Extending Narrowband Descriptions and Optimal Solutions to
Broadband Sensor Arrays
Stephan Weiss
Centre for Signal & Image Processing, Department of Electronic & Electrical Engineering,
University of Strathclyde, Glasgow, Scotland
Keywords:
Array Processing, Polynomial Matrices, Matrix Factorisations.
Abstract:
This overview paper motivates the description of broadband sensor array problems by polynomial matrices,
directly extending notation that is familiar from the characterisation of narrowband problems. To admit opti-
mal solutions, the approach relies on extending the utility of the eigen- and singular value decompositions, by
finding decompositions of such polynomial matrices. Particularly the factorisation of parahermitian polyno-
mial matrices — including space-time covariance matrices that model the second order statistics of broadband
sensor array data — is important. The paper summarises recent findings on the existence and uniqueness of
the eigenvalue decomposition of such parahermitian polynomial matrices, demonstrates some algorithms that
implement such factorisations, and highlights key applications where such techniques can provide advantages
over state-of-the-art solutions.
1 INTRODUCTION
When processing signals obtained from an M-element
sensor array in a data vector x[n], where n is the dis-
crete time index, information on e.g. the angle of ar-
rival of sources is contained in the delay with which
different signals arrive at sensors. In the narrowband
case, this delay is sufficiently expressed by a phase
shift, information on which can be found in e.g. the
instantaneous covariance matrix R = E

x[n]x
H
[n]

of the sensor signals, where E
{
·
}
is the expectation
operator and {·}
H
the Hermitian transpose operator.
Many narrowband array problems therefore are ba-
sed on this covariance matrix R, and optimum be-
amforming and direction finding methods are often
subsequently based on factorisations — typically the
eigenvalue decomposition (EVD) — of R (Schmidt,
1986) or equivalently the singular value decomposi-
tion (SVD) of the data matrix (Moonen and de Moor,
1995).
In the broadband case, explicit delays must be
considered instead of phase shifts. These lags can be
capture by the second order statistics via the space-
time covariance matrix R[τ] = E

x[n]x
H
[n −τ]

,
which includes a discrete lag parameter τ. Since R[τ]
contains auto- and cross-correlation terms of x[n], it
inherits the symmetry R[τ] = R
H
[−τ]. When taking
the z-transform, the resulting cross spectral density
(CSD) matrix R(z) =
∑
τ
R[τ]z
−τ
satisfies the para-
hermitian property R(z) = R
P
(z), where the para-
hermitian operation R
P
(z) = R
H
(1/z
∗
) involves Her-
mitian transposition and time reversal(Vaidyanathan,
1993).A matrix R(z) that satisfies the parahermitian
property is called a parahermitian matrix.
While the polynomial matrix notation R(z) per-
mits the formulation of broadband problems, the uti-
lity of the EVD does not naturally extend from the
narrowband to the broadband case. If a constant simi-
larity transform is applied to R(z) or R[τ], the CSD or
space-time covariance matrices can generally only be
diagonalised for one single coefficient or lag. There-
fore an extension of the EVD to polynomial matrices
is required in order to provide solutions for broadband
problem formulations. For this purpose, (McWhirter
et al., 2007; McWhirter and Baxter, 2004) have de-
fined a polynomial EVD that can approximately di-
agonalise R(z) for all its coefficients, with recently
analysis providing the underpinning theory on the ex-
istence of polynomial eigenvalues and -vectors, and
the ambiguity of the latter.
Over the past decade, a number of algorithms have
emerged that implement a polynomial EVD (McW-
hirter et al., 2007; Redif et al., 2011; Tohidian et al.,
2013; Corr et al., 2014c; Redif et al., 2015; Wang
et al., 2015a), and also triggered a range of applica-
tions in the area of filter banks (Redif et al., 2011;
Weiss S.
Extending Narrowband Descriptions and Optimal Solutions to Broadband Sensor Arrays.
DOI: 10.5220/0007239500000000
In Proceedings of the 8th Inter national Joint Conference on Pervasive and Embedded Computing and Communication Systems (PECCS 2018), pages 5-14
ISBN: 978-989-758-322-3
Copyright
c
 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

