Abstract
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.