Active noise control (ANC) - is a method for reducing unwanted sound by the addition of a second sound generated by an ANC controller to cancel the first. The two sounds cancel each other out through an effect that is called destructive interference.
This report discusses the task of designing an ANC system for headphones.
According to the control structure, controllers can be categorized into the feedforward (FF) and feedback (FB) types. The former uses the signal from the reference microphone, while the latter relies on the error signal coming from the error microphone. If an ANC system includes both controllers, it is usually called a hybrid system.
We consider the classical hybrid ANC sructure:
Overview of the alternative structures can be found in the article 1.
The controller synthesis problem can be formulated as a constrained optimization problem. The problem is formulated based on a dataset of measured impulse responses of a headphone system.
Typically, ANC systems use a linear, time-invariant or adaptive, digital filter with a finite impulse response (FIR) or an infinite impulse response (IIR).
Compared to analog controllers, one problem of a digital controller is that an extra delay is introduced. Also, in the case of headphones, the distance between the reference microphone and the speaker does not exceed 1-1.5 cm. Both problems motivate the use of higher sampling frequency to reduce the latency, but it comes with the cost of increased computational burden, making the use of FIR-based controllers impractical.
This problem could be alleviated with the use of warped FIR filters or cascades of biquad filters (2nd order IIR), the latter of which is particularly advantageous for commercial products since the controller complexity could be greatly reduced and the battery lifetime could be increased.
The primary advantage of an IIR-based controller is its reduced computational complexity. Each biquad filter only needs 5 multiplications and 4 additions per sample to calculate its output, and the total number of multiplications or additions is less than 100. This is several orders of magnitude less than for FIR-based controllers.
Based on this reduced computational complexity, the sampling frequency could be greatly increased up to 192 kHz or 384 kHz (typically used in commercial chips) so that a much smaller controller delay could be obtained. Consequently, a better noise reduction performance could be expected due to enhanced system causality.
To derive the transfer function of a hybrid ANC system, let us represent it as a signal flow graph:
List of abbreviations:
| Abbreviation | Definition |
|---|---|
| ERP | Error reference point (error microphone location) |
| DRP | Drum reference point |
| REF | Reference microphone |
| PP | Primary path |
| SP | Secondary path |
| FF | Feedforward controller |
| FB | Feedback controller |
Designing a controller necessitates obtaining accurate measurements of the plant to be controlled. In (feedback) active noise control, the plant includes multiple electro-acoustic components and the acoustic path inside the ear. This path from the ANC output to the error microphone output is commonly referred to as the secondary path (SP).
The SP combines the properties of the digital-analog (D/A) and analog-digital (A/D) converters, analog filters, power amplifier, loudspeaker, microphone, and the acoustic path between the loudspeaker and error microphone.
Traditionally, an ANC system is tuned to produce optimal noise cancellation at the location of the error microphone, also called the error reference point (ERP). This is because most of the ANC systems rely upon monitoring the cancelled signal to work. To maximize noise cancellation, an ideal placement of an error microphone would be at the eardrum. This point is often referred to as the drum reference point (DRP). However, that location is not practical or possible for many consumer devices. Thus, the ERP is used to provide a signal that is roughly indicative of the cancellation performance at the DRP, especially for lower frequencies (< 1 kHz). Also, sometimes it is considered acceptable to ignore any differences in noise cancellation between the ERP and the DRP.
