Filtered-x and Filtered-u LMS active noise control in ducts

Abstract

Prolonged noise exposure can cause problems related to hearing health, resulting in an attention deficit and eventually work accidents. This paper evaluates two methods of active noise control (ANC) in ducts: the Filtered-x LMS and a modified Filtered-u LMS algorithm that improves sound path identification by the optimization of the filters parameters. The ANC system is validated with the design and implementation of an experimental setup. A code was developed in MATLAB software language for microphone signal acquisition and speaker control. In general, the proposed control system obtained attenuations that depend on the type of noise evaluated (monotonal, multitonal, or broadband). The attenuation levels reach 23 dB for the monotonal test; however, in the tests for white noise, the system did not present pronounced attenuation.

Keywords

noise in buildings HVAC robust control active noise control

Introduction

In 1954, Beranek highlighted the urgent need to address noise from jet aircraft, which threatened the well-being of people near airports. Standards, such as OSHA noise regulations, Brazilian standard NR-15 (2014), NIOSH 1910-95, 1998, ACGIH (2003), or ISO 9612 (2009), specifies maximum noise limits and exposure times, so they do not affect hearing health (Elliot and Darlington, 1985). Usually, noise can be reduced at the source, along the path, or at the receiver, using passive or active control. Passive methods are cost-effective but limited in attenuation and frequency range. Active noise control (ANC), while more expensive, is effective in cases where passive methods fall short. Thus, ANC is reserved for applications where high performance justifies its cost.

Industrial ducts often produce harmful noise from narrowband tones (blade interactions), broadband turbulence, and mechanical vibrations, primarily in the 0–500 Hz range where ANC works best, according to Hansen et al. (2007) apud Oliveira (2012). In his study, Oliveira (2012) report the implementation of a mono-channel ANC system that achieved significant noise attenuation in ducts, up to 35 dB, under specific conditions.

The purpose of this paper is to develop a system to attenuate the noise in specific frequency ranges in ducts using microphones, speakers, and a control strategy. Passive methods generally struggle to reduce low-frequency noise, while active control permits a more substantial attenuation, according to Lessa (2010) and Gontijo (2006). Conversely, high-frequency noise is harder to control actively due to the need for extremely fast control loops. The paper presents a comparison between Filtered-x LMS (Fx-LMS) and a proposed novel Filtered-u LMS (Fu-LMS) algorithm that includes an optimization offline process to obtain the optimal parameters of the feedback $\hat{F} (z)$ filter into the adaptation process which reduces instabilities by 30% compared to standard Fu-LMS.

Noise control

Passive control requires no external energy and uses strategies like (i) design of sound-absorbing materials, (ii) architecture of environments for propagation control, (iii) barriers and enclosures, (iv) reduction of vibration in machines using damping adjustments, and (v) layout arrangement of machines and even use of personal protective equipment.

On the other hand, ANC, first proposed and patented in USA by Paul Lueg (1936), is based on wave cancellation and requires energy. The most common strategies in ANC are Feedback and feedforward control, being the latter used in this work. In feedforward (or anticipative) control, a reference microphone detects incoming noise, while a secondary source (speaker) emits an anti-noise signal. An error microphone measures the combined effect to assess performance. The controller processes the reference signal to generate the anti-noise, accounting for delays and the transfer functions of both the primary path (noise source to error mic) and the secondary path (control speaker to error mic). Figure 1 illustrates this setup applied to duct noise attenuation.

Figure 1.

Active control system in a duct (feedforward).

ANC represents a significant advancement in managing unwanted sound, particularly within duct systems where low-frequency noise can be pervasive and disruptive. Traditional passive noise control measures often fall short when addressing low-frequency disturbances, thereby justifying the necessity for ANC solutions.

Studies indicate that ANC can effectively diminish sound levels in ducts, specifically within a frequency range of 200 to 400 Hz, achieving reductions of up to 14 dB or 17% as reported in Anachkova et al. (2023). Furthermore, the ongoing development of signal processing techniques, such as those utilized in mobile devices and aircraft, reinforces ANC’s importance in contemporary technological applications as discussed in Sultana et al. (2023).

Recent advancements highlight the implementation of Griffiths variable step-size algorithm (Griffiths, 1978) within the Fx-LMS framework, optimizing secondary path estimation, which is crucial for enhancing ANC performance. Attention should be drawn to better estimation of secondary path transfer function $S (z)$ since it is directly influenced by the performance of the ANC (Gaur and Gupta, 2016).

Narasimhan et al. (2010) demonstrated a 25 dB and 15 dB noise reduction for narrowband and broadband noise in ducts, respectively, with only 1.25% processing time, highlighting Fx-LMS’s efficiency, reliability, and key role in modern ANC systems.

