d(n) y (n) v (n) The secondary noise component v (n) due to the additive random noise is estimated through model filter S^(z) as: v^ (n) = s^(n)*v(n). The component of the error due to the original noise is: u(n) = d(n) s(n)*y(n) = = [p(n) s(n)*w(n)]*x(n) Notice that u(n) depends only on x(n), and therefore is uncorrelated with v(n). v(n) S(z) S^(z) u(n) v (n) v^ (n) f(n) v (n)u(n) Using u(n) the problem is re-stated as a standard identification problem, where u(n) has the role of a plant noise, uncorrelated with the driving signal v(n). 9. On-line secondary path modeling techniques 9

10 The optimal solution that minimizes E[f(n) 2 ] can be expressed as: S^ (z) = S v v(z) S vv (z) where S v v (z) is the cross-power spectrum between v(n) and v (n) (which is equal to the z-transform of the cross-correlation function r v v (k) = E[v (n)v(nk)]) S vv (z) is the autopower spectrum of v(n) Notice that: r v v (k) = E[v (n)v(nk)] = E[s(n)*v(n)v(nk)] = s(n)*e[v(n)v(nk)] = s(n)*r vv (k) which implies S v v (z) = S(z)S vv (z) and therefore: S^ (z) = S(z) The solution is not affected by the presence of u(n), although the algorithm convergence properties are. 9. On-line secondary path modeling techniques 10

11 The coefficients of the adaptive filter are updated with the algorithm: s^(n1) = s^(n) µ v(n)f(n) = s^(n) µ v(n)[v (n)v^ (n)] µ v(n)u(n) The expected value of s^(n) will converge to the optimal solution s(n), provided v(n) and u(n) are uncorrelated. In other words, Ε[ s^(n)] = 0 if v (n) = v^ (n) E[v(n)u(n)] = 0 This does not imply that the instantaneous values of s^(n) will be equal to s(n): the undesired term µ v(n)u(n) will slow down convergence. if the interference u(n) is much larger than the excitation signal v(n), the convergence rate of filter S^(z) will be very slow unfortunately, v(n) must be kept small because it appears also at the control zone, and therefore influences the achievable level of noise attenuation 9. On-line secondary path modeling techniques 11

12 Cancelling the interference u(n) Convergence of S^(z) can be improved by using the adaptive noise cancellation approach to cancel the component of e(n) that is correlated with x(n), i.e., u(n) = d(n) s(n)*y(n) = = [p(n) s(n)*w(n)]*x(n) An additional adaptive filter H^ (z) is added to the estimation scheme, with x(n) as reference signal. Experimental evidence shows that the convergence rate can be improved by a factor of ~30. Clearly, the technique does not apply to components of the interference that are uncorrelated with x(n). 9. On-line secondary path modeling techniques 12

13 x(n) P(z) d(n) e(n) W(z) y(n) S(z) y (n)v (n) S^(z) (copy) x (n) e(n) Random noise generator v(n) S^(z) v^ (n) f(n) e (n) H^ (z) e^(n) 9. On-line secondary path modeling techniques 13

14 Overall modeling algorithm The direct on-line modeling approach resulted in a biased secondary path estimate S^ (z) = S(z) P(z) W(z) Another algorithm aims at the elimination of the biasing term P(z)/W(z) by introducing another adaptive filter P^(z) to model P(z). The output of P^(z) is then used to cancel the disturbance d(n) that is the output of P(z). The complete ANC system uses three adaptive filters, W(z), P^(z) and S^(z) to perform simultaneously noise control and secondary path modeling. The overall modeling algorithm has the capability to model the secondary path without using an additional excitation signal: x(n) can be used as both a primary and secondary excitation signal by setting (off-line) W(z) = z L in this way the primary and secondary signals are sufficiently decorrelated so that P^(z) and S^(z) will converge to P(z) and S(z), respectively 9. On-line secondary path modeling techniques 14

15 x(n) P(z) d(n) e(n) S^(z) (copy) x (n) z L W(z) off-line online e(n) y(n) S(z) S^(z) e^(n) x(n) P^(z) f(n) 9. On-line secondary path modeling techniques 15