Weiss et al., 2006), beamforming (Redif et al., 2006;
Koh et al., 2009; Alrmah et al., 2011; Weiss et al.,
2013; Vouras and Tran, 2014; Weiss et al., 2015; Al-
zin et al., 2016), communications (Weiss et al., 2006;
Davies et al., 2007; Ta and Weiss, 2007a; Sandmann
et al., 2015; Ahrens et al., 2017), or generic theore-
tical problems such as blind source separation (Re-
dif et al., 2017) or spectral factorisation (Wang et al.,
2015b).
The aim of this paper is to provide an overview
over efforts in the area of polynomial matrix decom-
positions, and to offer some insight into the advanta-
ges that this may bring for two exemplified applica-
tions, Therefore, this paper is organised as followed.
Sec. 2 defined the space-time covariance matrix and
its parahermitian matrix factorisation and its polyno-
mial approximation. Sec. 3 provides an overview over
polynomial matrix EVD algorithms, which are then
applied to two problems: Sec. 4 demonstrates the use
of polynomial matrix techniques for angle of arrival
estimation, while Sec. 5 discussed the applications in
broadband beamforming. A conclusion and outlook
over related fields is provided in Sec. 6.
2 PARAHERMITIAN MATRIX
EVD
Based on a short discourse on space-time covariance
and its properties in Sec. 2.1, we define a parahermi-
tian matrix EVD in Sec. 2.2. Its polynomial approxi-
mation is discussed in Sec. 2.3.
2.1 Space-time Covariance and
Cross-spectral Density Matrices
A scenario where L independent sources with non-
negative, real-valued power spectral densities (PSD)
S
`
(z), ` = 1...L, contribute to M sensor measure-
ments x
m
[n], m = 1 . . . M, the space-time covariance
matrix of the vector x[n] = [x
1
[n]...x
M
[n]]
T
is
R[τ] = E

x[n]x
H
[n −τ]

. (1)
If the PSD of the `th source is generated by a sta-
ble and causal innovation filter F
`
(z) (Papoulis, 1991),
and H
m`
(z) describes the transfer function of the cau-
sal and stable system between the `th source and the
mth sensor, then
R(z) = H(z)



S
1
(z)
.
.
.
S
L
(z)



H
P
(z) (2)
with the element in the mth row and `th column of
H(z) : C → C
M×L
given by H
m`
(z), and S
`
(z) =
F
`
(z)F
P
`
(z) the `th element of the diagonal matrix of
source PSDs.
The factorisation (2) can include the source mo-
del matrix F (z) = diag
{
F
1
(z),...,F
L
(z)
}
: C →C
L×L
,
such that
R(z) = H(z)F (z)F
P
(z)H
P
(z) . (3)
The components of H(z) and the source model F (z)
are assumed to be causal and stable, and their entries
can be either polynomials or rational functions in z. In
the most general latter case, the CSD matrix R(z) in
(3) can be represented as a Laurent series that is ab-
solutely convergent and therefore analytic within an
annulus containing the unit circle (Girod et al., 2001).
Further, since the PSDs satisfy S
`
(z) = S
P
`
(z), it is evi-
dent from both (2) and (3) that R(z) = R
P
(z) and so
is parahermitian.
2.2 Parahermitian Matrix EVD
For an analytic R(z), the factorisation
R(z) = Q(z)Λ(z)Q
P
(z) (4)
is called the parahermitian matrix EVD (Weiss
et al., 2018). If evaluated on the unit cir-
cle, the EVD at every frequency Ω, R(e
jΩ
) =
Q(e
jΩ
)Λ(e
jΩ
)Q
H
(e
jΩ
) can exist with analytic factors
Q(e
jΩ
) and Λ(e
jΩ
) (Rellich, 1937). The reparamete-
risation z = e
jΩ
can lead to analytic factors Q(z) and
Λ(z) provided that the eigenvalues are selected ap-
propriately. This selection will be motivated by an
example.
Example for eigenvalues. Inspected on the unit circle,
consider the eigenvalues λ
1
(e
jΩ
) = 1 and λ
2
(e
jΩ
) =
1 + cosΩ. Potentially, both functions can be permu-
ted at any frequency, and still form valid eigenva-
lues as long as they retain a 2π-periodicity. Besides
the analytic selection λ
1
(e
jΩ
) and λ
2
(e
jΩ
) shown in
Fig. 1(a), an important alternative are spectrally ma-
jorised eigenvalues λ
0
1
(e
jΩ
) and λ
0
2
(e
jΩ
) in Fig. 1(b),
where spectral majorisation implies that λ
0
1
(e
jΩ
) ≥
λ
0
2
(e
jΩ
) ∀Ω (Vaidyanathan, 1998).
If on the unit circle eigenvalues have algebraic
multiplicities greater than one, as in Fig. 1 for Ω =
π
2
and Ω =
3π
2
, then only the analytic selection can lead
to analytic eigenvalues in Λ(z). In the case of spectral
majorisation, the region for absolute convergence is
restricted to the unit circle itself.
For the eigenvectors, the representation on the unit
circle can have an arbitrary phase response. Only if
both the eigenvalues in Λ(z) and the arbitrary phase
responses are selected as analytic, it is be guaranteed

