While studying Fourier transforms, I came across an ambiguous summation notation : $ k = ⟨N⟩ $ . It appears in this book.

What does the $' k = ⟨N⟩ '$ under the following summation/sigma notation mean for the terms that are being summed in this summation?

$$ \sum_{k=⟨N⟩} a_k $$

The ambiguous notation (under the sigma) is the following: $$ k = ⟨N⟩ $$

It has the variable of the index of the summation term on the left of the equal sign and left/right angle brackets around the 'N' variable on the right of the equal sign.

$$ $$

'k' takes values depending on 'N' but how? What is the meaning of the notation for the '$a_k$' terms that are being summed? Does it mean we sum the terms ''from the first to the 'N-th' '' term or 'from the 0-th to '(N-1)-th' '' term? Or something else?

Edit: More specifically, it appears in the pdf pages 658,660 in appendix D - transforms table rows of the online book.

  • 2
    $\begingroup$ What does $⟨N⟩$ represent? $\endgroup$
    – vitamin d
    Commented Dec 29, 2023 at 15:52
  • 3
    $\begingroup$ Please provide some context. Where have you found this formula? $\endgroup$
    – md2perpe
    Commented Dec 29, 2023 at 15:53
  • $\begingroup$ Maybe $k=⟨N⟩$ should read like $k\in\mathbb N$ ? But that would require the sequence to be absolutely convergent, as for example, when all the $a_k>0$. $\endgroup$ Commented Dec 29, 2023 at 17:23
  • $\begingroup$ @vitamind this is the question I am posting. It represents the values of k to do the summation over $a_k$ terms. $\endgroup$
    – spiros
    Commented Dec 29, 2023 at 18:22
  • $\begingroup$ @GyroGearloose this is a question about the meaning of the summation notation $ 𝑘 = ⟨𝑁⟩ $ I don't think it has to do with the convergence or divergence of the series. $\endgroup$
    – spiros
    Commented Dec 29, 2023 at 18:24

1 Answer 1


This notation is used in the Discrete-Time Fourier tables in pages 658-660. The definition is $$⟨N⟩:=\{0,1,2,...,N-1\}\Longrightarrow\sum_{k=⟨N⟩}=\sum_{k=0}^{N-1}.$$ This is unconventional notation and in my opinion makes things more confusing than they have to be.

I was able to find the meaning by comparing the results in the tables with results I know, for instance, Parseval Equality (here the discrete case) on page 660 is normally written like this and also on page 660 the definition of the Discrete-Time Fourier Transform is usually written like this.


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