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Homework Question:

Consider a signal $x[n]=\alpha e^{j \omega_{0} n}+\beta e^{j \omega_{1} n}+\gamma e^{j \omega_{2} n} .$ What is the length of impulse response $h[n]$ of a system (non-trivial) such that $x[n] * h[n]=0$ ?

I have written the form in frequency domain in terms of system function as $$H(\omega)\big(\alpha \delta\left(w-w_{0}\right)+\beta \delta\left(w-w_{1}\right)+\gamma \delta\left(w-w_{2}\right)\big)=0$$

Thus $H(\omega)$ must have zeroes at $\omega$ = $\omega_0,\omega_1,\omega_2$

I am not sure about the meaning of impulse response length and also how to proceed to find the same from here.

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    $\begingroup$ Do you know about FIR and IIR filter types, and which type can create poles and which can the zeros? $\endgroup$ – Fat32 Sep 23 '20 at 10:34
  • $\begingroup$ Actually, we haven't done filters yet in the class, so I am not aware about them. However is there any resources that might be helpful. We have covered Z-transforms though. $\endgroup$ – Anuranan Das Sep 23 '20 at 11:18
  • $\begingroup$ strange; you are not given FIR (finite impulse repsonse), IIR (infinite impulse repsonse) yet but you are asked about the length of an impulse response... May be you have missed some topics. To be honest to solve this question with an indepth understanding (which is not that deep though) you should know chapters 2,3, and 5 from Discrete-Time Signal Processing A.Oppenheim. (or similar chapters from another author's book). Just finding the length of the particular filter, the argument in Matt's answer (as a polynomial in z whose zeros are w0,w1,w2) is sufficient though... $\endgroup$ – Fat32 Sep 23 '20 at 11:35
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You correctly figured out that the system must have $3$ zeros. So you just have to define a corresponding transfer function $H(z)$ that satisfies

$$H(z_i)=0,\qquad z_i=e^{j\omega_i},\quad i\in\{0,1,2\}$$

In other words, what is the minimum number of coefficients of a polynomial with $3$ zeros? Note that you might need to provide two solutions: one general solution, and one with the restriction that the polynomial coefficients be real-valued (as is required for a real-valued system).

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  • $\begingroup$ Thank you. We can define a three degree polynomial in Z accordingly. But how to get the impulse response length from there? I am also a bit unclear about what impulse response length is. $\endgroup$ – Anuranan Das Sep 23 '20 at 10:28
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    $\begingroup$ @AnurananDas: The impulse response $h[n]$ is given by the polynomial coefficients: $H(z)=\sum_{n=0}^{N-1}h[n]z^{-n}$, and the length $N$ of the impulse response equals the number of coefficients, i.e., one plus the order of the polynomial. $\endgroup$ – Matt L. Sep 23 '20 at 11:20

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