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.

  • $\begingroup$ In a practical sense you can neither low pas filter with a Sinc nor to 25 KHz for 50 KHz sampling. You need infinite time duration to accomplish both tasks. This means you will trade filter complexity and delay for the actual maximum feasible passband with 50 KHz sampling. 20 KHz would be a good number and then review the analog filter complexity to achieve your anti aliasing targets and decide if you can go higher or not. I wasn’t sure if this would change the rest of your question so wanted to hear what you had to say about that before reading the rest. $\endgroup$ Oct 1 '20 at 2:58
  • $\begingroup$ Thank you. I do understand the practical limitations, a practical sinc filter is unrealisable from what I know. But I am trying to understand the theoretical limits, especially relating to transient scenarios. A simpler way to ask my question would be "Would convolving an ideal infinite time sinc used for band limiting at 25khz with itself would give me a result same as original or would it be more smeared"? I wrote down scenarios just to give additional context $\endgroup$ Oct 1 '20 at 3:13
  • $\begingroup$ I'm not sure I understand your question: do you have several identical filters in series? If that is the case, it doesn't work: a sinc filtered with itself is exactly the same sinc. In other words, two identical ideal LPFs behave exactly the same as one LPF. $\endgroup$
    – MBaz
    Oct 1 '20 at 14:55
  • $\begingroup$ Yep you're right. Thanks. I did a mistake. For infinite time band limiting sinc the Fourier transform will give a rectangular output. Multiplying unit rectangular function with itself will give same unit rectangular function. Only for truncated sinc would I start to enter into deviations. For my scenario I should try to evaluate convolution of perfect/imperfect sinc with a heaviside filter (or multiplication of their fft). Then I'll be able to understand how it behaves in transient instances (theoretically). $\endgroup$ Oct 1 '20 at 17:51

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