0
/4 /2 3 /4 5 /4 3 /2 7 /4 2
0
0.5
1
1.5
2
(a)
0
/4 /2 3 /4 5 /4 3 /2 7 /4 2
0
0.5
1
1.5
2
(b)
Figure 1: (a) Analytic vs (b) spectrally majorised selection
of eigenvalues.
that Q(z) is analytic as well. If enforcing spectral ma-
jorisation violates the analyticity of the eigenvalues,
then no analytic solution exists for the eigenvectors in
Q(z).
2.3 Polynomial Approximation
Analyticity is important when trying to design reali-
sable filters. Specifically, while the factors in (4) are
analytic and therefore absolutely convergent, they ge-
nerally form algebraic or even transcendental functi-
ons, i.e. are infinite in length and do not have a ratio-
nal representation (Weiss et al., 2018). Due to the ab-
solute convergence of these analytic functions, an ar-
bitrarily close approximation can be achieved by trun-
cating the Laurent series to sufficiently long Laurent
polynomials, whereby the term ‘polynomial’ implies
finite length.
The truncation of (4) leads to the polynomial EVD
or McWhirter decomposition
R(z) ≈
ˆ
Q(z)
ˆ
Λ(z)
ˆ
Q
P
(z) , (5)
which was postulated in (McWhirter et al., 2007), ba-
sed on a paraunitary factor
ˆ
Q(z), and a diagonal pa-
rahermitian
ˆ
Λ(z). All matrices — R(z),
ˆ
Q(z), and
ˆ
Λ(z) — are Laurent polynomials, ambiguity in the
ordering of the eigenvalues had been suppressed by
demanding spectral majorisation for
ˆ
Λ(z).
3 ALGORITHMS FOR
POLYNOMIAL MATRIX EVD
Even though eigenvalues and particularly eigenvec-
tors are not guaranteed to exist as analytic functions
in case of spectral majorisation, a number of algo-
rithms targetting the McWhirter decomposition (5)
have been created over the past decade (McWhirter
and Baxter, 2004; McWhirter et al., 2007; Tkacenko
and Vaidyanathan, 2006; Tkacenko, 2010; Redif
et al., 2011; Tohidian et al., 2013; Corr et al., 2014c;
Redif et al., 2015; Wang et al., 2015a). These all
share the restriction of considering the EVD of a pa-
rahermitian matrix R(z) whose elements are Laurent
polynomials, which may be enforced by estimating or
approximating R[τ] over a finite lag windwo (Redif
et al., 2011).
The approximation sign in the McWhirter decom-
position (5), highlighting the approximation by po-
lynomials, has been included in all subsequent al-
gorithm designs over the past decade. Even though
many algorithms can be proven to converge, in the
sense that they reduce off-diagonal energy of Γ (z) at
each iteration, see e.g. (McWhirter et al., 2007; Re-
dif et al., 2011; Corr et al., 2014c; Redif et al., 2015;
Wang et al., 2015a), there is no practical experience
yet where these algorithms could not find a practica-
ble factorisation.
Enforcing spectral majorisation in the case of an
algebraic multiplicity greater than one as shown in
Fig. 1 leads to eigenvalues that are not infinitely
differentiable and to eigenvectors with discontinui-
ties (Weiss et al., 2018). Since current PEVD al-
gorithms can be shown to either favour or can even
be proven to yield spectral majorisation (McWhirter
and Wang, 2016), they result in matrix factors with
high polynomial order to approximate the factors in
(5). Therefore, some mechanisms to curb the order
of these polynomial (Foster et al., 2006) and speci-
fically the paraunitary factors (Ta and Weiss, 2007b;
McWhirter et al., 2007; Corr et al., 2015c; Corr et al.,
2015d) have been suggested, which are generally ba-
sed on a truncation with limited error impact, and in
some cases judiciously exploit the arbitrary phase re-
sponse of the eigenvectors.
Current efforts in terms of algorithmic rese-
arch have targetted numerical efficiencies to enhance
the convergence speed of PEVD algorithms; these
e.g have exploited search space reductions (Corr et al.,
2014b; Corr et al., 2015b; Coutts et al., 2016c; Coutts
et al., 2017a), approximate EVD algorithms (Corr
et al., 2014a; Corr et al., 2015b; Corr et al., 2015a;
Coutts et al., 2016b), and matrix partitioning (Coutts
et al., 2016c; Coutts et al., 2017a). Also, (Tohidian
et al., 2013) have presented a frequency domain algo-
rithm which can favour analytic over spectrally ma-
jorised solutions (Coutts et al., 2017b; Coutts et al.,
2018). A further route of investigation is the impact
which estimation errors in the space-time covariance
matrix have on the accuracy of the factorisation (De-
laosa et al., 2018).

