By applying the Fourier synthesis equation (Eq. (2.133)) to Eq. (2.172) and using Theorem 3 in Table 2.2, demonstrate the validity of the modulation theorem (Theorem 7, Table 2.2).

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FOURIER TRANSFORM THEOREMS TABLE 2.2 Sequence x[n] yln] Fourier Transform X(elw) Y(elw) a X (el) + bY(ej) e jama X(eja) 1. ax[n] + by[n] 2. x[n - na] (na an integer) 3. elaon I[n] X(ew-wo) X(e-iw) X* (ei") if x[n] real. 4. x-n] dX(el) 5. пx(n] dw 6. x[n] y[n] X(e )Y(ei) 7. x{n]y[n] X(el)Y(ei(w-9))de 27 Parseval's theorem: |X(e")dw x[n]? 8. %3! 9. *(n]y"(r] = X(e")Y* (e!")da %3D