Design Framework for Spherical Microphone and Loudspeaker Arrays in a
Multiple-input Multiple-output System

Hai Morgenstern1,111e-mail<EMAIL_ADDRESS>Boaz Rafaely1, and Markus
Noisternig2

1Department of Electrical and Computer Engineering,

Ben Gurion University of the Negev, Beer- Sheva, 84105, Israel

2 Acoustic and Cognitive Spaces Group,

IRCAM, CNRS, Sorbonne University, UPMC Paris 6, UMR 9912, STMS,

75004 Paris, France

Running title: Design framework for spherical MIMO system

###### Abstract

Spherical microphone arrays (SMAs) and spherical loudspeaker arrays (SLAs)
facilitate the study of room acoustics due to the three-dimensional analysis
they provide. More recently, systems that combine both arrays, referred to as
multiple-input multiple-output (MIMO) systems, have been proposed due to the
added spatial diversity they facilitate. The literature provides frameworks
for designing SMAs and SLAs separately, including error analysis from which
the operating frequency range of an array is defined. However, such a
framework does not exist for the joint design of an SMA and an SLA that
comprise a MIMO system. This paper develops a design framework for MIMO
systems based on a model that addresses errors and highlights the importance
of a matched design. Expanding on a free-field assumption, errors are
incorporated separately for each array and error bounds are defined,
facilitating error analysis for the system. The dependency of the error bounds
on the SLA and SMA parameters is studied and it is recommended that parameters
should be chosen to assure matched operating frequency ranges of the arrays in
MIMO system design. A design example is provided, demonstrating the
superiority of a matched system over an unmatched system in the synthesis of
directional room impulse responses.

##  1 INTRODUCTION

Acoustic systems that employ an array (either a microphone array or a
loudspeaker array) have been shown to be advantageous in a variety of
applications compared to systems that employ a single microphone or a single
loudspeaker. In room acoustics, spherical microphone arrays (SMAs) have been
recently proposed for measuring directional room impulse responses (RIRs) [1,
2, 3, 4, 5, 6]. These responses facilitate a 3-D analysis of the sound field
surrounding an array and the computation of spatial room-acoustics measures,
such as the directivity index (DI) and directional diffusion. The latter make
it possible to study enclosures with reference to human auditory perception.
Similarly, loudspeaker arrays and directional loudspeakers have been proposed
for studying room acoustics, due to their ability to radiate acoustic energy
to selected directions in space [7, 8, 9]. This ability enables the excitation
of individual reflection paths and improves the separability of reflections
when analyzing an RIR, compared to an omnidirectional loudspeaker. In
particular, spherical loudspeaker arrays (SLAs) use a compact arrangement of
loudspeakers mounted on a sphere to produce complex radiation patterns in 3-D
space [10, 11, 12]. Employing a compact SLA with variable directivity has also
been shown to lead to perceivable differences in the sound field, also
affecting source localization [13]. In general, the performance of spherical
arrays, either an SMA or an SLA, is limited by errors introduced by the
following factors [14, 15]. The use of a finite number of elements in each
array results in spatial-aliasing errors, which impose limitations at high
frequencies. Also, inaccuracies in the modeling of the arrays and in the
positioning of the array elements introduce errors at all frequencies. While
these errors are typically larger at high frequencies (due to the shorter
wavelength), they also become dominant at low frequencies due to the
employment of various methods for enhancing spatial resolution, such as plane-
wave decomposition. These methods introduce ill-conditioning at low
frequencies [3], imposing performance limitations at these frequencies [16,
17, 14, 18]. An analysis of all errors can be used for defining an operating
frequency range (OFR) of an array, i.e., defining the operating range of
frequencies in which an array performs well without introducing significant
errors. This analysis facilitates the design of SLAs and SMAs separately, and
the design and implementation of beamforming methods using these arrays.

Until recently, SLAs and SMAs were mainly used independently. Systems that
include both, referred to as acoustic multiple-input multiple-output (MIMO)
systems, are relatively new, and the added spatial diversity that these
systems provide could potentially improve the spatial analysis of room
acoustics compared to systems with a single array. MIMO systems have been
previously proposed for room acoustics analysis and three-dimensional
auralization of concert halls [19, 20, 21]. Although these works motivate the
use of acoustic MIMO systems, they neither present a theoretical study of MIMO
systems, nor of their potential performance. An elaborate theoretical
formulation of MIMO systems was recently presented, including an example for
the use of such a system in spatial analysis based on RIRs [22, 23]. Moreover,
additional examples of methods for room acoustic analysis that use MIMO
systems were presented in recent conference papers [24, 25]. However, previous
works do not provide an integrated framework for designing the arrays which
constitute a MIMO system.

In this paper it is shown that a MIMO system may become unusable due to
mismatch in the OFRs of the SLA and SMA. To overcome this problem, a framework
for the design of MIMO systems is developed, based on a model that includes
errors from both of the arrays. Initially, an error-free model is presented in
sec. 2. The model is extended, in sec. 3, to account for errors due to
sampling at the arrays and for modeling errors. Error bounds are then derived
in sec. 4, which are then used in sec. 5 for defining the system and the
arrays’ OFRs in a design example. An analysis of these bounds shows that the
arrays’ OFRs should be matched to achieve a useful OFR for the MIMO system. A
method is proposed in sec. 6 for extending the OFR of a MIMO system. The
superiority of a matched system over an unmatched system in the generation of
directional RIRs is demonstrated in a simulation study in sec. 7. Lastly,
conclusions are outlined in sec. 8.

##  2 System model

A model of an acoustic MIMO system in free-field is presented in this section.
For further details regarding the model formulation, a study of its
properties, and its extension for systems positioned in enclosures, the reader
is referred to [22].

A MIMO system comprises an SLA with $S$ loudspeakers mounted around a rigid
sphere with a radius of $r_{L}$ and an SMA with $R$ microphones distributed
over a rigid sphere with a radius of $r_{M}$. Beamforming is applied to both
arrays with the aim of controlling the arrays’ directivity patterns. At the
SLA, this is achieved by weighting the SLA input signal, $s(k)$, where $k$ is
the wavenumber, with complex beamforming coefficients
$\gamma_{1}(k),..,\gamma_{S}(k)$ before driving the loudspeakers. At the SMA,
the microphones signals are weighted with complex beamforming coefficients
$\lambda_{1}(k),...,\lambda_{R}(k)$ and then summed to produce the system
output, $y(k)$. This is formulated as:

$\displaystyle y(k)$ $\displaystyle=$
$\displaystyle\tilde{\bm{\gamma}}^{\mathrm{T}}(k)\tilde{\bm{H}}(k)\tilde{\bm{\lambda}}(k)\,s(k),$
(1)

where

$\displaystyle\tilde{\bm{\gamma}}(k)$ $\displaystyle=$
$\displaystyle[\gamma_{1}(k),\gamma_{2}(k),...,\gamma_{S}(k)]^{\mathrm{T}},$
(2) $\displaystyle\tilde{\bm{\lambda}}(k)$ $\displaystyle=$
$\displaystyle[\lambda_{1}(k),\lambda_{2}(k),...,\lambda_{R}(k)]^{\mathrm{T}},$
(3)

and $(\cdot)^{\mathrm{T}}$ is the transpose operator. $\tilde{\bm{H}}(k)$ is
the $S\times R$ system transfer matrix whose elements hold the transfer
functions for all loudspeaker and microphone combinations. For a system block
diagram representing Eq. (1), the reader is referred to Fig. 1. Under free-
field conditions and a far field assumption, $\tilde{\bm{H}}(k)$ can have a
simple form [22]:

$\displaystyle\tilde{\bm{H}}(k)$ $\displaystyle=$
$\displaystyle\tilde{\bm{h}}_{L}(k)\tilde{\bm{h}}_{M}^{\mathrm{H}}(k),$ (4)

where $(\cdot)^{\mathrm{H}}$ is the complex transpose operator.
$\tilde{\bm{h}}_{L}(k)$ is the $S\times 1$ SLA transfer vector from the input
of all loudspeakers to the sound pressure at the center of the SMA, and
$\tilde{\bm{h}}_{M}(k)$ the $R\times 1$ SMA transfer vector from the sound
pressure at the center of the SMA to the output of all microphones [22].
$\tilde{\bm{h}}_{L}(k)$ can be decomposed as follows [22]:

$\displaystyle\tilde{\bm{h}}_{L}(k)$ $\displaystyle=$
$\displaystyle{\bm{Y}}_{L}\tilde{\bm{G}}(k)\tilde{\bm{\psi}}_{L}(k),$ (5)

with matrix ${\bm{Y}}_{L}$ depending on the angles of the loudspeakers, matrix
$\tilde{\bm{G}}(k)$ representing the radiation of the loudspeaker cap, and
$\tilde{\bm{\psi}}_{L}(k)$ relating the centers of the SLA and SMA. In
particular, ${\bm{Y}}_{L}$ is an $S\times(\tilde{N}_{L}+1)^{2}$ matrix, given
by:

$\displaystyle{\bm{Y}}_{L}$ $\displaystyle=$
$\displaystyle[\bm{y}_{\tilde{N}_{L}}(\bm{\eta}_{1}),\bm{y}_{\tilde{N}_{L}}(\bm{\eta}_{2}),...,\bm{y}_{\tilde{N}_{L}}(\bm{\eta}_{S})],$
(6)

with

$\displaystyle\bm{y}_{\tilde{N}_{L}}(\bm{\eta})$ $\displaystyle=$
$\displaystyle[Y_{0}^{0}(\bm{\eta}),Y_{1}^{-1}(\bm{\eta}),...,Y_{\tilde{N}_{L}}^{\tilde{N}_{L}}(\bm{\eta})]^{\mathrm{T}},$
(7)

being an $(\tilde{N}_{L}+1)^{2}\times 1$ vector that holds spherical harmonics
(SH), $\\{Y_{n}^{{m}}(\bm{\eta})\\}$, evaluated at elevation angle $\theta$
and azimuth angle $\phi$ with respect to the SLA center, written shortly as
$\bm{\eta}=(\theta,\phi)$. The SH order $\tilde{N}_{L}$ should tend to
infinity for an exact representation of the sound field [22]. However, in
practice a finite order is chosen, as will be discussed in more detail in sec.
5. $\bm{\eta}_{1},\bm{\eta}_{2},...,\bm{\eta}_{S}$ are the spatial angles that
point at the loudspeakers.

