(Solved): IIR versus FIR systems A significant difference between IIR and FIR discrete-time systems is stabi ...

IIR versus FIR systems A significant difference between IIR and FIR discrete-time systems is stability. Consider an IIR filter with the difference equation \[ \gamma_{1}[n]=x[n]-0.5 y_{1}[n-1] \] where \( x \mid n] \) is the input and \( \gamma_{1}|n| \) is the output. Then consider an FIR filter \[ \left.\gamma_{2} \mid n\right]=x[n]+0.5 x[n-1]+3 x[n-2]+x[n-5] \] where \( x[n] \) is the input and \( y_{2}[n] \) is the output. (a) Since to check the stability of these filters we need their impulse responses, find the impulse responses \( h_{1}[n] \) corresponding to the IIR filter by recursion, and \( h_{2}[n] \) corresponding to the FIR filter. (b) Use the impulse response \( h_{1}[n] \) to check the stability of the IIR filter. (c) Use the impulse response \( h_{2}[n] \) to check the stability of the FIR filter. (d) Since the impulse response of a FIR filter has a finite number of nonzero terms, would it be correct to say that FIR filters are always stable? Explain.