Time-Bandwidth Products are Unbounded Above

We have considered two lower bounds for the time-bandwidth product based on two different definitions of duration in time. In the opposite direction, there is no upper bound on time-bandwidth product. To see this, imagine filtering an arbitrary signal with an allpass filter.^B.4 The allpass filter cannot affect bandwidth $\Delta\omega$, but the duration $\Delta t$ can be arbitrarily extended by successive applications of the allpass filter.