Implementing IIR/FIR Filters

G( θ)
hi ()e jiθ –
N– 1
=
i = 0
Eqn. 7-18
The phase response, φ(θ), is found by taking the inverse
tangent of the ratio of the imaginary to real
components of G(θ) so that:
φθ ( )=tan 1 –
where: h(i) is implicitly assumed to be real
Eqn. 7-19
Intuitively, it can be reasoned that, for any pulse to
retain its general shape and relative timing (i.e.,
pulse width and time delay to its peak value) after
passing through a filter, the delay of each frequency
component making up the pulse must be the same
so that each component recombines in phase to reconstruct
the original shape. In terms of phase, this
implies that the phase delay must be linearly related
to frequency or:
φθ ( ) =
– τθ
Eqn. 7-20
This relationship is illustrated in Figure 7-2 in which
a pulse having two components, sin ωt and sin 2ωt,
passes through a linear-phase filter having a delay
of two cycles per Hz and a constant magnitude response
equal to unity. That is, the fundamental is
7-4 MOTOROLA
∑
N– 1
= ∑ [ hi () cosiθ–
jh() i siniθ]
i = 0
N– 1
– ∑ hi () siniθ
i =
--------------------------------------
0
N– 1
∑
hi () cosiθ
i = 0