I have a question with reference to this Table.
With even N, the frequency axis extremes should be $\pm$Fs/2, where Fs is the sampling frequency. However in the array we have only one value corresponding to the Nyquist frequency. Following the notation of the table in the first column, N=10, we have
[$c_{0}$ $c_{1}$ $c_{2}$ $c_{3}$ $c_{4}$ $c_{-5}$ $c_{-4}$ $c_{-3}$ $c_{-2}$ $c_{-1}$]
which after fftshift should become
[$c_{-5}$ $c_{-4}$ $c_{-3}$ $c_{-2}$ $c_{-1}$ $c_{0}$ $c_{1}$ $c_{2}$ $c_{3}$ $c_{4}$ ]
The $c_{-5}$ value corresponds to the Nyquist frequency. So how is our double sided frequency spectrum symmetric when N is even? For real valued function it will be zero, but is there a reason that MATLAB calculates the negative frequency first as shown in the Table?
Example: A_even=[ 0 0 0 1 1 1 1 0 0 0],
N=10
B=fft(A_even)'
C=fftshift(B)
Then C is equal to
$$\begin{matrix} 0.0000 + 0.0000i\\ -0.3090 - 0.9511i\\ 0.4271 + 0.5878i\\ 0.8090 + 0.5878i\\ -2.9271 - 0.9511i\\ 4.0000 + 0.0000i\\ -2.9271 + 0.9511i\\ 0.8090 - 0.5878i\\ 0.4271 - 0.5878i\\ -0.3090 + 0.9511i\\ \end{matrix}$$
This implies $c_{-5}$ = 0.0000 + 0.0000i and $c_{0}$= 4.0000+0000i Thanks.