Prediction Filter Design for Active Noise
Cancellation Headphones
Markus Guldenschuh
Abstract—Digital active noise control (ANC) for
headphones usually has to predict the noise because
of the latency of AD-conversion. In adaptive feedback
ANC, the prediction is based on the noise that entered
the headphone. This noise is low pass filtered due to the
physical barrier of the ear cups. In this paper, this low
pass characteristic is exploited to define a prediction
filter which does not require real-time updates. For
broad band noises, the prediction filter performs better
than adaptive prediction methods like the least mean
squares (LMS) algorithm or iterated one-step predictions.
I. Introduction
Active noise control (ANC) systems sense, process and
play-back noise such that it interferes destructively with
the ambient noise. In headphones, ANC is a very efficient
and powerful protection against loud sound levels. It cancels ambient noises directly at the user’s ear and reduces
the need for loud music play-back.
There are analogue and digital realizations of ANC
headphones, whereas digital applications have the advantage that the inherent filters can be adapted to changing
conditions [1], [2]. However, the latency of conventional
audio converters severely limits the ANC performance [3].
In these cases, the adaptive filter has to predict the noise to
compensate for the delay. In the feedback ANC approach,
the prediction is based on noise that actually entered
the headphone. This has two advantages. Firstly, ANC
is independent from the direction of incident noise and
also works in diffuse sound-fields [4], [5]. And secondly,
the upper frequencies of the entered noise are damped by
the ear cup. This low-pass characteristic is advantageous
when it comes to signal prediction [6].
In literature, prediction is mostly done by different kinds
of the least mean squares (LMS) algorithm [7], [8], [9] or by
iterated one-step-ahead predictions [10]. Both algorithms
are based on sequential updates of the prediction filter
and account for changes in the noise-signal characteristics.
Since most environmental noises such as traffic noises
are broadband, the signal characteristic of the penetrated
noise mainly depends on the passive damping of the ear
cups.
In this paper, we therefore suggest the design of a
prediction filter that is only based on this passive damping
characteristic. The proposed filter does not require runtime coefficient updates which makes its application simple, economical and robust, and still it yields equal - if not
better - ANC results than the adaptive methods.
The structure of digital feedback ANC is reviewed in the
following section and the prediction filter design is outlined
in section III. Section IV shows simulation results and
compares the proposed prediction with LMS and iterated
one-step-ahead predictions before the conclusion and an
outlook are given in the last section.
II. Digital Feedback ANC
Headphones with feedback ANC, as depicted in Fig.1,
only require one microphone inside each ear cup. This
microphone measures the residual error signal e(t) which is
the superposition of the entered noise x(t) and the playedback anti noise y(t). With omitted time dependency,
e = x + y.
u(t)
x(t)
P (s)
attenuation by the ear cup
e(t)
+
y(t)
S(s)
x̂(n)
prediction unit -
DAC z −N ADC z −N
z −2N
e(n)
Ŝ(z)
- ŷ(n)
+
Figure 1: Digital feedback ANC: An estimate of the noise
inside the ear cup x̂(n) is used as input for the prediction
unit. The inverted output is played back to cancel the
entered noise x(t).
Since y should equal x with an inverted phase, an
estimate of x is required. This estimate x̂ can be obtained
by subtracting the played-back anti-noise from the sensed
residual error, x̂ = e− ŷ. The actual anti-noise y is digitally
unavailable. This is because the digital noise cancellation
signal is analogue converted and modified by the loudspeaker transfer-function S(s), also called secondary path.
Thus, an estimate ŷ is generated by means of a secondarypath measure Ŝ(z).
The resulting noise estimate x̂ is delayed by the digital
to analogue converter (ADC) and the final anti-noise y
will be delayed once again by a DAC and the groupdelay
of S(s). To compensate for these delays, the prediction
unit tries to predict future samples. If ADC and DAC each
have a latency of N samples, at least 2N samples have to
be predicted. The group delay of S(s) usually stays below
50 µs for most of the used bandwidth and can be neglected
compared to the latency of conventional audio codecs.
