Proc. of the EAA Joint Symposium on Auralization and Ambisonics, Berlin, Germany, 3-5 April 2014
EVALUATING THE ACCURACY OF THE AMBISONIC REPRODUCTION OF MEASURED
SOUNDFIELDS
Sam Clapp
Graduate Program in Architectural Acoustics
Rensselaer Polytechnic Institute
Troy, New York, USA
Anne Guthrie
Arup Acoustics
New York, NY
Jonas Braasch
Graduate Program in Architectural Acoustics
Rensselaer Polytechnic Institute
Troy, New York, USA
Ning Xiang
Graduate Program in Architectural Acoustics,
Rensselaer Polytechnic Institute
Troy, New York, USA
ABSTRACT
A spherical microphone array can encode a measured soundfield
into its spherical harmonic components. Such an array will be sub-
ject to limitations on the highest spherical harmonic order it can
encode and encoding accuracy at different frequencies. Ambison-
ics is a system designed to reproduce the spherical harmonic com-
ponents of a measured or virtual soundfield using multiple loud-
speakers. In ambisonic systems, the size of the sweet spot is wave-
length dependent, and thus decreases in size with an increase in
frequency. This paper examines how to reconcile the limitations
of the recording and playback stages to arrive at the optimum am-
bisonic decoding scheme for a given spherical array design. In
addition, binaural models are used to evaluate these systems per-
ceptually.
1. INTRODUCTION
Spherical microphone arrays have been studied extensively for beam-
forming [1, 2], source localization, and other applications. Like-
wise, much theoretical and practical work has been done on the
optimum methods for ambisonic decoding [3, 4, 5]. Both systems
use the concept of spherical harmonics: spherical microphone ar-
rays can decompose a soundfield into its spherical harmonic com-
ponents, while ambisonic decoding can reconstruct the spherical
harmonic components of a soundfield for a listener.
The performance of spherical microphone arrays is frequency-
dependent, affected primarily by the array’s size and number of
sensors. In many applications, such as beamforming and direction-
of-arrival (DOA) estimation, a narrow frequency band is used,
where the array’s performance is optimum. However, restricting
oneself to a narrow frequency band is untenable when presenting
auditory scenes to listeners. Thus, we require a way to utilize infor-
mation from outside of the optimum band, where higher spherical
harmonic orders might not be accurately decomposed, but lower
orders are.
Much of the literature on ambisonics deals with the reproduc-
tion of simulated soundfields, where the exact DOA of every sound
event is known, and the restrictions on the highest spherical har-
monic order come only from the number of channels in the loud-
speaker array.
This paper examines a method for determining mixed-order
ambisonic decoding schemes determined by the constraints of the
two systems, as well as a way to evaluate the perceptual accuracy
of these schemes using binaural models.
2. SPHERICAL HARMONICS
The homogeneous wave equation is given in its general form by:
∇2p=1
c2
∂2p
∂t2.(1)
If expressed in spherical coordinates, solutions can be obtained
through a separation of variables, yielding sets of functions for the
radial, azimuthal, and elevation components [6]. Solutions to the
azimuthal component are given by either sine and cosine terms or
complex exponentials, while solutions to the elevation component
are given by the associated Legendre functions (Pm
n(x)) in the co-
sine of the elevation angle. Combining these two components (and
adding a normalization term) yields the (complex-valued) expres-
sion for spherical harmonics:
Ym
n(θ, φ) = s2n+ 1
4π
(n−m)!
(n+m)!Pm
n(cos θ)eimφ.(2)
(The real-valued expression uses sine and cosine terms for the az-
imuthal component.)
The spherical harmonics form an orthonormal basis on the
sphere:
Z2π
0Zπ
0
Ym0
n0(θ, φ)∗Ym
n(θ, φ) sin θ dθ dφ =δnn0δmm0.(3)
3. SPHERICAL MICROPHONE ARRAY PROCESSING
3.1. Spherical Harmonic Decomposition
When a plane wave impinges upon a rigid sphere, the sphere will
radiate a spherical wave whose intensity will vary as a function
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Proc. of the EAA Joint Symposium on Auralization and Ambisonics, Berlin, Germany, 3-5 April 2014
0.1 0.5 1 5 10
−40
−35
−30
−25
−20
−15
−10
−5
0
ka
Amplitude (dB)
n=0
n=1
n=2
n=3
n=4
n=5
n=6
Figure 1: Modal amplitude for orders zero through six, as a func-
tion of ka, a quantity that relates the wavelength of the incoming
plane wave to the radius of the spherical array.