As demonstrated in several studies, including Wu and Bai (2000), the use of the Fu-LMS algorithm in environments with continuous noise sources from fan operations yields substantial reduction in undesirable acoustic emissions. Furthermore, integrating passive measures associated to Fu-LMS has validated its capabilities across varied frequency ranges, establishing this ANC strategy a robust solution for low-frequency noise attenuation (Maillard and Guigou-Carter, 2000).

In the literature survey presented by (Kuo and Morgan, 1999), they state that although the Fx-LMS algorithm is widely used and fundamental, it performance is hindered by slow convergence due to its reliance on finite impulse response filtering, particularly in short acoustic ducts. In contrast, the Fu-LMS algorithm improves upon this limitation, integrating concepts such as variable step sizes and error smoothing filters to enhance stability and fast convergence. This adaptability is crucial, as demonstrated by its application to various disturbance signals, which have shown the good performance over the Fx-LMS approach as reported in (Linh, 2022). Ultimately, advancements in these algorithms, including the inclusion of robust tracking behavior and the ability to mitigate interference from backward acoustic waves, highlight their suitability for real-time application in automotive and industrial problems (Ophen and Berkhoff, 2012).

Theoretical Basis

The sound pressure level (SPL) is evaluated on a logarithmic scale representing the ratio between the mean square value of the sound pressure signal history and the minimum audible pressure fluctuation, p₀ = 20 μPa (Gerges, 2000). For a point-like source emitting noise at a particular power SWL (dB), in all directions and without any interference, the sound pressure level SPL (dB) at a certain distance from this source can be calculated by

S P L = S W L_{p o i n t} + 10 \log_{10} \frac{Q}{4 π r^{2}} [d B],

(1)

where r is the distance from the sound source to the SPL measured point, Q is the percentage of the effective area crossed by sound, durectivity (Q = 1, omnidirectional propagation). For point-like sources on hard surface, the value of

Q

becomes 2. In the case of ducts, waves propagate approximately flat (without attenuation or absorption), and therefore equation (2) yields

S P L = S W L_{p o i n t} [d B]

(2)

In this way, theoretically, there would be no loss, and the noise propagates without attenuation for long distances, but in practice, other phenomena attenuate noise at long distances like curves of the ducts, air outlets obstacles, or cross-section reductions.

The transfer function between control speaker and the error microphone is called secondary path $S (z)$ . This is the path in which the anti-noise propagates directly to error microphone, but the output signal is also picked up by reference microphone. As a guarantee of significant attenuation results, it is of utmost importance that the estimation of the secondary path $\hat{S} (z)$ is obtained. When it is assumed that the characteristics of the secondary path $S (z)$ are invariant in time, one can use the offline modeling method during a training phase to estimate $\hat{S} (z)$ .

The transfer function between the reference microphone and the error microphone is called the primary path $P (z) .$ It is in the primary path $P (z)$ that the noise is most influenced by the propagation time and undergoes attenuation or amplification, as it is the longest path taken by the sound in the entire duct. The feedback path (or acoustic feedback) is the path between the signal that comes out of control speaker (the one that emits the anti-noise signal) and is captured by reference microphone. This signal, as it is an acoustic feedback, generates instabilities in the system. Therefore, its transfer function, as well as that of $S (z)$ , must also be estimated and used in the ANC algorithm. In the Fx-LMS algorithm this path $F (z)$ does not enter the code, so an adaptive filter $W (z)$ finds it more difficult to converge, because the values of $W (z)$ end up being fed back with looped signals. The adaptive Fu-LMS tries to attenuate this behavior.

An adaptive filter aims to adjust its parameters, so that they minimize an objective function (Duboc, 2015). This function has as variables the reference signal $x (n)$ , the signal coming from the noise source $d (n)$ , and the output signal from the filter $y (n)$ and $e (n)$ , which is the error at discrete time steps. Figure 2(a) shows the diagram for this application.

Figure 2.

(a) Adaptive filter diagram for system identification and (b) offline estimate of the feedback path $\hat{F} (z)$ .

The objective function to be minimized is usually the mean square error (MSE) (Haykin, 1986; Haykin and Widrow, 2003). To minimize the error, the adaptive algorithm aims to adapt its coefficients through FIR or IIR filters. Thus, the transfer function of the adaptive filter is close to the transfer function of the unknown system, making it possible to simulate propagation from this physical system by the coefficients of the adaptive filter $W (z)$ .

Active noise control Filtered-x LMS (feedforward)

The classic closed-loop feedback LMS control only supports one error sensor (microphone 2), which feeds the control logic. In turn, control logic triggers the actuator (speaker 2) (Haykin, 1986; NUNEZ IJC, 2005). In the traditional least mean square (LMS) method with adjustable filter coefficient $W (z)$ , instability in the control system may happen. This is due to the presence of the secondary path $S (z)$ , which filters the sound emitted by the actuator (and adds some noise). The original noise and the anti-noise are ultimately captured by the error microphone with some uncertainty (Elliot and Darlington, 1985). The Fx-LMS algorithm, proposed by Widrow in 1981, improves noise cancellation over standard LMS, especially when ignoring the unknown system response (Mayyas and Aboulnasr, 2002). Figure 3(a) shows its block diagram.

