Problem 2 (a)a[n is a real-valued DT signal whose DTFT for -T<w< T is given by X(e) -Wo tWM wo t WM ac[n]=an] cos[won Using table 5.2 and taking Fourier transform of coswonI DTFTicoswonJ=T>8( 2rl)+6(+ DT) 0 Using the multiplication property from table 5.1, Zc(eu) is the periodic convolution of X(e) and DTFT(cos[won) over period 2m and then scaled by 2. We take one period, from to of DT won) and do regular convolution with X(e) Centered at w=0, we get the superposition of two X(eu) scaled by 3. Zc(eu)is shown below for the interval-丌to丌 Ze(e) 12� Problem 2 (a) x[n] is a real-valued DT signal whose DTFT for −π < ω < π is given by X(ejω) 2 ω0 = 3π,ωM = π ✟✟✟✟ 4 ✟✟✟✟ ❍❍ 1 ✟✟✟ ❍❍❍❍❍❍❍❍❍ ω −π 0 ω0 π −ω0 − ωM −ω0 −ω0 + ωM ω0 − ωM ω0 + ωM Let zc[n] = x[n] cos[won] Using table 5.2 and taking Fourier transform of cos[won], +∞ DT FT {cos[won]} = π {δ(w − wo − 2πl) + δ(w + wo − 2πl)} l=−∞ DT FT {cos[won]} −π −ω0 0 ω0 π π π ω Using the multiplication property from table 5.1, Zc(ejw) is the periodic convolution on]} over period 2π and then scaled by 1 . We take one 2π of X(ejw) and DT FT {cos[w period, from −π to π, of DT FT {cos[won]} and do regular convolution with X(ejw). 1 Centered at w = 0, we get the superposition of two X(ejw) scaled by 2 . Zc(ejw) is shown below for the interval −π to π. Zc(ejw) 1 = 1 π × 2π 2 11π 5π 5π 11π −π− 12 − 12 −ωM 0 ωM 12 12 π ω 5