Show that for x[n] real, property 7 in Table 2.1 follows from property 1 and that properties 8 ? 11 follow from property 7.

SYMMETRY PROPERTIES OF THE FOURIER TRANSFORM TABLE 2.1 Sequence x[n] Fourier Transform X(el") 1. x*[n] 2. r*|-n] X*(e-j") X*(eju) 3. Re(x[n]} Xe(el") (conjugate-symmetric part of X(e) 4. jIm{x[n]} X,(ej") (conjugate-antisymmetric part of X(ei")) 5. x.[n] XR(ei") = Ref X(el)} (conjugate-symmetric part of x[n]) %3! iX;(e/") = jJm( X(e lu)) 6. xo(n] (conjugate-antisymmetric part of x[n]) The following properties apply only when x[n] is reat: X(e) = X"(e-i") (Fourier transform is conjugate symmetric) 7. Any real x[n] 8. Any real x[n] XR(eja) = XR(e-j) (real part is even) X;(el") = - X(e-i") (imaginary part is odd) 9. Any real x[n] |X(ej")| = 1X(e ja) (magnitude is cven) X(el) = -X(e-te) (phase is odd) 10. Any real x[n] %3D 11. Any real x[n] 12. xeln] (even part of x[n]) XR(e") 13. xon] (odd part of x[n]) jX;(ei")