Figure 3.

(a) Block diagram for the Filtered-x LMS (feedforward) control (modified from NUNEZ IJC, 2005; Bjarnason, 1995) and (b) block diagram of Filtered-u LMS control applied to ANC.

The controller’s reference input signal is $x (n)$ (from microphone 1, noise source), $n$ means the discrete-time instants, $d (n)$ is the sound signal from the noise source after the primary path, and $y (n)$ is the sound signal measure by microphone 2 after the secondary path (sent by speaker 2). $e (z)$ is the error signal (measured by microphone 2), $W (z)$ represents the weights of the digital filter (linear filter), $P (z)$ is the impulsive response between the noise source and the error microphone (primary path), $S (z)$ is the impulsive response between the actuator (control speaker) and the error microphone (secondary path). Finally, $\hat{S} (z)$ is the estimation of the impulse response (transfer function) between the actuator (speaker 2) and the error microphone (secondary path).

In a simplified way, the error $e (n)$ is evaluated from the convolution of the impulse response of the secondary path $S (z)$ and the weights of the digital filter $W (z)$ by

e (n) = d (n) - s (n) * [w^{t} (n) \cdot x (n)],

(3)

where

w (n)

are the elements of the vector,

d (n)

is the reference signal vector at the instant

n

x (n) = {[{x (0)}_{n}, \dots {x (n - M + 1)}_{n}]}^{T}

, and * means the convolution between the time signal and the weight vector and represents the order of the digital filter.

Assuming an error function of mean type $E (n) = μ [e^{2} (n)]$ and taking a gradient descending algorithm for minimization, then the correction of the weights of the filter $W (z)$ may be defined as

w (n + 1) = w (n) - [μ (n) / 2] \nabla E (n),

(4)

where

μ (n)

is an adaptive step factor and

\nabla E (n)

is the gradient of the error function. Several techniques can be used to evaluate this gradient. So, using the gradient of the instantaneous quadratic mean error, the error equation concerning

w (n)

is derived as follows:

\nabla E (n) = - 2 [s (n) * x (n)] . e (n)

(5)

Substituting equation (5) into equation (4) results in the equation for updating the Fx-LMS coefficients:

w (n + 1) = w (n) - μ (n) [s (n) * x (n)] e (n) .

(6)

The adaptation step

μ (n)

(which can be constant or adaptive) has an influence on the stability and convergence of the algorithm according to NUNEZ IJC (2005). It can be estimated through the equation proposed by Widrow et al. (1981):

0 < μ (n) < \frac{1}{(M + 1) \cdot σ 2 (n)},

(7)

where

σ^{2} (n)

is the power of the input signal. The most common value used for this power is 10% of the maximum value (NUNES, 1999), that is

μ (n) = \frac{0.1}{(M + 1) \cdot σ^{2} (n)} .

(8)

Since the transfer function of the secondary path

S (z)

is not known, it is necessary an estimate from an additional filter, thus obtaining the estimated transfer function of the secondary path

\hat{S} (z)

. We set the filtered reference signal

x^{'} (n)

S (n) * x (n)

, which can also be obtained from

\hat{S} (z)

, so

x^{'} (n) = \hat{s} (n) * x (n),

(9)

such that

\hat{s} (n)

is the estimated response to the impulse of S ̂(z). In other words, the steps for the Fx-LMS algorithm can be placed in the pseudocode indicated in Table 1.

Table 1.

Pseudocode of the Filtered-x LMS algorithm (modified from NUNEZ IJC, 2005; Bjarnason, 1995).

Step 1: Initialization:

Filter coefficients:

w {(n)}_{0 = 0}

M + 1

coefficients;

Signal Power recursive evaluation for:

σ^{2} (n) = 1

;

Step 2: Measurement of

x (n)

and

e (n)

(mic1 e mic2);

Step 3: Definition of α for signal power recursive evaluation;

Step 4: Evaluation of FIR filter output:

y (n) = \sum_{k = 0}^{M - 1} [{w (k)}_{n} \cdot x (n - k)]

Step 5: Evaluation of filtered input:

x^{'} (n) = \sum_{k = 0}^{M - 1} [{\hat{s} (k)}_{n} \cdot x (n - k)]

Step 6: Signal Power recursive evaluation:

σ^{2} (n) = α {x^{'}}^{2} (n) + (1 - α) \cdot σ^{2} (n - 1)

Step 7: Evaluate the adaptation step:

μ (n) = 0.1 / [(M + 1) {\cdot σ}^{2} (n)]

Step 8: Update weighting coefficients:

w (n + 1) = w (n) - μ (n) \cdot [\hat{s} (n) * x (n)] \cdot e (n)

Step 9: Update variable

n = n + 1

and return to step 2.

