Summation limits in DFT Assume a discrete time signal $(x_n)$ is given. Some texts define the DFT as
$$ X[k] = \sum_{n=-N}^N x_n\exp\left(\frac{-2\pi j k n}{N}\right) $$
while others define it as
$$X[k] = \sum_{n=0}^{N-1} x_n\exp\left(\frac{-2\pi j k n}{N}\right) $$
or
$$X[k] = \sum_{n=1}^N x_n\exp\left(\frac{-2\pi j k n}{N}\right)$$
or many other seemingly random variants with different indices. So what is the true index limits for DFT? Or is there no unique consensus and any random author is allowed to introduce a new one?
If $N$ is the DFT length, the summation is always over $N$ points. So under this assumption, the first formula doesn't make any sense. Note that the term $e^{-j2\pi nk/N}$ is $N$-periodic in $k$ and $n$, so summing over one period always gives the same value, regardless of the starting index. With the additional assumption that $x[n]$ is periodically continued, i.e., $x[n]=x[n+N]$, you can compute the given sum over any interval of length $N$. A notation emphasizing this fact is
$$X[k]=\sum_{&lt;N&gt;}x[n]e^{-j2\pi nk/N}\tag{1}$$