(Solved):   Using bilinear transform, we want to design a first order IIP digital low pass filter with ...

 

Using bilinear transform, we want to design a first order IIP digital low pass filter with \( -3 \mathrm{~dB} \) cutoff frequency at \( 1500 \mathrm{~Hz} \) assuming a sampling rate of \( 8000 \mathrm{samples} / \mathrm{sec} \). We want to use a first order Butter (note \( a_{0}=1 \) ) (note \( a_{0}=1 \) ) b) State the frequency response \( H\left(e^{j w}\right) \) for the designed filter. Calculate and approximately plot the magnitude response \( \left|H\left(e^{j \omega}\right)\right| \). Calculate the magnitude response \( \left|H\left(e^{j \omega}\right)\right| \) at the cutoff frequency. Check whether it fulfills the \( -3 \mathrm{~dB} \) design spec. Hint 1: A first order Butterworth analog low pass filter with \( -3 \mathrm{~dB} \) cutoff frequency at \( \Omega_{c} \mathrm{rad} / \mathrm{s} \) has the transfer function: \[ H(s)=\frac{1}{1+\frac{s}{\Omega_{c}}} \] Hint 2: Bilinear Transformation \[ H(z)=\left.H(s)\right|_{s=\frac{2}{T_{d}} \frac{1-z^{-1}}{1+z^{-1}}} \]