4 APPLICATION I: ANGLE OF
ARRIVAL ESTIMATION
As a first application example, this section visits an-
gle of arrival estimation. Sec. 4.1 first defines steer-
ing vectors, which together with the instantaneous co-
variance matrix are exploited in the multiple signal
classification (MUSIC) algorithm (Schmidt, 1986)
in Sec. 4.2. Broadband angle of arrival estimation
techniques are briefly touched in on Sec. 4.3, with the
polynomial broadband generalisation of narrowband
MUSIC outlined in Sec. 4.2.
4.1 Steering Vector
If a source illuminates an M-element array from an
elevation ϑ and azimuth angle ϕ, we assume that dif-
ferent delays τ
m
, m = 1 . . . M, are experienced as the
wavefront travels across the array. To describe these
sensor signals, a vector
s
ϑ,ϕ
[n] =
1
√
M




f [n −τ
1
]
f [n −τ
2
]
.
.
.
f [n −τ
M
]




, (6)
contains an ideal fractional delay filter f [n −τ], crea-
ting a delay of τ ∈ R samples (Laakso et al., 1996),
with n ∈ Z the discrete time index. Thus, given a
source signal u[n] and neglecting attenuation, its con-
tribution to the sensor signal vector x[n] is
x[n] = s
ϑ,ϕ
[n] ∗u[n] . (7)
The lag values τ
m
on the r.h.s. of (6) depend on the
elevation ϑ and azimuth ϕ of the source via tau
m
=
k
T
ϑ,ϕ
r
m
, where k
ϑ,ϕ
is the source’s slowness vector
pointing in the direction of propagation, and r
m
is the
position vector of the mth sensor.
The z-transform of s
ϑ,ϕ
[n],
s
ϑ,ϕ
(z) =
∞
∑
n=−∞
s
ϑ,ϕ
[n]z
−n
, (8)
is here called a broadband steering vector. By evalua-
ting the broadband steering vector s
ϑ,ϕ
(z) : C → C
M
on the unit circle, z = e
jΩ
, and for a particular fre-
quency Ω
0
we can also derive a narrowband steering
vector s
ϑ,ϕ,Ω
0
= s
ϑ,ϕ
(z)|
z=e
jΩ
0
.
4.2 Narrowband MUSIC
A classic angle of arrival estimation techniques is the
multiple signal classification (MUSIC) algorithm. It
builds on the instantaneous covariance matrix R =
E