Using Mason's gain formula, one can derive the transfer function:
that models the system's output
Graph determinant:
Paths and co-factors:
-
$P_1 = G_\text{DRP}$ ,$\Delta_1 = 1 + \mathrm{FB} \cdot \mathrm{SP}_\text{ERP}$ -
$P_2 = G_\text{REF} \cdot \mathrm{FF} \cdot \mathrm{SP}_\text{DRP}$ ,$\Delta_2 = 1$ -
$P_3 = -G_\text{ERP} \cdot \mathrm{FB} \cdot \mathrm{SP}_\text{DRP}$ ,$\Delta_3 = 1$
Transfer function:
Introducing the primary path (PP) responses as
and
the transfer function can be rewritten as:
Given this transfer function, one can define the sensitivity function for a whole system - the ratio of noise with ANC ON and ANC OFF:
Let us assume that
Under this assumption, designing a hybrid ANC controller, - minimization of the sensitivity function
Let us consider a feedback control system with a linear time-invariant (LTI) stable plant
The objective is to synthesize a linear time-invariant controller
This can be achieved by minimizing the sensitivity function, i.e., the closed-loop transfer function:
In the case of feedback ANC, the plant is
The design goal of minimizing the sensitivity function involves reducing its magnitude,
Beyond performance, the sensitivity function is closely related to system robustness. The peak magnitude of the sensitivity function is a widely used indicator of robustness to variations or uncertainties in the plant model. Smaller values (typically constrained to be less than 6 dB) signify a more robust system.
Additionally, the sensitivity function characterizes the so-called waterbed effect (see below), a fundamental constraint in feedback control systems, including ANC. This phenomenon dictates that efforts to suppress the sensitivity function (attenuate noise) in one frequency band inevitably lead to an increase in its magnitude (noise amplification) in other frequency bands. Therefore, the design process must carefully balance the desired level of noise attenuation within the control bandwidth against acceptable levels of noise amplification outside this band.
In summary, the optimization problem for ANC controller synthesis is to find a shape for
The following facts can be helpful for designing an ANC system for headphones.
- First, the human auditory system is less sensitive at low frequencies. See the equal-loudness contours for details.
- Second, a headphone system attenuates ambient noise in the high frequency region, that is, the noise is already partially attenuated by passive sound isolation.
Based on these assumptions, the objective function for controller design can be adjusted to put less emphasis on certain frequency regions (low and high frequencies). This can be done by using a weighted sensitivity function obtained by applying frequency weighting. A suitable frequency weighting function can be defined based on properties of the human ear (such as A-weighting) as well as the desired noise attenuation band.
Below is an illustration of weighting functions used for FF and FB controller design.
In 2, a method for designing an ANC controller, represented as an FIR filter, was proposed. The coefficients of the filter were found by solving a convex optimization problem. Despite the convenience of the problem formulation and the uniqueness of the solution, practical implementations of FIR controllers are usually restricted by their computational complexity. Therefore, IIR filters are usually preferred for implementing ANC controllers. Also, a cascade of 5-10 biquad filters can approximate a prescribed frequency response with reasonable accuracy, required for building a competitive ANC system. Here, cascades of biquad filters are used as the feedback and feedforward controllers, similar to the most commercial ANC headphones currently available on the market (2024).
Instead of the general form represented with poles and zeros, several kinds of parametric biquad filters are considered here. See 3 for details.
According to the Nyquist stability criterion, a closed-loop system is stable iif the Nyquist curve of the open-loop transfer function does not encircle the point
-
Stability margin is the shortest distance from the Nyquist curve to the critical point
$(-1, 0)$ . -
Gain margin (GM) is a factor by which the gain of a stable system is allowed to increase before the system becomes unstable. It is the inverse of the distance between the origin and the point between
$-1$ and$0$ where the Nyquist curve crosses the real axis. -
Phase margin (PM) is the amount of phase lag that a system can tolerate before becoming unstable. It is defined as the difference between
$-180$ degrees and the phase angle of the open-loop transfer function at the frequency where the magnitude is$1$ ($0$ $\mathrm{dB}$ ).
The gain margin and phase margins on the Nyquist plot:
In a practical situation a phase margin of 45° and a gain margin of 6 dB are often used.
