Ambiguous summation/sigma notation $\sum_{k=⟨N⟩} a_k$

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.

• What does $⟨N⟩$ represent? Commented Dec 29, 2023 at 15:52
• Please provide some context. Where have you found this formula? Commented Dec 29, 2023 at 15:53
• 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 $a_k>0$. Commented Dec 29, 2023 at 17:23
• @vitamind this is the question I am posting. It represents the values of k to do the summation over $a_k$ terms. Commented Dec 29, 2023 at 18:22
• @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. Commented Dec 29, 2023 at 18:24

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.