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DSP in Loudspeakers
By Francis Rumsey
Staff Technical Writer
igital signal processing is used
increasingly in loudspeakers to
compensate for a range of linear
and nonlinear distortion processes that
typically arise. DSP can also be used in
crossover design and for controlling
the spatial radiation characteristics of
loudspeakers or loudspeaker arrays.
This requires the detailed understanding and modeling of the acoustical
deficiencies and behavior of transducers and cabinets. In this article we
summarize a number of papers
describing recent research in this field;
they were all presented at the AES
32nd International Conference, held in
Denmark in September 2007.
D
ACOUSTIC AND TRANSDUCER
CONSIDERATIONS FOR DSP
LOUDSPEAKERS
Marshall Buck, in the paper “Acoustic
and Transducer Considerations for
DSP Loudspeakers,” describes the frequency response of a loudspeaker as
the single most important parameter
related to reproduction quality and the
one most susceptible to treatment
using DSP. He points out that by using
DSP a loudspeaker can be given a
peak deviation in its frequency
response of less than 1 dB on axis. But
how flat does it need to be before it is
audibly perfect? Referring to the work
of Toole and Olive, he suggests that a
resonant peak of less than 0.5 dB can
in fact be audible if it is relatively
broad. There are a number of distortion processes that can affect the
response by a number of decibels
when designing a DSP loudspeaker.
Buck explains that the resonances
and band limitation of loudspeaker
drivers can be relatively easily
addressed with DSP, but that the more
problematic distortions arise from
cabinet resonances, thermal effects,
directional variation, power compres-
sion, limiting, and protection. He
suggests some approaches to transducer selection and cabinet design that
may help to overcome these problems.
For example, the use of a metal
diaphragm can help to minimize the
effects of heat induced stiffness
change. However, such a diaphragm
can have a very considerable peak in
its response in the stop band (see Fig.
1), which can be controlled by using a
digital crossover with a high order (at
least 8th order). He also shows that
response smoothing in measurement
(such as 1/20th octave) gives a good
idea of the octave-to-octave balance in
the frequency response and suggests
that a flat response in this analysis may
be sufficient to deliver a perceptually
perfect performance, making further
smoothing using DSP unnecessary. In
other words, flattening every minute
detail in a loudspeaker’s response may
not be needed for perceptually high
sound quality. Buck concludes with the
view that one may be tempted to use
DSP to artificially flatten the response
of an otherwise mediocre loudspeaker,
but this may be only part of the
picture, leaving other sources of error
uncontrolled.
MODELING LOUDSPEAKER
NONLINEARITIES
Finn Agerkvist, in his AES 32nd paper
of this title, discusses the different
nonlinear processes that take place in
loudspeakers and looks at different
ways of simulating and compensating
for them using signal processing. ➥
File: C:\MLS\ADATA\ZIPA\SEAS\012-01.FRQ 11-2-2000 11:18 AM
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Message: Seas Magnesium Cone Woofer 60 Degrees F vs. 76 Degrees F
Fig. 1. Response of magnesium cone woofer showing two different temperatures
(solid and dotted lines), illustrating the peak in the response and the very small
difference in temperature behavior (courtesy Buck).
J. Audio Eng. Soc., Vol. 56, No. 1/2, 2008 January/February
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Fig. 2. Three main loudspeaker driver nonlinearities plotted with respect to different input power levels. A fourth, shown bottom
right, is the derivative of the inductance. These plots are for a 5-inch woofer (courtesy Agerkvist).
He explains that it is generally
accepted that three factors are of primary importance in determining loudspeaker driver nonlinearity: the force
factor, Bl (the magnetic field strength
times the length of the voice coil); the
voice coil inductance, Le; and the suspension compliance, C m . Using a
system developed by Klippel (see
below), Agerkvist is able to plot the
changes in these parameters at different power levels, as shown in Fig. 2.