$\displaystyle\tilde{\bm{G}}(k)$ $\displaystyle=$
$\displaystyle\text{diag}[g_{0}(kr_{L}),g_{1}(kr_{L}),...,g_{\tilde{N}_{L}}(kr_{L})]$
(8)

is an $(\tilde{N}_{L}+1)^{2}\times(\tilde{N}_{L}+1)^{2}$ diagonal matrix that
holds coefficients $g_{n}(kr_{L})$ that depend on spherical Hankel and Bessel
functions, the SLA array type, its radius, and the size of the loudspeaker cap
[22], and

$\displaystyle\tilde{\bm{\psi}}_{L}(k)$ $\displaystyle=$
$\displaystyle\frac{e^{ikr_{0}}}{r_{0}}\bm{y}_{\tilde{N}_{L}}(\bm{\eta}_{0}),$
(9)

is an $(\tilde{N}_{L}+1)^{2}\times 1$ vector, with $\bm{\eta}_{0}$ being the
angles that point at the SMA center, where $i^{2}=-1$, and $r_{0}$ is the
distance between the centers of the arrays.

For the SMA, $\tilde{\bm{h}}_{M}(k)$ can be decomposed as follows:

$\displaystyle\tilde{\bm{h}}_{M}(k)$ $\displaystyle=$
$\displaystyle{\bm{Y}}_{M}^{*}\tilde{\bm{B}}(k)\tilde{\bm{\psi}}_{M},$ (10)

where $(\cdot)^{*}$ is the complex conjugate operator. ${\bm{Y}}_{M}$ is an
$R\times(\tilde{N}_{M}+1)^{2}$ matrix, given by:

$\displaystyle{\bm{Y}}_{M}$ $\displaystyle=$
$\displaystyle[\bm{y}_{\tilde{N}_{M}}(\bm{\beta}_{1}),\bm{y}_{\tilde{N}_{M}}(\bm{\beta}_{2}),...,\bm{y}_{\tilde{N}_{M}}(\bm{\beta}_{R})],$
(11)

with $\bm{y}_{\tilde{N}_{M}}(\bm{\beta})$, an $(\tilde{N}_{M}+1)^{2}\times 1$
defined as in Eq. (7), but for elevation and azimuth angles
$\bm{\beta}=(\xi,\omega)$ and SH order $\tilde{N}_{M}$. $\tilde{N}_{M}$ should
tend to infinity for an exact representation of the sound field [22]. As in
the case of the SLA, a finite order is chosen, as will be discussed in sec. 5.
$\bm{\beta}_{1},\bm{\beta}_{2},...,\bm{\beta}_{R}$ are the spatial angles that
point at the microphones.

$\displaystyle\tilde{\bm{B}}(k)$ $\displaystyle=$
$\displaystyle\text{diag}[b_{0}(kr_{M}),b_{1}(kr_{M}),...,b_{\tilde{N}_{M}}(kr_{M})]$

is an $(\tilde{N}_{M}+1)^{2}\times(\tilde{N}_{M}+1)^{2}$ diagonal matrix that
holds coefficients $b_{n}(kr_{M})$ that depend on spherical Hankel and Bessel
functions, the SMA array type, and its radius [22], and

$\displaystyle\tilde{\bm{\psi}}_{M}$ $\displaystyle=$
$\displaystyle\bm{y}_{\tilde{N}_{M}}(\bm{\beta}_{0}),$ (13)

is an $(\tilde{N}_{M}+1)^{2}\times 1$ vector, with $\bm{\beta}_{0}$ being the
angles that point at the SLA center. $\bm{\eta}_{0}$ and $\bm{\beta}_{0}$ are
referred to as the direction of radiation (DOR) and direction of arrival
(DOA), respectively. A schematic illustration of the system is presented in
Fig. 2. For ease of illustration, the coordinate system of the SMA has been
rotated.

An alternative representation of the system in the SH domain is simpler and
more appropriate for spatial analysis with spherical arrays [22]. Transfer
functions are formulated between $(N_{L}+1)^{2}$ SH channels of the SLA and
$(N_{M}+1)^{2}$ SH channels of the SMA, instead of between the $S$
loudspeakers and the $R$ microphones, respectively. $N_{L}$, the SH order of
the SLA, denotes the SH order that can be controlled with $S$ loudspeakers
and, therefore, typically maintains: $(N_{L}+1)^{2}\leq S$. Note that $N_{L}$
may be different than $\tilde{N}_{L}$, as defined earlier (cf. Eq. (5)).
Similarly for the SMA, $N_{M}$, the SMA SH order, denotes the SH order that
can be controlled with $R$ microphones, typically maintaining
$(N_{M}+1)^{2}\leq R$, and may be different to $\tilde{N}_{M}$ (cf. Eq. (10)).
The system output in this case is given by applying beamforming in the SH
domain as [22]:

$\displaystyle y(k)$ $\displaystyle=$
$\displaystyle\bm{\gamma}^{\mathrm{T}}(k)\bm{H}(k)\bm{\lambda}(k)\,s(k),$ (14)

where

$\displaystyle{\bm{\gamma}}(k)$ $\displaystyle=$
$\displaystyle[\gamma_{00}(k),\gamma_{1(-1)}(k),\gamma_{10},\gamma_{11}(k),...$
$\displaystyle...,\gamma_{N_{L}(N_{L}-1)}(k),\gamma_{N_{L}N_{L}}(k)]^{\mathrm{T}}\,\,\text{and}$
(15) $\displaystyle{\bm{\lambda}}(k)$ $\displaystyle=$
$\displaystyle[\lambda_{00}(k),\lambda_{1(-1)}(k),\lambda_{10}(k),\lambda_{11}(k),...$
(16)
$\displaystyle...,\lambda_{N_{M}(N_{M}-1)}(k),\lambda_{N_{M}N_{M}}(k)]^{\mathrm{T}},$

are the $(N_{L}+1)^{2}\times 1$ SLA beamforming coefficients vector and the
$(N_{M}+1)^{2}\times 1$ SMA beamforming coefficients vector, respectively.
$\bm{H}(k)$ is an $(N_{L}+1)^{2}\times(N_{M}+1)^{2}$ matrix given by:

$\displaystyle\bm{H}(k)$ $\displaystyle=$
$\displaystyle\bm{h}_{L}(k)\bm{h}_{M}^{\mathrm{H}}(k),$ (17)

where $\bm{h}_{L}(k)$ and $\bm{h}_{M}(k)$ are the $(N_{L}+1)^{2}\times 1$ SLA
and $(N_{M}+1)^{2}\times 1$ SMA transfer vectors, respectively, represented in
SH. $\bm{h}_{L}(k)$ is given as:

$\displaystyle\bm{h}_{L}(k)$ $\displaystyle=$
$\displaystyle\bm{G}(k)\bm{\psi}_{L}(k),$ (18)

with $\bm{G}(k)$ and $\bm{\psi}_{L}(k)$ defined as in Eqs. (8) and (9),
respectively, but for SH order $N_{L}$ instead of $\tilde{N}_{L}$. For the
SMA, $\bm{h}_{M}(k)$ is given by:

$\displaystyle\bm{h}_{M}(k)$ $\displaystyle=$
$\displaystyle\bm{B}(k)\bm{\psi}_{M},$ (19)

with $\bm{B}(k)$ and $\bm{\psi}_{M}$ defined as in Eqs. (2) and (13),
respectively, but for SH order $N_{M}$ instead of $\tilde{N}_{M}$.

##  3 Model extension - consideration of errors

The system model presented in the last section and in ref. [22] is useful;
however, it does not include the effects of spatial sampling and model
mismatch errors, which is necessary for this paper. Therefore, the model is
now extended such that it includes these errors. In particular, since
$\tilde{\bm{H}}(k)$ from Eq. (4) is represented as an outer product of the SLA
and SMA transfer vectors, the errors are added to each array separately.
First, model mismatch errors are incorporated in $\tilde{\bm{h}}_{L}(k)$ and
$\tilde{\bm{h}}_{M}(k)$ in sec. 3.1. Then, errors caused by spatial sampling
at the arrays are incorporated in secs. 3.2 and 3.3 for the SMA and SLA,
respectively. Finally, in sec. 3.4, normalization is applied to both of the
arrays, in an attempt to enhance their overall spatial resolution.

