Back to the theoretical side, consider again the DFT reconstruction formula in~(\ref{eq:fa:idft}) (or, equivalently, the ``machine'' in Figure~\ref{fig:fa:gen}); normally, the expression is defined for $0\leq n < N$ but, if the index $n$ is outside of this interval, we can always write $n = mN + i$ with $m \in\mathbb{Z}$ and $i= n \mod N$. With this,
In other words, due to the aliasing property of the complex exponential, the inverse DFT formula generates an infinite, \emph{periodic} sequence of period $N$; this should not come as a surprise, given the $N$-periodic nature of the basis vectors for $\mathbb{C}^N$. Similarly, the DFT analysis formula remains valid if the frequency index $k$ is allowed to take values outside the $[0, N-1]$ interval and the resulting sequence of DFT coefficients is also an $N$-periodic sequence.
The natural Fourier representation of periodic signals is called the Discrete Fourier Series (DFS) and its explicit analysis and synthesis formulas are identical to~(\ref{eq:fa:dft}) and~(\ref{eq:fa:idft}), modified only with respect to the range of the indexes, which now span $\mathbb{Z}$; the DFS represents a change of basis in the space of periodic sequences $\tilde{\mathbb{C}}^N$. Since there is no mathematical difference between the DFT and the DFS, it is important to remember that even in the space of finite-length signals everything is implicitly $N$-periodic.
\subsection{Circular shifts revisited}
In Section~\ref{sec:dt:operators} we stated that circular shifts are the ``natural'' way to interpret how the delay operator applies to finite-length signals; considering the inherent periodicity of the DFT/DFS, the reason should now be clear. Indeed, the delay operator is always well-defined for a periodic signal $\mathbf{\tilde{x}}$ and, given its DFS $\mathbf{\tilde{X}}$, the $k$-th DFS coefficient of the sequence shifted by $m$ is easily computed as
in other words, a delay by $m$ samples in the time domain becomes a linear phase shift by $-2\pi m/N$ in the frequency domain.
With a finite-length signal $\mathbf{x}$, for which time shifts are not well defined, we can still always compute the DFT, multiply the DFT coefficients by a linear phase shift and compute the inverse DFT. The result is always well defined and, by invoking the mathematical equivalence between DFT and DFS, it is straightforward to show that
which justifies the circular interpretation for shifts of finite-length signals.
\subsection{DFT of multiple periods}
All the information carried by a $N$-periodic discrete-time signal is contained in $N$ consecutive samples and therefore its complete frequency representation is provided by the DFS, which coincides with the DFT of one period. This intuitive fact is confirmed if we try to compute the DFT of $L$ consecutive periods:
The above results shows that the DFT of $L$ periods is obtained simply by multiplying the DFT coefficients of one period by $L$ and appending $L-1$ zeros after each one of them.
\subsection{Pushing the DFS to the limit}
In the next section we will derive a complete frequency representation for aperiodic, infinite-length signals, which is still missing; but right now, the DFS can help us gain some initial intuition if we imagine such signals as the limit of periodic sequences when the period grows to infinity. Consider an aperiodic, infinite-length and absolutely summable sequence $\mathbf{x}$; given any integer $N > 0$ we can always build an $N$-periodic sequence $\tilde{\mathbf{x}}_N$, with \index{periodization}
the convergence of the sum for all $n$ is guaranteed by the absolute summability of $\mathbf{x}$ (see also Example~\ref{exa:dt:periodization}). As $N$ grows larger, the copies that make up the periodic signal will be spaced further and further apart and, in the limit,
in the above, we have used the definition of $\tilde{\mathbf{x}}_N$ and exploited the fact that $e^{-j(2\pi/N)nk} = e^{-j(2\pi/N)(n+pN)k}$. Now, for every value of $p$ in the outer sum, the argument of the inner sum varies between $pN$ and $pN + N -1$ so that the double sum can be simplified to
it is immediate to see that, for every value of the period $N$, the DFS coefficients of $\tilde{\mathbf{x}}_N$ are given by regularly spaced samples\footnote{
Once again, please remark the usual duality between time and frequency: in Section~\ref{sec:fa:dft_elementary} we showed that a maximally compact time-domain signal possesses a maximally wide spectrum, and vice-versa; here we notice how increasing the period in time induces a smaller sampling interval in frequency.
} of $X(\omega)$ computed at multiples of $2\pi/N$:
Figure~\ref{fig:fa:dsf2dtft} shows some examples for different values of $N$. As $N$ grows large, the set of samples will grow denser in the $[0, 2\pi]$ interval; since, in the limit, $\tilde{\mathbf{x}}_N$ tends to $\mathbf{x}$, it appears that the frequency-domain representation for $\mathbf{x}$ is indeed the function $X(\omega)$; and, indeed, we will show this momentarily.
\caption{top row: original infinite-length, absolutely summable signal (left) and the function $X(\omega)$ defined in~(\ref{eq:fa:dtftAnteLitteram}); rows 2-5: periodized signal $\mathbf{\tilde{x}}_N$ and its DFS for increasing values of $N$; all DFS values coincide with samples of $X(\omega)$.}\label{fig:fa:dsf2dtft}