08 - Feedback ANC

5/28/2018 08 - Feedback ANC

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Luigi Piroddi Active Noise Control course notes (January 2012)

8. Feedback active noise control 2

(Adaptive) feedforward or (non-adaptive) feedback ANC?

Feedback control systems differ from feedforward systems in the manner in which

the control signal is derived.

Feedforward systems rely on some predictive measure of the incomingdisturbance to generate an appropriate canceling disturbance.

Feedback systems generate the control signal by processing the error signal,with the goal of attenuating the residual effects of the disturbance afterit has

passed (the reference sensor is not required).

one sensor less no acoustic feedback we do not need to worry about the low coherence between reference

and disturbance

feedback ANC cannot make any difference between the noise and theuseful signal measured by the error microphone: everything is attenuated

A feedforward system should be implemented whenever it is possible to obtain

a suitable reference signal, because the performance of an adaptive feedforward

system is, in that case, superior to a feedback system.


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Feedback ANC is required for applications in which it is not possible or practical

to measure or internally generate a coherent reference signal:

spatially incoherent noise generated from turbulence noise generated from many sources and propagation paths resonant response of an impulsively excited structure, where no

coherent reference signal is available

Unlike feedforward systems for which the physical system and controller can be

optimized separately, feedback systems must be designed by considering the

physical system and controller as a coupled system.For noise problems:

Adaptive feedforward control has been applied successfully to ducts,aircraft cabins and motor vehicle interiors and exteriors.

Feedback control has been applied successfully to ear defenders whereit is not easy to sample the incoming signal in advance, making it difficult

to generate an appropriate reference signal for a feedforward controller.


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Location of sensors and actuators

The physical arrangement of control sources and error sensors plays a very

important role in determining the effectiveness of an active control system.

Moving the locations of the control sources and sensors affects both system

controllability and stability: For feedforward systems, the physical system arrangement can

be optimized independently of the controller.

For feedback systems, the physical system arrangement is animportant part of the controller design. For duct applications: the microphone should be placed at the surface of the duct (zero airflow velocity) the loudspeaker should be placed in the centre of the transverse cross-section of the

duct to minimize the number of transverse modes


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Classical feedback ANC scheme

Noise

sourcePrimary noise

Feedback

ANC

Canceling

loudspeaker

Error

microphone

(n)

e(n)

duct

d

The sensor output is processed by an amplifier that has:

an overall gain higher than unity a 180 phase reversal

The system requires only one sensor (the acoustic secondary-to-reference

feedback problem is avoided).

The feedback control system

employs a secondary source

located in the vicinity of an errorsensor (to minimize loop delay).


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Problems:

limited broadband noise attenuation limited frequency range of operation (up to a few hundredHz) the presence of a delay from the secondary source to the error sensor

implies that only periodic signals can be completely canceled

possible instability caused by positive feedback at high frequencies the control system will oscillate when the combined loop delays are equivalent

to a 180 phase shift at a particular frequency and the overall gain is greater than unity

The smaller the distance between the error microphone and the secondary source:

the higher the critical frequency over which the system cannot work, but the less uniform the sound field generated by the secondary source

(due to physical limitations of conventional loudspeakers)

Therefore, the non-uniform near-field of the loudspeaker determines an upper

bound on the frequencies that can be canceled.To prevent system oscillation, the loop gain must be less than 1 before the critical

frequency is reached (gain roll-off).


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Improved feedback ANC system implementation

The working range of a feedback

ANC system can be greatly

extended by feeding the

loudspeakers output into a

partially closed volume (with

dimensions much smaller than

the wavelength of the highest

frequency to be controlled)

containing the microphone.

Primary noise

Feedback

ANC

Canceling

loudspeaker

Error microphone

(n)

e(n)

duct

In this way,

the near-field of the loudspeaker becomes more uniform the acoustic effect of the partially closed volume yields a well-behavedroll-off characteristic the microphone is automatically isolated from nearby reflecting surfaces

(and, in cases of highly hostile environments, it is placed in a safer zone)


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Reported applications of feedback ANC

Duct ANC system comprising two (electrically) independent feedback ANCsystems in cascade (Hong et al., 1987)

each feedback stage provides additional attenuation (although less than thesum of the individual actions due to the interaction between the two systems)

