Question

Consider the complex exponential Fourier series coefficients Cn for the signal below

with some fundamental frequency, w. 6sin(4t) + 4cos(8t)+12+2cos(32t).
Which of the following is the correct expressions for the coefficients C1 and C-1?

a) C1 = 3j, C-1 = -3j
b) C1 = 3j, C-1 = 3j
c) C1 = -3j, C-1 = -3j
d) C1 = -3j, C-1 = 3j ?

Answer 1

The Fourier series coefficients for the fundamental frequency of the given signal are C1 = 3j and C-1 = -3j, which corresponds to answer choice (a).

Step-by-step explanation:

The question is asking for the complex exponential Fourier series coefficients C1 and C-1 for the signal 6sin(4t) + 4cos(8t) + 12 + 2cos(32t) with a certain fundamental frequency ω. To find these coefficients, we must recognize that the sine and cosine terms can be represented using Euler's formula which relates complex exponentials to trigonometric functions. However, for the fundamental frequency, which should correspond to 4t in the sine term, its Fourier coefficients will be purely imaginary due to the sine function. Specifically, 6sin(4t) can be represented as 3ei4t - 3e-i4t. Thus, C1 = 3j and C-1 = -3j, making the correct answer (a).