Given the transfer function for a system with zero initial

Question

Signal and Systems Assignment

Problem 1- Topics: System Response, State Equations, Block Diagrams

Given the transfer function for a system with zero initial conditions:

H(s) = 1/(s²+10s+21)

a) Find the system differential equation.

b) Draw a Block Diagram realization of the system (notation: input x(t) = u_in(t)).

c) Define state variables X1 and X2, and write the system state equations.

d) Express the previous state equations in matrix form, clearly labeling matrices A, B, C, D.

e) Find the system impulse response using any method.

f) Find the system output, y(t) when the input is a unit step.

Problem 2- Topics:  Discrete Time Systems, FiR and IIR filter, step response, unit sample response, Pole-Zero Plot.

Given a causal, LSI, discrete time system is characterized by the following unit sample response:           

h[n]= 0.8ⁿu[n]

a) Find System Function H(z)= Z{h[n]} including R.O.C

b) Sketch Pole - Zero Plot including R.O.C region.

c) Difference equation:

d) Draw a Direct Form 1, block diagram realization of the system.

e) Find the unit-step response of the filter.

f) Is this filter FIR or IIR?  Why?

g) Is this filter stable? Why?