Time-variant acoustic front-end measurements ofactive noise cancellation headphones
Johannes FABRY, David HILKERT, Stefan LIEBICH, Peter JAX
Institute of Communication Systems, RWTH Aachen University, Germany, {fabry, hilkert, liebich,jax}@iks.rwth-aachen.de

Abstract
A robust design of active noise cancellation (ANC) headphones by means of digital signal processing requiresa deep understanding of the underlying acoustic front-end. Particularly knowledge about the primary path,
which is commonly defined as the transfer path between the outer and inner microphone of the headphone,and the secondary path, which is the transfer path between the loudspeaker and the inner microphone, isrequired to designing an ANC system. These paths may vary due to e.g. fitting of the headphone, thephysiology of the user or the direction of arrival of ambient noise.
In this contribution, we present the results of a measurement series. The objective is to examine acoustic
paths for their intra-person variances considering jaw movement, head rotation and refitting of the headphone,
as well as inter-person variance between different subjects. The measurements were conducted in an anechoic
chamber with 25 participants from age 21 to 61. Furthermore, the implications on the performance fortime-invariant ANC solutions will be considered and the benefit of calibrating the secondary path will be
investigated.

Keywords: Active noise control, device related transfer functions

1

INTRODUCTION

These days, a lot of applications and products based on active noise cancellation (ANC) technology
are emerging. Especially ANC headphones have risen in popularity over the last few years, due to theever-growing amount of environmental noise [12].
An implementation of complex adaptive algorithms is still not feasible, due to strict power and form-factor
constraints of consumer ANC headphones. Hence, most designs rely on time-invariant digital feed-forward(FF) filters and feed-back (FB) controllers. The optimal solution to feed-forward filters for ANC is givenby the causal Wiener filter, which is a well-studied topic [7, 3, 5, 4]. Also, numerous publications cover
the topic of feed-back controller design for ANC applications [11, 8]. In order to obtain a robust feed-backsystem, the uncertainties of the secondary path of the ANC headphone need to be considered.
Hence, to obtain a well-performing and robust design for ANC headphones a good understanding of theelectro-acoustic front-end is required. To receive this deep understanding, several measurement series
have previously been conducted by the authors, which focus, e.g., on the direction of arrival dependencyof ANC headphones [10] and the occlusion effect [9]. However, these measurements did not specificallyanalyze the influence of the user and use-cases on the transfer functions of an ANC headphone.
This contribution presents the results of a measurement series to identify how knowledge of individual
device related transfer functions (DRTFs) can improve the performance of ANC headphones. We measured
time-variant DRTFs of 25 female and male participants in the age of 21 to 61 years. To obtain a databasethat covers a wide spectrum of transfer functions, the participants were asked to go through a scripted
sequence of different jaw positions, head rotations and headphone fittings.
This article is structured as follows. In Sec. 2 we describe the functional principle of ANC headphones. We
then elaborate on the measurement setup, execution and post-processing in Sec. 3. In Sec. 4 we evaluate
the measurements. We conclude the paper by discussing the implications on the performance of ANC
headphones in Sec. 4.2.

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2

ACTIVE NOISE CANCELLATION

The goal of active noise cancellation (ANC) is to create a zone of silence in which ambient noise isattenuated. This is achieved by the principle of destructive interference of sound waves. In the case of
ANC headphones, the zone of silence is defined as the volume inside the auditory canal. To create ananti-noise, we require knowledge about the ambient noise. For that reason, ANC headphones are equippedwith at least one microphone that measures the ambient noise. For the measurement series presented in thispaper, we used the electro-acoustic front-end of the Bose QC20 headphones, without the ANC electronics.
The left half of Fig. 1 illustrates the headphone. It contains the reference microphone Mref , directed tothe opposing side of the head, the error microphone Merr , directed towards the eardrum, as well as theinternal loudspeaker Sint that emits the anti-noise and can also be used to playback multimedia signals.
The behavior of the headphone is characterized by two acoustic transfer functions. The primary path
P (z) describes the transfer function between Mref and Merr , whereas the secondary path S(z) describes
the transfer function between Merr and Sint . An acoustic feedback from Sint to Mref can be neglected forsealed headphones due to a very low magnitude of the transfer function.
The anti-noise is generated by applying a digital filter W (z) to the reference microphone signal x(n) in thecase of feed-forward ANC and by applying a digital controller K(z) to the error microphone signal e(n) in
the case of feed-back ANC. Hence, the overall transfer function of a feed-forward ANC system HFF (z) is
HFF (z) = P (z) − W (z)S(z),(1)

w = Ψ−1
ss ϕps .

(2)

whereas P (z) and S(z) also comprise the characteristics of the analog-to-digital and digital-to-analog
converters. According to the Wiener criterion, we find the impulse response w ∈ RL of length L of W (z)asHere, Ψss denotes the auto-correlation matrix of the secondary path and ϕps denotes the cross-correlation
vector between the primary and secondary path.
The transfer function HFB (z) of a feed-back ANC system as described in [11, 8] is given by
HFB (z) =1.
1 + S(z)K(z)(3)

In order to obtain a robust system, the uncertainties of the plant, here S(z), 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 S(z)S(z) = S(z) [1 + ∆(z)U (z)] ,(4)

Figure 1. Measurement setup with participant, frontal and lateral external speakers in distance l of centeras well as an illustration of the ANC headphone that was used.

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with |∆(z)| < 1. We choose the nominal path as the average over a set of J measured plant transfer
functions and calculate the uncertainty so that all plants from this set can be obtained from (4)S(z) =
U (z) =1 X
Sj (z)
J j=0
J−1

max

j∈[0..J−1](5)

Sj (z)
−1 .
S(z)

(6)

Rafaely shows in [11] that the plant uncertainty has strong implications on the performance of a feed-backcontroller. We will have a closer look at the plant uncertainty in Sec. 4.2.

3

TIME-VARIANT MEASUREMENT

The following sub-sections elaborate on the acquisition of the data by means of time-variant measurements.
We first describe the measurement setup. Then, we shortly introduce the topic of time-variant system
identification and finally give some insights into the required post-processing steps.
3.1 Setup
The measurements were conducted in a chamber that is anechoic down to a frequency of 200 Hz. The
right half of Fig. 1 illustrates the measurement setup. The participants were placed on a rotating chair inthe center of the chamber, facing the frontal external speaker Sfrnt (ϕ = 0). Two Neumann KH120 speaker
were positioned at a distance of l = 90 cm in front and right of the center of the head. The participants
wore a pair of Bose QC20 headphones. Fig. 2 shows the connection of the hardware components. We
connected a RME Fireface UCX soundcard via USB to the host PC. A PreSonus DigiMax DP88 wasconnected to the soundcard via ADAT and served as a pre-amplifier for the microphone signals.
As a ground truth, the acoustic transfer functions of each participant were first measured using an
exponential sweep. Afterwards, the time-variant measurement began and the participants were instructedto go through a scripted set of movements, as outlined in Fig. 3. The instructions were displayed on a
TV screen. The movements involve different jaw positions, head rotations and headphone fittings. The
head rotation is ϕ = 0 by default and increases to ϕ = 90◦ within the last segment, so that all directionsof arrival are covered with the two external speakers. Furthermore, the headphones were removed andrefitted four times, twice instructed as a loose fit and twice as a normal fit. The headphones were initiallyinserted with a normal fitting.

PC

USB

Soundcard

xfrnt (n)Sfrnt

xltrl (n)direct

ADAT

xint (n)HP left

HP right

Sltrl

yerr,l (n) yref,l (n) yref,r (n) yerr,r (n)Pre-amp

Figure 2. Digital and electro-acoustic connection between computer, soundcard, pre-amp as well as theloudspeakers and microphones

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Time [s]

150

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ϕ

=

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210

Figure 3. Sequence of positions instructed to the participants. Here,marks time-variant segments astransitions between different positions andmarks segments where the headphones are refitted.
3.2 Time-Variant System Identification
Fig. 2 indicates that we measure a total of 10 time-variant impulse responses. We obtain one impulse
response from each of the two external speakers to each of the four microphones as well as one impulse
response from each of the internal speakers to the respective error microphone. As an example, the block
diagram in Fig. 4 shows the multi-channel system model to identify the transfer functions from the external
speakers to the left reference microphone. The measured response yref,l (n) is given as a superposition of
the excitation signals convoluted with the respective impulse responses and a noise term vref,l (n)Tyref,l (n) = hT
ltrl,ref,l (n)xltrl (n) + hfrnt,ref,l (n)xfrnt (n) + vref,l (n),
T

(7)
T

with h(n) = [h0 (n), h1 (n), . . . , hL−1 (n)] and x(n) = [x(n), x(n − 1), . . . , x(n − L + 1)] . We use thenormalized least mean squares (NLMS) algorithm to estimate the time-variant impulse responses as
described in [1]. As an example, the update rule for the adaptive filter from the lateral external speakerto the left reference microphone is given byĥltrl,ref,l (n + 1) = ĥltrl,ref,l (n) + µwith the error signal

eref,l (n)xltrl (n)2kxltrl (n)k + kxfrnt (n)k

(8)

2,

T
eref,l (n) = yref,l (n) − ĥTltrl,ref,l (n)xltrl (n) − ĥfrnt,ref,l (n)xfrnt (n)
h
iT
h
iT
= hltrl,ref,l (n) − ĥltrl,ref,l (n) xltrl (n) + hfrnt,ref,l (n) − ĥfrnt,ref,l (n) xfrnt (n) + vref,l (n).

