Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 121

which is another form of Discrete Fourier Transform (compare ( 4.26 )) enabling for
transformation of the N 1 -element series in time domain into an equivalent N 1 -
element series in frequency domain (exact to the constant coefficient 2/N 1 ).

8.1.1.4 Correlation Method

The digital correlation consists in representing the considered real signal with a
series of mutually orthogonal functions [ 8 ]. For power-system protection appli-
cations such representation is mostly limited to the components of fundamental
frequency only. The expansion components can be obtained from ( 8.22 ) and ( 8.23 )
assuming calculation for the discrete frequency k = 1, which yields:

;

N 1 1

X

X 1C ¼ 2
N 1

n 2p
N 1

x ð n Þ cos ð 8 : 25 Þ

n ¼ 0

:

X

N 1 1

X 1S ¼ 2
N 1

n 2p
N 1

x ð n Þ sin ð 8 : 26 Þ

n ¼ 0

For a pure fundamental frequency signal the values resulting from ( 8.25 ) and
( 8.26 ) are:

X 1C ¼ X 1m cos ð u Þ;

ð 8 : 27 Þ

X 1S ¼ X 1m sin ð u Þ; ð 8 : 28 Þ

being constant, in contrast to the outputs of digital Fourier filters that are
time-depending functions. Nevertheless, also here the harmonic distortions (if
any) are completely rejected by the correlation algorithm and the obtained values
X 1C and X 1S are orthogonal, which enables calculation of signal magnitude, e.g.
with application of ( 8.50 ).
If the correlation algorithm is to be applied on-line, where time instant
n changes, it is more convenient to use the algorithms in the form:

;

N 1 1

X

X 1C ð n Þ¼ 2
N 1

x ð n m Þ cos ð n m Þ 2p
N 1

ð 8 : 29 Þ

m ¼ 0

:

X

N 1 1

X 1S ð n Þ¼ 2
N 1

x ð n m Þ sin ð n m Þ 2p
N 1

ð 8 : 30 Þ

m ¼ 0

The advantage of correlation approach is a possibility of obtaining simple
recursive algorithms. Observing that for the samples n and n - 1 the sums ( 8.29 )
and ( 8.30 ) differ only by two elements, one can derive the following recursive
relationships: