Introduction to Acoustics

516 Part D Hearing and Signal Processing
Part D 14.7
which, for the one-pole filter, is Φ(ω) = tan −1 (−ωτ).
The phase shift is zero at zero frequency, and approaches
90 ◦ at high frequency. This phase behavior is typical
of simple filters in that important phase shifts occur
in frequency regions where the magnitude shows large
attenuation.
14.7.2 Phase Delay and Group Delay
The phase shifts introduced by filters can be interpreted
as delays, whereby the output is delayed in time
compared to the input. In general, the delay is different
for different frequencies, and therefore, a complex
signal composed of several frequencies is bent out of
shape by the filtering process. Systems in which the delay
is different for different frequencies are said to be
dispersive.
Two kinds of delay are of interest. The phase delay
simply reinterprets the phase shift as a delay. The phase
delay Tϕ is given by Tϕ =−Φ(ω)/ω.Thegroup delay Tg
is given by the derivative Tg =−dΦ(ω)/dω.Phaseand
group delays for a one-pole low-pass filter are shown in
Fig. 14.7 together with the phase shift.
14.7.3 Resonant Filters
Resonant filters, or tuned systems, have an amplitude response
that has a peak at some frequency where ω = ωo.
Such filters are characterized by the resonant frequency,
ωo, and by the bandwidth, 2∆ω. The bandwidth
is specified by half-power points such that |H(ωo +
∆ω)| 2 ≈|H(ωo)| 2 /2and|H(ωo −∆ω)| 2 ≈|H(ωo)| 2 /2.
The sharpness of a tuned system is often quoted as
a Q value, where Q is a dimensionless number given
Phase shift (rad)
2
1
0
0
Tg
Tj
2 4 6 8 10
ùô
Fig. 14.7 The phase shift Φ for a one-pole low-pass filter
can be read on the left ordinate. The phase and group delays
can be read on the right ordinate
j
2ô
ô
by
Q = ωo/(2∆ω) . (14.105)
As an example, a two-pole low-pass filter with a resonant
peak near the angular frequency ωo is described by the
transfer function
H(ω) =
ω2 o
ω2 o − ω2 . (14.106)
+ jωωo/Q
14.7.4 Impulse Response
Because filtering is described as a product of Fourier
transforms, i. e., in frequency space, the temporal representation
of filtering is a convolution
�
y(t) = dt ′ h(t − t ′ )x(t ′ �
) = dt ′ h(t ′ )x(t − t ′ ) .
(14.107)
The two integrals on the right are equivalent.
Equation (14.107) is a special form of linear processor.
A more general linear processor is described by the
equation
�
y(t) = dt ′ h(t, t ′ )x(t ′ ) , (14.108)
where h(t, t ′ ) permits a perfectly general dependence on
t and t ′ . The special system in which only the difference
in time values is important, i. e. h(t, t ′ ) = h(t − t ′ ), is
a linear time-invariant system. Filters are time invariant.
A system that operates in real time obeys a further
filter condition, namely causality. A system is causal if
the output y(t) depends on the input x(t ′ ) only for t ′ < t.
In words, this says that the present output cannot depend
on the future input. Causality requires that h(t) = 0for
t < 0. For the one-pole corona, low-pass filter of (14.101)
the impulse response is
h(t) = 1
for t > 0 ,
τ e−t/τ
h(t) = 0 fort < 0 ,
h(t) = 1
2τ
for t = 0 . (14.109)
For the two-pole low-pass resonant filter of (14.106),
the impulse response is
ωo
h(t) = �
1 −[1/(2Q)] 2
� �
×sin ωot 1 −[1/(2Q)] 2
�
, t ≥ 0 ,
e− ωo
2Q t
h(t) = 0 , t < 0 . (14.110)