Does it Work?

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Does it Work?

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Does it Work?

It is straightforward to show that the “additive synthesis” reconstruction method of the previous section actually works exactly (in the periodic case) in the following sense:

  • The reconstructed signal $x(t)$ is band-limited to $[-\, i.e., its Fourier transform $X(\ is zero for all $\. (This is not quite true in the truncated case.)
  • The reconstructed signal $x(t)$ passes through the samples $x(t_n)$ exactly. (This is true in both cases.)

Is this enough? Are we done? Well, not quite. We know by construction that $x(t)$ is a band-limited interpolation of$x(t_n)$. But are band-limited interpolations unique? If so, then this must be it, but are they unique? The answer turns out to be yes, based on Shannon’s Sampling Theorem. The uniqueness follows from the uniqueness of the inverse Fourier transform. We still have two different cases, however, depending on whether we assume periodic extension or zero extension beyond the range $n\. In the periodic case, we have found the answer; in the zero-extended case, we need to use the sum-of-sincs construction provided in the proof of Shannon’s sampling theorem.

Why do the DFT sinusoids suffice for interpolation in the periodic case and not in the zero-extended case? In the periodic case, the spectrum consists of the DFT frequencies and nothing else, so additive synthesis using DFT sinusoids works perfectly. A sum of $N$DFT sinusoids can only create a periodic signal (since each of the sinusoids repeats after $N$ samples). Truncating such a sum in time results in all frequencies being present to some extent (save isolated points) from $\ to $\. Therefore, the truncated result is not band-limited, so it must be wrong.

It is a well known Fourier fact that no function can be both time-limited and band-limited. Therefore, any truly band-limited interpolation must be a function which has infinite duration, such as the sinc function $\ used in bandlimited interpolation by a sum of sincs. Note that such a sum of sincs does pass through zero at all sample times in the “zero extension” region.

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