Shifted discrete Fourier transformΒΆ
By taking the sampling points differently:
\[x_n
=
\frac{
2 n + 1
}{
2
}
\frac{
L
}{
N
},\]
we obtain
\[ \begin{align}\begin{aligned}F_k
&
=
\sum_{n = 0}^{N - 1}
f_n
\twiddle{- 2 \pi}{k \left( n + \frac{1}{2} \right)}{N},\\f_n
&
=
\frac{1}{N}
\sum_{k = 0}^{N - 1}
F_k
\twiddle{2 \pi}{k \left( n + \frac{1}{2} \right)}{N},\end{aligned}\end{align} \]
which is known as the shifted discrete Fourier transform and appears when dealing with symmetric signals.