In this problem you will be comparing some of the characteristics of analog and digital...

In this problem you will be comparing some of the characteristics of analog and digital implementations of the single-pole haw-pass analog system

(a) What is the gain at dc? At what radian frequency is the analog frequency response 3 dB down from its de value? At what frequency is the analog frequency reponse zero? At what time has the analog impulse response decayed to 1/e of its initial value?

(b) Give the digital system function 11(z) for the impulse-invariant design for this filter. What is the gain at dc? Give an expression for the 3-dB radian frequency. At what (real-valued) frequency is the response zero? How many samples are there in the unit sample time-domain response before it has decayed to 1/e of its initial value?

(c) "Prewarp" the parameter α and perform the bilinear transformation to obtain the digital system function H(z) from the analog design. What is the gain at dc? At what (real-valued) frequency is the response zero? Give an expression for the 3-dB radian frequency. How many samples in the unit sample time-domain response before it has decayed to 1/e of its initial value?