Phase speed of backward-time, central-space scheme

Question

I'm studying the backward-time, central-space (BTCS) scheme $$u_{j}^{n+1} + \frac{k}{2h}[u_{j+1}^{n+1} - u_{j-1}^{n+1}] = u_j^n,$$ for $k$ the step size in time and $h$ the step size in space. I have been asked to find the "phase speed" $\alpha(\theta)$ of the scheme when applied to the advection equation $u_t + u_x = 0$. I have the modified equation as $$u_t + u_x = \frac{k}{2}u_{xx} + \frac{3k^2 - h^2}{6}u_{xxx},$$ but my Fourier transforms are not working out to anything nice. What should I be doing here?