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

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$\hat{x}(\phi) := \sum_{j = -\infty}^{\infty} x_j e^{-ij\phi}$

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!