Given an infinite number of samples $(N)$, a higher (or lower) number of samples $(cN)$ can be derived using sinc interpolation followed by sampling. How can this be applied to finite length signals?
With $\mathrm{sinc}$ interpolation, one can derive a continuous-time signal as:
$$y(t) = \sum^{\infty}_{n=-\infty} y[n]\mathrm{sinc}\left({t\over T}-n\right)$$
- For a finite number of sample points, should (can) we consider the $x[n]$ in the picture as
$$y[n] = \begin{cases} x[n], & \text{if } n \in [0, N-1] \\0, & \text{otherwise} \end{cases} $$
- Or should $y[n]$ be considered as a periodic version of $x[n]$? (This link briefly addresses the same. The original stated form cannot be directly used with periodic signals)
$$y[n] = x\left[n\pmod N\right]$$
In the first consideration, outside the region $[0,\ N-1]$, if I understand correctly, the Gibb's phenomenon would result in a ringing effect. Would this completely invalidate any values predicted outside the non-zero region or is it only that the degree of inconsistency is high? (More specifically for points close to but just outside the boundary in the interpolated continuous-time signal)
I am interested to know whether the addition of zeros would pollute the input set of points during the interpolation stage.
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