If $y$ is the $DFT$ of a real sequence $x$ of length $n$; where $n$ is a power of two, show that $y_0$ and $y_{n/2}$ must be real.

Question

If $y$ is the $DFT$ (Discret Fourier Transform) of a real sequence $x$ of length $n$; where $n$ is a power of two, show that $y_0$ and $y_{n/2}$ must be real.

I have no idea how to answer this question

Answer 1

Recall that $$y_j = \sum_{k=1}^n x_k e^{-2\pi i jk/n}$$ Thus, $y_0 = \sum_{k=1}^n x_k \in\mathbb{R}$ which has nothing to do with the fact that $n$ is a power of 2 - it works for any real sequence $x$.

Next, we have $y_{n/2} = \sum_{k=1}^n x_k e^{-\pi i j} = \sum_{k=1}^n x_k(-1)^k\in\mathbb{R}$ which again doesn't need the fact that $n$ is a power of two - merely that $n$ is divisible by $2$.