# Z-transform of an FIR filter

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.

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

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

Why?

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

• What's your question ? – Hilmar Oct 12 '15 at 15:06

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

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

• 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? – panthyon Oct 12 '15 at 15:16
• @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 $z=0$. – Matt L. Oct 12 '15 at 15:32