Digital Signal Processing Filters Problem

Question

Hello I need some help with this question. $$𝑦[𝑛] = 𝑎_1𝑥[𝑛 − 𝑘_1] + 𝑎_2𝑥[𝑛− 𝑘_2] + 𝑏_1 𝑦[𝑛− 𝑘_3]$$

Set $a_1, a_2$, and $b_1$ to any positive or negative real numbers of your choice. Set $k_1, k_2$, and $k_3$ to any positive integer numbers of your choice ($k_1$ can be zero). Present this equation with the numbers you have chosen. Present the transfer function $𝐻(𝑧)$ of your filter

I have chosen the values so my equation looks like this $$y[𝑛] = 0.4𝑥[𝑛] + 0.6𝑥[𝑛− 1] + 0.8𝑦[𝑛− 2]$$

I have worked out $H(z)$ to be $$\frac{0.4 + 0.6z^{-1}}{1-0.8z^{-2}}$$ is this correct?

My questions are:

• Is this filter an FIR or IIR type?
• How do you determine the poles and zero?
• How do you determine if the filter is stable or unstable?

Thanks!

Answer 1

I have worked out H(z) to be (0.4 + 0.6z^-1) / (1-0.8z^-2) is this correct?

Close but one detail is wrong

Is this filter an FIR or IIR type?

Look at the pole locations

How do you determine the poles and zero?

Poles are the roots of denominator polynomial. Zeros are the roots of the numerator polynomial.

How do you determine if the filter is stable or unstable?

Look at the pole locations

These questions are about as fundamental as it gets in DSP. They are also incredibly simple, in fact, so simple that it's hard to give a hint. If it takes you more than a few seconds to answer this, you need to adjust the way your are studying. I strongly recommend to not proceed to any more advanced topic until you very comfortable solving this type of problems. It's fundamental to whatever comes next.