III. Noise Signal Prediction
Assuming that a correct estimate of x is given, x̂(n)
equals x(n − N). With this, the prediction unit builds a
weighted sum of the available past samples to predict the
future noise sample
x(n + N) =
L−1
X
wi x(n − N − i),
(1)
recursively calculate the prediction filter
e(n)x̂
,
(4)
x̂T x̂
where µ is the step size parameter which determines the
speed of convergence. In the LMS algorithm, the convergence speed has to be held rather low to keep the recursion
stable.
However, the steady update of the coefficients is unnecessary if the main noise characteristic always stays the
same. And in our ANC application, this is the case because
the upper frequencies of the outside noise u (as in Fig. 1)
will always be damped by the physical barrier P (s) of the
ear cup. This passive attenuation can be written down as
a convolution operation
w(n + 1) = w(n) + µ
x = Pu,
i=0
with L being the prediction order and wi the coefficients
of a linear prediction filter. The residual error e follows as
e(n) = x(n) − wT Sx(n − D),
(2)
where w is the vector of filter coefficients, S is a convolution matrix of the secondary-path impulse response and
x is a signal vector starting from D = 2N samples in the
past. The minimum of the expected squared error leads
to the well known Wiener-Hopf equation [11] and to the
solution of the optimal prediction filter
T −1
wopt = (SRx S )
(3)
r,
where Rx is the autocorrelation matrix of the latest
available noise samples


r0
r1
. . . rL−1
 r1
r0
. . . rL−2 


Rx =  .
..
.. 
..
 ..
.
. 
.
rL−1
rL−2
...
r0
and r is a vector of autocorrelation elements starting from
lag D


rD
 rD+1 


r =  . .
.
 . 
rD+L−1
The matrix inversion in eq. (3) can be avoided with the
Levinson-Durbin algorithm [12], but only if D = 1. The
resulting one-step-ahead predictor can still be used for a
delay of 2N samples when the linear prediction of eq. (1)
is iterated 2N times. However, since the prediction filter is
stable, the recursive one-step-ahead prediction converges
to zero. Thus, the filter outputs have to be amplified to
get a reasonable multi-sample prediction.
To avoid this problem, gradient search algorithms are
often used for prediction problems with more than one
sample delay. Gradient search algorithms like the normalized LMS use the noise estimate and the error signal to
where P is a convolution matrix of a low-pass impulse
response that simulates the passive attenuation. With this,
the autocorrelation matrix Rx follows to
Rx = PRu PT .
(5)
When u has a flat spectrum, its autocorrelation matrix
Ru reduces to an identity matrix and Rx solely depends
on P. Thus for broadband noises, the calculation of the
optimal prediction filter (eq.(3)) simplifies to
wopt = (SPPT ST )−1 p,
(6)
where p is column number D of the low-pass autocorrelation matrix PPT . This equation allows for an a priori
filter design where no real-time calculation of the filter
coefficients is needed because all required data (S(ω) and
P (ω)) can be measured and designed in advance. The
prediction filter that follows from this a priori calculation
can be used as a fixed prediction filter in the ANC
headphone.
IV. Simulation Results
The following simulations are based on measurement
data from a prototype headphone. Its passive damping
and the low-pass filter that is used to derive the prediction
filter are displayed in Fig.2. The pass-band of the low-pass
filter is chosen to be narrower than the one of the ear-cup
damping. On the one hand, this increases the prediction
error for the upper frequencies, but on the other hand it
leads to better prediction in the low frequency band.
In order to reduce computational load, the DSP might
be sampled down to 24kHz. Lower sampling frequencies
are not advisable because they would reduce the spectral
information of the passive attenuation. Common audio
codecs have a talk-through latency of about 170µs at
192kHz. At 24kHz this corresponds to a total delay of
approximately 4 samples.
In the first simulation, the prediction filter is used for
aeroplane noise and its ANC performance is compared
70
−10
65
noise x
fix. pred. filter
LMS
1−step pred.