Recent advances by Chen et al. (2025) show an active noise control (HANC) system, integrating an adaptive active noise control (AANC) module based on the Fx-LMS algorithm and an audio-balance control circuit (ABCC) to mitigate low-frequency noise pollution caused by industrial and military machinery activities. In this case, the ABCC could enhance voice clarity and protect users from excessive impulse noise, achieving noise reduction up to 21.8 dB.

The adaptive Filtered-u LMS algorithm

In order to counteract the acoustic feedback problem, a new approach to Fx-LMS is proposed in this paper, the adaptive algorithm Fu-LMS. $F (z)$ is estimated and used to update the coefficients of the $W (z)$ digital filter, reducing possible problems of instabilities and lags in the system. In Figure 3(b), $u (n)$ is the new reference signal captured by MIC1 (previously $x (z)$ ). Typically, Fu-LMS exhibits faster convergence rates than Fx-LMS, particularly in environments with rapidly changing noise characteristics, non-stationary noise conditions (what cannot be dealt by Fx-LMS.

Looking at the block diagram in Figure 3(b), it can be seen that the reference signal $u (n)$ is first captured by the reference microphone MIC1, followed by the primary path $P (z)$ , becoming $d (n)$ , and captured by error microphone MIC2. Thus, when $u (n)$ is captured by reference microphone, it becomes $x (n)$ and is filtered by the transfer function of the estimated secondary path $\hat{S} (z)$ , resulting in the filtered reference signal $x^{'} (n)$ . The update of the coefficients of the adaptive filter $W (n)$ is given by means of the reference signal $x (n)$ , the filtered reference signal $x^{'} (n)$ , and the error signal $e (n)$ .

As soon as the output signal $y (n)$ is generated by the algorithm, it passes through the transfer function of the estimated feedback path $\hat{F} (z)$ , thus reducing the contribution to the acoustic feedback captured by the microphone.

When the output signal $y (n)$ is passing by the secondary path $S (z)$ , it becomes the anti-noise signal $y^{'} (n)$ . Then, $d (n)$ and $y^{'} (n)$ are picked up by error microphone resulting in the error signal $e (n)$ . The whole process is repeated (control cycle) until the lowest error signal value is found $e (n)$ . Then it is necessary to estimate the feedback path $F (z)$ to insert its transfer function in the implementation of the Fu-LMS block diagram.

To estimate the transfer function of $F (z)$ , the process is similar to that described previously for offline estimation of the secondary path $\hat{S} (z)$ . Figure 2(b) shows this process that is performed simultaneously for both $\hat{F} (z)$ and $\hat{S} (z)$ by a single white noise generator.

The LMS error is used to update this estimated $\hat{S} (z)$ and $\hat{F} (z)$ paths. The main difference from the proposed Fu-LMS algorithm to traditional ones resides in obtaining offline $\hat{S} (z)$ and $\hat{F} (z)$ based on an acquisition for a wideband white noise. Then the LMS algorithm is optimized offline using parameters $M$ (length of the filter) and $μ_{S}$ (adaptive step) to improve performance and accuracy (final error to the measured path signals) by an optimization process. This has shown a considerable improvement in the overall behavior for both Fx-LMS and Fu-LMS algorithms, since secondary path and the acoustic feedback are better represented by the filters as will be presented in section 4.

Filtered-u LMS algorithm has also been investigated by Jiang et al. (2024) which used it assisted by a neural network (NN), presenting a superior noise cancellation performance and faster convergence. However, its widespread adoption has been limited by stability challenges inherent in adaptive infinite impulse response (IIR) filters. They address this limitation by implementing an equation error (EE method to stabilize the IIR filter’s pole locations.

Materials and methods

The experiments used a Realtek ALC662 sound card for ANC, with 16-bit resolution, ±1.5 V limits, four input/output channels, and a 48 kHz sampling rate per channel. While higher-end sound cards (24-bit/192 kHz) offer better performance, the ALC662 provides a cost-effective solution for basic ANC applications.

Electret microphones (AOM-6738L, PuiAudio) were used as sensors due to their affordability, low sensitivity to vibration/humidity, and flat frequency response in the target range. Two 5″ triaxial speakers (Selenium 5TR5A, 8Ω, 12W RMS) served as actuators. The test bench consisted of 150 mm PVC ducts and a 45° Y-joint. Figure 4 shows the setup for evaluating Filter-x and Filter-u LMS ANC performance with monotone, multitone, and broadband noise.

Figure 4.

(a) Test rig for ANC in ducts and (b) response curve for speaker 5TR5A and microphone AOM-6738L-R.

Speakers were insulated with soft foam. Tests conducted during COVID lockdown ensured minimal lab noise interference. Measured SNR was 43.42 dB (mic1) and 16.89 dB (mic2). Gain curves for mics and speakers are shown in Figure 4(b).

Results and discussions

As previously described, the secondary and feedback transfer function estimators are first obtained in an experiment apart from the main experiment. Figure 5 shows the graph for estimating the secondary $\hat{S} (z)$ and Feedback $\hat{F} (z)$ paths.

