By means of a specific filter design procedure, a nonideal continuous-time lowpass filte...

By means of a specific filter design procedure, a nonideal continuous-time lowpass filter with frequency response , by the use of frequency scaling. The frequency response of the resulting filter is then of the form

where a is an appropriate scale factor

(a) Determine the scale factor a such that has a cutoff frequency of ω_c .

(b) Determine the impulse response

(c) Determine the step response

(d) Determine and sketch the rise time of the new filter as a function of its cutoff frequency to ω_c.

This is one illustration of the trade-off between time-domain and frequency-domain characteristics. In particular, as the cutoff frequency de-creases, the rise time tends to increase.