Orders 1 to 5 on a fractional delay of 0.4 samples
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 one-sample interval.
Figure
shows the
corresponding phase delays. As discussed in §4.2.10, the
amplitude response of every odd-order case is constrained to be zero at
half the sampling rate when the delay is half-way between integers,
which this example is near. As a result, the curves for the two
even-order interpolators lie above the three odd-order interpolators at
high frequencies in
Fig.
. It is
also interesting to note that the 4th-order interpolator, while showing
a wider ``pass band,'' exhibits more attenuation near half the sampling
rate than the 2nd-order interpolator.
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In the phase-delay 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 odd-order phase delay curves look generally better
in Fig. than
both of the even-order phase delays. Recall from
Fig.
that the
two even-order amplitude responses outperformed all three odd-order
cases. This illustrates a basic trade-off between gain accuracy and
delay accuracy. The even-order 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 half-sample
delay), but they pay for that with a relatively inferior phase-delay
performance at high frequencies.
Next Section:
Order 4 over a range of fractional delays
Previous Section:
Special Cases