Question

VA1.For the block diagram shown on the left, use frequency domain methods to obtain the impulse response h(t) of the system. 2

Answer 1

Now using partial fraction method - H(s) = A/(s - 1)^2 + B/(s - 1) equation (2) A = (s - 1)^2 middot H(s) Vertical-bar _s = 1 A = (s - 1)^2 middot s/(s - 1)^2 Vertical-bar _s = 1 = 1 B = d/ds [H(s) middot (s - 1)^2] Vertical-bar _s = 1 B = d/ds [s] = 1 formula for finding partial fraction if �n� roots are same. Ex. H(s) = s/(s + 1)^n A = 1/angle n d^n/ds^n [H(z) (s + 1)^n]_s = -1 Now put A & B in (2), we get - H(s) = 1/(s - 1)^2 + 1/(s - 1) h(t) = t^n - 1 middot e^+ at/angle n - 1 u(t) + e^+ at middot u(t) h(t) = [t middot e^t/1 + e^t]u(t) h(t) = (e^t + t middot e^t) middot u(t) Ans \r\n In s - domain - Y(s) = X(s)/s + 2/s Y(s)/1 - 1/s^2 middot Y(s)/1 Y(s) = X(s)/s + 2/s middot Y(s) - 1/s^2