Figure 5.

Experimentally obtained (a) $\hat{S} (z)$ and (b) $\hat{F} (z)$ .

The graph shows the target signal in blue (white noise) and the output signal of the filter that is being identified in orange. The yellow line graph represents the difference between the two former signals. In both cases the filter takes approximately 2 seconds to adapt the weights of the secondary path $\hat{S} (z)$ . The RMS value of the residual error obtained is about $1 \times 10^{- 3}$ V.

In the optimization process, it consists on a 20 s prior time acquisition and an offline optimization process to find the best filter length $M$ and adaptive step $μ_{S}$ used in the LMS algorithm, which resulted for the tested duct. For optimization, it used the non-gradient-based MATLAB fminsearch algorithm (Nelder–Mead simplex algorithm).

The constraints in the optimization for the two parameters were [256; 4096] for filter length M and [0.01; 1.0] for $μ_{S}$ . The optimization problem, for the secondary path $S_{z} (z)$ , is stated in equation (10):

\begin{array}{l} \arg \min_{M, μ_{s}} R M S (M, μ_{s}) = \sqrt{\frac{1}{N} \sum_{n = 1}^{N} {[\hat{s} (n) - s (n)]}^{2}}, \\ S u b j e c t t o 256 \leq M \leq 4096, M \in Z \\ 0.01 \leq μ_{s} \leq 1.0 \end{array}

(10)

The best parameters for the optimization resulted in $M = 256$ and $μ_{S} = 0.1$ for the monotonal tests, $M = 512$ and $μ_{S} = 0.06$ for the multitonal noise tests, and $M = 1024$ and $μ_{S} = 0.04$ for the wideband noise tests.

Active noise attenuation in multiple tone frequency using Fx-LMS and Fu-LMS algorithms

In this section, the efficiency of the Fx-LMS and Fu-LMS ANC was evaluated using monotonal noise (120 Hz, 180 Hz, 240 Hz, 250 Hz, 350 Hz, and 500 Hz) and multitonal noise (120 Hz, 180 Hz, 240 Hz, 250 Hz, 350 Hz, and 500 Hz). In all experiments, the sound emitting and sampling frequency was set to 48 kHz. Five replications of the experiments were performed to quantify the uncertainty, but only one of the tests will be presented.

Monotone frequency of 120 Hz using Fx-LMS and Fu-LMS algorithms

The frequency of 120 Hz was chosen in order to evaluate the behavior of the ANC in the presence of a typical noise of electric rotary motors together with the electronic noise of 60 Hz (electrical network). Values below this frequency could not be perfectly generated due to the frequency response curve of the used loud speakers. Figure 6(a) shows the output of the code generated in MATLAB, 2000 for the 120 Hz monotonal noise test (without ANC), and Figure 6(b) shows the corresponding graph with the Fx-LMS and Fu-LMS systems (with ANC).

Figure 6.

Frequency spectra for monotonal noise (120 Hz): (a) without ANC and (b) with ANC—Fx-LMS and Fu-LMS.

Two more accentuated peaks are noticed, the first at 60 Hz and the second at 120 Hz. The first peak probably comes from the electronic part of the system. The ANC obtained attenuation at the peak of 120 Hz of approximately 9.5 dB(Lin) and did not present large amplifications in the harmonics in the output signal (spectrum with ANC). The standard deviation for the set of replication experiments was 1.42 dB. In this test, the ANC obtained very good performance at the frequency of interest, reaching a reduction of approximately 22 dB in the octave band corresponding to 125 Hz. However, the control presented unwanted amplifications in some harmonics, as can be seen in Figure 6(b).

It is suggested that the performance gain of Fu-LMS in relation to Fx-LMS is linked to the addition of the estimated feedback path

\hat{F} (z)

in the algorithm’s operating steps. Table 2 summarizes these tests. This table and the followings ones will show two distinct situations, Fx-LMS (rows 2–4) and Fu-LMS (rows 5–7). Row 1 lists conditions, octave bands, and SPL attenuation. Rows 2/5 show uncontrolled dB levels, rows 3/6 show Fx-LMS/Fu-LMS results, and rows 4/7 display noise reduction per band. Column 10 summarizes total SPL differences. In that Table and following ones, the bold numbers represent the final obtained attenuation in SPL for the ANC strategies being compared.

Table 2.

Attenuation test results for monotone noise (120 Hz) without/with ANC—Fx-LMS (5th replication) dB and without/with ANC—Fu-LMS (5th replication) dB.