Active electronic muffler for automobile exhaust noise control (Taki, 1993) the noise attenuation performance is approximately equal to that of a passive

muffler, but a great reduction of exhaust back pressure is obtained, providing

a 510 % increase of engine power

Noise attenuation in transport aircraft cabin (Legrain and Goulain, 1988) a loudspeaker is mounted at one end of the cabin and microphones are located

at passenger ear level along the fuselage

Broadband noise control in an active headset (Carme,1988; Veit, 1988; Wheeler andSmeatham, 1992) the confined space minimizes phase shift and acoustic noise travel time between

the headsets loudspeaker and a closely spaced miniature microphone;

more than 10 dBattenuation is achieved at the ear drum below 3 kHz


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Analysis of a general feedback ANC system

+

d(n)

S(z)

W(z)e(n)y(n)

H(z)

d(n) = primary noise (at the error location)

e(n) = residual noise measured by theerror sensor

y(n) = secondary control signal

W(z) =controller transfer function

S(z) = secondary path transfer function

Under steady state conditions:

E(z) =1

1+S(z)W(z)D(z)

The error goes to 0 as the magnitude of the loop gain |S(ej)W(ej)| approaches

infinity. Therefore, significant noise reduction can be achieved by designing the

controller to have a large gain over the frequency band of interest.

H(z) = closed loop transfer function from the

primary noise to the error signal


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In the ideal case S(z) = 1, arbitrary noise reduction can be obtained by employing

a constant controller W(z) = with large gain.

In practice, the frequency response of S(z) is never perfectly flat and free of phase

shift:

The response of the secondary source introduces a considerable phase shift. The physical path from the secondary source to the error sensor introduces some

delay, due to propagation time.

Using a constant controller, as the phase shift in the secondary path approaches

180, the control system may become unstable (if the loop gain is greater thanunity at the corresponding frequency) see, e.g., Bode stability criterion

Define

L(e

j

) = S(e

j

)W(e

j

) =ML()e

jL()

where

ML() = |L(ej)| L() = L(ej)


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Then:

|1 + S(ej

)W(ej

)|2= 1 +ML()

2+ 2ML()cos(L())

The design of a feedback ANC involves finding a W(z) such thatML() is

maximized while 180 < L() < 180, ensuring at the same time the following

two constraints: causality of W(z) stability ofH(z) open-loop gain |ML()| < 1 at 180 phase shift

Remarks: If not compensated for stability at high frequencies, the ANC system will

diverge into uncontrolled oscillation initiated by low level noise.

Additional filters may be introduced into W(z) to compensate for the phaseshift of S(e

j

), thereby increasing the ANC bandwidth. High level of noise attenuation is obtained for periodic or very low-frequency noise. The gain-bandwidth limitation may be reduced by using cascaded feedback stages.


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The waterbed effect

Bode (1945) showed that the integral of the log sensitivity functionH(z)

(which is linearly proportional to the disturbance attenuation) in dB, calculated

over the whole frequency range, must be zero. constraint onH(z)

This performance limitation of feedback controllers implies that we can obtain good attenuation (small |H(ejT)|) only in a limited

bandwidth (where the disturbance has significant energy), but

we must allow for small enhancements elsewhere(typically, where the disturbance has little energy)

This disturbance enhancement can be made arbitrarily small for minimum-phase

plants, but not for non-minimum-phase ones.

This is known as the waterbed effect.Notice that this limitation is not shared by feedforward control systems (using a

time-advanced reference signal), that can achieve broadband attenuation of a

disturbance without any out-of-band enhancements.


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Closed loop rejection of harmonic disturbances for narrowband ANC

Thez-transform of e(n) is given by:

E(z) =1

1+L(z)D(z)

whereL(z) = S(z)W(z)

+

d(n)

S(z)

W(z)e(n)y(n)

H(z)

Since

1

1+L(ej

)

1

|L(ej

)| c

1 > c

the control system must be designed so that:

closed loop stability is guaranteed c>> (otherwise no attenuation can be achieved) |L(j)| A(level of attenuation)


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Since thez-transform of the disturbance signald(n) = cos(nTS) is equal to

D(z) =z2 cos(TS)z

z2 2cos(TS)z + 1

,

if the controller is stabilizing and includes a factorz2 2cos(TS)z + 1 at the

denominator (internalmodel principle), perfect rejection of the disturbance isobtained at steady state. In fact, defining:

W(z) =NW(z)

DW(z)=

NW(z)

(z22cos(TS)z+1)DW(z)

and S(z) =NS(z)

DS(z)

one obtains:

E(z) =1

1+L(z)D(z) =

1

1 +NW(z)NS(z)

(z22cos(TS)z+1)DW(z)DS(z)

z

2 cos(TS)z

z2 2cos(TS)z + 1

=

=(z

2cos(TS)z)DW(z)DS(z)

NW(z)NS(z)+DW(z)DS(z)


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Since the poles of functionE(z) are all in |z| < 1 (stabilizing regulator), e(n) 0 in

view of the final value theorem.