of the incident wave’s angle of incidence and wavelength (in re-
lation to the radius of the sphere). It is shown in [6] and [7] that
by solving the wave equation with the appropriate boundary con-
ditions, we arrive at an expression for the pressure at a point on
a rigid sphere of radius a(denoted by its angular position (θ, φ))
due to a plane wave incident from (θi, φi)with amplitude P0and
wavenumber k= 2πf/c:
p(θ, φ, ka) = 4πP0
∞
X
n=0
inbn(ka)
n
X
m=−n
Ym
n(θ, φ)Ym
n(θi, φi)∗,
(4)
with
bn(ka) = jn(ka)−j0
n(ka)
h(1)0
n(ka)h(1)
n(ka),(5)
a quantity referred to as the modal amplitude, which is shown for
several orders in Fig. 1. The slope of each curve is 3 dB per octave
multiplied by the order.
Now, if we apply a weighting factor Wto each point on the
sphere:
Wm0
n0(θ, φ, ka) = Ym0
n0(θ, φ)∗
4πin0bn0(ka),(6)
and then integrate over the entire sphere, we can use the orthonor-
mality of the spherical harmonics (Eq. 3) to yield the following
result:
Z2π
0Zπ
0
Wm0
n0(θ, φ, ka)p(θ, φ, ka) sin θ dθ dφ =
P0Ym0
n0(θi, φi)∗.(7)
Thus, we can determine the spherical harmonic components of the
incident plane wave. However, evaluating the integral in Eq. 7 re-
quires a continuous spherical transducer, and in practice, we must
sample the pressure at Qdiscrete points on the sphere (denoted by
their angular positions (θq, φq)), leading to the following summa-
tion:
Q
X
q=1
Wm0
n0(θq, φq, ka)p(θq, φq, ka)Cm
n(θq, φq)≈
P0Ym0
n0(θi, φi)∗,(8)
where Cm
nare the quadrature coefficients. Using a nearly uniform
sampling scheme, we can estimate the spherical harmonic compo-
nents up to order Nsuch that Q≥(N+ 1)2.
1 2 3 4 5
16
31.5
63
125
250
500
1k
2k
4k
8k
Sphere Radius (cm)
Frequency (Hz)
WNG = −20 dB
1 2 3 4 5
500
1k
2k
4k
8k
16k
Sphere Radius (cm)
Frequency (Hz)
Aliasing Frequency
7th Order
6th Order
5th Order
4th Order
3rd Order
2nd Order
1st Order
Figure 2: Left side: frequency at which WNG = -20 dB as a func-
tion of sphere radius for a 64-channel array at different spherical
harmonic orders. Right side: frequency at which aliasing errors
occur as a function of sphere radius for different spherical har-
monic orders.
3.2. Error Sensitivity
Spherical microphone arrays lend themselves well to beamform-
ing, a set of techniques for multi-channel sensors that allow for
spatial filtering [2, 8, 9, 10]. These techniques involve solving for
a set of weighting factors that are applied to each channel on the
array, allowing the array to “look” in a particular direction at a par-
ticular frequency. The robustness of the beamformer is given by a
quantity known as the white noise gain (WNG):
WNG(θ0, φ0, θq, φq, ka) = 10 log10 |dTW|2
WHW,(9)
where dTis a column vector of the microphone pressure values
due to a plane wave impinging on the sphere from some direction
(θi, φi), and Wthe sensor weights to look in that direction. WNG
represents the array’s sensitivity to noise and microphone position-
ing errors, with negative values representing an amplification and
positive values representing an attenuation of (spatially uncorre-
lated) white noise. A spherical array is most sensitive to these
types of errors at lower frequencies, where higher order spherical
harmonic components are low in level and must be amplified con-
siderably. The lefthand portion of Fig. 2 shows the frequencies
at which WNG = -20 dB for a 64-channel array of varying radius.
This value can be used as a guideline to determine the lowest fre-
quency at which a certain order of spherical harmonic components
can be accurately measured by a given array.
3.3. Aliasing
At higher frequencies, the aliasing of higher order spherical har-
monic components into lower orders becomes an issue. We can
see from Eq. 4 that a plane wave is not spherical harmonic order-
limited and from Eq. 5 that higher-order components become more
prominent at higher frequencies. Thus, aliasing errors affect the ar-
ray for frequencies such that ka > N, where ais the radius of the
sphere and Nis the highest spherical harmonic order that can be
measured by the array [11]. This frequency is shown in the right-
hand portion of Fig. 2 for a range of sphere radii and spherical
harmonic orders, with arrays of smaller radii and higher order (i.e.
