(Solved):   A causal discrete-time linear time-invariant system defined by its inp ...

 

A causal discrete-time linear time-invariant system defined by its input-output given as \[ y[n]+(0.5) y[n-1]=x[n] \] where \( x[n] \) and \( y[n] \) are the input and output signals respectively. a) Compute the z-domain transfer function \( H(z) \). b) Determine the poles and zeros, and find the ROC. Draw the pole-zero plot c) Find the system impulse response \( h[n] \). d) Is this system BIBO stable?. Justify your answer. e) Sketch the block diagram of this system using multipliers, adders, and unit f) Compute the step response \( s[n] \) of the system, that is, the output when unit step function \( x[n]=u[n] \). g) Find the frequency response \( H\left(e^{j \omega}\right) \) of the system. h) Compute the squared-magnitude response \( \left|H\left(e^{j \omega}\right)\right|^{2}=H\left(e^{j \omega}\right) \) \( \omega=0, \frac{\pi}{2}, \pi[\mathrm{rad}] \) and plot its variation for \( 0 \leq \omega \leq \pi \). Is this system a a high-pass filter, or a bandpass filter?