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In case of DFT where we have following $$ Y_{k}=\frac{1}{N}\sum_{n=0}^{N-1}y_ne^{-j{\frac{2\pi nk}{N}}} $$

  • What happens in case when we change summation, e.g., increase or decrease number of $n$ that we are summation over?
  • How does the rest of equation change? For example if we decide to sum over $NM$ (if $M<1$ we have windowing and if $M>1$ we have zero padding so I do not assume adding any new info data just scaling the duration of the signal) samples would we rewrite the latter equation like this \begin{equation} Y_{k}=\sum_{n=0}^{MN-1}y_ne^{-j{\frac{2\pi nk}{MN}}}\tag{1} \end{equation} or this \begin{equation} Y_{k}=\sum_{n=0}^{MN-1}y_ne^{-j{\frac{2\pi nk}{N}}}\quad \tag{2} \end{equation}
  • And what is the explanation behind. Also what is the appropriate scaling when summation limits change?
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If the length of your data is $NM$ long (regardless of $N$), your DFT should be $NM$ long, so the correct equation is (1).

The explanation is simple: how long is your data? Use that length for the DFT.

You may want to define: $$ y'_n = \left \{ \begin{array}{cl} y_n, & 0 \le n \le N-1\\ 0, & n \ge N\\ \end{array} \right . $$ and operate on that because $y_n$ for $n\ge N$ is not defined.

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  • $\begingroup$ And what is then approporiate scaling-$\frac{1}{MN}$ or $\frac{1}{N}$? For example if I zeropadd my signal would I scale then over all samples (where part of them are essentially zero),i.e.,$\frac{1}{MN}$ or just over ones which are actually representing data ,i.e., $\frac{1}{N}$? $\endgroup$
    – Cali
    Jul 14, 2016 at 7:49
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    $\begingroup$ The scalings are a matter of choice. I'd go with $\frac{1}{MN}$, but some forms of the DFT and its inverse prefer $\frac{1}{\sqrt{MN}}$ on both the forward and inverse transforms. $\endgroup$
    – Peter K. ♦
    Jul 14, 2016 at 12:07
  • $\begingroup$ But if I scale with $\frac{1}{MN}$ that means that amplitude of zeropadded signal would be different compared to original signal? If I am adding zeros why would my amplitude change? $\endgroup$
    – Cali
    Jul 14, 2016 at 12:38
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    $\begingroup$ Actually, some forms of analysis wrap longer data (by summing) around a shorter FFT. This sharpens some forms of frequency estimation (at the cost of other artifacts). It's "sort-of" the opposite of zero-padding. $\endgroup$
    – hotpaw2
    Jul 14, 2016 at 20:27
  • $\begingroup$ @hotpaw2 Yes, you can certainly do that. As you suggest, it'll give you something akin to "time-aliasing" but can be useful if the problem supports it. $\endgroup$
    – Peter K. ♦
    Jul 14, 2016 at 20:28

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