Análise Espectral de Sinais Periódicos

Four graphs of xt versus t are shown top left is a rectangular pulse from T to 2T1 and 2T1 to T top right is a triangular waveform from 2π to 2π peaking at xt2 bottom left is a sawtooth wave from 3π to 3π oscillating between 1 and 1 bottom right is a square wave from 3π to 3π oscillating between 1 and 1 Graph titled Fase do espectro Sinal 1 showing phase angle of Cn rad versus n from 0 to 10 values alternate between 0 and approximately 157 Graph titled Fase do espectro Sinal 3 showing phase angle of Cn rad versus n from 0 to 10 values alternate between 0 and approximately 157 positive and negative peaks Magnitude do espectro Sinal 3 Cn n Magnitude do espectro Sinal 1 Cn n xt 0 para 2T1 t 0 1 para 0 t 2T1 periódica de período 4T1 Logo a0 1 2T1 from 2T1 to 2T1 xt dt 1 2T1 from 0 to 2T1 dt 1 an 1 2T1 from 2T1 to 2T1 xt cosnπt 2T1 dt 1 2T1 from 0 to 2T1 cosnπt 2T1 dt 12T1 2T1nπ sen nπ t2T102T1 sempre nulo nZ sen nπ nπ 0 bₙ 12T1 from 2T1 to 2T1 Xt sennπ t2T1 dt 12T1 from 0 to 2T1 sen nπ t2T1 dt 12T1 2T1nπ cosnπ t2T102T1 1 cosnπ nπ Se n é par cos nπ 0 on seja bₙ 0 Se n for ímpar cosnπ 1 on seja bₙ 2nπ Logo Xt 12 from n0 to 22n1π sen 2n1 π t 2T1 Temos Xt 2π t 2 para π t 0 2π t 2 para 0 t π periódico de período 2π Temos uma função par on seja Xt Xt então bₙ 0 para todo n inteiro Logo a₀ 1π from π to π Xt dt 2π from 0 to π 2π t 2 dt 2π 2π t²2 2t0π 2 aₙ 1π from π to π Xt cos n t dt 2π from 0 to π 2π t 2 cos n t dt 2π from 0 to π 2π t cos n t 2π from 0 to π 2 cos n t dt 4π² π n sen nπ cos nπ 1 n² 2π 2 sen nπ n 4n² π² cos nπ 1 aₙ 0 se n par 8 n² π² se n ímpar Xt 1 from n0 to 8 2n1² π² cos n t Temos uma função ímpar an0 Temos Xt 1 para 0 t π 1 para π t 0 periódica período 2π bm 1π from π to π Xt sennt dt 2π from 0 to π sennt dt 2π 1 cosnπn bm 0 para n par 4nπ para n ímpar Xt Σ from n0 to 42n1π sennt Image contains only a graph of a periodic function xt with a triangular wave pattern no transcribable text Temos uma função ímpar logo an0 para todo n Xt t para π t π periódica de período 2π Temos bm 1π from π to π Xt sennt dt 2π from 0 to π t sennt dt 2π sennπ nπ cosnπn2 cosnπn Xt Σ from n0 to 1mnπ sennt Fase do espectro Sinal 4 Cnrad n Magnitude do espectro Sinal 4 Cn n Fase do espectro Sinal 2 Cnrad n Magnitude do espectro Sinal 2 Cn n