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The discrete system has poles at points $z_{1,2} = 0.8e^{\pm{i\pi /6}}$ and $z_{3,4}=0.8e^{\pm{i\pi/2}}$, and two-fold zeros at $1$ and $-1$. The task is to determine the impulse response of the system. The wording is weird, but I assume this means the transfer function is $$H(z) = \frac{(z-1)(z+1)}{(z-z_1)(z-z_2)(z-z_3)(z-z_4)}$$

The impulse response must then be $$h(t) = Z^{-1}\{H(z)\}$$

Based on the given poles, I can say that the impulse response is a decaying oscillation of sorts. So this is a FIR filter. Is it possible to determine the impulse response from this information without resorting to partial fractions? ( which is the next step I would take here )

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    $\begingroup$ This is not a FIR filter. FIR filters do not have poles (to be precise, FIR filters have as many poles as they have zeros. But all these poles are located at the origin). This is an IIR filter since it has poles not located at the origin. You should use partial fractions, and pay attention to the fact that you have complex conjugate poles. This will help. Good luck! $\endgroup$
    – Jdip
    Commented Oct 10, 2022 at 15:52
  • $\begingroup$ Also note that there's an unknown gain factor in your transfer function. Poles and zeros only determine the transfer function up to an unknown multiplicative constant. $\endgroup$
    – Matt L.
    Commented Oct 10, 2022 at 16:57
  • $\begingroup$ Does the term "synthetic division of polynomials" ring any bells? This is clearly a homework problem -- do you have an idea of what form the instructor wants the solution in? BTW: when you have the answer, ask yourself if the impulse response actually goes to zero and stays there. $\endgroup$
    – TimWescott
    Commented Oct 11, 2022 at 2:16

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