Practical Consideration and Implementation for Avoiding Saturation of Large
Amplitude Active Noise Control
Dongyuan Shi(1) , Woon-Seng Gan(2) , Bhan Lam(3) , Shulin Wen(4)(1-4) School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore(1-4) DSHI003@e.ntu.edu.sg, EWSGAN@ntu.edu.sg, bhanlam@ntu.edu.sg, alicia.wen@ntu.edu.sg

Abstract
The saturation distortion of the power amplifier is a common and intractable issue for most audio devices,let alone for the active noise control (ANC) system. The effect of amplitude and phase distortion, caused by
the saturated audio amplifier, on the adaptive algorithm in ANC is elaborated in this paper. To overcome thisissue, the previously proposed the two-gradient FxLMS algorithm is applied, which efficiently suppresses theamplifier saturation and has as the same computational complexity as the FxLMS algorithm. The performanceof the 2GD-FxLMS algorithm is experimentally validated for the first time in the control of both periodic andbroadband signals.
Keywords: Saturation distortion, 2GD-FxLMS, Output constraint, Active noise control.

1

INTRODUCTION

Active noise control (ANC), which generates anti-noise waves that destructively interfere with the unwanted
noise [1], is widely deployed in headphones [2], and other areas [3–9]. Similar to other audio devices, overdriving the audio amplifier to saturation usually results in degradation in active noise reduction performance andsound quality. Moreover, the amplitude and phase distortions caused by the saturated amplifier often result in
the divergence of the adaptive algorithm.
Saturation distortion often happens when the output signal exceeds the output-power limits and hence, overdrives
the amplifier [10]. Be specific. Saturation distortion is usually composed of the amplitude and phase distortion,
which play the central role in the nonlinear property of the secondary path in ANC. Accordingly, the nonlinear
adaptive algorithm seems to be an effective solution in overcoming the saturation distortion incurred in the ANC
system [11]. In general, when the power amplifiers are partially overdriven, the typical nonlinear algorithm
either using a truncated Volterra series [12] or a functional link artificial neural network (FLANN) [13, 14]
can improve the noise reduction performance and convergence speed. However, the enormous computational
expense of the nonlinear ANC algorithms always undermines their practicality. More importantly, when themagnitude of the optimal control signal (desired control signal) largely exceeds the threshold of the amplifier,neither the linear nor nonlinear ANC algorithms provide sufficient power to cancel the disturbance [10, 15].
The residual disturbance signal continuously accumulates in the control filter’s coefficients and may result in
overflow. Therefore, the nonlinear adaptive algorithm may not be the best solution for the saturation distortionin ANC when the amplifier is severely overdriven [16].
An alternative approach is to confine the output power of the audio amplifier to prevent it from being overdriven,and to maintain a specified power budget [17–19]. For instance, the clipping algorithm can truncate the partof the output signal above the limitation of the amplifier but at the expense of its stability and convergence
speed [16]. Some of the latest clipping algorithms, which include the improved method, the rescale algorithm,
using the projection method, which not only restricts the overdriven signal within the output constraint, butalso rescales the weight of control filter to stabilize the algorithm [16, 20]. The leaky-type FxLMS algorithm
introduces a leakage or penalization factor in the cost function to constrain the control effort and stabilize theadaptation [21]. However, these algorithms do not impose a specified constraint and only adjusts the leakage
factor by trial and error to meet the power budget requirement. Therefore, a two-gradient FxLMS (2GD-FxLMS)

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Figure 1. A periodic signal with amplitude distortion. Figure 2. A periodic signal with amplitude distortion.
algorithm, which imposes a specified output constraint with the same computational complexity as conventional
FxLMS, was recently proposed [22]. It uses the Hemstitching method [23] to switch the gradient direction of
the FxLMS algorithm to avoid the saturation distortion when the output signal violates the constraint and finallyconverges to the optimal control solution within the output constraint. Hence, this paper seeks to experimentally
validate the previously proposed 2GD-FXLMS algorithm.

