It is no longer uniform. It's counter-intuitive (unless you've really internalised the Fourier transform and/or the Shannon-Hartley theorem) but a pure sine wave stops being a pure sine wave if you key it on and off and occupies progressively more bandwidth as the keying rate increases.
An even less intuitive result is that you can decode a signal that is weaker than the noise floor if the data rate is sufficiently low and/or the bandwidth is sufficiently high. This has practical applications in amateur modes like JT65, ultra-wideband communications and even GPS.
You can see it happening in! [1] is a waterfall display (time is vertical axis, frequency is horizontal) of a few CW signals and compare the harsh braodband clicks on the right to the nice dotted lines on the left. That kind of broadband noise happens when your signal goes from on to off too fast (or something else like just not generating a clean sine wave). If your radio can shape your keying to have a little ramp-up/ramp-down you get a much cleaner looking signal like those on the left.
The noise is effectively AM, since you are modulating the signal from 0 to full amplitude, and with the very fast amplitude change you get what looks like characteristic AM signal with a center carrier and symmetric sidebands.
An even less intuitive result is that you can decode a signal that is weaker than the noise floor if the data rate is sufficiently low and/or the bandwidth is sufficiently high. This has practical applications in amateur modes like JT65, ultra-wideband communications and even GPS.