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

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

(b) Give the digital system function H(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 a 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?

H,(s) = eh,t) = e s+a.