Octave. Freq. Band	15.6 Hz	31.3 Hz	62.5 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	SPL (dB)
Without Fx-LMS	44.3	47.55	73.02	88.32	53.87	49.82	47.97	49.89	88.5
With ANC Fx-LMS	45.37	48.29	72.9	77.76	55.63	52.76	49.26	50.77	79.0
Attenuation Fx-LMS	1.07	0.74	−0.12	−10.56	1.76	2.94	1.29	0.88	−9.5
Without ANC Fu-LMS	41.6	49.99	72.97	88.14	52.88	49.31	47.92	49.79	88.3
With ANC Fu-LMS	42.79	49.27	72.89	64.8	55.32	53.14	48.31	50.52	73.7
Attenuation Fu-LMS	1.19	−0.72	−0.08	−23.34	2.44	3.83	0.39	0.73	−14.6

It is noted in the test with the Fx-LMS the system was able to attenuate the 125 Hz frequency band by 10.56 dB and in the 500 Hz band it amplified 2.94 dB in absolute SPL values.

The Fu-LMS algorithm showed better performance, reaching an attenuation of approximately 23 dB in the 125 Hz band. There were amplifications in the 15.6 Hz, 250 Hz, 500 Hz, 1 kHz, and 2 kHz bands. However, the absolute overall value in SPL with ANC system in this test remained lower than the without ANC system in both algorithms. The standard deviation in the set of five independent replications was 2.41 dB.

The remaining of the tests performed in the other monotonal frequencies are summarized and presented in the following. Table 3 shows the results of attenuation series of five replications in the case of single frequency 180 Hz, with/without Fx-LMS and with/without Fu-LMS.

Table 3.

Attenuation test results for monotone noise (180 Hz) without/with ANC Fx-LMS (1st replication) dB and without/with ANC – Fu-LMS (1st replication) dB.

Octave. Freq. Band	15.6 Hz	31.3 Hz	62.5 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	SPL (dB)
Without ANC Fx-LMS	50.36	51.17	54.21	62.75	98.57	54.59	52.19	52.35	98.6
With ANC Fx-LMS	44.9	53.71	48.83	62.13	93.17	63.14	61.83	55.09	93.2
Attenuation Fx-LMS	−5.46	2.54	−5.38	−0.62	−5.4	8.55	9.64	2.74	−5.4
Without ANC Fu-LMS	47.21	59.98	72.8	59.85	95.48	52.39	49	50.25	95.5
With ANC Fu-LMS	46.38	49.87	73.02	59.31	77.85	81.47	76.48	67.13	84.4
Attenuation Fu-LMS	−0.83	−10.11	0.22	−0.54	−17.63	29.08	27.48	16.88	−11.1

The ANC with Fx-LMS obtained attenuations in the 15.5 Hz, 62.5 Hz, 125 Hz, and 250 Hz bands, and in the other bands there were amplifications resulting in an absolute value of attenuation in SPL of 5.4 dB (the standard deviation for the five independent tests was 0.56 dB). As with the Fx-LMS, the sharpest amplifications are present in the 500 Hz, 1 kHz, and 2 kHz bands. The ANC system with Fu-LMS achieved its best performance in the 250 Hz band reaching approximately 17 dB of attenuation, outperforming the Fx-LMS algorithm by 12.23 dB (the standard deviation for the overall attenuation in the five independent tests was 0.53 dB, that are considered low for the level of attenuation).

Table 4 shows the data from test 1 and test 5, respectively, for the monotone noise attenuation tests with a frequency of 250 Hz, using the Fx-LMS and with Fu-LMS algorithms. In this case, both algorithms presented unwanted amplifications in most bands, but in general a good attenuation was obtained in the target band referring to the 250 Hz, being 9.1 dB for Fx-LMS and higher for Fu-LMS, 13.2 dB. The standard deviation for the overall attenuation in the five independent tests was 0.88 dB for the Fx-LMS and 0.83 dB for the Fu-LMS algorithm, that are considered low for the obtained level of attenuation.

Table 4.

Attenuation test results for monotone noise (250 Hz) without/with ANC Fx-LMS (1st replication) dB and without/with ANC—Fu-LMS (5th replication) dB.

Octave. Freq. Band	15.6 Hz	31.3 Hz	62.5 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	SPL (dB)
Without ANC Fx-LMS	48.91	52.13	51.99	53.33	98.68	53.49	48.84	49.99	98.7
With ANC Fx-LMS	46.97	53.77	54.21	59.28	89.61	61.74	55.63	55.21	89.6
Attenuation Fx-LMS	−1.94	1.64	2.22	5.95	−9.07	8.25	6.79	5.22	−9.1
Without ANC Fu-LMS	45.36	52.49	53.57	47.71	99.18	51.13	48.22	50.31	99.2
With ANC Fu-LMS	50.42	57.08	54.94	53.66	85.94	51.38	48.34	50.26	86.0
Attenuation Fu-LMS	5.06	4.59	1.37	5.95	−13.24	0.25	0.12	−0.05	−13.2

Table 5 shows the data for the monotonal attenuation test using ANC with Fx-LMS (test 5) and Fu-LMS (test 4 out of 5). The two algorithms obtained attenuations and amplifications in different frequency bands, with the maximum attenuation value of 11.95 dB in the 250 Hz band with the Fu-LMS and maximum amplification of 5.35 dB in the 500 Hz band with the Fx-LMS. The standard deviation for the overall attenuation in the five independent tests was 0.18 dB for the Fx-LMS and 1.14 dB for the Fu-LMS algorithm. Again, the variability in the replication of the experiments is considered very low for the level of attenuation.

