Assume, I am trying to use a 50khz sampling rate for some signal. I don't know what the spectrum of input signal is, so I will have to low pass manually with sinc (theoretical best low pass) to band limit it to 25khz.
Ignoring noise for now, assume my input signal is a 5khz signal starting at time = 5s, all values before that 5s is 0. Now to sample this, I would theoretically be passing this over a sinc low pass filter at 25khz and then sampling right?
At steady state (time = infinity) when I apply this low pass my output will just be the same sine at 5khz but with same amplitude. At steady state, no matter how much time I apply this low pass I'll still get a 5khz sine with same amplitude, because I have a train of 5khz sines preceding and succeeding it.
But during transients, ie at 5s areas I will be getting this sine smeared due to low pass right? In these transient scenarios, I am thinking every instance of such low passing will smear it further.
Would convolving a sinc used for band limiting at 25khz with itself would give me a result that explains this smearing? An ideal infinite bandwidth sinc would just have it's time domain property as a rectangular pulse, so multiplying them with each other will still return a rectangular pulse. Convolving two identical infinite bandwidth sincs will give back a result that's the same as the original sinc. Bandwidth is infinite and all transients are preserved, so it makes sense. In fact I'm getting back just the original signal.
But for a band limited sinc, the time domain property would also have Gibbs phenomenon on top of the rectangular pulse right, and if I multiply it with itself I'll actually get it even more smeared is my understanding. If I convolve two identical band limited sinc (limited at say 25khz) would the result be the same as original or would it be more smeared?
Practical scenarios of non infinite time is likely to have even more deviations, but for now I would like to know how the response would be for the kind of signal and process described above.
Note: there is no jump discontinuity. The sine just starts from time 5s, the value at 5s is 0. Till 5s the whole signal is a dc value of 0, and after 5 second it becomes a 5khz sine having value 0 at 5seconds.