x[n]x
H
[n]

, provided that x[n] contains narrow-
band data. By means of an EVD, R is separated
into a signal plus noise subspace, characterised by
large eigenvalues in Λ
s
∈R
R×R
, and a noise only sub-
space, characterised by small remaining eigenvalues
in Λ
n
∈ R
(M−R)×(M−R)
:
R =

Q
s
Q
⊥
s


Λ
s
0
0 Λ
n

Q
H
s
Q
⊥,H
s
.

(9)
The matrix Q
s
spans the signal plus noise subspace,
which is an orthogonalisation of R contributing, line-
arly independent sources. The columns of its comple-
ment, Q
⊥
s
, span the noise only subspace.
The fact that the steering vector of any of the R
linearly independent sources must be orthogonal to
the noise subspace spanned by Q
⊥
s
is exploited in the
MUSIC algorithm by probing the noise subspace with
steering vectors, such that
ρ(ϑ,ϕ) = kQ
⊥,H
s
s
ϑ,ϕ,Ω
s
k
−1
2
(10)
= s
H
ϑ,ϕ,Ω
s
Q
⊥
s
Q
⊥,H
s
s
ϑ,ϕ,Ω
s
, (11)
where Ω
s
is the narrowband frequency. The product
under the norm in (11) take on very small values if the
steering vector s
ϑ,ϕ,Ω
s
belongs to a valid source and
therefore is orthogonal to Q
⊥
s
. The MUSIC spectrum
ρ is the reciprocal of this value, i.e. returns large va-
lues if s
ϑ,ϕ,Ω
s
matches the steering vector of a source.
4.3 Broadband Approaches
Angle of arrival estimation techniques have been ge-
neralised to broadband signals. Recent works such
as (Souden et al., 2010) are restricted to single-source
scenarios. Early successful approaches have used the
coherent signal subspace approach (Wang and Ka-
veh, 1985; Wang and Kaveh, 1987; Hung and Kaveh,
1988), where effectively an array is pre-steered such
that the source appears at broadside, and can be tre-
ated as a narrowband signal as all contributions are
aligned. This however requires approximate know-
ledge from which direction a source illuminates an
array before the precise angle of arrival can be esti-
mated.
4.4 Polynomial MUSIC
Using the polynomial broadband approach, the sub-
space decomposition in (9) can be applied to the po-
lynomial EVD, and leads to a partitioning of the po-
lynomial modal matrix,
Q(z) =
h
Q
s
(z) Q
⊥
s
(z)
i
, (12)

