Let us look at some simple consequences of these properties:

a) What can we say about the Fourier transform of an even signal x(t) (with Fourier transform X(f) ) ?

x(-t) has Fourier transform X(-f). As x(t) is real, x(t) = x(-t), implying, X(f) = X(-f).

Thus, the Fourier transform of an even signal is even. Similarly, you can show the Fourier transform of an odd signal is odd.

b) What can we say about the Fourier transform of a real signal x(t), with Fourier transform X(f) ?

If x(t) is real,

Thus the Fourier transform of a real signal is Conjugate Symmetric.