No-Go Theorem for Linear Systems on Bounded Bandlimited Signals

In this paper we analyze the existence of efficient bandpass-type systems for the space of bounded bandlimited signals. Here efficient means that the system fulfills the following properties: every output signal contains only frequencies within the passband; every input signal that has only frequencies within the passband is not disturbed by the system; and the system is stable. Without using any further assumptions, such as time-invariance, we prove that a linear realization cannot exist. Moreover, we show that a nonlinear realization is possible. It is well-known that every signal with finite energy can be split into two signals with finite energy, each of which contains a different part of the spectrum. Surprisingly, this does not hold for the space of bounded bandlimited signals. It is shown that there exist bounded bandlimited signals that cannot be split in the above way. These results can be of relevance for all applications where filters are used and the peak value of the signals is decisive, e.g., the design of efficient power amplifiers in wireless communication systems. The no-go results in this paper are helpful to better understand the signal space of bounded bandlimited signals and the limits of signal processing operations on this space.