Finding waveforms from Fourier Coefficient Values

[PJV9126]

Homework Statement

Find the waveforms x1(t) and x2(t) (expressed in a simple form) that are consistent with the sets of Fourier coefficient values provided below. Assume that the period T is equal to 1/10 seconds in both cases

(a-1): a(0) = 6, a(-1) = 3 + 4j, a(1) = 3 - 4j, and a(k) = 0 for k != -1,0,+1

(a-2): a(0) = 4, a(3) = 2e^(jPi/4), a(-3) = 2e^(-jPi/4), and a(k) = 0 for k != -3, 0 , +3

Homework Equations

x(t) = Sum[a_k*e^jkω_0*t]

a_k = 1/T integral x(t)*e^-jkω_0*t dt

The Attempt at a Solution

The fundamental frequency can be calculated using the period given. This would be ω = 2∏/T = 2*∏*10 = 20∏. From this I believe we can plug this into the second equation along with the value of k given and set it equal to whatever that value of k was. I am then running into the issue of solving it (assuming that is correct). I do not know what to do with the x(t) term, along with the integral bounds.

Sorry for the sloppy equations, I am new here and haven't had a chance to check out Latex yet fyi.

 

[LCKurtz]

Homework Statement

Find the waveforms x1(t) and x2(t) (expressed in a simple form) that are consistent with the sets of Fourier coefficient values provided below. Assume that the period T is equal to 1/10 seconds in both cases

(a-1): a(0) = 6, a(-1) = 3 + 4j, a(1) = 3 - 4j, and a(k) = 0 for k != -1,0,+1

(a-2): a(0) = 4, a(3) = 2e^(jPi/4), a(-3) = 2e^(-jPi/4), and a(k) = 0 for k != -3, 0 , +3

Homework Equations

x(t) = Sum[a_k*e^jkω_0*t]

a_k = 1/T integral x(t)*e^-jkω_0*t dt

The Attempt at a Solution

The fundamental frequency can be calculated using the period given. This would be ω = 2∏/T = 2*∏*10 = 20∏. From this I believe we can plug this into the second equation along with the value of k given and set it equal to whatever that value of k was. I am then running into the issue of solving it (assuming that is correct). I do not know what to do with the x(t) term, along with the integral bounds.

Sorry for the sloppy equations, I am new here and haven't had a chance to check out Latex yet fyi.

You don't have to do any integrals. These are finite Fourier series, and they each have only 3 terms. Just write them out and simplify to sines and cosines and you will have your functions.