0
$\begingroup$

QUESTION

Compute the Z-transform of $y[n] = x[n] + 2x[n-1]$. and find the poles and zeros.

I just bombed an interview where I couldn't do this (because I have no grounding in fundamentals and have worked all from cookbooks). I'm not too bummed because I would have been working at something I wasn't prepared for. You get one guess who would ask this kind of question in a first phone interview.

ANSWER (30 minutes post interview):

$x[n-1] = z^{-1}$

$1 + 2z^{-1}$.

Why?

Because the z-transform of an FIR is the coefficients of the impulse response multiplied by the delays.

$\endgroup$
1
  • 2
    $\begingroup$ What's your question ? $\endgroup$ – Hilmar Oct 12 '15 at 15:06
3
$\begingroup$

If you know (or better: understand) the basic $\mathcal{Z}$-transform relation

$$x[n-k]\Longleftrightarrow z^{-k}X(z)$$

then from the given time domain equation you immediately get

$$Y(z)=X(z)+2X(z)z^{-1}=X(z)(1+2z^{-1})$$

Since the transfer function $H(z)$ is the ratio of $Y(z)$ and $X(z)$, you get

$$H(z)=1+2z^{-1}$$

$\endgroup$
2
  • $\begingroup$ can you modify this to show how you would find the poles and zeros? this has no poles and one zero correct? is it at z^-1 = -2? $\endgroup$ – panthyon Oct 12 '15 at 15:16
  • $\begingroup$ @panthyon: For the zero you must solve $1+2z^{-1}=0$, which indeed gives you $z=-2$. The filter also has one pole at $z=0$, because the transfer function becomes infinity at $z=0$. $\endgroup$ – Matt L. Oct 12 '15 at 15:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.