###  3.1 Model mismatch

Model-mismatch errors are caused by several factors. These include
uncertainties in the positions and frequency responses of loudspeakers and
microphones, inaccuracies in the model represented by the values of
$g_{n}(kr_{L})$ and $b_{n}(kr_{M})$ from Eqs. (8) and (2), respectively (which
may arise, for example, when the assumption of a rigid sphere is violated for
either the SLA or the SMA), and errors caused by computation with finite
numerical precision (such as in sound cards). While such errors can be modeled
accurately, given information on the sources of the errors, a generalized
additive error model is employed in this paper. Such a model facilitates a
matched design of a MIMO system that is independent of the exact model of the
errors, since these may vary for different conditions and systems. Moreover,
since the effect of model-mismatch errors has been shown to be similar to the
effect of transducer noise [14], this general error model also facilitates a
study of the system robustness against transducer noise.

Error vectors are added to the SMA and SLA transfer vectors, represented in
the spatial domain. At the SLA, the SLA transfer vector with error is written,
for simplicity, using the same notation as the SLA transfer vector without
error. $\tilde{\bm{h}}_{L}(k)$ from Eq. (5) is therefore updated to:

$\displaystyle\tilde{\bm{h}}_{L}(k)$ $\displaystyle=$
$\displaystyle{\bm{Y}}_{L}\tilde{\bm{G}}(k)\tilde{\bm{\psi}}_{L}(k)+\tilde{\bm{n}}_{L}(k),$
(20)

where $\tilde{\bm{n}}_{L}(k)$ is the $S\times 1$ SLA error vector, given by:

$\displaystyle\tilde{\bm{n}}_{L}(k)$ $\displaystyle=$
$\displaystyle[n^{L}_{1}(k),n^{L}_{2}(k),....,n^{L}_{S}(k)]^{\mathrm{T}},$
(21)

in which $n^{L}_{1}(k),n^{L}_{2}(k),...n^{L}_{S}(k)$ are assumed to be
independent and identically distributed (i.i.d.). In particular, the variances
of these errors can be set with respect to the average power of the
loudspeaker transfer functions; this will be discussed in sec. 5. Similarly
for the SMA, the notation of $\tilde{\bm{h}}_{M}(k)$ is kept for simplicity
and Eq. (10) is updated to:

$\displaystyle\tilde{\bm{h}}_{M}(k)$ $\displaystyle=$
$\displaystyle{\bm{Y}}_{M}\tilde{\bm{B}}(k)\tilde{\bm{\psi}}_{M}+\tilde{\bm{n}}_{M}(k),$
(22)

where $\tilde{\bm{n}}_{M}(k)$ is the $R\times 1$ SMA error vector given by:

$\displaystyle\tilde{\bm{n}}_{M}(k)$ $\displaystyle=$
$\displaystyle[n^{M}_{1}(k),n^{M}_{2}(k),....,n^{M}_{R}(k)]^{\mathrm{T}},$
(23)

in which $n^{M}_{1}(k),n^{M}_{2}(k),...n^{M}_{R}(k)$ are, as in the case of
the SLA, assumed to be i.i.d.

###  3.2 Spatial aliasing at the SMA

Spatial sampling may introduce aliasing errors that are non-negligible at high
frequencies [14]. For microphone arrays, spatial sampling is the process in
which the microphones that comprise an array sample the sound field at the
array surface. In particular for SMAs, an important feature of sampling is to
facilitate the computation of the spherical Fourier transform (SFT) of the
sound field on the array’s surface, which gives its SH representation. The
computation of the SFT is performed by applying sampling weights to the SMA
elements [1, 14, 34]. Applying the weights on $\tilde{\bm{h}}_{M}(k)$ from Eq.
(22), a new SMA transfer vector $\hat{\bm{h}}_{M}(k)$, represented in the SH
domain, can be defined be as:

$\displaystyle\hat{\bm{h}}_{M}(k)$ $\displaystyle=$
$\displaystyle\bm{\alpha}_{M}\left[{\bm{Y}}_{M}\tilde{\bm{B}}(k)\tilde{\bm{\psi}}_{M}+\tilde{\bm{n}}_{M}(k)\right],$
(24)

where

$\displaystyle\bm{\alpha}_{M}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\alpha_{1,00}^{M}&...&\alpha_{R,00}^{M}\\\
\vdots&\ddots&\vdots\\\
\alpha_{1,N_{M}N_{M}}^{M}&...&\alpha_{R,N_{M}N_{M}}^{M}\end{bmatrix}$ (25)

is an $(N_{M}+1)^{2}\times R$ matrix that holds the SMA sampling weights,
leading to $\hat{\bm{h}}_{M}(k)$ that is of dimensions $(N_{M}+1)^{2}\times
1$. When sampling weights that are ideal for SH order $N_{M}$ are chosen, the
following identity holds [14]:

$\displaystyle\bm{\alpha}_{M}{\bm{Y}}_{M}$ $\displaystyle=$
$\displaystyle[\bm{I}_{M}\,\,\bm{\epsilon}_{M}],$ (26)

where $\bm{I}_{M}$ is the $(N_{M}+1)^{2}\times(N_{M}+1)^{2}$ identity matrix
and $\bm{\epsilon}_{M}$ is an
$(N_{M}+1)^{2}\times[(\tilde{N}_{M}+1)^{2}-(N_{M}+1)^{2}]$ matrix that holds
the SMA aliasing function up to order $\tilde{N}_{M}$ [14]. Using this
identity and the definition from Eq. (19), Eq. (24) can be rewritten as:

$\displaystyle\hat{\bm{h}}_{M}(k)$ $\displaystyle=$
$\displaystyle[\bm{I}_{M}\,\,\bm{\epsilon}_{M}]\tilde{\bm{B}}(k)\tilde{\bm{\psi}}_{M}+\bm{\alpha}_{M}\tilde{\bm{n}}_{M}(k)$
(27) $\displaystyle=$
$\displaystyle[\bm{I}_{M}\,\,{\bm{0}}_{M}]\tilde{\bm{B}}(k)\tilde{\bm{\psi}}_{M}+[\tilde{\bm{0}}_{M}\,\,\bm{\epsilon}_{M}]\tilde{\bm{B}}(k)\tilde{\bm{\psi}}_{M}+$
$\displaystyle+\bm{\alpha}_{M}\tilde{\bm{n}}_{M}(k)$ $\displaystyle=$
$\displaystyle\bm{B}(k)\bm{\psi}_{M}+\bm{\epsilon}_{M}\tilde{\bm{z}}_{M}(k)+\bm{\alpha}_{M}\tilde{\bm{n}}_{M}(k),$

where ${\bm{0}}_{M}$ and $\tilde{\bm{0}}_{M}$ are matrices of zeros with the
same dimensions as those of $\bm{\epsilon}_{M}$ and $\bm{I}_{M}$,
respectively, and $\tilde{\bm{z}}_{M}(k)$ is defined:

$\displaystyle\tilde{\bm{z}}_{M}$ $\displaystyle=$
$\displaystyle[{\bm{0}}_{M}^{\mathrm{H}}\,\,\tilde{\bm{I}}_{M}]\tilde{\bm{B}}(k)\tilde{\bm{\psi}}_{M},$
(28)

with $\tilde{\bm{I}}_{M}$ being the
$[(\tilde{N}_{M}+1)^{2}-(N_{M}+1)^{2}]\times[(\tilde{N}_{M}+1)^{2}-(N_{M}+1)^{2}]$
identity matrix. $\tilde{\bm{z}}_{M}(k)$, therefore, represents coefficients
of the SMA transfer vector for orders higher than $N_{M}$. These orders are
projected into the first $(N_{M}+1)^{2}$ orders via multiplication with the
SMA aliasing coefficients in $\bm{\epsilon}_{M}$. This is referred to as the
spatial aliasing of orders higher than $N_{M}$. Finally, note that after
applying the sampling weights, beamforming will be applied to
$\hat{\bm{h}}_{M}(k)$ in Eq. (24) using $\bm{\lambda}$, and not
$\tilde{\bm{\lambda}}$. In practice, sampling weights and beamforming are
sometimes applied in a single stage, by applying beamforming vector
$\tilde{\bm{\lambda}}=\bm{\alpha}_{M}^{H}{\bm{\lambda}}$ directly to
$\tilde{\bm{h}}_{M}(k)$ from Eq. (22).

