Fourier transform of even/odd parts of a complex signal

1. Is Fourier transform even or odd?
2. How do you find the odd and even functions of a Fourier series?
3. What is the impact of even or odd signal on Fourier series coefficients?
4. What is the DFT of imaginary and odd signal?

Is Fourier transform even or odd?

Theorem 5.6 The Fourier transform of an odd function is odd. dt. = −F(−s). The Fourier transform of the even part (of a real function) is real (Theorem 5.3):

How do you find the odd and even functions of a Fourier series?

A function is called even if f(−x)=f(x), e.g. cos(x). A function is called odd if f(−x)=−f(x), e.g. sin(x).

What is the impact of even or odd signal on Fourier series coefficients?

Effect of Odd Symmetry on Fourier Series

If f(t) possesses odd symmetry, it can be shown that f(t)cos(nω₀t) and f(t)sin(nω₀t) have odd and even symmetries, respectively. Therefore, for an odd signal, all the a_n coefficients are zero.

What is the DFT of imaginary and odd signal?

Similarly, if a signal is odd and real, then its DTFT is odd and purely imaginary. This follows from Hermitian symmetry for real signals, and the fact that the DTFT of any odd signal is imaginary.