Application of ICAMM to Impact-Echo Testing - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 112

Fig. 5.4 Wave propagation
scheme proposed for an
inspection by impact-echo
using four sensors. The path
between the point flaws and
the sensors is depicted only
for a few focuses

the sum of F þ 1 contributions due to the F þ 1 focuses that are present. More-
over, if we call vector V ij the contribution to V i , due to focus j, we can write

V i ¼ X V ij ¼ X

F F

B ij F j ; ð 5 : 9 Þ

j ¼ 0 j ¼ 0

where B ij is a diagonal matrix that has the vector formed by the samples of the
frequency response between focus j and sensor i at its main diagonal, and where F j
is a vector formed by the samples of the complex spectrum of f j ð n Þ . On the other
hand, the focus signals F j j ¼ 1...F that model the defect response can be
expressed in terms of the impact excitation F 0

F j ¼ M 0j F 0 j ¼ 1...F ; ð 5 : 10 Þ

where M 0j is a diagonal matrix that has the vector formed by the samples of the
frequency response between the impact point and focus j at its main diagonal.
Combining Eqs. ( 5.9 ) and ( 5.10 ) and defining M 00 ¼ I, we can write

V i ¼ X

F

H ij F 0 H ij ¼ B ij M 0j ; ð 5 : 11 Þ

j ¼ 0

where H ij is a diagonal matrix that has the vector formed by the samples of the
frequency response modelling the path between the impact point and focus j, and
between the focus j and sensor i at its main diagonal.
The dimensionality of all the vectors and matrices involved in Eq. ( 5.11 )
depends on the size of the DFT used. For example, in the application of this work,
a size of 1024 is considered since it is large enough to both capture the transients
of the signals at the used frequency sampling and to have adequate frequency
resolution. Such long vectors imply a great computational burden and convey a lot