Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 42

X ð jx Þ¼ X ð z Þj z ¼ exp ð jxT S Þ ¼ X

n ¼1

x ð n Þ exp ð jnxT S Þ: ð 4 : 21 Þ

n ¼1

It is worth noticing that since exp ð jxT S Þ¼ exp ½ j ð xT S þ 2kp Þ ¼ exp ½ j ð x þ
kx S Þ T S ; the substitution and the spectrum of sampled signal X * (jx) are periodic in
frequency domain with the period equal to the sampling angular frequency. This
feature is essential for determination of the sampling frequency for the signal at
hand, which is thoroughly explained in Chap. 2 .
The inverse transformation (from the frequency to discrete time domain) is
expressed by:

p = T S

Z

x ð n Þ¼ 1
2p

X ð jx Þ exp ð jnxT S Þ dx :

ð 4 : 22 Þ

p = T S

4.7 Discrete Fourier Transform

A starting point for determination of the equations of direct and inverse DFT can
be Z transform of sampled signal x(n) having limited number of N elements with
all remaining elements equal to zero. The Z transform of such a signal is then
given by the equation:

X ð z Þ¼ X

N 1

x ð n Þ z n : ð 4 : 23 Þ

n ¼ 0

If numerical calculations are made, there is no possibility to determine the
transform for all frequencies but only for certain limited, chosen set of values. It is
known from sampling theorem that one turn on Z plane is equivalent to passing
frequency range equal to sampling frequency. Taking that into account and
substituting:

z ¼ exp ð jxT S Þ¼ exp ð jX Þ

ð 4 : 24 Þ

for a set of equally spaced angular frequencies along the unit circle:

X k ¼ k 2p
N

; k ¼ 1...N ð 4 : 25 Þ

and calculating frequency response at these frequencies yields the transformation
in the form:

where k ¼ 0 ; 1 ; ... ; N 1 :

X ð k Þ¼ X

N 1

x ð n Þ exp jnk 2p
N

ð 4 : 26 Þ

n ¼ 0