16 Off-line initialization Assume that the physical systems P(z) and S(z) can be modeled by FIR filters of orders L and M, respectively. Then, the error signal can be expressed as: e(n) = p(n) T x(n) s(n) T y(n) = h(n) T u(n) where p(n) = [p 0 (n) p 1 (n) p L1 (n) ] T is the impulse response vector of P(z) at time n x(n) = [x(n) x(n1) x(nl1)] T s(n) = [s 0 (n) s 1 (n) s M1 (n) ] T is the impulse response vector of S(z) at time n y(n) = [y(n) y(n1) y(nm1)] T h(n) = p(n) s(n) u(n) = x(n) y(n) 9. On-line secondary path modeling techniques 16

17 During the off-line initialization period, W(z) = z L so that y(n) = x(nl) and u(n) = x(n) y(n) = [x(n) x(n1) x(nlm1)]t. Thus, the expression e(n) = h(n) T u(n) actually represents one unknown FIR of order LM excited by a signal x(n) and with response e(n), in which the two subsystems are combined. With analogous settings, the combined output of the adaptive filters P^(z) and S^(z) can be written as: e^(n) = p^(n) T x(n) s^(n) T y(n) = h^(n) T u(n) where p^(n) = [p^0(n) p^1(n) p^l1(n)] T s^(n) = [s^0(n) s^1(n) s^m1(n)] T h^(n) = p^(n) s^(n) 9. On-line secondary path modeling techniques 17

18 The FIR filter H^ (z) is used to identify the unknown filter H(z) using the single excitation signal x(n), by minimization of the mean-square of the estimation error f(n) = e(n) e^(n) Assuming that x(n) is a persistently exciting signal, H^ (z) will converge to H(z) by using the algorithm: ^(n1) h = h^(n) µu(n)f(n), which is equivalent to: p^(n1) = p^(n) µx(n)f(n), s^(n1) = s^(n) µy(n)f(n) = s^(n) µx(nl)f(n), 9. On-line secondary path modeling techniques 18

19 On-line operation After the identification of filters P^(z) and S^(z), the secondary signal is switched from the output of an L-delay block to the actual filter W(z). Then W(z), P^(z) and S^(z) are all updated simultaneously (using the output of W(z) as excitation signal for P^(z) and S^(z)). F(z) = E(z) E^(z) = = [P(z)S(z)W(z)]X(z) [P^(z)S^(z)W(z)]X(z) = = {[P(z)P^(z)] [S(z)S^(z)]W(z)}X(z) The adaptive filter minimizes the residual error e(n), while P^(z) and S^(z) minimize the signal f(n). Ideally, f(n) will converge to 0, i.e., to the condition: [P(z)P^(z)] [S(z)S^(z)]W(z) = 0 9. On-line secondary path modeling techniques 19

20 This equation has many solutions, but, since both filters have been initialized to their optimal values, the algorithm is able to track variations of the two functions P(z) and S(z), provided that only one of the two changes at a time. Experimental simulations have shown that the algorithm is able to track slow changes of P(z) and S(z), whereas a re-tuning stage (replacing W(z) with z L again) is necessary to track drastic changes. Comparison with the additive random noise technique: in the overall modeling technique, the use of an additive random noise, which affects the overall ANC system performance in the on-line operation, is avoided in the additive random noise technique, persistent excitation conditions can be ensured, while in the overall modeling technique x(n) may not provide sufficient excitation over the entire band in the overall modeling technique, since the estimate S^(z) may change also as a result of variations of x(n) rather than actual secondary path variations, the copy of S^(z) used as a prefilter for the Fx algorithm has to be constantly adjusted while the noise canceling performance of the two approaches is equivalent, the convergence of the additive noise technique is faster 9. On-line secondary path modeling techniques 20

21 Multiple-channel secondary path modeling: the interchannel coupling effect On-line modeling of K M secondary paths for a multiple-channel ANC system is more difficult than for a single-channel case, since the error signal e m (n) from the mth sensor is a mixture of signals coming from the primary path P m (z) and secondary paths S mk (z), k = 1, 2,, K. This is called the interchannel coupling effect. For this reason off-line secondary path modeling involves activating one secondary source at a time injecting a random noise signal to identify the M paths from the secondary source to the M error sensors. The identification process must be repeated K times to cover all the secondary sources and estimate all K M secondary paths. 9. On-line secondary path modeling techniques 21

22 Consider the portion of a ANC system depicted in the figure, including two secondary signals and one of the two error signals. An additive random noise is added to the secondary signals to excite the secondary path systems. y 1 (n) u 1 (n) S 11 (z) d 1 (n) S^11(z) Random noise generator v(n) f 1 (n) e 1 (n) S^12(z) y 2 (n) u 2 (n) S 12 (z) e^(n) 9. On-line secondary path modeling techniques 22

23 From the perspective of the estimation of S^11(z), the modeling error used by the algorithm is the difference f 1 (n) = u(n) s^11(n)*v(n) v(n) S^11(z) f 1 (n) u(n) where u(n) = d 1 (n) s 11 (n)*[y 1 (n)v(n)] s 12 (n)*[y 2 (n)v(n)] s^12(n)*v(n) The filter will converge to S^11 (z) = S uv(z) S vv (z) where S uv (z) is the cross-power spectrum between u(n) and v(n) S vv (z) is the autopower spectrum of v(n) 9. On-line secondary path modeling techniques 23

24 Assuming v(n) to be a zero mean signal, uncorrelated with y 1 (n), y 2 (n) and d 1 (n) the cross-power spectrum can be computed as: S uv (z) = [S 11 (z) S 12 (z) S^12(z)] S vv (z) which implies S^11 (z) = S 11 (z) [S 12 (z) S^12(z)] Therefore, when the first error sensor picks up noise components through multiple secondary paths (S 11 (z) and S 12 (z)), the estimate S^11(z) is biased by the crosscoupled secondary paths S 12 (z) and S^12(z). The desired result S^11(z) = S 11 (z) is obtained only if S 12 (z) = S^12(z). Since S^11(z) and S^12(z) are adapted at the same time, there is no unique solution for either filter. Notice that the cross-coupling effect occurs both in the on-line and off-line modeling phases. 9. On-line secondary path modeling techniques 24

25 Multiple-channel on-line modeling algorithms y 1 (n) Random noise generator Random noise generator y 2 (n) v 1 (n) v 2 (n) S 11 (z) S^11(z) S^12(z) S 12 (z) f 1 (n) e 1 (n) d(n) Consider now a ANC system with two uncorrelated random noises, v 1 (n) and v 2 (n) added to the secondary signals y 1 (n) and y 2 (n), respectively, to excite the secondary path systems. We assume that v 1 (n) and v 2 (n) are mutually uncorrelated and also uncorrelated with y 1 (n), y 2 (n) and d(n). We have: S^11 (z) = S uv 1 (z) S v1 v 1 (z) 9. On-line secondary path modeling techniques 25

26 The modeling error used by the algorithm is the difference between a signal u(n) = d 1 (n) s 11 (n)*[y 1 (n) v 1 (n)] s 12 (n)*[y 2 (n) v 2 (n)] s^12(n)*v 2 (n) and the output of the adaptive filter S^11(z), i.e., f 1 (n) = u(n) s^11(n)*v 1 (n) This time, since v 1 (n) and v 2 (n) are mutually uncorrelated, we obtain: S uv1 (z) = S 11 (z)s v1 v 1 (z) which implies S^11 (z) = S 11 (z) Therefore, the inter-channel coupling problem can be solved by using K independent random noise generators (one for each secondary source). However, the cost may be too high for some ANC applications. 9. On-line secondary path modeling techniques 26

27 The inter-channel delay approach y 1 (n) Random noise generator y 2 (n) v(n) z L v(nl) S 11 (z) S 21 (z) S^11(z) S^21(z) S^12(z) S^22(z) S 12 (z) S 22 (z) d 1 (n) d 2 (n) e 1 (n) f 1 (n) f 2 (n) e 2 (n) A single noise source can be employed with an inter-channel delay to decouple the excitation signals. In the ANC system depicted in the figure: d 1 (n), d 2 (n) are the primary noises y 1 (n), y 2 (n) are the secondary signals e 1 (n), e 2 (n) are the error signals v(n) is the internally generated random noise, uncorrelated with d 1 (n), d 2 (n), y 1 (n), y 2 (n) Filters S^11(z) and S^12(z) are updated to minimize f 1 (n), while S^21(z) and S^22(z) are updated to minimize f 2 (n). 9. On-line secondary path modeling techniques 27

