Delay Lines and Effects
55
effective delay length
at the points where the phase response equates the multiples of 2. At these
frequencies, the components reappear in phase every round trip in the loop,
thus reinforcing their amplitude at the output. The phase delay at frequency
f
k
is therefore the effective delay length at that frequency, that is the length
of an ideal (linear phase) delay line that gives the same k-th resonance. Fig. 1
shows a phase curve and its crossings with multiples of 2 giving a distribution
of resonances.
arg(H)
2
4
6
1
2
3
Figure 1: Graphical construction to find the series of resonances produced by
a linear block in a feedback loop. The slope of the dashed lines indicates the
phase delay at each resonance frequency.
3.2.1
FIR Interpolation Filters
The easiest and most intuitive way to obtain a variable-length delay is to linearly
interpolate the output of the line with the content of its preceding cell in the
memory buffer. This corresponds to using the first-order FIR filter
H
l
(z) = c
0
+ c
1
z
-1
.
(7)
Given a certain phase delay
ph
0
= -
1
0
arctan
-c
1
sin
0
c
0
+ c
1
cos
0
(8)
that has to be obtained at a given frequency
0
, the following formulas give the
coefficient values:
c
0
+ c
1
= 1
c
1
=
1
1+
sin (0)
tan (ph0 0)
-cos (
0
)
ph
0
,
(9)
where the approximation is valid in the low-frequency range. The first of the (9)
is needed in order to normalize the low-frequency response to one. In the special
case that c
0
= c
1
=
1
2
(averaging filter) the phase is linear and the delay is of
half a sample. Unfortunately, the magnitude response of this interpolator is
lowpass with a zero at the Nyquist frequency. Fig. 2 shows the magnitude,
phase, and phase delay responses for several first-order linear interpolators. We
can see that the phase is linear in most of the audio range, but the magnitude
varies from the allpass to the lowpass with a zero at the Nyquist rate. When
the interpolator is inserted within a feedback loop, its lowpass behavior can be