One of the ways to define a constraint is the following:
where
The smaller values of
Forbidden area (gray) in the complex plane:
Another similar constraint can be defined based on a parabola in the complex plane:
Forbidden area (gray) in the complex plane:
Finally, physical systems generally have some high frequency plant uncertainty. For instance, a slight change in the system delay could cause a large phase variation in the high-frequency band. Thus, the open-loop gain should be small enough above
where
According to Bode's sensitivity integral theorem, there is a fundamental trade-off, known as the waterbed effect, between decreasing (below unity) the magnitude of the sensitivity function in some frequency regions and getting it increased (above unity) at the other frequency regions.
Therefore, to avoid excessive noise amplification (
This constraint defines a circular region with a radius
and ensures that the magnitude of the sensitivity function is below
In summary, given the frequency response of the plant (secondary path), a feedback controller can be designed by solving a constrained optimization problem defined in the frequency domain. The problem includes the objective function defined by the squared magnitude of the weighted sensitivity function, and the constraints outlined before. The problem is solved wrt the controller parameters
Here,
The objective function and the constraints were formulated based on the average PP and SP responses computed from the dataset of measurements collected from the headphone system.
Since continuous frequency responses cannot be directly handled by numerical optimization algorithms, they must be discretized. This involves evaluating the SP response, the controller response, and the weighting function at a finite set of discrete frequency points
A critical aspect of this discretization is ensuring that the behavior of the system between these sample points remains acceptable. The number of frequency points,
To mitigate this tra 8000 de-off between accuracy and computational burden, non-uniformly discretized spectra offer a promising approach 4. Instead of a uniform spacing of frequency points, this technique employs a variable density of points. More points are allocated to frequency regions deemed critical for the ANC system's performance or stability, while fewer points are used in less critical regions.
The optimization problem described above is highly non-convex since the controller has an IIR structure.
Consequently, metaheuristics are often used to solve such problems. These algorithms aim to find a global (rather than local) optimum, and although they have no guarantee of good performance, they have been found to perform acceptably in many use cases. Examples of such algorithms include evolutionary algorithms, particle swarm optimization, etc.
To use off-the-shelf metaheuristics available in open-source Python packages, the original optimization problem should be formulated as an unconstrained problem. This is done by adding the constraints as additional penalty terms to the original objective function, similar to 5.
The key concept of the FF ANC system is that an upstream reference microphone is used to capture the noise (disturbance) before it reaches the control source (loudspeaker) and the desired cancellation point (error microphone). Based on the time-advanced information about the noise from the reference microphone, the FF ANC controller generates an antinoise signal that is equal in amplitude but opposite in phase to the disturbance signal, which is then emitted by the loudspeaker. In the ideal case, i.e., perfect cancellation, the ANC controller should correspond to the optimal transfer function of the FF ANC filter
where the primary path
In practice, designing a feedforward ANC controller is technically similar to designing a feedback controller. In both cases one needs to minimize the magnitude of the sensitivity function
However, in the case of FF ANC, the problem is usually unconstrained because the stability constraints are not required.
We used the data from the PANDAR database of acoustic paths 6, which provides impulse responses measured for multiple persons equipped with a Bose QC20 in-ear headphone.
See https://www.bose.com/pressroom/bose-quietcomfort-20-acoustic-noise-cancelling-headphones for details.
First, let us verify the stability of the resulting feedback controller.
The stability of a closed-loop negative feedback system is assessed by applying the Nyquist stability criterion to the Nyquist plot of the open-loop system (i.e. the same system without its feedback loop). A Nyquist plot is a parametric plot of a frequency response. In Cartesian coordinates, the real part of the transfer function is plotted on the x-axis while the imaginary part is plotted on the y-axis. The frequency is swept as a parameter, resulting in one point per frequency. To determine closed-loop stability, the plot is analyzed for encirclements of the critical point
The Nyquist plot of the open-loop transfer function corresponding to the obtained feedback controller:
The Nyquist curve, shown as a dotted line, does not penetrate the forbidden region (gray) ensuring that the system is stable.
