In Section 2.9, we stated a number of Fourier transform theorems without proof. Using the Fourier syntheses or analyses equations ? (Eqs. (2.133) and (2.134)), demonstrate the validity of Theorems 1 ? 5 in Table 2.2.

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x[n] = . | x(elu)elndw, (2.133) X(ei") = *[n]e=jum_ (2.134) Sequence x[n] y[n] Fourier Transform X(ejw) Y(ei") 1. ax[n] + by[n] aX(el@) + bY(elw) e- juma X (ej@) 2. x[n – na] (na an integer) 3. ejmon x[n] X (e(w-wo)) X(e-j) X* (ej") if x[n] real. 4. x[-n] dX(el®) 5. пх[n] dw X (el")Y(ej") 6. x[n] * y[n] 7. x[n]y[n] X (el® )Y(ej(w-e))d® 27 Parseval's theorem: I lx[n]? = |X(e]"f°dw 2л 9. ) x[n]y*[ri] = X(el")Y*(el")dw