(9)

To obtain a good tracking behavior of the time-variant impulse responses, we need to choose a suitable
excitation signal. Perfect sequences (PSEQs) have proven to be a superior excitation signal for tracking
vref,l (n)+hltrl,ref,l (n)hfrnt,ref,l (n)Actual paths

yref,l (n)

+

+

−

ĥltrl,ref,l (n)−eref,l (n)ĥfrnt,ref,l (n)Estimates

xfrnt (n)xltrl (n)Figure 4. System identification block diagram at the example of the transfer functions to the left reference
microphone.

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time-variant impulse responses [1]. This is due to their perfectly flat magnitude spectrum. The autocorrelation function ϕxx (λ) of a perfect sequence of length M is equal to one for λ = iM with i ∈ N andzero otherwise. The tracking behavior of PSEQs has been analyzed extensively in [6]. Furthermore, aperfect sequence has been introduced as a noise signal, which has a uniformly distributed group delay anda flat magnitude spectrum. We have a maximum of three signals that arrive at the same microphone. To
prevent the three excitation signals from interfering, we design a perfect noise signal of the length 3L. The
signal is circularly shifted by L so that the mutual shift for the frontal, lateral and internal loudspeakersignal is L. This way, the first L values of all cross-correlation functions of the excitation signals are zero.
3.3 Post-Processing
To compensate for remaining flaws and to structure the measured data during evaluation, we introduce
some processing steps. The Neumann KH120 speakers have a frequency range starting from 52 Hz, andalso the anechoic chamber is not reflection free for such low frequencies. Furthermore, the pre-amps might
add a DC offset. Therefore, we apply a Butterworth high-pass filter with a passband frequency of 50 Hz to
all microphone signals y(n) as well as the respective excitation signals x(n) used for the NLMS algorithm
as in (9).
As we are solely interested in the acoustic transfer functions of the primary path P (z) and the secondary
path S(z) without the influence of the electronic back-end, we need to compensate the transfer functionsof the back-end. These transfer functions were measured beforehand, as indicated by the direct pathin Fig. 2. They are used to deconvolute the microphone signals by means of spectral division. Now, toobtain P (z) and S(z), we use the time-variant estimates from the NLMS algorithm. The secondary path
S(z, n) = Hint,err (z, n) at time n is given by the transfer function from the left or right internal speakerto the respective error microphone. As the primary path is the transfer function from Mref to Merr , weneed to perform a spectral division. The primary path for, e.g., the lateral external speaker is given by
P (z, n) =Hltrl,err (z, n).
Hltrl,ref (z, n)(10)

If the denominator transfer function has a notch, this will lead to an unnatural amplification at certain
frequencies in the primary path. When such primary paths are used for the design of a feed-forward filter,
the amplifications result in an annoying resonating sound [2]. We compensate these notches by analyzing
the resulting primary path p(n), as Fig. 5 shows. In case of a notch, the primary path’s impulse response
does not decay. We analyze the latter part of the primary path using a zero-padded Hann window q(n),and we use the phase of the denominator to generate the compensation spectrum
C(z) = α |F {p(n) · q(n)}| ej∠Hltrl,ref (z,n) ,b) impulse responses in dB−50

Href (z)C(z)
P (z)Pc (z)
7

8
Frequency [kHz]Magnitude [dB]

Magnitude [dB]a) magnitude spectra
0

(11)

−20

p(n)
pc (n)

−40
−60
−80
−100

500

1,000
SamplesFigure 5. Notch compensation by analyzing p(n) with a Hann window.

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1,500

q
2
with the Fourier transform F {·}, the scaling factor α = M/ kqk and the FFT length M . We thenobtain the transfer function of the corrected primary path as
Pc (z, n) =Hltrl,err (z, n).
Hltrl,ref (z, n) + C(z)(12)

In order to save only the transfer functions that convey new and reliable information, we define a setof rules which must be fulfilled before saving a set of impulse responses. The short-time power of theerror signals needs to be below −50 dB. Furthermore, for the time-invariant segments from Fig. 3 we save
only a single set of transfer functions where the short-time power of the error signal is smallest. For thetime-variant and refitting segments we save a new set of transfer functions if the Euclidean distance toany respective transfer function that was saved last exceeds −40 dB.

