Figures
4.15 and
4.16 show
amplitude response and
phase delay, respectively, for 4thorder
Lagrange interpolation
evaluated over a range of requested delays from
to
samples
in increments of
samples. The amplitude response is ideal (flat
at 0
dB for all frequencies) when the requested delay is
samples
(as it is for any integer delay), while there is maximum
highfrequency attenuation when the
fractional delay is half a sample.
In general, the closer the requested delay is to an integer, the
flatter the amplitude response of the Lagrange interpolator.
Figure 4.15:
Amplitude responses,
Lagrange interpolation, order 4, for the range of requested delays
, with thrown in as well (see next plot
for why). From bottom to top, ignoring the almost invisible split
in the bottom curve, the curves represent requested delays
, and . Then, because the curve for
requested delay is the same as the curve for delay
, for
, the same curves, from top to bottom,
represent requested delays
and
(which is nearly indistinguishable from ).

Figure 4.16:
Phase delays, Lagrange
interpolation, order 4, for the range of requested delays
, and additionally .

Note in Fig.
4.16 how the phasedelay jumps
discontinuously, as a function of delay, when approaching the desired
delay of
samples from below: The top curve in
Fig.
4.16 corresponds to a requested delay of 2.5
samples, while the next curve below corresponds to 2.499 samples. The
two curves roughly coincide at low frequencies (being exact at
dc),
but diverge to separate integer limits at half the
sampling
rate. Thus, the ``capture range'' of the integer 2 at half the
sampling rate is numerically suggested to be the halfopen interval
.
Next Section: Order 5 over a range of fractional delaysPrevious Section: Orders 1 to 5 on a fractional delay of 0.4 samples