The simplest regulator structure that contains the cancelling factor is

W(z) = kz (z b)

z2

2cos(T)z + 1

which is called notch filter.

The real zero bof the transfer function is generally introduced to reduce the phase

loss determined by the poles on the unit circle.

Choosing b= cos(T) centers the zero at the same frequency of the poles.


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-50

0

50

100

150

200

Magnitude(dB)

10-2

10-1

100

101

102

103

-90

-45

0

45

90

Phase(deg)

Bode Diagram

Frequency (rad/sec)

Regulator

transfer

function


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-200

-150

-100

-50

0

Magnitude(dB)

10-2

10-1

100

101

102

103

-90

-45

0

45

90

Phase(deg)

Bode Diagram

Frequency (rad/sec)

Closed

loop

sensitivity

transferfunction

(S(z) = 1)


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Problems:

The regulator is at the stability limit. As the relative degree of the systems transfer function increases it gets

more difficult to ensure stability (check the asymptotes of the root locus).

Stability is even harder to obtain if the controlled system adds resonances(and anti-resonances) at frequencies near to those of the filter.

The precise positioning of the notch is critical.The root locus is typically employed in the design process.


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Singularities of the loop transfer function on the unit circle

The control of resonant systems (e.g., vibration control of mechanical structures

or acoustic control of enclosures) involves loop transfer functions with multiple

singularities on the unit circle or near to it:

resonances and anti-resonances due to the vibrational/acoustic normalmodes of the controlled system

unit circle poles introduced by the controller for harmonic disturbance rejectionIf the low damped poles and zeros of the loop transfer function are alternated

on the unit circle, it can be shown (e.g., with the root locus approach) that thestabilization of the system is always possible, even in the presence of significant

system perturbations (robust stability).

On the contrary, if such property does not hold, stabilization is very critical or

impossible.The correct placement of the controller singularities on the unit circle is therefore

crucial.


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Case 1:L(z) with one couple of poles on (or near to) the unit circle

-1 0 1

-1

-0.5

0

0.5

1

rel. degree of L(z) = 0

Real Axis

Ima

ginaryAxis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

OK(always) OK

(but the gainmust be small)


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Case 2:L(z) with two couples of poles on (or near to) the unit circle

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Ima

ginaryAxis

OK

(stabilizable)

not OK(not

stabilizable)


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Adaptive feedforward ANC vs. non-adaptive feedback ANC

Control

algorithm

Advantages Disadvantages

adaptive

feedforward

ANC

error signal driven to 0 large stability bounds precise modeling not required

transient suppression isdifficult

coherent reference signalrequirednon-adaptive

feedback ANC

no reference microphone no acoustic feedback active damping providestransient signal suppression relatively simple control

algorithm

stability not guaranteed non-selective attenuation small delay requirementconstrains control setting modeling uncertainty

reduces robustness

limited cancellation overlimited bandwidth


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Single-channel adaptive ANC using a feedback-generated reference

A simple idea to recover some of the results found for adaptive feedforward ANC

consists in using feedback to estimate the primary noise, in order to use this

estimate as a mock reference signal for the ANC filter.

This technique can be viewed as an adaptive feedforward system that synthesizes

its reference signal based only on the adaptive filter output and the error signal.

+

d(n)

S(z)

e(n)y(n)

S^(z)

+

+x(n) = d^(n)

W(z)

D(z) =E(z) + S(z)Y(z)

X(z) =D(z) =E(z) + S^(z)Y(z)

Y(z) =W(z)

1S^(z)W(z)

E(z)

This scheme is very similar to the Internal Model Control (IMC) scheme.


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Interpretation of the IMC-like feedback ANC as a feedforward ANC

E(z) =1

1+S(z) W(z)

1S^(z)W(z)

D(z) =1S^(z)W(z)

1+[S(z)S^(z)]W(z)

D(z)

Under ideal conditions where S^(z)

= S(z) (and thereforex(n) = d(n))

we obtain:

E(z) = [1 S(z)W(z)]D(z)

and the feedback control scheme can

be seen to correspond to an adaptive

feedforward ANC system.

S(z)

+

d(n)

S(z)

e(n)y(n)W(z)

x(n)

LMSx(n)


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Filter adaptation with the FxLMS algorithm

The adaptive filter W(z) can be adapted using the FxLMS algorithm, employing

another instance of S^(z) to compensate for the secondary path.

x(n) = d^(n) =

= e(n) + m=0

M1

s^

m(n)y(nm)

y(n) = l=0

L1

wl(n)x(nl)

wl(n+1) = wl(n) + x(nl)e(n)

x(n) = m=0

M1s^m(n)x(nm)

+

d(n)

S(z)

e(n)y(n)

S^(z)

+

+

d^(n)

W(z)x(n)

S^(z)

LMSx(n)


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Example: active control of road noise in cars (Elliott and Sutton, 1996)

A comparison is made between adaptive feedforward ANC and adaptive feedbackANC, developed on the basis of the IMC method.

