Consider the system whose z-transform is given by Is this system stable? Explain
ID: 1797734 • Letter: C
Question
Consider the system whose z-transform is given by Is this system stable? Explain your answer. Suppose the system is excited by an input x(n) = u(n). Write the the output z-transform Y(z) = H(z)X(z) in terms of polynomials in z (instead of z-1) and then by examining the partial fractions of (or otherwise), find the output sequence y(n). (Note that the Partial Fractions approach is exactly the same as you have done for the Laplace Transform - except that the terms you get in the partial fractions will match known z-transforms such as the one above.)Explanation / Answer
A system is stable if the poles of the equation is with in unit circle then
here pole f H(z) is at z= .5 hence it is stable
X(z) = u[n]z^-n from 0 to infinity
X(z) = 1/(1-z^-1)
Y(z) = 1/(1-Z^-1)*1/(1-.5Z^-1)
Y(z) = 2/(1-Z^-1) - 1/(1-.5Z^-1)
Y(z) = 2u[n]-.5^nu[n]
Y(z) = (2-.5^n)u[n]