Question

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

(a) \(\frac{2}{1+0.5z^{-1}}\)

(b) \(\frac{0.5}{1+2z^{-1}}\)

(c) \(\frac{0.5}{1-2z^{-1}}\)

(d) \(\frac{2}{1-0.5z^{-1}}\)

This question was addressed to me in exam.

My query 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 (d) \(\frac{2}{1-0.5z^{-1}}\)

The best I can explain: Given difference equation of the system is y(n)=0.5y(n-1)+2x(n)

On applying z-transform on both sides we get, Y(z)=0.5z^-1Y(z)+2X(z)

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