In this example the largest change is
observed in the driver’s compliance,
which increases overall with power
level but also changes shape to some
extent. Agerkvist points out that it is
difficult to test nonlinearities such as
these because it requires that the loudspeaker is driven at very high power
levels, which can run the risk of
destroying the drive unit. For this reason it is desirable to have curve fitting
techniques that extrapolate well to levels that might be unsafe for physical
testing.
66
One of the problems with high-order
polynomial expansions for curve
fitting in these cases is that they tend to
deviate from the physical behavior of
real loudspeakers, so it is important to
have a good knowledge of how loudspeakers really behave in order that
such oddities can be spotted. One
example is given, based on the knowledge that Bl is usually maximum when
the cone is at rest and drops off rather
fast once the magnet gap is not filled
with the coil. For this reason it may be
desirable to fit the expansion to the
inverse of the Bl curve or to use a set
of localized Gaussian expansion functions that can be summed together.
This also seems to work quite well for
fitting the compliance curve. For
fitting the inductance curve one needs
a function that has a negative slope
around zero and stable values for
extreme positive and negative conditions, for example. One possibility
here is to use the sigmoid function,
because the standard polynomial
expansion does not seem to be particularly suitable.
It was found that one way of
measuring the compliance for the
purposes of modeling is to look at the
dissipated power. This is because it is
understood that any heating of the
voice coil will spread to the spider (the
flexible device that holds the cone in
its correct suspended place), and any
heating of this suspension is likely to
affect its flexibility. However, when
modeling distortion profiles of
modeled and real units, the simulations
did not match perfectly and it was
suggested that other factors might also
be important, such as the rms velocity
and/or displacement of the suspension.
The best prediction using power dissipation data was found at high power
levels, where presumably the voice
coil heating had a more substantial
effect on the spider compliance, but at
lower levels the prediction was less
satisfactory.
In his paper “Optimal Design of
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Fig. 3. Adaptive nonlinear control of a loudspeaker (Figs. 3–6 courtesy Klippel)
Fig. 4. Force factor Bl(x) and Stiffness Kms(x) versus voice coil displacement of loudspeaker under test
Loudspeakers with Nonlinear
Control,” Klippel discusses the practical benefits of such technology. The
general structure of an adaptive nonlinear control system for a loudspeaker
is shown in Fig. 3. A detector is used
to measure the loudspeaker parameters
by looking at the voltage and current
characteristics of electrical signals at
the loudspeaker terminal. These are
diagnosed and used to control a preprocessing system that treats the signal
fed to the loudspeaker appropriately.
Apparently it is possible to do this
diagnosis using any ordinary audio
signal having sufficient bandwidth and
J. Audio Eng. Soc., Vol. 56, No. 1/2, 2008 January/February
amplitude (for further discussion of
this matter see a summary of the paper
by Pedersen and Rubak, below).
During the initial identification of
those parameters the working range of
the driver is determined. Klippel shows
an interesting example (Fig. 4) in
which the stiffness of a particular ➥
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A
B
Fig. 5. (A) Positive and negative displacement values of an example driver with single tones varied in frequency and voltage.
(B) Equivalent reduction in level of fundamental tone due to this compression effect.
5-inch automotive audio driver is
asymmetrical: negative displacement
produces a higher restoring force than
positive displacement. One of the difficulties of dealing with a phenomenon
like this using DSP for correction is
that the system may attempt to correct
for such asymmetry by producing a
DC displacement of the voice coil,
which moves it away from the Bl
maximum thereby introducing intermodulation distortion.
Klippel shows that voice coil
displacement is an interesting
phenomenon to study, as illustrated in
the example in Fig. 5A. The voice coil
displacement is shown on the y axis
against the frequency of a single tone
on the x axis for a number of equally
spaced voltage levels. However,
although the voltages are equally
spaced, the displacement values are
not, suggesting that the loudspeaker
becomes less sensitive at higher signal
levels and exhibits a certain dynamic
compression, as shown in Fig. 5B. A
most interesting point is that if one
observes the probability distribution
function (PDF) of voice coil displacement for music signals, as shown in
Fig. 6, the voice coil spends the majority of its time at zero and relatively
little at the extremes; whereas for a
sine tone the opposite is true. He
Fig. 6. Probability distribution functions of cone displacement for music and a
50-Hz sine tone
68
suggests that the heating of the voice
coil in such a case for typical music
signals is of little concern, whereas in
the case of sustained tones it can’t be
neglected.
