Question

According to Time shifting property of z-transform, if X(z) is the z-transform of x(n) then what is the z-transform of x(n-k)?

(a) z^kX(z)

(b) z^-kX(z)

(c) X(z-k)

(d) X(z+k)

This question was addressed to me in an interview for job.

This intriguing question comes from Properties of Z Transform topic in division Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

Right option is (b) z^-kX(z)

Explanation: According to the definition of Z-transform

X(z)=\(\sum_{n=-\infty}^{\infty} x(n) z^{-n}\)

=>Z{x(n-k)}=\(X^1(z)=\sum_{n=-\infty}^{\infty} x(n-k) z^{-n}\)

Let n-k=l

=> X^1(z)=\(\sum_{l=-\infty}^{\infty} x(l) z^{-l-k}=z^{-k}.\sum_{l=-\infty}^{\infty} x(l) z^{-l}= z^{-k}X(z)\)