−80
−60
−40
−20
0
20
40
60
80
0
0.2
0.4
0.6
0.8
1
0
20
40
Ω/π
ϑ/◦
S
PSS
(ϑ,e
jΩ
)/[dB]
(a)
Figure 2: Polynomial MUSIC result for an 8 element li-
near array illuminated by 3 broadband sources (Weiss et al.,
2013).
where the R ≤M columns of Q
s
(z) contain the eigen-
vectors spanning the signal plus noise subspace, and
Q
⊥
s
(z) its complement.
Based on this subspace decomposition of R(z),
the polynomial MUSIC algorithm in (Alrmah et al.,
2011; Alrmah et al., 2012; Weiss et al., 2013; Alrmah
et al., 2014) provide a simple generalisation of (11) to
polynomial matrices, such that
ρ(ϑ,ϕ,z) = s
P
ϑ,ϕ
(z)Q
⊥
s
(z)Q
⊥,P
s
(z)s
ϑ,ϕ
(z) . (13)
The implementation of broadband steering vectors
can be achieved with filters of reasonable order if
windowing (Selva, 2008) or other schemes such as
in (Alrmah and Weiss, 2013; Alrmah et al., 2013) are
employed. The result of the polynomial MUSIC al-
gorithm in (13) is a power spectral density-type term,
which can either be evaluated in terms of its total
energy, thus depending on the angle of arrival only,
or additionally resolve frequency.
Example. An example for an M = 8 element li-
near array illuminated by a mixture of three mutually
uncorrelated Gaussian sources of equal power,
• ϑ
1
= −30
◦
, active over range Ω ∈ [
3π
8
; π],
• ϑ
2
= 40
◦
, active over range Ω ∈ [
π
2
; π], and
• ϑ
3
= 20
◦
, active over range Ω ∈ [
2π
8
;
7π
8
],
is shown in Fig. 2. When using PEVD algorithms,
the accuracy of the result depends on the accuracy
of the PEVD decomposition, with enhanced diago-
nalisation leading to improved results (Alrmah et al.,
2012; Coutts et al., 2017c).
5 APPLICATION II: BROADBAND
BEAMFORMING
As an example for beamforming, we review the nar-
rowband definition of the minimum variance distor-
tionless response (MVDR) beamformer in Sec. 5.1
and standard broadband extensions in Sec. 5.2, with
its generalised polynomial formulation for the broad-
band case in Sec. 5.3. The polynomial approach is
then demonstrated to generalise the Capon beamfor-
mer as well as a generalised sidelobe canceller (GSC)
in Secs. 5.4 and 5.5.
5.1 Narrowband MVDR
In beamforming, the aim is to isolate signals emit-
ted by spatially separated sources by spatial filte-
ring. This is achieved by creating constructive and de-
structive interference based on measurements obtai-
ned from M sensors, gathered in a data vector x[n] ∈
C
M
. In the narrowband case, recalling the steering
vector definition from Sec. 4.1, the alignment can
be achieved by complex multipliers, since only the
phase requires to be adjusted. The output of a narro-
wband beamformer therefore consists of a weighted
sum of the sensor contributions, e[n] = w
H
x[n], where
w ∈ C
M
contains the weights of the beamformer.
In the presence of interference, the aim of a mini-
mum variance beamformer is to minimise the output
power σ
2
e
= E
{
y[n]y
∗
[n]
}
= w
H
Rw,
min
w
w
H
Rw (14)
s.t. s
H
ϑ
s
,ϕ
s
,Ω
s
w = f , (15)
where R = E

x[n]x
H
[n]

is the instantaneous covari-
ance matrix and the trivial solution is discouraged by
imposing a gain constraint f in look direction (ϑ
s
,ϕ
s
)
at the narrowband operating frequency Ω
s
.
Direct constrained optimisation of the MVDR
problem via Lagrange multipliers leads to the Ca-
pon beamformer, see e.g. (Stoica et al., 2003; Lorenz
and Boyd, 2005). Alternatively, the generalised side-
lobe canceller projects that data onto an unconstrained
subspace, where standard unconstrained optimisation
techniques such the least mean squares or recursive
least squares algorithms can then solve the MVDR
problem (Widrow and Stearns, 1985; Haykin, 2002).
5.2 Broadband MVDR
In order to spatially filter broadband signals, explicit
delays must be resolved, such that each sensor has to
be followed by a tap delay line or finite impulse re-
sponse filter in order to be able to constructively or
destructively align signals. If a filter with temporal
length L is employed, then the data vector needs to
be extended to dimension ML, and include both spa-
tial and temporal samples (Buckley, 1987; Van Veen
and Buckley, 1988; Liu and Weiss, 2010). Subse-
quently, with a space-time covariance matrix of di-
mension ML ×ML, the output power of the MVDR
problem can be defined.
The constraint equation can be straightforwardly
extended to the broadband case if the look direction

