Symmetry and periodicity of frequency-shifted discrete Fourier transform

1. What is the frequency shifting property of Fourier transform?
2. Is discrete Fourier transform symmetric?
3. What is symmetry property of Fourier transform?
4. Does DFT assume periodicity?

What is the frequency shifting property of Fourier transform?

One of the most significant properties of the Fourier transform is modulation. Its application to signal transmission is fundamental in communications. That is, X(Ω) is shifted to frequencies Ω₀ and −Ω₀, and multiplied by 0.5.

Is discrete Fourier transform 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.

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.

Does DFT assume periodicity?

In many signal processing books, it is claimed that the DFT assumes the transformed signal to be periodic (and that this is the reason why spectral leakage for example may occur).