A further interesting observation regarding the bandlimitedness of
exp(a*t), which kind of confirms my previous conjecture. We are
considering a "periodic exp", which is a sawtooth, whose segments are
exponential rather than linear.
Consider the amplitudes of "unit" aliasing residuals (residuals for the
discontinuity functions obtained by successful integration of the Dirac
delta). Bandlimited Dirac delta is sinc(pi t/T), where sinc(t)=sin(t)/t.
Integrating successfully with respect to t we notice that the amplitudes
of the unit residuals fall off as (T/pi)^N/N. Where (T/pi)^N is just
obtained form a standard property of integration of a horizontally
stretched function and 1/N is obtained from the biggest term of the
formula for the Nth antiderivative of Si(t).
The derivatives of exp(a*t) fall off as a^N.
Thus, the amplitudes of the residuals needed for exp(a*t) discontinuity
bandlimiting fall off as (a*T/pi)^N/N. Therefore, if a*T<pi the sum of
the residuals will converge and exp(a*t) can be considered bandlimited,
otherwise not.
Notice that the same considerations can be applied to sin(a*t), which
fully coincides with the known condition for sin(a*t) being bandlimited:
sin(2*pi*f*t) is bandlimited iff f<1/2T
a=2*pi*f
a*T<pi iff 2*pi*f*T<pi iff 2*f*T<pi iff f<1/2T
Thus, it seems exp(a*t) is "bandlimited" exactly under the same condition.
--
Vadim Zavalishin
Reaktor Application Architect
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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