is towards broadside for a linear array. If the look
direction is off-broadside, or the array elements are
not arranged in a line, then either correction by pre-
steering is required to create a virtual linear array
with broadside look direction, or more complica-
ted constraint formulations are required (Godara and
Sayyah Jahromi, 2007; Somasundaram, 2013).
5.3 Polynomial MVDR Formulation
If the M-element vector w[n] contains the M fil-
ters following each sensor, then its z-transform
w(z) •—◦ w[n] enables to formulate the broadband
MVDR problem as (Weiss et al., 2015)
min
w(z)
I
|z|=1
w
P
(z)R(z)w(z)
dz
z
(16)
s.t. s
P
(ϑ
s
,ϕ
s
,z)w(z) = F(z) , (17)
where s(ϑ
s
,ϕ
s
,z) is the broadband steering vector
discussed in Sec. 4.1 that defines the beamformer’s
look direction. In the following sections, both the Ca-
pon and GSC polynomial formulations will be defi-
ned.
5.4 Polynomial Capon Beamformer
If we extend the constraint equation in (17) to include
N known interferers at angles of arrival (ϑ
i,n
,ϕ
i,n
, n =
1...N, then
C(z)w(z) = f (z) , (18)
with
C(z) =





s
P
(ϑ
s
,ϕ
s
,z)
s
P
(ϑ
i,1
,ϕ
i,1
,z)
.
.
.
s
P
(ϑ
i,N
,ϕ
i,N
,z)





(19)
f(z) =




F(z)
0
.
.
.
0




, (20)
then in a first step a polynomial Capon beamfor-
mer requires a pseudo-inverse of the polynomial con-
straint matrix C(z) to yield v(z) = C
†
(z)f(z). The
inversion of such a polynomial pseudo-inverse is
e.g. addressed in (Nagy and Weiss, 2017; Nagy and
Weiss, 2018).
With this extended constraint equation, the Capon
beamformer is given by (Alzin et al., 2016)
w
opt
(z) =
R
−1
(z)v(z)
˜v(z)R
−1
(z)v(z)
. (21)
w
P
q
(z)
B(z)
w
P
a
(z)
+
−
d[n]
e[n]
y[n]
x[n]
u[n]
Figure 3: Polynomial generalised sidelobe canceller with
quiescent beamformer w
q
(z), blocking matrix B(z), and
adaptive multichannel filter w
a
(z).
This formulation is a direct polynomial extension of
the narrowband formulation. The inversion of the
cross spectral density matrix R(z) can be accomplis-
hed via a polynomial EVD and the inversion of the
polynomial eigenvalues as discussed in (Weiss et al.,
2010).
5.5 Polynomial Generalised Sidelobe
Canceller
The GSC addresses the MVDR problem by for-
ming a beam in look direction irrespective of any
unknown structured interference. This quiescent be-
amformer w
q
(z) is the solution to the constraint equa-
tion — either (17), or, in the case of known interfe-
rers, (18). In order to remove the remaining interfe-
rence, a blocking matrix B(z) passes all signal com-
ponents orthogonal to w
q
(z), and therefore contains
the remaining interference only in its output u[n] in
Fig. 3. Thereafter, an adaptive noise canceller (Wi-
drow and Stearns, 1985; Haykin, 2002) can remove
the remaining interference from the quiescent beam-
former output d[n], thereby minimising the output po-
wer E
{
e[n]e
∗
[n]
}
.
The construction of w
q
(z) is such that its order
(and therefore computational complexity) is determi-
ned by the accuracy that is required of the fractio-
nal delay filters (Laakso et al., 1996; Selva, 2008).
The blocking matrix can then be determined by po-
lynomial matrix completion from a polynomial EVD
of w
q
(z)w
P
q
(z) (Weiss et al., 2015). Its computatio-
nal complexity is determined by the accuracy of the
PEVD and the desired suppression of leakage of the
signal of interest. In general, this order is significantly
lower (by at least a factor of L) compared to tap-
delay-line implementation (Buckley, 1987; Van Veen
and Buckley, 1988; Liu and Weiss, 2010) with off-
broadside constraints (Godara and Sayyah Jahromi,
2007).
The computational advantage of the polynomial
GSC is based on the fact that the complexities for
w
q
(z), B(z) and w
a
(z) are decoupled, while in the
case of a standard time domain broadband beamfor-