60
−20
dB
dB
0
P
−30
2nd order LP
55
50
−40
45
2
3
10
10
40
Hz
2
3
10
Figure 2: Transfer function P (ω) of the passive attenuation
by the ear cup and the second order low-pass that is used
for the prediction filter.
with adaptive methods like LMS and iterated one-step linear prediction. Fig. 3 shows the spectrum of the aeroplane
noise and the spectra of the residual noises after ANC.
80
noise x
fix. pred. filter
LMS
1−step pred.
75
Hz
Figure 4: Engine noise x that entered the headphone
and the residual noises after ANC with the three compared methods. Again, the fixed prediction filter performs
slightly better, although it is designed a priori and does
not change during runtime.
the third simulation in Fig. 5, pink noise was filtered
with a forth order Butterworth bandpass filter with cut-off
frequencies at 100 and 270 Hz.
70
65
60
60
55
55
50
noise x
fix. pred. filter
updated filter
LMS
1−step pred.
65
dB
dB
70
10
50
2
3
10
10
Hz
45
40
Figure 3: Spectrum of the aeroplane noise x that entered
the headphone and the residual noises after ANC with
the proposed prediction filter, an LMS filter and after
iterated 1-step ahead prediction. The fixed prediction filter
performs best between 100 and 1000 Hz. The deteriorated
ANC below 80 Hz comes due to the long group delay of
the loudspeaker in the low frequency band.
Above 2000 Hz, the LMS algorithm and the iterated
one-step-ahead prediction produce a lower error than the
proposed prediction filter, but the ANC performance of
the proposed filter between 100 and 1000 Hz is superior
to the other two methods. As the passive noise attenuation
above 1000 Hz is already very pronounced, the ANC of the
fixed predictor is preferable.
In the second simulation (Fig. 4), the noise of an accelerating engine is used for the ANC comparison. The fixed
prediction filter still leads to a slightly better performance,
although the engine-noise is narrowband.
Only for very narrowband noises, the adaptive methods
yield better results than the fixed prediction filter. For
35
2
3
10
10
Hz
Figure 5: Narrowband noise x and the residual noises
after ANC with the compared methods. Here, the fixed
prediction filter needs to be updated with a noise autocorrelation. The resulting updated 4th order prediction filter
yields the best ANC again.
For this narrowband noise, an update of the prediction
filter would be required to improve the ANC. This update
can be done by inverting the autocorrelation matrix as in
eq. (3). For the update in Fig. 5, the first 150 available
noise samples are taken to perform an autocorrelation.
From this autocorrelation, only four samples (beginning
form time lag 0) are taken for the autocorrelation matrix
Rx and four further samples, starting from lag 2D = 4,
are taken for r. Thus an inversion of a 4x4 matrix is
required to calculate the prediction filter coefficients. This
updated 4th order prediction filter again performs better
than the iterated one-step-ahead prediction and the LMS
prediction.
If narrowband signals are expected, sparse updates via
matrix inversion are therefore a strategy that does not
consume too much processing power and which leads
to better results than LMS or iterated one-step-ahead
predictions.
V. Conclusion and Outlook
This paper presents a new predictive solution for the
biggest problem in digital ANC headphones: The unavoidable delay of the secondary path and the AD/DA
conversion. We show that a main part of the noise characteristic is determined by the passive attenuation due
to the ear cups. With this information, we design an a
priori prediction filter that does not need any real-time
coefficient updates and is therefore very economical. To the
knowledge of the authors, this elegant prediction solution,
elaborated by [6, Jain & Ranganath], has never been
suggested for the feedback ANC applications. Simulations
show that this filter reaches better results than adaptive
prediction methods like the LMS or an iterated one-stepahead linear prediction for usual environmental noises such
as aeroplane or engine noises. The proposed prediction filter might also be used in hybrid ANC systems as proposed
in [13], [5], [14], where it is expected to lead to improved
ANC results too.
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