Delay Lines and Effects
57
with these architectures is immediate and free from complications.
3.2.2
Allpass Interpolation Filters
Another widely used technique to obtain the fractional part of a desired delay
length makes use of unitmagnitude IIR filters, i.e., allpass filters. Since the mag
nitude of these filters is constant there is no frequencydependent attenuation,
a property that can never be ensured by FIR filters. The simplest allpass filter
has order one, and it has the following transfer function:
H
a
(z) =
c + z
1
1 + cz
1
.
(10)
In order to make sure that the filter is stable, the coefficient c has to stay within
the unit circle. Moreover, if we stick with real coefficients, c belongs to the real
axis. The phase delay given by the filter (10) is shown in fig. 3 for several values
of the coefficient c. It is clear that the phase delay is not as flat as in the case
of the FIR interpolator, depicted in fig. 2.
0
1
2
3
3
2.5
2
1.5
1
0.5
0
frequency [rad/sample]
phase [rad]
0
1
2
3
0
0.5
1
1.5
2
2.5
frequency [rad/sample]
phase delay [samples]
Figure 3: Phase response and phase delay of a firstorder allpass filter for the
values of the coefficient c = 1.998k/17  0.999, k = 0, . . . , 16
that, at frequencies close to dc, the phase response of (10)
takes the approximate form
H() 
sin ()
c + cos ()
+
c sin ()
1 + c cos ()

1  c
1 + c
,
(11)
where the first approximation is obtained by replacing the argument of the
arctan with the function value and the second approximation, valid in an even
smaller neighborhood, is obtained by approximating sin x with x and cos x with
1. The phase and group delay around dc are
ph
()
gr
()
1  c
1 + c
.
(12)
Therefore, the filter coefficient c can be easily determined from the desired low
frequency delay as
c =
1 
ph
(0)
1 +
ph
(0)
.
(13)
2
The proof of (11) is left to the reader as a useful exercise.