Performance variations with respect to the systems delay are studied.

This delay is due to: the physical propagation time from loudspeaker to microphone the processing time of the digital controller the delays through the anti-aliasing and reconstruction filters

The feedforward control system operates with reference signals derived from sixaccelerometers.

The controller used six FIR filters with 128 coefficients operating at a sample rate

of 1 kHz.

A single FIR filter having 128 coefficients is used to implement the W(z) filter in

the feedback controller at a sample rate of 1 kHz.


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The following figures report:

the spectrum of the measured pressure inside a small car driven at60 km/hover a rough road surface (solid curve)

the residual spectrum predicted after using a feedforward/feedbackcontrol system assuming the plant response to be a pure delay

(1 msdelay = dashed line, 5 msdelay dotted line).

The attenuations achieved are relatively insensitive to delays in this range.

The main factor limiting the performance of the feedforward controller is the fact

that the multiple coherence between the reference signals and the disturbance

signal is less than unity.

The feedback control system does not use any reference signals (and therefore is

not limited by low coherence problems) and the disturbance is being cancelled by

a filtered version of the disturbance itself.


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Feedforward control system

L i i Pi ddi A i N i C l (J 2012)


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Feedback control system

L i i Pi ddi A ti N i C t l t (J 2012)


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Remarks:

The predictability of the disturbance over a time scale equal to the delay ofthe plant is expected to affect the feedback control system performance.

In fact, the performance is significantly degraded by the larger delay. For the 1 msdelay, the spectrum of the residual error is flat, indicating that

it is almost white. With the 5 msdelay, the residual error has a more colored spectrum, but the

sharp peaks in the original disturbance, which correspond to more predictable

components in the primary pressure signal, have been largely attenuated.

The overall attenuation falls to less than 1 dBfor a 5 msdelay.

For very short plant delays, the feedback controller can achieve a higher attenuationat the error microphone than the feedforward controller, although the error

microphone would have to be so close to the loudspeaker to achieve these small

delays that the acoustic effect of control would not be very widespread.



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In any case, the variation of attenuation with delay is much dependent on thestatistical properties of the disturbance. If the disturbance were a pure tone, for example, both feedforward and feedback control

systems could, in principle, give infinite attenuations at a single microphone, regardless of

the plant delay.

If, on the other hand, the disturbance were white noise, the feedback system would beunable to achieve any attenuation if there were a finite delay in the plant.

The residual error below ~20Hzafter control in the feedback case is somewhathigher than before. This increase is due to amplification of the disturbance by the

control system and may overload the loudspeaker used as the secondary source.



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Waveform synthesis using a feedback-generated reference

Noise

sourcePrimary noise

Canceling

loudspeaker

Error

microphone

e(n)

duct

PLL

y(n)S^(z)

++

d^(n)

W(z)

Synchronization pulse

The regenerated reference signalx(n) = d^(n) is used to synthesize a low frequencycomponent locked at the fundamental driving frequency of the primary noise

source, which is then used as an input to the waveform synthesizer W(z).



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Multiple-channel adaptive feedback ANC:

K1 system (1 error sensor)

+

d(n)

S1(z)

e(n)y1(n)

x(n)K1

Feedback

ANC SK(z)

...

Reference

signalestimator

yK(n)



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e(n) = d(n) k=1

K

sk(n)*yk(n)

where sk(n), k= 1, 2, , K, are the impulse responses of the secondary paths Sk(z)y

k(n), k= 1, 2, , K, are the secondary signals of the adaptive filters W

k(z)

x(n) = d^(n) = e(n) +

k=1

K

s^

k(n)*yk(n)

The FxLMS algorithm is used to minimize the error signal e(n) by adjusting the

weight vector for each adaptive filter Wk(z):

wk(n+1) = wk(n) + xk(n)e(n)

where

xk(n) = s^k(n)*x(n)



35/43

g ( y )


Example of a 21 adaptive feedback ANC system

+

LMS

LMS

S^

1(z)

S^2(z)

W2(z)

W1(z)y1(n)

y2(n)

x1(n)

x2(n)

x(n)

S^

1(z)

S^

2(z)

e(n)e(n)

+

+



36/43

g ( y )


Multiple-channel adaptive feedback ANC:

KMsystem (multiple error sensors)If a single error signal does not capture all the desired spatial and frequency

components, the general KMscheme can be adopted (Merror sensors and

Ksecondary sources).

