Why is this DFT of a real symmetric signal resulting in complex valued coefficients?

1. Can DFT be complex?
2. Why is DFT symmetric?
3. What is DFT coefficient?
4. Why is the DFT mirrored?

Can DFT be complex?

However, the complex DFT projects the input signal on exponential basis functions (Euler's formula connects these two concepts). When the input signal in the time domain is real valued, the complex DFT zero-fills the imaginary part during computation (That's its flexibility and avoids the caveat needed for real DFT).

Why is DFT symmetric?

And without going into mathematical details, DFT of real valued function is symmetric, i.e. resultant Fourier function has both real and imaginary parts which are mirror images with respect to 0 frequency component.

What is DFT coefficient?

DFT coefficients, Xk, give amplitudes and phases of complex sinusoids at integer frequencies k, from 0 to N−1, that sum to the original signal x[n], comprised of N points.

Why is the DFT mirrored?

Because both the positive and negative frequency sinusoids are 90 degrees out of phase and have the same magnitude, they will both respond to real signals in the same way.