Taxa de Nyquist e Séries de Fourier - Lista de Exercícios M1

Lista de atividade para a M1 Determine as taxas de Nyquist para os sinais a seguir a xt 1 cos 2000πt sen 4000πt b xt sinc 50πt c xt sinc2 100πt Considere o sinal analógico xt 3 cos 50πt 10 sin 300πt cos 100πt a Determine a taxa de amostragem mínima para evitar aliasing 1 a fN 2B 4000 Hz B Maior frequência do espectro do sinal 4000π 2π 2000 Hz b sinc 50πt 2π rectw100π π 100π kw w π2 π2 π sinc σt 2 2π rect wσ σ 50π σ 100π logo o espectro varia entre 50π e 50π fN 2B 50 Hz B 50 π 2π 25 Hz c sinc2 100 π t 1 π rect w 400π 100 π 2 100 π 200 π B 200 π 2 π 100 Hz fN 2B 200 Hz 2 fN 2B 300 Hz B 300 2 150 Hz Gráfico A Y 3 X 10 5 5 10 Gráfico B Y π X 3π 2π π 0 π 2π 3π 3 a0 15 px dx 15 3 dx 3 0 3 x5 0 15 155 3 3b a0 1π from π to π fx dx 1π from π to 0 x dx from 0 to π x dx 1π x22 from 0 to π x22 from 0 to π 1π 0 π22 π22 0 1π π22 π22 π an 1π from π to π fx cos mπxπ dx 1π from π to 0 x cosmx dx from 0 to π x cosmx dx x cosmx dx x sinmxm sinmxm dx ux dv cosmx dx du1 v sinmxm x sinmxm cosmxn2 sinmxn dx cosmxn2 an 1π x sinmxn cosmxn2 from π to 0 x sinmxn cosmxn2 from 0 to π 1π 1n2 cosπnn2 cosmπn2 1n2 1π 2 cosmπ 2 n2 2 πn2 1n 1 bn 1π from π to π fx sinmπxπ dx 1π from π to 0 x sinmx dx from 0 to π x sinmx dx x sinmx dx x cosmxn cosmxn dx ux dv sinmx dx du1 v cosmxn x cosmxn sinmxn2 cosmxn dx sinmxn2 bn 1π x cosmxn sinmxn2 from π to π sinmxn2 x cosmxn from 0 to π 1π π cosπnn π cosmπ n 0 Série de fouriers fx π2 Σ from n1 to 2nπ 1n 1 cosmx 4 a0 1π from π to π ex dx 1π eπ eπ 1π eπ 1eπ an 1π from π to π ex cosmx dx 1π m ex sinmx ex cosmx m2 1 from π to π ex cosmx dx ex sinmxn ex sinmxn dx u ex dv cosmx dx du ex dx v sinmxn ex sinmxn ex cosmxn2 1n2 ex cosmx dx ex sinmx dx ex cosmxn2 ex cosmxn2 dx u exn dv sinmx dx du exn dx v cosmxn ex cosmx n2 ex cosmxn2 dx ex cosmx dx 1n2 ex cosmx n ex sinmx ex cosmx n2 n2 1 ex cosmx dx n ex sinmx ex cosmx n2 1 an 1π eπ cosπm1 n2 1 eπ cosπm1 n2 1 1π eπ 1m n2 1 1m eπ m2 1 bm 1π from π to π ex sinmx dx 1π ex sinmx m ex cosmx n2 1ππ ex sinmx dx ex cosmx n ex cosmx n dx u ex dv sinmx dx du ex dx v cosmxn ex cosmx n ex sinmx 1n2 ex sinmx dx ex cosmx dx n ex sinmx n2 ex sinmx dx m2 u ex n dv cosmx dx du ex n dx v sinmx m ex sinmx 1 1m2 ex sinmx m2 ex cosmx m ex sinmx m ex sinmx m ex cosmx n2 1 km 1π m eπ cosmπ n2 1 m eπ cosπm n2 1 1π m cosmπ m eπ cosmπ eπ n2 1 n2 1 1π m 1m eπ n2 1 m eπ 1m n2 1 Série de Fourier fx e2π 1 2π eπ from n1 to e2π 11n cosmxπ eπ n2 1 ne2π 11n sinmx π eπ n2 1 FA T sin c β T A T Fsin c β T A T rect β T