28 The update equations are as follows: s^11 (n1) = s^11(n) µv(n)f 1 (n) s^21 (n1) = s^21(n) µv(n)f 2 (n) s^12 (n1) = s^12(n) µv(nl)f 1 (n) s^22 (n1) = s^22(n) µv(nl)f 2 (n) The z-transform of f 1 (n) has the following expression: F 1 (z) = D 1 (z) [S 11 (z) S^11(z)] V(z) [S 12 (z) S^12(z)] V(z)z L = = D 1 (z) [S 11 (z) S^11(z) S 12 (z)z L S^12(z)z L ] V(z) By exploiting the incorrelation between successive portions of noise, we obtain the desired solutions S^ij(z) = S ij (z). 9. On-line secondary path modeling techniques 28

29 Multiple-channel overall modeling algorithm The overall modeling algorithm can also be extended to multiple-channel ANC systems. Given a J K M ANC system, K1 adaptive filters for each of the M error signals are necessary: K adaptive filters S^mk(z) to model the corresponding secondary paths S mk (z) 1 adaptive filter P^ m(z) to cancel the highly correlated disturbance d m (n) from the primary noise source The modeling results S^mk(z) are signal dependent and a unique solution cannot be guaranteed. 9. On-line secondary path modeling techniques 29

30 Integrated ANC-audio system A multiple-channel ANC system can be integrated with an existing audio system, e.g., in a passenger compartment of a car. audio source l(n) r(n) y 1 (n) ANC system y 2 (n) e 2 (n) audio signal canceler e 1 (n) The audio signal becomes an interference for the ANC system which is reduced by means of an adaptive noise canceler with the audio signal as the reference signal. noise source primary noise enclosure e 2 (n) e 1 (n) The adaptive filter used for the audio interference cancellation will converge to the secondary path, and thus provides on-line secondary path modeling. 9. On-line secondary path modeling techniques 30

31 Integrated ANC-audio system: single channel scheme noise source x(n) P(z) d(n) e(n) S^(z) (copy) x (n) W(z) y(n) u(n) v(n) S(z) S^(z) y (n) audio source v(n) f(n) e (n) 1 9. On-line secondary path modeling techniques 31

32 The optimal solution for the audio interference cancellation filter is: S^ (z) = S ev(z) S vv (z) Assuming that the audio signal and the secondary noise are uncorrelated, we have: S ev (z) = S(z)S vv (z) so that the optimal solution becomes: S^ (z) = S(z) Therefore, provided that the audio signal is persistently exciting and uncorrelated with the primary noise, the interference cancellation filter can model the secondary path. After convergence of the interference cancellation filter we have: E (z) = E(z) S^ (z)v(z) = D(z) S(z) [Y(z) V(z)] S^ (z)v(z) = D(z) S(z)Y(z) as if the audio interference was completely absent. Then, the usual adaptation of the ANC filter becomes possible. 9. On-line secondary path modeling techniques 32

33 Some benefits of the audio interference cancellation algorithm are: the difference signal e (n) is a good estimate of the true residual noise error of the ANC system, which is employed by the Fx algorithm to update the filter W(z) on-line modeling of the secondary path transfer function is possible without the addition of external random noise (the audio signal is normally sufficiently exciting) however, this implies that the audio signal must be playing something (e.g., music) for the ANC system to work (at least for the time required to train the interference cancellation filter) indeed, music has sufficient exciting properties over the bandwidth involved in the ANC system a large step size may be used for the adaptation of the ANC filter, since the e (n) is not corrupted by a high volume audio signal 9. On-line secondary path modeling techniques 33

34 Notice that in a real enclosure application a multiple-channel ANC algorithm would be necessary, as well as a multiple-channel interference cancellation algorithm. Interference cancellation and on-line modeling are inherently more difficult in the multiple-channel case: the left and right audio signals may be partially correlated and that would cause problems in identifying the cross-coupled secondary paths the interchannel decoupling delay technique cannot be used because that would destroy the stereo effect of the desired signals consequently, off-line techniques must be used for multiple-channel systems (possibly continuing adaptation on-line in the assumption that only one path changes at a time) 9. On-line secondary path modeling techniques 34