###  3.3 Spurious harmonics at the SLA

Spatial sampling by loudspeakers may introduce errors that are non-negligible
at high frequencies, as in the case of SMAs. Applying sampling weights to the
loudspeakers that comprise an SLA has been proposed in ref. [37], in order to
calculate beamforming coefficients that produce a desired sound field
represented using SH functions. In this work, sampling at the SLA facilitates
a representation of the transfer vectors from the SLA input in the SH domain
to the sound pressure at the center of the SMA (instead of from loudspeakers
to the sound pressure at the center of the SMA). Given $\tilde{\bm{h}}_{L}(k)$
from Eq. (20), a new SLA transfer vector $\hat{\bm{h}}_{L}(k)$ can be defined
as:

$\displaystyle\hat{\bm{h}}_{L}(k)$ $\displaystyle=$
$\displaystyle\bm{\alpha}_{L}\left[{\bm{Y}}_{L}\tilde{\bm{G}}(k)\tilde{\bm{\psi}}_{L}(k)+\tilde{\bm{n}}_{L}(k)\right],$
(29)

where

$\displaystyle\bm{\alpha}_{L}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\alpha_{1,00}^{L}&...&\alpha_{S,00}^{L}\\\
\vdots&\ddots&\vdots\\\
\alpha_{1,N_{L}N_{L}}^{L}&...&\alpha_{S,N_{L}N_{L}}^{L}\end{bmatrix}$ (30)

is an $(N_{L}+1)^{2}\times S$ matrix that holds the SLA sampling weights.
Similar to the SMA, when sampling weights are chosen that are ideal for order
$N_{L}$, the following identity holds [14]:

$\displaystyle\bm{\alpha}_{L}{\bm{Y}}_{L}$ $\displaystyle=$
$\displaystyle[\bm{I}_{L}\,\,\bm{\epsilon}_{L}],$ (31)

where $\bm{I}_{L}$ is an $(N_{L}+1)^{2}\times(N_{L}+1)^{2}$ identity matrix
and $\bm{\epsilon}_{L}$ is an
$(N_{L}+1)^{2}\times[(\tilde{N}_{L}+1)^{2}-(N_{L}+1)^{2}]$ matrix that holds
the SLA spatial aliasing function up to order $\tilde{N}_{L}$. Using this
identity and the definition in Eq. (18), Eq. (29) can be rewritten as:

$\displaystyle\hat{\bm{h}}_{L}(k)$ $\displaystyle=$
$\displaystyle[\bm{I}_{L}\,\,\bm{\epsilon}_{L}]\tilde{\bm{G}}(k)\tilde{\bm{\psi}}_{L}(k)+\bm{\alpha}_{L}\tilde{\bm{n}}_{L}(k)$
(32) $\displaystyle=$
$\displaystyle[\bm{I}_{L}\,\,{\bm{0}}_{L}]\tilde{\bm{G}}(k)\tilde{\bm{\psi}}_{L}+[\tilde{\bm{0}}_{L}\,\,\bm{\epsilon}_{L}]\tilde{\bm{G}}(k)\tilde{\bm{\psi}}_{L}+$
$\displaystyle+\bm{\alpha}_{L}\tilde{\bm{n}}_{L}(k)$ $\displaystyle=$
$\displaystyle\bm{G}(k)\bm{\psi}_{L}(k)+\bm{\epsilon}_{L}\tilde{\bm{z}}_{L}(k)+\bm{\alpha}_{L}\tilde{\bm{n}}_{L}(k),$

where ${\bm{0}}_{L}$ and $\tilde{\bm{0}}_{L}$ are matrices of zeros with the
same dimensions as those of $\bm{\epsilon}_{L}$ and $\bm{I}_{L}$,
respectively, and $\tilde{\bm{z}}_{L}(k)$ is defined:

$\displaystyle\tilde{\bm{z}}_{L}$ $\displaystyle=$
$\displaystyle[{\bm{0}}_{L}^{\mathrm{H}}\,\,\tilde{\bm{I}}_{L}]\tilde{\bm{G}}(k)\tilde{\psi}_{L},$
(33)

with $\tilde{\bm{I}}_{L}$ being the
$[(\tilde{N}_{L}+1)^{2}-(N_{L}+1)^{2}]\times[(\tilde{N}_{L}+1)^{2}-(N_{L}+1)^{2}]$
identity matrix. $\tilde{\bm{z}}_{L}(k)$, therefore, represents coefficients
of the SLA transfer vector for orders higher than $N_{L}$. We refer to these
coefficients as spurious, or artificial, harmonics, and they can also be
viewed as the equivalent of spatial aliasing in SMAs. While spatial aliasing
in SMAs refers to the aliasing of higher-order SH coefficients into low-order
SH coefficients due to sampling, in SLAs, higher order SH are generated due to
the sampling of a low-order function on a sphere. Low-order harmonics of a
continuous function on a sphere are projected to these spurious, higher-order
harmonics via multiplication with $\bm{\epsilon}_{L}$. Due to the resemblance
of the spurious harmonics to spatial aliasing, results from the SMA literature
can be directly applied to SLAs. For example, spatial aliasing is reduced by
the SMA through the attenuation of $b_{n}(kr_{M})$ from Eq. (2) for $n>kr_{M}$
[14]. In a similar manner, spurious harmonics are also reduced when radiated
to the far field through the attenuation of $g_{n}(kr_{L})$ from Eq. (8), for
$n>kr_{L}$. Similarly, methods proposed for minimizing spatial aliasing, such
as those described in refs. [17, 38], can potentially be adapted and employed
for minimizing the energy of spurious harmonics. Finally, note that after
applying the sampling weights, beamforming will be applied to
$\hat{\bm{h}}_{L}(k)$ in Eq. (32) using ${\bm{\gamma}}$, and not
$\tilde{\bm{\gamma}}$. Similarly to SMAs, sampling weights and beamforming are
sometimes applied in a single stage, by applying
$\tilde{\bm{\gamma}}=\bm{\alpha}_{L}^{H}{\bm{\gamma}}$ directly to
$\tilde{\bm{h}}_{L}(k)$ from Eq. (20). In particular, this approach for
formulating $\tilde{\bm{\gamma}}$ yields the same beamforming vector as in
other methods proposed in the literature for SLAs [39, 9, 40, 35].

###  3.4 System normalization

Normalization can be applied in order to allow for a description of the
measured sound field that is independent of the arrays’ physical construction,
facilitating a representation of that sound field using plane waves [3]. In
particular, it was shown in ref. [22] that normalization leads to improved
performance in room acoustics applications. We consider it to be important to
incorporate normalization when developing a framework for a joint design of
the arrays in a MIMO system; normalization is widely used in spherical array
processing, but its employment introduces ill-conditioning at low frequencies.
Hence, its investigation at the system level is important.

Normalization is applied to the arrays of the MIMO system; at the SLA the SH
channels are normalized by the functions $g_{n}(kr_{L})$ from Eq. (8), while
at the SMA, SH channels are normalized by $b_{n}(kr_{M})$ from Eq. (2). In
practice, this is performed together with beamforming, by applying a
normalized beamforming vector
$\bm{v}(k)=[\bm{G}(k)^{-1}]^{\mathrm{H}}\bm{\gamma}(k)$ to
$\hat{\bm{h}}_{L}(k)$ from Eq. (32) for the SLA, and by applying vector
$\bm{u}(k)=[\bm{B}(k)^{-1}]^{\mathrm{H}}\bm{\lambda}(k)$ from Eq. (27) for the
SMA. In this work, normalization is formulated inherently in the system model
for deriving error bounds that are beamforming-independent in the following
section. Therefore, normalized system transfer vectors and matrix are defined.
For the SLA, an $(N_{L}+1)^{2}\times 1$ normalized transfer vector is defined
using $\hat{\bm{h}}_{L}(k)$ from Eq. (32) as:

$\displaystyle\hat{\bm{\psi}}_{L}(k)$ $\displaystyle=$
$\displaystyle\bm{G}(k)^{-1}\hat{\bm{h}}_{L}(k)$ (34) $\displaystyle=$
$\displaystyle\bm{\psi}_{L}(k)+\bm{z}_{L}(k)+\bm{n}_{L}(k),$

where $\bm{\psi}_{L}(k)$ is the error-free transfer vector given in Eq. (18),

$\displaystyle{\bm{z}}_{L}(k)$ $\displaystyle=$
$\displaystyle\bm{G}(k)^{-1}\bm{\epsilon}_{L}\tilde{\bm{z}}_{L}(k),\,\,\text{and}$
(35) $\displaystyle{\bm{n}}_{L}(k)$ $\displaystyle=$
$\displaystyle\bm{G}(k)^{-1}\bm{\alpha}_{L}\tilde{\bm{n}}_{L}(k).$ (36)

Similarly, for the SMA, an $(N_{M}+1)^{2}\times 1$ normalized transfer vector
is defined using $\hat{\bm{h}}_{M}(k)$ from Eq. (27) as:

$\displaystyle\hat{\bm{\psi}}_{M}(k)$ $\displaystyle=$
$\displaystyle\bm{B}(k)^{-1}\hat{\bm{h}}_{M}(k)$ (37) $\displaystyle=$
$\displaystyle\bm{\psi}_{M}+\bm{z}_{M}(k)+\bm{n}_{M}(k),$

where $\bm{\psi}_{M}$ is the error-free transfer vector given in Eq. (19),

$\displaystyle{\bm{z}}_{M}(k)$ $\displaystyle=$
$\displaystyle\bm{B}(k)^{-1}\bm{\epsilon}_{M}\tilde{\bm{z}}_{M}(k),\,\,\text{and}$
(38) $\displaystyle{\bm{n}}_{M}(k)$ $\displaystyle=$
$\displaystyle\bm{B}(k)^{-1}\bm{\alpha}_{M}\tilde{\bm{n}}_{M}(k).$ (39)

Given $\hat{\bm{\psi}}_{L}(k)$ and $\hat{\bm{\psi}}_{M}(k)$, an
$(N_{L}+1)^{2}\times(N_{M}+1)^{2}$ normalized transfer matrix of the system is
defined as:

$\displaystyle\hat{\bm{\Psi}}(k)$ $\displaystyle=$
$\displaystyle\hat{\bm{\psi}}_{L}(k)\hat{\bm{\psi}}^{\mathrm{H}}_{M}(k).$ (40)

Finally, an $(N_{L}+1)^{2}\times(N_{M}+1)^{2}$ error-free normalized transfer
matrix is defined as:

$\displaystyle\bm{\Psi}(k)$ $\displaystyle=$
$\displaystyle\bm{\psi}_{L}(k)\bm{\psi}_{M}^{\mathrm{H}},$ (41)

facilitating the formulation of errors in the next section.

