Serie de Fourier - Representacao de Sinais Periodicos em Tempo Continuo

Representação de sinais periódicos de tc em série de Fourier terçafeira 24 de outubro de 2023 1339 Combinação linear de exponenciais harmonicamente relacionados Seja o conceito de periodicidade xct xctT t tempo T período fundamental ω0 2πT 2πf f 1T f0 frequência em Hertz Exemplo xct ejω0t T 2πω0 Podemos criar um sinal harmonicamente relacionado da seguinte forma ϕkt ejkω0t k 0 1 2 ϕ0t 1 etc DC ω0 2πT ϕkt ejk 2πT t ϕ1t ϕ2t ϕ3t são frequências múltiplos de ω0 Então xct k to ak ejkω0t é periódica com período T k 1 Primeiro harmônico ou fundamental k 0 Constante k 2 Harmônicos de ordem superior k N nésimo harmônico Audacity Logo xct k to ak ejkω0t Representação de xct por uma série de Fourier Exemplo Seja xct k3 to 3 ak ejkω0t ω02π radianoss a01 a1 a1 14 a2 a2 12 a3 a3 13 Então cos ωt 12 ejωt ejωt xct 1 12x ejωt ejωt 12 ej2ωt ej2ωt 13 ej3ωt ej3ωt 22 cos2πt senωt 12i ejωt ejωt Componente DC Nível médio Para sinais reais xct xct sinal complexo conjugado Logo xct sum from k to of ak ejkω0t Substituindo k k xct sum from k to of a k ejkω0t Como xct é real então xct sum from k to of ak ejkω0t xct sum from k to of ak ejkω0t ak ak ou ak ak Logo xct sum from k to 1 of a k ejkω0t a0 sum from k1 to of ak ejkω0t xct sum from k1 to of ak ejkω0t a0 sum from k1 to of ak ejkω0t ak ak real xct sum from k1 to of ak ejkω0t a0 sum from k1 to of ak ejkω0t ak ak xct a0 sum from k1 to of ak ejkω0t ak ejkω0t ejθ ejθ 2 cos θ ejθ ejθ 2j sen θ xct a0 sum from k1 to of 2 Re ak ejkω0t ak Ak ejθk xct a0 2 sum from k1 to of Re Ak ej kω0t θk xct a0 2 sum from k1 to of Ak cos kω0t θk I xct a0 2 sum from k1 to of Ak cos kω0t θk realperiódico ou ak Bk j Ck cartesiana II xct a0 2 sum from k1 to of Bk cos kω0t Ck sen kω0t III xct sum from k to of an ejkω0t Sinal periódico expresso em série de Fourier Sinal periódico expresso em série de Fourier Determinar a representação de um sinal periódico em série de Fourier Como determinar os coeficientes an da série de Fourier xct sum from k to of ak ejkω0t xct ejnω0t sum from k to of ak ejnω0t ejkω0t integral from 0 to T of xct ejnω0t dt integral from 0 to T of sum from k to of ak ejnω0t ejkω0t dt integral from 0 to T of xct ejnω0t dt integral from 0 to T of sum from k to of ak ejknω0t dt ₀ᵀ rct ej n ω₀ t dt k to αk ₀ᵀ ej n ω₀ t ej k ω₀ t dt ₀ᵀ ej n k ω₀ t dt T k n 0 cc α₀ α₁₀₉ ₀ᵀ rct ej n ω₀ t dt an T an 1T ₀ᵀ rct ej n ω₀ t dt αk 1T ₀ᵀ rct ej k ω₀ t dt Logo a série de Fourier para sinais periódicos é definida por Equação de síntese rct k to an ej k ω₀ t ω₀ 2π T Equação de análise an 1T ₀ᵀ rct ej k ω₀ t dt coeficientes da série ou coeficientes espectrais a₀ 1T ₀ᵀ rct dt valor médio cc Ver os exemplos no livro Trigonometric Fourier series TFS 1 Synthesis equation xt a₀ k1 to ak cosk ω₀ t k1 to bk sink ω₀ t 440 2 Analysis equations ak 2T₀ t₀ to t₀T₀ x t cosk ω₀ t dt for k1 441 bk 2T₀ t₀ to t₀T₀ x t sink ω₀ t dt for k1 442 a₀ 1T₀ t₀ to t₀T₀ xt dt dc component 443 Example 41 Trigonometric Fourier series of a periodic pulse train A pulsetrain signal xt with a period of T₀3 seconds is shown in Fig 45 Determine the coefficients of the TFS representation of this signal Solution In using the integrals given by Eqns 441 442 and 443 we can start at any arbitrary time instant t₀ and integrate over a span of 3 seconds Applying Eqn 443 with t₀0 and T₀3 seconds we have a₀ 13 ₀₀ 1 dt ₁³ 0 dt 13 The fundamental frequency is f₀ 1T₀ 13 Hz and the corresponding value of ω₀ is ω₀ 2π f₀ 2π3 rads Using Eqn 441 we have ak 23 ₀¹ 1 cos2π k t 3 dt ₁³ 0 cos2π k t 3 dt sin2π k 3 π k for k12 Finally using Eqn 442 we get bk 23 ₀¹ 1 sin2π k t 3 dt ₁³ 0 sin2π k t 3 dt 1 cos2π k 3 π k for k12 Using these coefficients in the synthesis equation given by Eqn 440 the signal xt can now be expressed in terms of the basis functions as xt 13 k1 to sin2π k 3 π k cos2π k t 3 k1 to 1 cos2π k 3 π k sin2π k t 3 Refer to Pages 272 through 278 Section 422 Eqns 440 through 446 Examples 41 and 42 Figs 45 and 46 Largest harmonic to include in the approximation 4 k Include ak bk 1 1 07979 04775 2 0 01778 02997 3 1 25981E17 0 4 0 00899 01154 5 1 00561 00915 6 0 25981E17 0 7 1 01304 00862 8 0 0 0 9 1 25981E17 0847 10 0 0 0347 11 1 25981E17 0147 12 0 00072 00867 13 1 00867 00976 14 0 0 00841 15 1 10E16 0 x t and Trig Fourier Series Approximation Finiteharmonic approximation x4t Refer to Pages 272 through 278 Section 422 Eqns 440 through 446 Examples 41 and 42 Figs 45 and 46 Largest harmonic to include in the approximation 16 k Include ak bk 1 1 03333 04775 2 0 02757 0475 3 1 25981E17 0 4 0 00899 01154 5 1 004851 00915 6 0 25981E17 0 7 1 000854 00848 8 0 00767 0 9 1 00416 00569 10 0 00927 00437 11 1 25981E17 01347 12 0 0 01367 13 1 0 00022 14 0 0 01341 15 1 40E16 0 x t and Trig Fourier Series Approximation Finiteharmonic approximation x16t