Generally there are two distortion
systems at work in parallel: a linear
system that acts as a second- or higherorder high-pass filter obeying traditional Thiele-Small parameters, and a
nonlinear system that comes into play
more at higher amplitudes of cone
displacement. In the latter case,
Klippel explains, when the nonlinear
control feedback loop comes into play
there can be some unusual effects
because the distortion components
affect their own generation process. He
concludes his paper by making the
important point that one should
concentrate the design of the passive
driver in areas that cannot be corrected,
such as vibration of the cone or mechanism, directivity, size/weight, cost of
manufacture, and efficiency.
IDENTIFYING LINEAR
LOUDSPEAKER PARAMETERS
In their paper “Musical TransducerLess Identification of Linear Loudspeaker Parameters” Pedersen and
Rubak explain that linear loudspeaker
parameters such as resonance frequency, damping factor, and voice coil
resistance can drift over time more
substantially than nonlinear parameters such as force factor, compliance,
and voice coil inductance as a function
of coil position. They drift as a result
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Fig. 7. General structure of an FIR system identification with a loudspeaker. The
error signal: e(n) = d(n) – y(n)); d(n) is the desired response, loudspeaker current;
y(n) is the output of the FIR filter (courtesy Pedersen and Rubak).
of temperature, humidity, aging, and
production spread. In order to identify
both linear and nonlinear parameters,
a process known as system identification is employed, whereby the loudspeaker voltage and current are used
to measure the current state of the
driver. As music is often the main signal used to drive the loudspeaker, they
tried an approach that uses music as
the identification test signal. (System
identification in general is a process of
mathematical analysis that aims to
model dynamic systems on the basis
of measured input–output data.)
As shown in Fig. 7, an adaptive FIR
(finite impulse response) filter is used
to match the loudspeaker parameters in
order that the least mean square of the
error signal is minimized. The authors
explain that the length of the FIR filter
should be selected so that the truncation error will be sufficiently low—this
also determines the frequency resolution of the filter. When greater accuracy is needed, the frequency range of
the identification process can be
limited, for example to frequencies
only a little above the bass resonance
frequency. Data to describe the loudspeaker can come either from a
measurement or a model. The authors
show examples based on two Simulink
models, one linear and the other
nonlinear, of a 6.5-inch bass/mid-range
loudspeaker unit. First the filter was
optimized using the linear model with
white noise as a test signal. A process
of trial and error was used to choose
the best resolution for the process,
leading to a sampling frequency of
4 kHz and a filter length of M=200.
This was done to avoid the possibility
of one particular frequency component
of a musical signal having an undue
J. Audio Eng. Soc., Vol. 56, No. 1/2, 2008 January/February
influence on the outcome. The estimates of driver resonant frequency and
voice coil resistance both turned out to
be very close to those of the real
driver, with an error of around one
percent or less, whereas the damping
factor was slightly less well predicted,
with an error of nearly six percent. The
last parameter was believed to be in
error because of an over-simplified
fitting function for the damping factor.
Subsequently, the authors compared
the results from their model using
white noise with those derived from a
music signal. As far as the resonance
frequency results were concerned, the
results were similar to those obtained
with white noise but the dampingfactor results had a higher spread at
different loudspeaker input levels.
They suggest that the voice coil resistance is more difficult to estimate with
music because it rarely has substantial
energy below 20 Hz. This can be dealt
with by employing a small DC offset
in the test signal, which can provide
the necessary information.
In discussing the prediction and
modeling of nonlinear loudspeaker
parameters toward the end of their ➥
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on an FIR filter was found to be effective for identification of some loudspeaker parameters when one is
dealing with small cone displacements.
