Question

What is the z-transform of the signal defined as x(n)=u(n)-u(n-N)?

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

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

(c) \(\frac{1+z^{-N}}{1+z^{-1}}\)

(d) \(\frac{1-z^{-N}}{1-z^{-1}}\)

I have been asked this question in final exam.

The query is 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 (d) \(\frac{1-z^{-N}}{1-z^{-1}}\)

Easiest explanation: We know that \(Z{u(n)}=\frac{1}{1-z^{-1}}\)

And by the time shifting property, we have Z{x(n-k)}=z^-k.Z{x(n)}

=>Z{u(n-N)}=\(z^{-N}.\frac{1}{1-z^{-1}}\)

=>Z{u(n)-u(n-N)}=\(\frac{1-z^{-N}}{1-z^{-1}}\).