Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 82

parameters of frequency response of FIR filters may appear, which are not satisfied
with use of typical known filters. In such cases it is necessary to realize the full
synthesis of FIR filters, which means in general calculating their impulse response
(i.e. coefficients) from assumed frequency response.
The are many methods allowing to realize this task. Two of them are presented
here, [ 6 , 7 ]:

• the method applying complex Fourier series, together with special smoothing
filter windows, and
• the method applying Fast Fourier Transform (FFT), which can be used for any
arbitrary frequency response of the filter.

6.3.1 Application of Complex Fourier Series

It is well known from general theory that sampling process results in periodic
frequency response with a period equal to sampling angular frequency x S .This
periodic function (frequency response of sampled signal) can be described using
infinite complex Fourier series given by a pair of equations:

H i ð X Þ¼ X

1

h i ð n Þ exp ð jnX Þ; ð 6 : 41 Þ

n ¼1

Z

p

h i ð n Þ¼ 1
2p

H i ð X Þ exp ð jnX Þ dX ; ð 6 : 42 Þ

p

where H i ð X Þ is a filter frequency response, h i ð n Þ is a filter impulse response and its
coefficients, simultaneously.
This pair of equations and the second one especially can be used for direct
synthesis of FIR filters. However, some additional operations are needed, since the
Fourier series is infinite, but the filter designed is to be finite, having the number of
coefficients h i ð n Þ limited to N. With the method considered it is realized by ade-
quate cutting of infinite series of coefficients h i ð n Þ and leaving only N of them. The
operation can be considered as a product of this series of infinite coefficients and
rectangular window w ð n Þ :

h ð n Þ¼ h i ð n Þ w ð n Þ;
where w ð n Þ¼ 1 for 0 n N 1 :
0 for remaining n

ð 6 : 43 Þ

The result of this cutting is frequency response of the real filter H ð X Þ (having
N coefficients), different from an ideal filter H i ð X Þ ( 6.42 ). This real frequency
response can be calculated using equation: