Proof that DFT is symmetric

1. Is DFT symmetric?
2. Is A Fourier transform Symmetric?
3. Why is the DFT mirrored?
4. What is symmetry property of Fourier transform?

Is DFT symmetric?

Symmetry Property of Discrete-Time Fourier Transform

i.e., the real part of DTFT Xr(ω) is an even function of 𝜔, i.e., it has even symmetry property. Therefore, the imaginary part of DTFT Xi(ω) is an odd function of 𝜔, i.e., it has odd symmetry property.

Is A Fourier transform Symmetric?

When we take the the Fourier Transform of a real function, for example a one-dimensional sound signal or a two-dimensional image we obtain a complex Fourier Transform. This Fourier Transform has special symmetry properties that are essential when calculating and/or manip- ulating Fourier Transforms.

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.

What is symmetry property of Fourier transform?

Symmetry Properties

Represent x(t) as the sum of an even function and an odd function (recall that any function can be represented as the sum of an even part and an odd part). x(t)=xo(t)+xe(t) Express the Fourier Transform of x(t), substitute the above expression and use Euler's identity for the complex exponential.