2

PROBLEM FORMULATIONThis practical implementation is targeted at the audio amplifier, which is natively restricted by its electrical properties, especially its maximum output power. If the output signal exceeds its maximum allowance, the amplifier
becomes saturated, resulting in amplitude and phase distortion, deteriorating the noise reduction performanceand stability of ANC.
2.1 Amplitude distortion
Amplitude distortion arises when the part of the signal, whose amplitude is above the threshold (the maximum
output amplitude of the amplifier), is clipped. For instance, we consider an audio amplifier with unity gain andthe amplifier threshold of Vthr . If an input signal x(t) = A sin(ω0t) has an amplitude of A larger than Vthr , theamplifier’s output signal is clipped toππVthr , ω0t + 2kπ ∈ [ 2 − ∆t, 2 + ∆t](1)y(t) = A sin(ω0t), otherwise
3π−Vthr , ω0t + 2kπ ∈ [ 3π
−
∆t,
+
∆t]
2
2
as shown in Fig. 1, where ∆t = π2 − arcsin (Vthr /A) and k ∈ Z. Its Fourier series can be rewritten asy(t) =−2∆t + π + sin(2∆t)
4 cos(∆t) sin3 (∆t)
A sin(ω0t) +A sin(3ω0t) + . . .
π
3π

(2)

It is clear that harmonic components are generated as a result of amplitude distortion. Figure. 2 illustrates
a single-frequency adaptive noise canceller with a saturated amplifier model whose amplitude threshold isVthr . The secondary path transfer function denoted by Hs (z) has a gain of As . The noise disturbance d(n) =D sin(ω0 n) has an amplitude of D at the single frequency ω0 . The reference signal x(n) = sin(ω0 n + θr ) is of
the same frequency but at a different initial phase θr as the disturbance. As the amplitude of the disturbance
decides the status of the audio amplifier [10], there is no distortion when the amplitude is D ∈ [0, AsVthr ]. When6906

D ∈ (AsVthr , 4AsπVthr ], the amplifier is moderately overdriven, and the disturbance can be entirely canceled at thefundamental harmonic, but some harmonics are also generated. Once the amplitude becomes D ∈ ( 4AsπVthr , ∞),the disturbance is only partially suppressed. The residual signals will drive the control filter coefficients unboundedly until it overflows.
2.2 Phase distortion
In many applications, the saturation of the amplifier often results in phase distortion, which in turns contributes
to additional phase shift (phase error) in the secondary path. The disturbance is rewritten as its Fourier series
[24]
∞

∞

d(n) =∑ Dk ejkω0 n∑ Dk =

(3)

k=−∞

k=−∞

where Dk represents the amplitude of the kth harmonic. The output signal of the control filter is given by
y(n) =

∞

∞

k=−∞

k=−∞

∑ Y k = ∑ Yk ejkω0 n .

Hence, the square error of the noise canceller is stated to
(
)
(
∞

J=E

∑ kEk k

2

k=−∞

∞

=E

∑

(4)



2
Dk − Hs ejkω0 Y k

)
(5)

k=−∞

where E{·} denotes the expectation operation. According to the least mean square (LMS) error method, thenew output is derived as
Y k (n + 1) = Y k (n) + µHs∗ ejkω0 Ek (n)(6)
where µ is the step size. When the amplifier enters the saturation, a phase error θa is brought into the secondary
path. So, (6) is rewritten as
Y k (n + 1) = Y k (n) + µHs∗ ejkω0 ejθa Ek (n).
(7)
Substituting the optimal solution Yko into (7) yields
h

 
i 
o
o
Y k (n + 1) −Y k = 1 − µejθa Hs∗ ejkω0 Hs ejkω0
Y k (n) −Y k .
Hence, to guarantee the convergence of (7), the phase error should be
hµ

 
i
Hs∗ ejkω0 Hs ejkω0 .
|θa | < arccos
2

(8)

(9)

If the phase error, caused by the amplifier’s saturation, exceeds this bound, it may deteriorate the system’s
stability.

