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The following sentences are from the footnote of page 240 in R.C. Gonzalez's Digital Image Processing (3rd):

An important special case is when a function that extends from -Inf to +Inf is band-limited and periodic. In this case, the function can be truncated and still be band-limited, provided that the truncation encompass- es exactly an integral number of periods. A single truncated period (and thus the function) can be repre- sented by a set of discrete samples satisfying the sampling theorem, taken over the truncated interval.

Isn't these contradicting to the band-limiting theorem that a bandlimited signal cannot be also timelimited?

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  • $\begingroup$ No, there is no contradiction. The details why are detailed in this post: dsp.stackexchange.com/questions/52288/…. Due to the conditions of being periodic (and therefore discrete in frequency) and being sampled in time (and therefore also periodic in frequencies) make this so. Also note 'can be represented', and with this in mind read the post that I linked. $\endgroup$ Oct 2, 2018 at 3:56
  • $\begingroup$ Thank you for the linked answer. According to my understanding, does it mean that the extra information how the original function is like and how the truncating is done are necessary for reconstructing the original signal? $\endgroup$
    – 8cold8hot
    Oct 2, 2018 at 5:38
  • $\begingroup$ No extra information is needed; for a finite sampled sequence it is implied that it is equivalent to only one possible periodic infinite sequence, the sequence whose period is the base function, or if the base function is itself repeating, the base within that. Robert stated it very well. $\endgroup$ Oct 2, 2018 at 11:10

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there is some choice of words i would do differently.

yes truncating the infinite bandlimited function makes it time-limited and then band-unlimited. but truncating it to an integer number of periods still allows one to fully describe it with discrete samples.

a periodic function can be described with a Fourier series. the Fourier coefficients will virtually entirely describe the periodic function.

a bandlimited and periodic function can be completely described with a finite set of Fourier coefficients. also a bandlimited and periodic function can be completely described by sampling a single period with a sufficient number of samples that are spaced closer in time than $\frac{1}{2B}$ where $B$ is the bandlimit.

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  • $\begingroup$ Thanks a lot for the answer. These are more clear than the original statement. $\endgroup$
    – 8cold8hot
    Oct 2, 2018 at 5:30

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