Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 30

(b) Fourier integral that can be applied to analyze aperiodic (decaying) signals

which are also called energy signals, i.e. which have limited energy

E ¼ Z

1

x 2 ð t Þ dt\ 1:

1

The Fourier integral can also be applied to periodic signals being power
signals when impulse Dirac function is used. It is said that Fourier integrals
of these signals exist in distribution meaning. The so called sifting feature of
delta function is utilized here, which can be shortly expressed in the fol-
lowing way:

Z

1

x ð t Þ d ð t s Þ dt ¼ x ð s Þ:

1

(c) Laplace transform that is applied for continuous and sub-continuous signals

as well as for solving of the differential and partial differential equations
which arise in many engineering problems and describe the majority of linear
time invariant dynamical systems. The Laplace transform is closely related
to the Fourier transform that is equivalent to evaluating the bilateral Laplace
transform with complex argument s = jx. This relationship between the
Laplace and Fourier transforms is often used to determine the frequency
spectrum of a signal or dynamical system.

(d) Z transform that delivers frequency domain representation of a discrete time

domain signal, being a sequence of real or complex numbers. The Z trans-
form can be considered as a discrete equivalent of the Laplace transform.

(e) Fourier transform of the discrete signal that is applied for analysis of signals

with non-zero values only at discrete (usually equi-distant) time instants. The
signals can be treated either directly as discrete or as being an effect of
continuous signals sampling. This transformation is in a sense equivalent to
the complex Fourier series that is used for analysis of periodic signals, while
similarity refers here to periodicity in frequency domain.

(f) Discrete Fourier Transform and its fast version—Fast Fourier Transform

that can be applied for discrete signals with limited number of samples
0 \ n \ N. Utilization of DFT is possible for aperiodic signals only and its
result is also a periodic function in frequency domain. Unlike traditional
Fourier transform where discrete time domain signal is transformed into
continuous frequency domain function, in case of the DFT both signal and its
transform have discrete representation in time and frequency domains,
respectively.