Why are the negative frequencies of the DFT symmetrically reflected at the nyquist to the positive frequencies?

Question

I was playing around with plotting DFT and realized that the negative frequencies are symmetric to the positive frequencies reflected at the nyquist.

• Plot shown for the signal $f(x) = \cos(\frac{\pi}{2}x - \frac{\pi}{2}) + 2\cos(\pi x+ \frac{\pi}{2})$.

So why are the negative frequencies symmetrically distributed case for every DFT?

Possible approaches

• I understand that the negative frequencies are required for the complete inverse DFT even though leaving them out will not change the result of the DFT though this does not explain the symmetry
• Since for even $N$ the nyquiest is at $N/2$ the following should be true: $\exp(-\frac{j2 \pi n}{N}(\frac{N}{2}+1)) = \exp(-\frac{j2 \pi n}{N}(\frac{N}{2}-1))$

Looking at the statement above:

• $\exp(-\frac{j2 \pi n}{N}(\frac{N}{2}+1)) = \exp(-j \pi n -\frac{j2 \pi n}{N}) = \exp(- j \pi n) \cdot \exp(-\frac{j2 \pi n}{N}) = [-1]^n \cdot \exp(-\frac{j2 \pi n}{N})$

should be equal to

• $\exp(-\frac{j2 \pi n}{N}(\frac{N}{2}-1)) = \exp(-j \pi n + \frac{j2 \pi n}{N}) = \exp(- j \pi n) \cdot \exp(\frac{j2 \pi n}{N}) = [-1]^n \cdot \exp(\frac{j2 \pi n}{N})$

Answer 1

The DFT $X[k]$ of $x[n]$ is

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{1}$$

If $x[n]$ is real-valued, the complex conjugate mirrored DFT coefficients are

$$\begin{align}X^*[N-k]&=\sum_{n=0}^{N-1}x[n]e^{j2\pi n(N-k)/N}\\&=\sum_{n=0}^{N-1}x[n]e^{j2\pi n}e^{-j2\pi nk/N}\\&=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\\&=X[k]\end{align}$$

So for real-valued $x[n]$, $X[k]=X^*[N-k]$ holds, i.e., all DFT bins $X[k]$ for $k>\lfloor N/2\rfloor$ are redundant.

This makes sense because we need only $N$ real-valued numbers to represent the $N$ (real) values $x[n]$, and the real and imaginary parts of a complex DFT coefficient represent two numbers. For complex-valued sequences $x[n]$, there is generally no redundancy in the DFT coefficients.

Answer 2

I wrote my own DFT in C++ (plotted in python) and tested 7.5*cos(5*2πt)+2.5*cos(8*2πt) with a sample rate of 1000Hz and a 2 second duration. t is then the array([0., 0.001, 0.002, …, 1.999])

Here are the spikes at 5,8 and at 992, 995…

Focusing on the 5Hz comp, I plotted two cosine waves (for my given sampling rate) at frequencies of 5 and 995. It was great to see they are the same! They both should then produce a spike.

norm_time = np.array(range(len(t)))/len(t)
plt.scatter(t,np.exp(-2j*np.pi*10*norm_time).real)
plt.scatter(t,np.exp(-2j*np.pi*1990*norm_time).real,marker='.')
# In this ex the freq step is .5Hz, so coefficients 10 and 1990 are used to get freq 5 and 995. 

To me, the identical higher frequency wave is more like an aliasing effect and not the critical “negative” frequency it's referred to.

So instead of testing frequencies from 0-1000, I adjusted my DFT to test frequencies from -500 to 500.

Now we have spikes at 5 and -5 which look a lot more like some negative frequencies to me. Feels silly to even test this below, since of course they will align.

plt.scatter(t,np.exp(-2j*np.pi*10*norm_time).real)
plt.scatter(t,np.exp(-2j*np.pi*-10*norm_time).real,marker='.')

Just for good measure, here is my inv Fourier working on my -500 to 500 DFT and original signal.

Maybe it is safe to say that the higher frequencies in a DFT are just aliased discrete waves that equal the negative frequencies (for a given sampling rate)