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With sampling modeled as a multiplication with a dirac comb, we can recover the original signal perfectly with a low pass filter given that the sample-rate criterion is satisfied.

What about a more practical case where the signal is sampled with a pulse train, and not an impulse train? E.g. rectangular pulse train with varying duty cycle (99.999% duty cycle could represent an ideal CCD pixel array). Is there a way to recover the original signal perfectly? Any pointing directions to reading about this matter would be appreciated, the only term I could find is "aperture error" but it does not point to the generalized theoretical case. Thanks

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  • $\begingroup$ You mean that the actual sampled signal is something like $x_n = \int_{n T_s - T_w}^{n T_s} x(\tau) d\tau$, where $T_s$ is the sampling interval and $T_w$ is an integration window? $\endgroup$
    – TimWescott
    Dec 9, 2022 at 22:35

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When the signal is sampled with a pulse train, we multiply in time the signal with the pulse which is the same as convolving in frequency the Fourier Transform of the signal with the Fourier Transform of the pulse. The Fourier Transform of a pulse of width $T$ seconds is a Sinc with the first null in frequency at $1/T$ Hz. This convolution in frequency will "smear" the spectrum, and will be inconsequential if the width of the pulse is significantly less than the bandwidth of the signal.

However what is more likely in actual implementation is “aperture uncertainty” which doesn’t mean that digitally what I described above actually occurs— for every aperture there will only be one resulting sample that will be saved as the representative sample for that duration in time. The aperture uncertainty refers to time variation from sample to sample (jitter) in the sample that is used and results in a noise source that limits the achievable SNR for that data converter. The uncertainty will be a white noise source (if indeed independent from sample to sample) similar to the quantization noise which is also white, and will therefore degrade the achievable precision (reflected as part of the “Effective Number of Bits” or ENOB on the ADC datasheet.) This is a limit of the ADC itself and cannot externally be improved, but sampling clock jitter from the sampling clock itself will independently add with this and cause even further degradation in SNR if not carefully controlled.

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  • $\begingroup$ thanks. what about the case of high duty cycle sampling ? is there any mathematical framework for perfect signal reconstruction ? $\endgroup$ Dec 9, 2022 at 18:11
  • $\begingroup$ Sorry I had continued the post but didn’t save- Please reread the update above $\endgroup$ Dec 9, 2022 at 21:02

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