Table 5.

Attenuation test results for monotone noise (350 Hz) without/with ANC Fx-LMS (5th replication) dB and without/with ANC—Fu-LMS (4th replication) dB.

Octave. Freq. Band	15.6 Hz	31.3 Hz	62.5 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	SPL (dB)
Without ANC Fx-LMS	42.39	47.52	51.05	51.15	97.41	61.6	48.34	49.98	97.4
With ANC Fx-LMS	47.7	47.85	49.95	48.57	86.12	66.95	48.63	50.01	86.2
Attenuation Fx-LMS	5.31	0.33	−1.1	−2.58	−11.29	5.35	0.29	0.03	−11.2
Without ANC Fu-LMS	45.7	53.57	54.82	52.3	100.9	64.34	48.68	50.06	100.9
With ANC Fu-LMS	47.45	57.41	51.89	49.17	88.95	64.96	50.89	52.57	89.0
Attenuation Fu-LMS	1.75	3.84	−2.93	−3.13	−11.95	0.62	2.21	2.51	−11.9

Table 6 shows the data for the Fx-LMS algorithm together with the data for the Fu-LMS algorithm for the last test at 500 Hz for monotone frequency. The system obtained its best attenuation for the case using the Fu-LMS, reaching an attenuation in absolute SPL values of 12.8 dB, thus surpassing the 5.5 dB of the Fx-LMS algorithm. It is worth mentioning that in the test with the Fu-LMS there was an amplification of 16.2 dB in the 2 kHz band; however, it did not have a great influence on the total final value of attenuation. A very low variability in the replication of the five experiments was obtained, with a standard deviation for the overall attenuation of 0.75 dB for the Fx-LMS and 0.85 dB for the Fu-LMS algorithm.

Table 6.

Attenuation test results for monotone noise (500 Hz) without/with ANC Fx-LMS (2nd replication) dB and without/with ANC Fu-LMS (5th replication) dB.

Octave. Freq. Band	15.6 Hz	31.3 Hz	62.5 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	SPL (dB)
Without ANC Fx-LMS	58.95	54.23	48.52	47.79	56.57	92.99	50.61	50.24	93
With ANC Fx-LMS	42.88	47.37	45.77	45.87	56.47	87.51	50.52	50.34	87.5
Attenuation Fx-LMS	−16.07	−6.86	−2.75	−1.92	−0.1	−5.48	−0.09	0.1	−5.5
Without ANC Fu-LMS	45.77	50.43	53.23	46.41	55.88	93.16	50.06	50.69	93.2
With ANC Fu-LMS	46.54	49.56	51.27	45.62	56.02	80.15	49.99	66.89	80.4
Attenuation Fu-LMS	0.77	−0.87	−1.96	−0.79	0.14	−13.01	−0.07	16.2	−12.8

Multiple frequencies using Fx-LMS and Fu-LMS

In the test for multiple frequencies using Fx-LMS and Fu-LMS algorithms (120 Hz, 180 Hz, and 240 Hz), the algorithms obtained similar performances in their attenuations, in Table 7 the Fx-LMS algorithm shows a small difference in relation to the Fu-LMS algorithm. The standard deviations in the overall attenuation in the replication were 0.31 dB and 0.53 dB for the Fx-LMS and Fu-LMS, respectively.

Table 7.

Attenuation test results for multiple tone noise (120 Hz, 180 Hz, and 240 Hz) without/with ANC—Fx-LMS (2nd replication) dB and without/with ANC—Fu-LMS (2nd replication) dB.

Octave. Freq. Band	15.6 Hz	31.3 Hz	62.5 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	SPL (dB)
Without ANC Fx-LMS	42.71	45.69	48.61	57.17	99.86	54.07	48.15	50.07	99.9
With ANC Fx-LMS	46.18	47.32	48.65	59.19	92.93	53.9	50.11	51.28	92.9
Attenuation Fx-LMS	3.47	1.63	0.04	2.02	−6.93	−0.17	1.96	1.21	−7.0
Without ANC Fu-LMS	45.3	48.3	73.87	81.38	93.54	50.81	48.47	49.96	93.8
With ANC Fu-LMS	44.28	48.26	74.22	76.87	85.78	50.41	47.87	49.87	86.6
Attenuation Fu-LMS	−1.02	−0.04	0.35	−4.51	−7.76	−0.4	−0.6	−0.09	−7.2

Here, the point to be noted is that both algorithms obtained the absolute attenuation values in the 250 Hz band, reaching 6.93 dB for the Fx-LMS and 7.76 dB for the Fu-LMS.

