Question

A periodic signal x(t) is represented with a Polar Form Fourier Series having the following coefficients: C

0
​
=4,C
2
​
=2,θ
2
​
=−
4
3π
​
,C
3
​
=4,θ
3
​
=
3
π
​
Determine the coefficients of the exponential Fourier Series form for this same signal: (Enter integer values only) a
0
​
=
a
1
​
=
a
2
​
=
​

<
<
degrees; a
−2
​
=
​

degrees; a
−1
​
=
degrees;
​

Answer 1

To determine the coefficients of the exponential Fourier Series form for the given periodic signal x(t), we can use the following formulas:

a_n = C_n * cos(n * θ_n)
b_n = C_n * sin(n * θ_n)

Given the coefficients C_0 = 4, C_2 = 2, θ_2 = -4π/3, C_3 = 4, and θ_3 = 3π, we can calculate the coefficients of the exponential Fourier Series as follows:

a_0 = C_0 = 4
a_1 = a_-1 = 0 (since there are no coefficients for n = 1 or n = -1)
a_2 = C_2 * cos(2 * θ_2) = 2 * cos(2 * (-4π/3)) = 2 * cos(-8π/3)
a_-2 = C_2 * cos(-2 * θ_2) = 2 * cos(-2 * (-4π/3)) = 2 * cos(8π/3)
a_3 = C_3 * cos(3 * θ_3) = 4 * cos(3 * 3π) = 4 * cos(9π)
a_-3 = C_3 * cos(-3 * θ_3) = 4 * cos(-3 * 3π) = 4 * cos(-9π)

Therefore, the coefficients of the exponential Fourier Series form for the given signal x(t) are:
a_0 = 4
a_1 = a_-1 = 0
a_2 = 2 * cos(-8π/3)
a_-2 = 2 * cos(8π/3)
a_3 = 4 * cos(9π)
a_-3 = 4 * cos(-9π)

Please note that the actual values of the coefficients will depend on the specific values of π and θ provided in the question, but the general procedure outlined above should be followed.