zer0square
Newbie level 2
A sequence $x[n]$ is the output of a linear time-invariant system whose input is $s[n]$. This system is described
by the difference equation
$(1.1)$ $$x[n]=s[n]-e^{-8\alpha}s[n-8]$$
$$\alpha>0$$
a) Find the system function
$$H(z)=X(z)/S(z)$$
and plot its poles and zeros in the z-plane. Indicate the region of convergence.
b) We wish to recover $s[n]$ from $x[n]$ with a linear time-invariant system. Find the system function
$$H_2(z)=Y(z)/X(z)$$
such that $y[n]=s[n]$. Find all possible regions of convergence for $H_2(z)$, and for each, tell whether or not the system is causal and/or stable.
c) Find all possible choices for the impulse response $h_2[n]$ such that
$$y[n]=h_2[n]*x[n]=s[n]$$
d) For all choices determined in Part (c), demonstrate, by explicitly evaluating the convolution in $(1.1)$, that
when $s[n]=\delta[n]$, then $y[n]=\delta[n]$.
I am having problems defining the region of convergence in the first 2 parts and the third one say to find all the possible choices but I can only think of one and part d) is related to those.
by the difference equation
$(1.1)$ $$x[n]=s[n]-e^{-8\alpha}s[n-8]$$
$$\alpha>0$$
a) Find the system function
$$H(z)=X(z)/S(z)$$
and plot its poles and zeros in the z-plane. Indicate the region of convergence.
b) We wish to recover $s[n]$ from $x[n]$ with a linear time-invariant system. Find the system function
$$H_2(z)=Y(z)/X(z)$$
such that $y[n]=s[n]$. Find all possible regions of convergence for $H_2(z)$, and for each, tell whether or not the system is causal and/or stable.
c) Find all possible choices for the impulse response $h_2[n]$ such that
$$y[n]=h_2[n]*x[n]=s[n]$$
d) For all choices determined in Part (c), demonstrate, by explicitly evaluating the convolution in $(1.1)$, that
when $s[n]=\delta[n]$, then $y[n]=\delta[n]$.
I am having problems defining the region of convergence in the first 2 parts and the third one say to find all the possible choices but I can only think of one and part d) is related to those.
