I understand that a pure sine wave of infinite duration occupies no bandwidth, i.e. it is only the modulation of a carrier that gives it sidebands. Does the exact timing of a sudden modulation make any difference?
For simple on-off keying of a signal, it looks like https://dsp.stackexchange.com/a/24604/18819 is giving an equation to determine the relative power away from a carrier:
The spectrum of your windowed sinusoid is just a shifted sinc function
…and the magnitude of this function is based only on the length of the windowed signal. In all discussions I've seen (e.g. CW bandwidth described) they discuss things like keying envelopes and such to reduce the splattering of significant power away from the intended carrier frequency. They don't discuss the precise timing of the envelope.
I'm wondering: does it make any difference at which phase of its cycle I "key" a sine wave on or off?
The Wikipedia article on Zero-crossing circuitry explains:
…the switching device will "wait" to switch on until the output AC wave reaches its next zero point. This is useful when sudden turn-on in the middle of a sine-wave half cycle could cause undesirable effects like high frequency spikes for which the circuit or the environment is not expected to handle gracefully
Is this just a practical concern (i.e. non-ideal components in a real-world circuit) or does the spectrum actually change if I make a "chopped-off sinewave" from 0º to 360º, vs. a section from 90º to 450º? What about from 0º to 180º vs. 90º to 270º?! [Update: to be clearer, maybe I'm really meaning in each case e.g. "from 90º to (360n+270)º" for some relatively large n so that the signal lasts for a bit between start/end.]
In both of those pairings, the signal which starts and ends at 0 magnitude certainly seems "cleaner" than a signal suddenly starting like in the last example at +1 and ending suddenly at -1. But can this difference be demonstrated mathematically? Each pair starts/ends the sine wave "instantaneously" and they both keep it running for the same duration, so are they equivalent as far as "splatter"?