Transforming Finite Series: Solving with Z-Transform?

[ElijahRockers]

Homework Statement

Let ## x_j = \begin{Bmatrix}
{1, 0 \leq j \leq N-1} \\
{0, else} \\
\end{Bmatrix}
##

Show that ##\hat{x}(\phi) = \frac{e^{-i\frac{N-1}{2}\phi}sin(\frac{N}{2}\phi)}{sin(\frac{1}{2}\phi)}##

Homework Equations

[/B]
##\hat{x}(\phi) := \sum_{j = -\infty}^{\infty} x_j e^{-ij\phi}##

The Attempt at a Solution

So I get ##\hat{x}(z) = \sum_{j = 0}^{N-1} z^{-j} = \sum_{0}^{N-1} (\frac{1}{z})^{j}##

I believe this is a geometric series, with sum ##\hat{x}(z) = \frac{1-z^{-N}}{1-z^{-1}} = \frac{1-e^{-iN\phi}}{1-e^{-i\phi}}##

Of course this makes the assumption that ##|\frac{1}{z}| < 1 ## which I am not entirely sure about. Any tips appreciated.

 

[Orodruin]

Yes, it is a geometric series. Since it is finite you do not need the condition on z for convergence.

What remains is to recast it on the form given in the assignment. I suggest working backwards if you have problems.

 

[ElijahRockers]

Sorry, I should have added that's where I am stuck. I can play around with ##e^{ix} = cos(x) + isin(x)## but it doesn't seem to be getting any closer to the final answer.

 

[Orodruin]

This is why I suggest working backwards. How can you express sin(x) in terms of exponents of ix?

 

[ElijahRockers]

This is why I suggest working backwards. How can you express sin(x) in terms of exponents of ix?

Got it! Thank you!