Fig. 8. Topology of a biquad IIR filter in direct form I, with optional error feedback
(EFB). The graph shows the self-generated noise of such a filter when using singleand double-precision (48 bit) processing (Figs. 8 and 9 courtesy Thaden et al.).
paper, the authors note that a simple
model of compliance based only on a
function of the cone position seems to
be insufficient. Also they note that
system identification can sometimes
come up with the wrong parameters,
particularly if the test signal is not suitable for the algorithm concerned. The
design of such things needs to take into
account signal-to-noise ratio, clipping,
and displacement of the loudspeaker.
A system-identification model based
Fig. 9. Simplified principle of complex EQ FIR filter coefficient generation for a midrange loudspeaker. (1) Measured loudspeaker response is (2) inverted to form a
compensation response, which is combined (3) with the crossover filter characteristic
to lead to a target response for the FIR filter. An inverse FFT process (4) is used to
create a time domain response that can be turned into a suitable FIR filter.
70
DSP-BASED LOUDSPEAKER
MANAGEMENT SYSTEM
In another AES 32nd Conference
paper, Thaden et al. describe a loudspeaker management system based on
signal processing, comparing the relative merits of IIR (infinite impulse
response) and FIR filters. IIR filters
are like conventional analog filters,
they explain, with their built-in nonminimum phase response and other
typical “analog” characteristics. However, they are simple to implement and
most typical filter characteristics (such
as high-pass, low-pass with Butterworth, Bessel, or Linkwitz-Riley
forms) can be built from IIR building
blocks based on biquad filters, as
shown in Fig. 8. Such filters can be
used for both crossover design and
equalization of loudspeakers, and
higher-order filters can be built by
simply cascading biquads. However,
the authors also point out that the
noise and distortion performance of
IIR filters can be poorer than that of
FIR filters because of the feedback
loops involved, particularly when
using fixed-point processing. The
authors suggest that 48-point data and
arithmetic paths provide a good
dynamic range, whereas it is only necessary to use 24-bit precision for the
filter coefficients. They found that
48-bit integer processing (fixed point)
produced considerably less noise and
distortion than 32-bit floating point
processing for medium and high signal
levels.
The advantages of FIR filters lie in
the possibility for creating filters with a
much wider range of phase characteristics, as well as the chance of making
one FIR filter that can replace a whole
chain of IIR filters. While it is indeed
possible to combine both types of filter
in one design, it is apparently more
common to employ a one-FIR-filterdoes-it-all approach. In this way one
can design an FIR filter that deals with
the equalization and crossover functions
together. One of the advantages of
using linear-phase FIR filters for
crossovers is that they can be designed
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process for generating the appropriate
coefficients for equalizing and crossing
over a typical loudspeaker are shown in
Fig. 9.
Fig. 10. Conventional, left, and pairwise symmetric three-way loudspeaker, right,
(Figs. 10 and 11 courtesy Horbach and Keele)
to avoid any of the problematic cancellations and dips in the frequency
response that can otherwise arise in the
crossover range. One problem with FIR
filter design is that the processing
power required to implement complex
filters across the entire audio frequency
range is considerable. This is because
the length and number of coefficients of
an FIR filter that are needed to get good
frequency selectivity are proportional to
the frequency. For this reason different
bands tend to be operated at different
sampling frequencies, resulting in a
multirate filter design. One of the side
effects of multirate filtering is that
different frequency bands end up
having different group delays; also the
signal-to-noise ratio becomes worse as
the sampling rate drops. The typical
Fig. 11. Approximation of a crossover filter using an FIR filter
J. Audio Eng. Soc., Vol. 56, No. 1/2, 2008 January/February
DIGITAL CROSSOVER FILTERS
In two AES 32nd papers dealing with
the application of linear digital
crossover filters to pair-wise symmetric multiway loudspeakers, Horbach
and Keele consider both the control of
off-axis frequency response and
beamwidth or polar pattern of the
loudspeaker. They start by pointing
out that it is desirable to have a loudspeaker with a uniform and smooth
off-axis response, as this is a feature
that is widely accepted as giving rise
to high-quality sound. There are various ways of attempting to achieve this,
including the use of a large number of
drive units in a long curved array, but
these are not always practical and
have various drawbacks. Quite commonly engineers will modify the
parameters of multiway crossovers in
an attempt to “voice” a loudspeaker
for optimum perceived sound quality.
