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.
Best Answer
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.