Question

What is the z-transform of the signal x(n)=na^nu(n)?

(a) \(\frac{(az)^{-1}}{(1-(az)^{-1})^2}\)

(b) \(\frac{az^{-1}}{(1-(az)^{-1})^2}\)

(c) \(\frac{az^{-1}}{(1-az^{-1})^2}\)

(d) \(\frac{az^{-1}}{(1+az^{-1})^2}\)

I have been asked this question during an interview.

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

Answer 1

The correct option is (c) \(\frac{az^{-1}}{(1-az^{-1})^2}\)

The explanation: We know that Z{a^nu(n)}=\(\frac{1}{1-az^{-1}}\)=X(z) (say)

Now the z-transform of na^nu(n)=\(-z\frac{dX(z)}{dz} = \frac{az^{-1}}{(1-az^{-1})^2}\)