A causal first-order digital filter is described by the system function (a) Ske...

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

(a) Sketch the direct form I and direct form II realizations 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 is equal to 1.

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

(e) In a specific 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 lowpass or a better highpass filter).