##  4 Error formulation

Expressions for the errors and error bounds are formulated in this section,
based on the MIMO transfer matrices defined in the previous section. Employing
a matrix-based formulation, errors and error bounds that are independent of
beamforming at the arrays are derived.

The total system error $\delta(k)$ is defined using matrices
$\hat{\bm{\Psi}}(k)$ and ${\bm{\Psi}}(k)$ from Eqs. (40) and (41),
respectively, as:

$\displaystyle\delta(k)$ $\displaystyle=$
$\displaystyle\frac{||{\bm{\Psi}}(k)-\hat{\bm{\Psi}}(k)||}{{||\bm{\Psi}}(k)||},$
(42)

where $||(\cdot)||$ denotes the standard 2-norm. Substituting
$\hat{\bm{\Psi}}(k)$ and ${\bm{\Psi}}(k)$ and employing the Cauchy-Schwarz
inequality, $\delta(k)$ can be shown to be bounded by:

$\displaystyle\delta(k)$ $\displaystyle\leq$
$\displaystyle\frac{1}{||\bm{\psi}_{L}(k)\bm{\psi}_{M}^{\mathrm{H}}||}\times$
(43)
$\displaystyle\times\Bigg{(}\bigg{|}\bigg{|}\bm{\psi}_{L}(k)\bm{z}_{M}^{\mathrm{H}}(k)\bigg{|}\bigg{|}+\bigg{|}\bigg{|}\bm{\psi}_{L}(k){\bm{n}}_{M}^{\mathrm{H}}(k)\bigg{|}\bigg{|}+\bigg{|}\bigg{|}{\bm{z}}_{L}(k)\bm{\psi}_{M}^{\mathrm{H}}\bigg{|}\bigg{|}+\bigg{|}\bigg{|}{\bm{n}}_{L}(k)\bm{\psi}_{M}^{\mathrm{H}}\bigg{|}\bigg{|}+$
$\displaystyle+\bigg{|}\bigg{|}{\bm{z}}_{L}(k){\bm{z}}_{M}^{\mathrm{H}}(k)\bigg{|}\bigg{|}+\bigg{|}\bigg{|}{\bm{z}}_{L}(k){\bm{n}}_{M}^{\mathrm{H}}(k)\bigg{|}\bigg{|}+\bigg{|}\bigg{|}{\bm{n}}_{L}(k){\bm{z}}_{M}^{\mathrm{H}}(k)\bigg{|}\bigg{|}+\bigg{|}\bigg{|}{\bm{n}}_{L}(k){\bm{n}}_{M}^{\mathrm{H}}(k)\bigg{|}\bigg{|}\Bigg{)}.$

Note that all matrices in the last equation are outer products of two vectors.
For such matrices, the 2-norm is equal to the product between the 2-norms of
the two vectors that compose them; i.e., for vectors $\bm{q}$ and $\bm{w}$,
the following holds:

$\displaystyle||\bm{q}\,\bm{w}^{\mathrm{H}}||$ $\displaystyle=$
$\displaystyle||\bm{q}||\,||\bm{w}||,$ (44)

which can be shown directly using the definition of the matrix norm. Using
this identity, $\delta(k)$ can be shown to be bounded by:

$\displaystyle\delta(k)$ $\displaystyle\leq$
$\displaystyle\mathit{a}_{L}(k)+\mathit{m}_{L}(k)+\mathit{a}_{M}(k)+\mathit{m}_{M}(k)+\left[\mathit{a}_{L}(k)+\mathit{m}_{L}(k)\right]\left[\mathit{a}_{M}(k)+\mathit{m}_{M}(k)\right],$
(45)

where

$\displaystyle\mathit{a}_{L}(k)$ $\displaystyle=$
$\displaystyle\frac{||\bm{z}_{L}(k)||}{||\bm{\psi}_{L}(k)||},$ (46)
$\displaystyle\mathit{m}_{L}(k)$ $\displaystyle=$
$\displaystyle\frac{||\bm{n}_{L}(k)||}{||\bm{\psi}_{L}(k)||},$ (47)
$\displaystyle\mathit{a}_{M}(k)$ $\displaystyle=$
$\displaystyle\frac{||\bm{z}_{M}(k)||}{||\bm{\psi}_{M}||},\,\,\text{and}$ (48)
$\displaystyle\mathit{m}_{M}(k)$ $\displaystyle=$
$\displaystyle\frac{||\bm{n}_{M}(k)||}{||\bm{\psi}_{M}||}.$ (49)

are referred to as the spurious-harmonics, the SLA model-mismatch, the
spatial-aliasing, and the SMA model-mismatch error bounds, repsectively. The
first two terms from Eq. (45) are referred to as the SLA error bounds since
they depend only on the SLA; An SLA-only error can be defined as:

$\displaystyle\delta_{L}(k)$ $\displaystyle=$
$\displaystyle\frac{||\bm{\psi}_{L}(k)-\hat{\bm{\psi}}_{L}(k)||}{||\bm{\psi}_{L}(k)||}.$
(50)

By substituting $\bm{\psi}_{L}(k)$ and $\hat{\bm{\psi}}_{L}(k)$ from Eqs. (18)
and (34), respectively, $\delta_{L}(k)$ can be shown to bounded by the sum of
the SLA-related error terms from Eq. (45):

$\displaystyle\delta_{L}(k)$ $\displaystyle\leq$
$\displaystyle\mathit{a}_{L}(k)+\mathit{m}_{L}(k).$ (51)

Similarly, $\mathit{a}_{M}(k)$ and $\mathit{m}_{M}(k)$ are referred to as the
SMA error bounds since they depend only on the SMA. Defining an SMA-only error
as:

$\displaystyle\delta_{M}(k)$ $\displaystyle=$
$\displaystyle\frac{||\bm{\psi}_{M}-\hat{\bm{\psi}}_{M}(k)||}{||\bm{\psi}_{M}(k)||},$
(52)

and substituting $\bm{\psi}_{M}(k)$ and $\hat{\bm{\psi}}_{M}(k)$ from Eqs.
(19) and (37), respectively, $\delta_{M}(k)$ can be shown to be bounded by:

$\displaystyle\delta_{M}(k)$ $\displaystyle\leq$
$\displaystyle\mathit{a}_{M}(k)+\mathit{m}_{M}(k).$ (53)

Finally, the last four terms from the bottom line of Eq. (45) are referred to
as the joint SLA-SMA errors. These errors are given as products of the SLA and
SMA errors. Having defined the seperate SLA and SMA errors, $\delta(k)$ from
Eq. (45) can be bounded by:

$\displaystyle\delta(k)$ $\displaystyle\leq$
$\displaystyle\delta_{L}(k)+\delta_{M}(k)+\delta_{L}(k)\delta_{M}(k).$ (54)

It is thus sufficient to bound the SLA and SMA errors in order to ensure that
the energy of the total system error is bounded, justifying a system design
based on the error bounds of the separate arrays.

##  5 Matched design of an SMA and an SLA

The error bounds defined in the previous section facilitate a general system
design, in which multiple beamforming vectors can be applied to both arrays.
The design is conducted through an error analysis for MIMO systems with
different system and error parameters. System parameters include the array
types (i.e., rigid-sphere, open-sphere, etc.), the number of elements in each
array and the arrays’ radii. The error parameters are used for generating
model mismatch errors. Guidelines for system design and the definition of a
matched MIMO system are given in this section through a design example.

A design example is given for two MIMO systems that vary in only one of the
system parameters: the SMA radius. The first system, SYS1, is comprised of an
SLA and an SMA with equal radii of $r_{L}=r_{M}=0.2\,$m. Both arrays are
rigid-sphere arrays. $S=144$ loudspeakers are uniformaly distributed over the
surface of the SLA, which yields an order of $N_{L}=8$. The size of the
loudspeakers’ cap size, which is required for calculating coefficients
$g_{n}(kr_{L})$ from Eq. (8), is set to correspond to a 2”-diameter
loudspeaker. As for the SMA, $R=162$ microphones are distributed using a
Gaussian distribution, which yields the same SH order as that of the SLA,
i.e., $N_{M}=8$. A distance of $r_{0}=1\,$m is set between the centers of the
arrays. The orientations of the arrays are set such that the SMA is located at
the direction of the north pole with respect to the SLA coordinate system
(i.e., $(\theta_{0},\phi_{0})=(0,0)$), and vice versa, i.e.,
$(\xi_{0},\omega_{0})=(0,0)$. For the second system, SYS2, the SMA radius is
set to $r_{M}=0.04\,$m, while all the other system parameters are the same as
in SYS1.

