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Why discreteness in time / frequency domain dictates periodicity in the other frequency / time domian?

  • For example the DTFT is perodic in frequency?
  • Why it doesn't contain all the frequencies?
  • Why the DTFS contain only finite number of frequencies?

NOTE:

I've read the derivation but sometimes i see it's like a definition or a claim in the case of DTFS.

  • How we prove that equation of DTFS ? $$ x[n] = \sum_{k=0}^{N-1} a_k e^{jl\omega_o n} $$
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This is, essentially, what the sampling theorem is about. Uniform sampling in one domain (e.g. the time domain) causes periodic extension in the reciprocal domain (e.g. frequency domain).

The reason why is that the sampling function is a periodic function which means it can be represented as a Fourier series

$$\begin{align} \operatorname{III}_T(t) \ &\triangleq\ \sum_{k=-\infty}^{\infty} \delta(t - k T) \\ &= \sum_{n=-\infty}^{+\infty} c_n e^{j 2 \pi n \frac{t}{T}} \\ \end{align}$$

where the Fourier coefficients are

$$ \begin{align} c_n\, & = \frac{1}{T} \int_{t_0}^{t_0 + T} \operatorname{III}_T(t) e^{-j 2 \pi n \frac{t}{T}}\, dt \quad ( -\infty < t_0 < +\infty ) \\[4pt] & = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \operatorname{III}_T(t) e^{-j 2 \pi n \frac{t}{T}}\, dt \\[4pt] & = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \delta(t) e^{-j 2 \pi n \frac{t}{T}}\, dt \\ & = \frac{1}{T} e^{-j 2 \pi n \frac{0}{T}} \\[4pt] & = \frac{1}{T} \ . \end{align} $$

Using the definition of Fourier Transform most common with electrical engineering

$$ X(f) \triangleq \int\limits_{-\infty}^{+\infty} x(t) e^{-j 2 \pi f t} \ dt $$ $$ x(t) = \int\limits_{-\infty}^{+\infty} X(f) e^{+j 2 \pi f t} \ df $$

when sampling $x(t)$

$$\begin{align} x_\text{s}(t) &\triangleq x(t) \cdot T \cdot \operatorname{III}_T(t) \\ &= x(t) \cdot T \sum_{k=-\infty}^{\infty} \delta(t - k T) \\ &= T \sum_{k=-\infty}^{\infty} x(t) \ \delta(t - k T) \\ &= T \sum_{k=-\infty}^{\infty} x(kT) \ \delta(t - k T) \\ \end{align}$$

which shows how $x(t)$ is converted to samples $x(kT)$ and this is also true:

$$\begin{align} x_\text{s}(t) &\triangleq x(t) \cdot T \cdot \operatorname{III}_T(t) \\ &= x(t) \cdot T \cdot \sum_{n=-\infty}^{+\infty} \frac1T e^{j 2 \pi n \frac{t}{T}} \\ &= \sum_{n=-\infty}^{+\infty} x(t) \ e^{j 2 \pi n \frac{t}{T}} \\ \end{align}$$

the resulting Fourier transform is

$$ X_\text{s}(f) = \sum_{n=-\infty}^{+\infty} X\left(f - \frac{n}{T} \right) $$

which is periodic in the frequency domain with period $\frac{1}{T}$.

Because of the Duality of the Fourier Transform, if can be also shown that sampling in the frequency domain causes periodicity in the time domain, which is essentially all that Fourier series is about.

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  • $\begingroup$ Thanks for your effort, partially answered my question, i read from "Oppenhieum signals and systems" and he proved all of the transforms before the sampling theorem,but it was periodic of 2 PI not 1/T maybe he normalized it, anyway why the DTFS contains only finite frequencies? he justified that by the fact that discrete exponentials when the frequency increases it repeats itself after 2PI so we will take only a finite frequencies, i;m not fully convinced with DTFS and proving the transform without using the sampling method. $\endgroup$ – Mohamed Osama Sep 9 '16 at 2:06
  • $\begingroup$ well, i hadn't dealt with the $2 \pi$ scaling issue. that has to do with different definitions of the Fourier Transform and different definitions of the quantity we call "frequency" (as in "regular frequency" $f$ vs. "angular frequency" $\omega$). you should learn the difference between the two and that will explain all $2 \pi$ related questions. otherwise i fully answered the question. $\endgroup$ – robert bristow-johnson Sep 9 '16 at 2:43
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Regarding your first statement, discreteness in time or frequency domain does not always dictate periodicity in the other domain. Only certain kinds of discrete sets do. For instance, a set of impulses that are spaced apart by relatively irrational values will not be periodic in the other domain.

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  • $\begingroup$ The restrictions on the properties of those sets of impulses that do produce periodicity in the other domain may hint at the answer to your following questions. $\endgroup$ – hotpaw2 Sep 9 '16 at 5:29
  • $\begingroup$ if the spaces between the impulses are irrational then they are considered continuous then not discrete ? $\endgroup$ – Mohamed Osama Sep 9 '16 at 12:12
  • $\begingroup$ the issue is uniformity of spacing between the samples. if the spacing between samples, $T$, are all $\sqrt{2}$ units of time, then the spectrum is periodic in frequency. uniform sampling in one domain causes periodic extension in the reciprocal domain. $\endgroup$ – robert bristow-johnson Sep 9 '16 at 13:55

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