3

TWO-GRADIENT FILTERED-X LEAST MEAN SQUARES (2GD-FXLMS) ALGORITHM

To confine the output power of the amplifier while maximumly reducing the disturbance, we define a new costfunction for the active control problem as


2

L−1

min J (w) = E  d(n) − ∑ sl wT (n − l)x(n − l) 
w

l=0

h

s.t. g (w) = E wT (n)x(n)

i
2

(10)

≤ ρ2

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Table 1. Pseudocode of 2GD-FxLMS algorithm
Two gradients FxLMS algorithm (2GD-FxLMS)
Input: Reference signal x(n) and error signal e(n).
Output: Clipped output signal yout (n).
Step 1: y(n) = wT (n)x(n);x0 (n) = ŝT (n)x(n).
Step 2: If |y(n)| ≤ ρw(n + 1) = w(n) + µ1 e(n)x0 (n);elsew(n + 1) = w(n) − µ2 y(n)x(n).
Step 3: If y(n) > ρ, yout (n) = ρelse if y(n) < −ρ, yout (n) = −ρelse yout (n) = y(n).

where | · | denotes the absolute value. sl is the lth coefficient of the secondary path Hs (z) of length L, and x(n)and w(n) represent the reference signal and the control filter’s weight vector, respectively. ρ 2 is the constraint
on the power of the output signal. According to the Kuhn-Tucker condition [25], the (10) has only one optimal
solution as
wo = (λo Rxx + Rx0 x0 )−1 Pdx0
(11)where Rxx and Rx0 x0 represent the autocorrelation matrix of the reference x(n) and filtered reference
L−1

x0 (n) = ∑ sl x(n − l)

(12)

l=0

and Pdx0 denotes the cross-correlation vector of E [d(n)x0 (n)]. λo is the Lagrangian factor given byλo =E {y0o (n)[d(n) − y0o (n)]}
≥ 0,
ρ2

(13)

where the optimal anti-noise isL−1

y0o (n) = ∑ sl wTo x(n − l).

(14)

l=0

To achieve the optimal solution of (10), we utilize the hemstitching method to derive the recursive
formulai ofh2
the control filter. When the output power of the control is within the constraint (E[y(n)2 ] = E wT (n)x(n) ≤ρ 2 ), the weight update is given by
1∇J(w)w(n + 1) = w(n) − µ1.
(15)2 ||∇J(w)||hi2
In contrast, when the output power exceeds the constraint (E wT (n)x(n) > ρ 2 ), the weight update changes
to
∇g(w)
1
(16)
w(n + 1) = w(n) − µ22 ||∇g(w)||6908

Figure 3. The error signal (top) and control signal Figure 4. The active noise control air duct with the(bottom) of the 2GD-FxLMS algorithm with an out- Class-AB (TDA2030A) audio amplifier.
put constraint of 0.8.
where || · || denotes the Euclidean distance, and µ1 and µ2 are the step sizes. To further reduce the computation,
we approximated the gradient in (15) and (16) by their instantaneous values as(∇J(w) = −2e(n)x0 (n)(17)∇g(w) = 2y(n)x(n).
By substituting (17) into (15) and (16), and replacing the normalized gradient and average power constraint
with the instantaneous gradient and amplitude constraint, the update equation is expressed asw(n + 1) = w(n) + µ1 e(n)x0 (n)(18)

when the output signal |y(n)| = |wT (n)x(n)| ≤ ρ, whilew(n + 1) = w(n) − µ2 y(n)x(n)(19)

when |y(n)| > ρ. To guarantee that the output signal falls within the constraint, the output signal is clipped
y(n) > ρρyout (n) = −ρ y(n) < −ρ(20)y(n) others.
From (18) and (19), the algorithm operates with two different gradients, giving rise to its name: two gradientsFxLMS (2GD-FxLMS). The 2GD-FxLMS uses the first term of (17) (as the same as conventional FxLMS)
in updating the control filter to cancel the disturbance, when the output signal is within the output constraint.
However, when the output violates the constraint, the algorithm adopts the second term of (17) to minimize thepower of the output signal.