Next, given the synthesized controllers, the noise reduction performance of the resulting ANC system can be assessed. The practical performance assessment involves measuring the acoustic noise at the DRP under two conditions: with the ANC system turned off and with the ANC system turned on.
The relative attenuation (RA) is a widely used metric for quantifying the performance of an ANC system in the frequency domain. It is a ratio of the power spectral density (PSD) of the error microphone signal with an ANC system turned on to the same PSD with ANC turned off, typically expressed in decibels (dB). A negative value of RA at a particular frequency indicates that the ANC system is reducing the noise power at that frequency. Conversely, a positive value indicates noise amplification.
Assuming that the system behaves linearly, the ratio of these PSDs is approximately equal to the squared magnitude of the sensitivity function
Any significant discrepancies between the
Below is the result of running the ANC system simulation.
Here, the curve labeled as FF+FB corresponds to the hybrid system, while the other curves show individual performance of each controller.
Ultimately, the synthesis of an effective ANC controller is an iterative process. It often involves meticulous tuning of weighting functions in the objective, careful definition of constraint boundaries, and repeated optimization runs to navigate the complex trade-offs inherent in feedback control.
For practical design of an ANC system, several issues must be considered, such as the measured data uncertainty and specifics of headphone fit.
The resulting ANC system depends on the secondary path and the primary path measurements. Both the secondary path and the primary path can vary according to the placement of the earpiece in the ear, the shape of the ear canal, the level of leakage, etc. The figure below shows variations of the secondary path responses estimated from different persons and a dummy head6.
In order to obtain a robust system, uncertainties of the plant (secondary path) need to be considered. The multiplicative uncertainty model is a common choice. It describes the plant by means of a multiplicative deviation from a nominal path 578.
Multiplicative uncertainty model:
Here,
...work in progress!
In the discussion above, the ANC performance was generally evaluated at the error microphone location, aka the error reference point (ERP). However, the residual error signal at the ERP might be different from the one perceived by the human ear at the eardrum, aka the drum reference point (DRP). There are studies where the ERP vs. DRP mismatch is addressed for an ANC system design 9.
...work in progress! Contributions are welcome!
Please note that some images, code snippets, and text excerpts have been sourced from various publicly available materials and are not original to the repository's contributors. The owner apologizes for the lack of citations for these materials.
Footnotes
-
Benois, P.R., Nowak, P., Zölzer, U. (2018). Hybrid Active Noise Control Structures: A Short Overview. ↩
-
Rafaely, B., Elliott, S. J. (1999). $H_2/H_{\infty}$ Active Control of Sound in a Headrest: Design and Implementation. ↩
-
Nercessian, S., Sarroff, A. M., Werner, K. J. (2021). Lightweight and Interpretable Neural Modeling of an Audio Distortion Effect Using Hyperconditioned Differentiable Biquads. ↩
-
Klatt, W. A., Burger, M., Martin, R., Puder, H. (2024). Filter Synthesis for Robust Feedback Active Noise Control using Non-Uniformly Discretized Fourier Spectra. ↩
-
Hua, Y., Peng, L. (2024). Uncertainty Constraint on Headphone Secondary Path Function for Designing Cascade Biquad Feedback Controller with Improved Noise Reduction Performance. ↩ ↩2
-
Liebich, S., Fabry, J., Jax, P., Vary, P. (2019). Acoustic Path Database for ANC In-Ear Headphone Development. ↩ ↩2
-
Benois, P.R., Zölzer, U. (2019). Psychoacoustic Optimization of a Feedback Controller for Active Noise Cancelling Headphones. ↩
-
Klatt, W. A., Martin, R. (2025). Control Filter Design with Convex-Set-Based Uncertainty Model for Robust Feedback Active Noise Control. ↩
-
An, F., Wu, Q., Liu, B. (2022). Feedback Controller Optimization for Active Noise Control Headphones Considering Frequency Response Mismatch between Microphone and Human Ear. ↩