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Proc. of the EAA Joint Symposium on Auralization and Ambisonics, Berlin, Germany, 3-5 April 2014
with a greater number of channels) offering higher thresholds for
aliasing.
4. AMBISONICS
4.1. Basic Decoding
Ambisonics is a system that uses multiple loudspeakers to synthe-
size a soundfield, where the gains of the loudspeakers are deter-
mined based on the spherical harmonic expansion of the soundfield
[12, 3, 4, 5].
Let us start with a plane wave of wavenumber kiand incident
from the direction (θi, φi), defined by its spherical harmonic coef-
ficients as:
p=eikir= 4π
∞
X
n=0
injn(kr)
n
X
m=−n
Ym
n(θ, φ)Ym
n(θi, φi)∗.
(10)
If we want to synthesize this plane wave with Lplane wave sources
located at (θl, φl)for 1≤l≤L, each with an amplitude of wl,
then our expression for the synthesized field ˆpis:
ˆp= 4π
∞
X
n=0
injn(kr)
n
X
m=−n
Ym
n(θ, φ)
L
X
l=1
wlYm
n(θl, φl)∗.
(11)
Then we solve for each nand min order to have Eq. 11 be equal
to Eq. 10:
L
X
l=1
wlYm
n(θl, φl)∗=Ym
n(θi, φi)∗.(12)
These amplitudes wlcan be solved for up to order N, subject to
the requirement that L≥(N+1)2for 3-D arrays and L≥2N+1
for horizontal arrays. This is known as basic decoding.
4.2. Wavefield Error Analysis
One method of evaluating the quality of the reconstruction is by
calculating the normalized radial error between the synthesized
sound field ˆpand the sound field being reconstructed, p, given by:
¯(kr) = R2π
0Rπ
0|p−ˆp|2sin θ dθ dφ
R2π
0Rπ
0|p|2sin θ dθ dφ .(13)
This quantity is a function of the distance from the center of the ar-
ray as related to the wavelength one is examining, expressed as kr,
and is shown for 1st through 7th order (moving from left to right)
in the top portion of Fig. 3. A given value of kr will be a larger dis-
tance from the center of the array for a lower frequency (i.e. longer
wavelength) than for a higher frequency. Thus, the sweet spot will
be larger for lower frequencies than for higher frequencies.
One way to evaluate the performance of ambisonic systems in
terms of human perception is to examine the frequency at which
the radius of the sweet spot (as determined by some threshold in
the normalized radial error) becomes smaller than the radius of
the average human head (given in [4] as 8.9 cm). As the order of
ambisonic reproduction increases, so will this frequency, as shown
in the bottom portion of Fig. 3, for several different thresholds of
normalized radial error. Thus, as higher spherical components are
added to the reconstruction, higher frequencies are reconstructed
accurately at the two ears.
0 2 4 6 8 10 12
ï30
ï20
ï10
0
Distance from Center (kr)
Normalized Radial Error (dB)
1234567
500
1k
2k
4k
Ambisonic Reconstruction Order
Threshold Frequency (Hz)
ï10 dB
ï15 dB
ï20 dB
Figure 3: The top shows normalized radial error (in dB) as a func-
tion of distance from the center of the array (in units of kr), from
1st to 7th order basic decoding (moving left to right), after Ward
and Abhayapala[13] and Poletti[4]. The bottom shows the fre-
quencies at which the -10, -15, and -20 dB error thresholds occur
at a radius of 8.9 cm, the size of a typical human head.
4.3. Max-rEDecoding
For a given order of ambisonic reproduction, there will be some
frequency above which the area of the sweet spot will be smaller
than the size of a typical human head. Thus, audio content above
this frequency will not be simulated accurately at the listener’s
ears. In [14], Gerzon proposed that high frequency localization
can be predicted by the direction of the energy vector at the center
of the array, ˆrE, as given in:
rEˆrE=PL
l=1 w2
lˆul
PL
l=1 w2
l
,(14)
where wlis the gain of the lth loudspeaker radiating sound from a
position denoted by the vector ˆulfrom the loudspeaker’s position
to the center of the array. The magnitude of the vector, rE, repre-
sents the concentration of source energy in the desired direction,
and thus the accuracy of high-frequency localization from that di-
rection. In order to maximize this value, one can apply correcting
gains (g0, g1, ..., gN)to each order of spherical harmonics (i.e. to
the right side of Eq. 12.) Methods for calculating the correcting
gains are given in [3]. This is known as max-rEdecoding.
