3.2 Symmetry properties of the fourier transform

When $x\left(t\right)$ is real, the Fourier transform has conjugate symmetry , $X(-j\Omega )=X{\left(j\Omega \right)}^{*}$ . It is not hard to see this:

where the second equality uses the definition of a Riemann integral as the limiting case of a summation, and the fact that the complex conjugate of a sum is equal to the sum of the complex conjugates. The third equality used the fact that the complex conjugate of a product is equal to the product of complex conjugates.

Letting $X\left(j\Omega \right)=a\left(j\Omega \right)+jb\left(j\Omega \right)$ , it follows that

and

Equating [link] and [link] gives $a\left(j\Omega \right)=a(-j\Omega )$ and $b(-j\Omega )=-b\left(j\Omega \right)$ , which implies that the real and imaginary parts of $X\left(j\Omega \right)$ have even and odd symmetry, respectively. A consequence of this is that $\left|X,\left(,j,\Omega ,\right)\right|=\left|X,{\left(j\Omega \right)}^{*}\right|=\left|X,(,-,j,\Omega ,)\right|$ , that is, the magnitude of the Fourier transform has even symmetry. It can similarly be shown that the phase of the Fourier transform has odd symmetry.