The total system error, the SLA and SMA errors, and the error bounds are
calculated for both systems, for frequencies between $f_{min}=30\,$Hz and
$f_{max}=10\,$kHz. Array transfer and error vectors are initially generated
for calculating these errors and error bounds. $\bm{\psi}_{L}(k)$ and
$\bm{\psi}_{M}$ are generated as in Eqs. (18) and(19), respectively, given the
system configuration. $\hat{\bm{\psi}}_{L}(k)$ and $\hat{\bm{\psi}}_{M}$ from
Eqs. (34) and (37), respectively, are generated using $\bm{z}_{L}(k)$,
$\bm{n}_{L}(k)$, $\bm{z}_{M}(k)$, and $\bm{n}_{M}(k)$. $\bm{z}_{L}(k)$ and
$\bm{z}_{M}(k)$ are generated as in Eqs. (35) and (38), respectively. This
requires choosing $\tilde{N}_{L}$ and $\tilde{N}_{M}$ for determining the
dimensions of $\bm{\epsilon}_{L}$, $\tilde{\bm{z}}_{L}(k)$,
$\bm{\epsilon}_{M}$, and $\tilde{\bm{z}}_{M}(k)$. For simplicity, a single SH
order $\tilde{N}=\tilde{N}_{L}=\tilde{N}_{M}$ was used. $\tilde{N}$ was chosen
in accordance with $f_{max}$ and the arrays’ radii, based on a criterion
proposed in Ref. [14], and was set as:

$\displaystyle\tilde{N}$ $\displaystyle=$
$\displaystyle\left\lceil{\text{max}(r_{M},r_{L})\frac{2\pi
f_{max}}{c}}\right\rceil+2,$ (55)

yielding $\tilde{N}=39$, where $\left\lceil{\cdot}\right\rceil$ is the ceiling
operator. This order was chosen so that the contribution of orders higher than
$\tilde{N}$, which are not taken into account when calculating the spatial
aliasing and spurious harmonics errors, will be negligible compared to that of
orders lower than $\tilde{N}$. In particular, two additional orders were added
to the last equation, compared to the criterion developed in [14]; this was
intended to ensure that non-modelled spatial-aliasing and spurious-harmonics
errors will be significantly lower than $-40\,$dB, a value which will later be
used for determing the variance of model-mismatch errors. $\bm{n}_{L}(k)$ and
$\bm{n}_{M}(k)$ are generated as in Eqs. (36) and (39), respectively. For this
it is necessary to initially generate zero-mean normally-distributed i. i. d.
errors for constructing $\tilde{\bm{n}}_{L}(k)$ and $\tilde{\bm{n}}_{M}(k)$
from Eqs. (21) and (23), respectively. The variance of the errors in
$\tilde{\bm{n}}_{L}(k)$ is set constant over frequency, and $40\,$dB lower
than the average power of the elements in $\tilde{\bm{h}}_{L}(k)$ from Eq. (5)
evaluated at $1\,$kHz. Similarly for the SMA, the variance of the errors in
$\tilde{\bm{n}}_{M}(k)$ is set constant over frequency, and $40\,$dB lower
than the average power of the elements in $\tilde{\bm{h}}_{M}(k)$ from Eq.
(10) evaluated at $1\,$kHz. Given the array transfer and error vectors, the
total system error, the SLA error, and the SMA error are calculated as in Eqs.
(42), (51), and (52), respectively, and the SLA and SMA error bounds are
calculated as in Eqs. (46)-(49). Finally, for smooth errors and error bounds,
$\delta(k)$, $\delta_{L}(k)$, $\delta_{M}(k)$, $\mathit{m}_{L}(k)$, and
$\mathit{m}_{M}(k)$ are calculated for thirty different realisations of
$\tilde{\bm{n}}_{L}(k)$ and $\tilde{\bm{n}}_{M}(k)$, and then averaged.

Errors and error bounds are studied for the arrays of a MIMO system, and the
SLA and SMA OFRs are defined. Figs. 3 and 4 present the errors error bounds of
the SLA and SMA, respectively, that comprise SYS1\. For both arrays, the
model-mismatch error bounds are dominant at low frequencies due to
normalization, as explained in sec. 3.4. At high frequencies, the spurious-
harmonics and spatial aliasing error bounds are seen to be dominant in Figs. 3
and 4, respectively. The total SLA error, $\delta_{L}(k)$, is also plotted on
Fig. 3 with a 2 dB vertical offset added to this error for convenience of
illustration, in order to avoid the overlap between the curves in the figure.
$\delta_{L}(k)$ converges with the SLA model-mismatch error bound at low
frequencies, and with the spurious-harmonics error bound at high frequencies.
This shows that $\delta_{L}(k)$ is bounded by the superposition of
$\mathit{m}_{L}(k)$ and $\mathit{a}_{L}(k)$, as suggested by Eq. (51).
Similarly for the SMA, $\delta_{M}(k)$ is plotted in Fig. 4 with a 2 dB
vertical offset, showing that it is bounded by the sum of errors as in Eq.
(53). $\delta_{L}(k)$ and $\delta_{M}(k)$ can be used for defining OFRs for
the SLA and the SMA, respectively, and for a chosen threshold of tolerated
error, denoted by $\sigma$. These OFRs are defined as:

$\displaystyle\mathit{O}_{L}$ $\displaystyle=$
$\displaystyle\\{k:\delta_{L}(k)\leq\sigma\\}$ and
$\displaystyle\mathit{O}_{M}$ $\displaystyle=$
$\displaystyle\\{k:\delta_{M}(k)\leq\sigma\\},$ (57)

for the SLA and SMA, respectively. For example, with $\sigma=0\,$dB, the SLA
and SMA OFRs are calculated to be $900\,$Hz-$5\,$kHz and
$1.2\,$kHz$\,-3\,$kHz, respectively. Moreover, the low- and high- frequency
bounds of $\mathit{O}_{L}$ can be calculated as the frequencies at which
$\mathit{m}_{L}(k)=\sigma$ and $\mathit{a}_{L}(k)=\sigma$, respectively, since
$\delta_{L}(k)$ is bounded by $\mathit{m}_{L}(k)$ at low frequencies and
$\mathit{a}_{L}(k)$ at high frequencies, as explained earlier. Similarly, the
low- and high- frequency bounds of $\mathit{O}_{M}$ can be calculated using
$\mathit{m}_{M}(k)$ and $\mathit{a}_{M}(k)$.

The total error of a MIMO system is studied, and a system OFR is defined.
Figs. 5 and 6 present the SLA, SMA, and total system errors for SYS1 and SYS2,
respectively. In particular, a 2 dB vertical offset is added to the total
system error, $\delta(k)$, in both figures for convenience of illustration.
For SYS1, $\delta(k)$ converges with the SMA total error when
$\delta_{M}(k)\leq 0$. The latter shows that: (i) it is sufficient to bound
the SLA and SMA errors in order to ensure bounded energy of the total system
error, as implied in Eq. (54); and (ii) when the total error of the SMA is
more significant than that of the SLA, or vice versa, $\delta(k)$ can be
bounded by the dominant array error, which is $\delta_{M}(k)$ in this example.
In Fig. 6, the total system error of SYS2 exhibits values higher than $0\,$dB
at all frequencies since at least one error, of either the SLA or the SMA, is
greater than or equal to zero at all simulated frequencies. This is a
consequence of the smaller radius of the SMA comprising SYS2, which results in
a shift of the SMA error bounds to higher frequencies compared to those of
SYS1\. For each system, an OFR, $\mathit{O}$, can be defined using $\delta(k)$
as

$\displaystyle\mathit{O}$ $\displaystyle=$
$\displaystyle\\{k:\delta(k)\leq\sigma\\}.$ (58)

With $\sigma=0\,$dB, the OFR of SYS1 is calculated to be $1.2\,$kHz -$3\,$kHz,
and an empty set is calculated for SYS2\. Since $\delta(k)$ is dominated by a
single array error for different frequencies, $\mathit{O}$ can also be
calculated as the intersection between $\mathit{O}_{L}$ and $\mathit{O}_{M}$
for thresholds $\sigma\leq 0\,$dB as:

$\displaystyle\mathit{O}$ $\displaystyle=$
$\displaystyle\mathit{O}_{L}\cap\mathit{O}_{M}.$ (59)

This is also demonstrated in Figs. 5 and 6; the SLA and SMA OFRs, calculated
using $\sigma=0\,$dB, are indicated by the shaded areas. Their intersection,
which is visibly darker in the figure, indicates the system OFR.

The fact that the system OFR is defined by the intersection of the SLA and SMA
OFRs motivates the definition of a matched MIMO system. A matched MIMO system
is defined as a system for which the OFR of one array is a subset of the OFR
of the other array. The system OFR in this case, $\mathit{\hat{O}}$, maintains
the following condition:

$\displaystyle\mathit{\hat{O}}$ $\displaystyle=$
$\displaystyle\left\\{({{O}_{L}},{{O}_{M}}):\,\,{{O}_{L}}\subseteq\mathit{O}_{M}\,\,\text{or}\,\,{{O}_{M}}\subseteq\mathit{O}_{L}\right\\}.$
(60)

SYS1 is therefore considered a matched system, as Eq. (60) is maintained in
this case.

The matching of the SLA and SMA in a MIMO system may not be unique. If the OFR
of one array is a proper subset of the other arrays’ OFR, i.e.,
$\mathit{O}_{L}\subset\mathit{O}_{M}$ or
$\mathit{O}_{M}\subset\mathit{O}_{L}$, then there are multiple ways to match
the arrays’ OFRs and to maintain Eq. (60). A practical criterion is proposed
for matching the arrays’ OFRs. Selecting the parameters of the system such
that:

$\displaystyle r_{M}N_{L}$ $\displaystyle=$ $\displaystyle r_{L}N_{M},$ (61)

aligns the spurious-harmonics and spatial-aliasing error bounds of the SLA and
SMA, respectively [14]. This is because the frequency that maintains
$kr_{L}=N_{L}$ and $kr_{M}=N_{M}$, which is considered as the upper operating
frequency limit in the literature [14], is the same for both arrays if Eq.
(61) is maintained. This criterion is proposed since it is independent of the
power of model-mismatch errors at the arrays, which may vary for different
conditions and systems (such as the use of different equipment, for example).
Note that Eq. (61) is maintained for SYS1, while it is not maintained for
SYS2\. As described earlier, the matching of a system may not be unique.
Therefore, not maintaining Eq. (61) does not necessarily imply that a system
is not matched.

