(Solved): Problem 4 (30 points) We want to build a discrete-time low pass filter with the cutoff irequency of ...

Problem 4 (30 points) We want to build a discrete-time low pass filter with the cutoff irequency of \( \omega_{\mathrm{m}}=\pi / 3 \). Assuming that it is possible to build ideal filters, answer the following: (a) (6 points) What should be the frequency response \( H\left(e^{2}\right) \) of this filter? Draw the magnitude and the phase responses of this filter. (b) (6 points) What is the response of this filter for the input signal? \[ x[n]=\cos \left(\frac{\pi}{4} n\right)+j \sin \left(\frac{\pi}{2} n\right)+e^{j \frac{\pi}{4} n} \] (c) \( \left(9\right. \) points) We build a second filter with the frequency response \( H_{2}\left(e^{j}\right)=1-H\left(e^{j \omega}\right) \) where \( H\left(e^{j \omega}\right) \) is the frequency of the ideal low pass filter form part (a). What kind of a filter is this new filter? What is its response to the input signal \( x[n] \) from part (b)? (d) (9 points) We build a third system by cascading the two system from part (a) with a high pass filter with a cutoff frequency of \( \vec{\omega}_{\mathrm{co}_{2}}=\pi / 6 \). What kind of a filter is this third filter? What is its response to the input signal \( x[n] \) from part (b)?