Question

If X+(z) is the one sided z-transform of x(n), then what is the one sided z-transform of x(n-k)?

(a) z^-k X^+(z)

(b) z^k X^+(z^-1)

(c) z^-k \([X^+(z)+\sum_{n=1}^k x(-n)z^n]\); k>0

(d) z^-k \([X^+(z)+\sum_{n=0}^k x(-n)z^n]\); k>0

I have been asked this question in an interview for job.

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

Answer 1

Correct choice is (c) z^-k \([X^+(z)+\sum_{n=1}^k x(-n)z^n]\); k>0

The explanation is: From the definition of one sided z-transform we have,

Z^+{x(n-k)}=\(z^{-k}[\sum_{l=-k}^{-1} x(l) z^{-l}+\sum_{l=0}^{\infty} x(l)z^{-l}]\)

=\(z^{-k}[\sum_{l=-1}^{-k} x(l) z^{-l}+X^+ (z)]\)

By changing the index from l to n= -l, we obtain

Z^+{x(n-k)}=\(z^{-k}[X^+(z)+\sum_{n=1}^k x(-n)z^n]\) ;k>0