##  6 Extension of the system OFR

If the OFR of the MIMO system is not sufficiently large it can be extended by
employing a lower SH order at one of the arrays. Employing a lower SH orders
leads to lowered spatial resolution of the array, on one hand; however, it
also leads to lower levels of the array errors and error bounds, thus
increasing the width of the array’s OFR and, potentially, the width of the
system OFR.

A criterion for selecting a lower SH order is developed, based on the
criterion from Eq. (61) for matching a system. However, given that the arrays’
radii are fixed, and that $N_{L}$ and $N_{M}$ are integers, it may not be
possible to maintain this condition exactly. Therefore, a SH order is computed
for minimizing $|r_{M}N_{L}-r_{L}N_{M}|$, which aligns the spurious-harmonics
and spatial-aliasing error bounds [14]. In particular, since SH orders higher
than $N_{L}$ and $N_{M}$ cannot be employed at the SLA and SMA, respectively,
a SH order

$\displaystyle\check{N}_{L}$ $\displaystyle=$
$\displaystyle\underset{\check{N}_{L}\leq N_{L}}{\text{arg
min}}\,|r_{M}\check{N}_{L}-r_{L}N_{M}|$ (62)

is employed at the SLA if $r_{M}N_{L}>r_{L}N_{M}$, or, if
$r_{M}N_{L}<r_{L}N_{M}$, a SH order

$\displaystyle\check{N}_{M}$ $\displaystyle=$
$\displaystyle\underset{\check{N}_{M}\leq N_{M}}{\text{arg
min}}\,|r_{M}N_{L}-r_{L}\check{N}_{M}|$ (63)

is employed at the SMA. For example, Eq. (63) gives $\check{N}_{M}=2$ for
SYS2.

Errors and error bounds are formulated for SYS2, employing $\check{N}_{M}=2$.
Initially the SMA transfer vectors are calculated. $\bm{\psi}_{M}$ are
generated, as in Eq. (19), but using $\check{N}_{M}=2$, instead of $N_{M}$.
Generating $\hat{\bm{\psi}}_{M}$ from Eq. (37) requires the calculation of
$\bm{n}_{M}(k)$ and $\bm{z}_{M}(k)$, using the same $\tilde{N}$ as in the
previous section. $\bm{n}_{M}(k)$ requires the calculation of a modified
$\bm{\alpha}_{M}$ (cf. Eq. (39)). A modified matrix,
$\check{\bm{\alpha}}_{M}$, is defined up to the new SH order by truncating the
first $(\check{N}_{M}+1)^{2}$ rows of $\bm{\alpha}_{M}$, from the previous
section. $\bm{z}_{M}(k)$ requires the calculation of a modified
$\bm{\epsilon}_{M}$ (cf. Eq. (38)). A modified matrix,
$\check{\bm{\epsilon}}_{M}$, is defined similarly to Eq. (26), but using
$\check{\bm{\alpha}}_{M}$, and is given by using the following equation:

$\displaystyle\check{\bm{\alpha}}_{M}{\bm{Y}}_{M}$ $\displaystyle=$
$\displaystyle[\check{\bm{I}}_{M}\,\,\check{\bm{\epsilon}}_{M}],$ (64)

where $\check{\bm{I}}_{M}$ is an
$(\check{N}_{M}+1)^{2}\times(\check{N}_{M}+1)^{2}$ identity matrix. Note that
other methods can be considered for defining modified sampling weights, and
employing such methods may alter the formulated SMA errors. These may include
numerical designs suitable for $\check{N}_{M}$, and methods for aliasing
cancellation [17, 38], and are out of the scope of this work. Given the
modified matrices, the SMA error and the total system error were calculated as
in Eqs. (52) and (42), respectively, and the SMA error bounds were calculated
as in Eqs. (48) and (49). Moreover, for smooth errors and error bounds,
$\delta(k)$, $\delta_{M}(k)$, and $\mathit{m}_{M}(k)$ were averaged over
thirty different realisations of $\tilde{\bm{n}}_{L}(k)$ and
$\tilde{\bm{n}}_{M}(k)$, as in the previous section.

The errors of SYS2 are analyzed and its OFR is calculated for SH order
$\check{N}_{M}=2$ at the SMA. Fig. 7 presents the total system error for the
modified SH order, denoted $\check{\delta}(k)$, and for comparison,
${\delta}(k)$ of the original system from Fig. 6 is also presented. Employing
order $\check{N}_{M}$ is seen to result in a lower level of the total error
bound, $\check{\delta}(k)$, compared to ${\delta}(k)$. In particular, using
$\sigma=0\,$dB, a modified OFR is calculated at $900\,$Hz-$5.6\,$kHz,
demonstrating the effect of employing a lower SH order at one of the arrays on
the system OFR. Finally, note that a SH order higher than $\check{N}_{M}$ can
also be selected and, while employing a higher order may lead to an unmatched
system, it is expected to improve the SMA spatial resolution.

##  7 Beamforming examples

The importance of MIMO system design and the implications of system matching
on system performance and robustness against errors are studied in this
section in the context of an analysis of directional RIRs. Beamforming is
applied to the arrays in order to produce directional RIRs, with the aim of
filtering a single reflection from the early refelction segment of an RIR.
Positioning errors and transducer noise are also simulated in order to study
the robustness against errors.

###  7.1 Setup

An example analysis is provided for MIMO systems positioned in a room. As in
sec. 5, two systems are studied, which vary in only one system parameter,
which is the SMA radius. The first system, SYS1, is comprised of an SLA and an
SMA with equal radii of $r_{L}=r_{M}=0.2\,$m. $S=36$ loudspeaker units are
distributed uniformly on the surface of the SLA, which yields an order of
$N_{L}=4$. The size of the loudspeakers’ cap size is set to correspond to a
3”-diameter loudspeaker. As for the SMA, $R=50$ microphones are distributed
using a Gaussian distribution, which yields $N_{M}=4$. For the second system,
SYS2, all system parameters are the same as for SYS1 except for the radius of
the SMA, which is set to $r_{M}=0.04\,$m. Both systems are positioned in a
room with dimensions of $(25,15,10)\,$m and a reverberation time of
approximately $750\,$ms. For both systems, transfer functions were simulated
using MCRoomSim software [42], with a sampling frequency of $f_{s}=48$kHz, and
with the SMA and the SLA positioned at $(15,8,3)\,$m and $(10,4,1.5)\,$m,
respectively. The transfer functions were arranged in matrix form, as in Eq.
(1). Instead of generating model mismatch errors using the general error model
employed in sec. 3.1, in this section positioning errors were simulated for
both arrays directly. Positioning errors were generated using a uniform
distribution on the range $[-1^{\circ},1^{\circ}]$, for modelling arrays with
a rotating mechanism, which typically encounter higher positioning errors
compared to fixed arrays. These errors were then added to the angles of the
loudspeaker and microphone positions. In addition to the positioning errors,
transducer noise was also simulated for both arrays, with errors generated
using a zero-mean Gaussian distribution. At the SLA, errors were added to the
input signals of the different loudspeakers, which are given by products of
the SLA input signal, $s(k)$, and the corresponding beamforming coefficient.
The variance of the errors was set equal over frequency and for all
loudspeakers, to produce a 25 dB average SNR for the loudspeakers at frequency
1 kHz. For a given loudspeaker, the SNR is defined as the power of the
corresponding input signal divided by the variance of the simulated errors. In
particular, the SNR is defined with respect to the input signals that include
beamforming coefficients of the non-normalized system in the space domain.
This is becasue sampling weights and normalization were applied with
beamforming in a single step in the following sections. At the SMA, errors
were added to the input signals of the microphones. The variance of the errors
was set equal over frequency and for all microphones, to produce a 25 dB
average SNR for the microphones at 1 kHz. Finally, all responses were band-
pass filtered to a range of $300\,$Hz-$1.9\,$kHz, as the arrays are expected
to have significant errors outside this frequency band. Using the method from
sec. 5, the OFR was calculated to be $550\,$Hz-$1.7\,$kHz for SYS1 using
$\sigma=0\,$dB, and the system is matched since it maintains Eq. (60). For
SYS2 an empty set is calculated for the OFR using $\sigma$. The same band-pass
filter was applied to the responses of both systems so as to match the
systems’ temporal resolution. For reference, the first $100\,$ms of the
normalized RIR between an omnidirectional loudspeaker and an omnidirectional
microphone is presented in Fig. 8. This RIR is obtained by taking the element
of the first row and first column in $\bm{\Psi}(k)$, and then employing an
inverse FFT. A typical RIR is composed of a sum of reflections for the direct
sound and for sound reflected off the room walls. Each reflection is
associated with a DOR around the SLA, a DOA around the SMA, and a time-delay
(TD). For the simulated setup, the DORs, DOAs, and TDs of the direct sound and
the first six reflections were calculated analytically and are summarized in
Table 1. The TDs are also plotted on Fig. 8 as vertical dashed lines.

###  7.2 Maximum DI beamformer

The first pair of directional RIRs is synthesised for the maximum DI
beamformer [31]. This beamformer is designed to maximize an array’s DI, and
since this index depends on the SH order employed at an array (and not on the
array’s radius), the same DI is calculated for the SLAs and the SMAs in both
systems. Since the DI is a measure for the directivity of an array, the
performance of both systems is considered to be equal in this example.
Therefore, a study of directional RIRs synthesized using this beamfomer serves
to compare the robustness against errors of the two systems.

