A causal first-order digital filter is described by the system function

(a) Sketch the direct form I and direct form II realization of this filter and find the corresponding difference equations

(b) For a = 0.5 and b = -0.6, sketch the pole-zero pattern. Is the system stable? Why?

(c) For a = 0.5 and b = -0.5, determine b₀. So that the maximum value of |H(ω)| is equal to 1.

(d) Sketch the magnitude response | H(ω)| and the phase response < H(ω) of the filter obtained in part (c).

(e) In a specified application it is known that a = 0.8. Does the resulting filter amplify high frequencies or low frequencies in the input? Choose the value of b so as to improve the characteristics of this filter (i.e., make it a better low pass or a better high pass filter).