An alternative, proposed by Horbach
and Keele, is to use a DSP-based
crossover with a loudspeaker array
that has a single central tweeter and
pairs of lower-frequency units
arranged on either side, as shown in
Fig. 10. The idea is that only one or
two pairs of speakers are operating at
a time, although the single central
tweeter operates on its own, which
allows the method to be applied to
both equally and nonequally spaced
drivers. The aim is to specify a
crossover response shape that forces a
flat response at an arbitrary off-axis
angle. The authors found that if this is
done the response at other off-axis
angles is also reasonably flat.
The bandpass crossover filter the
authors arrived at through their process
of development, considered by examining primarily the far-field response
of the loudspeaker, tends to have a
shape like that shown in Fig. 11. It has
an unusual pointed-top shape that rolls
off very steeply to either side of a socalled critical frequency at which only
one pair of drivers is energized. The
spacing of the speakers and the
crossover frequencies are related so
that the spacing is a constant ➥
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DSP in Loudspeakers
THE P ROC E EDINGNDS OF
THE AES 32 ND
IN T E R NATIONA L C ONFERE NCE
DSP for
LOUDSPEAKERS
DSP for Loudspeakers is now more relevant than ever. This is
effectively demonstrated by these 21 scientific papers, which
describe how digital signal processing really can make the
difference in loudspeaker design and usage. The proceedings
hold 235 pages of state-of-the-art technology in this
important research field.
Purchase online at
www.aes.org/publications/conf.cfm
Also available as a
downloadable PDF (11.3
MB).
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For more information
email Donna Vivero at
DMV@aes.org or
call +1 212 661 8528, ext. 42
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distance apart in terms of the acoustic
wavelength at the critical frequency of
the crossover. The filter heavily attenuates the signal at frequencies above
and below the critical frequencies of
adjacent drivers. Between the critical
frequencies only two pairs of speakers
operate.
One of the beneficial side effects of
this design is that the vertical polar
pattern is also maintained relatively
consistently over the frequency ranges
of different drivers, so that the pattern
when only one pair of sources is active
is relatively similar to that which arises
between the critical frequencies when
more than one pair is active. It is
suggested that the spacings of the
drivers at their critical frequencies
should be in the range of 0.4 to 0.6 of
one wavelength in order to achieve this.
A spacing of 0.5 wavelength is near
optimum and gives a polar pattern with
no side lobes and a vertical beam width
of around 84 degrees. The authors found
that simplifying the design technique, so
as to restrict the flattened off-axis
response to the 6-dB-down level
compared with on axis, defined the
beamwidth, and that forcing the 6-dBdown off-axis frequency response to be
flat forced the beamwidth to be constant
in the same frequency region.
SUMMARY
Based on the summary of these papers,
it can be seen that there is considerable
potential these days to compensate for
both linear and nonlinear characteristics
of loudspeakers by using digital signal
processing. One might be able to
employ drive units previously considered unusable or inappropriate and use
signal processing to make the resulting
sound more acceptable. The relative
costs of improving drive units and of
signal processing are beginning to reach
the point where it may be more economical to employ signal processing
than to improve the mechanics or
acoustics of the physical loudspeaker,
although there are still things that cannot be adequately dealt with using signal processing. Such techniques may
lead to the possibility for more compact
transducers that have previously
unheard of sonic performance, although
the laws of physics naturally still limit
what can be achieved in this way.
J. Audio Eng. Soc., Vol. 56, No. 1/2, 2008 January/February