Question

Find the Z-transform of the anticausal sequence x(n) = {4,2,3,-1,-2,1}. (1 as the reference variable)

(a) 4z^5 + 2z^4 + 3z^3 – z^2 – 2z + 1

(b) 4z^-5 + 2z^-4 + 3z^-3 -z^-2 – 2z^-1 + 1

(c) -4z^5 – 2z^4 – 3z^3 + z^2 + 2z – 1

(d) -4z^-5 – 2z^-4 – 3z^-3 + z^-2 + 2z^-1 – 1

I got this question during a job interview.

This interesting question is from Properties of Z-Transforms in section Z-Transform and Digital Filtering of Signals and Systems

Answer 1

Right choice is (a) 4z^5 + 2z^4 + 3z^3 – z^2 – 2z + 1

To explain I would say: Given sequence values are :

x(-5)=4, x(-4)=2, x(-3)=3, x(-2)=-1, x(-1)=-2, x(0)=1

We know that

\(X(Z) = \sum\limits_{n=-∞}^{∞} x(n) z^{-n}\)

X(Z) = x(-5) z^5 + x(-4) z^4 + x(-3) z^3 + x(-2) z^2 + x(-1)z + x(0)

X(Z) = 4z^5 + 2z^4 + 3z^3 – z^2 – 2z + 1.