Question

3, Use CTFS properties to determine which signal with period T = 4 has 0 CTFS coefficients, for k ck = 0 and for k ior k co 64

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Answer 1

Given that Use CTFS properties to determine which signal with period T = 4 has CTFS coefficients, for k notequalto 0 C_k = 1/-j(2Pi)^3 k^3 ((-j)^3 -1)^3 and for k = 0 c_0 = 1/64 According to the exponential power Series, x(t) = Sigma^infinity_n = -infinity C_n e^jnw_0 t Here we consider k So, therefore x(t) = Sigma^infinity_k = -infinity C_k e^jkw_0 t equation (A) Given that T = 4 w_0 = 2Pi/T = 2Pi/4 = Pi/2 therefore w_0 = Pi/2 Now substitute, w_0 value in equation (A) \r\n X(t) = Sigma^infinity_k = -infinity C_k e^jk Pi/2 t i.e. X(t) = C_0 e^j(0) Pi/2 t + Sigma^infinity_k = -infinity C_k e^jkPi/2 t x(t) = 1/64 + Sigma^infinity_k = -infinity 1/-j(2Pi)^3 k^3 ((-f)^k - 1)^3 e^jkPi/2 t therefore X(t) = 1/64 + Sigma^infinity_k = -infinity k notequalto 0 ((-j)^k - 1)^3/-j(2Pi)^3 k^3 e^jkPi/2 t equation (B) The required signal with period T = 4 is equation (B)