Question

[4 Mar (c) Consider the following periodic function, defined as: fO) = 7? - ?, - <t<T and f(t) = f(t + 27) (0 ) State the period, P. [1 Marks) ( 11) Sketch a graph of f(t). [2 marks] State if f(t) is either even or odd, or neither. (1 Marks) (iv) Which Fourier coefficients are zero and why? [1 Marks) (v) Compute do [2 marks] (vi) Compute the non-zero Fourier coefficients. [5 Marks) (vii) Write down the Fourier series representation of f(t), up to and including the term for n = 3. [1 Marks)

Answer 1

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aos 1 HEIM 3 ) 17 an -- &ltl wont dt -U all (+2+² ) cosnt at 1² cosnt at - Joc ĪT t² contat TI 1² cosnt at 2 72 sinnt n 712 10-0) t²coon't dt op Apply biparts and solving 2t coint) (n27-2) -2) sinnt n? af costiti n3 + 8) - { ***+) by Sinnitab ☺ b3 sino 31 (-1) a 27 (6-1)"seo)

% an = 2. 0 - 2. CT (-1)' 으 n2 01 an + (-1) Vii) ft) - ao 도 an cosnt + n) n 4 (-1) cosnt n2 +2 b 1 Elor n 4 cost = + 지 티에 n2 T CODE 티이 6 n=3. + 노 Cos3t . . ft) 392 2 3|| 2. + 4cost - Cost +4 Cost 6 4.