![](bg4.png)
FFT if finally all spectral components are required.
However, as shall be seen, given the need to only
identify a low number of signatures the method offers
significant advantages, both with respect to numerical
efficiency as also in estimating confidence intervals.
Prior to computing the spectrum it is necessary to
take measures so as to limit the errors associated with
the Gibbs phenomena. The classical approach is to
use windowing (Press et al., 2002; Lindquist, 1988),
however, a new method based on polynomial approx-
imation was introduced recently, see (O’Leary and
Harker, 2011) for derivations and exact nature of the
computation. The signal is projected onto the orthog-
onal complement of a set of Gram polynomial basis
functions of low degree. The Gram polynomial ba-
sis functions up to degree d are also used to form the
columns of a matrix G
d
, the projection P onto the or-
thogonal complement can be computed as,
P = I − G
d
G
T
d
, (5)
where I is the unit matrix.
The spectrum with reduced spectral leakage is
now computed as,
s = F
I − G
d
G
T
d
y. (6)
It should be noted that, H , F
I − G
d
G
T
d
is once
again a matrix. In this manner reducing the Gibbs
effect has added no additional numerical complexity,
since H can be computed a-priori.
3.1 Signature Identification
The calibration procedure takes advantage of super-
position, in the assumption that the noise of the ma-
chine and the sound from the resonator can be re-
garded as additive signals. The i
th
resonator alone is
artificially activated while the machine is not running.
The signal from the accelerometer is acquired and the
corresponding spectrum is computed,
s
i
= H y. (7)
This procedure is repeated for each resonator yield-
ing a set of n initial spectral signatures one for each
resonator. Unfortunately, due to the complicated me-
chanical forms and internal acoustic reflections within
the form these vectors are not fully orthogonal.
The orthogonality of the signatures is achieved by
projecting them onto their mutual orthogonal comple-
ments. Given n signatures this is computed as,
ˆs
i
=
(
I −
∑
k
s
k
s
T
k
!)
s
i
∀ k ∈ [1. .. n],k 6= i. (8)
To support understanding it is helpful to formulate
this computation for two signatures,
ˆs
1
= s
1
− s
2
s
T
2
s
1
=
I − s
2
s
T
2
s
1
(9)
ˆs
2
= s
2
− s
1
s
T
1
s
2
=
I − s
1
s
T
1
s
2
. (10)
This computation yields a set of n orthogonalized sig-
natures ˆs
i
, which are complex vectors each of length
N. These are then used in the signature matching pro-
cess. The matrix of signatures S is formed by con-
catenating the individual orthogonalized vectors and
dividing them by their norm,
S =
h
ˆs
1
|ˆs
1
|
,.. .,
ˆs
n
|ˆs
n
|
i
. (11)
In this manner the matrix S has a unitary norm. Con-
sequently, S
+
, which is discussed next, has also a uni-
tary norm.
The orthogonalization process described by Equa-
tions 9 and 10 have worked well with the sensors used
in the experiment presented in this paper. However,
other experiments suggest that a diagonalisation of
the matrix of signatures, using singular value decom-
position, yields an even better separation of the sig-
nals with a lower cross sensitivity. This issue, how-
ever, is still the subject of further investigation.
3.2 Signature Matching
To support understanding it is helpful to take a more
fundamental look at the nature of the system and the
computation being performed. The excitation of the
pin can be approximated as a dirac pulse, in this case
the signatures correspond to the impulse response of
the resonators. For simplicity the orthogonalization
process is not considered now. The impulse response
corresponds to the first eigenfunction of the differ-
ential equation describing the dynamics of the res-
onator. Given a measurement of the resonator’s re-
sponse, with the addition of noise, the task is to per-
form de-convolution of the measured signal with the
response of the differential equation. This is funda-
mentally an inverse problem. In a lose sense it is
equivalent to inversion of the stochastic differential
equation for the system.
An algebraic approach to the computation has
been chosen since the Moore-Penrose pseudo-
inverse (Golub and Van Loan, 1996) provides a least
square approximation for the inversion of a rectangu-
lar matrix,
S
+
,
S
T
S
−1
S
T
, (12)
which ensures,
S
+
S = I. (13)
Computing S
+
is akin to inverting the eigenfunctions
s
i
contained in S which describe the differential equa-
tions of the resonators.
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