Question

2TT sinn (1) a) Let x1 [n] = πη Find the Discrete Time Fourier transform of this signal and plot it with all its critical values. (you can use only transform tables from the book). b) Now, define xzlv) = (**) GHS) Using transform properties, find the Discrete Time Fourier transform of x2[n] and plot it with all its critical values. In your calculations be sure to show your steps ! 2TT sinn sinn sinwon c) Let y[n] [( ) * (sinwon) where * indicates πη πη convolution and ſwol St. On the Discrete Time Fourier transform plot of y[n], find the minimum value of w, such that y[n] = x2[n].

Answer 1

we know that ideal digital low pass filter w -oc we TT h(0) = q (reco) eion de sekian WC il e do [roch seon 2n 2010 - h = sin (nWc) In sin (Wen) D.T.ET an wo tWc cice m, In] = Sine) Tin -20) 20 *166) rect levering

© 22 [n] sin (21) sin (7) Tin Trn Xz let. Ny [n sin (41) DT. ET Th slu Elth 2₂ [n] - X2 [ano] = x, Cotto) x. [n]. Xy Ch] fecio) * x3 lekeo] imedial 7,3 Ce son * abst Elb Fasch pescenes X2 cebol -170 35 - 31 35 39 35 w 170 35 1787 to 35 limits 170) 35. -2 1/2 - 13 to 24/7 + 1 when two unequal. rectangle signals are convolved then their resultant Signal would be a trapezium Area property! Y = X*H =) Ay --- An An

Area. under 40 7 Xo Cesto) = (17(497 Area under Xz (eng) = (1) (-2) = 21 Area under x₂ Coilo) = Axi - h (611 2x 1 / 2 (35 =h b 135, of E 1617 35 + 199 1407 357 b 411 An b 7 A . An. Ada Ny 47 40 20 h h= 21 +xz. 21/ W + -30 37 -1771 35 190 35 35 35

6 g [n]= (sin (2011) Sin ( * sin(n) Tin ТИ TO y (9) = x2 (n) * sin (won) TTD ylema) = x, Celine ). Komente اقے وني* let na (n) DITET sinloon) Th Iti w wo Axceses) 27/7 >w - -30 171 35 35 Х 35 3n 35 xulow) 3 000 WO yleio). X2(e) if ol> 1717 35 if (Wolz ş y (n)= aż (n) 177 35 i minimum value of wo = 171 35