Question

What is the unit sample response of the system described by the difference equation y(n)=0.5y(n-1)+2x(n)?

(a) 0.5(2)^nu(n)

(b) 2(0.5)^nu(n)

(c) 0.5(2)^nu(-n)

(d) 2(0.5)^nu(-n)

This question was posed to me in examination.

This interesting question is from Rational Z Transform topic in portion Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

Right option is (b) 2(0.5)^nu(n)

To explain I would say: By applying the z-transform on both sides of the difference equation given in the question we obtain,

\(\frac{Y(z)}{X(z)}=\frac{2}{1-0.5z^{-1}}\)=H(z)

By applying the inverse z-transform we get h(n)=2(0.5)^nu(n).