4.4. Binaural Cue Analysis
The localization accuracy of different decoding schemes can be
examined using a model of the auditory system used previously in
[15] to evaluate the localization of various stereo recording tech-
niques. The auditory periphery is simulated with Head-Related
Transfer Functions (HRTFs). The behavior of the basilar mem-
brane and the hair cells is simulated with a gammatone filter bank
with 72 bands and half-wave rectification. ITD analysis is per-
formed using an interaural cross-correlation in each frequency band.
ILD analysis is performed using an array of excitation/inhibition
(EI) cells. This model allows for the creation of maps that corre-
171
Proc. of the EAA Joint Symposium on Auralization and Ambisonics, Berlin, Germany, 3-5 April 2014
late ITD and ILD values (expressed in milliseconds and decibels,
respectively) with azimuthal directions.
This allows for another way to evaluate the accuracy of am-
bisonic reproduction - by calculating errors in the ITD and ILD
cues rather than errors in the reproduced wavefield, as detailed ear-
lier. First, the spherical harmonic signals are decoded to a virtual
24-channel horizontal loudspeaker array (with equiangular spac-
ing, putting each channel 15 degrees apart in azimuth from its
neighbors.) The signals reaching each ear are obtained by con-
volving the loudspeaker feeds with the appropriate HRTFs and
summing over all loudspeakers. Plane waves can then be simu-
lated from a variety of directions, and the ITD and ILD cues can
be compared to the natural condition.
From these maps we can then look at average binaural cue er-
ror as a function of frequency by averaging over azimuthal angles
and also over a number of different HRTF catalogs, as measured in
[16]. This is shown in Fig. 4 for 1st through 7th order ambisonic
rendering. The dotted lines indicate the average error across all
frequency bands. The results are in line with what we would ex-
pect from an analysis of the wavefields:
1. Cue errors are less at lower frequencies
2. The higher the order, the higher the frequency at which sig-
nificant cue errors occur
3. The higher the order, the lower the average cue error across
all frequency bands
5. BAND-SPLITTING FILTERS
As discussed in the previous section, it is possible to achieve more
perceptually accurate decoding by using basic decoding in lower
frequencies and max-rEdecoding in higher frequencies. This ne-
cessitates a crossover network to transition between the different
decoding schemes. One popular crossover is the Linkwitz-Riley
(LR) crossover [17]. A second-order LR crossover is formed from
cascading two first-order Butterworth filters, a fourth-order LR
crossover from two second-order Butterworth filters, etc. The ben-
efit of the LR crossover is that the low- and high-pass portions
are phase-matched, and the magnitude of each portion is -6 dB at
the crossover frequency, leading to a unity gain for the sum. The
rolloff is equal to 6 dB per octave multiplied by the order.
Depending on the characteristics of the particular systems be-
ing used, one might want to decode the signals in three or more
bands, requiring multiple crossovers. Putting two crossovers too
close to one another in frequency results in a summed signal whose
magnitude deviates noticeably from unity gain. These problems
can be avoided by placing crossovers far enough apart in frequency
so that the magnitudes of the low-pass and high-pass components
are equal at a maximum value of -20 dB.
6. PROCESSING MEASURED SOUNDFIELDS FOR
PLAYBACK
In this section, four hypothetical spherical microphone arrays are
considered: two 16-channel and two 64-channel rigid spherical ar-
rays, each set with radii of 2.5 and 5 cm. These arrays utilize a
nearly-uniform sampling scheme, with the positions of the sen-
sors given in [18]. Therefore, the 16-channel arrays are capable
of decomposing the spherical harmonics up to third order, the 64-
channel arrays up to seventh order.
Fig. 5 shows the White Noise Gain thresholds (the lower line
corresponding to a WNG value of -30 dB, the upper to -20 dB) and
aliasing frequencies, plotted together with the thresholds for tran-
sitioning from basic to max-rEdecoding. (Note that the decoding
thresholds are based on the size of a typical human head, and thus
are not affected by the properties of the microphone array.) Over-
laid on these plots are the decoding schemes chosen based on the
principles outlined previously, namely:
1. Do not utilize spherical harmonic components below a WNG
of -20 to -30 dB
2. Do not decode above the spherical harmonic aliasing fre-
quency
3. Use max-rEdecoding at frequencies above which the nor-
malized radial error reaches -10 to -20 dB
4. Separate multiple crossovers by at least 20 dB.
As before, where we compared different orders of ambison-
ics, we can evaluate the accuracy of the ITD and ILD cues of these
various decoding schemes (of “mixed" order) across multiple fre-
quency bands, as shown in Fig. 6 (with the plots of 1st and 5th
order ambisonics shown for reference). The average errors across
all frequency bands shown in Table 1.
