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I am trying to intuitively understand the mystery of why the DFT of a complex vector with real values produces apparent symmetry in the magnitude plot (see the second plot here for example). In DFT $X[k]$ of a $N$-dimensional complex vector with real entries $x[t]$, the relation $|X[k]|=|X[N-k]|$ holds for each frequency $k=0,1,2,\dots N-1$. It should be the case that the frequencies $k$ and $N-k$ should trace the same sinusoid, right?

My reasoning is that $e^{i\frac{2\pi kt}{N}}$ ($t=0,1,2,3,\dots$) traces a unit circle in counterclockwise direction on the complex plane while $e^{i\frac{2\pi (N-k)t}{N}}$ ($t=0,1,2,3,\dots$) traces a unit circle in clockwise direction on the complex plane. It is evident if you look at real parts of $e^{i\frac{2\pi kt}{N}}$ and $e^{i\frac{2\pi (N-k)t}{N}}$ for each $t$:

(Assume $N=6$ and $k=1$)

$\cos\left(\frac{2\pi k}{N}\cdot0\right)=\cos\left(\frac{2\pi (N-k)}{N}\cdot0\right)=1$

$\cos\left(\frac{2\pi k}{N}\cdot1\right)=\cos\left(\frac{2\pi (N-k)}{N}\cdot1\right)=\frac12$

$\cos\left(\frac{2\pi k}{N}\cdot2\right)=\cos\left(\frac{2\pi (N-k)}{N}\cdot2\right)=-\frac12$

etc...

How do I call this weird phenomenon? Aliasing?

Also, I am wondering how do I graph this result? This is my attempt: https://www.desmos.com/calculator/w6gl0cqwwa and this is not what I had in mind. Frequencies $k$ and $N-k$ should trace the same sinusoid

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1 Answer 1

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Intuition is in the eyes of the beholder but this property pops out directly from the definition of the DFT

$$X[k] = \sum_{n=0}^{N-1}x[n] \cdot e^{-j2\pi\frac{kn}{N}} $$

First of all, it follows directly from the definitions that both $x[n]$ and $X[k]$ are periodic with $N$, i.e. $X[k] = X[k + mN], m \in \mathbb{Z}$

Hence we have

$$X[N-k] = X[-k] = \sum_{n=0}^{N-1}x[n] \cdot e^{j2\pi\frac{kn}{N}}$$

If the input is real, i.e. $x[n] \in \mathbb{R}$ then we see directly that

$$X[-k] = X^{*}[k]$$

The DFT at a negative frequency is the complex conjugate of the value at the positive frequency (and vice ver

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