Question

Let x(t)= (1+5sin(1πt))⋅2cos(4πt) .

Find the Fourier series coefficients ak by inspection, do not integrate.

Hint:use the Euler expansion for the sine and cosine to rewrite x(t) in the form of the Fourier series synthesis equation
x(t)=∑[infinity]ₖ₌−[infinity] ₐₖₑʲᵏ²πᶠ⁰ᵗ .

Answer 1

The question involves finding the Fourier series coefficients for the given function using the Euler expansion technique, transforming it into an exponential Fourier series for easy coefficient extraction by inspection.

Step-by-step explanation:

The question concerns finding the Fourier series coefficients ak by inspection for the function x(t) = (1+5sin(1πt))·2cos(4πt). To find the coefficients, we can use the Euler expansion for the sine and cosine functions. Euler's formula states that eiθ = cos(θ) + i sin(θ), and it helps us rewrite trigonometric functions into their exponential forms. By doing so, we can rewrite x(t) to match the form of the Fourier series, x(t) = ∑ [infinity]ₖ₀₀ [infinity] akei k 2πf0t, where ak are the Fourier series coefficients, which can now be determined by inspection from the rewritten expression of x(t).