Directional RIRs are generated, analyzed, and compared for both systems. These
responses were synthesized using normalized beamforming vectors

$\displaystyle\bm{\gamma}(k)$ $\displaystyle=$
$\displaystyle\bm{y}_{N_{L}}(\bm{\eta}_{5})\,\,\text{and}$ (65)
$\displaystyle\bm{\lambda}(k)$ $\displaystyle=$
$\displaystyle\bm{y}_{N_{M}}(\bm{\beta}_{5})$ (66)

for the SLA and SMA, respectively. These vectors were preprocessed to account
for sampling and normalization, and were then applied to the simulated system
matrix, as in Eq. (1). An inverse FFT was then employed for the resulting
system outputs. In particular, look directions $\bm{\eta}_{5}$ and
$\bm{\beta}_{5}$ were selected, which correspond to the DOR and DOA of the
fifth reflection from Table 1. Fig. 9 presents the directional RIRs for SYS1
and SYS2\. For ease of comparison, each response in this figure is normalized
by its maximum absolute value. A dominant peak is evident at the TD of the
fifth reflection for SYS1\. For SYS2, on the other hand, energy is distributed
for all times in the response, implying that a significant error is introduced
to the system output in this case. A system error is defined, similarly to in
Eq. (42), but including beamforming at the arrays, as:

$\displaystyle\Upsilon(k)$ $\displaystyle=$
$\displaystyle\frac{|\bm{\lambda}^{\mathrm{H}}(k)\bm{\Psi}(k)\bm{\gamma}(k)-\bm{\lambda}^{\mathrm{H}}(k)\hat{\bm{\Psi}}(k)\bm{\gamma}(k)|}{|\bm{\lambda}^{\mathrm{H}}(k)\bm{\Psi}(k)\bm{\gamma}(k)|}.$
(67)

Fig. 10 presents $\Upsilon(k)$ for both systems. For smooth curves,
$\Upsilon(k)$ was averaged over thirty realizations of positioining errors and
transducer noise for both errors. It is evident that for SYS1, $\Upsilon(k)$
is lower than $0\,$dB for low frequencies (within the frequency range defined
by the band-pass filter). SYS2, however, encounters errors higher than $0\,$dB
at low frequencies, which explains the low-frequency error component in Fig.
9. In summary, this beamforming example demonstrated that a matched system is
more robust to errors, compared to an unmatched system, when the directivity
(or performance) of the arrays in both systems, is equal.

###  7.3 Maximum white-noise gain beamformer

The second pair of directional RIRs is synthesised for the maximum white-noise
gain (WNG) beamformer [31]. This beamformer is designed to maximize an array’s
WNG. Since the WNG depends on the number of elements in an array (and not on
the array’s radius) [14, 11], the same value is calculated for the SLAs and
the SMAs in both systems. Since the WNG is considered to be a measure for an
array’s robustenss against errors, system robustness to errors is considered
to be equal for both arrays in this example. Therefore, a study of directional
RIRs synthesized using this beamfomer serves to compare the performance of the
two systems.

Directional RIRs are generated, analyzed, and compared for both systems. These
responses were synthesized using normalized beamforming vectors

$\displaystyle\bm{\gamma}(k)$ $\displaystyle=$
$\displaystyle\frac{\bm{G}^{\mathrm{H}}(k)\bm{G}(k)\bm{y}_{N_{L}}(\bm{\eta}_{5})}{\sum_{n=0}^{N_{L}}\frac{2n+1}{4\pi}|g_{n}(kr_{L})|^{2}}\,\,\text{and}$
(68) $\displaystyle\bm{\lambda}(k)$ $\displaystyle=$
$\displaystyle\frac{\bm{B}(k)\bm{B}^{\mathrm{H}}(k)\bm{y}_{N_{M}}(\bm{\beta}_{5})}{\sum_{n=0}^{N_{M}}\frac{2n+1}{4\pi}|b_{n}(kr_{M})|^{2}}$
(69)

for the SLA and SMA, respectively. As in the previous example, the beamfroming
coefficient vectors were preprocessed to account for sampling and
normalization, and were then applied to the simulated system matrix, as in Eq.
(1), and an inverse FFT was employed for the resulting system outputs. Fig. 11
presents the directional RIRs for SYS1 and SYS2\. In this case, no error
component is evident in the directional RIRs of both systems. This can be
explained by the similar errors $\Upsilon(k)$ for the systems, as seen in Fig.
12. Moreover, only one significant peak is evident for SYS1 in Fig. 11, while
an additional peak is seen at the TD of the direct sound in the response of
SYS2\. This can be explained by the SMA directivity of SYS1, which is higher
than that of SYS2, as further illustrated below. To evaluate the directivity
of the SMAs, the SMA beam pattern is plotted for SYS1 and SYS2 in Figs. 13 and
14 for $1.1\,$kHz. For ease of comparison, the beampatterns power at the fifth
DOA, i.e., $\bm{\beta}_{5}$, is normalized to $0\,$dB for both figures, and a
dynamic range of $30\,$dB is used. Moreover, on the figures, ‘X’- and
‘O’-marks are plotted to indicate the DOA of the fifth reflection and that of
the direct sound, respectively. Comparing the beampattern of the two systems
shows that SYS1 is more directive than SYS2 at the evaluated frequency.
Moreover, at the DOA of the direct sound, the power of the beampattern is
calculated at $-25.11\,$dB and $-7.63\,$dB for SYS1 and SYS2, respectively,
explaining the significant difference in the attenuation of the direct sound
illustrated in Fig. 11.

In conclusion, in this example it was shown that a matched system has improved
performance, compared to an unmatched system, when considering directivity and
robustness.

##  8 Conclusions

A framework for designing an SLA and an SMA in a MIMO system was presented,
based on the formulation of a system model that includes errors due to spatial
sampling and model mismatch at both arrays. Using this framework, a matched
system was defined, for which the OFRs of the SMA and the SLA are matched. The
superiority of a matched system over an unmatched system was demonstrated in
several simulation examples, in which spatial filtering was applied to the
systems in order to synthesize directional RIRs. The development of improved
methods for analysis and synthesis of sound fields using MIMO systems is
proposed for future work.

##  9 ACKNOWLEDGMENTS

This research was supported by The Israel Science Foundation (Grant No.
146/13) and the Chateaubriand fellowship.

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reflection | TD [sec] | DOR $(\theta,\phi)$ [deg] | DOA $(\xi,\omega)$ [deg]
---|---|---|---
Direct sound | 0.0192 | (76.82, 38.66) | (103.18, 218.66)
1 | 0.0228 | (125.10, 38.66) | (125.10, 218.66)
2 | 0.0382 | (83.42, 292.62) | (96.58, 247.38)
3 | 0.0401 | (109.09, 292.62) | (109.09, 247.38)
4 | 0.0489 | (22.45, 38.66) | (22.45, 218.66)
5 | 0.0546 | (85.41, 74.48) | (94.59, 105.52)
6 | 0.0560 | (103.54, 74.48) | (103.54, 105.52)
Table 1: TDs, DORs, and DOAs of the direct sound and the first six reflections
in the normalized RIR, presented in Fig. 8.

Figure 1: System block diagram with SLA beamforming coefficients,
$\gamma_{1}(k),..,\gamma_{S}(k)$, and SMA beamforming coefficients,
$\lambda_{1}(k),...,\lambda_{R}(k)$. Figure 2: Diagram of the SLA and the SMA,
with Cartesian coordinate systems $(x,y,z)$ and
$(x^{\prime},y^{\prime},z^{\prime})$, respectively.
$\bm{\eta}_{0}=(\theta_{0},\phi_{0})$ and
$\bm{\beta}_{0}=(\xi_{0},\omega_{0})$ are the DOR and DOA, respectively.
Figure 3: (Color online) SLA errors and error bounds for SYS1. Figure 4:
(Color online) SMA errors and error bounds for SYS1.

Figure 5: (Color online) SLA, SMA, and system total error for SYS1\.
$\mathit{O}_{L}$ and $\mathit{O}_{M}$, the SLA and SMA OFRs, respectively,
calculated using $\sigma=0$, are indicated by the shaded areas. Figure 6: Same
as Fig. 5 but for SYS2. Figure 7: (Color online) System errors for SYS2:
${\delta}(k)$, before system matching (using $N_{M}$), and
$\check{\delta}(k)$, after system matching (using $\check{N}_{M}=2$). The
modified OFR, calculated with $\sigma=0$ after the matching of the system, is
indicated by the shaded area. Figure 8: (Color online) RIR simulated for
omnidirectional loudspeaker and microphone. Vertical red-dashed lines indicate
the time delays of direct sound and the first six reflections, listed in Table
1.

Figure 9: (Color online) Directional RIRs synthesized by employing the maximum
DI beamformer to the arrays of both systems. Figure 10: (Color online) Total
system error when the maximum DI beamformers were applied to the arrays of
both systems. Figure 11: (Color online) Directional RIRs synthesized by
employing the maximum WNG beamformer to the arrays of both systems. Figure 12:
(Color online) Total system error when maximum WNG beamformers were applied to
the arrays of both systems.

Figure 13: SMA beampattern corresponding to beamforming vector $\bm{u}$ from
Eq. (69) for SYS1, calculated at a frequency of $1.1\,$kHz. ‘X’- and ‘O’-marks
indicate the DOA of the fifth reflection and of the direct sound,
respectively. Figure 14: Same as Fig. 13 but for SYS2.