Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 66

H ð z Þ¼ X

N 1

a ð k Þ z k : ð 6 : 3 Þ

k ¼ 0

The frequency response of the FIR filter is obtained by substituting e ð jxTs Þ for

z in Eq. 6.3 , which for FIR filter yields:

H ð jx Þ¼ X

N 1

a ð k Þ e jkxT S :

ð 6 : 4 Þ

k ¼ 0

The other advantage of FIR filters is a possibility to design the filters having
linear phase (versus frequency), allowing to obtain a pair of orthogonal filters
easily.
It can be proved that FIR filter has linear phase when its impulse response is
either even or odd function of k, which may be written in the form:

a ð k Þ¼ a ð N 1 k Þ: ð 6 : 5a Þ

a ð k Þ¼ a ð N 1 k Þ: ð 6 : 5b Þ

It may also be proved that each of the filters fulfilling the conditions ( 6.5 ) has
linear phase and, moreover, that the phase difference between their phase equals
p = 2 for any frequency.
The frequency response of a FIR filter can also be given in the form:

H ð jX Þ¼ X

N 1

a ð k Þ exp ð jkX Þ; ð 6 : 6 Þ

k ¼ 0

where X ¼ xT S ¼ 2pf = f S is a ratio of angular frequency to sampling frequency.
The frequency transfer function ( 6.6 ) can be analyzed in order to prove the filter
phase linearity as well as orthogonality for a pair of filters fulfilling the symmetry
conditions. Rearranging ( 6.6 ) one gets:

H ð jX Þ¼ X

N 1

a ð k Þ exp ð jkX Þ

k ¼ 0

¼ X

N = 2 1

f a ð k Þ exp ½ jkX þ a ð N 1 k Þ exp ½ j ð N 1 k Þ X g

k ¼ 0

and then:

X X X

N = 2 1

N 1
2

N 1
2

H ð jX Þ¼ exp j a ð k Þ exp j k

k ¼ 0

)

X

N 1
2

exp j k ð 6 : 8 Þ