Extended Instantaneous Spectral Analysis (E-ISA) for Advanced Signal Processing

This work is devoted to introducing an extended form of the instantaneous spectral analysis (ISA) technique recently reported. ISA is based on the extension of Euler’s formula
$e^{jx}=\cos (x)+j\sin (x)$
to fractional powers of
$j$
of the special form
$e^{j^{a}x}$
with
$a=2^{(2-m)}$
and integer values of
$m$
. Here, we generalize the technique to the case where the exponent
$a$
in the extended Euler’s formula is an arbitrary fraction
${m}/{n}$
, where
$m$
and
$n$
are integers with no common factors. Hence, we propose an extended form of ISA (E-ISA), which is not restricted to the values of
$a= 0.5, 0.25, 0.125\ldots$
Compared to the standard Fourier transform (FT), E-ISA is shown to be more suitable in dealing with nonergodic signal sources having nonstationary spectra. In addition, the bandwidth utilized by E-ISA is much smaller than the corresponding one in standard FT. Application of E-ISA to synthetic data, an electrocardiogram signal, and nonstationary current time series data from a submerged electrochemical microplasma system are demonstrated.