Solving for the Real and Imaginary Number

[Tan Thom]

Homework Statement

Find the 4th Coefficient in a sample of 4 discrete time Fourier Series coefficients in a real time valued periodic sequence. k = 0,1,2,3

a_k = {3, 1-2j, -1, ?}

Homework Equations

[/B]

The Attempt at a Solution

Step 1: (1-2j)e^(j*.5pi*n) +a_3 e ^ -(j*.5pi*n) + 3 + (-1)^(n+1)

Step 2: [(1-2j)(cos (pi/2)n + j sin (pi/2)n) + a_3 (cos (pi/2)n)-j sin (pi/2)n]

Solving for a_3 in the final step is where I am confused.

 

[Blanci1]

I had a lot of experience with such things and the question doesn't seem to be fully formed to me. Perhaps there is some extra info somewhere? Perhaps the solution revolves around making sure that the resulting function will be real valued as stated in question. That sets constraints between coefficients.
However only summing over positive k values Will make it imposible to get a purely real signal. Normally that would happen when a(-k) is the complex conjugate of a(k). Or should it be using a discrete Fourier sin series, summing over real basis functions rather tan over exp(ikx) complex functions? Though you wouldn't need complex coeffs in that case. Need to see the full question I think.