Figure
shows the
amplitude responses of
Lagrange interpolation, orders 1 through 5, for
the case of implementing an
interpolated delay line of length
samples. In all cases the interpolator follows a
delay line of
appropriate length so that the interpolator coefficients operate over
their central onesample interval.
Figure
shows the
corresponding
phase delays. As discussed in §
4.2.10, the
amplitude response of every oddorder case is constrained to be zero at
half the
sampling rate when the delay is halfway between integers,
which this example is near. As a result, the curves for the two
evenorder interpolators lie above the three oddorder interpolators at
high frequencies in
Fig.
. It is
also interesting to note that the 4thorder interpolator, while showing
a wider ``pass band,'' exhibits more attenuation near half the
sampling
rate than the 2ndorder interpolator.
Figure 4.13:
Amplitude
responses, Lagrange interpolation, orders 1 to 5, for an
interpolated delay of samples. From the bottomright corner
along the right edge, the curves represent orders 1,3,5,4,2.

Figure 4.14:
Phase
delays, Lagrange interpolation, orders 1 to 5, for an interpolated
delay of samples. From bottom to top, the curves represent
orders 2,4,1,3,5.

In the phasedelay plots of
Fig.
, all cases
are exact at frequency zero. At half the sampling rate
they all give 5 samples of delay.
Note that all three oddorder phase delay curves look generally better
in Fig.
than
both of the evenorder phase delays. Recall from
Fig.
that the
two evenorder amplitude responses outperformed all three oddorder
cases. This illustrates a basic tradeoff between gain accuracy and
delay accuracy. The evenorder interpolators show generally less
attenuation at high frequencies (because they are not constrained to
approach a gain of zero at half the sampling rate for a halfsample
delay), but they pay for that with a relatively inferior phasedelay
performance at high frequencies.
Next Section: Order 4 over a range of fractional delaysPrevious Section: Special Cases