In filter design, it is often possible and convenient to transform a lowpass filter to a...

In filter design, it is often possible and convenient to transform a lowpass filter to a highpass filter and vice versa. With H(s) denoting the transfer function of the original filter and G(s) that of the transformed filter, one such commonly used trans-formation consists of replacing s by 1/s; that is,

(a) For H(s) = .1/(s + 1/2), sketch | H( jω) | and | G(jω) |.

(b) Determine the linear constant-coefficient differential equation associated with H(s) and with G(s).

(c) Now consider a more general case in which H(s) is the transfer function associated with the linear constant-coefficient differential equation in the general form

Without any loss of generality, we have assumed that the number of derivatives N is the same on both sides of the equation, although in any particular case, some of the coefficients may be zero. Determine H(s) and G(s).

(d) From your result in part (c), determine, in terms of the coefficients in eq. (P9.63-1), the linear constant-coefficient differential equation associated with G(s).