angle of arrival ϑ /[
◦
]
20log
10
|A(ϑ,e
jΩ
)| / [dB]
Ω
2π
Figure 4: Gain response of polynomial GSC for M = sen-
sors in a linear array and look direction ϑ
s
= 30
◦
, in depen-
dency of the angle of arrival and normalised angular fre-
quency.
mer (Buckley, 1987; Van Veen and Buckley, 1988;
Liu and Weiss, 2010), all quantities are linked to
the tap delay line length L . Additionally, for off-
broadside look directions without pre-steering, con-
straints generally have to be defined in the frequency
domain. As a result, the gain response is only tied
down at isolated frequencies, while the broadband
constraint in (17) preserved coherence across the
spectrum.
Example. Fig. 4 shows the gain response of an
adapted beamformer for a linear array with M = 8
sensors with look direction ϑ
s
= 30
◦
, with interfe-
rence by three broadband jammers. The gain in look
direction is preserved, while spatial nulls are placed
in the directions of the interfering sources over the
frequency ranges of these jammers.
6 CONCLUSIONS
This paper has summarised some of the developments
in the area of polynomial matrix factorisations and
their application in particular to broadband array pro-
blems. Many of these problems can be straightfor-
wardly formulated as a simple extension from the
classical narrowband case to a broadband scenario
when utilising polynomial matrix notation. The solu-
tion, in the narrowband case often reliant on decom-
positions such as the EVD or SVD, has its broadband
equivalent in the parahermitian — or if approximated
— the polynomial EVD, for which several mature al-
gorithms exist (see e.g. pevd-toolbox.eee.strath.ac.uk
for Matlab implementations and examples). Even
though the focus of this paper has been on parahermi-
tian or polynomial EVD, the polynomial approach
can also be extended to other linear algebraic factori-
sations such as the SVD (Foster et al., 2010; McWhir-
ter, 2010), the QR decomposition (Foster et al., 2010;
Coutts et al., 2016a) or the generalised EVD (Corr
et al., 2016).
Generally, the advantage of polynomial matrix
methods as opposed to DFT-based approaches is ge-
nerally that they preserve coherence between fre-
quency bins. This has lead to the exploration of a
number of applications besides the angle of arrival
and beamforming examples summarised on this pa-
per. Successful applications have, for example, target-
ted for example in denoising-type (Redif et al., 2006)
or decorrelating array pre-processors (Koh et al.,
2009), transmit and receive beamforming across bro-
adband MIMO channels (Davies et al., 2007; Ta and
Weiss, 2007a; Sandmann et al., 2015; Ahrens et al.,
2017), broadband angle of arrival estimation (Alrmah
et al., 2011; Weiss et al., 2013), optimum subband
partitioning of beamformers (Vouras and Tran, 2014),
filter bank-based channel coding (Weiss et al., 2006)
or broadband blind source separation (Redif et al.,
2017). In some cases the polynomial approach can
enable solutions that otherwise have been unobtaina-
ble: e.g. the design of optimal compaction filter banks
beyond the two channel case (Redif et al., 2011).
It is hoped that this overview paper can inspire the
use of these methods to a wider range of applications.
ACKNOWLEDGEMENTS
I would like to very grateful acknowledge the im-
mense help and input from a number of collaborators,
in particular John McWhirter, who initiated this field
of research, as well as Ian Proudler, Jennifer Pestana,
Malcolm Macleod, Soydan Redif, Jamie Corr, Fraser
Coutts, Chi Hieu Ta, Mohamed Alrmah, Connor De-
laosa, Ahmed Alzin, Amr Nagy, and Zeliang Wang.
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