The Shannon upper bound places a limit on the error-free information transmission rate (capacity) of a noisy channel. It has stood for over sixty years, and underlies both theoretical and practical work in the telecommunications industry. This upper bound arises from the Shannon-Hartley law, which has two parameters: the available bandwidth and the signal-to-noise power ratio. However, aside from these explicit parameters, the Shannon-Hartley law also rests on certain assumptions. One of these is that the channel is linear: recent work has shown that nonlinear channels are not limited by the Shannon upper bound. A second assumption, arising from the mathematical tools used in its proof, is that signals are periodic. Surprisingly, the capacity limit associated with non-periodic signals has not previously been examined. Here we show, both theoretically and by construction, that the use of non-periodic signals, based on complex spirals, allows the Shannon upper bound to be extended.