As mentioned in §3.6, every audio signal can be regarded as infinitely differentiable due to the finite bandwidth of human hearing. That is, given any audio signal , its Fourier transform is given by
For the Fourier transform to exist, it is sufficient that be absolutely integrable, i.e., . Clearly, all audio signals in practice are absolutely integrable. The inverse Fourier transform is then given by
Because hearing is bandlimited to, say, kHz, sounds identical to the bandlimited signal
where . Now, taking time derivatives is simple (see also §C.1):
Since the length of the integral is finite, there is no possibility that it can ``blow up'' due to the weighting by in the frequency domain introduced by differentiation in the time domain.
A basic Fourier property of signals and their spectra is that a signal cannot be both time limited and frequency limited. Therefore, by conceptually ``lowpass filtering'' every audio signal to reject all frequencies above kHz, we implicitly make every audio signal last forever! Another way of saying this is that the ``ideal lowpass filter `rings' forever''. Such fine points do not concern us in practice, but they are important for fully understanding the underlying theory. Since, in reality, signals can be said to have a true beginning and end, we must admit that all signals we really work with in practice have infinite-bandwidth. That is, when a signal is turned on or off, there is a spectral event extending all the way to infinite frequency (while ``rolling off'' with frequency and having a finite total energy).^{E.2}
In summary, audio signals are perceptually equivalent to bandlimited signals, and bandlimited signals are infinitely smooth in the sense that derivatives of all orders exist at all points in time .