Question

P-7.4 Determine the DTFT of each of the following sequences sin(02570 5δ[n] (a) hi[n] 0.2tn sin(0.1zn) (b) hIn-sin(0.4mn) _ sin(0.1πη) (b) h2[n]= 0.1πη 0.1 πη sin(0.47(n-8)) sin(0.17(n-8)) π(n-8) (c) haln]--π(n-8)

this is the answer of a).. i got confued how got 5-5{ ..... here. can anyone please explain me how we got that

Answer 1

Inverse Fourier transform of H(e^jw) = {1 Vertical-bar w Vertical-bar < w_c w_c < Vertical-bar w Vertical-bar < pi F^-1[H(e^jw)] = h(n) = 1/2pi integral^w_c_-w_c (1) e^+ jwn dw = 1/2pi(jn)[e^jwn]^w_c_-w_c = 1/(2j)pi n [e^jw_c n -e^-jw_c n] - 1/pi n [e^jw_c n - e^-jw_cn/2j = 1/pi n sin (w_c n) = sp_n(w_c n)/pi n FT{sin (w_c n)/pi n} = {1, Vertical-bar w Vertical-bar < w c 0, w_c < Vertical-bar w Vertical-bar < pi equation (1) DTFT {delta (n)} = sigma_n = 0 [1]e^-jwn = e^-jwn Vertical-bar _n = 0 = e^0 = 1 equation (2) Apply (1) and (2), a) R_1(n) = 5 delta (n) - sin(0.25 Hn)/0.2 pi n DTFT [h_1(n)] = DTFT[5 delta (n)] - DTFT [sin (0.25 pi n)/0.2 pi n] = DTFT [delta(n)] - 1/0.2 {DTFT [sin (0.25 pi n)/pi n]} = 5(1) - 5 DTFT {sin (w_c n)/pi n} where w_c = 0.25 pi = 5- 5 {1, Vertical-bar w Vertical-bar < 0.25 pi 0, 0.25 pi < Vertical-bar w Vertical-bar < pi = {5 - 5, Vertical-bar w Vertical-bar < 0 25 pi 5 - 0, 0.25 pi