Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 35

and the magnitude of this spectrum is equal:

sin ð p Þ
p

j X ð jx Þ j ¼ T

and has zero points at p = kp, i.e. for angular frequencies x k ¼ 2pk = T (see
Fig. 4.2 b).

As it was mentioned above all periodic signals can be represented and analyzed
with Fourier series, while the aperiodic ones—with use of Fourier integral. For
practical applications it would be convenient and favorable if the same technique
could be utilized for possibly wide class of signals. Since the Fourier series must
not be applied for non-periodic signals, one may eventually try applying the
Fourier integral for periodic (power) signals. Such an extension can be reached
through appropriate substitution of the complex exponential function (component
of the Euler's equations for sin, cos) with the delta Dirac function. Taking
advantage of the sifting feature of delta Dirac function one may arrive at:
Z

1

d ð x x 0 Þ exp ð jxt Þ dx ¼ exp ð jx 0 t Þ; ð 4 : 5 Þ

1

and by comparison of ( 4.5 ) with ( 4.4 ) one gets:

F 1 f 2pd ð x x 0 Þg ¼ exp ð jx 0 t Þ; i.e. F f exp ð jx 0 t g¼ 2pd ð x x 0 Þ;
where F{} denotes simple Fourier transform ( 4.3 ) and F -1 {} stands for inverse
transform ( 4.4 ).

Example 4.3 Determine the Fourier transform of the periodic function A cos ð x 0 t Þ:
Solution With application of the Euler's substitutions one obtains:

x ð t Þ¼ A cos ð x 0 t Þ¼ A

2 ½ exp ð jx 0 t Þþ exp ð jx 0 t Þ:
Now, applying the formula ( 4.5 ) to both exponential components one gets the
sought Fourier transform:

F f x ð t Þg ¼ X ð jx Þ¼ pA ½ d ð x x 0 Þþ d ð x þ x 0 Þ:

The signal and its transform are presented in Fig. 4.3 .

4.4 Laplace Transform

The continuous signal x in time domain can be transformed into frequency domain
by calculation of the integral: