A discrete-time lowpass filter is to be designed by applying the impulse invariance meth...

A discrete-time lowpass filter is to be designed by applying the impulse invariance method to a continuous-time Butterworth filter having magnitude-squared function

The specifications for the discrete-time system are those of Example 7.2, i.e.,

Assume, as in that example, that aliasing will not be a problem; i.e., design the continuous-time Butterworth filter to meet passband and stopband specifications as determined by the desired discrete-time filter.

(a) Sketch the tolerance bounds on the magnitude of the frequency response, of the continuous-time Butterworth filter such that after application of the impulse invariance method (i.e., h[n] = T_dh_c(nT_d )), the resulting discrete-time filter will satisfy the given design specifications. Do not assume that T_d = 1 as in Example 7.2.

(b) Determine the integer order N and the quantity such that the continuous-time Butterworth filter exactly meets the specifications determined in part (a) at the passband edge.

(c) Note that if Td = 1, your answer in part (b) should give the values of N and obtained in Example 7.2. Use this observation to determine the system function H_c(s) for and to argue that the system function H(z) which results from impulse invariance design with is the same as the result for Td = 1 given by Eq. (7.17).