KM

adaptive

filters

Reference

signal

estimator

FxLMS

K

y(n)

K(n)

1(n)

eM(n)

e1(n)

... ...

Noise source

Enclosure

e(n)

M

M

x(n)



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This system includes:

MKsecondary paths Smk(z) from the kth secondary source to the mth error sensor KMadaptive filtersWkm(z)

The synthesized reference signals are expressed as:

xm(n) = em(n) + k=1

K

smk(n)*yk(n), m= 1, 2, ,M

where

s^mk(n) is the impulse response of the secondary path estimate S^mk(z)yk(n) is the kth secondary signal, computed as

m=1

M

wkm(n)*xm(n)

wkm(n) is the impulse response of the adaptive filter Wkm(z)The filter weights are adapted using the (JKM) FxLMS algorithm, withJ=M.



38/43


Example of a 22 multiple-channel adaptive feedback ANC scheme

+

S^

11(z)

e1(n)

e2(n)

y1(n)

FxLMS

S^21(z)

S^

22(z)

S^

12(z)

y2(n)

W11(z)

W21(z)

e1(n)x1(n)

x2(n)

+

+

+

+

+

+

+

+e2(n)

+

W12(z)

W22(z)



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Hybrid ANC systems

Noise

sourcePrimary noise

Canceling

loudspeaker

Error

microphone

(n) e(n)

duct

++

Reference

microphone

(n)

Feedback

ANC

Feedforward

ANC

The feedforward ANC system uses two sensors, the reference sensor to measure theprimary disturbance, and the error sensor to monitor the ANC system performance.

The adaptive feedback ANC uses only the error sensor and cancels only thepredictable part of the primary noise

The feedforward ANC attenuates the primary noise correlated withx(n), while thefeedback ANC cancels the narrowband components of the primary noise not

observed by the reference sensor.



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Hybrid ANC with FIR feedforward ANC

+

LMSS^(z)

P(z)

x(n)

A(z)y(n)

S(z)

x(n)

LMSS^(z)

C(z)

d^

(n) d^(n)

S^(z)

e(n)

e(n)

e(n)

e(n)

++

+

+

W(z)

d(n)



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The combined controller has two reference inputs:

x(n) from the reference sensor d^(n), the estimated primary signal

Filtered versions of these signals are used to adapt the coefficients of two FIR

filters (A(z) and C(z)).Then, the secondary signal is generated as:

y(n) =a(n)Tx(n) +c(n)

Td^(n)

wherea(n) = [ a0(n) a1(n) aL1(n) ]Tis the weight vector ofA(z) at time nx(n) = [x(n) x(n1) x(nL+1) ]Tc(n) = [ c

0(n) c

1(n) c

L1(n) ]

Tis the weight vector of C(z) at time n

d^(n) = [ d^(n) d^(n1) d^(nL+1) ]T



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The weights are updated with the FxLMS algorithm:

a(n+1) =a(n) + x(n)e(n)c(n+1) =c(n) + d^(n)e(n)

wherex(n) andd^(n) are the filtered reference signal vectors:

x(n) = s^(n)*x(n)d^(n) = s^(n)*d^(n)

Simulation results show that the hybrid ANC system can achieve:

15 dBadditional attenuation with respect to the purely feedforward FxLMS scheme 3 dBadditional attenuation with respect to the feedback ANC algorithm

A hybrid scheme can also be realized where the feedforward part uses IIR filters

adapted with the Filtered-u Recursive LMS algorithm.



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Comparison of hybrid and non-hybrid ANC structures

FeedforwardANC

Adaptivefeedback ANC

Hybrid ANC(FIR)

Hybrid ANC(IIR)

Filter order moderate high low low

Spectral

capability

broadband and

narrowband

narrowband

only

broadband and

narrowband

broadband and

narrowband

Plant noise not canceled good

cancellation

good

cancellation

good

cancellation

Noise field

coherence

coherent only coherent or

incoherent

coherent or

incoherent

coherent or

incoherent

Hybrid ANC allows to use a lower-order filter to achieve the same performance.

Feedback ANC cancels only the predictable part of the primary noise, whereas

hybrid ANC achieves also broadband noise cancellation.

Feedforward ANC works well if the primary signal is highly correlated with thereference signal: this condition may not be met if the reference sensor picks up

plant noise or if the primary noise and the reference sensors are not coherent.

08 - Feedback ANC

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