State and prove symmetry property of DTFT.

The Symmetry Property of the Discrete Time Fourier Transform (DTFT) states that if x[n] is a sequence, then its DTFT X(e^jω) and the DTFT of its complex conjugate x*[n] are related as follows:

X(e^-jω) = X*(e^jω)

Proof:

Let's start with the definition of the DTFT:

X(e^jω) = Σ x[n] e^-jωn

where the sum is over all n.

Now, let's consider the complex conjugate of X(e^jω):

X*(e^jω) = Σ x*[n] e^jωn

where again the sum is over all n.

Now, let's replace e^jω with e^-j(-ω):

X*(e^jω) = Σ x*[n] e^-j(-ω)n

This is exactly the definition of the DTFT of x*[n] evaluated at -ω. Therefore, we have shown that:

X(e^-jω) = X*(e^jω)

This completes the proof of the symmetry property of the DTFT.