Consider two real sequences with time- origin marked by the bold value, $${x_1}\left[ n \right] = \left\{ {1,\,2,\,3,\,0} \right\}\,,\,{x_2}\left[ n \right] = \left\{ {1,\,3,\,2,\,1} \right\}$$ Let $${X_1}(k)$$ and $${X_2}(k)$$ be 4-point DFTs of $${x_1}\left[ n \right]$$ and $${x_2}\left[ n \right]$$, respectively. Another sequence $${X_3}(n)$$ is derived by taking 4-ponit inverse DFT of $${X_3}(k)$$= $${X_1}(k)$$$${X_2}(k)$$. The value of $${x_3}\left[ 2 \right]$$

The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is ℎ[n]. If ℎ[0] =1, we can conclude.

For the discrete-time system shown in the figure, the poles of the system transfer function are located at

The result of the convolution $$x\left( { - t} \right) * \delta \left( { - t - {t_0}} \right)$$ is