Symmetry and periodicity of frequency-shifted discrete Fourier transform The unitary discrete Fourier transform (DFT) of a sequence of numbers $x_n$ to $X_k,$ with integer $0 \le n &lt; N$ and $0 \le k &lt; N,$ can be defined as:
$$X_k = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} x_n e^{-2\pi ikn/N}\tag{1}$$
and the inverse discrete Fourier transform (IDFT) as:
$$x_n = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X_k e^{2\pi ikn/N}\tag{2}$$
If $x_n$ is modulated (multiplied) by a unit-magnitude zero-phase complex sinusoid $e^{-2\pi ibn/N}$ before DFT, and the IDFT output is demodulated (divided) by the same, then we get another transform pair from the family of generalized discrete Fourier transforms, parameterized by the constant $b:$
$$
\begin{aligned}
X_k(b) &amp;= \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} x_ne^{-2\pi ikn/N}e^{-2\pi ibn/N}\\
&amp;= \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} x_ne^{-2\pi i(k+b)n/N}
\end{aligned}
\tag{3}
$$
$$
\begin{aligned}
x_n &amp;= e^{2\pi ibn/N}\frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X_k(b)\cdot e^{2\pi ikn/N} \\
&amp;= \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X_k(b)\cdot e^{2\pi i(k+b)n/N}
\end{aligned}
\tag{4}
$$
Whereas DFT samples frequencies $2\pi k/N,$ the frequency-shifted transform samples frequencies $2\pi (k+b)/N.$ This can be visualized on the Z-plane:
Figure 1. Z-plane representation, showing the unit circle, of the frequencies sampled by A) DFT and B) the frequency-shifted DFT, with $N = 12$ and $b = 1/3.$ The shift $2\pi b/N$ appears as the angle between the real axis and the vector from origin to one of the sampled frequencies.
Question: What are the time-domain symmetry and periodicity properties of such frequency-shifted Fourier transforms when extending $n$ beyond $0\le n&lt;N$ in the inverse transform, and how does this depend on the parameter $b?$
For DFT (or with $b = 0$) the time-domain extension has period $N$ with no time-domain symmetry imposed by the transform.
One version that's useful is $b=\frac{1}{2}$ 

If you have an input sequence of N real numbers, the output sequence can also be represented as N real numbers: Two for DC and Nyquist and 2*(N/2-1) for N/2-1 complex values. Mixing real and complex numbers makes real time processing a bit more awkward.

Using the transform you suggest with $b=\frac{1}{2}$ results in exactly N/2 complex numbers which allows for more efficient code on SIMD real time processors. $b=\frac{1}{2}$  maintains complex conjugate symmetry. Instead of, say, sampling -10,0,10,20,30, .. you sample at -150,50,50,150,250,... avoiding DC and Nyquist.