4

IMPLEMENTATION OF ALGORITHM

To use the 2GD-FxLMS algorithm for avoiding amplifier saturation, the following procedure is implemented.

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Figure 5. Error signals’ power spectrum of FxLMS Figure 6. Error signals’ power spectrum of FxLMS
and 2GD-FxLMS for the tonal noise.
and 2GD-FxLMS for the broadband noise.
Stage 1: Obtain the threshold ρ 2 of the power amplifier by experimental measurement or referring to its manualbook.
Stage 2: Implement 2GD-FxLMS as described in Table 1. It is noted that a 2GD-FxLMS, which consists ofthe control filter with M taps and the secondary path estimate with L taps, requires 2M + L + 1 multiplications
and 2M + L − 2 additions, which is the same as the conventional FxLMS.
Stage 3: Determine µ1 as the conventional FxLMS. Facing severe saturation distortion, adopt a large µ2 to exert
a tight restraint on the output power. However, if higher noise reduction with slight distortion is acceptable, asmall µ2 is more suitable.

5

SIMULATION RESULTS

In the simulation, the primary and secondary paths were measured from an air duct. The sampling rate andprimary noise’s frequency were set to 16 kHz and 1 kHz, respectively. The step size µ1 and µ2 in 2GDFxLMS were both chosen as 5 × 10−5 , and the output constraint was set 0.8. Fig. 3 validates that 2GD-FxLMS
can confine the output signal within the constraint while reducing the disturbance to a certain extent.

6

EXPERIMENT RESULTS

A real-time implementation based on the feedforward FxLMS and 2GD-FxLMS algorithms in a NI PXIe 8880
controller was also carried out in a small noise chamber, as shown in Fig. 4. The audio amplifier TDA2030A

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had a threshold of 750 mV. The NI PXIe-6368 module accomplished analog input, signal conditioning, andoutput of the system, with a sampling rate of 16 kHz. The length of the control filter and the secondary pathestimate were chosen as 512 taps and 256 taps, respectively.
To understand the distortion created by the audio amplifier, we define the total harmonic distortion (THD) asq
V22 +V32 +V42 + · · ·
(21)
THD =
V1
where Vn (n = 1, 2, · · · ) is root mean square (RMS) voltage of the nth harmonic and n = 1 is the fundamental
frequency.
In the experiment, the two primary noise signals were tested: a 1 kHz tone, and a broadband signal from 500
Hz to 1.4 kHz. For the tonal noise, as shown in Fig. 5, FxLMS reduces the disturbance from 58 dB to 43 dB
but cannot prevent the saturation distortion, resulting in a total harmonic distortion (THD) of 174%. In contrast,
2GD-FxLMS not only mitigate the disturbance to 53 dB but also prevents the audio amplifier from distortion
with THD of 0.6%. For broadband noise, 2GD-FxLMS also suppresses the high-frequencies distortion better
than FxLMS, as illustrated in Fig. 6.

7

CONCLUSION

This paper analyzed the amplitude and phase distortions created by the saturated audio amplifier and theireffects on the performance of an ANC system. To deal with this issue, the previously proposed 2GD-FxLMS
algorithm was applied. It has the same computational complexity as the conventional FxLMS and effectively
suppresses the saturation distortion by restricting the output power of the audio amplifier. To validate the 2GDFxLMS algorithm experimentally, it was implemented on an NI PXIe platform in a small noise chamber. The
experiments verify that the algorithm can successfully mitigate the tonal and broadband noise, while preventing
the audio amplifier from distortion. Further research could investigate the influence of the two step-sizes on the
2GD-FxLMS algorithm.

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