4

MEASUREMENT RESULTS

In this section, we discuss the results of the time-variant measurement series. The following plots are
based on over 10, 000 individual paths for 25 participants. All paths are calculated with a filter length
L = 2048 at a sample rate of fs = 48 kHz.
4.1 Nominal Paths
Fig. 6 shows the nominal secondary paths in a) as the average over all segments with normal fitting foreach subject and in b) as the average over all subjects for each use-case. It is evident that S(z) strongly
depends on the geometry of the subjects ear canal. We further see that nominal path changes notably forthe segments headphones out, where the headphones are not inserted into the ears, and loose fit, where theheadphones are inserted loosely into the ears. The drop in magnitude at frequencies below 1 kHz for theuse-case headphones out is due to the lower acoustic impedance for which the internal headphone driver
does not provide enough power. On the other hand, the nominal secondary path is independent of theyaw position.
a) subjectsb) use-cases

Magnitude [dB]

20
0
−20
−40

headphones outloose fit
102

103
Frequency [Hz]

104

102

103
Frequency [Hz]104

Figure 6. Nominal secondary paths S(z) in a) averaged over all segments with instructed normal fitting
for different subjects and in b) averaged over all subjects for different segments.
Similar results can be observed when evaluating the primary path. Fig. 7 shows the averaged magnitude
spectra of the primary paths of the right headphone and the lateral external loudspeaker, again averaged
over use-cases and subjects. We see that the use-cases headphones out and loose fit result in a boost at
frequencies between 400 Hz and 4 kHz. This is due to the reduced passive insulation. For ϕ = 90◦ weexperience additional attenuation of frequencies above 500 Hz and especially the peak at 4 kHz is reduced.
We notice that different yaw positions do also not affect the primary path.

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Magnitude [dB]a) subjectsb) use-cases

0

Sweep
headph. out
loose fit
ϕ = 90◦

−20
−40

102

103
Frequency [Hz]

104

102

103
Frequency [Hz]104

Figure 7. Averaged magnitude spectra of the primary paths P (z) for the right headphone and the lateralexternal loudspeaker in a) averaged over all use-cases with normal fitting for different subjects and in b)
averaged over all subjects for different use-cases.
4.2 Implications on the Performance of Active Noise Cancellation Headphones
Rafaely derived an explicit relationship between the uncertainty of the plant in a feedback ANC system andthe possible performance [11]. From Sec. 4.1 we know that the secondary path varies strongly for different
users and not so much for different use-cases. Fig. 8 reinforces this statement. It shows the percentiles ofthe multiplicative uncertainty according to (6) in steps of 10% from 10% to 100% for a) the nominal pathbased on multiple subjects and b) an individual nominal path for each subject. We see that the maximum
uncertainty for both cases is similar, nonetheless, the 90% percentile for the intra-subject uncertainty isalready much smaller than for the inter-subject uncertainty. Hence, calibrating S(z) once for a user may
result in a very accurate representation of the actual path and can drastically improve the performance ofboth feed-back and feed-forward ANC [11, 4]. Also, calibrating the secondary path can simply be doneby using the internal loudspeaker. The primary path on the other hand varies quite strongly, especiallyfor different directions-of-arrival, which has been examined in more detail in [10]. Calibrating P (z) for
multiple discrete angles is feasible but requires a dedicated measurement setup. With a good primary pathmeasurement the performance of feed-forward ANC systems can be drastically improved as shown in [4].
b) intra-subject uncertainty

20
0
−20
−40
−60

102

103
Frequency [Hz]

104

102

103
Frequency [Hz]

104

100
80
60
40
20
0

Percentile

Uncertainty [dB]a) inter-subject uncertainty

Figure 8. Multiplicative uncertainty of the secondary path in decibels for a) the nominal path based on
all use-cases and subjects and b) individual nominal paths for each subject based on all use-cases.

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5

CONCLUSION

In this contribution we present the results of a measurement series of time-variant device-related impulse
responses of an active noise cancellation (ANC) headphone. The focus of this study is to see how therelevant acoustic transfer functions change for different subjects and use-cases. The measurements wereconducted using perfect sequences (PSEQs) and the normalized least mean squares (NLMS) algorithm. We
further propose a method for compensating notches when calculating impulse responses based on spectral
division. The results based on over 10, 000 acoustic transfer functions show that an individualization of
the primary and secondary path is a promising measure to improve the performance of time-invariant
ANC systems. This is because the intra-person variance of the acoustic transfer functions is much lower
compared to the inter-person variance. Furthermore, we investigate the multiplicative uncertainty, whichplays an important role in designing robust feed-back systems, to underline this statement.

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