Table 1: Average ITD and ILD cue error and aliasing frequency
for each spherical microphone array.
Array
Avg. ITD
error (ms)
Avg. ILD
error (dB)
Aliasing
Frequency (Hz)
a=2.5 cm, Q=16 0.24 4.3 6551
a=5 cm, Q=16 0.21 4.1 3275
a=2.5 cm, Q=64 0.20 4.1 15285
a=5 cm, Q=64 0.17 3.8 7643
There are two main points that we can gather from Fig. 6. The
first is that increasing the number of channels for a spherical mi-
crophone array of a given radius is a “win-win” scenario: higher
spherical harmonic orders become available, the WNG threshold
frequencies for lower orders are pushed lower in frequency, and
the aliasing frequency moves higher, yielding a wider frequency
range for reconstruction. Thus, we see improvements in both lo-
calization accuracy and bandwidth. Of course, increasing the num-
ber of channels will increase the cost and complexity of building
the array. In addition, going from a 3rd order to a 7th order micro-
phone array does not yield the same gains in localization accuracy
as moving from 3rd order to 7th order ambisonic reproduction of
a simulated soundfield, as the highest order components are some-
times available for only 1 or 2 octaves before encountering the
aliasing frequency.
Increasing the spherical array radius, however, involves a trade-
off. On the positive side, the WNG thresholds move lower, mak-
ing higher order spherical harmonic components available at lower
frequencies. However, the aliasing frequency also moves lower,
meaning that we are trading localization accuracy for bandwidth.
7. CONCLUSION
The common basis in spherical harmonics makes ambisonics a nat-
ural fit for reproducing soundfields measured with spherical mi-
crophone arrays. Incorporating higher order spherical harmonic
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Proc. of the EAA Joint Symposium on Auralization and Ambisonics, Berlin, Germany, 3-5 April 2014
31.5 63 125 250 500 1k 2k 4k 8k
0.1
0.2
0.3
0.4
0.5
0.6
Frequency (Hz)
ITD Error (ms)
ITD Errors by Frequency Band
1st
2nd
3rd
4th
5th
6th
7th
31.5 63 125 250 500 1k 2k 4k 8k
1
2
3
4
5
6
7
8
Frequency (Hz)
ILD Error (dB)
ILD Errors by Frequency Band
Figure 4: ITD and ILD error by frequency band for 1st through 7th order ambisonics, with average error over all frequency bands shown
by the dotted lines.
components offers the opportunity for more precise localization,
but at the same time introduces complexities at both the recording
stage and the playback stage that need to be dealt with. The goal of
this paper is to illuminate the sources of those issues and develop
a framework to resolve them, particularly with respect to auditory
perception.
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63
250
1k
4k
16k
Frequency (Hz)
a = 2.5 cm, Q = 16
123
63
250
1k
4k
16k
Order
Frequency (Hz)
a = 5 cm, Q = 16
a = 2.5 cm, Q = 64
1 2 3 4 5 6 7
Order
a = 5 cm, Q = 64
WNG Thresholds
Decoding Thresholds
Aliasing Frequency
1st Order Basic
2nd Order Max-rE
1st Order Basic
2nd Order Basic
3rd Order Max-rE
1st Order Basic
2nd Order Basic
4th Order Max-rE
1st Order Basic
2nd Order Basic
5th Order Max-rE
Figure 5: WNG thresholds and aliasing frequencies for 4 spherical arrays, plotted together with basic/max-rEambisonic decoding thresh-
olds, overlaid with decoding schemes.
31.5 63 125 250 500 1k 2k 4k 8k
0.1
0.2
0.3
0.4
0.5
0.6
ITD Errors by Frequency Band
Frequency (Hz)
ITD Error (samples)
a = 2.5 cm, Q = 16
a = 5 cm, Q = 16
a = 2.5 cm, Q = 64
a = 5 cm, Q = 64
31.5 63 125 250 500 1k 2k 4k 8k
1
2
3
4
5
6
7
8
ILD Errors by Frequency Band
Frequency (Hz)
ILD Error (dB)
Figure 6: ITD and ILD errors for the 4 different decoding schemes, with the errors for 1st and 5th order ambisonic decoding shown for
reference.
174