Table 8 presents the data from the second attenuation test for multiple frequency noise (250 Hz, 350 Hz, and 500 Hz). As in the previous test, both algorithms obtained their highest attenuations in the 250 Hz band.

Table 8.

Attenuation test results for multiple tone noise (250 Hz, 350 Hz, and 500 Hz) without/with ANC—Fx-LMS (4th replication) dB and without/with ANC—Fu-LMS (2nd replication) dB.

Octave. Freq. Band	15.6 Hz	31.3 Hz	62.5 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	SPL (dB)
Without ANC Fx-LMS	44.29	51.88	48.62	46.88	97.85	80.66	48.76	50.45	97.9
With ANC Fx-LMS	43.97	49.64	53.38	46.48	92	80.41	48.37	50.27	92.3
Attenuation Fx-LMS	−0.32	−2.24	4.76	−0.4	−5.85	−0.25	−0.39	−0.18	−5.6
Without ANC Fu-LMS	44.45	51.72	73.21	54.58	99.16	81.63	48.59	49.89	99.3
With ANC Fu-LMS	47.21	53.81	73.09	54.51	91.12	81.61	49.98	51.02	91.6
Attenuation Fu-LMS	2.76	2.09	−0.12	−0.07	−8.04	−0.02	1.39	1.13	−7.7

The absolute values of attenuation in SPL for this test were 5.6 dB and 7.7 dB, respectively, for Fx-LMS and Fu-LMS. The highest amplification happened at the 62.5 Hz band, reaching 4.76 dB. The standard deviation in the overall attenuation in the replication were 0.26 dB and 1.85 dB for the Fx-LMS and Fu-LMS algorithm, respectively.

White noise attenuation using Fx-LMS and Fu-LMS

The following are the results for the tests carried out using white noise from 120 to 750 Hz as a white noise source. Table 9 shows the data from the last tests carried out for white noise attenuation. Both cases of ANC algorithm did not present significant attenuation, even amplifying some frequency bands.

Table 9.

Attenuation test results for white noise (120 to 750 Hz) without/with ANC—Fx-LMS (5th replication) dB and without/with ANC—Fu-LMS (4th replication) dB.

Octave. Freq. Band	15.6 Hz	31.3 Hz	62.5 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	SPL (dB)
Without ANC Fx-LMS	52.94	53.76	50.26	84.47	96.01	92.56	78.29	49.75	97.8
With ANC Fx-LMS	60.46	52.74	57.45	84.8	95.94	92.39	78.09	50.43	97.8
Attenuation Fx-LMS	7.52	−1.02	7.19	0.33	−0.07	−0.17	−0.2	0.68	0.0
Without ANC Fu-LMS	54.11	50.61	53.02	79.36	90.6	86.73	74.15	50.2	92.4
With ANC Fu-LMS	44.88	48.67	54.15	80.32	90.22	86.7	74.15	50	92.2
Attenuation Fu-LMS	−9.23	−1.94	1.13	0.96	−0.38	−0.03	0	−0.2	−0.2

It is noted that the ANC did not result in good performance for attenuation when the type of noise was a wide frequency range. A probable cause may be limitations on the acquisition card for this wide frequency band. The standard deviation in the overall attenuation in the replication were 1.47 dB and 0.89 dB for the Fx-LMS and Fu-LMS algorithm, respectively.

Conclusions

This work proposes designing an ANC system using the Fx-LMS and Fu-LMS algorithms for ANC in ducts. It was developed a low cost test rig (based on a previous work by Vanset, 2016) consisting of a computer, PVC ducts, two microphones, two speakers, and a sound card to simulate/acquire noise in different frequency bands: single tones, multiple tones, and wideband for ducts (120∼750 Hz). In addition to ANC, the system allowed measuring the noise levels with accuracy.

While the ANC developed system achieved attenuations up to 23 dB, several tests demonstrated significant noise reduction. The Fu-LMS algorithm, which incorporates the feedback path as a control parameter, outperformed the Fx-LMS in most scenarios. Both algorithms performed well in monotonal and multitonal cases. However, as expected, neither algorithms showed significant attenuation for wideband white noise.

Future improvements include online secondary path adaptation and test rig improvements. Optimizing duct materials, components, and tube length may enhance tuning and reduce resonance. Insulated speaker enclosures could improve wavefront quality and minimize noise interference.

Footnotes

Acknowledgements

The authors are thankful for CAPES and CNPq for the support during the research.

ORCID iD

Herbert Martins Gomes

Authors’ contributions

R.R.B.S.: Writing, data accumulation, visualization, validation, data curation, software, and writing—review and editing. H.M.G.: Conceptualization, supervision, investigation, and writing—review and editing. A.M.L.: Writing, data curation, visualization, validation, and writing—review and editing.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was partially funded by the Conselho Nacional de Desenvolvimento Científico e Tecnológico—Brasil (CNPq)—Brasil (No. 304626-2021-0) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—CAPES, 001.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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